Handbook of Materials Behavior ModelsVOLUME I Deformations of Materials

This Page Intentionally Left Blank

Handbook of Materials Behavior ModelsVOLUME I Deformations of Materials

EDITORJEAN LEMAITREUniversit# Paris 6 LMT-Cachan Cachan Cedex France

ACADEMIC PRESSA Harcourt Science and Technology Company

San Diego San Francisco New York London Sydney Tokyo

Boston

This book is printed on acid-flee paper. Copyright 9 2001 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida, 32887-6777. ACADEMIC PRESS A Division of Harcourt, Inc. 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA http ://www.academicpress, com Academic Press Harcourt Place, 32 Jamestown Road, London, NW1 7BY, UK http ://www.aca demicpress, corn Library of Congress Catalog Number: 2001089698

Set International Standard Book Number: 0-12-443341-3 Volume 1 International Standard Book Number: 0-12-443342-1 Volume 2 International Standard Book Number: 0-12-443343-X Volume 3 International Standard Book Number: 0-12-443344-8

Printed in the United States of America 01 02 03 04 05 MB 9 8 7 6 5 4 3 2 1

CONTENTS

Foreword (E. van der Giessen) Introduction (J. Lemaitre) Contributors

VOLUME I

DEFORMATIONS OF MATERIALS1

Chapter 1 Background on mechanics of materials Chapter 2 Elasticity, viscoelasticity Chapter 3 Chapter 4 Chapter 5 VOLUME II Chapter 6 Chapter 7 Chapter 8 VOLUME III Chapter 9 Yield limit Plasticity Viscoplasticity FAILURES OF MATERIALS Continuous damage Cracking and fracture Friction and wear MULTIPHYSICS BEHAVIORS Multiphysics coupled behaviors

69 125 195 299

409 537 673

793 955 1073 1179

Chapter 10 Composite media, biomaterials Chapter 11 GeomaterialsINDEX

vi

Contents

CHAPTER

1

Background on mechanics of materials1.1 Background on modelingJ. Lemaitre

1.2 Materials and process selectionY. Brechet

15 30

1.3 Size effect on structural strengthZ. Bazant

CHAPTER

271 75 84 91 95

Elasticity, viscoelasticity2.1 Introduction to elasticity and viscoelasticityJ. Lemaitre

2.2 Background on nonlinear elasticityR. W. Ogden

2.3 Elasticity of porous materialsN. D. Cristescu

2.4 Elastomer modelsR. W. Ogden

2.5 Background on viscoelasticityK. Ikegami

2.6 A nonlinear viscoelastic model based on fluctuating modesR. Rahouadj, C. Cunat

107 117

2.7 Linear viscoelasticity with damageR. Schapery

CHAPTER

3127

Yield limit3.1 Introduction to yield limitsJ. Lemaitre

Contents

vii 129 137 155 166

3.2 Background on isotropic criteriaD. Drucker

3.3 Yield loci based on crystallographic textureP. Van Houtte

3.4 Anisotropic yield conditionsM. Zyczkowski

3.5 Distortional model of plastic hardeningT. Kurtyka

3.6 A generalized limit criterion with application to strength, yielding, and damage of isotropic materialsH. Altenbach

175 187

3.7 Yield conditions in beams, plates, and shellsD. Drucker

CHAPTER

4197

Plasticity4.1 Introduction to plasticityJ. Lemaitre

4.2 Elastoplasticity of metallic polycrystals by the self-consistent modelM. Berveiller

199

4.3 Anisotropic elastoplastic model based on crystallographic textureA. M. Habraken, L. Duchr A. Godinas, S. Cescotto

204

4.4 Cyclic plasticity model with nonlinear isotropic and kinematic hardening: No LIKH modelD. Marquis

213

4.5 Muhisurface hardening model for monotonic and cyclic response of metalsZ. Mroz

223

4.6 Kinematic hardening rule with critical state of dynamic recoveryN. Ohno

232

viii 4.7 4.8 4.9 Kinematic hardening rule for biaxial ratchetingH. Ishikawa, K. Sasaki

Contents

240 247 255 265

Plasticity in large deformationsY E Dafalias

Plasticity of polymersJ. M. Haudin, B. Monasse

4.10 Rational phenomenology in dynamic plasticityJ. R. Klepaczko

4.11 Conditions for localization in plasticity and rate-independent materialsA. Benallal

274 281

4.12 An introduction to gradient plasticityE. C. Aifantis

CHAPTER 5

Viscoplasticity5.1 5.2 Introduction to viscoplasticityJ. Lemaitre

301

A phenomenological anisotropic creep model for cubic single crystalsA. Bertram, J. Olschewski

303

5.3

Crystalline viscoplasticity applied to single crystalsG. Cailletaud

308

5.4

Averaging of viscoplastic polycrystalline materials with the tangent self-consistent modelA. Molinari

318 326

5.5 5.6

Fraction models for inelastic deformationJ. E Besseling

Inelastic compressible and incompressible, isotropic, small-strain viscoplasticity theory based on overstress (VBO)E. Krempl, K. Ho

336

Contents

ix

5.7

An outline of the Bodner-Partom (BP) unified constitutive equations for elastic-viscoplastic behaviorS. Bodner

349

5.8

Unified model of cyclic viscoplasticity based on the nonlinear kinematic hardening ruleJ. L. Chaboche

358 368 377

5.9

A model of nonproportional cyclic viscoplasticityE. Tanaka

5.10 Rate-dependent elastoplastic constitutive relationsE Ellyin

5.11 Physically based rate- and temperature-dependent constitutive models for metalsS. Nemat-Nasser

387 398

5.12 Elastic-viscoplastic deformation of polymersE. M. Arruda, M. BoyceCHAPTER

6411 413

Continuous damage6.1 6.2 6.3 Introduction to continuous damageJ. Lemaitre

Damage-equivalent stress-fracture criterionJ. Lemaitre

Micromechanically inspired continuous models of brittle damageD. Krajcinovic

417 421 430

6.4 6.5 6.6

Anisotropic damageC. L. Chow, Y. Wei

Modified Gurson modelV. Tvergaard, A. Needleman

The Rousselier model for porous metal plasticity and ductile fractureG. Rousselier

436 446

6.7

Model of anisotropic creep damageS. Murakami

Contents

6.8 6.9

Multiaxial fatigue damage criteriaD. Sauci

453

Muhiaxial fatigue criteria based on a muhiscale approachK. Dang Van

457

6.10 A probabilistic approach to fracture in high cycle fatigueE Hild

464 472

6.11 Gigacycle fatigue regimeC. Bathias

6.12 Damage mechanisms in amorphous glassy polymers: CrazingR. Schirrer

488 500 513

6.13 Damage models for concreteG. Pijaudier-Cabot, J. Mazars

6.14 Isotropic and anisotropic damage law of evolutionJ. Lemaitre, R. Desmorat

6.15 A two-scale damage model for quasi-brittle and fatigue damageR. Desmorat, J. Lemaitre

525

7 Cracking and .fractureCHAPTER

7.1 7.2 7.3 7.4 7.5

Introduction to cracking and fractureJ. Lemaitre

539 542 549 558 566

Bridges between damage and fracture mechanicsJ. Mazars, G. Pijaudier-Cabot

Background on fracture mechanicsH. D. Bui, J. B. Leblond, N. Stalin-Muller

Probabilistic approach to fracture: The Weibull modelE Hild

Brittle fractureD. Franc~ois

Contents

xi 577 582

7.6 7.7 7.8

Sliding crack modelD. Gross

Delamination of coatingsH. M. Jensen

Ductile rupture integrating inhomogeneities in materialsJ. Besson, A. Pineau

587

7.9

Creep crack growth behavior in creep ductile and brittle materialsT. Yokobori Jr.

597 611

7.10 Critical review of fatigue crack growthT. Yokobori

7.11 Assessment of fatigue damage on the basis of nonlinear compliance effectsH. Mughrabi

622

7.12 Damage mechanics modeling of fatigue crack growthX. Zhang, J. Zhao

633 645

7.13 Dynamic fractureW. G. Knauss

7.14 Practical applications of fracture mechanics: Fracture controlD. Broek

661

CHAPTER 8

Friction a n d w e a r8.1 8.2 8.3 Introduction to friction and wearJ. Lemaitre

675 676 700

Background on friction and wearY. Berthier

Models of frictionA. Savkoor

xii 8.4 Friction in lubricated contactsJ. FrCne, T. Cicone

Contents

760

8.5 A thermodynamic analysis of the contact interface in wear mechanicsH. D. Bui, M. Dragon-louiset, C. Stolz

768

8.6 Constitutive models and numerical methods for frictional contactM. Raous

777 787

8.7 Physical models of wear, prediction of wear modesK. Kato

CHAPTER 9

Multiphysics coupled behavior9.1 Introduction to coupled behaviorsJ. Lemaitre

795

9.2 Elastoplasticity and viscoplasticity coupled with damageA. Benallal

797

9.3 A fully coupled anisotropic elastoplastic damage modelS. Cescotto, M. Wauters, A. M. Habraken, Y. Zhu

802 814 821

9.4 Model of inelastic behavior coupled to damageG. Z. Voyiadjis

9.5 Thermo-elasto-viscoplasticity and damageP. Perzyna

9.6 High-temperature creep deformation and rupture modelsD. R. Hayhurst

835

9.7 A coupled diffusion-viscoplastic formulation for oxidasing multiphase materialsE. P. Busso

849

Contents

xiii 856

9.8 9.9

Hydrogen attackE. van der Giessen, S. Schl6gl

Hydrogen transport and interaction with material deformation: Implications for fractureP Sofronis

864 875

9.10 Unified disturbed state constitutive modelsC. S. Desai

9.11 Coupling of stress-strain, thermal, and metallurgical behaviorsT. Inoue

884

9.12 Models for stress-phase transformation couplings in metallic alloysS. Denis, P Archambault, E. Gautier

896 905

9.13 Elastoplasticity coupled with phase changesE D. Fischer

9.14 Mechanical behavior of steels during solid-solid phase transformationsJ. B. Leblond

915

9.15 Constitutive equations of a shape memory alloy under complex loading conditionsM. Tokuda

921 928 944

9.16 Elasticity coupled with magnetismR. Billardon, L. Hirsinger, E Ossart

9.17 Physical aging and glass transition of polymersR. Rahouadj, C. Cunat

CHAPTER 1 0

Composite media, biomaterials10.1 Introduction to composite mediaJ. Lemaitre

957 959

10.2 Background on micromechanicsE. van der Giessen

xiv 10.3 Nonlinear composites" Secant methods and variational boundsP. Suquet

Contents

968 984 996 1004

10.4 10.5 10.6 10.7

Nonlocal micromechanical modelsJ. Willis

Transformation field analysis of composite materialsG. Dvorak

A damage mesomodel of laminate compositesP Ladev~ze

Behavior of ceramix-matrix composites under thermomechanical cyclic loading conditionsE A. Leckie, A. Burr, E Hild

1015

10.8

Limit and shakedown analysis of periodic heterogeneous mediaG. Maier, V. Carvelli, A. Taliercio

