KOICHI HASHIGUCHIYUKI YAMAKAWA

Introduction to FINITE STRAIN THEORY for CONTINUUM ELASTO-PLASTICITY

WILEY SERIES IN COMPUTATIONAL MECHANICS

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HASHIGUCHIYAMAKAWA

Introduction to FINITE STRA

IN TH

EORY

for CON

TINU

UM

ELASTO

-PLASTICITY

KOICHI HASHIGUCHI, Kyushu University, JapanYUKI YAMAKAWA, Tohoku University, Japan

Introduction to FINITE STRAIN THEORY for CONTINUUM ELASTO-PLASTICITY

Elasto-plastic deformation is frequently observed in machines and structures, hence its prediction is an important consideration at the design stage. Elasto-plasticity theories will be increasingly required in the future in response to the development of new and improved industrial technologies. Although various books for elasto-plasticity have been published to date, they focus on infi nitesimal elasto-plastic deformation theory. However, modern computational techniques employ an advanced approach to solve problems in this fi eld and much research has taken place in recent years into fi nite strain elasto-plasticity. This book describes this approach and aims to improve mechanical design techniques in mechanical, civil, structural and aeronautical engineering through the accurate analysis of fi nite elasto-plastic deformation.

Introduction to Finite Strain Theory for Continuum Elasto-Plasticity presents introductory explanations that can be easily understood by readers with only a basic knowledge of elasto-plasticity, showing physical backgrounds of concepts in detail and derivation processes of almost all equations. The authors address various analytical and numerical fi nite strain analyses, including new theories developed in recent years, and explain fundamentals including the push-forward and pull-back operations and the Lie derivatives of tensors.

Key features:• Comprehensively explains fi nite strain continuum mechanics and explains the fi nite

elasto-plastic constitutive equations• Discusses numerical issues on stress computation, implementing the numerical algorithms

into large-deformation fi nite element analysis • Includes numerical examples of boundary-value problems• Accompanied by a website (www.wiley.com/go/hashiguchi) hosting computer programs for the

return-mapping and the consistent tangent moduli of fi nite elasto-plastic constitutive equations

Introduction to Finite Strain Theory for Continuum Elasto-Plasticity is an ideal reference for research engineers and scientists working with computational solid mechanics and is a suitable graduate text for computational mechanics courses.

WILEY SERIES IN COMPUTATIONAL MECHANICS

WILEY SERIES IN COMPUTATIONAL

MECHANICS

INTRODUCTION TOFINITE STRAIN THEORYFOR CONTINUUMELASTO-PLASTICITY

WILEY SERIES IN COMPUTATIONAL MECHANICS

Series Advisors:

Rene de BorstPerumal NithiarasuTayfun E. TezduyarGenki YagawaTarek Zohdi

Introduction to Finite Strain Theory forContinuum Elasto-Plasticity

Hashiguchi and Yamakawa October 2012

Nonlinear Finite Element Analysis ofSolids and Structures: Second edition

De Borst, Crisfield, Remmersand Verhoosel

August 2012

An Introduction to MathematicalModeling: A Course in Mechanics

Oden November 2011

Computational Mechanics ofDiscontinua

Munjiza, Knight and Rougier November 2011

Introduction to Finite Element Analysis:Formulation, Verification and Validation

Szabo and Babuška March 2011

INTRODUCTION TOFINITE STRAIN THEORYFOR CONTINUUMELASTO-PLASTICITY

Koichi HashiguchiKyushu University, Japan

Yuki YamakawaTohoku University, Japan

A John Wiley & Sons, Ltd., Publication

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Library of Congress Cataloging-in-Publication Data

Hashiguchi, Koichi.Introduction to finite strain theory for continuum elasto-plasticity / Koichi Hashiguchi, Yuki Yamakawa.

p. cm.Includes bibliographical references and index.ISBN 978-1-119-95185-8 (cloth)

1. Elastoplasticity. 2. Strains and stresses. I. Yamakawa, Yuki. II. Title.TA418.14.H37 2013620.1′1233–dc23

2012011797

A catalogue record for this book is available from the British Library.

Print ISBN: 9781119951858

Typeset in 10/12pt Times by Aptara Inc., New Delhi, India

Contents

Preface xi

Series Preface xv

Introduction xvii

1 Mathematical Preliminaries 11.1 Basic Symbols and Conventions 11.2 Definition of Tensor 2

1.2.1 Objective Tensor 21.2.2 Quotient Law 4

1.3 Vector Analysis 51.3.1 Scalar Product 51.3.2 Vector Product 61.3.3 Scalar Triple Product 61.3.4 Vector Triple Product 71.3.5 Reciprocal Vectors 81.3.6 Tensor Product 9

1.4 Tensor Analysis 91.4.1 Properties of Second-Order Tensor 91.4.2 Tensor Components 101.4.3 Transposed Tensor 111.4.4 Inverse Tensor 121.4.5 Orthogonal Tensor 121.4.6 Tensor Decompositions 151.4.7 Axial Vector 171.4.8 Determinant 201.4.9 On Solutions of Simultaneous Equation 231.4.10 Scalar Triple Products with Invariants 241.4.11 Orthogonal Transformation of Scalar Triple Product 251.4.12 Pseudo Scalar, Vector and Tensor 26

1.5 Tensor Representations 271.5.1 Tensor Notations 271.5.2 Tensor Components and Transformation Rule 271.5.3 Notations of Tensor Operations 28

vi Contents

1.5.4 Operational Tensors 291.5.5 Isotropic Tensors 31

1.6 Eigenvalues and Eigenvectors 361.6.1 Eigenvalues and Eigenvectors of Second-Order Tensors 361.6.2 Spectral Representation and Elementary Tensor Functions 401.6.3 Calculation of Eigenvalues and Eigenvectors 421.6.4 Eigenvalues and Vectors of Orthogonal Tensor 451.6.5 Eigenvalues and Vectors of Skew-Symmetric Tensor and Axial

Vector 461.6.6 Cayley–Hamilton Theorem 47

1.7 Polar Decomposition 471.8 Isotropy 49

1.8.1 Isotropic Material 491.8.2 Representation Theorem of Isotropic Tensor-Valued Tensor Function 50

1.9 Differential Formulae 541.9.1 Partial Derivatives 541.9.2 Directional Derivatives 591.9.3 Taylor Expansion 621.9.4 Time Derivatives in Lagrangian and Eulerian Descriptions 631.9.5 Derivatives of Tensor Field 681.9.6 Gauss’s Divergence Theorem 711.9.7 Material-Time Derivative of Volume Integration 73

