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IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
IUTAM BOOKSERIESVolume 24
Series Editors
GML Gladwell University of Waterloo Waterloo Ontario CanadaR Moreau INPG Grenoble France
Editorial Board
J Engelbrecht Institute of Cybernetics Tallinn EstoniaLB Freund Brown University Providence USAA Kluwick Technische Universitt Vienna AustriaHK Moffatt University of Cambridge Cambridge UKN Olhoff Aalborg University Aalborg DenmarkK Tsutomu IIDS Tokyo JapanD van Campen Technical University Eindhoven Eindhoven
The NetherlandsZ Zheng Chinese Academy of Sciences Beijing China
Aims and Scope of the Series
The IUTAM Bookseries publishes the proceedings of IUTAM symposia under theauspices of the IUTAM Board
For other titles published in this series go tohttpwwwspringercomseries7695
Meinhard Kuna Andreas RicoeurEditors
of Fatigue Damageand Fracture in SmartMaterials
Proceedings of the IUTAM Symposium onMultiscale Modelling of Fatigue Damageand Fracture in Smart Materials held inFreiberg Germany September 1ndash4 2009
123
IUTAM Symposiumon Multiscale Modelling
EditorsMeinhard KunaTU Bergakademie FreibergInst Mechanik und FluiddynamikLampadiusstr 409596 FreibergGermanyMeinhardKunaimfdtu-freibergde
Andreas RicoeurUniversitat KasselInst MechanikMonchebergstr 734109 KasselGermanyricoeuruni-kasselde
ISSN 1875-3507 e-ISSN 1875-3493ISBN 978-90-481-9886-3 e-ISBN 978-90-481-9887-0DOI 101007978-90-481-9887-0Springer Dordrecht Heidelberg London New York
Library of Congress Control Number 2010938713
c Springer Science+Business Media BV 2011No part of this work may be reproduced stored in a retrieval system or transmitted in any form or byany means electronic mechanical photocopying microfilming recording or otherwise without writtenpermission from the Publisher with the exception of any material supplied specifically for the purposeof being entered and executed on a computer system for exclusive use by the purchaser of the work
Cover design SPi Publisher Services
Printed on acid-free paper
Springer is part of Springer Science+Business Media (wwwspringercom)
Preface
Multi-functional materials as piezoelectricferroelectric ceramics magnetostrictiveand shape memory alloys are gaining increasing applications as sensors actuatorsor smart composite materials systems for promising high tech areas One primaryproblem is however that these functional materials suffer from various mechanicalandor electromagnetical degradation mechanisms as fatigue damage and fractureAs a consequence of field coupling effects fabrication processes and service loadssmart materials systems are exposed to high mechanical andor electromagneticalfield concentrations under internal and external loading of static cyclic and dy-namic type For this reason the investigation of fatigue damage and fracture playsa decisive role for the optimum design reliability and durability of smart materi-als systems Thus the topic of the symposium represents an active internationalresearch area in mechanics of materials From the experience and investigationsduring the last decade it has become evident that progress in this scientific disci-pline is only possible if material models are based on the true physical nature ofthe phenomena and if theoretical predictions are verified by skilful experimentsTherefore the scientific challenges can only be solved by
A multi-scale modelling at several length scales from atomistic to macroscopiclevel
An interdisciplinary cooperation between solid mechanics materials science andphysics
To promote the international scientific exchange in this important field in 2006the General Assembly of IUTAM approved the proposal to host this symposium inFreiberg Germany and appointed the International Scientific Committee The IU-TAM Symposium (GA 06-16) ldquoMultiscale Modelling of Fatigue Damage and Frac-ture in Smart Materials Systemsrdquo was held on September 1ndash4 2009 at TechnischeUniversitat Bergakademie Freiberg Germany organized by the Institute of Mechan-ics and Fluid Dynamics
This symposium stands in a line with former symposia on related topics heldunder the auspices of IUTAM in 2000 at Magdeburg and in 2004 at Beijing Thehelpful assistance of the International Scientific Committee to communicate thesymposium and to recommend invited speakers is thankfully appreciated
v
vi Preface
According to the rules and tradition of IUTAM the aim of the Symposium is tobring together internationally leading researchers working in the area of smart ma-terials The goal is to exchange recent scientific results to discuss new achievementand actual problems in an open and frank atmosphere The organizers were happyto welcome a lot of outstanding scientists in this field from all parts of the globeas well as many young researchers In total there were 44 participants coming from14 countries Australia (1) Austria (3) Belgium (1) Canada (1) China (8) GreatBritain (1) France (2) Germany (19) Israel (1) Italy (1) Japan (2) Slovakia (1)Ukraine (1) and USA (2) The scientific program covered 35 invited oral contribu-tions presented in 10 sessions
The following main topics have been addressed during the symposium
Development of computational methods for coupled electromechanical fieldanalysis especially extended adaptive and multi-level finite element techniquesin combination with boundary elements
Constitutive modeling of smart materials with coupled electric magnetic ther-mal and mechanical fields especially of nonlinear dissipative hysteresis behaviorMajor trend is the development of micromechanical models Especially for fer-roelectric materials and shape memory alloys the simulation of microstructure(domain switching martensitic transformation etc) are of paramount concern
Further understanding and modeling of fracture and fatigue in piezoelectric andferroelectric ceramics especially the modeling of fracture process zone and ofelectric boundary conditions at crack faces Applications of phase field simula-tion and configurational mechanics
Reliability and durability of sensors and actuators under in service loading byalternating mechanical electrical and thermal fields The role of interface cracksbetween layers and in thin films is addressed
Experimental methods to measure fracture strength and to investigate fatiguecrack growth in ferroelectric materials under electromechanical loading It hasbeen pointed out that complicated theoretical predictions have to be contrastedand verified by skilful experiments
New ferroelectric materials compounds and composites with enhanced straincapabilities
The chairman and its organizing team tried to make this IUTAM symposium notonly a successful scientific meeting but an outstanding social event too Manythanks are due to Prof Dr A Ricoeur who carried the main workload in organizingall details of this symposium
The Technische Universitat Bergakademie Freiberg is located in the East GermanFederal State Saxony Besides Dresden and Chemnitz TU Bergakademie Freibergis considered as the ldquosmallestrdquo but ldquosmartestrdquo among these Technical UniversitiesIts history started with the discovery of silver in the middle ages The ldquoMiningAcademyrdquo Freiberg was founded by the Saxon King in 1765 in order to promote thetechnologies in surveying mining and metallurgy Thus TU Bergakademie Freibergpossesses a long and famous tradition as one of the oldest montanistic universi-ties in the world Nowadays TU Bergakademie Freiberg is established as modern
Preface vii
Technical University focusing mainly on Geoscience Resources Materials ScienceEnergy and Environmental Technologies
Freiberg has one of the largest and most splendid mineralogical exhibitionsldquoTerra Mineraliardquo in the world hosted in the old castle ldquoFreudensteinrdquo The par-ticipants of the symposium enjoyed the visit very much
Freiberg 2009 Meinhard KunaChairman of the Symposium
International Scientific Committee
Kuna M (Freiberg chair Germany)Gabbert U (Magdeburg Germany)Huber JE (Oxford Great Britain)Mai YW (Sydney Australia)McMeeking RM (Santa Barbara USA)Rajapakse N (Burnaby Canada)Shindo Y (Sendai Japan)Yang W (Hangzhou IUTAM representative China)
Local Organizing Committee at TU Bergakademie Freiberg
Meinhard Kuna (Chair Germany)Andreas Ricoeur (Germany)
Sponsors
International Union of Theoretical and Applied MechanicsTechnische Universitat Bergakademie FreibergSpringer Verlag
Contents
A Fracture Criterion for Piezoelectric Material 1Leslie Banks-Sills and Yael Motola
What Do We Know About Surface Charges on Cracksin Ferroelectric Ceramics 9Andrea R Engert Frank Felten Hans Jelittoand Gerold A Schneider
Effects of Electric Field and Poling on Fatigue Behaviorof PZT Ceramics with Single-Edge Crack by Three-PointBending 21Yasuhide Shindo and Fumio Narita
Shape Memory Alloys Material Modeling and Device FiniteElement Simulations 33Ferdinando Auricchio Michele Conti Simone Morgantiand Alessandro Reali
Effective Computational Methods for the Modelingof Ferroelectroelastic Hysteresis Behavior 43Artem S Semenov Albrecht C Liskowsky Peter Neumeisterand Herbert Balke
Finite Element Simulation of the Non-remanent StrainingFerroelectric Material Behaviour Based on the ElectrostaticScalar Potential Convergence and Stability 55Stephan Roth Peter Neumeister Artem S Semenovand Herbert Balke
Constitutive Behavior of Nano-particle Ferroelectric Ceramics 67Li Yu Shouwen Yu and Dietmar Gross
xi
xii Contents
An Optimization-Based Computational Modelfor Polycrystalline Ferroelastics 79Faxin Li
Modeling of Domain Structure Evolution in FerroelectricMaterials 89Ralf Muller Bai Xiang Xu David Schrade and Dietmar Gross
Micromechanical Simulation of Ferroelectric DomainSwitching at Cracks 101Qun Li Marco Enderlein and Meinhard Kuna
A Phenomenological Constitutive Model for FerroelectricCeramics and Ferromagnetic Materials 111Sven Klinkel and Konrad Linnemann
The Concept of Material Forces in NonlinearElectro-elastostatics 123Duc Khoi Vu and Paul Steinmann
Permeable Interfacial Crack in Electrostrictive Materials 133Cun-Fa Gao and Yiu-Wing Mai
Some Numerical Studies with X-FEM for CrackedPiezoelectric Media 141Eric Bechet and Meinhard Kuna
Singularity Analysis of Electro-mechanical Fields in AngularlyInhomogeneous Piezoelectric Composites Wedges 153Jian-Shan Wang Xiaoqiao He and Qing-Hua Qin
Crack Propagation Simulations in Piezoelectric Structureswith an Efficient Adaptive Finite Element Tool 163Łukasz Janski Peter Steinhorst and Meinhard Kuna
Periodic Set of the Interface Cracks with LimitedElectric Permeability 175VV Loboda and SV Kozinov
Interfacial Delamination of PZT Thin Films 189Fulin Shang Yabin Yan and Takayuki Kitamura
Mechanical Behavior of Thin Film Comprised of SculpturedNano-elements 197Takayuki Kitamura Takashi Sumigawa and Taisuke Sueda
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
IUTAM BOOKSERIESVolume 24
Series Editors
GML Gladwell University of Waterloo Waterloo Ontario CanadaR Moreau INPG Grenoble France
Editorial Board
J Engelbrecht Institute of Cybernetics Tallinn EstoniaLB Freund Brown University Providence USAA Kluwick Technische Universitt Vienna AustriaHK Moffatt University of Cambridge Cambridge UKN Olhoff Aalborg University Aalborg DenmarkK Tsutomu IIDS Tokyo JapanD van Campen Technical University Eindhoven Eindhoven
The NetherlandsZ Zheng Chinese Academy of Sciences Beijing China
Aims and Scope of the Series
The IUTAM Bookseries publishes the proceedings of IUTAM symposia under theauspices of the IUTAM Board
For other titles published in this series go tohttpwwwspringercomseries7695
Meinhard Kuna Andreas RicoeurEditors
of Fatigue Damageand Fracture in SmartMaterials
Proceedings of the IUTAM Symposium onMultiscale Modelling of Fatigue Damageand Fracture in Smart Materials held inFreiberg Germany September 1ndash4 2009
123
IUTAM Symposiumon Multiscale Modelling
EditorsMeinhard KunaTU Bergakademie FreibergInst Mechanik und FluiddynamikLampadiusstr 409596 FreibergGermanyMeinhardKunaimfdtu-freibergde
Andreas RicoeurUniversitat KasselInst MechanikMonchebergstr 734109 KasselGermanyricoeuruni-kasselde
ISSN 1875-3507 e-ISSN 1875-3493ISBN 978-90-481-9886-3 e-ISBN 978-90-481-9887-0DOI 101007978-90-481-9887-0Springer Dordrecht Heidelberg London New York
Library of Congress Control Number 2010938713
c Springer Science+Business Media BV 2011No part of this work may be reproduced stored in a retrieval system or transmitted in any form or byany means electronic mechanical photocopying microfilming recording or otherwise without writtenpermission from the Publisher with the exception of any material supplied specifically for the purposeof being entered and executed on a computer system for exclusive use by the purchaser of the work