1025 1037 1048 1057

10.9

Flow-induced anisotropy in short-fiber compositesA. Poitou, E Meslin

10.10 Elastic properties of bone tissueStephen C. Cowin

10.11 Biomechanics of soft tissueS. C. Holzapfel

CHAPTER 1 1

Geomaterials11.1 11.2 11.3 Introduction to geomaterialsJ. Lemaitre

1075 1076

Background of the behavior of geomaterialsE Darve

Models for compressible and dilatant geomaterialsN. D. Cristescu

1084

Contents

XV

11.4 11.5

Behavior of granular materialsI. Vardoulakis

1093

Micromechanically based constitutive model for frictional granular materialsS. Nemat-Nasser

1107 1118

11.6 11.7

Linear poroelasticityJ. W. Rudnicki

Nonlinear poroelasticity for liquid nonsaturated porous materialsO. Coussy, P. Dangla

1126

11.8

An elastoplastic constitutive model for partially saturated soilsB. A. Schrefler, L. Simoni

1134

11.9

Sinfonietta classica: A strain-hardening model forsoils and soft rocksR. Nova

1146

11.10 A generalized plasticity model for dynamic behavior of sand, including liquefactionM. Pastor, O. C. Zienkiewicz, A. H. C. Chan

1155 1164

11.11 A critical state bounding surface model for sandsM. T. Manzari, Y. E Dafalias

11.12 Lattice model for fracture analysis of brittle disordered materials like concrete and rockJ. G. M. van Mier

1171

Index

1179

FOREWORD

We know that there is an abundance of models for particular materials and for specific types of mechanical responses. Indeed, both the developers of models and their users sometimes criticize this situation, for different reasons. The presence of different models that attempt to describe the same material and response is due not only to the personal style of their inventors, but also to a desirable element of competition that drives the progress in the field. Given this situation, the selection of the proper constitutive model from all the available ones can be difficult for users or even materials modelers when they are not experts in the field. This Handbook is the first attempt to organize a wide range of models and to provide assistance in model selection and actual application. End-users will find here either potential models relevent for their application and ready to be used for the problem at hand, or an entrance to the specific technical literature for more details. Recognizing the breadth of the field as well as the unavoidable personal touch of each approach, Jean Lemaitre has chosen to include in this Handbook the writings of as many as 130 authors. Drawing on his wide experience developing and using constitutive models for many materials, he has addressed his worldwide network of colleagues, all experts in their pertinent subject, to accomplish this difficult task. Yet, even though the Handbook covers an unprecedented range of materials and types of behavior, it is only a sample of currently available models, and other choices would have been possible. Indeed, more choices will become possible as the development of novel and improved material models continues.

Erik van der Giessen Koiter Institute Delft Delft University of Technology The Netherlands xvi

INTRODUCTION

Why a Handbook of models? Handbooks are often compilations of characteristic numbers related to well-established laws or formulae that are ready to apply. In this case of the behavior of materials, no unique law exists for any phenomenon, especially in the range of nonlinear phenomena. This is why we use the term model instead of law. During the past thirty years many models have been proposed, each of them having its own domain of validity. This proliferation is partly due to advances in computers. It is now possible to numerically simulate the "in-service life" of structures subjected to plasticity, fatigue, crack propagation, shock waves and aging for safety and economy purposes. The time has come to try to classify, compare, and validate these models to help users to select the most appropriate model for their applications. How is the Handbook organized? All solid materials are considered, including metals, alloys, ceramics, polymers, composites, concrete, wood, rubber, geomaterials such as rocks, soils, sand, clay, and biomaterials. But the Handbook is organized first by phenomena because most engineering mesomodels apply to different materials. 9 In the first volume: "Deformation of Materials," the first chapter is an attempt to give general methodologies in the "art" of modeling with special emphasis, on domains of validity in order to help in the choice of models, in the selection of the appropriate materials for each specific application, and in the consideration of the so-called "size effect" in engineering structures. Chapter 2 to 5 deal, respectively, with elasticity and viscoelasticity, yield limit, plasticity, and viscoplasticity. 9 The second volume is devoted to "Failure of Materials": continuous damage in Chapter 6, cracking and fracture in Chapter 7, friction and wear in Chapter 8. 9 In the third volume "Multiphysics Behaviors" are assembled. The different possible couplings are described in Chapter 9. Chapters 10 and 11 are devoted to special classes of materials: composites andxvii

xviii

Introduction

geomaterials, respectively, because they each corresponds to a particular modeling typed and moreover to a self-organized community of people. 9 In each chapter the different sections written by different authors describe one model with its domain of validity, its background, its formulation, the identification of material parameters for as many materials as possible, some advice on implementation or use of the model, and some references. The order of the sections follows as much as possible from physical and micromechanical oriented models to more phenomenological and engineering oriented ones. How to use the Handbook? 9 Search by phenomena: This is the normal order of the Handbook described in the "Contents". 9 Search by model name: Unfortunately, not all models have a name, and some of them have several. Look in the list of contributors, where the names of all authors are given. 9 Search by type of application: Each chapter begins with a chapter introduction in which a few words are written on each section. If you do not find exactly what you are looking for, please remember that the best model is the simplest which gives you what you need and nothing more! In case of any difficulty, get in touch with the author(s), whose address is given after the title of each section. Some personal comments. This Handbook has been initiated by the editor of "Academic Press" who gave me much freedom to organize the book. It took me two years to prepare the contents, to obtain the agreement of more than 100 authors, to ask for manuscripts, to ask again and again (and again for some of them!) to review and to obtain the final material. It was an exciting experience for which all actors must be thanked: the editors Z. Ruder, G. Franklin, and M. Filion, all the authors who are still my friends, my colleagues and friends from the LMT-Cachan who often advised me on subjects and authors and particularly Erik van der Giessen, who helped me in the selection of the subjects, who corrected the chapter introductions, and who agreed to write the foreword, Catherine Genin who was so kind and so efficient with letters, fax, e-mail, telephone, disks and manuscripts and answered so many questions in order to obtain the materials in due time. I must also mention Annie, my wife, who accepted 117 articles on the table at home!

Merci d tous, Jean Lemaitre Septembre 2000

CONTRIBUTORS

Numbers in parentheses indicate the section of authors' contributions. ELIAS C. AIFANTIS (4.12), Aristotle University of Thessaloniki, Thessaloniki, 54006 Greece, and Michigan Technological University, Houghton, Michigan HOLM ALTENBACH (3.6), Fachbereich Ingenieurwissenschaften, MartinLuther-Universitat Halle-Wittenberg, D-06099 Halle (Saale), Germany E ARCHAMBAULT (9.12), Laboratoire de Science et GSnie des Mat~riaux et de M~tallurgie, UMR 7584 CNRS/INPL, Ecole des Mines de Nancy, Parc de Saurupt, 54042 Nancy Cedex, France ELLEN M. ARRUDA (5.12), Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan C. BATHIAS (6.11), Laboratoire de M~canique de la Rupture, CNAM/ITMA, 2 rue Conte, 75003 Paris, France ZDENEK P. BAZANT (1.3), Northwestern University, Evanston, Illinois, USA AHMED BENALLAL (4.11, 9.2), Laboratoire de M~canique et Technologie, ENS de Cachan/CNRS/Universit~ Paris 6, 61 avenue du President Wilson, 94235 Cachan, France ALBRECHT BERTRAM (5.2), Otto-von-Guericke-University Magdeburg, Universit/~tsplatz 2, 39106 Magdeburg, Germany YVES BERTHIER (8.2), Laboratoire de M~canique des Contacts, UMR CNRSINSA de Lyon 5514, Batiment 113, 20, Avenue Albert Einstein, 69621 Villeurbanne Cedex, Francexix

XX

Contributors

B. J. BESSON (7.8), Ecole des Mines de Paris, Centre des Mat~riaux, UMR CNRS 7533, BP 87, 91003 Evry Cedex, France J. E BESSELING (5.5), j.f.besseling@wbmt.tudelft.nl M. BERVEILLER (4.2), Laboratoire de Physique et M&anique des Mat~riaux, Ile du Saulcy, 57045 Metz Cedex, France RENt~ BILLARDON (9.16), ENS de Cachan/CNRS/Universit~ Paris 6, 61 avenue du President Wilson, 94235 Cachan Cedex, France SOL R. BODNER (5.7), Technion Israel Institute of Technology, Haifa 32000, Israel MARY C. BOYCE (5.12), Department of Mechanical Engineering, Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA YVES BRECHET (1.2), L.T.EC.M. BP75, Institut National Polytechnique de Grenoble, 38402 St Martin d'Heres Cedex, France DAVID BROEK (7.14), 263 Dogwood Lane, Westerville, Ohio, USA HUY DUONG BUI (7.3, 8.5), Laboratoire de M&anique des Solicles, Ecole Polytechnique, 91128 Palaiseau, France Electricit4 de France, R&D, Clamart, France ALAIN BURR (10.7), Laboratoire de Physico-Chimie Structurale et Macromol&ulaire, UMR 7615, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France ESTEBAN P. BUSSO (9.7), Department of Mechanical Engineering, Imperial College, University of London, London, SW7 2BX, United Kingdom GEORGES CAILLETAUD (5.3), Centre des Mat4riaux de l't~cole des Mines de Paris, UMR CNRS 7633, BP 87, F91003 Evry Cedex, France VALTER CARVELLI (10.8), Department of Structural Engineering, Technical University (Politecnico) of Milan, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy SERGE CESCOTTO (4.3, 9.3), D4partement MSM, Universit4 de Liege, 1, chemin des Chevreuils bfit.B52/3, 4000 Liege, Belgique J. L. CHABOCHE (5.8), O.N.E.R.A., DMSE, BP 72, 92322 ChStillon Cedex, France and LASMIS, Troyes University of Technology, BP 2060, 10010 Troyes Cedex, France A. H. C. CHAN (11.10), School of Engineering, University of Birmingham, United Kingdom