1.10 Variations and Rates of Geometrical Elements 741.10.1 Variations of Line, Surface and Volume 751.10.2 Rates of Changes of Surface and Volume 76

1.11 Continuity and Smoothness Conditions 791.11.1 Continuity Condition 791.11.2 Smoothness Condition 80

1.12 Unconventional Elasto-Plasticity Models 81

2 General (Curvilinear) Coordinate System 852.1 Primary and Reciprocal Base Vectors 852.2 Metric Tensors 892.3 Representations of Vectors and Tensors 952.4 Physical Components of Vectors and Tensors 1022.5 Covariant Derivative of Base Vectors with Christoffel Symbol 1032.6 Covariant Derivatives of Scalars, Vectors and Tensors 1072.7 Riemann–Christoffel Curvature Tensor 1122.8 Relations of Convected and Cartesian Coordinate Descriptions 115

3 Description of Physical Quantities in Convected Coordinate System 1173.1 Necessity for Description in Embedded Coordinate System 1173.2 Embedded Base Vectors 1183.3 Deformation Gradient Tensor 1213.4 Pull-Back and Push-Forward Operations 123

Contents vii

4 Strain and Strain Rate Tensors 1314.1 Deformation Tensors 1314.2 Strain Tensors 136

4.2.1 Green and Almansi Strain Tensors 1364.2.2 General Strain Tensors 1414.2.3 Hencky Strain Tensor 144

4.3 Compatibility Condition 1454.4 Strain Rate and Spin Tensors 146

4.4.1 Strain Rate and Spin Tensors Based on VelocityGradient Tensor 147

4.4.2 Strain Rate Tensor Based on General Strain Tensor 1524.5 Representations of Strain Rate and Spin Tensors in Lagrangian and

Eulerian Triads 1534.6 Decomposition of Deformation Gradient Tensor into Isochoric and

Volumetric Parts 158

5 Convected Derivative 1615.1 Convected Derivative 1615.2 Corotational Rate 1655.3 Objectivity 166

6 Conservation Laws and Stress (Rate) Tensors 1796.1 Conservation Laws 179

6.1.1 Basic Conservation Law 1796.1.2 Conservation Law of Mass 1806.1.3 Conservation Law of Linear Momentum 1816.1.4 Conservation Law of Angular Momentum 182

6.2 Stress Tensors 1836.2.1 Cauchy Stress Tensor 1836.2.2 Symmetry of Cauchy Stress Tensor 1876.2.3 Various Stress Tensors 188

6.3 Equilibrium Equation 1946.4 Equilibrium Equation of Angular Moment 1976.5 Conservation Law of Energy 1976.6 Virtual Work Principle 1996.7 Work Conjugacy 2006.8 Stress Rate Tensors 203

6.8.1 Contravariant Convected Derivatives 2036.8.2 Covariant–Contravariant Convected Derivatives 2046.8.3 Covariant Convected Derivatives 2046.8.4 Corotational Convected Derivatives 204

6.9 Some Basic Loading Behavior 2076.9.1 Uniaxial Loading Followed by Rotation 2076.9.2 Simple Shear 2156.9.3 Combined Loading of Tension and Distortion 220

viii Contents

7 Hyperelasticity 2257.1 Hyperelastic Constitutive Equation and Its Rate Form 2257.2 Examples of Hyperelastic Constitutive Equations 230

7.2.1 St. Venant–Kirchhoff Elasticity 2307.2.2 Modified St. Venant–Kirchhoff Elasticity 2317.2.3 Neo-Hookean Elasticity 2327.2.4 Modified Neo-Hookean Elasticity (1) 2337.2.5 Modified Neo-Hookean Elasticity (2) 2347.2.6 Modified Neo-Hookean Elasticity (3) 2347.2.7 Modified Neo-Hookean Elasticity (4) 234

8 Finite Elasto-Plastic Constitutive Equation 2378.1 Basic Structures of Finite Elasto-Plasticity 2388.2 Multiplicative Decomposition 2388.3 Stress and Deformation Tensors for Multiplicative Decomposition 2438.4 Incorporation of Nonlinear Kinematic Hardening 244

8.4.1 Rheological Model for Nonlinear Kinematic Hardening 2458.4.2 Multiplicative Decomposition of Plastic Deformation Gradient

Tensor 2468.5 Strain Tensors 2498.6 Strain Rate and Spin Tensors 252

8.6.1 Strain Rate and Spin Tensors in Current Configuration 2528.6.2 Contravariant–Covariant Pulled-Back Strain Rate and Spin Tensors

in Intermediate Configuration 2548.6.3 Covariant Pulled-Back Strain Rate and Spin Tensors in

Intermediate Configuration 2568.6.4 Strain Rate Tensors for Kinematic Hardening 259

8.7 Stress and Kinematic Hardening Variable Tensors 2618.8 Influences of Superposed Rotations: Objectivity 2668.9 Hyperelastic Equations for Elastic Deformation and Kinematic Hardening 268

8.9.1 Hyperelastic Constitutive Equation 2688.9.2 Hyperelastic Type Constitutive Equation for Kinematic Hardening 269

8.10 Plastic Constitutive Equations 2708.10.1 Normal-Yield and Subloading Surfaces 2718.10.2 Consistency Condition 2728.10.3 Plastic and Kinematic Hardening Flow Rules 2758.10.4 Plastic Strain Rate 277

8.11 Relation between Stress Rate and Strain Rate 2788.11.1 Description in Intermediate Configuration 2788.11.2 Description in Reference Configuration 2788.11.3 Description in Current Configuration 279

8.12 Material Functions of Metals 2808.12.1 Strain Energy Function of Elastic Deformation 2808.12.2 Strain Energy Function for Kinematic Hardening 2818.12.3 Yield Function 2828.12.4 Plastic Strain Rate and Kinematic Hardening Strain Rate 283