Cover design SPi Publisher Services
Printed on acid-free paper
Springer is part of Springer Science+Business Media (wwwspringercom)
Preface
Multi-functional materials as piezoelectricferroelectric ceramics magnetostrictiveand shape memory alloys are gaining increasing applications as sensors actuatorsor smart composite materials systems for promising high tech areas One primaryproblem is however that these functional materials suffer from various mechanicalandor electromagnetical degradation mechanisms as fatigue damage and fractureAs a consequence of field coupling effects fabrication processes and service loadssmart materials systems are exposed to high mechanical andor electromagneticalfield concentrations under internal and external loading of static cyclic and dy-namic type For this reason the investigation of fatigue damage and fracture playsa decisive role for the optimum design reliability and durability of smart materi-als systems Thus the topic of the symposium represents an active internationalresearch area in mechanics of materials From the experience and investigationsduring the last decade it has become evident that progress in this scientific disci-pline is only possible if material models are based on the true physical nature ofthe phenomena and if theoretical predictions are verified by skilful experimentsTherefore the scientific challenges can only be solved by
A multi-scale modelling at several length scales from atomistic to macroscopiclevel
An interdisciplinary cooperation between solid mechanics materials science andphysics
To promote the international scientific exchange in this important field in 2006the General Assembly of IUTAM approved the proposal to host this symposium inFreiberg Germany and appointed the International Scientific Committee The IU-TAM Symposium (GA 06-16) ldquoMultiscale Modelling of Fatigue Damage and Frac-ture in Smart Materials Systemsrdquo was held on September 1ndash4 2009 at TechnischeUniversitat Bergakademie Freiberg Germany organized by the Institute of Mechan-ics and Fluid Dynamics
This symposium stands in a line with former symposia on related topics heldunder the auspices of IUTAM in 2000 at Magdeburg and in 2004 at Beijing Thehelpful assistance of the International Scientific Committee to communicate thesymposium and to recommend invited speakers is thankfully appreciated
v
vi Preface
According to the rules and tradition of IUTAM the aim of the Symposium is tobring together internationally leading researchers working in the area of smart ma-terials The goal is to exchange recent scientific results to discuss new achievementand actual problems in an open and frank atmosphere The organizers were happyto welcome a lot of outstanding scientists in this field from all parts of the globeas well as many young researchers In total there were 44 participants coming from14 countries Australia (1) Austria (3) Belgium (1) Canada (1) China (8) GreatBritain (1) France (2) Germany (19) Israel (1) Italy (1) Japan (2) Slovakia (1)Ukraine (1) and USA (2) The scientific program covered 35 invited oral contribu-tions presented in 10 sessions
The following main topics have been addressed during the symposium
Development of computational methods for coupled electromechanical fieldanalysis especially extended adaptive and multi-level finite element techniquesin combination with boundary elements
Constitutive modeling of smart materials with coupled electric magnetic ther-mal and mechanical fields especially of nonlinear dissipative hysteresis behaviorMajor trend is the development of micromechanical models Especially for fer-roelectric materials and shape memory alloys the simulation of microstructure(domain switching martensitic transformation etc) are of paramount concern
Further understanding and modeling of fracture and fatigue in piezoelectric andferroelectric ceramics especially the modeling of fracture process zone and ofelectric boundary conditions at crack faces Applications of phase field simula-tion and configurational mechanics
Reliability and durability of sensors and actuators under in service loading byalternating mechanical electrical and thermal fields The role of interface cracksbetween layers and in thin films is addressed
Experimental methods to measure fracture strength and to investigate fatiguecrack growth in ferroelectric materials under electromechanical loading It hasbeen pointed out that complicated theoretical predictions have to be contrastedand verified by skilful experiments
New ferroelectric materials compounds and composites with enhanced straincapabilities
The chairman and its organizing team tried to make this IUTAM symposium notonly a successful scientific meeting but an outstanding social event too Manythanks are due to Prof Dr A Ricoeur who carried the main workload in organizingall details of this symposium
The Technische Universitat Bergakademie Freiberg is located in the East GermanFederal State Saxony Besides Dresden and Chemnitz TU Bergakademie Freibergis considered as the ldquosmallestrdquo but ldquosmartestrdquo among these Technical UniversitiesIts history started with the discovery of silver in the middle ages The ldquoMiningAcademyrdquo Freiberg was founded by the Saxon King in 1765 in order to promote thetechnologies in surveying mining and metallurgy Thus TU Bergakademie Freibergpossesses a long and famous tradition as one of the oldest montanistic universi-ties in the world Nowadays TU Bergakademie Freiberg is established as modern
Preface vii
Technical University focusing mainly on Geoscience Resources Materials ScienceEnergy and Environmental Technologies
Freiberg has one of the largest and most splendid mineralogical exhibitionsldquoTerra Mineraliardquo in the world hosted in the old castle ldquoFreudensteinrdquo The par-ticipants of the symposium enjoyed the visit very much
Freiberg 2009 Meinhard KunaChairman of the Symposium
International Scientific Committee
Kuna M (Freiberg chair Germany)Gabbert U (Magdeburg Germany)Huber JE (Oxford Great Britain)Mai YW (Sydney Australia)McMeeking RM (Santa Barbara USA)Rajapakse N (Burnaby Canada)Shindo Y (Sendai Japan)Yang W (Hangzhou IUTAM representative China)
Local Organizing Committee at TU Bergakademie Freiberg
Meinhard Kuna (Chair Germany)Andreas Ricoeur (Germany)
Sponsors
International Union of Theoretical and Applied MechanicsTechnische Universitat Bergakademie FreibergSpringer Verlag
Contents
A Fracture Criterion for Piezoelectric Material 1Leslie Banks-Sills and Yael Motola
What Do We Know About Surface Charges on Cracksin Ferroelectric Ceramics 9Andrea R Engert Frank Felten Hans Jelittoand Gerold A Schneider
Effects of Electric Field and Poling on Fatigue Behaviorof PZT Ceramics with Single-Edge Crack by Three-PointBending 21Yasuhide Shindo and Fumio Narita
Shape Memory Alloys Material Modeling and Device FiniteElement Simulations 33Ferdinando Auricchio Michele Conti Simone Morgantiand Alessandro Reali
Effective Computational Methods for the Modelingof Ferroelectroelastic Hysteresis Behavior 43Artem S Semenov Albrecht C Liskowsky Peter Neumeisterand Herbert Balke
Finite Element Simulation of the Non-remanent StrainingFerroelectric Material Behaviour Based on the ElectrostaticScalar Potential Convergence and Stability 55Stephan Roth Peter Neumeister Artem S Semenovand Herbert Balke
Constitutive Behavior of Nano-particle Ferroelectric Ceramics 67Li Yu Shouwen Yu and Dietmar Gross
xi
xii Contents
An Optimization-Based Computational Modelfor Polycrystalline Ferroelastics 79Faxin Li
Modeling of Domain Structure Evolution in FerroelectricMaterials 89Ralf Muller Bai Xiang Xu David Schrade and Dietmar Gross
Micromechanical Simulation of Ferroelectric DomainSwitching at Cracks 101Qun Li Marco Enderlein and Meinhard Kuna
A Phenomenological Constitutive Model for FerroelectricCeramics and Ferromagnetic Materials 111Sven Klinkel and Konrad Linnemann
The Concept of Material Forces in NonlinearElectro-elastostatics 123Duc Khoi Vu and Paul Steinmann
Permeable Interfacial Crack in Electrostrictive Materials 133Cun-Fa Gao and Yiu-Wing Mai
Some Numerical Studies with X-FEM for CrackedPiezoelectric Media 141Eric Bechet and Meinhard Kuna
Singularity Analysis of Electro-mechanical Fields in AngularlyInhomogeneous Piezoelectric Composites Wedges 153Jian-Shan Wang Xiaoqiao He and Qing-Hua Qin
Crack Propagation Simulations in Piezoelectric Structureswith an Efficient Adaptive Finite Element Tool 163Łukasz Janski Peter Steinhorst and Meinhard Kuna
Periodic Set of the Interface Cracks with LimitedElectric Permeability 175VV Loboda and SV Kozinov
Interfacial Delamination of PZT Thin Films 189Fulin Shang Yabin Yan and Takayuki Kitamura
Mechanical Behavior of Thin Film Comprised of SculpturedNano-elements 197Takayuki Kitamura Takashi Sumigawa and Taisuke Sueda
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
Meinhard Kuna Andreas RicoeurEditors
of Fatigue Damageand Fracture in SmartMaterials
Proceedings of the IUTAM Symposium onMultiscale Modelling of Fatigue Damageand Fracture in Smart Materials held inFreiberg Germany September 1ndash4 2009
123
IUTAM Symposiumon Multiscale Modelling
EditorsMeinhard KunaTU Bergakademie FreibergInst Mechanik und FluiddynamikLampadiusstr 409596 FreibergGermanyMeinhardKunaimfdtu-freibergde
Andreas RicoeurUniversitat KasselInst MechanikMonchebergstr 734109 KasselGermanyricoeuruni-kasselde
ISSN 1875-3507 e-ISSN 1875-3493ISBN 978-90-481-9886-3 e-ISBN 978-90-481-9887-0DOI 101007978-90-481-9887-0Springer Dordrecht Heidelberg London New York
Library of Congress Control Number 2010938713
c Springer Science+Business Media BV 2011No part of this work may be reproduced stored in a retrieval system or transmitted in any form or byany means electronic mechanical photocopying microfilming recording or otherwise without writtenpermission from the Publisher with the exception of any material supplied specifically for the purposeof being entered and executed on a computer system for exclusive use by the purchaser of the work
Cover design SPi Publisher Services
Printed on acid-free paper
Springer is part of Springer Science+Business Media (wwwspringercom)
Preface
Multi-functional materials as piezoelectricferroelectric ceramics magnetostrictiveand shape memory alloys are gaining increasing applications as sensors actuatorsor smart composite materials systems for promising high tech areas One primaryproblem is however that these functional materials suffer from various mechanicalandor electromagnetical degradation mechanisms as fatigue damage and fractureAs a consequence of field coupling effects fabrication processes and service loadssmart materials systems are exposed to high mechanical andor electromagneticalfield concentrations under internal and external loading of static cyclic and dy-namic type For this reason the investigation of fatigue damage and fracture playsa decisive role for the optimum design reliability and durability of smart materi-als systems Thus the topic of the symposium represents an active internationalresearch area in mechanics of materials From the experience and investigationsduring the last decade it has become evident that progress in this scientific disci-pline is only possible if material models are based on the true physical nature ofthe phenomena and if theoretical predictions are verified by skilful experimentsTherefore the scientific challenges can only be solved by
A multi-scale modelling at several length scales from atomistic to macroscopiclevel
An interdisciplinary cooperation between solid mechanics materials science andphysics
To promote the international scientific exchange in this important field in 2006the General Assembly of IUTAM approved the proposal to host this symposium inFreiberg Germany and appointed the International Scientific Committee The IU-TAM Symposium (GA 06-16) ldquoMultiscale Modelling of Fatigue Damage and Frac-ture in Smart Materials Systemsrdquo was held on September 1ndash4 2009 at TechnischeUniversitat Bergakademie Freiberg Germany organized by the Institute of Mechan-ics and Fluid Dynamics
This symposium stands in a line with former symposia on related topics heldunder the auspices of IUTAM in 2000 at Magdeburg and in 2004 at Beijing Thehelpful assistance of the International Scientific Committee to communicate thesymposium and to recommend invited speakers is thankfully appreciated
v
vi Preface