Contributors

xxi

C.L. CHOW (6.4), Department of Mechanical Engineering, University of Michigan-Dearborn TRAIAN CICONE (8.4), Dept. of Machine Elements and Tribology, Polytechnic University of Bucharest, Romania N.D. CRISTESCU (2.3), 231 Aerospace Building, University of Florida, Gainesville, Florida OLIVIER COUSSY (11.7), Laboratoire Central des Ponts et Chaussees, Paris, France STEPHEN C. COWIN (10.10), New York Center for Biomedical Engineering, School of Engineering, The City College, New York CHRISTIAN CUNAT (2.6, 9.17), LEMTA, UMR CNRS 7563, ENSEM INPL 2, avenue de la Foret-de-Haye, 54500 Vandoeuvre-les-Nancy, France PATRICK DANGLA (11.7), Laboratoire Central des Ponts et Chaussees, Paris, France FI~LIX DARVE (11.2), EINP Grenoble, L3S-BP 53 38041 Grenoble, France YANNIS E DAFALIAS (4.8, 11.11), Civil and Environmental Engineering, The George Washington University, Washington, D.C. S. DENIS (9.12), Laboratoire de Science et G~nie des Mat~riaux et de M~tallurgie, UMR 7584 CNRS/INPL, Ecole des Mines de Nancy, Parc de Saurupt, 54042 Nancy Cedex, France CHANDRA S. DESAI (9.10), Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, Arizona, USA RODRIGUE DESMORAT (6.14, 6.15), Universite Paris 6-LMS, 8, Rue du Capitaine Scott, F-75015 Paris, France MARTA DRAGON-LOUISET (8.5), Laboratoire de M~canique des Solides, Ecole Polytechnique, 91128 Palaiseau, France DANIEL C. DRUCKER (3.2, 3.7), Department of Aerospace Engineering, Mechanics Engineering Service, University of Florida, 231 Aerospace Building, Gainesville, Florida 32611 GEORGE J. DVORAK (10.5), Rensselaer Polytechnic Institute, Troy, New York L. DUCHENE (4.3), D6partement MSM, Universit8 de Liege, 1, chemin des chevreuils b~t.B52/3, 4000 Liege, Belgique FERNAND ELLYIN (5.10), Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada

xxii

Contributors

E D. FISCHER (9.13), Montanuniversit. at Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria DOMINIQUE FRAN~;OIS (7.5), 12cole Centrale de Paris, Chfitenay-Malabry, F92 295, France JEAN FRIS.NE (8.4), Laboratoire de M~canique des Solides, Universit~ de Poitiers, France E. GAUTIER (9.12), Laboratoire de Science et G~nie des Mat~riaux et de M~tallurgie, UMR 7584 CNRS/INPL, t~cole des Mines de Nancy, Parc de Saurupt, 54042 Nancy Cedex, France A. GODINAS (4.3), D~partement MSM, Universit~ de Liege, 1, chemin des Chevreuils bfit.B52/3, 4000 Liege, Belgium DIETMAR GROSS (7.6), Institute of Mechanics, TU Darmstadt, Hochschulstrasse 1, D 64289 Darmstadt ANNE MARIE HABRAKEN (4.3, 9.3), D~partement MSM, Universit~ de Liege, 1, chemin des Chevreuils b~t.B52/3, 4000 Liege, Belgique JEAN-MARC HAUDIN (4.9), CEMEF- BP 207, 06904 Sophia Antipolis, France D. R. HAYHURST (9.4), Department of Mechanical Engineering, UMIST, P 9 Box 88, Manchester M60 1QD, United Kingdom FRANCOIS HILD (7.4, 10.7), LMT-Cachan, 61 avenue du Pr4sident Wilson, F-94235 Cachan Cedex, France LAURENT HIRSINGER (9.16), ENS de Cachan/CNRS/Universit4 Paris 6, 61 avenue du Pr4sident Wilson, 94235 Cachan Cedex, France K. HO (5.6), Yeungnam University, Korea GERHARD A. HOLZAPFEL (10.11), Institute for Structural Analysis, Computational Biomechanics, Graz University of Technology, 8010 Graz, Austria KOZO IKEGAMI (2.5), Tokyo Denki University, Kanda-Nishikicho 2-2, Chiyodaku, Tokyo 101-8457, Japan TATSUO INOUE (9.11), Department of Energy Conversion Science, Graduate School of Energy Science, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, Japan HIROMASA ISHIKAWA (4.7), Hokkaido University, N13, W8, Kita-ku, Sapporo 060-8628, Japan

Contributors

xxiii

HENRIK MYHRE JENSEN (7.7), Department of Solid Mechanics, 404, Technical University of Denmark, DK-2800 Lyngby, Denmark KOJI KATO (8.7), Tohoku University, Aramaki-Aza-Aoba 01, Sendal 980-8579, Japan JANUSZ R. KLEPACZKO (4.10), Metz University, Laboratory of Physics and Mechanics of Materials, lie du Saulcy, 57045 Met7, France W. G. KNAUSS (7.13), California Institute of Technology, Pasadena, California DUSAN KRAJCINOVIC (6.3), Arizona State University, Tempe, Arizona E. KREMPL (5.6), Mechanics of Materials Laboratory, Rensselaer Polytechnic Institute, Troy, New YorkTADEUSZ KURTYKA (3.5), C E R N -

European Organization for Nuclear Research, CH-1211 Geneve 23, Switzerland

PIERRE LADEVI~ZE (10.6), LMT-Cachan, ENS de Cachan/CNRSAJniversit6 Paris 6, 61 avenue du President Wilson, 94235 Cachan Cedex, France FREDERICK A. LECKIE (10.7), Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, California J-B. LEBLOND (7.3, 9.14), Laboratoire de Mod~lisation en M~canique, Universit~ de Pierre et Marie Curie, Paris, France JEAN LEMAITRE (1.1, 2.1, 3.1, 4.1, 5.1, 6.1, 6.2, 6.14, 6.15, 7.1, 8.1, 9.1, 10.1, 11.1), Universit~ Paris 6, LMT-Cachan, 61, avenue du Pr6sident Wilson, F-94235 Cachan Cedex, France GIULIO MAIER (10.8), Department of Structural Engineering, Technical University (Politecnico) of Milan, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy DIDIER MARQUIS (4.4), Laboratoire de M~canique et Technologie, Ecole Normale Sup~rieure de Cachan, 61 avenue du President Wilson, 94230 Cachan, France MAJID T. MANZARI (11.11), Department of Mechanics, National Technical University of Athens, 15773, Hellas, and Civil and Environmental Engineering, University of California, Davis, California JACKY MAZARS (6.13, 7.2), LMT-Cachan, Ecole Normale Superieure de Cachan, 61, avenue du President Wilson, 94235 Cachan, France and L35-Institut National Polytechniquede Grenoble, F38041 Grenoble Cedex 9, France

xxiv

Contributors

FREDERIC MESLIN (10.9), LMT-Cachan, ENS de Cachan, Universit6 Paris 6, 61 avenue du Pr6sident Wilson, 94235 Cachan Cedex, France ALAIN MOLINARI (5.4), Laboratoire de Physique et M&anique des Mat6riaux, l~cole Nationale d'Ing4nieurs, Universit~ de Metz, Ile du Saulcy, 57045 MetzCedex, France BERNARD MONASSE (4.9), CEMEF- BP 207, 06904 Sophia Antipolis, France HAEL MUGHRABI (7.11), Universit~it Erlangen-Nfirnberg, Institut f~lr Werkstoffwissenschaften, Martensstr. 5, D-91058 Erlangen, Germany N. STALIN-MULLER (7.3), Laboratoire de M4canique des Solides, 12cole Polytechnique, 91128 Palaiseau, France Z. MROZ (4.5), Institute of Fundamental Technological Research, Warsaw, Poland SUMIO MURAKAMI (6.7), Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603 Japan ROBERTO NOVA (11.9), Milan University of Technology (Politecnico), Department of Structural Engineering, Milan, Italy A. NEEDLEMAN (6.5), Brown University, Division of Engineering, Providence, Rhode Island and Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, 2800 Lyngby, Denmark SIA NEMAT-NASSER (5.11, 11.5), Center of Excellence for Advanced Materials, Department of Mechanical and Aerospace Engineering, University of California, San Diego, California R. W. OGDEN (2.2, 2.4), Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom NOBUTADA OHNO (4.6), Department of Mechanical Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan J. URGEN OLSCHEWSKI (5.2), BAM-V2, Unter den Eichen 87, 12200 Berlin, Germany FLORENCE OSSART (9.16), ENS de Cachan/CNRS/Universit~ Paris 6, 61 avenue du President Wilson, 94235 Cachan Cedex, France M. PASTOR (11.10), Centro de Estudios y Experimentaci6n de Obras P~blicas and ETS de Ingenieros de Caminos, Madrid, Spain PIOTR PERZYNA (9.5), Institute of Fundamental Technological Research, Polish Academy of Sciences, Swir 21, 00-049 Warsaw, Poland

Contributors

XXV

GILLES PIJAUDIER-CABOT (6.13), Laboratoire de G~nie Civil de Nantes SaintNazaire, t~cole Centrale de Nantes, BP 92101, F-44321 Nantes Cedex 03, France A. PINEAU (7.8), ]~cole des Mines de Paris, Centre des Mat~riaux, UMR CNRS 7533, BP 87, 91003 Evry Cedex, France ARNAUD POITOU (10.9), LMT-Cachan, ENS de Cachan, Universit8 Paris 6, 61 avenue du President Wilson, 94235 Cachan Cedex, France RACHID RAHOUADJ (2.6, 9.17), LEMTA, UMR CNRS 7563, ENSEM INPL 2, Avenue de la For~t-de-Haye, 54500 Vandoeuvre-l~s-Nancy, France MICHEL RAOUS (8.6), Laboratoire de Mecanique et d'Acoustique, 31, chemin Joseph Aiguier, 13402 Marseille Cedex 20, France GILLES ROUSSELIER (6.6), EDF/R&D Division, Les Renardi~res, 77818 Moret-sur-Loing Cedex, France J. W. RUDNICKI (11.6), Department of Civil Engineering, Northwestern University, Evanston, Illinois, USA KATSUHIKO SASAKI (4.7), Hokkaido University, N13, W8, Kita-ku, Sapporo 060-8628, Japan A. R. SAVKOOR (8.3), Vehicle Research Laboratory, Delft University of Technology, Delft, The Netherlands R. A. SCHAPERY (2.7), Department of Aerospace Engineering and Engineering Mechanics, The University of Texas, Austin, Texas ROBERT SCHIRRER (6.12), Institut Charles Sadron, 6 rue Boussingault, F-67083 Strasbourg, France SABINE M. SCHLOGL (9.8), Koiter Institute Delft, Delft University of Technology, The Netherlands B. A. SCHREFLER (11.8), Department of Structural and Transportation Engineering, University of Padua, Italy L. SIMONI (11.8), Department of Structural and Transportation Engineering, University of Padua, Italy PETROS SOFRONIS (9.9), Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, 104 South Wright Street, Urbana, Illinois DARRELL SOCIE (6.8), Department of Mechanical Engineering, University of Illinois, Urbana, Illinois CLAUDE STOLZ (8.5), Laboratoire de M~canique des Solides, Ecole Polytechnique, 91128 Palaiseau, France

XXVi

Contributors

PIERRE M. SUQUET (10.3), LMA/CNRS, 31 Chemin Joseph Aiguier, 13402, Marseille, Cedex 20, France ALBERTO TALIERCIO (10.8), Department of Structural Engineering, Technical University (Politecnico) of Milan, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy EIICHI TANAKA (5.9), Department of Mechano-Informatics and Systems, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan MASATAKA TOKUDA (9.15), Department of Mechanical Engineering, Mie University, Kamihama 1515 Tsu 514-8507, Japan V. TVI~RGAARD (6.5), Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, 2800 Lyngby, Denmark K. DANG VAN (6.9), Laboratoire de Mechanique des Solid, l~cole Polytechnique, 91128 Palaiseau, France ERIK VAN DER GIESSEN (9.8, 10.2), University of Groningen, Applied Physics, Micromechanics of Materials, Nyenborgh 4, 9747 AG Groningen, The Netherlands P. VAN HOUTTE (3.3), Department MTM, Katholieke Universiteit Leuven, B-3000 Leuven, Belgium J. G. M. VAN MIER (11.12), Delft University of Technology, Faculty of Civil Engineering and Geo-Sciences, Delft, The Netherlands IOANNIS VARDOULAKIS (11.4), National Technical University of Athens, Greece GEORGE Z. VOYIADJIS (9.4), Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, Louisiana MICHAEL WAUTERS (9.3), MSM-1, Chemin des Chevreuils B52/3 4000 Liege, Belgium YONG WEI (6.4), Department of Mechanical Engineering, University of Michigan-Dearborn, USA J. R. WILLIS (10.4), Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom A. TOSHIMITSU YOKOBORI, JR. (7.9), Fracture Research Institute, Graduate School of Engineering, Tohoku University, Aoba 01 Aramaki, Aoba-ku Sendaishi 980-8579, Japan

