STRENGTH OF MATERIALS No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form orby any means. The publisher has taken reasonable care in the preparation of this digital document, but makes noexpressedorimpliedwarrantyofanykindandassumesnoresponsibilityforanyerrorsoromissions.Noliabilityisassumedforincidentalorconsequentialdamagesinconnectionwithorarisingoutofinformationcontained herein. This digital document is sold with the clear understanding that the publisher is not engaged inrendering legal, medical or any other professional services. STRENGTH OF MATERIALS GUSTAVO MENDES AND BRUNO LAGO EDITORS Nova Science Publishers, Inc. New York Copyright 2009 by Nova Science Publishers, Inc. Allrightsreserved.Nopartofthisbookmaybereproduced,storedinaretrievalsystemor transmittedinanyformorbyanymeans:electronic,electrostatic,magnetic,tape,mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or impliedwarrantyofanykindandassumesnoresponsibilityforanyerrorsoromissions.No liabilityisassumedforincidentalorconsequentialdamagesinconnectionwithorarisingoutof informationcontainedinthisbook.ThePublishershallnotbeliableforanyspecial, consequential,orexemplarydamagesresulting,inwholeorinpart,fromthereadersuseof,or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independentverificationshouldbesoughtforanydata,adviceorrecommendationscontainedin this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage topersonsorpropertyarisingfromanymethods,products,instructions,ideasorotherwise contained in this publication. Thispublicationisdesignedtoprovideaccurateandauthoritativeinformationwithregardtothe subjectmattercoveredherein.ItissoldwiththeclearunderstandingthatthePublisherisnot engagedinrenderinglegaloranyotherprofessionalservices.Iflegaloranyotherexpert assistanceisrequired,theservicesofacompetentpersonshouldbesought.FROMA DECLARATIONOFPARTICIPANTSJOINTLYADOPTEDBYACOMMITTEEOFTHE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Strength of materials / edited by Gustavo Mendes and Bruno Lago. p. cm. Includes bibliographical references. ISBN 978-1-61728-584-4 (E-Book) 1.Strength of materials. 2.Composite materials.I. Mendes, Gustavo. II. Lago, Bruno. TA405.S765 2009 620.1'12--dc22 2009013535 Published by Nova Science Publishers, Inc. New York CONTENTS Preface vii Chapter 1 High Temperature Mechanical Properties and Microstructure of Sic-Based Fibers under Severe Environments 1 Jianjun Sha Chapter 2 Ionomers as Candidates for Structural Materials61Daniel J. Klein Chapter 3 Failure of Layered Composites Subject to Impacts: Constitutive Modeling and Parameter Identification Issues 97Stefano Mariani Chapter 4 Current State of the Art of the Ceramic Composite Material BIOLOXdelta 133Meinhard Kuntz, Bernard Masson and Thomas Pandorf Chapter 5 Particle Modeling and Its Current Success in the Simulations of Dynamics Fragmentation of Solids 157G. Wang, A. Al-Ostaz, A.H.D. Cheng and P. Radziszewski Chapter 6 Non-Oriented Electrical Steels: Materials for Saving Energy and Conserving the Environment 183Taisei Nakayama Chapter 7 Influence of Luting Cement Application Technique on Quartz Fiber Post Regional Bond Strengths 217Camillo DArcangelo, Francesco De Angelis, Maurizio DAmario, Simone Zazzeroni, Mirco Vadini and Sergio Caputi Chapter 8 Microstructural Influence on Flexure Strength of a Ceromer Reinforced by Two Types of Fibers (Polyethylene and Glass) 233Silvana Marques Miranda Spyrides and Fernando Luiz Bastian Contentsvi Chapter 9 Influence on Strength Properties of Anisotropy Planes in Slates Samples in the NW of Spain 247M.A. Rodrguez-Sastre, M. Gutirrez-Claverol, M. Torres-Alonso and L. Calleja Index261 PREFACE Thestrengthofamaterialreferstothematerial'sabilitytowithstandanappliedstress without failure. The applied stress may be tensile, compressive, or shear. A material's strength is dependent on its microstructure. The engineering processes to which a material is subjected canalterthismicrostructure.Thisbookprovidesavarietyofmaterialstrengthresearch including an extensive overview on the state of the art ceramic composite material BIOLOX deltawhich,since2001,hassuccessfullyimplantedmorethan500,000artificialhipjoints. Duetotheuniquestrengthandtoughnessofthismaterial,theriskoffracturehasbeen substantiallyreducedwhencomparedtoconventionalceramicmaterials.Severaldifferent aspectsofionomerresearchfromaphysicalpropertystandpointisdiscussedaswell, includingthehistoryandcurrenttrendsinionomerresearchandadiscussiononthe immediateneedsinthisfield.Furthermore,particlemodeling(PM)asaninnovative particulatedynamicsbasedmodelingapproachisexaminedasarobusttoolforsimulating fracture problems of solids under extreme loading conditions, including situations of collapse, impact,blastingorhighstrainratetension/compression.Thisbookincludesresearchonthe ability of particle modeling to correctly predict dynamic fragmentation of materials with good accuracy. Ceramic-matrixcomposites(CMCs)havebeenconsideringaspotentialstructural materialsforadvancedenergy-generationsystemsandpropulsionsystems.SiCfiberswith lowoxygencontentandhighcrystallinity,whichderivedfrompolycarbonsilane,arethe backboneasreinforcementsinload-bearingCMCs.Forhightemperatureapplication,the mostdesiredcriticalpropertiesofSiCfibersarehighstrengthandstiffnessaswellasthe reliableretentionofthesepropertiesthroughouttheservicelifeofapplication.Lowfiber strength and thermal stability could result in low fracture toughness and accelerate sub-critical crackpropagationinCMCs.Thus,themechanicaldurabilityandmicrostructurestabilityof SiCfibersaremajorconcernsundersevereenvironments.Furthermore,inpracticalservice environments,rarelyisonedegradationmechanismoperative,butseveralmechanisms operatesimultaneously,leadingtotheenvironment-pertinentdegradationmechanismis complexfortheSiCmaterials.Inordertoenhancetheunderstandingofreliabilityand durabilityofCMCsappliedtohightemperatureandoxidativeenvironments,the investigationsonthehightemperaturemechanicalpropertiesandmicrostructureforSiC-based fibers subjected to severe environments were integrated into Chapter 1. Thefieldofionomersisanoftenoverlookedandunder-utilizedbranchofpolymer research. Although ionomers can be broadly described as a class of polymers that contain any Gustavo Mendes and Bruno Lagoviii numberofionicgroups,fromastructuralpropertystandpointonlyalowpercentofionic groups are necessary to impart significantly improved properties over the nonionic version of thesamepolymer.Currenttrends in thefield ofionomersarehighlyfocused on thefieldof fuelcelltechnology.Thereappearstobeasignificantholeremaininginthestudyof impartingstrengthtomaterialsusingionicgroups.Thisholeisverysignificantfroman industrialpointofview,andhasalargecommercialpotential.Thereareveryfew commercially available ionomers, which shows how little this field has been explored to date.Chapter2willfocusonseveralaspectsofionomerresearchfromaphysicalproperty standpoint: 1) A history of ionomer research, 2) Current trends in ionomer research - a) stand-alone polymers, b) nanocomposites, c) blends; 3) A commentary on the immediate needs in the field of ionomer research. Layeredcompositessubjecttoimpactscanfailbydelamination,i.e.bydebonding between laminae, if the stress waves cause damaging phenomena to take place mainly within theresin-enrichedinterlaminarphases.Tosimulatedelaminationatthestructurallevel, processesdissipatingenergyarelumpedontofictitiouszero-thicknessinterlaminarsurfaces, and softening interface constitutive laws are adopted to describe the progressive failure of the interlaminar phases. Since delamination occurs inside very narrow regions, results of experimental testing on wholecompositesneedtobeaccuratelyandreliablyfilteredtocalibratetheinterface constitutive laws. To this aim, Chapter 3 proposes a sigma-point Kalman filter approach. The performancesoftheproposedmethodology,intermsofconstitutiveparameterestimations anddynamicdelaminationtracking,areassessedthroughpseudoexperimentaltestingsona two-layercomposite,andrealtestingsonmulti-layerglassfiberreinforcedplastic composites. Anextensiveoverviewaboutthestateoftheartoftheceramiccompositematerial BIOLOXdeltaisgiven.Theuniquepropertiesrelyonawelldefinedaluminabasedfine compositemicrostructurewhich ismainlyachievedbyhigh temperaturesolidbodyreaction of the different ceramic phases during sintering. Zirconia comprises 17 % of the total volume. The tetragonal phase of zirconia is stabilized chemically and mechanically. The high strength and toughness of the material depend on transformation toughening of the zirconia which is clearly shown by various experimental results. The excellent mechanical properties are reproduced batch by batch with a very low scatter.AspresentedinChapter4,theoutstandingpropertiesofthematerialBIOLOXdelta support advantageous properties of the final product, e.g. ceramic hard-hard bearings for hip arthroplasty. The burst load of the components is significantly increased. It is shown that the design of the components is also very important for the reliability and the ultimate properties ofthesystem.Wearpropertiesatsevereconditionsaresignificantlyimprovedbyusingthe new composite material BIOLOXdelta in comparison to pure alumina.Phasetransformationofzirconiafromthetetragonaltothemonoclinicphasedueto hydrothermalagingisextensivelydiscussed.Duetotheparticulardistributionand stabilizationofthezirconiaparticlesinstableagingeffectsarenotpossibleinthismaterial. After very long time of accelerated aging conditions an increase of monoclinic phase is found however,itisshownthatdynamicandstaticpropertiesofBIOLOXdeltaarenot influenced by this effect. PrefaceixChapter5studiesparticlemodeling(PM),whichisaninnovativeparticulatedynamics basedmodelingapproach.Ithasbeendemonstratedasarobusttoolforsimulatingfracture problemsofsolidswithdynamicfragmentationunderextremeloadingconditions.These loadingconditionscanincludesituationsofcollapse,impact,blastingorhighstrainrate tension/compression, as well as thermally-induced breakage problems.Initially,PMwasdevelopedforthepurposeofmimickingthemicroscopicmaterial processatmacroscopiclevel.Thismethodcanbeconceptuallyillustratedbyfullydynamic particles(orquasi-particles)placedatthenodesofalatticenetworkwithoutexplicitly consideringtheirgeometricsize.Thepotentialcanbespecifiedforparticle-particle interactionsviaaxialsprings.Theoretically,PMisanupscaleofthemoleculardynamics (MD)modelapplicabletovariouslengthscaleproblems.Thisispossibleifaproper equivalent macroscopic potential is found, and, in case of lattice spacing decreasing to a few Angstroms, a MD model at zero Kelvin with, say, Leonard-Jones potential is recovered. In its current form, PM has been developed as a tool applicable to real engineering problems. TheadvantagesofPMovertheexistingdiscreteelementbasedmethodscanbe summarizedasfollows:(1)Sampleintheory.Fourconservative/equivalentrules(mass, potential energy, Youngs modulus and tensile/compression strength) are applied to preserve theequivalentmaterialproperties.(2)Easyforimplementation.Sincethephysicalsizeof eachparticleisignoredotherthanitsequivalentmass,thealgorithmofcodingaPM computation is fairly easy.CurrentresearchworkhasexhibitedthatPMisabletocorrectlypredictdynamic fragmentation of materials with a good accuracy. In modeling an epoxy plate with randomly distributed holes in tension, the PM result of the final crack pattern compared favorably with theassociateexperiment;forthesimulationsofimpactstudyoftwopolymericmaterials (nylon, 6-6 and vinyl ester) subject to a rigid falling indenter, the modeling results of resistant force,energy,deflectionanddropspeedofindentervs.timequantitativelyagreefairlywell with the according empirical observations.Electricalsteelsarethecorematerialsforelectricalmotorsortransformers.Those materialsformotorsareplayedanenergyconversionrollfromelectricitytomotion. However, energy losses are accompanied with this conversion. To minimize these losses is a key technology to conserve our environment. Numerousresearchesonthegrain-orientedelectricalsteelsreported.Thoseresearches especially for transformers are focused on the reducing the losses at supplying the electricity from power plants. On the other hand, home or industrial appliances are the power consuming devices, and the most effective point on the energy loss reduction. These home appliances are used small motors using non-oriented electrical steels. InChapter6,severalresearchesonthenon-orientedelectricalsteelsarediscussedand focused on the metallurgical control of the steels to reduce the core loss for generating waste heatsandmotorbuildinginnovationtechnologiesfordecreasingthebuildingfactorofthe core losses.Inthemetallurgicalpart,someadditiveelementsasphosphorus,aluminumand manganeseforimprovingmagneticpropertiesreviewed.Moreoversomecontaminating elementsasvanadium,titaniumandzirconiumarediscussedespeciallyforprecipitation studiesinthesteelshavebeendone.Theseprecipitationsareinhibitedthegraingrowthat final annealing or stress relief annealing. These inhibited small grains increase the core losses.Gustavo Mendes and Bruno Lagox Forstudyingmotorbuildingtechnologies,compressionstresseffect,shearingstress effect are discussed. Even though the best core materials are used for manufacturing motors, thosebuildingdeteriorationsmakeworseforthemotorefficiency.Therefore,those technologies are also important for reducing the carbon dioxide emission. The aim of Chapter 7 was to investigate regional root canal push-out bond strengths for a fiber-reinforced post system varying the application method of the luting agent.Recentlyextractedmaxillaryincisors(n=30)weresectionedtransversallyatthelabial cemento-enameljunction,andtherootstreatedendodontically.Followingpostspace preparations,fiber-reinforcedposts(EndoLight-Post;RTD)wereplacedusingadhesive systemandresincementprovidedbythemanufacturer.Threeequalgroups(n=10)were assessedaccordingtothetechniqueusedtoplacethelutingagentintopostspace:usinga lentulo spiral, applying the cement onto the post surface, injecting the material with a specific syringe.Each rootwasslicedinto threediscs(2mmthick)representingthe coronal,middle and apical part of the bonded fiber post. Push-out tests were performed for each specimen to measure regional bond strengths. Results were statistically analyzed using two-way ANOVA and Tukey tests ( = 0.05). All fractured specimens were observed using a scanning electron microscope to identify the types of failure. Theresultsindicatedthatbondstrengthvaluesweresignificantlyaffectedbythe applicationmethodoftheresincement(p20 nm)Low thermal stabilityLow strengthLow stiffnessImproved thermal stabilityModest oxidation resistanceIncreased elastic modulusEnhanced creep resistanceEnhanced oxidation resistanceFigure 3. Illustration of R&D of SiC-based Fibers.Toavoidthethermalinstabilitycausedbythedecompositionofoxycarbidephase(SiCxOy),in1990,anearlyoxygen-freeSiCfiber,Hi-Nicalon(Nippon-Carbon)wasdevelopedbymeltspinning,electronbeamcuringandpyrolysisofapolycarbosilaneprecursor(PCS)underanaerobicconditions[15-16].ThisfiberhadamuchhigherthermalstabilitythanthestandardNicalonfiberandwasviewedastherepresentativeofthesecondgeneration.However,theHi-NicalonfiberconsistsofnotonlySiCnanocrystals(averagecrystalsize:5nm)butalsoexcessoffreecarbonwhichaffectsitsoxidationandcreepresistance.To reduce the free carbon content and eventually improve the high temperature propertiesofthefibers,extensiveeffortshavebeendevotedtodevelopnearstoichiometricandhighcrystallizedSiCfibers.Theprecursorfibercanbesinteredathightemperaturesthatexcesscarbon and oxygen are lost as volatile species to yield polycrystalline and near-stoichiometricSiCfiber.ThesefibersareadvancedSiCfibersandgenerallycalledthethirdgenerationofSiCfibers(Figure3),includingHi-NicalontypeSfiber[17],TyrannoSAfibers[18]andSylramicSiCfiber[19].ThethirdgenerationofSiCfibersisoxygen-freeandnear-stoichiometric (atomic ratio: C/Si=1.001.08). Furthermore, their grain size is relatively large(20200 nm) and their thermal stability is excellent.ForenhancingtheenvironmentaldurabilityofCMCs,SiC-basedfiberswithhighcrystallinityandnearstoichiometrywouldbepreferential.Basedonthisstandingpoint,thefollowing SiC-based fibers were used for the work presented in this chapter (Table 1).Jianjun Sha 6Table 1. SiC-based fibers used for the work in this chapter and their propertiesprovided by manufactureSiC fiber C/SiOxygen(wt%)Strength (Gpa)Modulus(Gpa)Density (g/cm3)Diameter(m) HNL 1.39 0.5 2.8 270 2.74 14HNLS 1.05 0.2 2.6 420 3.1 12TySA 1.07 [u]e(13)whereis a model parameter that allows to match the slope of the softening branch justbeyond the attainment of the peak traction M.Sometimes, a smooth transition from the elastic regime to the softening one turns out tobe more representative of the actual interphase response. The nonlinear binding model,originallyproposedin[30, 31]formetalsandbimetalliccompoundsandlateradoptedalso in nonlinear fracture mechanics [20, 21, 32],allows to describe such smooth transi-tion through the following exponential (EXP) law (see Figure 2(c)):= K[u] exp_[u][u]e_(14)Besides the effective stiffness K and strength M, a full characterization of the nonlin-ear behavior of the interface has to match the fracture energy, or work of separation G. Interms of effective quantities, G is dened as the amount of energy required to annihilate theinteraction between the opening/sliding crack faces, i.e.:G =_0 d[u] (15)From a model calibration perspective, parameters Q and in (12) and (13) can be tunedtoaccuratelymatchtheactual G, sincetheydoaffectonlythesofteningbranchoftheinterface law. On the other hand, after having assigned K in (14), only [u]e can be adjusted:therefore, both M and Gcan not be accurately matched. In the exponential law, in fact, thefollowing constraint holds:KG2M= e (16)104 Stefano Marianie being the Nepero number. To avoid problems related to this ctitious constraint, a modi-ed exponential law is here formulated as follows (see Figure 3):= K[u] exp__ [u][u]e_q_(17)where q shows up as an additional constitutive parameter. In law (17): Kstill representsthe initial elastic stiffness; [u]eis a reference effective displacement discontinuity, whilepeak traction Mis attained at [ u] (1/q)1/q[u]e. The effective peak traction and fractureenergy are affected by q, according to:M=K[u]e_1q_1/qexp_1q_G =Kq[u]2ef_2q_(18)f being the gamma function. The dependence of M and G on the parameter q is depictedin Figure 4: it can be seen that Gis a monotonically decreasing function of q, whereas M islower-bounded by the value corresponding to q= 1. Having tuned K, this law thus allowsthe calibration of both M and G.All the above laws but the piecewise linear one, assume that the interaction betweenthe opening interface sides continues up to[u] , which seemsnot physical at themacroscale. To simulate delamination growth a breakdown threshold therefore needs to beintroduced [33, 34]:as soon as the current traction reduces to a small fraction (say 5-10%) of the peak value M, the interaction is suddenly assumed to vanish.When unloading fromthe tensile envelope occurs, i.e. when [ u] < 0, the above interfacemodels can be viewed as either reversible, if always belongs to the envelope (leading tointerface healing if softening has already started), or irreversible, if decreases following aradial path to the origin of the [u] plane. These two alternative constitutive assumptionslead to different entries in the interface tangent stiffness matrix E, linking rates of and[u] in the local n sj1 sj2 reference frame according to: = E[ u] (19)For additional details, readers are referred to [27, 28].2.3. Finite Element FormulationThe weak form of the equilibrium equations (1)-(3) reads:_\vTd =_\vT(b u) d +_vT d n