According to the rules and tradition of IUTAM the aim of the Symposium is tobring together internationally leading researchers working in the area of smart ma-terials The goal is to exchange recent scientific results to discuss new achievementand actual problems in an open and frank atmosphere The organizers were happyto welcome a lot of outstanding scientists in this field from all parts of the globeas well as many young researchers In total there were 44 participants coming from14 countries Australia (1) Austria (3) Belgium (1) Canada (1) China (8) GreatBritain (1) France (2) Germany (19) Israel (1) Italy (1) Japan (2) Slovakia (1)Ukraine (1) and USA (2) The scientific program covered 35 invited oral contribu-tions presented in 10 sessions
The following main topics have been addressed during the symposium
Development of computational methods for coupled electromechanical fieldanalysis especially extended adaptive and multi-level finite element techniquesin combination with boundary elements
Constitutive modeling of smart materials with coupled electric magnetic ther-mal and mechanical fields especially of nonlinear dissipative hysteresis behaviorMajor trend is the development of micromechanical models Especially for fer-roelectric materials and shape memory alloys the simulation of microstructure(domain switching martensitic transformation etc) are of paramount concern
Further understanding and modeling of fracture and fatigue in piezoelectric andferroelectric ceramics especially the modeling of fracture process zone and ofelectric boundary conditions at crack faces Applications of phase field simula-tion and configurational mechanics
Reliability and durability of sensors and actuators under in service loading byalternating mechanical electrical and thermal fields The role of interface cracksbetween layers and in thin films is addressed
Experimental methods to measure fracture strength and to investigate fatiguecrack growth in ferroelectric materials under electromechanical loading It hasbeen pointed out that complicated theoretical predictions have to be contrastedand verified by skilful experiments
New ferroelectric materials compounds and composites with enhanced straincapabilities
The chairman and its organizing team tried to make this IUTAM symposium notonly a successful scientific meeting but an outstanding social event too Manythanks are due to Prof Dr A Ricoeur who carried the main workload in organizingall details of this symposium
The Technische Universitat Bergakademie Freiberg is located in the East GermanFederal State Saxony Besides Dresden and Chemnitz TU Bergakademie Freibergis considered as the ldquosmallestrdquo but ldquosmartestrdquo among these Technical UniversitiesIts history started with the discovery of silver in the middle ages The ldquoMiningAcademyrdquo Freiberg was founded by the Saxon King in 1765 in order to promote thetechnologies in surveying mining and metallurgy Thus TU Bergakademie Freibergpossesses a long and famous tradition as one of the oldest montanistic universi-ties in the world Nowadays TU Bergakademie Freiberg is established as modern
Preface vii
Technical University focusing mainly on Geoscience Resources Materials ScienceEnergy and Environmental Technologies
Freiberg has one of the largest and most splendid mineralogical exhibitionsldquoTerra Mineraliardquo in the world hosted in the old castle ldquoFreudensteinrdquo The par-ticipants of the symposium enjoyed the visit very much
Freiberg 2009 Meinhard KunaChairman of the Symposium
International Scientific Committee
Kuna M (Freiberg chair Germany)Gabbert U (Magdeburg Germany)Huber JE (Oxford Great Britain)Mai YW (Sydney Australia)McMeeking RM (Santa Barbara USA)Rajapakse N (Burnaby Canada)Shindo Y (Sendai Japan)Yang W (Hangzhou IUTAM representative China)
Local Organizing Committee at TU Bergakademie Freiberg
Meinhard Kuna (Chair Germany)Andreas Ricoeur (Germany)
Sponsors
International Union of Theoretical and Applied MechanicsTechnische Universitat Bergakademie FreibergSpringer Verlag
Contents
A Fracture Criterion for Piezoelectric Material 1Leslie Banks-Sills and Yael Motola
What Do We Know About Surface Charges on Cracksin Ferroelectric Ceramics 9Andrea R Engert Frank Felten Hans Jelittoand Gerold A Schneider
Effects of Electric Field and Poling on Fatigue Behaviorof PZT Ceramics with Single-Edge Crack by Three-PointBending 21Yasuhide Shindo and Fumio Narita
Shape Memory Alloys Material Modeling and Device FiniteElement Simulations 33Ferdinando Auricchio Michele Conti Simone Morgantiand Alessandro Reali
Effective Computational Methods for the Modelingof Ferroelectroelastic Hysteresis Behavior 43Artem S Semenov Albrecht C Liskowsky Peter Neumeisterand Herbert Balke
Finite Element Simulation of the Non-remanent StrainingFerroelectric Material Behaviour Based on the ElectrostaticScalar Potential Convergence and Stability 55Stephan Roth Peter Neumeister Artem S Semenovand Herbert Balke
Constitutive Behavior of Nano-particle Ferroelectric Ceramics 67Li Yu Shouwen Yu and Dietmar Gross
xi
xii Contents
An Optimization-Based Computational Modelfor Polycrystalline Ferroelastics 79Faxin Li
Modeling of Domain Structure Evolution in FerroelectricMaterials 89Ralf Muller Bai Xiang Xu David Schrade and Dietmar Gross
Micromechanical Simulation of Ferroelectric DomainSwitching at Cracks 101Qun Li Marco Enderlein and Meinhard Kuna
A Phenomenological Constitutive Model for FerroelectricCeramics and Ferromagnetic Materials 111Sven Klinkel and Konrad Linnemann
The Concept of Material Forces in NonlinearElectro-elastostatics 123Duc Khoi Vu and Paul Steinmann
Permeable Interfacial Crack in Electrostrictive Materials 133Cun-Fa Gao and Yiu-Wing Mai
Some Numerical Studies with X-FEM for CrackedPiezoelectric Media 141Eric Bechet and Meinhard Kuna
Singularity Analysis of Electro-mechanical Fields in AngularlyInhomogeneous Piezoelectric Composites Wedges 153Jian-Shan Wang Xiaoqiao He and Qing-Hua Qin
Crack Propagation Simulations in Piezoelectric Structureswith an Efficient Adaptive Finite Element Tool 163Łukasz Janski Peter Steinhorst and Meinhard Kuna
Periodic Set of the Interface Cracks with LimitedElectric Permeability 175VV Loboda and SV Kozinov
Interfacial Delamination of PZT Thin Films 189Fulin Shang Yabin Yan and Takayuki Kitamura
Mechanical Behavior of Thin Film Comprised of SculpturedNano-elements 197Takayuki Kitamura Takashi Sumigawa and Taisuke Sueda
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
EditorsMeinhard KunaTU Bergakademie FreibergInst Mechanik und FluiddynamikLampadiusstr 409596 FreibergGermanyMeinhardKunaimfdtu-freibergde
Andreas RicoeurUniversitat KasselInst MechanikMonchebergstr 734109 KasselGermanyricoeuruni-kasselde
ISSN 1875-3507 e-ISSN 1875-3493ISBN 978-90-481-9886-3 e-ISBN 978-90-481-9887-0DOI 101007978-90-481-9887-0Springer Dordrecht Heidelberg London New York
Library of Congress Control Number 2010938713
c Springer Science+Business Media BV 2011No part of this work may be reproduced stored in a retrieval system or transmitted in any form or byany means electronic mechanical photocopying microfilming recording or otherwise without writtenpermission from the Publisher with the exception of any material supplied specifically for the purposeof being entered and executed on a computer system for exclusive use by the purchaser of the work
Cover design SPi Publisher Services
Printed on acid-free paper
Springer is part of Springer Science+Business Media (wwwspringercom)
Preface
Multi-functional materials as piezoelectricferroelectric ceramics magnetostrictiveand shape memory alloys are gaining increasing applications as sensors actuatorsor smart composite materials systems for promising high tech areas One primaryproblem is however that these functional materials suffer from various mechanicalandor electromagnetical degradation mechanisms as fatigue damage and fractureAs a consequence of field coupling effects fabrication processes and service loadssmart materials systems are exposed to high mechanical andor electromagneticalfield concentrations under internal and external loading of static cyclic and dy-namic type For this reason the investigation of fatigue damage and fracture playsa decisive role for the optimum design reliability and durability of smart materi-als systems Thus the topic of the symposium represents an active internationalresearch area in mechanics of materials From the experience and investigationsduring the last decade it has become evident that progress in this scientific disci-pline is only possible if material models are based on the true physical nature ofthe phenomena and if theoretical predictions are verified by skilful experimentsTherefore the scientific challenges can only be solved by
A multi-scale modelling at several length scales from atomistic to macroscopiclevel
An interdisciplinary cooperation between solid mechanics materials science andphysics
To promote the international scientific exchange in this important field in 2006the General Assembly of IUTAM approved the proposal to host this symposium inFreiberg Germany and appointed the International Scientific Committee The IU-TAM Symposium (GA 06-16) ldquoMultiscale Modelling of Fatigue Damage and Frac-ture in Smart Materials Systemsrdquo was held on September 1ndash4 2009 at TechnischeUniversitat Bergakademie Freiberg Germany organized by the Institute of Mechan-ics and Fluid Dynamics
This symposium stands in a line with former symposia on related topics heldunder the auspices of IUTAM in 2000 at Magdeburg and in 2004 at Beijing Thehelpful assistance of the International Scientific Committee to communicate thesymposium and to recommend invited speakers is thankfully appreciated
v
vi Preface
According to the rules and tradition of IUTAM the aim of the Symposium is tobring together internationally leading researchers working in the area of smart ma-terials The goal is to exchange recent scientific results to discuss new achievementand actual problems in an open and frank atmosphere The organizers were happyto welcome a lot of outstanding scientists in this field from all parts of the globeas well as many young researchers In total there were 44 participants coming from14 countries Australia (1) Austria (3) Belgium (1) Canada (1) China (8) GreatBritain (1) France (2) Germany (19) Israel (1) Italy (1) Japan (2) Slovakia (1)Ukraine (1) and USA (2) The scientific program covered 35 invited oral contribu-tions presented in 10 sessions
The following main topics have been addressed during the symposium
Development of computational methods for coupled electromechanical fieldanalysis especially extended adaptive and multi-level finite element techniquesin combination with boundary elements
Constitutive modeling of smart materials with coupled electric magnetic ther-mal and mechanical fields especially of nonlinear dissipative hysteresis behaviorMajor trend is the development of micromechanical models Especially for fer-roelectric materials and shape memory alloys the simulation of microstructure(domain switching martensitic transformation etc) are of paramount concern
Further understanding and modeling of fracture and fatigue in piezoelectric andferroelectric ceramics especially the modeling of fracture process zone and ofelectric boundary conditions at crack faces Applications of phase field simula-tion and configurational mechanics
Reliability and durability of sensors and actuators under in service loading byalternating mechanical electrical and thermal fields The role of interface cracksbetween layers and in thin films is addressed
Experimental methods to measure fracture strength and to investigate fatiguecrack growth in ferroelectric materials under electromechanical loading It hasbeen pointed out that complicated theoretical predictions have to be contrastedand verified by skilful experiments
New ferroelectric materials compounds and composites with enhanced straincapabilities
The chairman and its organizing team tried to make this IUTAM symposium notonly a successful scientific meeting but an outstanding social event too Manythanks are due to Prof Dr A Ricoeur who carried the main workload in organizingall details of this symposium
The Technische Universitat Bergakademie Freiberg is located in the East GermanFederal State Saxony Besides Dresden and Chemnitz TU Bergakademie Freibergis considered as the ldquosmallestrdquo but ldquosmartestrdquo among these Technical UniversitiesIts history started with the discovery of silver in the middle ages The ldquoMiningAcademyrdquo Freiberg was founded by the Saxon King in 1765 in order to promote thetechnologies in surveying mining and metallurgy Thus TU Bergakademie Freibergpossesses a long and famous tradition as one of the oldest montanistic universi-ties in the world Nowadays TU Bergakademie Freiberg is established as modern