Contributors

xxvii

TAKEO YOKOBORI (7.10), School of Science and Engineering, Teikyo University, Utsunomiya, Toyosatodai 320-2551, Japan XING ZHANG (7.12), Division 508, Department of Flight Vehicle Design and Applied Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China JUN ZHAO (7.12), Division 508, Department of Flight Vehicle Design and Applied Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China Y. ZHU (9.3), ANSYS Inc., Houston, Texas O. C. ZIENKIEWICZ (11.10), Department of Civil Engineering, University of Wales at Swansea, United Kingdom MICHA ZYCZKOWSKI (3.4), Cracow University of Technology, ul. Warszawska 24, PL-31155 Krak6w, Poland

This Page Intentionally Left Blank

CHAPTER

1

Background on Mechanics of Materials

This Page Intentionally Left Blank

SECTION

1.1

Background O I l ModelingJEAN LEMAITREUniversitY. Paris 6, LMT-Cachan, 61 avenue du Pr&ident Wilson, 94235 Cachan Cedex, France

Contents 1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Observations and Choice of Variables . . . . . . . . . 1.1.2.1 Scale of observation . . . . . . . . . . . . . . . . . . . 1.1.2.2 Internal Variables . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.1 State Potential . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.2 Dissipative Potential . . . . . . . . . . . . . . . . . . . 1.1.4 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.1 Qualitative Identification . . . . . . . . . . . . . . . 1.1.4.2 Quantitative Identification . . . . . . . . . . . . 1.1.5 Validity Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6 Choice of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.7 Numerical Implementation . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4 5 6 6 7 8 9 9 11 13 13 14

14

1.1.1

INTRODUCTION

M o d e l i n g , as has a l r e a d y b e e n said for m e c h a n i c s , m a y be c o n s i d e r e d "a science, a t e c h n i q u e , a n d an art." It is s c i e n c e b e c a u s e it is the p r o c e s s by w h i c h o b s e r v a t i o n s can be p u t in a logical m a t h e m a t i c a l f r a m e w o r k in o r d e r to r e p r o d u c e or s i m u l a t e r e l a t e d p h e n o m e n a . In m e c h a n i c s of m a t e r i a l s c o n s t i t u t i v e e q u a t i o n s relate l o a d i n g s as stresses, t e m p e r a t u r e , etc. to effects as strains, d a m a g e , fracture, wear, etc.Handbook of Materials BehaviorModels. ISBN 0-12-443341-3.Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.

4

Lemaitre

It is a technique because it uses tools such as mathematics, thermodynamics, computers, and experiments to build close form models and to obtain numerical values for the parameters that are used in structure calculations to predict the behavior of structures in the service or forming process, etc., safety and optimal design being the main motivations. It is an art because the sensibility of the scientist plays an important role. Except for linear phenomena, there is not unique way to build a model from a set of observations and test results. Furthermore, the mathematical structure of the model may depend upon its use. This is interesting from the human point of view. But it is sometimes difficult to select the proper model for a given application. The simplest is often the more efficient event, even if it is not the most accurate.

1.1.2

OBSERVATIONS AND CHOICE

OF VARIABLES First of all, in mechanics of materials, a model does not exist for itself; it exists in connection with a purpose. If it is the macroscopic behavior of mechanical components of structures that is being considered, the basic tool is the mechanics of continuous media, which deals with the following: 1. Strain, a second-order tensor related to the displacement ff of two points: 9 Euler's tensor ~ for small perturbations. 1~ ij -~ ( u i j -3t- uj , i )

(1)

In practice, the hypothesis of "small" strain may be applied if it is below about 10%. 9 Green-Lagrange tensor A (among others) for large perturbations, if F is the tangent linear transformation which transforms under deformation a point M0 of the initial configuration into M of the actual configuration.1 _v (M0) (2) _

(_vT_v - 1)

With _F the transpose of F. ~2. Stress, a second-order tensor dual of the strain tensor; its contracted product by the strain rate tensor is the power involved in the mechanical process.

1.1 Background on Modeling

9 Cauchy stress tensor _a for small perturbations, checking the equilibrium with the internal forces density f and the inertia forces pff,d2R

crijo + fi - oiii

with/ii - dt-T

(3)

9 Piola-Kirchoff tensor _S (among others) for large perturbations.

S _ - det(F_)~_F_- r3. Temperature T. These three variables are functions of the time t.

(4)

1.1.2.1

SCALE OF OBSERVATION

From the mathematical point of view, strains and stresses are defined on a material point, but the real materials are not continuous. Physically, strain and stress represent averages on a fictitious volume element called the representative volume element (RVE) or mesoscale. To give a subjective order of magnitude of a characteristic length, it can be 0.1 mm for metallic materials; 1 mm for polymers; 1 0 m m for woods; 100 mm for concrete. It is below these scales that observations must be done to detect the micromechanisms involved in modeling: 9 slips in crystals for plasticity of metals; 9 decohesions of sand particles by breaking of atomic bonds of cement for damage in concrete; 9 rupture of microparticles in wear; 9 etc. These are observations at a microscale. It is more or less an "art" to decide at which microscale the main mechanism responsible for a mesoscopic phenomenon occurs. For example, theories of plasticity have been developed at a mesoscale by phenomenological considerations, at a microscale when dealing with irreversible slips, and now at an atomic scale when modeling the movements of dislocations. At any rate, one's first priority is to observe phenomena and to select the representative mechanism which can be put into a mathematical framework

6

Lemaitre

of homogenization to give variables at a mesoscale compatible with the mechanics of continuous media.

1.1.2.2

INTERNALVARIABLES

When the purpose is structural calculations with sets of constitutive equations, it is logical to consider that each main mechanism should have its own variable. For example, the total strain _8 is directly observable and defines the external state of the representative volume element (RVE), but for a better definition of the internal state of the RVE it is convenient to look at what happens during loading and unloading of the RVE to define an elastic strain ee and a plastic strain e_P such ase P ~j - ~j + ~;

(5)

The elastic strain represents the reversible movements of atoms, and the plastic strain corresponds to an average of irreversible slips. All variables which define the internal state of the RVE are called internal variables. They should result from observations at a microscale and from a homogenization process: 9 isotropic hardening in metals related to the density of dislocations; 9 kinematic hardening related to the internal residual microstresses at the level of crystals; 9 damage related to the density of defects; 9 etc. How many do we need? As many as the number of phenomena taken into consideration, but the smallest is the best. Finally, the local state method postulates that the considered thermodynamic state is completely defined by the actual values of the corresponding state variables: observable and internal.

1.1.3 FORMULATIONThe thermodynamics of irreversible processes is a general framework that is easy to use to formulate constitutive equations. It is a logical guide for incorporating observations and experimental results and a set of rules for avoiding incompatibilities. The first principle is the energy balance: If e is the specific internal energy, p the density, co the volume density of internal heat produced by external

1.1 Background on Modeling

sources, and ~' The heat flux:fie. = cr ij ~ ij -Jr- (.0 - - q i , i

(6)

where the sommation convention of Enstein applies. The second principle states that the entropy production i must be larger or equal to the heat received divided by the temperature Ps >T ,i

(7)

If ~ = e - Ts is the Helmholtz specific free energy (this is the energy in the RVE which can eventually be recovered),

~,j~j - 0(~; + ~ir) -q~r~ > oT -

(8)

This is the Clausius-Duhem inequality, which corresponds to the positiveness of the dissipated energy and which has to be fulfilled by any model for all possible evolutions.

1.1.3.1 STATEPOTENTIALThe state potential allows for the derivation of the state laws and the definition of the associate variables or driving forces associated with the state variables VK tO define the energy involved in each phenomenon. Choosing the Helmholtz free energy ~, it is a function of all state variables concave with respect to the temperature and convex with respect to all other VK, 0 = O(~, T , f , f , . . . or in classical elastoplasticityO = O ( F , ~_p, r , . . . v ~ . . .)

VK...)

(9)

(10)

The state laws derive from this potential to ensure that the second principle is always fulfilled.a ij fiPij - ~_~ p

~ K "V K

T q~r,~ > 0

(11)

They are the laws of thermoelasticity

oo ~J - P oe~o~S ~-

(12)

(13)

OT

Lemaitre

The associated variables are defined by

00cr~j - p 0 ~ O0 AK -- p OVK

(14)

(15)

Each variable AK is the main cause of variation of the state variable VK. In other words, the constitutive equations of the phenomenon represented by VK will be primarily a function of its associated variable and eventually from others.VK -- g K ( . . . A K . . . )

(16)

They also allow us to take as the state potential the Gibbs energy dual of the Helmholtz energy by the Legendre-Fenchel transform ~* = ~* (_~,s,...AK...) (17)

or any combination of state and associated variables by partial transform.

1.1.3.2

DISSIPATIVE POTENTIAL

To define the gK function of the kinetic equations, a second potential is postulated. It is a function of the associate variables, and convex to ensure that the second principle is fulfilled. It can also be a function of the state variables but taken only as parameters. q) -- q~(K,...AK..., gracl T;e_.e, T , . . . V K . . . ) The kinetic laws of evolution of the internal state variables derive from.p _ Oq)

(18)

eij - Oaij

(19)

% _

&oOAK

(z0)

-

=

-

----------~

(21)

T

0grad T

Unfortunately, for phenomena which do not depend explicitly upon the time, this function (p is not differentiable. The flux variables are defined by the subdifferential of q~. If F is the criterion function whose the convex F - 0 is

1.1 B a c k g r o u n d o n M o d e l i n g

the indicatrice function of qo. (,o-- 0(,o - - o c

if if

F Cro and the stress at the change of slope. The scatter of size effect measurements within a practicable size range (up to 1:30) normally does not permit identifying more than one characteristic length (measurements of postpeak behavior are used for that purpose). Vice versa, when only the maximum loads of structures in the bridging region between plasticity and LEFM are of interest, hardly more than one characteristic length (namely, cf) is needed.

~

P,u

a

. . . .---r

JI

,>w D

FIGURE 1.3.6

Cohesive crack and distribution of crack-bridging cohesive stresses.

