Introduction to Finite Strain Theory for Continuum Elasto-Plasticity E-Book

Koichi Hashiguchi

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Series in Computational Mechanics
• Sprache: Englisch

Seitenzahl: 402

Veröffentlichungsjahr: 2012

Beschreibung

Comprehensive introduction to finite elastoplasticity, addressing various analytical and numerical analyses & including state-of-the-art theories

Introduction to Finite Elastoplasticity presents introductory explanations that can be readily understood by readers with only a basic knowledge of elastoplasticity, showing physical backgrounds of concepts in detail and derivation processes of almost all equations. The authors address various analytical and numerical finite 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.

As a foundation to finite strain theory, the authors begin by addressing the advanced mathematical and physical properties of continuum mechanics. They progress to explain a finite elastoplastic constitutive model, discuss numerical issues on stress computation, implement the numerical algorithms for stress computation into large-deformation finite element analysis and illustrate several numerical examples of boundary-value problems. Programs for the stress computation of finite elastoplastic models explained in this book are included in an appendix, and the code can be downloaded from an accompanying website. 

Leseprobe

Contents

Cover

Series

Title Page

Copyright

Preface

Series Preface

Introduction

Prominence of the finite strain elasto-plasticity theory

Chapter 1: Mathematical Preliminaries

1.1 Basic Symbols and Conventions

1.2 Definition of Tensor

1.3 Vector Analysis

1.4 Tensor Analysis

1.5 Tensor Representations

1.6 Eigenvalues and Eigenvectors

1.7 Polar Decomposition

1.8 Isotropy

1.9 Differential Formulae

1.10 Variations and Rates of Geometrical Elements

1.11 Continuity and Smoothness Conditions

1.12 Unconventional Elasto-Plasticity Models

Chapter 2: General (Curvilinear) Coordinate System

2.1 Primary and Reciprocal Base Vectors

2.2 Metric Tensors

2.3 Representations of Vectors and Tensors

2.4 Physical Components of Vectors and Tensors

2.5 Covariant Derivative of Base Vectors with Christoffel Symbol

2.6 Covariant Derivatives of Scalars, Vectors and Tensors

2.7 Riemann–Christoffel Curvature Tensor

2.8 Relations of Convected and Cartesian Coordinate Descriptions

Chapter 3: Description of Physical Quantities in Convected Coordinate System

3.1 Necessity for Description in Embedded Coordinate System

3.2 Embedded Base Vectors

3.3 Deformation Gradient Tensor

3.4 Pull-Back and Push-Forward Operations

Chapter 4: Strain and Strain Rate Tensors

4.1 Deformation Tensors

4.2 Strain Tensors

4.3 Compatibility Condition

4.4 Strain Rate and Spin Tensors

4.5 Representations of Strain Rate and Spin Tensors in Lagrangian and Eulerian Triads

4.6 Decomposition of Deformation Gradient Tensor into Isochoric and Volumetric Parts

Chapter 5: Convected Derivative

5.1 Convected Derivative

5.2 Corotational Rate

5.3 Objectivity

Chapter 6: Conservation Laws and Stress (Rate) Tensors

6.1 Conservation Laws

6.2 Stress Tensors

6.3 Equilibrium Equation

6.4 Equilibrium Equation of Angular Moment

6.5 Conservation Law of Energy

6.6 Virtual Work Principle

6.7 Work Conjugacy

6.8 Stress Rate Tensors

6.9 Some Basic Loading Behavior

Chapter 7: Hyperelasticity

7.1 Hyperelastic Constitutive Equation and Its Rate Form

7.2 Examples of Hyperelastic Constitutive Equations

Chapter 8: Finite Elasto-Plastic Constitutive Equation

8.1 Basic Structures of Finite Elasto-Plasticity

8.2 Multiplicative Decomposition

8.3 Stress and Deformation Tensors for Multiplicative Decomposition

8.4 Incorporation of Nonlinear Kinematic Hardening

8.5 Strain Tensors

8.6 Strain Rate and Spin Tensors

8.7 Stress and Kinematic Hardening Variable Tensors

8.8 Influences of Superposed Rotations: Objectivity

8.9 Hyperelastic Equations for Elastic Deformation and Kinematic Hardening

8.10 Plastic Constitutive Equations

8.11 Relation between Stress Rate and Strain Rate

8.12 Material Functions of Metals

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

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

Chapter 9: Computational Methods for Finite Strain Elasto-Plasticity