Preface vii
Technical University focusing mainly on Geoscience Resources Materials ScienceEnergy and Environmental Technologies
Freiberg has one of the largest and most splendid mineralogical exhibitionsldquoTerra Mineraliardquo in the world hosted in the old castle ldquoFreudensteinrdquo The par-ticipants of the symposium enjoyed the visit very much
Freiberg 2009 Meinhard KunaChairman of the Symposium
International Scientific Committee
Kuna M (Freiberg chair Germany)Gabbert U (Magdeburg Germany)Huber JE (Oxford Great Britain)Mai YW (Sydney Australia)McMeeking RM (Santa Barbara USA)Rajapakse N (Burnaby Canada)Shindo Y (Sendai Japan)Yang W (Hangzhou IUTAM representative China)
Local Organizing Committee at TU Bergakademie Freiberg
Meinhard Kuna (Chair Germany)Andreas Ricoeur (Germany)
Sponsors
International Union of Theoretical and Applied MechanicsTechnische Universitat Bergakademie FreibergSpringer Verlag
Contents
A Fracture Criterion for Piezoelectric Material 1Leslie Banks-Sills and Yael Motola
What Do We Know About Surface Charges on Cracksin Ferroelectric Ceramics 9Andrea R Engert Frank Felten Hans Jelittoand Gerold A Schneider
Effects of Electric Field and Poling on Fatigue Behaviorof PZT Ceramics with Single-Edge Crack by Three-PointBending 21Yasuhide Shindo and Fumio Narita
Shape Memory Alloys Material Modeling and Device FiniteElement Simulations 33Ferdinando Auricchio Michele Conti Simone Morgantiand Alessandro Reali
Effective Computational Methods for the Modelingof Ferroelectroelastic Hysteresis Behavior 43Artem S Semenov Albrecht C Liskowsky Peter Neumeisterand Herbert Balke
Finite Element Simulation of the Non-remanent StrainingFerroelectric Material Behaviour Based on the ElectrostaticScalar Potential Convergence and Stability 55Stephan Roth Peter Neumeister Artem S Semenovand Herbert Balke
Constitutive Behavior of Nano-particle Ferroelectric Ceramics 67Li Yu Shouwen Yu and Dietmar Gross
xi
xii Contents
An Optimization-Based Computational Modelfor Polycrystalline Ferroelastics 79Faxin Li
Modeling of Domain Structure Evolution in FerroelectricMaterials 89Ralf Muller Bai Xiang Xu David Schrade and Dietmar Gross
Micromechanical Simulation of Ferroelectric DomainSwitching at Cracks 101Qun Li Marco Enderlein and Meinhard Kuna
A Phenomenological Constitutive Model for FerroelectricCeramics and Ferromagnetic Materials 111Sven Klinkel and Konrad Linnemann
The Concept of Material Forces in NonlinearElectro-elastostatics 123Duc Khoi Vu and Paul Steinmann
Permeable Interfacial Crack in Electrostrictive Materials 133Cun-Fa Gao and Yiu-Wing Mai
Some Numerical Studies with X-FEM for CrackedPiezoelectric Media 141Eric Bechet and Meinhard Kuna
Singularity Analysis of Electro-mechanical Fields in AngularlyInhomogeneous Piezoelectric Composites Wedges 153Jian-Shan Wang Xiaoqiao He and Qing-Hua Qin
Crack Propagation Simulations in Piezoelectric Structureswith an Efficient Adaptive Finite Element Tool 163Łukasz Janski Peter Steinhorst and Meinhard Kuna
Periodic Set of the Interface Cracks with LimitedElectric Permeability 175VV Loboda and SV Kozinov
Interfacial Delamination of PZT Thin Films 189Fulin Shang Yabin Yan and Takayuki Kitamura
Mechanical Behavior of Thin Film Comprised of SculpturedNano-elements 197Takayuki Kitamura Takashi Sumigawa and Taisuke Sueda
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
Preface
Multi-functional materials as piezoelectricferroelectric ceramics magnetostrictiveand shape memory alloys are gaining increasing applications as sensors actuatorsor smart composite materials systems for promising high tech areas One primaryproblem is however that these functional materials suffer from various mechanicalandor electromagnetical degradation mechanisms as fatigue damage and fractureAs a consequence of field coupling effects fabrication processes and service loadssmart materials systems are exposed to high mechanical andor electromagneticalfield concentrations under internal and external loading of static cyclic and dy-namic type For this reason the investigation of fatigue damage and fracture playsa decisive role for the optimum design reliability and durability of smart materi-als systems Thus the topic of the symposium represents an active internationalresearch area in mechanics of materials From the experience and investigationsduring the last decade it has become evident that progress in this scientific disci-pline is only possible if material models are based on the true physical nature ofthe phenomena and if theoretical predictions are verified by skilful experimentsTherefore the scientific challenges can only be solved by
A multi-scale modelling at several length scales from atomistic to macroscopiclevel
An interdisciplinary cooperation between solid mechanics materials science andphysics
To promote the international scientific exchange in this important field in 2006the General Assembly of IUTAM approved the proposal to host this symposium inFreiberg Germany and appointed the International Scientific Committee The IU-TAM Symposium (GA 06-16) ldquoMultiscale Modelling of Fatigue Damage and Frac-ture in Smart Materials Systemsrdquo was held on September 1ndash4 2009 at TechnischeUniversitat Bergakademie Freiberg Germany organized by the Institute of Mechan-ics and Fluid Dynamics
This symposium stands in a line with former symposia on related topics heldunder the auspices of IUTAM in 2000 at Magdeburg and in 2004 at Beijing Thehelpful assistance of the International Scientific Committee to communicate thesymposium and to recommend invited speakers is thankfully appreciated
v
vi Preface
According to the rules and tradition of IUTAM the aim of the Symposium is tobring together internationally leading researchers working in the area of smart ma-terials The goal is to exchange recent scientific results to discuss new achievementand actual problems in an open and frank atmosphere The organizers were happyto welcome a lot of outstanding scientists in this field from all parts of the globeas well as many young researchers In total there were 44 participants coming from14 countries Australia (1) Austria (3) Belgium (1) Canada (1) China (8) GreatBritain (1) France (2) Germany (19) Israel (1) Italy (1) Japan (2) Slovakia (1)Ukraine (1) and USA (2) The scientific program covered 35 invited oral contribu-tions presented in 10 sessions
The following main topics have been addressed during the symposium
Development of computational methods for coupled electromechanical fieldanalysis especially extended adaptive and multi-level finite element techniquesin combination with boundary elements
Constitutive modeling of smart materials with coupled electric magnetic ther-mal and mechanical fields especially of nonlinear dissipative hysteresis behaviorMajor trend is the development of micromechanical models Especially for fer-roelectric materials and shape memory alloys the simulation of microstructure(domain switching martensitic transformation etc) are of paramount concern
Further understanding and modeling of fracture and fatigue in piezoelectric andferroelectric ceramics especially the modeling of fracture process zone and ofelectric boundary conditions at crack faces Applications of phase field simula-tion and configurational mechanics
Reliability and durability of sensors and actuators under in service loading byalternating mechanical electrical and thermal fields The role of interface cracksbetween layers and in thin films is addressed
Experimental methods to measure fracture strength and to investigate fatiguecrack growth in ferroelectric materials under electromechanical loading It hasbeen pointed out that complicated theoretical predictions have to be contrastedand verified by skilful experiments
New ferroelectric materials compounds and composites with enhanced straincapabilities
The chairman and its organizing team tried to make this IUTAM symposium notonly a successful scientific meeting but an outstanding social event too Manythanks are due to Prof Dr A Ricoeur who carried the main workload in organizingall details of this symposium
The Technische Universitat Bergakademie Freiberg is located in the East GermanFederal State Saxony Besides Dresden and Chemnitz TU Bergakademie Freibergis considered as the ldquosmallestrdquo but ldquosmartestrdquo among these Technical UniversitiesIts history started with the discovery of silver in the middle ages The ldquoMiningAcademyrdquo Freiberg was founded by the Saxon King in 1765 in order to promote thetechnologies in surveying mining and metallurgy Thus TU Bergakademie Freibergpossesses a long and famous tradition as one of the oldest montanistic universi-ties in the world Nowadays TU Bergakademie Freiberg is established as modern
Preface vii
Technical University focusing mainly on Geoscience Resources Materials ScienceEnergy and Environmental Technologies
Freiberg has one of the largest and most splendid mineralogical exhibitionsldquoTerra Mineraliardquo in the world hosted in the old castle ldquoFreudensteinrdquo The par-ticipants of the symposium enjoyed the visit very much
Freiberg 2009 Meinhard KunaChairman of the Symposium
International Scientific Committee
Kuna M (Freiberg chair Germany)Gabbert U (Magdeburg Germany)Huber JE (Oxford Great Britain)Mai YW (Sydney Australia)McMeeking RM (Santa Barbara USA)Rajapakse N (Burnaby Canada)Shindo Y (Sendai Japan)Yang W (Hangzhou IUTAM representative China)
Local Organizing Committee at TU Bergakademie Freiberg
Meinhard Kuna (Chair Germany)Andreas Ricoeur (Germany)
Sponsors
International Union of Theoretical and Applied MechanicsTechnische Universitat Bergakademie FreibergSpringer Verlag
Contents
A Fracture Criterion for Piezoelectric Material 1Leslie Banks-Sills and Yael Motola
What Do We Know About Surface Charges on Cracksin Ferroelectric Ceramics 9Andrea R Engert Frank Felten Hans Jelittoand Gerold A Schneider
Effects of Electric Field and Poling on Fatigue Behaviorof PZT Ceramics with Single-Edge Crack by Three-PointBending 21Yasuhide Shindo and Fumio Narita
Shape Memory Alloys Material Modeling and Device FiniteElement Simulations 33Ferdinando Auricchio Michele Conti Simone Morgantiand Alessandro Reali
Effective Computational Methods for the Modelingof Ferroelectroelastic Hysteresis Behavior 43Artem S Semenov Albrecht C Liskowsky Peter Neumeisterand Herbert Balke
Finite Element Simulation of the Non-remanent StrainingFerroelectric Material Behaviour Based on the ElectrostaticScalar Potential Convergence and Stability 55Stephan Roth Peter Neumeister Artem S Semenovand Herbert Balke
Constitutive Behavior of Nano-particle Ferroelectric Ceramics 67Li Yu Shouwen Yu and Dietmar Gross
xi
xii Contents
An Optimization-Based Computational Modelfor Polycrystalline Ferroelastics 79Faxin Li
Modeling of Domain Structure Evolution in FerroelectricMaterials 89Ralf Muller Bai Xiang Xu David Schrade and Dietmar Gross
Micromechanical Simulation of Ferroelectric DomainSwitching at Cracks 101Qun Li Marco Enderlein and Meinhard Kuna
A Phenomenological Constitutive Model for FerroelectricCeramics and Ferromagnetic Materials 111Sven Klinkel and Konrad Linnemann
The Concept of Material Forces in NonlinearElectro-elastostatics 123Duc Khoi Vu and Paul Steinmann
Permeable Interfacial Crack in Electrostrictive Materials 133Cun-Fa Gao and Yiu-Wing Mai
Some Numerical Studies with X-FEM for CrackedPiezoelectric Media 141Eric Bechet and Meinhard Kuna
Singularity Analysis of Electro-mechanical Fields in AngularlyInhomogeneous Piezoelectric Composites Wedges 153Jian-Shan Wang Xiaoqiao He and Qing-Hua Qin
Crack Propagation Simulations in Piezoelectric Structureswith an Efficient Adaptive Finite Element Tool 163Łukasz Janski Peter Steinhorst and Meinhard Kuna
Periodic Set of the Interface Cracks with LimitedElectric Permeability 175VV Loboda and SV Kozinov
Interfacial Delamination of PZT Thin Films 189Fulin Shang Yabin Yan and Takayuki Kitamura
Mechanical Behavior of Thin Film Comprised of SculpturedNano-elements 197Takayuki Kitamura Takashi Sumigawa and Taisuke Sueda
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
vi Preface
According to the rules and tradition of IUTAM the aim of the Symposium is tobring together internationally leading researchers working in the area of smart ma-terials The goal is to exchange recent scientific results to discuss new achievementand actual problems in an open and frank atmosphere The organizers were happyto welcome a lot of outstanding scientists in this field from all parts of the globeas well as many young researchers In total there were 44 participants coming from14 countries Australia (1) Austria (3) Belgium (1) Canada (1) China (8) GreatBritain (1) France (2) Germany (19) Israel (1) Italy (1) Japan (2) Slovakia (1)Ukraine (1) and USA (2) The scientific program covered 35 invited oral contribu-tions presented in 10 sessions