58

Ba~.ant

The crack band model, which is easier to implement and is used in commercial codes (e.g., DIANA, SBETA) [49], is, for localized cracking or fracture, nearly equivalent to the cohesive crack model ([BP], [97]), provided that the effective (average) transverse strain in the crack band is taken as ey = w / h where h is the width of the band. All that has been said about the cohesive crack model also applies to the crack band model. Width h, of course, represents an additional characteristic length, ~4 - - h. It matters only when the cracking is not localized but distributed (e.g., due to the effect of dense and strong enough reinforcement), and it governs the spacings of parallel cracks. Their minimum spacing cannot be unambiguously captured by the cohesive crack model.

1 . 3 . 7 . 9 SIZE EFFECT VIA NONLOCAL, GRADIENT~ OR DISCRETE ELEMENT MODELS The hypostatic feature of any model capable of bridging the power law size effects of plasticity and LEFM is the presence of some characteristic length, g. In the equivalent LEFM associated with the size effect law in Eq. 10, cf serves as a characteristic length of the material, although this length can equivalently be identified with 8CrOD in Wells-Cottrell or JenqShah models, or with the crack opening wf at which the stress in the cohesive crack model (or crack band model) is reduced to zero (for size effect analysis with the cohesive crack model, see [BP] and Ba~.ant and Li [251). In the integral-type nonlocal continuum damage models, ~ represents the effective size of the representative volume of the material, which in turn plays the role of the effective size of the averaging domain in nonlocal material models. In the second-gradient nonlocal damage models, which may be derived as an approximation of the nonlocal damage models, a material length is involved in the relation of the strain to its Laplacian. In damage simulation by the discrete element (or random particle) models, the material length is represented by the statistical average of particle size. The existence of g in these models engenders a quasi-brittle size effect that bridges the power-law size effects of plasticity and LEFM and follows closely Eq. 10 with ~rN = 0, as documented by numerous finite element simulations. It also poses a lower bound on the energy dissipation during failure, prevents spurious excessive localization of softening continuum damage, and eliminates spurious mesh sensitivity ([BP], ch. 13). These important subjects will not be discussed here any further because there exists a recent extensive review [18].

1.3 Size Effect on Structural Strength

59

1.3.7.10 NONLOCAL STATISTICALGENERALIZATIONOF THE WEIBULL

THEORY

Two cases need to be distinguished: (a) The front of the fracture that causes failure can be at only one place in the structure, or (b) the front can lie, with different probabilities, at many different places. The former case occurs when a long crack whose path is dictated by fracture mechanics grows before the m a x i m u m load, or if a notch is cut in a test specimen. The latter case occurs when the maximum load is achieved at the initiation of fracture growth. In both cases, the existence of a large FPZ calls for a modification of the Weibull concept: The failure probability P1 at a given point of the continuous structure depends not on the local stress at that point, but on the nonlocal strain, which is calculated as the average of the local strains within the neighborhood of the point constituting the representative volume of the material. The nonlocal approach broadens the applicability of the Weibull concept to the case of notches or long cracks, for which the existence of crack-tip singularity causes the classical Weibull probability integral to diverge at realistic m-values (in cleavage fracture of metals, the problem of crack singularity has been circumvented differently m by dividing the cracktip plastic zone into small elements and superposing their Weibull contributions [77]). Using the nonlocal Weibull theory, one can show that the proper statistical generalizations of Eq. 10 (with aR = 0 ) and Eq. 12 having the correct asymptotic forms for D---+ oo, D - + 0, and m - + oo are (Fig. 1.3.7): Case (a)" Case (b)" aN--

Bo'o(fl2maIm +/~r)-l/2r

f l - D/Do~ -- Db/D

(22) (23)

aN - o'0~"a/m(1 + r~l-'~a/m) 1/r

where it is assumed that rna< m, which is normally the case. The first formula, which was obtained for r = 1 by Ba~.ant and Xi [36] and refined for n ~ 1 by Planas, has the property that the statistical influence on the size effect disappears asymptotically for large D. The reason is that, for long cracks or notches with stress singularity, a significant contribution to the Weibull probability integral comes only from the FPZ, whose size does not vary much with D. The second formula has the property that the statistical influence asymptotically disappears for small sizes. The reason is that the FPZ occupies much of the structure volume. Numerical analyses of test data for concrete show that the size ranges in which the statistical influence on the size effect in case (a) as well as (b) would be significant do not lie within the range of practical interest. Thus the deterministic size effect dominates and its statistical correction in Eqs. 22 and

60z

Ba~.ant

o

0")

aD0 Z

log D

h. r

~ . _

m

b

log D

FIGURE 1.3.7 Scaling laws according to nonlocal generalization of Weibull theory for failures after long stable crack growth (top) or a crack initiation (right).

23 may be ignored for concrete, except in the rare situations where the deterministic size effect vanishes, which occurs rarely (e.g., for centric tension of an unreinforced bar).

1.3.8 OTHER SIZE EFFECTS1 . 3 . 8 . 1 HYPOTHESIS OF FRACTAL ORIGIN OF SIZE EFFECT The partly fractal nature of crack surfaces and of the distribution of microcracks in concrete has recently been advanced as the physical origin of the size effects observed on concrete structures. Bhat [38] discussed a possible role of fractality in size effects in sea ice. Carpinteri [43, 44], Carpinteri et al. [45], and Carpinteri and Chiaia [46] proposed the so-called multifractal scaling law (MFSL) for failures occurring at fracture initiation from a smooth

1.3 Size Effect on Structural Strength

61

surface, which readsGN =

v/A1 q-(A2/D)

(24)

where A1, A2= constants. There are, however, four objections to the fractal theory [11 ]: (i) A mechanical analysis (of either invasive or lacunar fractals) predicts a different size effect trend than Eq. 24, disagreeing with experimental observations. (ii) The fractality of the final fracture surface should not matter because typically about 99% of energy is dissipated by microcracks and frictional slips on the sides of this surface. (iii) The fractal theory does not predict how A1 and A2 should depend on the geometry of the structure, which makes the MFSL not too useful for design application. (iv) The MFSL is a special case of the second formula in Eq. 12 for r = 2, which logically follows from fracture mechanics;

A1 - EGf /cfg'(O)

A2 -- -EGfg"(O)/2cf[g'(O)] 3

(25)

[12]. Unlike fractality, the fracture explanation of Eq. 24 has the advantage that, by virtue of these formulae, the geometry dependence of the size effect coefficients can be determined.

1.3.8.2

BOUNDARY LAYER, SINGULARITY,

AND DIFFUSION Aside from the statistical and quasi-brittle size effects, there are three further types of size effect that influence the nominal strength: 1. The boundary layer effect, which is due to material heterogeneity (i.e., the fact that the surface layer of heterogeneous material such as concrete has a different composition because the aggregates cannot protrude through the surface), and to the Poisson effect (i.e., the fact that a plane strain state on planes parallel to the surface can exist in the core of the test specimen but not at its surface). 2. The existence of a three-dimensional stress singularity at the intersection of crack edge with a surface, which is also caused by the Poisson effect ([BP], Sec. 1.3). This causes the portion of the FPZ near the surface to behave differently from that in the interior. 3. The time-dependent size effects caused by diffusion phenomena such as the transport of heat or the transport of moisture and chemical agents in porous solids (this is manifested, e.g., in the effect of size on shrinkage and drying creep, due to size dependence of the drying half-time) and its effect on shrinkage cracking [96].

62 1.3.9 CLOSING REMARKS

Ba~.ant

Substantial though the recent progress has been, the undersl[anding of the scaling problems of solid mechanics is nevertheless far from complete. Mastering the size effect that bridges different behaviors on adjacent scales in the microstructure of material will be contingent upon the development of realistic material models that possess a material length (or characteristic length). The theory of nonlocal continuum damage will have to move beyond the present phenomenological approach based on isotropic spatial averaging, and take into account the directional and tensorial interactions between the effects causing nonlocality. A statistical description of such interactions will have to be developed. Discrete element models of the microstructure of fracturing or damaging materials will be needed to shed more light on the mechanics of what is actually happening inside the material and separate the important processes from the unimportant ones.

ACKNOWLEDGMENTPreparation of the present review article was supported by the Office of Naval Research under Grant N00014-91-J-1109 to Northwestem University, monitored by Dr. Yapa D. S. Rajapakse.

REFERENCES AND BIBLIOGRAPHY1. Argon, A. S. (1972). Fracture of composites, in Treatise of Materials Science and Technology, p. 79, vol. 1, New York: Academic Press. 2. Barenblatt, G. I. (1959). The formation of equilibrium cracks during brittle fracture. General ideas and hypothesis, axially symmetric cracks. Prikl. Mat. Mekh. 23 (3): 434-444. 3. Barenblatt, G. I. (1962). The mathematical theory of equilibrium cracks in brittle fracture. Advanced Appl. Mech. 7: 55-129. 4. Ba~.ant, Z. P. (1976). Instability, ductility, and size effect in strain-softening concrete. J. Engrg. Mech. Div., Am. Soc. Civil Engrs., 102: EM2, 331-344; disc. 103, 357-358, 775-777, 104, 501-502. 5. Ba~.ant, Z. P. (1984). Size effect in blunt fracture: Concrete, rock, metal.J. Engrg. Mech. ASCE 110: 518-535. 6. Ba~.ant, Z. P. (1992). Large-scale thermal bending fracture of sea ice plates. J. Geophysical Research, 97 (Cll): 17,739-17,751. 7. Ba~ant, Z. P. ed. (1992). Fracture Mechanics of Concrete Structures, Proc., First Intern. Conf. (FraMCoS-1), held in Breckenridge, Colorado, June 1-5, Elsevier, London (1040 pp.). 8. Ba~.ant, Z. P. (1993). Scaling laws in mechanics of failure. J. Engrg. Mech., ASCE 119 (9): 1828-1844.