j=1_j[v]T djv U0(20)where: v is the test function;v=Cv; Uis the trial solution space, collecting displace-ment eldsu continuous in \, possibly discontinuous along each jand fullling theboundarycondition(5)onu; U0istherelevant variationspace, withzeroprescribedFailure of Layered Composites Subject to Impacts 105Figure 3. effect of q on the modied exponential traction-displacement discontinuity law(17).(a) (b)Figure 4. modied exponential law (17). Effects of q on (a) the effective peak traction Mand on (b) the effective fracture energy G.displacements onu. In (20), in view of the assumed linearized kinematics, the relationj +j jfor each interface has been exploited.Allowing for the elastic bulk constitutive law (8), the following variational statement isarrived at:nd u U:_\ vT ud+_\vTE d +n

j=1_j[v]T dj=_\vTb d +_vT dv U0(21)106 Stefano MarianiNow, let the nite element approximation of the displacement and deformation elds in\ be (see [35] for the notation):u =uh(22) = Cuh= Buh(23)where matrix gathers the nodal shape functions, and vectoruhcollects the nodal dis-placements.If delamination is allowed to occur only along element boundaries, the discrete dis-placement jump eld can be written:[u]j= Bjuhj= 1, ..., n (24)Owing to the discrete interpolation elds dened above, the semi-discretized equationsof motion of the composite turn out to be:M uh+Kuh+n