The following main topics have been addressed during the symposium
Development of computational methods for coupled electromechanical fieldanalysis especially extended adaptive and multi-level finite element techniquesin combination with boundary elements
Constitutive modeling of smart materials with coupled electric magnetic ther-mal and mechanical fields especially of nonlinear dissipative hysteresis behaviorMajor trend is the development of micromechanical models Especially for fer-roelectric materials and shape memory alloys the simulation of microstructure(domain switching martensitic transformation etc) are of paramount concern
Further understanding and modeling of fracture and fatigue in piezoelectric andferroelectric ceramics especially the modeling of fracture process zone and ofelectric boundary conditions at crack faces Applications of phase field simula-tion and configurational mechanics
Reliability and durability of sensors and actuators under in service loading byalternating mechanical electrical and thermal fields The role of interface cracksbetween layers and in thin films is addressed
Experimental methods to measure fracture strength and to investigate fatiguecrack growth in ferroelectric materials under electromechanical loading It hasbeen pointed out that complicated theoretical predictions have to be contrastedand verified by skilful experiments
New ferroelectric materials compounds and composites with enhanced straincapabilities
The chairman and its organizing team tried to make this IUTAM symposium notonly a successful scientific meeting but an outstanding social event too Manythanks are due to Prof Dr A Ricoeur who carried the main workload in organizingall details of this symposium
The Technische Universitat Bergakademie Freiberg is located in the East GermanFederal State Saxony Besides Dresden and Chemnitz TU Bergakademie Freibergis considered as the ldquosmallestrdquo but ldquosmartestrdquo among these Technical UniversitiesIts history started with the discovery of silver in the middle ages The ldquoMiningAcademyrdquo Freiberg was founded by the Saxon King in 1765 in order to promote thetechnologies in surveying mining and metallurgy Thus TU Bergakademie Freibergpossesses a long and famous tradition as one of the oldest montanistic universi-ties in the world Nowadays TU Bergakademie Freiberg is established as modern
Preface vii
Technical University focusing mainly on Geoscience Resources Materials ScienceEnergy and Environmental Technologies
Freiberg has one of the largest and most splendid mineralogical exhibitionsldquoTerra Mineraliardquo in the world hosted in the old castle ldquoFreudensteinrdquo The par-ticipants of the symposium enjoyed the visit very much
Freiberg 2009 Meinhard KunaChairman of the Symposium
International Scientific Committee
Kuna M (Freiberg chair Germany)Gabbert U (Magdeburg Germany)Huber JE (Oxford Great Britain)Mai YW (Sydney Australia)McMeeking RM (Santa Barbara USA)Rajapakse N (Burnaby Canada)Shindo Y (Sendai Japan)Yang W (Hangzhou IUTAM representative China)
Local Organizing Committee at TU Bergakademie Freiberg
Meinhard Kuna (Chair Germany)Andreas Ricoeur (Germany)
Sponsors
International Union of Theoretical and Applied MechanicsTechnische Universitat Bergakademie FreibergSpringer Verlag
Contents
A Fracture Criterion for Piezoelectric Material 1Leslie Banks-Sills and Yael Motola
What Do We Know About Surface Charges on Cracksin Ferroelectric Ceramics 9Andrea R Engert Frank Felten Hans Jelittoand Gerold A Schneider
Effects of Electric Field and Poling on Fatigue Behaviorof PZT Ceramics with Single-Edge Crack by Three-PointBending 21Yasuhide Shindo and Fumio Narita
Shape Memory Alloys Material Modeling and Device FiniteElement Simulations 33Ferdinando Auricchio Michele Conti Simone Morgantiand Alessandro Reali
Effective Computational Methods for the Modelingof Ferroelectroelastic Hysteresis Behavior 43Artem S Semenov Albrecht C Liskowsky Peter Neumeisterand Herbert Balke
Finite Element Simulation of the Non-remanent StrainingFerroelectric Material Behaviour Based on the ElectrostaticScalar Potential Convergence and Stability 55Stephan Roth Peter Neumeister Artem S Semenovand Herbert Balke
Constitutive Behavior of Nano-particle Ferroelectric Ceramics 67Li Yu Shouwen Yu and Dietmar Gross
xi
xii Contents
An Optimization-Based Computational Modelfor Polycrystalline Ferroelastics 79Faxin Li
Modeling of Domain Structure Evolution in FerroelectricMaterials 89Ralf Muller Bai Xiang Xu David Schrade and Dietmar Gross
Micromechanical Simulation of Ferroelectric DomainSwitching at Cracks 101Qun Li Marco Enderlein and Meinhard Kuna
A Phenomenological Constitutive Model for FerroelectricCeramics and Ferromagnetic Materials 111Sven Klinkel and Konrad Linnemann
The Concept of Material Forces in NonlinearElectro-elastostatics 123Duc Khoi Vu and Paul Steinmann
Permeable Interfacial Crack in Electrostrictive Materials 133Cun-Fa Gao and Yiu-Wing Mai
Some Numerical Studies with X-FEM for CrackedPiezoelectric Media 141Eric Bechet and Meinhard Kuna
Singularity Analysis of Electro-mechanical Fields in AngularlyInhomogeneous Piezoelectric Composites Wedges 153Jian-Shan Wang Xiaoqiao He and Qing-Hua Qin
Crack Propagation Simulations in Piezoelectric Structureswith an Efficient Adaptive Finite Element Tool 163Łukasz Janski Peter Steinhorst and Meinhard Kuna
Periodic Set of the Interface Cracks with LimitedElectric Permeability 175VV Loboda and SV Kozinov
Interfacial Delamination of PZT Thin Films 189Fulin Shang Yabin Yan and Takayuki Kitamura
Mechanical Behavior of Thin Film Comprised of SculpturedNano-elements 197Takayuki Kitamura Takashi Sumigawa and Taisuke Sueda
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
Preface vii
Technical University focusing mainly on Geoscience Resources Materials ScienceEnergy and Environmental Technologies
Freiberg has one of the largest and most splendid mineralogical exhibitionsldquoTerra Mineraliardquo in the world hosted in the old castle ldquoFreudensteinrdquo The par-ticipants of the symposium enjoyed the visit very much
Freiberg 2009 Meinhard KunaChairman of the Symposium
International Scientific Committee
Kuna M (Freiberg chair Germany)Gabbert U (Magdeburg Germany)Huber JE (Oxford Great Britain)Mai YW (Sydney Australia)McMeeking RM (Santa Barbara USA)Rajapakse N (Burnaby Canada)Shindo Y (Sendai Japan)Yang W (Hangzhou IUTAM representative China)
Local Organizing Committee at TU Bergakademie Freiberg
Meinhard Kuna (Chair Germany)Andreas Ricoeur (Germany)
Sponsors
International Union of Theoretical and Applied MechanicsTechnische Universitat Bergakademie FreibergSpringer Verlag
Contents
A Fracture Criterion for Piezoelectric Material 1Leslie Banks-Sills and Yael Motola
What Do We Know About Surface Charges on Cracksin Ferroelectric Ceramics 9Andrea R Engert Frank Felten Hans Jelittoand Gerold A Schneider
Effects of Electric Field and Poling on Fatigue Behaviorof PZT Ceramics with Single-Edge Crack by Three-PointBending 21Yasuhide Shindo and Fumio Narita
Shape Memory Alloys Material Modeling and Device FiniteElement Simulations 33Ferdinando Auricchio Michele Conti Simone Morgantiand Alessandro Reali
Effective Computational Methods for the Modelingof Ferroelectroelastic Hysteresis Behavior 43Artem S Semenov Albrecht C Liskowsky Peter Neumeisterand Herbert Balke
Finite Element Simulation of the Non-remanent StrainingFerroelectric Material Behaviour Based on the ElectrostaticScalar Potential Convergence and Stability 55Stephan Roth Peter Neumeister Artem S Semenovand Herbert Balke
Constitutive Behavior of Nano-particle Ferroelectric Ceramics 67Li Yu Shouwen Yu and Dietmar Gross
xi
xii Contents
An Optimization-Based Computational Modelfor Polycrystalline Ferroelastics 79Faxin Li
Modeling of Domain Structure Evolution in FerroelectricMaterials 89Ralf Muller Bai Xiang Xu David Schrade and Dietmar Gross
Micromechanical Simulation of Ferroelectric DomainSwitching at Cracks 101Qun Li Marco Enderlein and Meinhard Kuna
A Phenomenological Constitutive Model for FerroelectricCeramics and Ferromagnetic Materials 111Sven Klinkel and Konrad Linnemann
The Concept of Material Forces in NonlinearElectro-elastostatics 123Duc Khoi Vu and Paul Steinmann
Permeable Interfacial Crack in Electrostrictive Materials 133Cun-Fa Gao and Yiu-Wing Mai
Some Numerical Studies with X-FEM for CrackedPiezoelectric Media 141Eric Bechet and Meinhard Kuna
Singularity Analysis of Electro-mechanical Fields in AngularlyInhomogeneous Piezoelectric Composites Wedges 153Jian-Shan Wang Xiaoqiao He and Qing-Hua Qin
Crack Propagation Simulations in Piezoelectric Structureswith an Efficient Adaptive Finite Element Tool 163Łukasz Janski Peter Steinhorst and Meinhard Kuna
Periodic Set of the Interface Cracks with LimitedElectric Permeability 175VV Loboda and SV Kozinov
Interfacial Delamination of PZT Thin Films 189Fulin Shang Yabin Yan and Takayuki Kitamura
Mechanical Behavior of Thin Film Comprised of SculpturedNano-elements 197Takayuki Kitamura Takashi Sumigawa and Taisuke Sueda
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
International Scientific Committee
Kuna M (Freiberg chair Germany)Gabbert U (Magdeburg Germany)Huber JE (Oxford Great Britain)Mai YW (Sydney Australia)McMeeking RM (Santa Barbara USA)Rajapakse N (Burnaby Canada)Shindo Y (Sendai Japan)Yang W (Hangzhou IUTAM representative China)
Local Organizing Committee at TU Bergakademie Freiberg
Meinhard Kuna (Chair Germany)Andreas Ricoeur (Germany)
Sponsors
International Union of Theoretical and Applied MechanicsTechnische Universitat Bergakademie FreibergSpringer Verlag
Contents
A Fracture Criterion for Piezoelectric Material 1Leslie Banks-Sills and Yael Motola
What Do We Know About Surface Charges on Cracksin Ferroelectric Ceramics 9Andrea R Engert Frank Felten Hans Jelittoand Gerold A Schneider
Effects of Electric Field and Poling on Fatigue Behaviorof PZT Ceramics with Single-Edge Crack by Three-PointBending 21Yasuhide Shindo and Fumio Narita
Shape Memory Alloys Material Modeling and Device FiniteElement Simulations 33Ferdinando Auricchio Michele Conti Simone Morgantiand Alessandro Reali
Effective Computational Methods for the Modelingof Ferroelectroelastic Hysteresis Behavior 43Artem S Semenov Albrecht C Liskowsky Peter Neumeisterand Herbert Balke
Finite Element Simulation of the Non-remanent StrainingFerroelectric Material Behaviour Based on the ElectrostaticScalar Potential Convergence and Stability 55Stephan Roth Peter Neumeister Artem S Semenovand Herbert Balke
Constitutive Behavior of Nano-particle Ferroelectric Ceramics 67Li Yu Shouwen Yu and Dietmar Gross
xi
xii Contents
An Optimization-Based Computational Modelfor Polycrystalline Ferroelastics 79Faxin Li
Modeling of Domain Structure Evolution in FerroelectricMaterials 89Ralf Muller Bai Xiang Xu David Schrade and Dietmar Gross
Micromechanical Simulation of Ferroelectric DomainSwitching at Cracks 101Qun Li Marco Enderlein and Meinhard Kuna
A Phenomenological Constitutive Model for FerroelectricCeramics and Ferromagnetic Materials 111Sven Klinkel and Konrad Linnemann
The Concept of Material Forces in NonlinearElectro-elastostatics 123Duc Khoi Vu and Paul Steinmann
Permeable Interfacial Crack in Electrostrictive Materials 133Cun-Fa Gao and Yiu-Wing Mai
Some Numerical Studies with X-FEM for CrackedPiezoelectric Media 141Eric Bechet and Meinhard Kuna
Singularity Analysis of Electro-mechanical Fields in AngularlyInhomogeneous Piezoelectric Composites Wedges 153Jian-Shan Wang Xiaoqiao He and Qing-Hua Qin
Crack Propagation Simulations in Piezoelectric Structureswith an Efficient Adaptive Finite Element Tool 163Łukasz Janski Peter Steinhorst and Meinhard Kuna
Periodic Set of the Interface Cracks with LimitedElectric Permeability 175VV Loboda and SV Kozinov
Interfacial Delamination of PZT Thin Films 189Fulin Shang Yabin Yan and Takayuki Kitamura
Mechanical Behavior of Thin Film Comprised of SculpturedNano-elements 197Takayuki Kitamura Takashi Sumigawa and Taisuke Sueda
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
Contents
A Fracture Criterion for Piezoelectric Material 1Leslie Banks-Sills and Yael Motola
What Do We Know About Surface Charges on Cracksin Ferroelectric Ceramics 9Andrea R Engert Frank Felten Hans Jelittoand Gerold A Schneider