Contents ix

8.13 On the Finite Elasto-Plastic Model in the Current Configuration by theSpectral Representation 284

8.14 On the Clausius–Duhem Inequality and the Principle of Maximum Dissipation 285

9 Computational Methods for Finite Strain Elasto-Plasticity 2879.1 A Brief Review of Numerical Methods for Finite Strain Elasto-Plasticity 2889.2 Brief Summary of Model Formulation 289

9.2.1 Constitutive Equations for Elastic Deformation and Isotropic andKinematic Hardening 289

9.2.2 Normal-Yield and Subloading Functions 2919.2.3 Plastic Evolution Rules 2919.2.4 Evolution Rule of Normal-Yield Ratio for Subloading Surface 293

9.3 Transformation to Description in Reference Configuration 2939.3.1 Constitutive Equations for Elastic Deformation and Isotropic and

Kinematic Hardening 2939.3.2 Normal-Yield and Subloading Functions 2949.3.3 Plastic Evolution Rules 2959.3.4 Evolution Rule of Normal-Yield Ratio for Subloading Surface 296

9.4 Time-Integration of Plastic Evolution Rules 2969.5 Update of Deformation Gradient Tensor 3009.6 Elastic Predictor Step and Loading Criterion 3019.7 Plastic Corrector Step by Return-Mapping 3049.8 Derivation of Jacobian Matrix for Return-Mapping 308

9.8.1 Components of Jacobian Matrix 3089.8.2 Derivatives of Tensor Exponentials 3109.8.3 Derivatives of Stresses 312

9.9 Consistent (Algorithmic) Tangent Modulus Tensor 3129.9.1 Analytical Derivation of Consistent Tangent Modulus Tensor 3139.9.2 Numerical Computation of Consistent Tangent Modulus Tensor 315

9.10 Numerical Examples 3169.10.1 Example 1: Strain-Controlled Cyclic Simple Shear Analysis 3189.10.2 Example 2: Elastic–Plastic Transition 3189.10.3 Example 3: Large Monotonic Simple Shear Analysis with

Kinematic Hardening Model 3209.10.4 Example 4: Accuracy and Convergence Assessment of

Stress-Update Algorithm 3229.10.5 Example 5: Finite Element Simulation of Large Deflection

of Cantilever 3269.10.6 Example 6: Finite Element Simulation of Combined Tensile,

Compressive, and Shear Deformation for Cubic Specimen 330

10 Computer Programs 33710.1 User Instructions and Input File Description 33710.2 Output File Description 340

x Contents

10.3 Computer Programs 34110.3.1 Structure of Fortran Program returnmap 34110.3.2 Main Routine of Program returnmap 34310.3.3 Subroutine to Define Common Variables: comvar 34310.3.4 Subroutine for Return-Mapping: retmap 34510.3.5 Subroutine for Isotropic Hardening Rule: plhiso 37710.3.6 Subroutine for Numerical Computation of Consistent Tangent

Modulus Tensor: tgnum0 377

A Projection of Area 385

B Geometrical Interpretation of Strain Rate and Spin Tensors 387

C Proof for Derivative of Second Invariant of Logarithmic-DeviatoricDeformation Tensor 391

D Numerical Computation of Tensor Exponential Function andIts Derivative 393

D.1 Numerical Computation of Tensor Exponential Function 393D.2 Fortran Subroutine for Tensor Exponential Function: matexp 394D.3 Numerical Computation of Derivative of Tensor Exponential Function 396D.4 Fortran Subroutine for Derivative of Tensor Exponential Function: matdex 400

References 401

Index 409

Preface

The first author of this book recently published the book Elastoplasticity Theory (2009) whichaddresses the fundamentals of elasto-plasticity and various plasticity models. It is mainlyconcerned with the elasto-plastic deformation theory within the framework of the hypoelastic-based plastic constitutive equation. It has been widely adopted and has contributed to theprediction of the elasto-plastic deformation behavior of engineering materials and structurescomposed of solids such as metals, geomaterials and concretes. However, the hypoelastic-based plastic constitutive equation is premised on the additive decomposition of the strain rate(symmetric part of velocity gradient) into the elastic and the plastic strain rates and the linearrelation between the elastic strain rate and stress rate. There is no one-to-one correspondencebetween the time-integrations of the elastic strain rate and stress rare and the energy may beproduced or dissipated during a loading cycle in hypoelastic equation. Therefore, the elasticstrain rate does not possess the elastic property in the strict sense so that an error may beinduced in large-deformation analysis and accumulated in the cyclic loading analysis. Anexact formulation without these defects in the infinitesimal elasto-plasticity theory has beensought in order to respond to recent developments in engineering in the relevant fields, suchas mechanical, aeronautic, civil, and architectural engineering.

There has been a great deal of work during the last half century on the finite strain elasto-plasticity theory enabling exact deformation analysis up to large deformation, as represented bythe epoch-making works of Oldroyd (1950), Kroner (1959), Lee (1969), Kratochvil (1971),Mandel (1972b), Hill (1978), Dafalias (1985), and Simo (1998). In this body of work themultiplicative decomposition which is the decomposition of the deformation gradient into theelastic and plastic parts, introducing the intermediate configuration obtained by unloading tothe stress free-state, was proposed and the elastic part is formulated as a hyperelastic relationbased on the elastic strain energy function. Further, the Mandel stress, the work-conjugateplastic velocity gradient with the Mandel stress, the plastic spin, and various physical quantitiesdefined in the intermediate configuration have been introduced. The physical and mathematicalfoundations for the exact finite elasto-plasticity theory were established by 2000 and theconstitutive equation based on these foundations – the hyperelastic-based plastic constitutiveequation – has been formulated after 2005. However, a textbook on the hyperelastic-basedfinite elasto-plasticity theory has not been published to date.