j=1Rj= F (25)where the mass matrix M, the bulk stiffness matrix K, the internal force vectors Rjandthe external load vector Fare, respectively:M=_\ TdK=_\BTEBdRj=_jBTj djF=_\Tb d +_T d(26)Smarter nite element formulations, like the extended or generalized ones [36,37], havebeen recently formulated to simulate mixed-mode crack growth in homogeneous solids,see e.g. [27, 38, 39]. These methodologies allow cracks to propagate not only along inter-element edges, but also inside the elements; possible constraints imposed by the mesh lay-out on crack trajectories, evidenced e.g. in [40], can be therefore alleviated. When deal-ing with delamination in layered continua, where debonding occurs only along the a-prioriknown interlaminar surfaces, crack description looks simple and the aforementioned featureof the extended nite element method looses much of its advantages.Failure of Layered Composites Subject to Impacts 1072.4. Time IntegrationInour previousworks[11, 15] it wasshownthat thetimeintegrationschemecanstrongly affect the stability of the ltering procedure and, therefore, the accuracy of modelparameter estimates.In case of impact loadings, which cause the propagation of shock waves inside the com-posite, the time marching algorithm has to dump the spurious high frequency oscillationslinked to space discretization. Otherwise, the numerically computed displacement and ve-locity elds do not prove reliable enough to be compared to the experimental data.We adopt here the explicitmethod [41, 42] to advance in time the solution of theequations of motion (25), see also [19]. After having partitioned the time interval of interestaccording to[t0tN] = Nt1i=0[titi+1],at the end of the generic time step[titi+1]thesolutionto(25)isobtainedaccordingtothefollowingpredictor-integrator-correctorsplitting:predictor: ui+1=ui + t ui + t2(12 ) ui(27) ui+1= ui + t(1 ) ui(28)where t = ti+1 ti;explicit integrator: ui+1= M1__Fi+1+ (1 +)__K ui+1 +

jRji+1__+__Kui +

jRji____(29)where: Fi+1+= F(ti+(1+)t); Rji=_jBTjidjandRji+1=_jBTj i+1dj;corrector:ui+1= ui+1 + t2 ui+1(30) ui+1= ui+1 + t ui+1(31)In the above equations, andare algorithmic coefcients. To get a non-oscillatoryvelocity eld, = 0.3 has been adopted in all the forthcoming simulations; furthermore, and have been nely tuned around the values allowing second-order accuracy in linearelasto-dynamics.To ensure accuracy of the ltering procedure, the time step size t has been always setso as to fulll the Courant condition in the bulk of the composite. Moreover, to speed upthe explicit integrator phase (29), the mass matrix M has been diagonalized by means of astandard row-sum lumping procedure [42].108 Stefano MarianiAccount taken of the explicit format of the integrator stage, the space-time discretizedequations of motion of the laminate (state equations) can be formally written:zi+1=___ui+1 ui+1 ui+1___= fzi(zi) (32)where z is the structural state vector, and mapping fzturns out to be nonlinear because ofthe softening interface behavior.3. Constrained Sigma-point Kalman Filtering3.1. Parameter Identication via Joint Kalman FilteringAccording to a standard methodology [43], the calibration of constitutive laws can bepursued by Kalman ltering if a state vectorx is obtained by joining the structural statevectorz (see Eq. 32) with a vector gathering all the model parameters to be tuned. Attime ti this can be written:xi=_zii_(33)While the current structural statezis always at least partially observed,model parame-ters to be identied can not be directly measured; by joiningz and, state tracking canconsistently improve model calibration.In case of irreversible constitutive laws, internal state variables must be gathered by xtoo, see e.g. [12, 19].Allowing for model and measurement errors, the state-space model describing the evo-lution within the time interval [titi+1] of the joint state vector and its link with observa-tions turns out to be:_xi+1= fi (xi) +viyi= Hxi +wi(34)where: y is the observation vector, which collects the measured components of the statevector; vistheprocessnoises; wisthemeasurementnoise. v, wareassumedtobeadditive, uncorrelated white and Gaussian processes, with zero mean and covariancesVand W[44, 45]. Since z is dened according to Eq. (32), the observation equation in (34)shows up as a linear relation betweenyandx. On the contrary, the interface behaviorrenders the evolution equation f nonlinear.By way of the EKF [12, 46], within the time step the nonlinear mapping f is expandedin Taylor series, up to the rst order, around the current estimates of the state vector andof model parameters. Bounds on the required accuracy of the initialization ofx, and onthe statistics of noisesv andw to assure lter stability were provided for linear systemsin [47] and, more recently, for nonlinear systems in [48]. Even in the absence of lterinstabilities, the softening response of the interlaminar surfaces does not always guaranteethe achievement of an accurate model calibration, see [15, 19].Failure of Layered Composites Subject to Impacts 109Table 1. Sigma-point Kalman lter.Initialization at t0: x0=E[x0]P0=E[(x0 x0) (x0 x0)T]At ti, for i = 0, ..., N1. Predictor phase: i,j= xi +i,jj = 0, ..., N i+1,j=fi( i,j) xi+1=N

j=0j i+1,jPi+1=Ri+1 +VwhereRi+1=N

j=0j_ i+1,j xi+1__ i+1,j xi+1_T2. Corrector phase: xi= xi+GUi_yi H xi_Pi=PiGUiHRiwhereGUi=RiHT_HRiHT+W_1To improve the results when nonlinearities become dominant, the SPKF has been re-cently proposed [16, 4951]. At the beginning of the time step, the probability distributionofx is deterministically sampled through a set of sigma-points j, j =0, ..., N. Thesesigma-points are then allowed to evolve according to the nonlinear mapping f. The statis-tics of x at the end of the time step are nally obtained through a proper weighted averagingscheme [18].This ltering procedure is detailed in Table 1, whereE[2] represents the ex-pected value of 2.The number of sigma-points and their location in the state vector space are accuratelychosen, so as to achieve high accuracy in the estimated probability distribution of x at theend of each time step; when compared to the EKF, a better performance of the SPKF, alsoin terms of model calibration, is therefore expected [16]. The enhanced accuracy of the110 Stefano MarianiSPKF is discussed next; even though these results have been already presented elsewhere,they are here collected to show how possible constraints on parameter estimates, not dealtwith by the standard SPKF, can be managed.3.2. Accuracy of a Constrained Sigma-point TransformationIn this Section we focus on the time interval[titi+1], but we avoid using indexesiand i + 1 to simplify the notation.Let xbearandomvector, featuringatthebeginningofthetimestepmean xandcovarianceP. Weconceivexasthesumofthemean xandazero-meandisturbancex=x x. If xundergoesanonlineartransformation, governedbyamappingfanalytic everywhere so that it can be expanded in Taylor series about x, at the end of thetime step we get:x= f(x) = f( x +x) =f( x) +

n=11n!Dnx(35)where, with a slight abuse in notation, the nth order term Dnx in the series expansion is:Dnx _Nx

=1fxx= xx_n(36)Nx being the number of components of the state vector x. Since the derivatives of f in (36)are evaluated at x = x, they are not random variables. The expected value of xthereforereads: x=E[x] =E[f( x)] +E_

n=11n!Dnx_=f( x) +

n=11n!E_Dnx(37)The relevant error covariance matrix is:P=E__x x_ _x x_T_=E___

n=11n!_DnxE_Dnx___

m=11m!_DmxE_Dmx__T__=

n=1

m=11n!1m!_E_DnxDmxT_E_DnxE_DmxT__(38)Now, let us suppose to sample the probability distribution of x through a set of sigma-points j, j = 0, ..., N, chosen around the current mean x according to: j= x +j, j = 0, ..., N(39)Failure of Layered Composites Subject to Impacts 111where the termsj need to be determined.Similarly to x, within the time step each sigma-point undergoes the transformation: j= f( j) = f( x) +

n=11n!Dnj(40)where:Dnj_Nx

=1fxx= xj_n(41)At the end of the time step, the information in the evolved sigma-points are collectedvia a weighted averaging procedure to obtain: xSPT=N

j=0j j=__N

j=0j__f( x) +

n=11n!__N

j=0jDnj__(42)where j are the weights of the sigma-point transformation relevant to the mean of x. Thecorresponding error covariance matrix is given by:PSPT=N

j=0j_ j xSPT__ j xSPT_T=N

j=0j

n=1

m=11n!1m!__DnjN

r=0rDnr____DmjN

s=0sDms__T(43)where jare the weights of the sigma-point transformation relevant to the covariance of x.If x is a Gaussian random vector, its probability distribution is symmetric with respectto the mean x; therefore, all the odd central moments Dnx, n = 1, 3, 5, ..., are zero. To becompliant with this condition, couples of sigma-points are symmetrically placed around x,according to [16]:___0=0k= +P1kk = 1, ...,N2N2+k= P1k(44)Here:PrepresentsthesquarerootofmatrixP, computede.g. throughaCholeskyfactorization; is a scaling parameter;1k is a unit vector aligned with component k in the112 Stefano Marianistate vector space. The series expansions (37) and (42) agree up to third order if:___

Nj=0j= 1

Nj=0jD1j=0

Nj=0jD2j=E_D2x

Nj=0jD3j=0(45)To simplify the matter, let us assume j= for j=1, ..., N; relations involving the rstand third order terms in (45) are then automatically satised. Relations involving the zerothand second order terms in (45) then furnish:_0 +N= 122= 1(46)A further condition to set 0, and can be furnished by matching the diagonal entries ofthe fourth order terms (kurtoses) in (37) and (42). This leads to [19]:0= 1 N6, =16, =3 (47)Here we propose an alternative condition to determine 0, and , partially exploitingthe features of the so-called scaled unscented transformation [52]. Let us assume that modelparameters have to satisfy the constraints:m M(48)where mand Mrespectively gather the minimum and maximum (if any) allowed valuesofmodelparameters. Thisrequirementmustbefullledbyeachsigma-point j, j =0, ..., N. For j = 0 the conditions (48) are automatically satised, since x (and, therefore,) is computed at the end of the previous time step by averaging sigma-points all fulllingthe constraints. Further,if=Bx, Bbeing a Boolean matrix,conditions (48) aresatised by all the sigma-points if: min_ mak,Mak_, k = 1, ..., N2; = 1, ..., N(49)where ak= BP1k, and N is the number of model parameters in . In the forthcomingexamples, we initially assume =3 (according to what reported in 47) and reduce itsvalue if necessary, according to relation (49).As for the error covariance matrix PSPT, by letting j= j= for j = 1, ..., N, weget:PSPT=