Effects of Electric Field and Poling on Fatigue Behaviorof PZT Ceramics with Single-Edge Crack by Three-PointBending 21Yasuhide Shindo and Fumio Narita
Shape Memory Alloys Material Modeling and Device FiniteElement Simulations 33Ferdinando Auricchio Michele Conti Simone Morgantiand Alessandro Reali
Effective Computational Methods for the Modelingof Ferroelectroelastic Hysteresis Behavior 43Artem S Semenov Albrecht C Liskowsky Peter Neumeisterand Herbert Balke
Finite Element Simulation of the Non-remanent StrainingFerroelectric Material Behaviour Based on the ElectrostaticScalar Potential Convergence and Stability 55Stephan Roth Peter Neumeister Artem S Semenovand Herbert Balke
Constitutive Behavior of Nano-particle Ferroelectric Ceramics 67Li Yu Shouwen Yu and Dietmar Gross
xi
xii Contents
An Optimization-Based Computational Modelfor Polycrystalline Ferroelastics 79Faxin Li
Modeling of Domain Structure Evolution in FerroelectricMaterials 89Ralf Muller Bai Xiang Xu David Schrade and Dietmar Gross
Micromechanical Simulation of Ferroelectric DomainSwitching at Cracks 101Qun Li Marco Enderlein and Meinhard Kuna
A Phenomenological Constitutive Model for FerroelectricCeramics and Ferromagnetic Materials 111Sven Klinkel and Konrad Linnemann
The Concept of Material Forces in NonlinearElectro-elastostatics 123Duc Khoi Vu and Paul Steinmann
Permeable Interfacial Crack in Electrostrictive Materials 133Cun-Fa Gao and Yiu-Wing Mai
Some Numerical Studies with X-FEM for CrackedPiezoelectric Media 141Eric Bechet and Meinhard Kuna
Singularity Analysis of Electro-mechanical Fields in AngularlyInhomogeneous Piezoelectric Composites Wedges 153Jian-Shan Wang Xiaoqiao He and Qing-Hua Qin
Crack Propagation Simulations in Piezoelectric Structureswith an Efficient Adaptive Finite Element Tool 163Łukasz Janski Peter Steinhorst and Meinhard Kuna
Periodic Set of the Interface Cracks with LimitedElectric Permeability 175VV Loboda and SV Kozinov
Interfacial Delamination of PZT Thin Films 189Fulin Shang Yabin Yan and Takayuki Kitamura
Mechanical Behavior of Thin Film Comprised of SculpturedNano-elements 197Takayuki Kitamura Takashi Sumigawa and Taisuke Sueda
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
xii Contents
An Optimization-Based Computational Modelfor Polycrystalline Ferroelastics 79Faxin Li
Modeling of Domain Structure Evolution in FerroelectricMaterials 89Ralf Muller Bai Xiang Xu David Schrade and Dietmar Gross
Micromechanical Simulation of Ferroelectric DomainSwitching at Cracks 101Qun Li Marco Enderlein and Meinhard Kuna
A Phenomenological Constitutive Model for FerroelectricCeramics and Ferromagnetic Materials 111Sven Klinkel and Konrad Linnemann
The Concept of Material Forces in NonlinearElectro-elastostatics 123Duc Khoi Vu and Paul Steinmann
Permeable Interfacial Crack in Electrostrictive Materials 133Cun-Fa Gao and Yiu-Wing Mai
Some Numerical Studies with X-FEM for CrackedPiezoelectric Media 141Eric Bechet and Meinhard Kuna
Singularity Analysis of Electro-mechanical Fields in AngularlyInhomogeneous Piezoelectric Composites Wedges 153Jian-Shan Wang Xiaoqiao He and Qing-Hua Qin
Crack Propagation Simulations in Piezoelectric Structureswith an Efficient Adaptive Finite Element Tool 163Łukasz Janski Peter Steinhorst and Meinhard Kuna
Periodic Set of the Interface Cracks with LimitedElectric Permeability 175VV Loboda and SV Kozinov
Interfacial Delamination of PZT Thin Films 189Fulin Shang Yabin Yan and Takayuki Kitamura
Mechanical Behavior of Thin Film Comprised of SculpturedNano-elements 197Takayuki Kitamura Takashi Sumigawa and Taisuke Sueda
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
Contents xiii
Propagation of SAW and PSAW in a SmartAlNDiamondrdquo-TiAl Structure 207LM Gao Ch Zhang Z Zhong C-P Fritzen X JiangH-J Christ and U Pietsch
Experimental Investigation and Theoretical Modelingof Piezoelectric Actuators Used in Fuel Injectors 219MS Senousy RKND Rajapakse and M Gadala
Analytical Homogenizations of Piezoceramic d15 ShearMacro-fibre Composites 229Ayech Benjeddou and Mohammed Al-Ajmi
Influence of the Load Dependent Material Propertieson the Performance of Multilayer Piezoelectric Actuators 243Hannes Grunbichler Josef Kreith Raul BermejoClemens Krautgasser and Peter Supancic
Roles of Micro-cracking and Phase Transition on ElectricFatigue for [001]-Oriented PbMg1=3Nb2=3O3-PbTiO3 SingleCrystals 255F Fang W Yang and X Luo
Multiscale Modeling of Electro-mechanically CoupledMaterials Homogenization Procedure and Computationof Overall Moduli 265Jorg Schroder and Marc-Andre Keip
A Boundary Element Method Coupled to Phase Fieldto Compute Ferroelectric Domains in Complex Geometries 277Kaushik Dayal and Kaushik Bhattacharya
Low Energy Periodic Microstructure in FerroelectricSingle Crystals 287Nien-Ti Tsou Ingo Munch and John E Huber
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
List of Participants
K Albe TU Darmstadt Institut fur MaterialwissenschaftFG MaterialmodellierungPetersenstraszlige 23 64287 Darmstadtalbemmtu-darmstadtde
T Antretter Institut fur Mechanik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriathomasantretterunileobenacat
H Balke Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyHerbertBalketu-dresdende
F Auricchio Dipartimento di Meccanica Strutturale Universitarsquo di PaviaVia Ferrata 1 27100 Pavia Italyauricchiounipvit
L Banks-Sills Dreszer Fracture Mechanics Laboratory School of MechanicalEngineering Tel Aviv UniversityRamat Aviv 69978 Israelbanksengtauacil
E Bechet Department of Aerospace and Mechanical EngineeringUniversite de LiegeChemin des Chevreuils 1 4000 Liege Belgiumericbechetulgacbe
A Benjeddou Supmeca Structures3 rue Fernand Hainault 93407 Saint Ouen CEDEX Francebenjeddousupmecafr
R Danzer Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriaisfkunileobenacat
K Dayal Civil and Environmental Engineering Carnegie Mellon University118J Porter Hall Pittsburgh PA 15213-3890 USAkaushikcmuedu
xv
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
xvi List of Participants
F Fang Tsinghua University School of Mechanical EngineeringBeijing 100084 Chinafangfmailtsinghuaeducn
CF Gao Nanjing University of Aeronautics and AstronauticsNanjing 210016 Chinacfgaonuaaeducn
L Gao Chair of Structural Mechanics Department of Civil EngineeringUniversity of SiegenPaul-Bonatz-Street 9-11 57076 Siegen Germanyczhanguni-siegende
D Gross Institute of Mechanics TU DarmstadtD-64289 Darmstadt Germanygrossmechanicstu-darmstadtde
H Grunbichler Institut fur Struktur- und Funktionskeramik Montanuniversitat LeobenPeter-Tunner-Straszlige 5 8700 Leoben Austriahannesgruenbichlermclat
C Hausler TU Bergakademie Freiberg IMFDLampadiusstr 4 09596 FreibergChristophHaeuslerimfdtu-freibergde
JE Huber Department of Engineering Science University of OxfordParks Road Oxford OX1 3PJ UKjohnhuberengoxacuk
Ł Janski Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyLukaszJanskiimfdtu-freibergde
S Klinkel Statik und Dynamik der Tragwerke Technische UniversitatKaiserslauternPaul-Ehrlich-Str 14 67663 Kaiserslautern Germanyklinkelrhrkuni-klde
B Kranz Fraunhofer-Institut fur Werkzeugmaschinen und Umformtechnik IWUNothnitzer Straszlige 44 81187 DresdenBurkhardkranziwufraunhoferde
M Kuna Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyMeinhardKunaimfdtu-freibergde
C Lexcellent Universite de Franche Comte Dept de Mecanique Appliquee24 Chemin de lrsquoEpitaphe 25030 Besancon Cedex FranceChristianlexcellentuniv-fcomtefr
F Li State Key Lab for Turbulence and Complex SystemsCollege of Engineering Peking UniversityBeijing 100871 Chinalifaxinpkueducn
Q Li Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg Germanyqunliimfdtu-freibergde
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
List of Participants xvii
C Linder Universitat Stuttgart Institute of Applied MechanicsPfaffenwaldring 7 70550 Stuttgartlindermechbauuni-stuttgartde
A Liskowsky Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden GermanyAlbrechtLiskowskytu-dresdende
VV Loboda Department of Theoretical and Applied MechanicsDniepropetrovsk National UniversitySt Kosakova 18 Building No 1449050 Dnipropetrovsk Ukrainelobodamaildsudpua
R McMeeking Department of Mechanical and Enviromental EngineeringUniversity of CaliforniaSanta Barbara CA 93106 USArmcmengineeringucsbedu
R Muller Technische Universitat Kaiserslautern Lehrstuhl fur TechnischeMechanikPostfach 3049 67653 Kaiserslauternramrhrkuni-klde
A Ricoeur Universitat Kassel Institut fur MechanikMonchebergstraszlige 7 34109 Kasselandreasricoeuruni-kasselde
J Rodel TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 64287 Darmstadtroedelceramicstu-darmstadtde
S Roth Technische Universitat Bergakademie Freiberg Institut fur Mechanikund FluiddynamikLampadiusstr 4 09599 Freiberg GermanyStephanRothimfdtu-freibergde
GA Schneider Technische Universitat Hamburg-Harburg Institut fur KeramischeHochleistungswerkstoffeDenickestraszlige 15 (K) 21073 Hamburggschneidertu-harburgde
J Schroder Universitat Duisburg-Essen Institut fur MechanikUniversitatsstraszlige 15 45117 Essenjschroederuni-duede
AS Semenov Technische Universitat Dresden Institut fur Festkorpermechanik01062 Dresden Germanyartemsemenovtu-dresdende
MS Senousy The University of British ColumbiaDepartment of Mechanical Engineering2054-6250 Applied Science LaneVancouver BC V6T 1Z4 Canadamsenousyinterchangeubcca
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
xviii List of Participants
F Shang Department of Engineering Mechanics School of AerospaceXirsquoan Jiaotong University28 West Xian-Ning Road Xirsquoan 710049 Chinashangflmailxjtueducn
Y Shindo Tohoku Universtiy Graduate School of EngineeringDepartment of Materials ProcessingSendai 980-8579 Japanshindomaterialtohokuacjp
DK Vu LTM University of ErlangenEgerlandstr 5 91058 Erlangen Germanyvultmuni-erlangende
M Wagner Ruhr-Universitat Bochum Institut fur Werkstoffe (WW) Fakultat furMaschinenbau Universitatsstraszlige 150 44801 Bochummartinwagnerruhr-uni-bochumgde
JS Wang Department of Mechanics Tianjin UniversityWeijin Road No 92 Nankai district Tianjin 300072 Chinawjs tju163com
S Wang Institute of Applied Mechanics Zhejiang UniversityHangzhon Zhejiang 310027 Chinajwzjueducn
K Webber TU Darmstadt MatGeo FG Nichtmetallisch-AnorganischeWerkstoffePetersenstraszlige 23 63287 Darmstadtwebberceramicstu-darmstadtde
S Yu Tsingua University Department of Mechanics School of AerospaceBeijing 100084 Chinayuswmailtsinghuaeducn
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
Scientific Program
Tuesday September 01
Opening and Welcome Address by the Chairmanof the Symposium M Kuna
Fracture Mechanics I (session leader T Kitamura)
Leslie Banks-Sills Yael Motola ldquoFailure of piezoelectric ceramicsrdquo
Gerold Schneider A Engert H Jelitto ldquoWhat do we know about crack surfacecharges and the difference between intrinsic and extrinsic process during fracturein ferroelectric ceramicsrdquo
Yasuhide Shindo Fumio Narita ldquoEffects of electric field and poling on fatiguebehavior of PZT ceramics with single-edge crack by three-point bendingrdquo
Daining Fang Yihui Zhang Guanzhong Mao Bin Liu ldquoElectric field inducedfatigue crack growth in ferroelectric ceramicsrdquo
Shape Memory Alloys I (session leader C Lexcellent)
Martin F-X Wagner Christian Grossmann Marcus Young Gunther EggelerldquoLocalized deformation mesoscopic phase interfaces and functional fatigue inpseudoelastic NiTi shape memory alloysrdquo
F Auricchio M Conti S Morganti A Reali U Stefanelli ldquoShape-memory alloyseffective 3D modelling computational aspects and biomedical device analysisrdquo
xix
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
xx Scientific Program
Ferroelectrics I (session leader R McMeeking)
Artem S Semenov Albrecht C Liskowsky Herbert Balke ldquoEffective computa-tional methods for the modeling of ferroelectroelastic hysteresis behaviorrdquo
Stephan Roth Peter Neumeister Artem Semenov Herbert Balke ldquoFinite elementsimulation of the non-remanent straining ferroelectric material behaviour based onthe electric scalar potential ndash convergence and stabilityrdquo
Li Yu Shouwen Yu Dietmar Gross ldquoConstitutive behaviour of nano-particle fer-roelectric ceramicsrdquo
Wednesday September 02
Ferroelectrics II (session leader J Huber)
Faxin Li ldquoAn optimization-based computational model for polycrystalline ferroe-lasticsrdquo
Ralf Muller David Schrade Baixiang Xu Dietmar Gross ldquoModelling of domainstructure evolution in ferroelectric materialsrdquo
Qun Li Marco Enderlein Meinhard Kuna ldquoMicrostructural FEM modeling ofdomain switching in tetragonalrhombohedral ferroelectricsrdquo
Sven Klinkel K Linnemann ldquoA phenomenological constitutive model for piezo-electric ceramics and magnetostrictive materialsrdquo