9.1 A Brief Review of Numerical Methods for Finite Strain Elasto-Plasticity

9.2 Brief Summary of Model Formulation

9.3 Transformation to Description in Reference Configuration

9.4 Time-Integration of Plastic Evolution Rules

9.5 Update of Deformation Gradient Tensor

9.6 Elastic Predictor Step and Loading Criterion

9.7 Plastic Corrector Step by Return-Mapping

9.8 Derivation of Jacobian Matrix for Return-Mapping

9.9 Consistent (Algorithmic) Tangent Modulus Tensor

9.10 Numerical Examples

Chapter 10: Computer Programs

10.1 User Instructions and Input File Description

10.2 Output File Description

10.3 Computer Programs

Appendix A: Projection of Area

Appendix B: Geometrical Interpretation of Strain Rate and Spin Tensors

Appendix C: Proof for Derivative of Second Invariant of Logarithmic-Deviatoric Deformation Tensor

Appendix D: Numerical Computation of Tensor Exponential Function and Its Derivative

D.1 Numerical Computation of Tensor Exponential Function

D.2 Fortran Subroutine for Tensor Exponential Function: matexp

D.3 Numerical Computation of Derivative of Tensor Exponential Function

D.4 Fortran Subroutine for Derivative of Tensor Exponential Function: matdex

References

Index

WILEY SERIES IN COMPUTATIONAL MECHANICS

Series Advisors:

René de Borst

Perumal Nithiarasu

Tayfun E. Tezduyar

Genki Yagawa

Tarek Zohdi

Introduction to Finite Strain Theory for Continuum Elasto-PlasticityHashiguchi and YamakawaOctober 2012Nonlinear Finite Element Analysis of Solids and Structures: Second editionDe Borst, Crisfield, Remmers and VerhooselAugust 2012An Introduction to Mathematical Modeling: A Course in MechanicsOdenNovember 2011Computational Mechanics of DiscontinuaMunjiza, Knight and RougierNovember 2011Introduction to Finite Element Analysis: Formulation, Verification and ValidationSzabó and BabuškaMarch 2011

This edition first published 2013 © 2013, John Wiley & Sons Ltd

Registered officeJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

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 2013 620.1′1233–dc23 2012011797

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

Print ISBN: 9781119951858

Preface

The first author of this book recently published the book Elastoplasticity Theory (2009) which addresses the fundamentals of elasto-plasticity and various plasticity models. It is mainly concerned 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 the prediction of the elasto-plastic deformation behavior of engineering materials and structures composed 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 linear relation between the elastic strain rate and stress rate. There is no one-to-one correspondence between the time-integrations of the elastic strain rate and stress rare and the energy may be produced or dissipated during a loading cycle in hypoelastic equation. Therefore, the elastic strain rate does not possess the elastic property in the strict sense so that an error may be induced in large-deformation analysis and accumulated in the cyclic loading analysis. An exact formulation without these defects in the infinitesimal elasto-plasticity theory has been sought in order to respond to recent developments in engineering in the relevant fields, such as 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 by the 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 the multiplicative decomposition which is the decomposition of the deformation gradient into the elastic and plastic parts, introducing the intermediate configuration obtained by unloading to the stress free-state, was proposed and the elastic part is formulated as a hyperelastic relation based on the elastic strain energy function. Further, the Mandel stress, the work-conjugate plastic velocity gradient with the Mandel stress, the plastic spin, and various physical quantities defined in the intermediate configuration have been introduced. The physical and mathematical foundations for the exact finite elasto-plasticity theory were established by 2000 and the constitutive equation based on these foundations – the hyperelastic-based plastic constitutive equation – has been formulated after 2005. However, a textbook on the hyperelastic-based finite elasto-plasticity theory has not been published to date.