Against this backdrop, the aim of this book is to give a comprehensive explanation of thefinite elasto-plasticity theory. First, the classification of elastoplasticity theories from the view-point of the relevant range of deformation will be given and the prominence of the hyperelastic-based finite elastoplasticity theory will be explained in Introduction. Exact knowledge of the

xii Preface

basic mechanical ingredients – finite strain (rate) tensors, the Lagrangian and Eulerian tensors,the objectivity of the tensor and the systematic definitions of pull-back and push-forwardoperations, the Lie derivative and the corotational rate – is required in the formulation of finitestrain theory. To this end, descriptions of the physical quantities and relations in the embedded(convected) coordinate system, which turns into the curvilinear coordinate system under thedeformation of material, are required, since their physical meanings can be captured clearlyby observing them in a coordinate system which not only moves but also deforms and rotateswith material itself. In other words, the essentials of continuum mechanics cannot be capturedwithout the incorporation of the general curvilinear coordinate system, although numerousbooks with ‘continuum mechanics’ in their title and confined to the rectangular coordinatesystem have been published to date. On the other hand, knowledge of the curvilinear coordinatesystem is not required for users of finite strain theory. It is sufficient for finite element analyzersusing finite strain theory to master the descriptions in the rectangular coordinate system andthe relations between Lagrangian and Eulerian tensors through the deformation gradient, forinstance. This book, which aims to impart the exact finite strain theory, gives a comprehensiveexplanation of the mathematical and physical fundamentals required for continuum solidmechanics, and provides a description in the general coordinate system before moving on toan explanation of finite strain theory.

In addition to the above-mentioned issues on the formulation of the constitutive equation, theformulation and implementation of a numerical algorithm for the state-updating calculation areof utmost importance. The state-updating procedure in the computational analysis of elasto-plasticity problems usually requires a proper algorithm for numerical integration of the rateforms of the constitutive laws and the evolution equations. The return-mapping scheme hasbeen developed to a degree of common acceptance in the field of computational plasticityas an effective state-updating procedure for elasto-plastic models. In the numerical analysisof boundary-value problems, a consistent linearization of the weak form of the equilibriumequation and use of the so-called consistent (algorithmic) elasto-plastic tangent modulustensor are necessary to ensure effectiveness and robustness of the iterative solution procedure.A Fortran program for the return-mapping and the consistent tangent modulus tensor whichcan readily be implemented in finite element codes is provided along with detailed explanationand user instructions so that readers will be able to carry out deformation analysis using thefinite elasto-plasticity theory by themselves.

Chapters 1–7 were written by the first author, Chapter 8 by both authors, and Chapters 9and 10 by the second author in close collaboration, and the computer programming andcalculations were performed by the second author based on the theory formulated by bothauthors in Chapter 8. The authors hope that readers of this book will capture the fundamentalsof the finite elasto-plasticity theory and will contribute to the development of mechanicaldesigns of machinery and structures in the field of engineering practice by applying thetheories addressed in this book. A reader is apt to give up reading a book if he encountersmatter which is difficult to understand. For this reason, explanations of physical concepts inelasto-plasticity are given, and formulations and derivations/transformations for all equationsare given without abbreviation for Chapters 1–8 for the basic formulations of finite straintheory. This is not a complete book on the finite elasto-plasticity theory, but the authors will bequite satisfied if it provides a foundation for further development of the theory by stimulatingthe curiosity of young researchers and it is applied widely to the analyses of engineering

Preface xiii

problems in practice. In addition, the authors hope that it will be followed by a variety ofbooks on the finite elasto-plasticity.

The first author is deeply indebted to Professor B. Raniecki of the Institute of FundamentalTechnological Research, Poland, for valuable suggestions and comments on solid mechanics,who has visited several times Kyushu University. His lecture notes on solid mechanics andregular private communications and advice have been valuable in writing some parts of thisbook. He wishes also to express his gratitude to Professor O.T. Bruhns of the Ruhr University,Bochum, Germany, and Professor H. Petryk of the Institute of Fundamental Technological Re-search, Poland, for valuable comments and for notes on their lectures on continuum mechanicsdelivered at Kyushu University. The second author would like to express sincere gratitude toProfessor Kiyohiro Ikeda of Tohoku University for valuable suggestions and comments onnonlinear mechanics. He is also grateful to Professor Kenjiro Terada of Tohoku Universityfor providing enlightening advices on numerical methods for finite strain elasto-plasticity. Healso thanks Dr. Ikumu Watanabe of National Institute for Materials Science, Japan, for helpfuladvices on numerical methods for finite strain elasto-plasticity. The enthusiastic support ofDr. Keisuke Sato of Terrabyte lnc., Japan, and Shoya Nakaichi, Toshimitsu Fujisawa, YosukeYamaguchi and Yutaka Chida at Tohoku University in development and implementation ofthe numerical code is most appreciated. The authors thank Professor S. Reese of RWTHAachen University, Professor J. Ihlemann and Professor A.V. Shutov of Chemnitz University,Germany, and Professor M. Wallin of Lund University, Sweden for valuable suggestions andfor imparting relevant articles on finite strain theory to the authors.

Koichi HashiguchiYuki Yamakawa

February 2012

Series Preface

The series on Computational Mechanics is a conveniently identifiable set of books coveringinterrelated subjects that have been receiving much attention in recent years and need to have aplace in senior undergraduate and graduate school curricula, and in engineering practice. Thesubjects will cover applications and methods categories. They will range from biomechanicsto fluid-structure interactions to multiscale mechanics and from computational geometry tomeshfree techniques to parallel and iterative computing methods. Application areas will beacross the board in a wide range of industries, including civil, mechanical, aerospace, au-tomotive, environmental and biomedical engineering. Practicing engineers, researchers andsoftware developers at universities, industry and government laboratories, and graduate stu-dents will find this book series to be an indispensible source for new engineering approaches,interdisciplinary research, and a comprehensive learning experience in computationalmechanics.

Over the last three decades, the finite strain theory for elasto-plasticity has been extensivelydeveloped to provide very precise descriptions of large elasto-plastic deformations. Thesetheoretical developments have been augmented with robust computational methods, whichprovide accurate solutions to the corresponding boundary-value problems. Deep mathematicaland physical knowledge of related continuum mechanics is required to learn the theory, whichis difficult for students, engineers and even researchers in the field of applied mechanics, tocapture in depth.

This book is one of the first introductory books to address finite strain elasto-plasticitytheory, where the mathematical and physical foundations are comprehensively described.In particular, the representation of physical quantities in convected (curvilinear) coordinatesystems is employed, which is required for the substantial interpretation of basic concepts,such as pull-back and push-forward operations and convected (Lie) derivatives, that is, thegeneral objective rate in continuum mechanics. Furthermore, all the mathematical derivationsand transformations of equations are shown without any abbreviation, explaining the numericalmethod for the finite elasto-plasticity in detail.