n=1

m=11n!1m!__N

j=0jDnjDmjT+ (0 +N 2)N

j=0jDnjN

k=0kDmkT__(50)Failure of Layered Composites Subject to Impacts 113In case of Gaussian random variables, independently of the value of 0, (38) and (50) agreeup to the third order. Weight 0can be set by matching part of the fourth order terms(specically those involving D2jD2kTin 50), thereby obtaining (see also [52]):0= 4 N2 2(51)In what precedes we have assumed the mappingfto be analytic everywhere in thexspace. Iffis not differentiable, low order terms of the Taylor series expansions of meanandcovariancegetaffected. Itisdifculttoquantifythediscrepancieswithrespecttothe analytic case, because they depend on whether the sigma-points sample the loci of non-differentiability. However, it can be generally said that the order of accuracy is detrimentallyaffected.4. ResultsToassesstheperformancesoftheproposedlteringapproachincalibratingthein-terlaminar constitutive law while detecting impact-induced delamination, we rst study asimple problem consisting of a two-layer composite stricken by a homogeneous impactor.Hence, two different impact tests on GRP composites [14, 53] are considered to mainlyshow the accuracy in detecting delamination in real-time.In all the cases, it is assumed that the contact between specimen and impactor is per-fect (i.e. distributed all over their approaching surfaces) and that failure of the laminateoccurs because of the propagation of dilatational plane waves in the through-the-thicknessdirection: the interlaminar surfaces are therefore subject to pure mode I loading.4.1. Pseudo-experimental TestingsAs a starting benchmark, a pseudo-experimental testing condition is conceived. Thepseudo-experimental response of the laminate to the impact loading has been computed byadding a white noise (of assigned variance) to sampled outcomes of nite element analyses.Eventhoughthisapproachissometimescriticized, beingnotclearwhetheroneistest-ing with it the ltering approach or its implementation, it helps in getting insights into theperformance of the lter in terms of stability and convergence rate. Indeed, calibration ofinterlaminar constitutive models may become difcult if delamination occurs almost instan-taneously: the lter has to be highly sensitive to model parameters to promptly react to theinformation conveyed by measurements. This requires a careful setting of lter parameters,likeP0 andV . It is worth mentioning that, to further complicate the problem, the effectsof the shape of the tensile envelope of interlaminar laws (at assigned strength and tough-ness) on the overall response of a laminate subject to impacts have not been thoroughlyunderstood yet: depending on the loading and boundary conditions, on the composite ge-ometry and on the stacking sequence, in some circumstances the shape affects the response,whereas in others it does not [54].The capabilities of the SPKF are therefore rst assessed through a simple test: a two-layer composite is stricken by a homogeneous impactor. The laminae and the impactor are114 Stefano MarianiFigure 5. impact on a two-layer composite. Space-time diagram (the vertical dashed linehere represents the possibly debonding surface when a brittle, homogeneous material issubject to the same impact).assumed to be isotropic and elastic, featuring Youngs modulus E= 10 GPa, Poissons ratio= 0.35 and mass density = 1500 kg/m3(see also [11]). Each lamina and the impactorare0.75 mm in thickness. Target mechanical properties of the interlaminar surfaces areassumed:2K=277.09 (N/mm3)M=75 (MPa)G =0.15 (N/mm)(52)According to the space-time diagram of Figure 5, failure can occur only along the interlam-inar surface because of the interaction of the two release waves propagating inwards fromthe free surfaces of impactor and specimen.Two different values of the velocityv of the impactor are considered. In a rst casev=10.19 m/s leads to the propagation in the through-the-thickness direction of a com-pressive/tensile wave of amplitude = 50 MPa, which does not cause interface failure. Ina second case v=20.38 m/s causes laminate failure, i.e. whole delamination, because ofthe propagation of a compressive/tensile wave of amplitude =100 MPa. Outcomes ofthe two tests are respectively reported in Figures 6 and 7, in terms of time evolution of thefree surface velocity velocity ur at the rear laminate surface, of the opening displacementdiscontinuity [u] and of the normal traction (here and in what follows the subscript n hasbeen dropped to simplify the notation). Results are shown for all the constitutive modelsdescribed in Section 2.2., having assumed q= 1 for the modied exponential interface law(17). For comparison purposes, the response of an interface-free specimen is reported too;in such a case, the purely elastic behavior of the material leads to the propagation of sharpfronts of a shock wave.If delamination is not incepted ( =50 MPa), the response is almost independent ofthe shape of the interface law. Only in the presence of an interface that behaves accordingto the exponential model, the pre-peak nonlinearity of the constitutive law (see Figure 2(a))Failure of Layered Composites Subject to Impacts 115(a)(b) (c)Figure 6. impact on a two-layer composite, = 50 MPa. Effects of the interlaminar laws onthe time evolution of (a) the free surface velocity ur, of (b) the displacement discontinuity[u] and (c) relevant traction along the interface.leads to a larger opening [u] of the interlaminar surface when subject to a tensile stress. Forany constitutive model, the signature of the interface is shown in Figure 6(a) by the delay inthe sudden changes of ur with respect to the reference, interface-free solution. This delay,which grows in time, is caused by the compliance of the interface, that is additional to thebulk one.In case of failure ( = 100 MPa), the free surface velocity ur is affected by the interfacemodel only when the waves,traveling across the interlaminar surface while softening istaking place,reach the rear surface;this occurs in the present case around 1s after theimpact, seeFigure7(a). Afterfailure, therearlaminadetachesfromthefrontoneandfreely ies off, as testied by the diverging[u] history in Figure 7(b). Part of the shockwaves then get conned inside the back lamina: this explains the subsequent doubling ofthe drops in the urevolution. It is worth noting that the time elapsed between softening116 Stefano Mariani(a)(b) (c)Figure 7. impact on a two-layer composite, = 100 MPa. Effects of the interlaminar lawson the time evolution of (a) the free surface velocity ur, of (b) the displacement discontinu-ity [u] and (c) relevant traction along the interface.inception (t =0.72 s) and whole failure of the interface (t =0.9 s), see Figure 7(c), isvery short; only the information on this failure event, constituting the so-called pull-backsignal (PBS), conveyed by the shock waves to the free laminate surface, can be used by thelter to calibrate the interface law.The effects on the PBS of the shape of the tensile envelope and of the strength Mandtoughness G values need to be assessed.In the absence of any dissipation mechanisms, inthis test the free surface velocity ur would drop to zero at t=0.917 s (see Figure 6(a));in case of delamination, the minimum attained velocity after the arrival of the unloadingtensile wave and the shape of the PBS do furnish information on the interface response. Tounderstand the roles of interface law, strength and toughness, the results of a parametricanalysis are shown in Figure 8: for any interface model, MandG are varied by 20%at most with respect to the target values (52). The piecewise linear and linear-exponentialFailure of Layered Composites Subject to Impacts 117(a) (b)(c) (d)(e) (f)Figure 8. impact on a two-layer composite, = 100 MPa. Effects of the interface strengthM(left column) and toughness G (right column) on the pull-back signal. (a-b) piecewiselinear law (12); (c-d) linear-exponential law (13); (e-f) exponential law (14).118 Stefano Marianilaws, having a common initial elastic phase in tension, lead to a common descending branchin the PBS. At variance, the local tangent stiffness of the exponential law in the hardeningphase is affected by Mand G: the slope of the descending branch of the PBS is thereforeaffected byMandG too. Independently of the interface law,Mturns out to affect thestarting stage of debonding, whereas G affects its tail and the time needed to complete it;this is clearly evidenced by the PBS, sinceMmodies the minimum attained velocity,whereas G inuences only the ascending branch with no effects on the pull-back velocity.From a model calibration perspective, it is clear that the SPKF has chances to improveparameter estimates only in the time interval0.9 t 1.2s. It is therefore hard togure out from the previous plots whether the effects of tensile envelope,MandG canbe actually interpreted by the lter to improve model calibration. The performances ofthe SPKF are hence tested here not only looking at parameter estimates, but also checkingits capability to detect whether a laminate is failing and, in case of delamination, where itactually takes place.Typical results of the ltering procedure are depicted in Figures 9 and 10; in this casethe pseudo-experimental data, which consist in the free surface velocity alone, have beensupposed very accurate, featuring a standard deviation =0.33 m/s (the measurementerror covariance matrixWbecomes scalar-valued, with entryW=2). As far as theprocess covariance matrixVis concerned, in case of pseudo-experimental testing it canbe assumed to be vanishing, since the lter employs the same structural model adopted toget the pseudo-experimental data. Components ofP0instead need to be nely tuned toenhance lter convergence [11, 15].Figure 9 shows the obtained estimates of M and Gas a function of their initial guess in x0 (here respectively denoted by M,0 and G0), for all the interface models. These estimatesevolve from M,0 and G0 once the PBS is processed by the lter; after that, they becomestationary. The tracked state of the laminate is shown in Figure 10 in terms of predictedinterfaceopening[u]andfreesurfacevelocity ur. Knowingthetargetresponseofthecomposite to the impact, here denoted by the dashed lines, allows to certify stability andconvergence of the SPKF, no matter if displacement is diverging in a part of the system andwhat kind of interface constitutive law has been adopted. It can be seen that estimates getenhanced as soon as the lter senses the PBS: in fact, the sudden changes in the estimate of[u] show up only while processing the PBS, starting from t = 0.9 s.These outcomes testify that the SPKF is very efcient in tracking the state of the lam-inate, i.e. in understanding whether the structure is failing or not. Model calibration isinstead less accurately accomplished: independently of the interface law, Mis quite pre-cisely estimated, provided the initial guess M,0 is not too far from the target value, whereasG can be hardly inferred.These conclusions are in agreement with the results of the para-metricanalysis: whileMaffectsthewholePBS, Gaffectsonlyitsascendingbranch.Therefore, ltering out from ur the effects of G alone turns out to be extremely difcult.In case of a much higher scattering of pseudo-experimental data ( =3.3 m/s),seeFigure 11, results loose accuracy as for the calibration task. Contrariwise, state trackingmaintain accuracy: even though measurements contain poor information, the SPKF is againable to provide the evolution of the free surface velocity in the PBS.Failure of Layered Composites Subject to Impacts 119(a) (b)(c) (d)(e) (f)Figure 9. impact on a two-layer composite, = 100 MPa (W=101m2/s2). Effects of theinitialization valuesM,0andG0on the converged estimates ofM(left column) andG(right column). (a-b) piecewise linear law; (c-d) linear-exponential law; (e-f) exponentiallaw.120 Stefano Mariani(a) (b)(c) (d)(e) (f)Figure 10. impact on a two-layer composite, = 100 MPa (W=101m2/s2). Evolution intime of interface opening [u] (left column) and free surface velocity ur (right column), andcomparison among tracked state (orange squares), actual state (dashed lines) and pseudo-experimental data (blue circles). (a-b) piecewise linear law; (c-d) linear-exponential law;(e-f) exponential law.Failure of Layered Composites Subject to Impacts 121(a) (b)(c) (d)(e) (f)Figure 11. impact on a two-layer composite, =100 MPa (W=10 m2/s2). Evolution intime of interface opening [u] (left column) and free surface velocity ur (right column), andcomparison among tracked state (orange squares), actual state (dashed lines) and pseudo-experimental data (blue circles). (a-b) piecewise linear law; (c-d) linear-exponential law;(e-f) exponential law.122 Stefano Mariani(a)(b)Figure 12. impact on a 7-layer composite [53]. (a) space-time diagram (the vertical dashedline here represents the possibly debonding surface when a brittle, homogeneous materialis subject to the same impact), and (b) experimentally measured free surface velocity.4.2. Actual Experimental TestingsTo nally check the performances of the SPKF when dealing with multi-layered com-posites, we consider two of the experiments reported in [53] and [14].In the rst experiment (experiment FY06001 in [53]), the specimen is a 7-layer com-posite plate; each lamina is 1.37 mm in thickness, and is made of a balanced 5-harness satinweave E-glass and LY564 epoxy. The wave speed in the through-the-thickness direction is3.34 km/s, while the mass density is =1885 kg/m3[53]. This laminate was subject to aplane impact, stricken by an aluminum impactor (12.5 mm thick) ying at velocity v= 71m/s. The relevant space-time diagram is shown in Figure 12, along with the free surfacevelocity prole measured via a velocity interferometer for any reector (VISAR).Account taken of the