Against this backdrop, the aim of this book is to give a comprehensive explanation of the finite 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 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-forward operations, the Lie derivative and the corotational rate – is required in the formulation of finite strain 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 the deformation of material, are required, since their physical meanings can be captured clearly by observing them in a coordinate system which not only moves but also deforms and rotates with material itself. In other words, the essentials of continuum mechanics cannot be captured without the incorporation of the general curvilinear coordinate system, although numerous books with ‘continuum mechanics’ in their title and confined to the rectangular coordinate system have been published to date. On the other hand, knowledge of the curvilinear coordinate system is not required for users of finite strain theory. It is sufficient for finite element analyzers using finite strain theory to master the descriptions in the rectangular coordinate system and the relations between Lagrangian and Eulerian tensors through the deformation gradient, for instance. This book, which aims to impart the exact finite strain theory, gives a comprehensive explanation of the mathematical and physical fundamentals required for continuum solid mechanics, and provides a description in the general coordinate system before moving on to an explanation of finite strain theory.

In addition to the above-mentioned issues on the formulation of the constitutive equation, the formulation and implementation of a numerical algorithm for the state-updating calculation are of utmost importance. The state-updating procedure in the computational analysis of elasto-plasticity problems usually requires a proper algorithm for numerical integration of the rate forms of the constitutive laws and the evolution equations. The return-mapping scheme has been developed to a degree of common acceptance in the field of computational plasticity as an effective state-updating procedure for elasto-plastic models. In the numerical analysis of boundary-value problems, a consistent linearization of the weak form of the equilibrium equation and use of the so-called consistent (algorithmic) elasto-plastic tangent modulus tensor 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 which can readily be implemented in finite element codes is provided along with detailed explanation and user instructions so that readers will be able to carry out deformation analysis using the finite elasto-plasticity theory by themselves.

Chapters 1–7 were written by the first author, Chapter 8 by both authors, and Chapters 9 and 10 by the second author in close collaboration, and the computer programming and calculations were performed by the second author based on the theory formulated by both authors in Chapter 8. The authors hope that readers of this book will capture the fundamentals of the finite elasto-plasticity theory and will contribute to the development of mechanical designs of machinery and structures in the field of engineering practice by applying the theories addressed in this book. A reader is apt to give up reading a book if he encounters matter which is difficult to understand. For this reason, explanations of physical concepts in elasto-plasticity are given, and formulations and derivations/transformations for all equations are given without abbreviation for Chapters 1–8 for the basic formulations of finite strain theory. This is not a complete book on the finite elasto-plasticity theory, but the authors will be quite satisfied if it provides a foundation for further development of the theory by stimulating the curiosity of young researchers and it is applied widely to the analyses of engineering problems in practice. In addition, the authors hope that it will be followed by a variety of books on the finite elasto-plasticity.

The first author is deeply indebted to Professor B. Raniecki of the Institute of Fundamental Technological Research, Poland, for valuable suggestions and comments on solid mechanics, who has visited several times Kyushu University. His lecture notes on solid mechanics and regular private communications and advice have been valuable in writing some parts of this book. 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 Research, Poland, for valuable comments and for notes on their lectures on continuum mechanics delivered at Kyushu University. The second author would like to express sincere gratitude to Professor Kiyohiro Ikeda of Tohoku University for valuable suggestions and comments on nonlinear mechanics. He is also grateful to Professor Kenjiro Terada of Tohoku University for providing enlightening advices on numerical methods for finite strain elasto-plasticity. He also thanks Dr. Ikumu Watanabe of National Institute for Materials Science, Japan, for helpful advices on numerical methods for finite strain elasto-plasticity. The enthusiastic support of Dr. Keisuke Sato of Terrabyte lnc., Japan, and Shoya Nakaichi, Toshimitsu Fujisawa, Yosuke Yamaguchi and Yutaka Chida at Tohoku University in development and implementation of the numerical code is most appreciated. The authors thank Professor S. Reese of RWTH Aachen University, Professor J. Ihlemann and Professor A.V. Shutov of Chemnitz University, Germany, and Professor M. Wallin of Lund University, Sweden for valuable suggestions and for imparting relevant articles on finite strain theory to the authors.

Koichi Hashiguchi Yuki Yamakawa February 2012

Series Preface

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

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

This book is one of the first introductory books to address finite strain elasto-plasticity theory, where the mathematical and physical foundations are comprehensively described. In particular, the representation of physical quantities in convected (curvilinear) coordinate systems 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, the general objective rate in continuum mechanics. Furthermore, all the mathematical derivations and transformations of equations are shown without any abbreviation, explaining the numerical method for the finite elasto-plasticity in detail.