In addition, a Fortran program for the stress-update algorithm based on the return-mappingscheme and the consistent (algorithmic) tangent modulus tensor, which can readily be imple-mented in finite element codes, is appended with a detailed explanation and user instructions,

xvi Series Preface

so that the readers will be able to implement the numerical analysis on the basis of the finiteelasto-plasticity theory without difficulty.

It is our hope that the readers of this book will contribute to the improvement of mechanicaldesign of machinery and structures in the field of engineering by adopting and widely usingthe basic ideas of the finite elasto-plasticity theory and the corresponding numerical methodsintroduced in this text.

Introduction

Prominence of the finite strain elasto-plasticity theory

This book addresses the finite strain elasto-plasticity theory, abbreviated as the finite elasto-plasticity. Then, the prominence of finite strain elasto-plasticity theory and the necessity ofits incorporation to deformation analysis is first reviewed by comparing with the infinitesimalelasto-plasticity theory, abbreviated as the infinitesimal elastoplasticity, prior to the detailedexplanation of the finite elastoplasticity in the subsequent chapters.

Elastoplasticity theory is classified from the view point of relevant range of deformation asfollows:

A) Infinitesimal elastoplasticityThe infinitesimal strain tensor defined by the symmetric part of displacement gradienttensor is additively decomposed into an elastic and a plastic parts, while there does notexist the distinction between the reference and the current configurations in which theinfinitesimal strain tensor is based. The spin of material is described by the skew-symmetricpart of the rate of displacement gradient tensor. This theory is limited to the descriptionof infinitesimal deformation and rotation. Meanwhile, for the elastic part, the hyperelasticconstitutive equation with a stored energy function can be formulated, which provides theone-to-one correspondence between the stress tensor and the infinitesimal elastic straintensor. Therefore, the return-mapping and the consistent (algorithmic) tangent modulustensor can be employed in numerical calculations under infinitesimal deformation androtation.

B) Finite elastoplasticityThe frameworks of the theories describing the finite elastoplastic deformation and rotationare classified further as follows:

B-1) Hypoelastic-based plasticityIt is premised on the following assumptions, which would be called the Hill–Riceapproach.

i) Deformation and rotation are described by the strain rate and the spin tensors,that is, the symmetric and the skew-symmetric parts, respectively, of velocitygradient tensor.

ii) The strain rate and the spin tensors are additively decomposed into elastic andplastic parts. Here, it should be noticed that the additive decomposition is de-rived from the multiplicative decomposition of the deformation gradient tensor

xviii Introduction

only when an elastic deformation is infinitesimal, whereas the multiplicativedecomposition is the rigorous approach for the exact partition of the deformationinto the elastic and the plastic parts.

iii) The elastic part of the strain rate tensor, that is, elastic strain rate tensor is relatedlinearly to an appropriate corotational rate of the Cauchy stress tensor. It fallswithin the framework of the so-called hypoelasticity (Truesdell, 1955).

iv) The elastic part of the spin tensor, that is, elastic spin tensor, which is given bythe subtraction of the plastic part of the spin tensor, that is, plastic spin tensor(Dafalias, 1985) from the continuum spin tensor, is regarded as the spin of sub-structure, that is, the substructure (corotational) spin tensor. Then, it is adoptedin corotational rates of stress tensor and tensor-valued internal variables. Theplastic spin tensor is formulated as the skew-symmetric part of the multiplicationof the stress tensor by the plastic strain rate tensor (Zbib and Aifantis, 1988).

v) All variables adopted in the constitutive relation are Eulrian tensors based in thecurrent configuration.

Constitutive behavior under finite deformation and rotation is properly describedby incorporating corotational rate tensors based on an appropriate substructure spintensor under the limitation of infinitesimal elastic deformation

However, the hypoelastic-based plasticity would possess following problems:

(1) Hypoelastic equation does not fulfill the complete integrability condition. There-fore, one-to-one correspondence between integrations of stress rate tensor andelastic strain rate tensor, that is, stress vs. strain relation cannot be obtained ingeneral. Then, the hypoelastic deformation remains even after a closed stresscycle. In addition, work done during a closed loading cycle may not be zero asan energy is produced or dissipated during a closed loading cycle even when aplastic deformation is not induced. Therefore, the elastic strain rate tensor doesnot possess the reversibility in the strict sense, so that the hypoelastic constitutiveequation is merely an elastic-like constitutive equation. This approach possessesthe pertinence on the premise that the elastic deformation is infinitesimal.

(2) Stress cannot be calculated directly from strain variable but it has to be calculatedby the time-integration of stress rate. Also, tensor-valued internal variables cannotbe calculated directly from the elastic strain-like variable but they have to becalculated by the time-integration of their rates. Here, note that rates of stressand tensor-valued internal variables in the current configuration are influencedby the rigid-body rotation of material. Consequently, their pertinent objectivecorotational rate tensors must be adopted and their pertinent time-integrationprocedures must be incorporated in order that unrealistic and/or impertinentcalculation such as oscillatory stress and tensor-valued internal variables arenot resulted. Proper time-integrations would be given by the time-integrationreflecting the convected derivative process (Simo and Hughes, 1998).

(3) The return-mapping and the consistent (algorithmic) tangent modulus tensor,which enable drastically accurate and efficient numerical calculations, cannotbe employed since they are premised on the exact evaluation of stress by thehyperelastic constitutive equation, while the plastic strain increment in the plas-tic corrector step is calculated based on the overstress from the yield surface.

Introduction xix

Nevertheless, the hypoelasticity with constant elastic moduli leads to the one-to-one correspondence between the time-integrations of objective stress rate andelastic strain rate tensors by performing proper time-integrations reflecting theconvected derivative process, so that the return-mapping and the consistent tan-gent modulus tensor can be employed under the finite deformation and rotation.On the other hand, it should be noticed that the automatic controlling function toattract the stress to the yield surface in a plastic deformation process is furnishedin the subloading surface model (Hashiguchi, 1989; Hashiguchi et al. 2012) aswill be described in Chapter 8, so that numerical analyses of materials, in whichelastic parameters in a hypoelastic equation are not constant, can be enforceddrastically by this function in the Euler forward-integration method.