high accuracy of the experimentally measured ur, the identica-tion procedure have furnished the results reported in Figure 13 in terms of time evolutionof estimates of M and G as a function of their initialization values within the domain:C= {50 M,0 250 (MPa), 0.5 G0 2.5 (N/mm)} (53)Failure of Layered Composites Subject to Impacts 123(a) (b)(c) (d)(e) (f)Figure 13. impact on a 7-layer composite [53]. Evolution in time of the estimated values ofM (left column) and G (right column). (a-b) piecewise linear law; (c-d) linear-exponentiallaw; (e-f) exponential law.124 Stefano Mariani(a)(b) (c)(d) (e)(f) (g)Figure 14. impact on a 7-layer composite [53]. Evolution in time of (a) free surface velocity ur and (b-g) estimated interface openings [u]1 [u]6.Failure of Layered Composites Subject to Impacts 125(a)(b)Figure 15. impact on a 11+11-layer composite [14]. (a) space-time diagram (the verticaldashed line here represents the possibly debonding surface when a brittle, homogeneousmaterial is subject to the same impact), and (b) experimentally measured free surface ve-locity.Converged estimates ofMare in good agreement with the spall strength of 119.5 MPareported in [53]; on the other hand, nal estimates of G are well representative for this kindof composites. Figure 14 reports the estimated state of the specimen: the capability to trackthe measured free surface velocity and to foresee delamination along the third interlaminarsurface away from the impact plane, is evidenced. This latter result, allowing also for wavedispersioncausedbyinterlaminarsurfacesandinnerinhomogeneitiesofthecomposite,well agrees with the state-space diagram of Figure 12(a).In the second experiment (experiment 1 in [14]), a GRP specimen,7.02 mm thick, isbacked by another GRP plate, 6.91 mm thick; both laminates are made of 11 plies. Thewave speed in the through-the-thickness direction now amounts to 3.19 km/s, and the massdensity to = 1867 kg/m3[14]. The specimen is stricken by a 5-layer GRP yer, 2.96 mmin thickness, ying at velocity v= 85 m/s. The corresponding space-time diagram, and thefree surface velocity prole measured via a VISAR are reported in Figure 15. Because ofthe test set-up, the release waves interact causing delamination inside the back plate.126 Stefano Mariani(a) (b) (c)(d) (e) (f)(g) (h) (i)(j) (k)Figure 16. Impact on a 11+11-layer composite [14]. Evolution in time of (a) free surfacevelocity ur and (b-k) estimated interface openings [u]1 [u]10 in the back plate.Results of the ltering process are reported in Figure 16 is terms of tracked free surfacevelocity and estimated displacement jumps along all the interfaces inside the back plate(sequence starts at the specimen-back plate contact surface). These estimations turn outonce again to be independent of the interface law and of the initialization values of M andFailure of Layered Composites Subject to Impacts 127G inside the domain:C= {25 M,0 100 (MPa), 0.1 G0 0.6 (N/mm)} (54)While the free surface velocity is accurately tracked, delamination is foreseen to take placealong the 7-th interlaminar surfaces, in agreement with the results of [11, 14]. As far asmodel calibration is concerned, outcomes turn out to be qualitatively in agreement withthose already reported for the previous tests.5. ConclusionIn this Chapter we have addressed some issues related to constitutive modeling and pa-rameter identication in nite element simulations of layered composites subject to impacts.Assuming the impact energy to be high enough to cause damage spreading inside the inter-laminar resin-enriched phases, but not high enough to result in penetration of the impactoraccompanied by intralaminar damage, a numerical scheme for structural-level analyses hasbeen revised. Within this scheme the laminae are assumed to behave elastically, whereasdissipation mechanisms are lumped onto zero-thickness interlaminar surfaces. Along theseinterlaminar surfaces strength reduction, eventually leading to delamination, is governed bysoftening interface constitutive laws linking tractions to displacement jumps.Interface lawsare known tobe difcult to calibrate, since direct testing ona singleinterlaminar phase can not be devised. Here we have offered a sigma-point Kalman lteringapproach to estimate uncertain model parameters of the aforementioned interface laws. Thistechnique outperforms most of the customarily adopted ones, since it efciently deals withnonlinearities, whicharearesultofinterlaminarstrengthdegradationinthecaseunderstudy.The performances of the lteringprocedure have beenassessedthroughpseudo-experimental testings on a two-layer composite, and through real testings on multi-layerglass ber reinforced plastic composites. It has been shown that the state of the composite,including delamination inception and growth, is always tracked with a noteworthy level ofaccuracy. Due to the fast failure events, model calibration is instead less accurately per-formed and sometimes requires initialization values of uncertain model parameters to beappropriately chosen to avoid biased estimates.References[1] S. Abrate. Impact on composite structures. Cambridge University Press, 1998.[2] H.D. Espinosa, S. Dwivedi, P.D. Zavattieri, and G. Yuan. A numerical investigation ofpenetration in multilayered material/structure systems. International Journal of Solidsand Structures, 35:29753001, 1998.[3] O. Allix and P. Ladev eze. Interlaminar interface modelling for the prediction of de-lamination. Composite Structures, 22:235242, 1992.128 Stefano Mariani[4] A. Corigliano. Formulation, identication and use of interface elements in the numer-ical analysis of composite delamination. International Journal of Solids and Struc-tures, 30:27792811, 1993.[5] A. Corigliano and O. Allix. Some aspects of interlaminar degradation in composites.Computer Methods in Applied Mechanics and Engineering, 185:203224, 2000.[6] A. Corigliano and S. Mariani. Simulation of damage in composites by means of inter-face models: parameter identication. Composites Science and Technology, 61:22992315, 2001.[7] G. Alfano and M.A. Criseld. Finite element interface models for the delaminationanalysis of laminated composites: mechanical and computational issues. InternationalJournal for Numerical Methods in Engineering, 50:17011736, 2001.[8] A. Corigliano. Damage and fracture mechanics techniques for composite structures. InI. Milne, R.O. Ritchie, and B. Karihaloo, editors, Comprehensive Structural Integrity,volume 3, chapter 9, pages 459539. Elsevier Science, 2003.[9] A. Corigliano and S. Mariani. Parameter identication of a time-dependent elastic-damage interface model for the simulation of debonding in composites. CompositesScience and Technology, 61:191203, 2001.[10] A. Corigliano and S. Mariani. Identication of a constitutive model for the simula-tion of time-dependent interlaminar debonding processes in composites. ComputerMethods in Applied Mechanics and Engineering, 191:18611894, 2002.[11] S. Mariani and A. Corigliano. Impact induced composite delamination: state and pa-rameter identication via joint and dual extended Kalman lters. Computer Methodsin Applied Mechanics and Engineering, 194:52425272, 2005.[12] S. Bittanti, G. Maier, and A. Nappi. Inverse problems in structural elastoplasticity:a Kalman lter approach. In A. Sawczuk and G. Bianchi, editors, Plasticity today.Modelling, Methods and Applications, pages 311329. Elsevier Applied Science Pub-lishers, 1984.[13] A. Corigliano, S. Mariani, and B. Orsatti. Identication of Gurson-Tvergaard materialmodel parameters via Kalman ltering technique - I. Theory. International Journal ofFracture, 104:349373, 2000.[14] H.D. Espinosa, S. Dwivedi, and H.-C. Lu. Modeling impact induced delamination ofwoven ber reinforced composites with contact/cohesive laws. Computer Methods inApplied Mechanics and Engineering, 183:259290, 2000.[15] A. Corigliano and S. Mariani. Parameter identication in explicit structural dynamics:performance of the extended Kalman lter. Computer Methods in Applied Mechanicsand Engineering, 193:38073835, 2004.Failure of Layered Composites Subject to Impacts 129[16] S.J. Julier,J. Uhlmann,and H.F. Durrant-Whyte. A new method for the nonlineartransformation of means and covariances in lters and estimators. IEEE Transactionson Automatic Control, 45:477482, 2000.[17] T. Lefebvre, H. Bruyninckx, and J. De Schutter. Kalman lters for nonlinear systems:a comparison of performance. International Journal of Control, 77:639653, 2004.[18] K. Ito and K. Xiong. Gaussian lters for nonlinear ltering problems. IEEE Transac-tions on Automatic Control, 45:910927, 2000.[19] S. Mariani and A. Ghisi. Unscented Kalman ltering for nonlinear structural dynam-ics. Nonlinear Dynamics, 49:131150, 2007.[20] A. Corigliano, S. Mariani, and A. Pandol. Numerical modeling of rate-dependentdebonding processes in composites. Composite Structures, 61:3950, 2003.[21] A. Corigliano, S. Mariani, and A. Pandol. Numerical analysis ofrate-dependentdynamic composite delamination. Composites Science and Technology, 66:766775,2006.[22] R. Lund and J. Fish. Space-time multiscale laminated theory. International Journalof Multiscale Computational Engineering, 2:2647, 2004.[23] S. Mariani, A. Pandol, and R. Pavani. Coupled space-time multiscale simulations ofdynamic delamination tests. Materials Science, 23:509519, 2005.[24] R.R. Talreja Y.W. Kwon, D.H. Allen, editor. Multiscale Modeling and Simulation ofComposite Materials and Structures. Springer, 2007.[25] G.T. Camacho and M. Ortiz. Computational modelling of impact damage in brittlematerials. International Journal of Solids and Structures, 33:28992938, 1996.[26] Y. ArunRoyandR.H. Dodds. Simulationof ductilecrackgrowthinthinallu-minum panels using 3-Dsurface cohesive elements. International Journal of Fracture,110:2145, 2001.[27] S. Mariani and U. Perego. Extended nite element method for quasi-brittle fracture.International Journal for Numerical Methods in Engineering, 58:103126, 2003.[28] C. Comi and S. Mariani. Extended nite element simulation of quasi-brittle fracturein functionally graded materials. Computer Methods in Applied Mechanics and Engi-neering, 196:40134026, 2007.[29] S. Li, M.D. Thouless, A.M. Waas, J.A. Schroeder, and P.D. Zavattieri. Mixed-modecohesive-zone models for fracture of an adhesively bonded polymer matrix composite.Engineering Fracture Mechanics, 73:6478, 2006.[30] J.H. Rose, J. Ferrante, and J.R. Smith. Universal binding energy curves for metals andbimetallic interfaces. Physical Review Letters, 47:675678, 1981.130 Stefano Mariani[31] J.H. Rose, J.R. Smith, and J. Ferrante. Universal features of bonding in metals. Phys-ical Review B, 28:18351845, 1983.[32] X.-P. Xu and A. Needleman. Numerical simulation of fast crack growth in brittlesolids. Journal of the Mechanics and Physics of Solids, 42:13971434, 1994.[33] A.NeedlemanandA.J.Rosakis. Theeffectofbondstrengthandloadingrateonthe conditions governing the attainment of intersonic crack growth along interfaces.Journal of the Mechanics and Physics of Solids, 47:24112449, 1999.[34] C.M. Landis, T. Pardoen, and J.W. Hutchinson. Crack velocity dependent toughnessin rate dependent materials. Mechanics of Materials, 32:663678, 2000.[35] O. C. Zienkiewicz and R. L . Taylor. The nite element method, volume II. McGraw-Hill, London, 1989.[36] N. Moes, J. Dolbow, and T. Belytschko. A nite element method for crack growthwithoutremeshing. InternationalJournalforNumericalMethodsinEngineering,46:131150, 1999.[37] T. Strouboulis, K. Copps, and I. Babuska. The generalized nite element method.Computer Methods in Applied Mechanics and Engineering, 190:40814193, 2001.[38] G.N. Wells and L.J. Sluys. A new method for modelling cohesive cracks using niteelements. International Journal for Numerical Methods in Engineering, 50:26672682, 2001.[39] C. Comi, S. Mariani, and U. Perego. An extended FE strategy for transition fromcontinuum damage to mode I cohesive crack propagation. International Journal forNumerical and Analytical Methods in Geomechanics, 31:213238, 2007.[40] M. Jirasek and T. Zimmerman. Embedded crack model: II. Combination with smearedcracks. International Journal for Numerical Methods in Engineering, 50:12911305,2001.[41] H.M. Hilber, T.J.R. Hughes, and R.L. Taylor. Improved numerical dissipation for timeintegration algorithms in structural dynamics. Earthquake Engineering and StructuralDynamics, 5:283292, 1977.[42] T.J.R. Hughes. The nite element method. Linear static and dynamic nite elementanalysis. Dover Publications, 2000.[43] L. Ljung. System identication. Theory for the user. Prentice Hall, second edition,1999.[44] A. Gelb. Applied optimal estimation. The M.I.T. Press, 1974.[45] E.A. Wan and A.T. Nelson. Dual extended Kalman lter methods. In S. Haykin,editor, Kalman ltering and neural networks, pages 123173. John Wiley & Sons,Inc., 2001.Failure of Layered Composites Subject to Impacts 131[46] R. E. Kalman and R. S. Bucy. New results in linear ltering and prediction theory.ASME Journal of Basic Engineering, 83D:95108, 1961.[47] L. Ljung. Asymptotic behavior of the extended Kalman lter as a parameter estimatorfor linear systems. IEEE Transactions on Automatic Control, AC-24:3650, 1979.[48] K. Reif, S. Gunther, E. Yaz, and R. Unbehauen. Stochastic stability of the discrete-time extended Kalman lter. IEEE Transactions on Automatic Control, 44:714728,1999.[49] S.J. Julier and J.K. Uhlmann. A general method for approximating nonlinear trans-formations of probability distributions. Technical report, RRG, Department of Engi-neering Science, University of Oxford, 1996.[50] S.J. Julier, J. Uhlmann, and H.F. Durrant-Whyte. A new approach for ltering nonlin-ear systems. In Proceedings of the American Control Conference, pages 16281632,Seattle, June 1995.