Shape Memory Alloys II (session leader F Auricchio)
Christian Lexcellent R Laydi V Taillebot P Malecot ldquoPrediction of the phasetransformation zone around the crack tip of a shape memory alloy exhibiting anasymmetry between tension and compressionrdquo
Thomas Antretter Wolfgang Pranger Thomas Waitz Franz D Fischer ldquoMartensitemorphologies in nanostructured NiTi shape memory alloys ndash energetic considera-tionsrdquo
Fracture Mechanics II (session leader L Banks-Sills)
Duc Khoi Vu Paul Steinmann ldquoThe concept of material forces and the coupledboundary-finite element method in electroelastostaticsrdquo
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
Scientific Program xxi
Cun-Fa Gao Yiu-Wing Mai ldquoStresses in an electrostrictive solid containing an el-liptic cavityrdquo
Eric Bechet Meinhard Kuna ldquoSome numerical experiments about cracked piezo-electric mediardquo
Qing-Hua Qin XQ He JS Wang ldquoSingularity analysis of electro-mechanicalfields in angularly inhomogeneous piezoelectric composites wedgesrdquo
Thursday September 03
Fracture Mechanics III (session leader Y Shindo)
Jie Wang Marc Kamlah ldquoThree dimensional finite element modeling of nonlinearfracture of ferroelectric materialsrdquo
Ł Janski P Steinhorst M Kuna ldquoCrack propagation simulations in piezoelectricstructures with an efficient adaptive finite element toolrdquo
Volodymyr Loboda SV Kozinov ldquoPeriodic set of interface cracks with limitedelectric permeabilityrdquo
Fulin Shang Yabin Yan Takayuki Kitamura ldquoInterfacial delamination of PZT thinfilmsrdquo
Nano (session leader J Schroder)
Karsten Albe Paul Erhart ldquoModelling of point defects in ferroelectric materialsrdquo
Takayuki Kitamura Takashi Sumigawa Taisuke Sueta ldquoMechanical behaviour ofthin film comprised of sculptured nano-elementsrdquo
Actuators (session leader J Rodel)
LM Gao Ch Zhang Z Zhong C-P Fritzen X Jiang H-J Christ U PietschldquoPropagation of SAW and PSAW in a Smart AlNDiamond -TiAl Structurerdquo
MS Senousy RKND Rajapakse M Gadala ldquoModeling of thermo-electromechanical response of PZT stack actuators used in fuel injectorsrdquo
Ayech Benjeddou Mohammad Al-Ajmi ldquoAnalytical homogenization of piezoce-ramic shear macro-fibre compositesrdquo
Hannes Grunbichler Raul Bermejo Peter Supancic Robert Danzer ldquoInfluence ofthe load dependent material properties on the performance of multilayer piezoelec-tric actuatorsrdquo
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
xxii Scientific Program
Technical Visit Laboratories of the host Institute of Mechanicsand Fluid Dynamics
Friday September 04
Ferroelectrics III (session leader S Yu)
Robert McMeeking SMA Jimenez ldquoModels for actuation failure and tearing ofelectroactive materialsrdquo
Fei Fang W Yang X Luo ldquoIn-situ observations of multi-phase coexistence andpolarization rotation under electric loadings for Pb(Mg1=3Nb2=3O3-PbTiO3singlecrystals at the morphotropic phase boundaryrdquo
Jorg Schroder Marc-Andre Keip ldquoMultiscale modeling of electromechanically cou-pled materials homogenization procedure and computation of overall modulirdquo
Kaushik Dayal Kaushik Bhattacharya ldquoA boundary element method coupled tophase field to compute ferroelectric domains in complex geometriesrdquo
Jurgen Rodel Kyle Webber Emil Aulbach Wook Jo Thorsten Leist ldquoTemperature-dependent ferroelasticity of ferroelectricsrdquo
Nien-Ti Tsou Ingo Munch John Huber ldquoLow energy periodic microstructure inferroelectric single crystalsrdquo
Closing Address M Kuna
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
A Fracture Criterion for Piezoelectric Material
Leslie Banks-Sills and Yael Motola
Abstract A fracture criterion for piezoelectric ceramics is proposed which is basedupon the energy release rate and two phase angles determined from the ratio be-tween the intensity factors It is assumed that the crack plane is at an angle to thepoling direction within a symmetry plane of the body The special cases of a crackperpendicular and parallel to the poling direction are presented This criterion wasimplemented with experimental results from the literature in which the crack faceswere perpendicular to the poling direction and the applied electric field Excellentagreement was found between the fracture curve and the experimental results
1 Introduction
Piezoelectric ceramics are in widespread use as sensors and actuators in smart struc-tures despite the absence of a fundamental understanding of their fracture behaviorPiezoceramics are brittle and susceptible to cracking As a result of the importanceof the reliability of these devices there has been tremendous interest in studying thefracture and failure behavior of such materials To understand failure mechanisms ofpiezoelectric materials and maintain the stability of cracked piezoelectric structuresoperating in an environment of combined electro-mechanical loading analysis of itsmechanical and electrical behavior is a prerequisite
For cracks in piezoelectric material there are three stress intensity factors KI KII and KIII representing the usual three deformation modes and an electric fluxdensity intensity factor KIV Depending on the applied load and electric fieldas well as the poling direction relative to the crack faces various modes are excitedIn this paper a fracture criterion presented in Section 2 is proposed based uponthe energy release rate and two phase angles determined from the ratio between the
L Banks-Sills () and Y MotolaDreszer Fracture Mechanics Laboratory School of Mechanical EngineeringTel Aviv University Ramat Aviv 69978 Israele-mail banksengtauacil sphera4ugmailcom
1M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 c Springer Science+Business Media BV 2011
IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
1
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
2 L Banks-Sills and Y Motola
intensity factors It is assumed that the crack plane is at an angle to the poling di-rection within a symmetry plane of the body Four point bend fracture tests carriedout in [1] were analyzed in [2] to obtain the non-zero intensity factors KI and KIV These values are used to obtain a fracture criterion for this material (PIC-151) whenthe crack faces are perpendicular to the poling direction and the electric field Theseresults are presented in Section 3
2 Fracture Criterion
A mixed mode fracture criterion is developed for piezoelectric ceramics This crite-rion is based upon the energy release rate and one or two phase angles determinedfrom the ratio between the intensity factors It is assumed that the crack plane is atan angle to the poling direction with poling within a symmetry plane of the bodyThe special cases of a crack perpendicular and parallel to the poling direction arepresented as well
A general expression for the energy release rate for piezoelectric ceramics wasderived in [3] as
G D 1
2kT L1 k (1)
where k is the intensity factor vector defined as
kT D ŒKII KI KIII KIV (2)
where the superscript T represents transpose The matrix L is one of the BarnettndashLothe tensors [4] whose components are related to material properties Based onEq 1 the energy release rate for a crack at an angle to the poling direction withpoling within a symmetry plane is obtained as
G D 1
2
OL122
OK2I C OL1
11OK2
II C OL144
OK2IV C 2 OL1
12OKI OKII
C2 OL114
OKII OKIV C 2 OL124
OKI OKIV
(3)
Out-of-plane loading is omitted here The parameters OL111 OL1
12 OL114 OL1
22 OL124
and OL144 in Eq 3 are given by
OL111 D EA
2L111L
OL112 D EA
2L112L
OL114 D EAe26L
114L
OL122 D EA
2L122L
OL124 D EAe26L
124L
OL144 D e226L
144L (4)
where L1ij are elements of the matrix L1 EA and e26 are the Youngrsquos modulus
in the poling direction and a piezoelectric coupling coefficient respectively andL is a characteristic length of the problem The BarnettndashLothe tensor L1 is ill-conditioned Hence it was normalized as
OL1 D VL1V (5)
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
A Fracture Criterion for Piezoelectric Material 3
where
V D
2666664
EA
pL 0 0 0
0 EA
pL 0 0
0 0 GTpL 0
0 0 0 e26pL
3777775
(6)
and GT is the shear modulus perpendicular to the poling direction In this way thediagonal and off-diagonal elements are the same order of magnitude [5] It is worthmentioning that the units of OL1
ij are Nm Finally the intensity factors in Eq 3 arenormalized according to
OKI D KI
EApL
OKII D KII
EApL
OKIV D KIV
e26pL (7)
The energy release rate G may be rewritten as
G D 1
2OL122
OK2I
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(8)
Thus it is possible to define
GI 1
2OL122
OK2I (9)
When the crack propagates G D Gc so that
Gc D GIc
1C 2
OL124
OL122
OKIVOKI
C 2OL112
OL122
OKIIOKI
C 2OL114
OL122
OKII OKIVOK2I
COL144
OL122
OK2IV
OK2I
COL111
OL122
OK2II
OK2I
(10)
where
GIc D 1
2OL122
OK2Ic (11)
To obtain GIc values of GI from Eq 9 are obtained at failure for each test andaveraged as will be discussed in Section 3 Introducing two phase angles
D tan1 OKIVOKI D tan1 OKII
OKI (12)
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
4 L Banks-Sills and Y Motola
Equation 10 becomes
Gc D GIc
1C 2
OL124
OL122
tan C 2OL112
OL122
tan C 2OL114
OL122
tan tan COL144
OL122
tan2
COL111
OL122
tan2
(13)
This is a three-dimensional failure surface for the case in which the crack faces areat an angle to the poling direction and in which the critical energy release rate Gc isa function of the phase angles and
For a crack perpendicular to the poling direction the component OL114 is zero
In addition for symmetric applied loading and electric field OKII was found to benegligible [2] Therefore for this case there is only one non-zero phase angle given in Eq 121 leading to the failure curve
Gc D GIc
1C 2
OL124
OL122
tan COL144
OL122
tan2
(14)
For a crack perpendicular to the poling direction there is coupling between the firstand fourth modes of fracture This coupling is expressed by the second term in theparentheses of the right hand side of Eq 14
For a crack parallel to the poling direction OL124 is zero In addition if both the
applied loading and electric field are symmetric then OKII is negligible (implying D 0) and there is nearly no coupling between modes I and IV (see [2]) Thusthe failure curve for a crack parallel to the poling direction with symmetric appliedloading and electric field may be found based on Eq 13 as
Gc D GIc
1C
OL144
OL122
tan2
(15)
3 Fracture Tests
Test were reported in [1] on four-point bend specimens (see Fig 1) fabricated fromthe piezoelectric ceramic PIC-151 In these experiments the crack faces were per-pendicular to the poling direction and both mechanical loads and electric fields wereapplied The dimensions of the specimens are L D 24 S1 D 20 S2 D 10 mm thewidth W D 4 mm and the thickness B D 3 mm (see Fig 1) The samples wereplaced in a Fluorinert-liquid to prevent electric sparks this material is dielectricallyisotropic with the permittivity a D 1750
Using the same boundary conditions applied in the tests [1] numerical analyseswere carried in [2] These included the finite element method and the interaction
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
A Fracture Criterion for Piezoelectric Material 5
Fig 1 Four-point bend specimen and an enlargement of the notch [1]
Fig 2 The intensity factor OKIV versus OKI from analyses of experimental results from [1]
energy or M -integral for obtaining the intensity factors The Fluorinert-liquidwithin the notch was modeled as a dielectric isotropic material Average valuesof OKI and OKIV given in Eqs 71 and 73 respectively with application of dif-ferent loads and electric fields for all notch lengths were determined In Eqs 7EA D 602 GPa as calculated according to material properties given in [1] ande26 D 120 Cm2 Results for OKII were negligible in comparison to the other in-tensity factors
Values of OKIV are plotted versus those of OKI and presented in Fig 2 It is ob-served that values of OKI are relatively constant with respect to OKIV The averagevalue of OKI obtained for all points is 144 105 This leads to an average value
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
6 L Banks-Sills and Y Motola
Fig 3 Fracture curve and experimental results of [1]
of the mode I stress intensity factor KI D 086 MPap
m and KIV ranges between046 103 and 082 103 Cm3=2