B-2) Hyperelastic-based plasticityThe hyperelastic-based plasticity has been formulated to overcome the above-mentioned problems in the hypoelastic-based plasticity by incorporating the fol-lowing notions.

1. Deformation itself (not rate) is decomposed into the elastic and the plastic partsto provide the one-to-one correspondence between elastic deformation and stress,although the strain rate is decomposed into them in the hypoelastic-based plas-ticity. It has been materialized by the decomposition of the deformation gradienttensor into the elastic and the plastic parts in the multiplicative form, that is, themultiplicative decomposition by Kroner (1959), Lee and Liu (1967), Lee (1969),etc., while the elasto-plasticity based on this decomposition would be called theMandel–Lee approach. Therein, the intermediate configuration is incorporated,which is attained by unloading to the stress free state along the hyperelastic defor-mation. Then, the deformation gradient is multiplicatively decomposed into theplastic deformation gradient tensor induced in the process from the initial to theintermediate configuration and the elastic deformation gradient tensor induced inthe process from the intermediate to the current configuration.

Formulations by principal values based on the Hencky strain, that is, principallogarithmic strain have been studied by a lot of workers (cf. Simo and Meschke,1993; Borja and Tamagnini, 1998; Tamagnini et al., 2002; Borja et al., 2001;Rosati and Valoroso, 2004, Raniecki and Nguyen, 2005; Yamakawa et al., 2010).However, they have been limited to the description of isotropic materials in whichthe principal directions of stress, elastic strain and plastic strain rate coincide witheach other leading to the co-axiality.

Then, the formulation based on the following notions has been developed, bywhich general elastoplastic constitutive behavior can be described accurately overthe finite deformation and rotation.

2. Both of the elastic and the plastic deformation gradient tensors are the two-point tensors, the one of the two base vectors of which lives in the intermediateconfiguration which is independent of the superposition of rigid-body rotation.Therefore, constitutive relation is formulated originally by tensor variables in theintermediate configuration.

3. Elastic deformation is described by an elastic strain tensor based on the elas-tic deformation gradient tensor and it is related to the stress tensor by the

xx Introduction

hyperelastic constitutive equation possessing the elastic strain energy function.Then, the one-to-one correspondence between stress tensor and elastic strain ten-sor holds and the work done during a closed stress cycle is zero exactly when theplastic strain rate is not induced.

4. In addition, the plastic deformation gradient tensor is further decomposed into theenergy-storage part causing the variation of substructure and the energy-dissipativepart causing the slip between substructures (Lion, 2000). Then, the kinematichardening variable, that is, the back stress is also formulated in relation to theelastic strain-like variable induced by the energy-storage part as a hyperelastic-like equation possessing a potential energy function of the variable.

5. Stress and back stress are formulated in the intermediate configuration and cal-culated directly from the elastic strain and the elastic strain-like variable of thekinematic hardening without performing the time-integrations of stress rate andback stress rate. Therefore, material rotation is independent of a rigid-body ro-tation and it would not be influenced severely be a plastic deformation on thecalculation of stress and back stress.

6. Plastic flow rule is given by the relationship of the plastic velocity gradient tensor tothe Mandel stress tensor (Mandel, 1972b) in the intermediate configuration, wherethese tensors fulfill the plastic work-conjugacy. Concurrently, the yield surface isdescribed by the Mandel stress tensor which is obtained by the pull-back of theKirchhoff stress tensor or the push-forward of the second Piola–Kirchhoff stresstensor.

7. In addition to the above-mentioned issues on the formulation of constitutiveequation, the formulation and implementation of a numerical algorithm for thestate-updating calculation are of utmost importance. The state-updating procedurein the computation of elastoplasticity problems usually requires a proper algorithmfor numerical integration of the rate forms of the constitutive laws and the evolutionequations. Responding to the formulation of hyperelastic-based plastic constitutiveequation, the return-mapping scheme has achieved a degree of common acceptancein the field of computational plasticity as an effective state-updating procedurefor elastoplastic models. In the numerical analysis of boundary-value problems, aconsistent linearization of the weak form of equilibrium equation and a use of theso-called consistent elasto-plastic tangent modulus tensor are necessary to ensureeffectiveness and robustness of the iterative solution procedure.

The hyperelastic-based plasticity may be called the finite strain elastoplasticity.On the other hand, the hypoelastic-based plasticity should be called merely the finitedeformation elastoplasticity, since it does not use the elastic strain itself but it isbased on the velocity gradient.

The hyperelastic-based plasticity will be explained exhaustively in this book. A cer-tain amount of advanced mathematical knowledge is required to capture the essentialsof continuum mechanics and to formulate constitutive equations in the framework ofthe finite strain theory. In order to capture the meanings of physical quantities andrelations exactly, it is indispensable to describe them in the embedded coordinatesystem which not only moves but also deforms and rotates with material itself, whichbelongs to the general curvilinear coordinate system. Then, the incorporation of the

Introduction xxi

curvilinear coordinate system is one of the distinctions of the mathematical method-ology in the finite strain theory from that in the infinitesimal strain theory. The conciseexplanation of vector and tensor analysis in the curvilinear coordinate system will begiven in chapter 2 in addition to the analysis in the Cartesian coordinate system inchapter 1 as the preliminary to the study of finite strain theory.

1Mathematical Preliminaries

Advanced mathematical knowledge is required to learn the finite elasto-plasticity theory. Thebasics of vector and tensor analyses are described in preparation for the explanations of ad-vanced theory in later chapters. Various representations of tensors, for example, the eigenvaluesand principal directions, the Cayley–Hamilton theorem, the polar and spectral decompositions,the isotropic function and various differential equations, the time-derivatives, and the integra-tion theorems are described. Component descriptions of vectors and tensors in this chapterare limited to the normalized rectangular coordinate system, that is, the rectangular coordi-nate system with unit base vectors, while the terms orthogonal, orthonormal and Cartesianare often used instead of rectangular. However, the derived tensor relations hold even in thegeneral curvilinear coordinate system of the Euclidian space described in later chapters.