[51] E.A. Wan and R. van der Merwe. The unscented Kalman lter. In S. Haykin, editor,Kalman ltering and neural networks, pages 221280. John Wiley & Sons, Inc., 2001.[52] S.J. Julier. The scaled unscented transformation. In Proceedings of the AmericanControl Conference, volume 6, pages 45554559, Anchorage, May 2002.[53] F.Yuan, L.Tsai, V.Prakash, A.M.Rajendran, andD.P.Dandekar. Spallstrengthofglassberreinforcedpolymercomposites. InternationalJournalofSolidsandStructures, 44:77317747, 2007.[54] G. Alfano. On the inuence of the shape of the interface law on the application ofcohesive-zone models. Composites Science and Technology, 66:723730, 2006.In: Strength of MaterialsISBN: 978-1-60741-500-8 Editors: G. Mendes and B. Lago, pp. 133-155 2009 Nova Science Publishers, Inc. Chapter 4 CURRENT STATE OF THE ART OF THE CERAMIC COMPOSITE MATERIAL BIOLOXDELTA Meinhard Kuntz1, Bernard Masson2 and Thomas Pandorf1 1 CeramTec AG, Plochington, Germany 2 CeramTec AG, Pechabou, France Abstract Anextensiveoverviewaboutthestateoftheartoftheceramiccompositematerial BIOLOXdeltaisgiven.Theuniquepropertiesrelyonawelldefinedaluminabasedfine compositemicrostructurewhichismainlyachievedbyhightemperaturesolidbodyreaction of the different ceramic phases during sintering. Zirconia comprises 17 % of the total volume. The tetragonal phase of zirconia is stabilized chemically and mechanically. The high strength and toughness of the material depend on transformation toughening of the zirconia which is clearly shown by various experimental results. The excellent mechanical properties are reproduced batch by batch with a very low scatter.TheoutstandingpropertiesofthematerialBIOLOXdeltasupportadvantageous properties of the final product, e.g. ceramic hard-hard bearings for hip arthroplasty. The burst loadofthecomponentsissignificantlyincreased.Itisshownthatthedesignofthe components is also very important for the reliability and the ultimate properties of the system. Wearpropertiesatsevereconditionsaresignificantlyimprovedbyusingthenewcomposite material BIOLOXdelta in comparison to pure alumina.Phasetransformationofzirconiafromthetetragonaltothemonoclinicphasedueto hydrothermalagingisextensivelydiscussed.Duetotheparticulardistributionand stabilizationofthezirconiaparticlesinstableagingeffectsarenotpossibleinthismaterial. After very long time of accelerated aging conditions an increase of monoclinic phase is found however,itisshownthatdynamicandstaticpropertiesofBIOLOXdeltaarenot influenced by this effect. 1. Introduction Since2001morethan500.000artificialhipjointswithcomponentsofthenewhigh performanceceramiccompositeBIOLOXdeltahavebeensuccessfullyimplantedona Meinhard Kuntz, Bernard Masson and Thomas Pandorf134 global basis. Due to the unique strength and toughness of this material the risk of fracture has been substantially reduced when compared to conventional ceramic materials.TheoutstandingpropertiesofBIOLOXdeltarelyoncomplexreinforcingmechanisms. Therefore, it is necessary to assess if reinforcement is maintained throughout the life-time of the artificial joint which is anticipated to exceed more than 20 years. Furthermore, it is shown that the challenging production of BIOLOXdelta is reproduced at a high quality from batch to batch.Within the scope of this technical contribution the composite ceramic BIOLOXdelta is extensivelydescribedandanalyzed.Thecompositionandthematerialpropertiesare presentedbasedondataofregularproductionlots.Itisshownthattheadvantageous properties of this material are based on the reinforcing mechanisms which are activated due to the unique composition of this material.Theparticulareffectofmonoclinicphasetransformationandhydrothermalagingis describedindetailbasedongeneralmechanismsandspecificanalysisofphase transformationinBIOLOXdelta.Furthermore,experimentaldataareprovidedwhich describethelongtermpropertiesofthematerial,inparticularwithrespecttohydrothermal phase transformation of zirconia in combination with wear and cyclic load conditions. 2. International Material Standards BIOLOXdelta is a modern ceramic composite material for biomedical applications. The maincomponentsofthecompositearealuminaandzirconia.ThereareISOstandards available for bioceramics of high purity alumina (ISO 6474 - 1) and high purity zirconia (ISO 13356).However,thesestandardsarenotdirectlyapplicableforthecompositionof BIOLOXdelta. The ISO organisation is already on the way to prepare a new standard which isapplicableforBIOLOXdeltaandothersimilarcompositematerials(ISO6474-2).The new standard will be released presumably in 2010.Meanwhile,itishelpfultoreferandcomparethepropertiesofBIOLOXdeltatothe draftofthenewISO6474-2andtheotherinternationalmaterialstandardswhichare applicable for related high purity bioceramics. ISO 6474 1Implants for Surgery Ceramic Materials -Part 1: Ceramic Materials Based on High Purity Alumina The current version of this standard was released in 1994. The material properties which aredefinedherereflecttypicalpropertiesofhighqualitypurealumina.Strengthand toughnesswhicharerequiredaccordingtothisstandardaresignificantlylowerthanthose which are available with the composite material BIOLOXdelta.Today, theexperts oftheISOworking group agreethatsomedetailsofthecurrentISO 6474donotrepresentthestateoftheart.Thus,anewversionofISO6474-1isbeing preparedwhichisalreadypublishedasaDraftInternationalStandard(ISO/DIS).Most technicaldetailsinthisversionarealreadyfinallyimplemented.Somedetails(fracture Current State of the Art of the Ceramic Composite Material BIOLOXdelta135toughness, microstructure) are still under discussion. Presumably, the final release of the new standard will be in 2009.Comment: There is a similar ASTM standard F 603 for the same application and material type. The required material properties are comparable. ISO 6474 2. Implants for Surgery Ceramic Materials -Part 2: Composite Materials Based on a High Purity Alumina Matrix with Zirconia Reinforcement Astandardwhichisapplicableforaluminazirconiacompositematerialsisunder preparation.Suchcompositematerialsaredistinguishedinthosewherethemainphaseis alumina(ZTA=zirconiatoughenedalumina)andthosewherezirconiaisthedominating phase(ATZ=aluminatoughenedzirconia).Bothmaterialtypesareavailableforbiomedical applications.Thebasicphysicalproperties(e.g.hardness,thermalconductivity)ofacompositeare primarilyderivedfromthemainphase.Itisthususefultodescribealuminabasedzirconia toughenedmaterialsparalleltopurealuminamaterials.Consequently,thenewstandardhas been proposed as part 2 of the established standard ISO 6474 for pure alumina. This concept hasbeendiscussedattheISOTC150meetingsandwasapprovedbytheinternational experts.ThenewstandardISO6474 - 2isacceptedasaWorkingDraft.Itcoversallmaterial properties (except of Youngs modulus) which are defined in ISO 6474 - 1. Additionally, the specificsubjectsofhydrothermalagingandradioactivity,whicharerelevantforzirconia toughened materials is also included.ISO 6474 2 is defined such that a broad range of inorganic compositions are included. A composition of 60 wt. % alumina and 10 wt. % zirconia is required. Other ingredients are allowed. The new standard is not exclusively designated for BIOLOXdelta.So far there is no ASTM standard for alumina zirconia composite biomaterials available.ISO 13 356. Implants for Surgery Ceramic Materials Based on Yttria-Stabilized Tetragonal Zirconia (Y-TZP) This standard was revised and published in 2008 as an International Standard (ISO).In contrast to pure alumina, zirconia as a ceramic material can not be produced without a significantamountofothersubstancesforphasestabilization.Severalelementsareknown whichareapplicable.ISO 13356isonlyfocussedonYttriaasthestabilizingelement.A specific range of Y content is predetermined. It should be noted that the typical range of Y in purezirconiamaterialscanbedifferenttotherequiredamountofYinaluminazirconia composites. This issue is thoroughly discussed in chapter 4.Pure zirconia bioceramics can be applied either for biomedical bearings (e.g. hip or knee) orfordentalapplications.ISO13356doesnotdistinguishbetweenthesedifferent applications.As a specific issue of zirconia a test for accelerated hydrothermal aging is required. Meinhard Kuntz, Bernard Masson and Thomas Pandorf136 Table1givesanoverviewoftherequiredpropertiesofthe3standards.Thematerial properties according to the latest revised versions are chosen Table 1. Required material properties according to the ISO standards 6474 1 (pure alumina), ISO 6474 2 (alumina zirconia composite) and ISO 13 356 (zirconia). ISO StandardISO 6474 - 1ISO 13 356ISO 6474 2 MaterialUnitPure AluminaZirconiaAlumina Zirconia Average Bulk density 3,94 g/cm3 6,00 g/cm3 98,7 % Chemical Composition wt%Al2O3 99,7MgO 0,2 Impurities 0,1 ZrO2+HfO2+Y2O3 99,0 Y2O3 4,5 6,0 HfO2 5 Al2O3 0,5 Others 0,5 Al2O360 - 90ZrO2+HfO210 - 30Additives 10 Impurities 0,2 Grain SizemMV 2,5 SD 40 % MV 0,4Al2O3 MV 1,5 ZrO2MV 0,6 SD 40 % StrengthWeibull Modulus (4 pt bending) MPa 500 8 800 1000 10 Youngs modulusGPa 380Fracture Toughness MPam 2,5 4,0 Hardness HV1GPa 18 17 Wear ResistanceInfoInfo Cyclic fatigue limit No failure at 200 MPa No failure at 320 MPa No failure at 400 MPa Amount of monoclinic phase % 20 Accelerated Aging 25 % monocl. phasestrength decrease not more than 20% Accomplish requirements described above RadioactivityBq / kg 200 100 Note: The values of ISO 6474 1 & 2 are not finally fixed at the date of this publication.3. Description of BIOLOXdelta BIOLOXdelta is an alumina based composite ceramic. Approximately 80 vol.-% of the matrixconsistoffinegrainedhighpurityaluminawhichisverysimilartothewellknown materialBIOLOXforte(ISO6474).Asitisthecaseinanyothercompositematerial,the basicphysicalpropertieslikestiffness,hardness,thermalconductivityetc.aremainly predeterminedfromthedominatingphase.Itwasthebasicideaforthedevelopmentofthe newmaterialtopreserveallthedesirablepropertiesofBIOLOXforte-asanexcellent Current State of the Art of the Ceramic Composite Material BIOLOXdelta137bioceramicwithmorethan30yearsclinicalexperience-buttoincreaseitsstrengthand toughness.Thesepropertiesaresubstantiallyimprovedbyimplementationofreinforcingelements. Figure 1 shows the microstructure of BIOLOXdelta. Figure 1. Microstructure of BIOLOXdelta. Two reinforcing components are integrated into BIOLOXdelta. 17 vol.-% of the matrix consistoftetragonalzirconiaparticles.Theaveragegrainsizeofthezirconiaisaround 0.27 m. As a further reinforcing element, approximately 3 vol.-% of the matrix are built by platelet shaped crystals of the ceramic composition strontium aluminate. The platelets stretch to a maximum length of approximately 5 m with an aspect ratio of 5 10. The reinforcing ability of these ingredients is explained below.Additionallytothereinforcingcomponents,therearealsostabilizingelementsdopedto thematerial.Chromiumisaddedwhichissolubleinthealuminamatrixandincreasesthe hardness of the composite. The minor amount of chromium is the reason for the pink color of thematerial,seeFigure2.Furthermore,someyttriumisaddedtothecompositewhichis solvedinthezirconiaandsupportsthestabilizationofthetetragonalphase.InTable2,the composition is given: Table 2. Raw material specification for BIOLOXdelta IngredientFormulaWeight percent Yttrium oxideY2O3 Chromium oxideCr2O3 Strontium oxideSrO 1,4 2,0 Zirconium oxideZrO224,0 25,5 Other oxides TiO2, MgO, SiO2, CaO, Fe2O3, Na2O, K2O < 0,22 AluminaAl2O3balance Alumina matrix Zirconia Platelets Meinhard Kuntz, Bernard Masson and Thomas Pandorf138 During thermal treatment of the material the ingredients are transformed to the particular compositionwiththe3components.Thebasictransformationequationsareknownas follows: 3 2 3 2 32 O Cr O Y YCrO + (1) 2 3ZrO SrO SrZrO + (2) Cr O Al O Cr O Al :3 2 3 2 3 2 + (3) 19 12 3 2: 6 O Cr SrAl Cr O Al SrOx x + (4) Y ZrO O Y ZrO :2 3 2 2 + (5) InTable3thevolumefractionsofthefinalproductsaccordingtoequations (3) (5) are given. Table 3. Components of the final composite BIOLOXdelta Component of the compositeFormulaVolume percent Alumina, doped with ChromiaAl2O3:Cr80 % Zirconia with Y-stabilizationZrO2:Y17% Strontiumaluminate (minor Cr-content)SrAl12-xCrxO193 % Figure 2. Ball heads and inserts of BIOLOXdelta. Current State of the Art of the Ceramic Composite Material BIOLOXdelta1394. Reinforcing Mechanism on BIOLOXDelta Benefit of Phase Transformation The reinforcing elements, in particular zirconia, substantially increase fracture toughness andstrengthofthematerial.Fracturetoughness(KIC)isameasurefortheabilityofthe material to withstand crack extension. Strength (c) is defined as the maximum stress within a structure at failure of the component.The correlation of strength and toughness is given in the fundamental equation of fracture mechanics: Y a Kcc IC = (6) whereacisthesizeofatypicalcriticaldefectinthematerialandYtheshapefactor. Consequently,whenthefracturetoughnessofthealuminaisincreasedalsothestrengthis directly improved. This basic principle is the concept of the development of BIOLOXdelta. Themicrostructureisdesignedinordertoprovideanoptimumofresistanceagainstcrack extension.Thebenefitincrackresistancewhichisobtainedfromincorporatingzirconiaintoan aluminamatrix(asshowninFigure3)iswellknowninthescienceofhighperformance ceramics. Figure3.ReinforcingmechanisminBIOLOXdeltaatcrackinitiationandpropagation.Yellow particles represent tetragonal zirconia. Color change to red indicatesmonoclinic phase transformation. Arrows show the region of compressive stresses due to phase transformation.The figure represents a realistic part of the microstructure. The gray particles refer to the aluminamatrix,yellowtotetragonalzirconia.Thephasetransformationofzirconiais indicatedbythechangetoredcolor.Inthecaseofsevereoverloadingcrackinitiationand crackextensionwilloccur.Hightensilestressesinthevicinityofthecracktiptriggerthe tetragonaltomonoclinicphasetransformationofthezirconiaparticles.Theaccompanied volumeexpansionleadstotheformationofcompressivestresseswhichareefficientfor blocking the crack extension.The model as represented in Figure 3 has also been verified experimentally. Pezzotti et.al. [PezTBP]analyzedthemonoclinicphasetransformationinthevicinityofanartificialcrack tip as shown in Figure 4. 