In addition, a Fortran program for the stress-update algorithm based on the return-mapping scheme and the consistent (algorithmic) tangent modulus tensor, which can readily be implemented in finite element codes, is appended with a detailed explanation and user instructions, so that the readers will be able to implement the numerical analysis on the basis of the finite elasto-plasticity theory without difficulty.

It is our hope that the readers of this book will contribute to the improvement of mechanical design of machinery and structures in the field of engineering by adopting and widely using the basic ideas of the finite elasto-plasticity theory and the corresponding numerical methods introduced 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 elastoplasticity. Then, the prominence of finite strain elasto-plasticity theory and the necessity of its incorporation to deformation analysis is first reviewed by comparing with the infinitesimal elasto-plasticity theory, abbreviated as the infinitesimal elastoplasticity, prior to the detailed explanation of the finite elastoplasticity in the subsequent chapters.

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

A.Infinitesimal elastoplasticity

The infinitesimal strain tensor defined by the symmetric part of displacement gradient tensor is additively decomposed into an elastic and a plastic parts, while there does not exist the distinction between the reference and the current configurations in which the infinitesimal strain tensor is based. The spin of material is described by the skew-symmetric part of the rate of displacement gradient tensor. This theory is limited to the description of infinitesimal deformation and rotation. Meanwhile, for the elastic part, the hyperelastic constitutive equation with a stored energy function can be formulated, which provides the one-to-one correspondence between the stress tensor and the infinitesimal elastic strain tensor. Therefore, the return-mapping and the consistent (algorithmic) tangent modulus tensor can be employed in numerical calculations under infinitesimal deformation and rotation.

B.Finite elastoplasticity

The frameworks of the theories describing the finite elastoplastic deformation and rotation are classified further as follows:

B-1Hypoelastic-based plasticity

It is premised on the following assumptions, which would be called the Hill–Rice approach.

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 velocity gradient tensor.

ii. The strain rate and the spin tensors are additively decomposed into elastic and plastic parts. Here, it should be noticed that the additive decomposition is derived from the multiplicative decomposition of the deformation gradient tensor only when an elastic deformation is infinitesimal, whereas the multiplicative decomposition is the rigorous approach for the exact partition of the deformation into the elastic and the plastic parts.

iii. The elastic part of the strain rate tensor, that is, elastic strain rate tensor is related linearly to an appropriate corotational rate of the Cauchy stress tensor. It falls within 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 by the 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 substructure, that is, the substructure (corotational) spin tensor. Then, it is adopted in corotational rates of stress tensor and tensor-valued internal variables. The plastic spin tensor is formulated as the skew-symmetric part of the multiplication of 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 the current configuration.

Constitutive behavior under finite deformation and rotation is properly described by incorporating corotational rate tensors based on an appropriate substructure spin tensor 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. Therefore, one-to-one correspondence between integrations of stress rate tensor and elastic strain rate tensor, that is, stress vs. strain relation cannot be obtained in general. Then, the hypoelastic deformation remains even after a closed stress cycle. In addition, work done during a closed loading cycle may not be zero as an energy is produced or dissipated during a closed loading cycle even when a plastic deformation is not induced. Therefore, the elastic strain rate tensor does not possess the reversibility in the strict sense, so that the hypoelastic constitutive equation is merely an elastic-like constitutive equation. This approach possesses the pertinence on the premise that the elastic deformation is infinitesimal.

2. Stress cannot be calculated directly from strain variable but it has to be calculated by the time-integration of stress rate. Also, tensor-valued internal variables cannot be calculated directly from the elastic strain-like variable but they have to be calculated by the time-integration of their rates. Here, note that rates of stress and tensor-valued internal variables in the current configuration are influenced by the rigid-body rotation of material. Consequently, their pertinent objective corotational rate tensors must be adopted and their pertinent time-integration procedures must be incorporated in order that unrealistic and/or impertinent calculation such as oscillatory stress and tensor-valued internal variables are not resulted. Proper time-integrations would be given by the time-integration reflecting 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, cannot be employed since they are premised on the exact evaluation of stress by the hyperelastic constitutive equation, while the plastic strain increment in the plastic corrector step is calculated based on the overstress from the yield surface. Nevertheless, the hypoelasticity with constant elastic moduli leads to the one-to-one correspondence between the time-integrations of objective stress rate and elastic strain rate tensors by performing proper time-integrations reflecting the convected derivative process, so that the return-mapping and the consistent tangent modulus tensor can be employed under the finite deformation and rotation. On the other hand, it should be noticed that the automatic controlling function to attract the stress to the yield surface in a plastic deformation process is furnished in the subloading surface model (Hashiguchi, 1989; Hashiguchi et al. 2012) as will be described in Chapter 8, so that numerical analyses of materials, in which elastic parameters in a hypoelastic equation are not constant, can be enforced drastically by this function in the Euler forward-integration method.