1.1 Basic Symbols and Conventions

An index appearing twice in a term is summed over the specified range of the index. Forinstance, we may write

urvr =n∑

r=1

urvr, Tirvr =n∑

r=1

Tirvr, Trr =n∑

r=1

Trr, (1.1)

where the range of index is taken to be 1, 2, . . . , n. The indices used repeatedly are arbitrary,and thus they are called dummy indices: note that urvr = usvs and Tirvr = Tisvs, for example.This is termed Einstein’s summation convention. Henceforth, repeated indices refer to thisconvention unless specified by the additional remark ‘(no sum)’.

The symbol δi j(i, j = 1, 2, 3) defined in the following equation is termed the Kroneckerdelta:

δi j = 1 for i = j, δi j = 0 for i �= j (1.2)

from which it follows that

δirδr j = δi j, δii = 3. (1.3)

Introduction to Finite Strain Theory for Continuum Elasto-Plasticity, First Edition. Koichi Hashiguchi and Yuki Yamakawa.C© 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd.

2 Introduction to Finite Strain Theory for Continuum Elasto-Plasticity

Furthermore, the symbol εi jk defined by the following equation is called the alternating (orpermutation) symbol or Eddington’s epsilon or Levi-Civita ‘e’ tensor:

εijk =⎧⎨

⎩

1 for cyclic permutation of ijk from 123,

−1 for anticyclic permutation of ijk from 123,

0 for others.(1.4)

1.2 Definition of Tensor

In this section the definition of an objective tensor is given and, based on it, the criteria for agiven physical quantity to be a tensor and its order are provided.

1.2.1 Objective Tensor

Let the set of nm functions be described as T (p1, p2, . . . , pm) in the coordinate system{O, xi} with the origin O and the axes xi in n-dimensional space, where each of the in-dices p1, p2, . . . , pm takes a numberical value 1, 2, . . . , n. This set of functions is defined asthe mth-order tensor in n-dimensional space, if the set of functions is observed in the othercoordinate system {O, x∗

i } with the origin O and the axes x∗i as follows:

T ∗(p1, p2, . . . , pm) = Qp1q1 Qp2q2 · · · Qpmqm T (q1, q2, . . . , qm) (1.5)

or

T ∗(p1, p2, . . . , pm) = ∂x∗p1

∂xq1

∂x∗p2

∂xq2

· · · ∂x∗pm

∂xqmT (q1, q2, . . . , qm), (1.6)

provided that only the directions of axes are different but the origin is common and relativemotion does not exist. Here, Qi j in equation (1.5) is defined by

Qi j = ∂x∗i

∂x j(1.7)

which fulfills

QirQjr = δi j (1.8)

because of

QirQjr = ∂x∗i

∂xr

∂xr

∂x∗j

.

Denoting T (p1, p2, . . . , pm) by the symbol Tp1 p2...pm , equation (1.6) is expressed as

T ∗p1 p2...pm

= Qp1q1 Qp2q2 · · · Qpmqm Tq1q2...qm (1.9)

that is,

T ∗p1 p2...pm

= ∂x∗p1

∂xq1

∂x∗p2

∂xq2

· · · ∂x∗pm

∂xqmTq1q2...qm . (1.10)

Mathematical Preliminaries 3

Noting that

Qp1r1 Qp2r2 · · · Qpmrm T ∗p1 p2...pm

= Qp1r1 Qp2r2 · · · QpmrmQp1q1 Qp2q2 · · · Qpmqm Tq1q2...qm

= (Qp1r1 Qp1q1 )(Qp2r2 Qp2q2 ) · · · (Qpmrm Qpmqm )Tq1q2...qm

= δr1q1δr2q2 · · · δrmqmTq1q2...qm

with equation (1.8), the inverse relation of equation (1.9) is given by

Tr1r2···rm = Qp1r1 Qp2r2 · · · Qpmrm T ∗p1 p2···pm

. (1.11)

Indices put in a tensor take the dimension of the space in which the tensor exists. Thenumber of indices, which is equal to the number of operators Qij, is called the order of tensor.For instance, the transformation rule of the first-order tensor, that is, the vector vi, and thesecond-order tensor Ti j are given by

v∗i = Qirvr, vi = Qriv∗

rs

T ∗i j = QirQjsTrs, Ti j = QriQs jT ∗

rs

}. (1.12)

Consequently, in order to prove that a certain quantity is a tensor, one needs only to showthat it obeys the tensor transformation rule (1.9) or that multiplying the quantity by a tensorleads to a tensor by virtue of the quotient rule described in the next section.

The coordinate transformation rule in the form of equation (1.9) or (1.11) is called theobjective transformation. A tensor obeying the objective transformation rule even betweencoordinate systems with a relative rate of motion, that is, relative parallel and rotational veloci-ties, is called an objective tensor. Vectors and tensors without the time-dimension, for example,force, displacement, rotational angle, stress and strain, are objective vectors and tensors. On theother hand, time-rate quantities, for example, rate of force, velocity, spin and the material-timederivatives of physical quantities, for example, stress and strain, are not objective vectors andtensors in general; they are influenced by the relative rate of motion between coordinate sys-tems. Constitutive equations of materials have to be formulated in terms of objective tensors,since material properties are not influenced by the rigid-body rotation of material and thereforemust be described in a form independent of the coordinate systems, as will be explained inSection 5.3.

Tensors obviously fulfill the linearity

Tp1 p2···pm (Gp1 p2···pl + Hp1 p2···pl ) = Tp1 p2···pm Gp1 p2···pl + Tp1 p2···pm Hp1 p2···pl

Tp1 p2···pm (aAp1 p2···pl ) = aTp1 p2···pmAp1 p2···pl

}, (1.13)

where a is an arbitrary scalar variable. Therefore, the tensor plays the role of linearly trans-forming one tensor into another and thus it is also called a linear transformation. The operationthat lowers the order of a tensor by multiplying it by another tensor is called contraction.

Denoting by e∗1, e∗

2, . . . , e∗m the unit base vectors of the coordinate axes x∗

1, x∗2, . . . , x∗

m, thequantity Qi j in equation (1.7) is represented in terms of the base vectors as follows:

Qi j = e∗i · e j, (1.14)

4 Introduction to Finite Strain Theory for Continuum Elasto-Plasticity

noting that

Qi j = ∂x∗i

∂x j= e∗

i · ∂x∗s

∂x je∗

s = e∗i · ∂xs

∂x jes = e∗

i · δ jses,

where the coordinate transformation operator Qi j is interpreted as

Qi j ≡ cos(angle between e∗i and e j) (1.15)

which fulfills equations (1.8), that is,

QirQjr = QriQr j = δi j (1.16)

which can be also verified by

QirQjr = (e∗i · er)(e∗

j · er) = e∗i · (e∗

j · er)er = δi j.