1 m Meinhard Kuntz, Bernard Masson and Thomas Pandorf140 AsitisdemonstratedinFigure3thisreinforcingmechanismisfullyactivatedwithina regionofafewmicrometers.Forthemacroscopicperformanceofthematerialitisvery importantthatimmediatelyatthebeginningofcrackinitiationalsothereinforcing mechanismsareactivated.RegardingFigure3oneshouldkeepinmindthattheaverage distancebetween thereinforcing zirconiaparticles isapprox. 0,3m,i.e.similartothe grain size. Thus, the reinforcement is activated immediately when any microcrack is initiated. The reinforcing ability of zirconia particles is a consequence of the phase transformation, i.e. the spontaneous change from the tetragonal to the monoclinic phase [Han00]. The phase transformationisaccompaniedbyavolumechangeof4%ofthezirconiaparticle. Spontaneous phase transformation is a well known principle in material science. For example, the properties of high performance steels also rely on phase transformation from austenite to martensite. Raman Spectroscopy, G. Pezzotti. Figure 4. Monoclinic phase transformation in the vicinity of an artificial crack tip. Itshouldbeemphasizedthattheabilityofphasetransformationisthepreconditionfor anybenefitofthezirconiawithinthematerial.Thecompositeisdesignedsuchthatphase transformation occurs when it is needed, i.e. to prevent microcrack initiation and propagation at a high mechanical stress level.Experiment: What Happens when Phase Transformation Is Suppressed? Ithasbeenshownexperimentallythattheabilityofzirconiaphasetransformationin BIOLOXdelta is necessary for the excellent mechanical properties. The experiment has been designed such that the experimental material was identically produced to BIOLOXdelta but with a significant higher amount of Y2O3. Yttria is known for stabilizing the tetragonal phase ofzirconia.Consequently,inthecaseofatoohighamountofyttria,theabilityofphase transformation is suppressed. This has been shown in the experiment.TheexperimentalmaterialW3530hasbeenproducedequivalentlytotheproductionof BIOLOXdelta.InFigure5itisshownthatthemicrostructureisidentical.InTable4the properties of the two materials are compared. Grey intensity proportional to monoclinic phase content Current State of the Art of the Ceramic Composite Material BIOLOXdelta141 BIOLOXdeltaExperimental material W3530 Figure 5. Microstructure of regular BIOLOXdelta and experimental material W 3530 Table 4. Comparison of regular BIOLOXdelta and high stabilized experimental material Material propertiesBIOLOXdeltaW3530 Ratio Y2O3 / ZrO2 [mol %]1,33,0 Final density [g/cm3]4,374,38 Grain size [m]0,540,55 Strength [MPa]1392777 Hardness [HV1]17571747 Monoclinic phase content [%]5 1 Fracture toughness6,55,1 The basic properties of regular BIOLOXdelta and the experimental material W3530 are identical, i.e. microstructure, density, grain size and hardness. The fundamental difference is the ratio of Y2O3/ZrO2. In BIOLOXdelta the amount of yttria is significantly lower. As can beseenfromthedata,thehigheramountofyttriaintheexperimentalmaterialleads-as expected-toalowercontentofmonoclinicphase,becausephasetransformationis suppressed.Asaconsequencethefracturetoughnessandthestrengthoftheexperimental material are muchlowerthanthe properties ofBIOLOXdelta.In particular, thestrength of theexperimentalmaterialW3530isonly55%ofthenormalstrengthofBIOLOXdelta. From this result it is immediately clear that the ability of phase transformation is necessary to obtainahighperformancecompositeceramic.Thephasetransformationcanbeeasily suppressedbychemicalstabilization(usingyttria).However,suppressingphase transformation means loosing the excellent mechanical properties of the material. Meinhard Kuntz, Bernard Masson and Thomas Pandorf142 Stabilization of the Zirconia Tetragonal Phase InBIOLOXdeltathecontentofyttriahasbeenoptimizedduringthematerial development.ItshouldbenotedthattheY2O3/ZrO2-ratioislowerthaninnormalpure zirconiamaterials(3Y-TZP),becausethestabilizationofthetetragonalphasein BIOLOXdeltaisalsoinfluencedbymechanicalstabilization,i.e.theembeddingof zirconia in the stiff alumina matrix. 2 4 6 8500temperature [C]Y O[mol %]2 3cubtet + cubmon + cub montet100015002000 Figure 6. Background of stabilizing effects: Y-doping and embedding of zirconia particles in a matrix. As can be seen in Figure 6 doping of Y2O3 into ZrO2 shifts the temperature of tetragonal to-monoclinicphasetransformationtowardslowertemperatures.Thus,dopingwithY meanschemicalstabilization.AtY2O3contentlower10%thestablephaseatroom temperature is monoclinic. So additionally the embedding of the zirconia particles in a matrix aswellassurfacestressesinthesmallparticlesalsoactasstabilizingmechanisms.The surrounding material will oppose the transformation and it is the strain energy that is involved in this constraint that allows the tetragonal phase to be retained at room temperature [Gre89]. This effect is referred to as mechanical stabilization 5. Material Production and Properties BIOLOXdeltaisacomparativelycomplexcompositematerialwhere4different ingredientsaremixedduringpowderpreparationandundergosolidphasetransformationat hightemperaturetreatmentasexplainedinchapter2.Theabilityofreproducingsucha materialinhighquantitieswithexcellentqualitybatchbybatchisthekeyqualificationof CeramTec as manufacturer.It is the purpose of this section both to summarize the importantmaterial properties and to elucidate the reproducibility of the production.The important production and analytical steps are as follows: Current State of the Art of the Ceramic Composite Material BIOLOXdelta143 Figure 7. Schematic description of processing and material data generation of a single batch. As shown in Figure 7 the material data are obtained for every powder batch. In Table 5 the material data as obtained for all powder batches in 2007 are summarized.Meinhard Kuntz, Bernard Masson and Thomas Pandorf144 Table 5. Material properties of BIOLOXdelta batches in 2007 Density 4-pt. Bending StrengthWeibull ModulusHardnessHV 10 Monoclinic contentGrain Size Alumina Fracture Toughnessg/cm3MPaGPa%mMPa m Average4,37141114,917,24,40,546,4 Std. Dev.0,007503,10,091,10,0270,20 The physical background of these parameters should be discussed. There are parameters whicharealmostinvariableduetothephysicalnatureoftheproperty,inparticulardensity andhardness.Inthenormalcaseofregularproductionthereisonlyverylittlescatterwith these data. However, it is important to analyze these parameters for every batch because any deviation of the expected results would indicate an insufficient production lot.On the other hand, it is worth highlighting the low alumina grain size and the very low scatter of this value. During sintering and final densification of any ceramic the particles build adensematrixbutsimultaneouslygraingrowthalsooccurs.Itisthegoalofadequate sinteringtoachievefulldensitybuttosuppressgraingrowth,asafinemicrostructureis necessary for good mechanical properties. Obviously, the sintering of BIOLOXdelta is very well reproduced batch by batch. The grain size is low in comparison to pure alumina because the dispersed zirconia particles prevent grain growth of the alumina matrix.The fracture toughness is a measure for the reinforcing mechanisms in the material. As described in chapter 3 the high fracture toughness depends on the transformation mechanism ofthezirconiaparticles.Obviouslyalsothisvalueshowsverylowscatter.Theaverage fracture toughness is 6,4 MPam. In contrast, the overstabilized material W3530 described inchapter 3hasasignificantlowerfracturetoughnessK1c5,1 MPam.Thus,itcanbe derivedfromtheevaluationoffracturetoughnessfrombatchtobatchthatthedesired transformation toughening is working properly.As explained in chapter 3 the fracture toughness should be discussed in context with the monoclinic phase content which is determined on a polished flat surface of a specimen. The monoclinic phase content [in %] as obtained from the regular X-ray diffraction is relative to thetotalzirconiacontent,nottothetotalvolumeofthematerial.Thus,inanycasethe monoclinicphasecontentofthetotalmaterialcanbedeterminedbysimplyreferringtothe zirconia fraction of 17 vol.%. Example: 10% monoclinic phase content is equivalent to 1,7% relative to the total volume of the material. Accordingtothematerialsspecificationthemonoclinicphasecontentafterpolishing (intrinsic monoclinic phase content) is 10% of ZrO2 which is regularly determined by X-ray diffraction. The sensitivity of this technique is around 1% monoclinic phase content. As it is evidentfromthe datathe average intrinsicmonoclinic contentofZrO2of 4,4%isabovethe sensitivitylimit.Thisindicatesagainthatthesoundmaterialisindispositionofphase transformation.NotethatthemonocliniccontentoftheexperimentalmaterialW3530is below the sensitivity limit.Incontrasttotheotherparametersdiscussedabovethestrengthofthematerialshows significantlyhigherscatter.Itisimportanttounderstandthatfailureinceramicsisalways Current State of the Art of the Ceramic Composite Material BIOLOXdelta145triggeredbyimperfectionsofthemicrostructure.Highperformanceceramicscanonlybe achievedwhenthenaturaldefectsinthematerialareverysmall.Typicalrelevant imperfections in BIOLOXdelta are within a range of 5 50m. Accordingly, the scatter of naturaldefectsizedirectlymatchesthescatterofstrengthwhichisdescribedbythe Weibulls modulus. A high modulus indicates low scatter. For the high performance material BIOLOXdeltaaWeibullsmodulusof7istoleratedinthespecification.Ascanbeseen from the data the normal scatter of the strength is much lower (i.e. higher modulus).6. Correlation of Material and Component Properties In chapter 5 the extensive efforts of analyzing the material properties of BIOLOXdelta batchbybatchhavebeendiscussed.Thesematerialpropertiesaredeterminedaccordingto ISO6474whichisapplicableforpurealuminaandcurrentlybeingextendedforalumina-composite materials such as BIOLOXdelta (ISO 6474-2). This type of data is very familiar for evaluation of the performance of ceramics. In this chapter it is intended to discuss shortly how these material data correlate to component properties, e.g. the strength of ball heads and inserts.Ingeneral,theperformanceofanysystemdependsontheintrinsicmaterialproperties, the design and manufacturing quality of the components and the system, the external load and theparticularenvironment,andfinallythequalityofmountingandinstallation.Theuseof highperformancematerialsinevitablypromotestheperformanceofasystem-however,the otherfactorsmaybeevenmoredecisiveforthesuccessofasystem.Thesecomplex correlationsmustbenecessarilyevaluatedbydesignanalysis,modeling,simulations,risk analysis and many other tools. In order to eliminate any influences of design features most of the material testing has been performed using 4-point bending bars.In Figure 9 the setup of the regular 4-point bending test as recommended in ISO 6474-2 andthebursttestaccordingtoISO7206areshown.Thebendingtestrevealstheintrinsic strength of the material whereas the burst test is designed in order to simulate the in-vivo load of ceramic ball heads. 4-Point Bending TestBurst Test Figure 8. Schematic set-up of 4 point bending test and ball head burst test. Meinhard Kuntz, Bernard Masson and Thomas Pandorf146 The strength measured with the bending test is to a first approach - an intrinsic material parameter1,whereastheburstloadasobtainedfromthebursttestdependsonthematerials strength and the design of the ball head and the taper. This is directly shown by comparison of strengthandburstloadofdifferentballheadsmadeofBIOLOXforteandBIOLOXdelta, see Table 6.Table 6. Comparison of strength and burst load of BIOLOXforte and BIOLOXdelta ParameterTest / DesignUnitBIOLOXforteBIOLOXdelta ratiodelta / forteStrength 4-point bending MPa62014002,3 Burst load28-12/14 LkN54851,6 Burst load36-12/14 MkN1101311,2 Allburstloadsarefarabovetherequiredvalueof46kN.Themaximuminvivoloadat worst case conditions is approximately 10 kN.ThedataofthebursttestsgiveninTable6areobtainedfromballheadswithidentical geometry,Titesttaperandthesametestsetup.Thus,theadvantageofBIOLOXdeltaball heads in the burst load only comes from the higher strength of the material in comparison to thepurealuminaBIOLOXforte.ThestrengthofBIOLOXdeltaismorethantwicethe strengthofBIOLOXforte,whereastheratiooftheburststrengthvaluesislower.Thisis explainedbytheductiledeformationoftheTitaperduringthebursttestwhichsteadily increasesthecontactareaoftheconicalboreoftheceramicballheadandthemetaltaper. However, the burst load of identical ball heads is always higher when a high strength material is used.It is also seen that the burst strength strongly depends on the ball head size. A larger ball headshowsa higher burst load.Itis concluded that thebenefit ofusingahighperformance materialishigherwhenappliedtoachallengingdesign,e.g.aballheadwithlowerwall thickness.Nevertheless,theuseofthehighstrengthmaterialalwaysincreasesthesafety margin of the component. 7. Wear Performance of BIOLOXDelta At normal wear conditions (e.g. standard wear simulator) the wear of a hard-hard couple ofBIOLOXdeltaisidenticaltotheexcellent performance ofthewellprovenpurealumina BIOLOXforte.Thereisonlyaminordifferenceinhardnessofthesebothmaterialswhich does not compromise the normal wear behavior.However,asignificantadvantageofBIOLOXdeltaisidentifiedinthecaseofworst casewearsimulationasshowninFigure9.Inthisexperimentmicroseparationofceramic ball head and insert during each load cycle has been simulated which leads to highly localized