The fracture criterion in Eq 14 is applied using the intensity factors in Table 6of [2] and presented in Fig 2 From those values and with OKII D 0 the critical en-ergy release rate Gc and the phase angle were calculated using Eqs 3 and 121for each test It was found that some of the values of the critical energy releaserate are negative or close to zero hence these values are not physically plausibleIt is possible that this phenomenon occurs because of a large domain switchingzone in the vicinity of the notch tip This aspect of the problem will be examinedin the future The calculated points are plotted as the colored symbols in Fig 3omitting those points in which Gc 0 In addition Eq 14 is plotted as the solidcurve in Fig 3 Values below this curve are safe for those above it failure isexpected The value of GIc used for this curve is given in Eq 11 The value ofOKIc is found as the average of the values of OKI at failure in Fig 2 excluding
those for which Gc is negative or close to zero Its value is 146 105 which isslightly different from the value shown in Fig 2 which includes all of the testsWith OL1
22 D 81511010 Nm a value of GIc D 86 Nm is found Excellent agree-ment is observed between the experimental results and the curve As a result of thecoupling between the first and fourth modes of fracture which is expressed by thesecond term in parentheses of the right hand side of Eq 14 the fracture curve inFig 3 is not symmetric with respect to Recall that the crack faces are perpendic-ular to the poling direction Furthermore it should be emphasized that the apparentfracture toughness GIc should not be used to predict catastrophic failure Only themixed mode fracture curve presented in Eq 14 and Fig 3 should be used to predictfailure
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
A Fracture Criterion for Piezoelectric Material 7
4 Conclusions
A fracture criterion has been presented for piezoelectric material for poling at anangle to the crack plane and within a material symmetry plane It is based uponthe energy release rate and one or two phase angles If mode III deformation wasadded andor poling would be in a more general direction the criterion could easilybe generalized with the addition of third phase angle Experimental results [1] wereused in a special case of the criterion for which the crack is perpendicular to thepoling direction Excellent agreement was observed between the fracture curve andthe experimental results Nevertheless some values of the critical energy releaserate were negative or close to zero These points were neglected as being physicallyunreasonable Future study is required to understand this behavior The criterionpresented may be used for other poling directions
References
1 Jelitto H Kessler H Schneider GA Balke H (2005) Fracture behavior of poled piezoelectricPZT under mechanical and electrical loads J Euro Cer Soc 25749ndash757
2 Motola Y Banks-Sills L Fourman V (2009) On fracture testing of piezoelectric ceramics Int JFract 159167ndash190 DOI 101007s10704-009-9392-x
3 Suo Z Kuo CM Barnett DM Willis JR (1992) Fracture mechanics for piezoelectric ceramicsJ Mech Phys Solids 40739ndash765
4 Barnett DM Lothe J (1975) Dislocations and line charges in anisotropic piezoelectryc insula-tors Physica Status Solidi (b) 67105ndash111
5 Banks-Sills L Motola Y Shemesh L (2008) The M-integral for calculating intensity factors ofan impermeable crack in a piezoelectric material Engng Fract Mech 75901ndash925
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
What Do We Know About Surface Chargeson Cracks in Ferroelectric Ceramics
Andrea R Engert Frank Felten Hans Jelitto and Gerold A Schneider
Abstract The present work investigates the static and time dependent electricpotential distribution around cracks in a poled ferroelectric ceramic by Kelvin ProbeForce Microscopy (KFM) In a first step a Vickers indentation crack in poled leadzirconate titanate (PZT) was subjected to static electric fields of up to 500 Vmm inpoling direction and the potential distribution around the crack was measured In asecond step the polarity of the applied voltage was reversed against the poling di-rection during the measurement of the potential Using a simple model an effectivedielectric constant of the crack as well as the surface charge density on the crackface were calculated as a function of the distance from the crack tip the applied fieldand the time The results are discussed with reference to free charges on the cracksurface electrically induced domain switching at the crack tip and crack bridging
1 Introduction
Ferroelectric materials are used amongst other applications in electro-mechanicaltransducer applications converting mechanical forces into an electrical potential(direct piezoelectric effect) or vice versa (inverse piezoelectric effect) Duringtheir lifetime ferroelectric ceramics must be capable to operate under long-termelectro-mechanical loading While they continue to find increasing use their frac-ture mechanics are still investigated due to their low fracture toughness and complexmaterial behavior This work contributes to the understanding of the effect of elec-trical loading on fracture in ferroelectric ceramics
AR Engert H Jelitto and GA Schneider ()Technische UniversitRat Hamburg-Harburg Institut fRur keramische HochleistungswerkstoffeHamburg Germanye-mail gschneidertu-harburgde wwwtu-harburgdegk
F FeltenRobert Bosch GmbH Corporate Sector Research and Advance EngineeringApplied Research 1 ndash Materials 70049 Stuttgart Germany
M Kuna and A Ricoeur (eds) IUTAM Bookseries 24
DOI 101007978-90-481-9887-0 2 c Springer Science+Business Media BV 2011
9IUTAM Symposium on Multiscale Modelling of FatigueDamage and Fracture in Smart Materials
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
10 AR Engert et al
A crack filled with air in a dielectric material with a high permittivity is a voidwith a relative dielectric constant of one in a matrix with a dielectric constantthat can be up to three orders of magnitude higher If an electrical field is ap-plied normal to the crack surface it will penetrate the crack The crack shapethe crack opening displacement and the ratio of the permittivities of the crackinterior and the ceramic will determine to what extent the electric flux throughthe crack will be diluted and increased ahead of the tip For cracks in a ferro-electric ceramic ie in an electromechanically coupled material this leads to anon-linear boundary value problem In the literature three approaches are usedto describe cracks in ferroelectric materials assuming impermeable boundaries(impermeable crack) permeable boundaries with an infinite dielectric constant ofthe crack (permeable crack) and permeable boundaries with a dielectric constantof the crack higher than one (semi-permeable crack) Theoretical works eg fromBalke et al [1] using the capacitor model of Hao and Shen [2] have shown thatthe latter results in the most realistic representation of the electric field around thecrack However the capacitor model proved to be energetically not consistent [3]In consequence Landis [45] derived the so called ldquoenergetically consistentrdquo bound-ary conditions along the crack faces that include electrostatic tractions which arecaused by an electric field perpendicular to the crack faces closing the crack Hisimproved model that also includes surface charges on the crack faces and an elec-trical discharge model within the crack predicts that an electrical load increases thecritical mechanical load for fracture Except of the permeable crack model all othertheoretical approaches predict that the electric field applied perpendicular to thecrack plane increases the critical mechanical load for crack growth [6] These theo-retical predictions could not be observed experimentally Eg Jelitto et al [7ndash9] andHausler [10] show that DC electric fields of 13 of the coercive field only influencethe critical loads for crack growth to a very small extent
Haug and McMeeking [11] showed that screening charges on the crack surfacewould change the fracture behavior of the crack dramatically Schneider et al [12]already successfully used Kelvin Probe Force Microscopy (KFM) to study thepotential distribution around an electrically loaded Vickers crack in PZT From theresults an effective dielectric constant was calculated for the crack tip On the basisof a Griffith crack it was further shown that a higher dielectric constant of the crackreduces the dependency of the crack tip energy release rate on an applied elec-tric field The present work is a continuation of the work presented by Schneideret al [12] It takes into account not only a single applied electric field but differentfield strengths up to 500 Vmm Besides the potential distribution is not only an-alyzed at the crack tip but to a crack length of around 60m The objective wasto analyze whether or not the crack could also be described by a single effectivedielectric constant farther away from the tip and at different applied electric fieldsThis part of the work is described in more detail in the PhD thesis from Felten [13]In addition the time response of the electric potential distribution upon a reversalof the electric field against the polarization direction was studied The motivationfor this experiment was the assumption that free charges on the crack surface would
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
What Do We Know About Surface Charges on Cracks in Ferroelectric Ceramics 11
not immediately change after the reversal of the applied voltage The objective ofall experiments was to verify the theoretical assumption of screening charges on thecrack surface trying to quantify and characterize them
2 Samples Experimental Setup and Methods
KFM is a scanning probe technique which allows mapping the surface poten-tial of a sample versus the in-plane coordinates x and y in a two-pass techniqueDuring the first pass the topography is recorded in tapping-mode In the secondpass the so-called interleave scan the topography is retraced in a set lift heightand the contact potential difference is measured As the tip travels over the sam-ple surface in lift-mode the tip and the cantilever experience a force wherever thepotential on the surface is different from the potential of the tip The force is nul-lified by varying the voltage of the tip so that the tip is at the same potential asthe region of the sample surface underneath it This voltage is plotted versus thein-plane x-y-coordinates creating the surface potential image The principle of theKFM-mode is described eg by Kalinin and Bonnell [14] the resolution of KFM isdiscussed eg by Jacobs et al [15]
The experiments were performed in a similar way as described by Schneideret al [12] 270m thin plates of a commercial soft PZT (Vibrit1100 JohnsonMatthey) covered with thin gold electrodes on both sides were cut into samplesof 1 2 cm2 The plates were poled at room temperature with an electric fieldof 2 kVmm The coercive field extracted from a hysteresis measurement (roomtemperature 2 kVmm maximum field 83 mHz) is around 700 Vmm Accordingto the data sheet the dielectric constant copy33 of the poled material is 4500 [16]One thin 4 cm 260m face was polished on a semi-automatic polishing ma-chine (Saphir550 ATM GmbH) using diamond suspension down to a grain size of025 m (DP-Suspension P on MD-NAP Struers GmbH) The materialrsquos grain sizemeasured with an optical microscope is around 3m After polishing a Vickers in-dent (05 kg 10 s Vickers hardness tester 3212 preceding model of ZVH10 Zwick)was induced in the middle of the surface producing cracks parallel to the surfaceedges (Fig 1) The advantage of the indentation methodology was its easy controland simplicity concerning the experimental performance Vickers indenters producetwo basic types of crack systems radial-median and lateral After the diamond isremoved local residual stress states hold the cracks open [17 p 249ff]
The sample was fixed in a custom made holder and aligned under a ScanningProbe Microscope (SPM) in a way that the cantilever was perpendicular to the ap-plied field and would not electrostatically interact with the indent The voltagewas externally applied to the sample For the measurements with constant fieldstwo separate power supplies were used (HCN-35-12500 MCN-35-1250 FuG) Forthe measurements where the field was reversed a bipolar power supply with twoports and a fast response (PZD700 large signal bandwidth up to 15 kHz TREKINC) was chosen The power supplies were controlled using a computer program(LabView R National Instruments) Therewith the voltage was applied in a way that
![multiscale fatigue revised - Rensselaer Polytechnic Institute · · 2006-03-09The Gurson-Tvergaard-Needleman (GTN) [25] model serves asa basis for modeling of mi-crovoid nucleation,](https://img.pdfslide.net/doc/110x75/5aee78a67f8b9a4556912028/multiscale-fatigue-revised-rensselaer-polytechnic-institute-gurson-tvergaard-needleman.jpg)