1.3 Size Effect on Structural Strength

63

9. Ba~.ant, Z. P (1997a). Fracturing truss model: Size effect in shear failure of reinforced concrete. J. Engrg. Mech., ASCE 123 (12): 1276-1288. 10. Ba~.ant, Z. P. (1997b). Scaling of quasibrittle fracture: Asymptotic analysis. Int. J. Fracture 83 (1): 19-40. 11. Ba~.ant, Z. P (1997c). Scaling of quasibrittle fracture: Hypotheses of invasive and lacunar fractality, their critique and Weibull connection. Int. J. Fracture 83 (1): 41-65. 12. Ba~.ant, Z. P. (1998). Size effect in tensile and compression fracture of concrete structures: Computational modeling and design. Fracture Mechanics of Concrete Structures (3rd Int. Conf., FraMCos-3, held in Gifu, Japan), H. Mihashi and K. Rokugo, eds., Aedificatio Publishers, Freiburg, Germany, 1905-1922. 13. Ba~.ant, Z. P. (1999). Structural stability. International Journal of Solids and Structures 37 (200): 55-67; special issue of invited review articles on Solid Mechanics edited by G. J. Dvorak for U.S. Nat. Comm. on Theor. and Appl. Mech., publ. as a book by Elsevier Science, Ltd. 14. Ba~.ant, Z. P. (1999). Size effect. International Journal of Solids and Structures 37 (200): 69-80; special issue of invited review articles on Solid Mechanics edited by G. J. Dvorak for U.S. Nat. Comm. on Theor. and Appl. Mech., Publ. as a book by Elsevier Science, Ltd. 15. Ba~.ant, Z. P. (1999). Size effect on structural strength: A review. Archives of Applied Mechanics, pp. 703-725, vol. 69, Berlin: Ingenieur-Archiv, Springer Verlag. 16. Ba~.ant, Z. P. (2000). Scaling laws for brittle failure of sea ice, Preprints, IUTAM Syrup. on Scaling Laws in Ice Mechanics (Univ. of Alaska, Fairbanks, June), J. P Dempsey, H. H. Shen, and L. H. Shapiro, eds., Paper No. 3, pp. 1-23. 17. Ba~.ant, Z. P., and Cedolin, L. (1991). Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories (textbook and reference volume), New York: Oxford University Press. 18. Ba~.ant, Z. P, and Chen, E.-P. (1997). Scaling of structural failure. Applied Mechanics Reviews, ASME 50 (10): 593-627. 19. Ba~.ant, Z. P, Daniel I. M., and Li, Zhengzhi (1996). Size effect and fracture characteristics of composite laminates. ASME J. of Engrg. Materials and Technology 118 (3): 317-324. 20. Ba~.ant, Z. P, and Kazemi, M. T. (1990). Size effect in fracture of ceramics and its use to determine fracture energy and effective process zone length. J. American Ceramic Society 73 (7): 1841-1853. 21. Ba~.ant, Z. P., and Kazemi, M. T. (1991). Size effect on diagonal shear failure of beams without stirrups. ACI Structural Journal 88 (3): 268-276. 22. Ba~.ant, Z. P, Kim, J.-J. H., Daniel, I. M., Becq-Giraudon, E., and Zi, G. (1999). Size effect on compression strength of fiber composites failing by kink band propagation. Int. J. of fracture (special issue on Fracture Scaling, edited by Z. P. Ba~.ant and Y. D. S. Rajapakse) (June), 95: 103-141. 23. Ba~.ant, Z. P., and Kim, J.-J. H. (1998). Size effect in penetration of sea ice plate with partthrough cracks. I. Theory. J. of Engrg. Mech., ASCE 124 (12): 1310-1315; II. Results, ibid., 1316-1324. 24. Ba~.ant, Z. P., Kim, J.-J. H., Daniel, I. M., Becq-Giraudon, E., and Zi, G. (1999). Size effect on compression strength of fiber composites failing by kink band propagation. Int. J. of Fracture, in press. 25. Ba~,ant, Z. P, and Li, Yuan-Neng (1997). Stability of cohesive crack model: Part I n Energy principles. Tran. ASME, J. Applied Mechanics 62: 959-964; Part II n Eigenvalue analysis of size effect on strength and ductility of structures, ibid. 62: 965-969. 26. Ba~.ant, Z. P., and Li, Yuan-Neng (1997). Cohesive crack with rate-dependent opening and viscoelasticity: I. Mathematical model and scaling. Int. J. Fracture 86 (3): 247-265.

64

Ba~-ant

27. Ba~.ant, Z. P., Lin, E-B., and Lippmann, H. (1993). Fracture energy release and size effect in borehole breakout. Int. Journal for Numerical and Analytical Methods in Geomechanics 17: 1-14. 28. Ba~.ant, Z. P., and Novhk, D. (2000). Probabilistic nonlocal theory for quasibrittle fracture initiation and size effect. I. Theory, and II. Application. J. Engrg. Mech., ASCE 126 (2): 166-174 and 175-185. 29. Ba~.ant, Z. P., and Novhk, D. (2000). Energetic-statistical size effect in quasibrittle materials. ACI Materials Journal 97 (3): 381-392. 30. Ba~.ant, Z. P., and Oh, B.-H. (1983). Crack band theory for fracture of concrete. Materials and Structures (RILEM, Paris) 16: 155-177. 31. Ba~.ant, Z. P., and Pfeiffer, P. A. (1987). Determination of fracture energy from size effect and brittleness number. ACI Materials J. 84: 463-480. 32. Ba~.ant, Z. P., and Planas, J. (1998). Fracture and Size Effect in Concrete and Other Quasibrittle Materials, Boca Raton, Florida: CRC Press. 33. Ba~.ant, Z. P., and Pfeiffer, P. A. (1987). Determination of fracture energy from size effect and brittleness number. ACI Materials J. 84: 463-480. 34. Ba~.ant, Z. P., and Vitek, J. L. (1999). Compound size effect in composite beams with softening connectors. I. Energy approach, and II. Differential equations and behavior. J. Engrg. Mech., ASCE 125 (11): 1308-1314 and 1315-1322. 35. Ba~.ant, Z. P., and Rajapakse, Y. D. S., ed. (1999). Fracture Scaling, Dordrecht: Kluwer Academic Publishers (special issue of Int. J. Fracture), (June), 95: 1-433. 36. Ba~.ant, Z. P., and Xi, Y. (1991). Statistical size effect in quasi-brittle structures: II. Nonlocal theory. ASCE J. Engineering Mechanics 117 (11): 2623-2640. 37. Beremin, E M. (1983). A local criterion for cleavage fracture of a nuclear pressure vessel steel. Metallurgy Transactions A, 14: 2277-2287. 38. Bhat, S. U. (1990). Modeling of size effect in ice mechanics using fractal concepts. Journal of Offshore Mechanics and Arctic Engineering 112: 370-376. 39. Budiansky, B. (1983). Micromechanics. Computers and Structures 16 (1-4): 3-12. 40. Budiansky, B., Fleck, N. A., and Amazigo, J. C. (1997). On kink-band propagation in fiber composites. J. Mech. Phys. Solids 46 (9): 1637-1635. 41. Carpinteri, A. (1986). Mechanical damage and crack growth in concrete, Dordrecht, Boston: Martinus Nijhoff B Kluwer. 42. Carpinteri, A. (1989). Decrease of apparent tensile and bending strength with specimen size: Two different explanations based on fracture mechanics. Int. J. Solids Struct. 25 (4): 407-429. 43. Carpinteri, A. (1994a). Fractal nature of material microstructure and size effects on apparent mechanical properties. Mechanics of Materials 18: 89-101. 44. Carpinteri, A. (1994b). Scaling laws and renormalization groups for strength and toughness of disordered materials. Int. J. Solids and Struct. 31: 291-302. 45. Carpinteri, A., Chiaia, B., and Ferro, G. (1994). Multifractal scaling law for the nominal strength variation of concrete structures, in Size Effect in Concrete Structures (Proc., Japan Concrete Institute International Workshop, held in Sendai, Japan, 1993), pp. 193-206, M. Mihashi, H. Okamura and Z.P. Ba~.ant eds., London, New York: E & FN Spon. 46. Carpinteri, A., and Chiaia, B. (1995). Multifractal scaling law for the fracture energy variation of concrete structures, in Fracture Mechanics of Concrete Structures (Proceedings of FraMCoS-2, held at ETH, Zfirich), pp. 581-596, E H. Wittmann, Freiburg: Aedificati6 Publishers. 47. Carter, B. C. (1992). Size and stress gradient effects on fracture around cavities. Rock Mech. and Rock Engng. (Springer) 25 (3): 167-186. 48. Carter, B. C., Lajtai, E. Z., and Yuan, Y. (1992). Tensile fracture from circular cavities loaded in compression. Int. J. Fracture 57: 221-236.

1.3 Size Effect on Structural Strength

65

49. Cervenka, V., and Pukl, R. (1994). SBETA analysis of size effect in concrete structures, in Size Effect in Concrete Structures, pp. 323-333, H. Mihashi, H. Okamura, and Z. P. Ba~.ant, eds., London: E & FN Spon. 50. Cottrell, A. H. (1963). Iron and Steel Institute Special Report 69: 281. 51. da Vinci, L. (1500s) - - s e e The Notebooks of Leonardo da Vinci (1945), Edward McCurdy, London (p. 546); and Les Manuscrits de L~onard de Vinci, trans, in French by C. RavaissonMollien, Institut de France (1881-91), vol. 3. 52. Dempsey, J. E, Adamson, R. M., and Mulmule, S. V. (1995a). Large-scale in-situ fracture of ice. in Fracture Mechanics of Concrete Structures, vol. 1 (Proc., 2nd Int. Conf. on Fracture Mech. of Concrete Structures [FraMCoS-2], held at ETH, Ziirich), pp. 575-684, E H. Wittmann, ed., Freiburg: Aedificati6 Publishers. 53. Dempsey, J. E, Adamson, R. M., and Mulmule, S. V. (1999). Scale effect on the in-situ tensile strength and failure of first-year sea ice at Resolute, NWR. Int. J. Fracture (special issue on Fracture Scaling, Z. P. Ba~.ant and Y. D. S. Rajapakse, eds.), 95: 325-345. 54. Dempsey, J. P., Slepyan, L. I., and Shekhtman, I. I. (1995b). Radial cracking with closure. Int. J. Fracture 73 (3): 233-261. 55. Dugdale, D. S. (1960). Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8: 100-108. 56. Evans, A. G. (1978). A general approach for the statistical analysis of multiaxial fracture. J. Am. Ceramic Soc. 61: 302-308. 57. Fr~chet, M. (1927). Sur la loi de probabilit~ de l'~cart maximum. Ann. Soc. Polon. Math. 6: 93. 58. Fisher, R. A., and Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest and smallest member of a sample. Proc., Cambridge Philosophical Society 24: 180-190. 59. Frankenstein, E. G. (1963). Load test data for lake ice sheet. Technical Report 89, U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, New Hampshire. 60. Frankenstein, E. G. (1966). Strength of ice sheets. Proc., Conf. on Ice Pressures against Struct.; Tech. Memor. No. 92, NRCC No. 9851, Laval University, Quebec, National Research Council of Canada, Canada, pp. 79-87. 61. Freudenthal, A. M. (1956). Physical and statistical aspects of fatigue, in Advances in Applied Mechanics, pp. 117-157, vol. 4, New York: Academic Press. 62. Freudenthal, A. M. (1956). Statistical approach to brittle fracture, Chapter 6 in Fracture, vol. 2, pp. 591-619, H. Liebowitz, ed., New York: Academic Press. 63. Freudenthal, A. M., and Gumbell, E. J. (1956). Physical and statistical aspects of fatigue, in Advances in Applied Mechanics, pp. 117-157, vol. 4, New York: Academic Press. 64. Galileo, Galilei Linceo (1638). Discorsi i Demostrazioni Matematiche intorno/L due Nuove Scienze, Elsevirii, Leiden; English trans, by T. Weston, London (1730), pp. 178-181. 65. Gettu, R., Ba~.ant, Z. P., and Karr, M. E. (1990). Fracture properties and brittleness of highstrength concrete. ACI Materials Journal 87 (Nov.-Dec.): 608-618. 66. Griffith, A. A. (1921). The phenomena of rupture and flow in solids. Phil. Trans. 221A: 179-180. 67. Haimson, B. C., and Herrick, C. G. (1989). In-situ stress calculation from borehole breakout experimental studies. Proc., 26th U.S. Syrup. on Rock Mech., 1207-1218. 68. Hillerborg, A., Mod~er, M., and Petersson, P. E. (1976). Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research 6: 773-782. 69. Iguro, M., Shiyoa, T., Nojiri, Y., and Akiyama, H. (1985). Experimental studies on shear strength of large reinforced concrete beams under uniformly distributed load. Concrete