B-2Hyperelastic-based plasticity

The hyperelastic-based plasticity has been formulated to overcome the above-mentioned problems in the hypoelastic-based plasticity by incorporating the following notions.

1. Deformation itself (not rate) is decomposed into the elastic and the plastic parts to provide the one-to-one correspondence between elastic deformation and stress, although the strain rate is decomposed into them in the hypoelastic-based plasticity. It has been materialized by the decomposition of the deformation gradient tensor into the elastic and the plastic parts in the multiplicative form, that is, the multiplicative decomposition by Kroner (1959), Lee and Liu (1967), Lee (1969), etc., while the elasto-plasticity based on this decomposition would be called the Mandel–Lee approach. Therein, the intermediate configuration is incorporated, which is attained by unloading to the stress free state along the hyperelastic deformation. Then, the deformation gradient is multiplicatively decomposed into the plastic deformation gradient tensor induced in the process from the initial to the intermediate configuration and the elastic deformation gradient tensor induced in the process from the intermediate to the current configuration. Formulations by principal values based on the Hencky strain, that is, principal logarithmic 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 which the principal directions of stress, elastic strain and plastic strain rate coincide with each other leading to the co-axiality. Then, the formulation based on the following notions has been developed, by which general elastoplastic constitutive behavior can be described accurately over the 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 intermediate configuration which is independent of the superposition of rigid-body rotation. Therefore, constitutive relation is formulated originally by tensor variables in the intermediate configuration.

3. Elastic deformation is described by an elastic strain tensor based on the elastic deformation gradient tensor and it is related to the stress tensor by the hyperelastic constitutive equation possessing the elastic strain energy function. Then, the one-to-one correspondence between stress tensor and elastic strain tensor holds and the work done during a closed stress cycle is zero exactly when the plastic strain rate is not induced.

4. In addition, the plastic deformation gradient tensor is further decomposed into the energy-storage part causing the variation of substructure and the energy-dissipative part causing the slip between substructures (Lion, 2000). Then, the kinematic hardening variable, that is, the back stress is also formulated in relation to the elastic 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 calculated directly from the elastic strain and the elastic strain-like variable of the kinematic hardening without performing the time-integrations of stress rate and back stress rate. Therefore, material rotation is independent of a rigid-body rotation and it would not be influenced severely be a plastic deformation on the calculation of stress and back stress.

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

7. In addition to the above-mentioned issues on the formulation of constitutive equation, the formulation and implementation of a numerical algorithm for the state-updating calculation are of utmost importance. The state-updating procedure in the computation of elastoplasticity problems usually requires a proper algorithm for numerical integration of the rate forms of the constitutive laws and the evolution equations. Responding to the formulation of hyperelastic-based plastic constitutive equation, the return-mapping scheme has achieved a degree of common acceptance in the field of computational plasticity as an effective state-updating procedure for elastoplastic models. In the numerical analysis of boundary-value problems, a consistent linearization of the weak form of equilibrium equation and a use of the so-called consistent elasto-plastic tangent modulus tensor are necessary to ensure effectiveness 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 finite deformation elastoplasticity, since it does not use the elastic strain itself but it is based on the velocity gradient. The hyperelastic-based plasticity will be explained exhaustively in this book. A certain amount of advanced mathematical knowledge is required to capture the essentials of continuum mechanics and to formulate constitutive equations in the framework of the finite strain theory. In order to capture the meanings of physical quantities and relations exactly, it is indispensable to describe them in the embedded coordinate system which not only moves but also deforms and rotates with material itself, which belongs to the general curvilinear coordinate system. Then, the incorporation of the curvilinear coordinate system is one of the distinctions of the mathematical methodology in the finite strain theory from that in the infinitesimal strain theory. The concise explanation of vector and tensor analysis in the curvilinear coordinate system will be given in chapter 2 in addition to the analysis in the Cartesian coordinate system in chapter 1 as the preliminary to the study of finite strain theory.