1 Due to the statistical nature the strength also slightly depends on the specimen size and the stress distribution. Size effectscanbemathematicallybalanced.Inordertoobtainresultswhichcanbedirectlycomparedtoeach other the standardized set-up of the bending test should be used.Current State of the Art of the Ceramic Composite Material BIOLOXdelta147forces at the contact area. This experimental setup was supposed to simulate e.g. low tension ofthesofttissueaftersurgeryasitisdiscussedfrequentlybyorthopedicexperts.Itwas concludedfrom various retrievals that in some cases a well defined stripe-shaped area shows amoreintensewornsurfacethanthenormalsurfaceoftheballhead.Thisphenomenonis known as stripe wear.Intheexperiment,heavywearconditionsweresimulatedduetothehighlylocalized contactarea.Itwasfoundafter5miomicroseparationloadcyclesthatthewearvolumeof BIOLOXdeltacouples(bothceramicballheadandinsertweremadeofBIOLOXdelta) was 7 times lower than that with the coupling made of pure alumina BIOLOXforte [Cla05]. The wear rate of the mixed couplings (either ball head or insert made of BIOLOXdelta, the remaining made of pure alumina BIOLOXforte) ranged between those values. It is important tomentionthatduetothesmalldifferentialhardnessbetweenBIOLOXforteand BIOLOXdeltacanbecombinedinaceramic-ceramiccouplingwithoutrunningtheriskof excessive wear or other adverse effects. Figure 9. Wear performance of BIOLOXdelta and BIOLOXforte at simulated micro separation. ObviouslyBIOLOXdeltashowsanexcellentstripeweartolerance.Theanalysisas given in [Cla05] shows that in the stripe wear region the monoclinic phase content is strongly increased.Thisindicatesthemechanismbehindtheexcellentstripeweartoleranceof BIOLOXdelta. As a first approach it is assumed that at these special test conditions a very highlocalizedstressactsinthecontactareawhichmaybeabletointroducedamag