66

Ba~.ant

Library International, Japan Soc. of Civil Engrs. No. 5: 137-154. (translation of 1984 article in Proc. JSCE). 69a. Irwin, G. R. (1958). Fracture. In Handbuck der Physik 6 (Fl~gge, ed.) Springer Verlag, Berlin, 551-590. 70. Jenq, Y. S., and Shah, S. P. (1985). A two parameter fracture model for concrete. J. Engrg. Mech. ASCE, 111 (4): 1227-1241. 71. Kani, G. N. J. (1967). Basic facts concerning shear failure. ACI Journal, Proceeding 64 (3, March): 128-141. 72. Kaplan, M.E (1961). Crack propagation and the fracture concrete, ACIJ. 58, No. 11. 73. Kerr, A. D. (1996). Bearing capacity of floating ice covers subjected to static, moving, and oscillatory loads. Appl. Mech. Reviews, ASME 49 (11): 463-476. 74. Kesler, C. E., Naus, D. J., and Lott, J. L. (1971). Fracture mechanics - - Its applicability to concrete, Proc. Int. Conf. on the Mechanical Behavior of Materials, pp. 113-124, vol. 4, Kyoto, The Soc. of Mater. Sci. 75. Kittl, P., and Diaz, G. (1988). Weibull's fracture statistics, or probabilistic strength of materials: State of the art. Res Mechanica. 24: 99-207. 76. Kittl, P., and Diaz, G. (1990). Size effect on fracture strength in the probabilistic strength of materials. Reliability Engrg. Sys. Saf. 28: 9-21. 77. Lei, Y., O'Dowd, N. P., Busso, E. P., and Webster, G. A. (1998). Weibull stress solutions for 2-D cracks in elastic and elastic-plastic materials. Int. J. Fracture 89: 245-268. 78. Leicester, R. H. (1969). The size effect of notches. Proc., 2nd Australasian Conf. on Mech. of Struct. Mater., Melbourne, pp. 4.1-4.20. 79. Li, Yuan-Neng, and Ba~.ant, Z. P. (1997). Cohesive crack with rate-dependent opening and viscoelasticity: II. Numerical algorithm, behavior and size effect. Int. J. Fracture 86 (3): 267-288. 80. Li, Zhengzhi, and Ba~.ant, Z. P. (1998). Acoustic emissions in fracturing sea ice plate simulated by particle system. J. Engrg. Mech. ASCE 124 (1): 69-79. 81. Lichtenberger, G. J., Jones, J. w., Stegall, R. D., and Zadow, D. W. (1974). Static ice loading tests, Resolute Bay m Winter 1973/74. APOA Project No. 64, Rep. No. 745B-74-14, (CREEL Bib. No. 34-3095), Rechardson, Texas: Sunoco Sci. and Technol. 82. Mariotte, E. (1686). TraitS. du mouvement des eaux, posthumously edited by M. de la Hire; Engl. trans, by J. T. Desvaguliers, London (1718), p. 249; also Mariotte's collected works, 2nd ed., The Hague (1740). 83. Marti, P. (1989). Size effect in double-punch tests on concrete cylinders. ACI Materials Journal 86 (6): 597-601. 84. Mihashi, H. (1983). Stochastic theory for fracture of concrete, in Fracture Mechanics of Concrete, pp. 301-339, E H. Wittmann, ed., Amsterdam: Elsevier Science Publishers. 85. Mihashi, H, and Izumi, M. (1977). Stochastic theory for concrete fracture. Cem. Concr. Res. 7: 411-422. 86. Mihashi, H., Okamura, H., and Ba~ant, Z. P., eds., (1994). Size Effect in Concrete Structures (Proc., Japan Concrete Institute Intern. Workshop held in Sendai, Japan, Oct. 31-Nov. 2, 1995), London, New York: E & FN Spon. 87. Mihashi, H., and Rokugo, K., eds. (1998). Fracture Mechanics of Concrete Structures (Proc., 3rd Int. Conf., FraMCoS-3, held in Gifu, Japan), Freiburg: Aedificatio Publishers. 88. Mihashi, H., and Zaitsev, J. W. (1981). Statistical nature of crack propagation, Section 4-2 in Report to RILEM TC 50 ~ FMC, E H. Wittmann, ed. 89. Mulmule, S. V., Dempsey, J. P., and Adamson, R. M. (1995). Large-scale in-situ ice fracture experiments. Part II: Modeling efforts. Ice Mechanics ~ 1995 (ASME Joint Appl. Mechanics

1.3 Size Effect on Structural Strength

67

90. 91.

92. 93.

94.

95. 96.

97. 98. 99. 100.

101.

102. 103. 104.

105. 106. 107. 108. 109.

and Materials Summer Conf., held at University of California, Los Angeles, June), AMD-MD '95, New York: Am. Soc. of Mech. Engrs. Nesetova, V., and Lajtai, E. Z. (1973). Fracture from compressive stress concentration around elastic flaws. Int. J. Rock Mech. Mining Sci. 10: 265-284. Okamura, H., and Maekawa, K. (1994). Experimental study of size effect in concrete structures, in Size Effect in Concrete Structures, pp. 3-24, H. Mihashi, H. Okamura, and Z. P. Ba~.ant, eds., London: E & FN Spon (Proc. of JCI Intern. Workshop held in Sendai, Japan, 1993). Peirce, E T. (1926). J. Textile Inst. 17: 355. Petersson, P.E. (1981). Crack growth and development of fracture zones in plain concrete and similar materials. Report TVBM-1006, Div. of Building Materials, Lund Inst. of Tech., Lund, Sweden. Planas, J., and Elices, M. (1988). Conceptual and experimental problems in the determination of the fracture energy of concrete. Proc. Int. Workshop on Fracture Toughness and Fracture Energy, Test Methods of Concrete and Rock. Tohoku Univ., Sendai, Japan, pp. 203-212. Planas, J., and Elices, M. (1989). In Cracking and Damage, pp. 462-476, J. Mazars and Z. P. Ba~.ant, eds., London: Elsevier. Planas, J., and Elices, M. (1993). Drying shrinkage effects on the modulus of rupture, in Creep and Shrinkage of Concrete (Proc., 5th Int. RILEM Symp., Barcelona), pp. 357-368, Z. P. Ba~.ant and I. Carol, eds., London: E & FN Spon. Planas, J., Elices, M., and Guinea, G. V. (1983). Cohesive cracks vs. nonlocal models: Closing the gap. Int. J. Fracture 63 (2): 173-187. Reinhardt, H. W. (1981). Massstabseinfluss bei Schubversuchen im Licht der Bruchmechanik. Beton and Stahlbetonbau (Berlin), No. 1, pp. 19-21. RILEM Recommendation (1990). Size effect method for determining fracture energy and process zone of concrete. Materials and Structures 23: 461-465. Rosen, B. W. (1965). Mechanics of composite strengthening, Fiber Composite Materials, Am. Soc. for Metals Seminar, Chapter 3, American Society for Metals, Metals Park, Ohio, pp. 37-75. Ruggieri, C., and Dodds, R. H. (1996). Transferability model for brittle fracture including constraint and ductile tearing effects - - in probabilistic approach, Int. J. Fracture 79: 309-340. Sedov, L. I. (1959). Similarity and Dimensional Methods in Mechanics, New York: Academic Press. Selected Papers by Alfred M. Freudenthal (1981). Am. Soc. of Civil Engrs., New York. Shioya, Y., and Akiyama, H. (1994). Application to design of size effect in reinforced concrete structures, in Size Effect in Concrete Structures (Proc. of Intern. Workshop in Sendai, 1993), pp. 409-416, H. Mihashi, H. Okamura, and Z. P. Ba~ant, eds., London: E & FN Spon. Slepyan, L.I. (1990). Modeling of fracture of sheet ice. Izvestia AN SSSR, Mekh. Tverd. Tela 25 (2): 151-157. Sodhi, D. S. (1995). Breakthrough loads of floating ice sheets. J. Cold Regions Engrg., ASCE 9 (1): 4-20. Tippett, L. H. C. (1925). On the extreme individuals and the range of samples. Biornetrika 17: 364. von Mises. R. (1936). La distribution de la plus grande de n valeurs. Rev. Math. Union Interbalcanique 1: 1. Walraven, J., and Lehwalter (1994). Size effects in short beams loaded in shear. ACI Structural Journal 91 (5): 585-593.

68

Ba~.ant

110. Walraven, J. (1995). Size effects: their nature and their recognition in building codes. Studi e Ricerche (Politecnico di Milano) 16: 113-134. 111. Walsh, P. E (1972). Fracture of plain concrete. Indian Concrete Journal 46, No. 11. 112. Walsh, P. E (1976). Crack initiation in plain concrete. Magazine of Concrete Research 28: 37-41. 113. Weibull, W (1939). The phenomenon of rupture in solids. Proc., Royal Swedish Institute of Engineering Research (Ingenioersvetenskaps Akad. Handl.) 153, Stockholm, 1-55. 114. Weibull, W (1949). A statistical representation of fatigue failures in solids. Proc., Roy. Inst. of Techn. No. 27. 115. Weibull, W. (1951). A statistical distribution function of wide applicability. J Appl. Mech., ASME, 18. 116. Weibull, W. (1956). Basic aspects of fatigue, in Proc., Colloquium on Fatigue, Stockholm: Springer-Verlag. 117. Wells, A. A. (1961). Unstable crack propagation in metals-cleavage and fast fracture. Syrup. on Crack Propagation. Cranfield, 1: 210-230. 118. Williams, E. (1957). Some observations of Leonardo, Galileo, Mariotte and others relative to size effect, Annals of Science 13: 23-29. 119. Wisnom, M. R. (1992). The relationship between tensile and flexural strength of unidirectional composite. J. Composite Materials 26: 1173-1180. 120. Wittmann, E H., ed. (1995). Fracture Mechanics of Concrete Structures (Proc., 2nd Int. Conf. on Fracture Mech. of Concrete and Concrete Structures [FraMCoS-2]), held at ETH, Zurich), pp. 515-534, Freiburg: Aedificatio Publishers. 121. Wittmann, E H., and Zaitsev, Yu.V. (1981). Crack propagation and fracture of composite materials such as concrete, in Proc., 5th. Int. Conf. on Fracture (ICF5), Cannes. 122. Zaitsev, J. W., and Wittmann, E H. (1974). A statistical approach to the study of the mechanical behavior of porous materials under multiaxial state of stress, in Proc. of the 1973 Symp. on Mechanical Behavior on Materials, Kyoto, Japan. 123. Zech, B., and Wittmann, E H. (1977). A complex study on the reliability assessment of the containment of a PWR, Part II. Probabilistic approach to describe the behavior of materials. in Trans. 4th Int. Conf. on Structural Mechanics in Reactor Technology, pp. 1-14, vol. H, J1/11, T. A. Jaeger and B. A. Boley, eds., Brussels: European Communities.