The transformation rule for base vectors is given by

ei = Qrie∗r , e∗

i = Qirer, (1.17)

noting that

ei = (ei · e∗r )e

∗r , e∗

i = (e∗i · er)er.

1.2.2 Quotient Law

There is a convenient law, referred to as the quotient law, which enables us to judge whetheror not a given quantity is tensor and to find its tensorial order as follows: If a set of functionsT (p1, p2, . . . , pm) becomes Bpl+1 pl+2···pm ((m − l)th-order tensor lacking the indices p1 ∼ pl)through multiplying it by Ap1 p2···pl (lth-order tensor (l ≤ m)), the set is an mth-order tensor.

(Proof ) This convention is proved by showing that the quantity T (p1, p2, . . . , pm) is anmth-order tensor when the relation

T (p1, p2, . . . , pm)Ap1 p2···pl = Bpl+1 pl+2···pm (1.18)

holds, which is described in the coordinate system {O-x∗i } as follows:

T ∗(p1, p2, . . . , pm)A∗p1 p2···pl

= B∗pl+1 pl+2···pm

. (1.19)

Here, Ap1 p2···pl is the lth-order tensor and Bpl+1 pl+2···pm is the (m − l)th-order tensor. Therefore,the following relation holds:

B∗pl+1 pl+2···pm

= Qpl+1rl+1Qpl+2rl+2 · · · Qpmrm Brl+1rl+2···rm

= Qpl+1rl+1Qpl+2rl+2 · · · Qpmrm T (r1, r2, . . . , rm)Ar1r2···rl

= Qpl+1rl+1Qpl+2rl+2 · · · Qpmrm︸ ︷︷ ︸T (r1, r2, . . . , rm) Qp1r1 Qp2r2 · · · Qpl rl︸ ︷︷ ︸

A∗p1 p2···pl

.

(l + 1 ∼ m) (1 ∼ l) (1.20)

Substituting equation (1.19) into the left-hand side of equation (1.20) yields

{T ∗(p1, p2, . . . , pm) − Qp1r1 Qp2r2 · · · Qpmrm T (r1, r2, . . . , rm)}A∗p1 p2···pl

= 0,

Mathematical Preliminaries 5

from which it follows that

T ∗(p1, p2, . . . , pm) = Qp1r1 Qp2r2 · · · Qpmrm T (r1, r2, . . . , rm). (1.21)

Equation (1.21) satisfies the definition of tensor in equation (1.5). Therefore, the quantityT (p1, p2, . . . , pm) is an mth-order tensor.

According to the proof presented above, equation (1.18) can be written as

Tp1 p2···pm Ap1 p2···pl = Bpl+1 pl+2···pm . (1.22)

For instance, if the quantity T (i, j) transforms the first-order tensor, that is, vector vi, to thevector ui by the operation T (i, j)v j = ui, one can regard T (i, j) as the second-order tensor Ti j.

1.3 Vector Analysis

In this section some basic rules for vectors are given which are required to understand therepresentation of tensors in the general coordinate system described in the next chapter.

1.3.1 Scalar Product

The scalar (or inner) product of the vectors a and b is defined by

a · b = ||a|| ||b|| cos θ = aibi, (1.23)

where θ is the angle between the vectors a and b, and || || means the magnitude, that is,

||v|| = √vivi = √

v · v. (1.24)

Here, the following relations hold for the scalar product:

a · b = b · a (commutative law), (1.25)

a · (b + c) = a · b + a · c (distributive law), (1.26)

s(a · b) = (sa · b) = a · (sb) = (a · b)s, (1.27)

(aa + bb) · c = aa · c + bb · c (1.28)

for arbitrary scalars s, a, b.The vector is represented in terms of components with base vectors as follows:

v = viei, (1.29)

where the components vi are given by the projection of v onto the base vector ei, that is, theirscalar product and thus it follows that

vi = v · ei, v = (v · ei)ei. (1.30)

6 Introduction to Finite Strain Theory for Continuum Elasto-Plasticity

1.3.2 Vector Product

The vector (or outer or cross) product of vectors is defined by

a × b = ||a|| ||b|| sin θn = aiei × b je j = εijkaib jek

= (a2b3 − a3b2)e1 + (a3b1 − a1b3)e2 + (a1b2 − a2b1)e3, (1.31)

where n is the unit vector which forms the right-handed bases (a, b, n) in this order. It followsfor base vectors from equation (1.31) that

ei × e j = εijkek. (1.32)

Here, the following equations hold for the vector product:

a × a = 0, (1.33)

a × b = −b × a, (1.34)

a × (b + c) = a × b + a × c (distributive law), (1.35)

s(a × b) = (sa × b) = a × (sb) = (a × b)s, (1.36)

(aa + bb) × c = aa × c + bb × c, (1.37)

‖a × b‖2 + (a · b)2 = (‖a‖‖b‖)2. (1.38)

1.3.3 Scalar Triple Product

The scalar triple product of vectors is defined by

[abc] ≡ a · (b × c) = εijkaib jck, (1.39)

fulfilling

[abc] = [bca] = [cab] = −[bac] = −[cba] = −[acb]. (1.40)

Denoting the vectors a, b, c by v1, v2, v3, it follows from equation (1.39) that

[viv jvk] = eijk[v1v2v3], (1.41)

noting the fact that the term on the right-hand side of this equation is +[v1, v2, v3], −[v1, v2, v3]and 0 when indices i, j, k are even and odd permutations and two of indices coincide with eachother, respectively.

Here, the following equations hold for the scalar triple product.

[eie jek] = εijk, (1.42)

[sa, b, c] = [a, sb, c] = [a, b, sc] = s[abc], (1.43)

[aa + bb, c, d] = a[acd] + b[bcd], (1.44)

[a × b, b × c, c × a] = [abc]2, (1.45)

[abc]d = [bcd]a + [adc]b + [abd]c, (1.46)

[a × b, c × d, e × f] = [abd][cef] − [abc][def]. (1.47)