CHAPTER

2

Elasticity and Viscoelasticity

This Page Intentionally Left Blank

SECTION

2.1

Introduction to Elasticity and ViscoelasticityJEAN LEMAITREUniversit~ Paris 6, LMT-Cachan, 61 avenue du Pr&ident Wilson, 94235 Cachan Cedex, France

For all solid materials there is a domain in stress space in which strains are reversible due to small relative movements of atoms. For many materials like metals, ceramics, concrete, wood and polymers, in a small range of strains, the hypotheses of isotropy and linearity are good enough for many engineering purposes. Then the classical Hooke's law of elasticity applies. It can be derived from a quadratic form of the state potential, depending on two parameters characteristics of each material: the Young's modulus E and the Poisson's ratio v. 1 ~k* -- 2---pAijkl(E'v)0"ij0"kl 0~t* l+v E v ~ crkka~j (1)

e~j - p 0a ~

ao

(2)

E and v are identified from tensile tests either in statics or dynamics. A great deal of accuracy is needed in the measurement of the longitudinal and transverse strains (6e ~ -+-10-6 in absolute value). When structural calculations are performed under the approximation of plane stress (thin sheets) or plane strain (thick sheets), it is convenient to write these conditions in the constitutive equation.9 Plane

stress

(033 --

0"13 -- 0"23 -- 0)"1 E v E 0

Igll 1 822 g12

Sym

1

0 l+v E

i lll~22

(3)

0"12

Handbook of Materials Behavior Models. ISBN0-12-443341-3.

Copyright 9 2001by AcademicPress.All rightsof reproductionin any formreserved.

71

729 Plane

Lemaitre

strain

(833 - - 813 - - 823 - - 0)"

0.220.12

-

i

Sym

2 + 2/~

2#

01i 1110822

(4)

812

vE 2 - - (1 + v ) ( 1 - 2v) with E -- 2(1 + v)

For orthotropic materials having three planes of symmetry, nine independent parameters are needed: three tension moduli El, E2, E3 in the orthotropic directions, three shear moduli G12, G23, G31, and three contraction ratios v12, v23, v31. In the frame of orthotropy:8111

v12E1 1

v13E1 23

_

0.11

0 001

0 00

0 00 0"22

E1 822

E2833--

E21

0"33(5)

E3 ] 823 2G23

01

0 0

0"23

Sym831 i

2G31 12G12

O"31

812

0"12 _

Nonlinear elasticity in large deformations is described in Section 2.2, with applications for porous materials in Section 2.3 and for elastomers in Section 2.4. Thermoelasticity takes into account the stresses and strains induced by thermal expansion with dilatation coefficient ~. For small variations of temperature 0 for which the elasticity parameters may be considered as constant:sO = l+v v E 0./j - ~ 0.khcS/j+ a06/j (6)

For large variations of temperature, E, v, and a will vary. In rate formulations, such as are needed in elastoviscoplasticity, for example, the

2.1 Introduction to Elasticity and Viscoelasticity

73

derivative of E, v, and c~ must be considered.

~v =

E

ev - g e ~ a v + ~0av + b-0

E

~v - F0

~ a v + N 0av 0

(7)Viscoelasticity considers in addition a dissipative phenomenon due to "internal friction," such as between molecules in polymers or between cells in wood. Here again, isotropy, linearity, and small strains allow for simple models. Quadratric functions for the state potential and the dissipative potential lead to either Kelvin-Voigt or Maxwell's models, depending upon the partition of stress or strains in a reversible part and in an irreversible part. They are described in detail for the one-dimensional case in Section 2.5 and recalled here in three dimensions. 9 Kelvin-Voigt model:

ffij = i~(~,kk -+- 02~'kk)(~ij _Jr_2/2(gij _+_Op~,ij )

(8)

Here 2 and/.z are Lame's coefficients at steady state, and 0x and 0~ are two time parameters responsible for viscosity. These four coefficients may be identified from creep tests in tension and shear. 9 Maxwell model:

9 1 + v (rij + giJ = E ~

-

-E

(rkk +

r2 /

aij

(9)

Here E and v are Young's modulus and Poisson's ratio at steady state, and rl and r2 are two other time parameters. It is a fluidlike model: equilibrium at constant stress does not exist. In fact, a more general way to write linear viscoelastic constitutive models is through the functional formulation by the convolution product as any linear system. The hereditary integral is described in detail for the one-dimensional case, together with its use by the Laplace transform, in Section 2.5.

'?'ij(t)--

fo'

Jijkl(t- "C) ~ dcrkl dr + ~-~Jijkl(t- "c)AO'Pk/p=l

(10)

[J(t)] is the creep functions matrix, and Ao-~l are the eventual stress steps. The dual formulation introduces the relaxation functions matrix JR(t)]

O'ij(t) --

/o t Rijkl(t-

"C) dC,kl dr, +p=l

RijklAgPkkl

(11)

When isotropy is considered the matrix, [J] and two functions:

[R] each reduce to

9 either J(t), the creep function in tension, is identified from a creep test at constant stress; J ( t ) = g ( t ) / r and K, the second function, from the

74creep function in shear. This leads to

Lemaitre

~,ij

-

-

( J + K) |

Dcrij _ K| (~ij Dz wz

(12)

where | stands for the convolution product and D for the distribution derivative, taking into account the stress steps. 9 or M(t), the relaxation function in shear, and L(t), a function deduced from M and from a relaxation test in tension R(t) = ~r(t)/~; L(t) = M ( R - 2 M ) / ( 3 M - R) ~0 - L |

D( ekk ) 6~j Dr

+ 2M |

~

D~3ij

dr

(13)

All of this is for linear behavior. A nonlinear model is described in Section 2.6, and interaction with damage is described in Section 2.7.

SECTION

2.2

Background on Nonlinear ElasticityR. W. OGDEN Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK

Contents 2.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Stress and Equilibrium . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Material Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Constrained Materials . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Boundary-Value Problems . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 76 77 78 80 81 82

2.2.1 VALIDITYThe theory of nonlinear elasticity is applicable to materials, such as rubberlike solids and certain soft biological tissues, which are capable of u n d e r g o i n g large elastic deformations. More details of the theory and its applications can be found in Beatty [1] and Ogden [3].

2.2.2 DEFORMATIONFor a continuous body, a reference configuration, denoted by ~r, is identified and 0 ~ r denotes the b o u n d a r y of ~ r . Points in ~ r are labeled by their position vectors X relative to some origin. The body is deformed quasistatically from ~ r SO that it occupies a new configuration, denoted ~ , withHandbook of Materials Behavior Models. ISBN 0o12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.

75

76

Ogden

boundary 0~. This is the current or deformed configuration of the body. The deformation is represented by the mapping Z::~r --~ ~ , SO that x = z(X) XC~r (1)

where x is the position vector of the point X in ~ . The mapping X is called the deformation from ~ r tO ~ , and Z is required to be one-to-one and to satisfy appropriate regularity conditions. For simplicity, we consider only Cartesian coordinate systems and let X and x, respectively, have coordinates X~ and x~, where ~, i C { 1, 2, 3}, so that xi--zi(X~). Greek and Roman indices refer, respectively, to ~ r and ~ , and the usual summation convention for repeated indices is used. The deformation gradient tensor, denoted E is given by F = Gradx

Fia = OXi/OXo~

(2)

Grad being the gradient operator in Nr. Local invertibility of Z and its inverse requires that 0 < J = det F < oo wherein the notation J is defined. The deformation gradient has the (unique) polar decompositions F = RU = VR (4) where R is a proper orthogonal tensor and U, V are positive definite and symmetric tensors. Respectively, U and V are called the right and left stretch tensors. They may be put in the spectral forms3 3

(3)

ui=1

| ul,I

v-

Zi=1

,vl,I | vl,I

where v (i) = Ru (i), i C {1,2,3}, 2i are the principal stretches, u (i) the unit eigenvectors of U (the Lagrangian principal axes), v (i) those of V (the Eulerian principal axes), and | denotes the tensor product. It follows from Eq. 3 that J =/~1,~2,~3 9 The right and left Cauchy-Green deformation tensors, denoted C and B, respectively, are defined by C = FTF = U 2 B = FF ~ = V 2 (6)

2.2.3Let

STRESS AND EQUILIBRIUM

Pr and p be the mass densities in Nr and N, respectively. The mass conservation equation has the form

Pr = pJ

(7)

2.2 Background on Nonlinear Elasticity

77

The Cauchy stress tensor, denoted g, and the nominal stress tensor, denoted S, are related by S = jF-lo The equation of equilibrium may be written in the equivalent forms div ~ + pb = 0 Div S + Prb = 0 (9) (8)

where div and Div denote the divergence operators in ~ and ~r, respectively, and b denotes the body force per unit mass. In components, the second equation in Eq. 9 is

OS~ic3X~

-t- IOrbi = 0

(10)

Balance of the moments of the forces acting on the body yields simply a t = ~, equivalently S TFT= FS. The Lagrangian formulation based on the use of S and Eq. 10, with X as the independent variable, is used henceforth.

2.2.4

ELASTICITY

The constitutive equation of an elastic material is given in the equivalent forms S - H(F) - - - ~

oqW

(F)

( ~ - G(F) --j-1FH(F)

(11)

where H is a tensor-valued function, defined on the space of deformation gradients E W is a scalar function of F and the symmetric tensor-valued function G is defined by the latter equation in Eq. 11. In general, the form of H depends on the choice of reference configuration and it is referred to as the response function of the material relative to Nr associated with S. For a given ~r, therefore, the stress in ~ at a (material) point X depends only on the deformation gradient at X. A material whose constitutive law has the form of Eq. 11 is generally referred to as a hyperelastic material and W is called a strain-energy function (or stored-energy function). In components, the first part of Eq. 11 has the form S~i = cgW/cgFi~, which provides the convention for ordering of the indices in the partial derivative with respect to E If W and the stress vanish in N'r, so that W(I) -- 0 OW OF (I) - O (12)

where I is the identity and O the zero tensor, then Nr is called a natural

configuration.

78

Ogden

Suppose that a rigid-body deformation x* = Qx + c is superimposed on the deformation x = z(X), where Q and c are constants, Q being a rotation tensor and c a translation vector. The resulting deformation gradient, F* say, is given by F* -- QF. The elastic stored energy is required to be independent of superimposed rigid deformations, and it follows that W(QF) = W(F) (13)

for all rotations Q. A strain-energy function satisfying this requirement is said to be objective. Use of the polar decomposition (Eq. 4) and the choice Q - - R ~ in Eq. 13 shows that W ( F ) = W(U). Thus, W depends on F only through the stretch tensor U and may therefore be defined on the class of positive definite symmetric tensors. We write

for the (symmetric) X = (SR + RTST)/2.

OW ou Biot stress tensor, whichT =

(14) is related to S by

2.2.5 MATER