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Designing engineering components that make optimal use of materials re-
quires consideration of the nonlinear characteristics associated with both
the manufacturing and working environments. The increasing availability
of computer software to simulate component behavior implies the need for a
theoretical exposition applicable to both research and industry. By present-
ing the topics nonlinear continuum analysis and associated ¬nite element
techniques in the same book, Bonet and Wood provide a complete, clear,
and uni¬ed treatment of these important subjects.
After a gentle introduction and a chapter on mathematical preliminar-
ies, kinematics, stress, and equilibrium are considered. Hyperelasticity for
compressible and incompressible materials includes descriptions in principal
directions, and a short appendix extends the kinematics to cater for elasto-
plastic deformation. Linearization of the equilibrium equations naturally
leads on to ¬nite element discretization, equation solution, and computer
implementation. The majority of chapters include worked examples and
exercises. In addition the book provides user instructions, program descrip-
tion, and examples for the FLagSHyP computer implementation for which
the source code is available free on the Internet.
This book is recommended for postgraduate level study either by those
in higher education and research or in industry in mechanical, aerospace,
and civil engineering.




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NONLINEAR CONTINUUM MECHANICS
FOR FINITE ELEMENT ANALYSIS




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NONLINEAR CONTINUUM
MECHANICS FOR FINITE
ELEMENT ANALYSIS



Javier Bonet Richard D. Wood
University of Wales Swansea University of Wales Swansea




iii
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge CB2 1RP, United Kingdom

CAMBRIDGE UNIVERSITY PRESS
The Edinburgh Building, Cambridge CB2 2RU, United Kingdom
40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, Melbourne 3166, Australia

c Cambridge University Press 1997


This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.

First published 1997


Printed in the United States of America

Typeset in Times and Univers


Library of Congress Cataloging-in-Publication Data
Bonet, Javier, 1961“
Nonlinear continuum mechanics for ¬nite element analysis / Javier
Bonet, Richard D. Wood.
p. cm.
ISBN 0-521-57272-X
1. Materials “ Mathematical models. 2. Continuum mechanics.
3. Nonlinear mechanics. 4. Finite element method. I. Wood.
Richard D. II. Title.
TA405.B645 1997
620.1 1 015118 “ dc21 97-11366
CIP

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

ISBN 0 521 57272 X hardback




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To Catherine, Doreen and our children




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CONTENTS




xiii
Preface

1 INTRODUCTION 1
1.1 NONLINEAR COMPUTATIONAL MECHANICS 1
1.2 SIMPLE EXAMPLES OF NONLINEAR STRUCTURAL BEHAVIOR 2
1.2.1 Cantilever 2
1.2.2 Column 3
1.3 NONLINEAR STRAIN MEASURES 4
1.3.1 One-Dimensional Strain Measures 5
1.3.2 Nonlinear Truss Example 6
1.3.3 Continuum Strain Measures 10
1.4 DIRECTIONAL DERIVATIVE, LINEARIZATION AND
EQUATION SOLUTION 13
1.4.1 Directional Derivative 14
1.4.2 Linearization and Solution of Nonlinear
Algebraic Equations 16

2 MATHEMATICAL PRELIMINARIES 21
2.1 INTRODUCTION 21
2.2 VECTOR AND TENSOR ALGEBRA 21
2.2.1 Vectors 22
2.2.2 Second-Order Tensors 26
2.2.3 Vector and Tensor Invariants 33



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2.2.4 Higher-Order Tensors 37
2.3 LINEARIZATION AND THE DIRECTIONAL DERIVATIVE 43
2.3.1 One Degree of Freedom 43
2.3.2 General Solution to a Nonlinear Problem 44
2.3.3 Properties of the Directional Derivative 47
2.3.4 Examples of Linearization 48
2.4 TENSOR ANALYSIS 52
2.4.1 The Gradient and Divergence Operators 52
2.4.2 Integration Theorems 54

3 KINEMATICS 57
3.1 INTRODUCTION 57
3.2 THE MOTION 57
3.3 MATERIAL AND SPATIAL DESCRIPTIONS 59
3.4 DEFORMATION GRADIENT 61
3.5 STRAIN 64
3.6 POLAR DECOMPOSITION 68
3.7 VOLUME CHANGE 73
3.8 DISTORTIONAL COMPONENT OF THE DEFORMATION
GRADIENT 74
3.9 AREA CHANGE 77
3.10 LINEARIZED KINEMATICS 78
3.10.1 Linearized Deformation Gradient 78
3.10.2 Linearized Strain 79
3.10.3 Linearized Volume Change 80
3.11 VELOCITY AND MATERIAL TIME DERIVATIVES 80
3.11.1 Velocity 80
3.11.2 Material Time Derivative 81
3.11.3 Directional Derivative and Time Rates 82
3.11.4 Velocity Gradient 83
3.12 RATE OF DEFORMATION 84
3.13 SPIN TENSOR 87
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3.14 RATE OF CHANGE OF VOLUME 90
3.15 SUPERIMPOSED RIGID BODY MOTIONS AND OBJECTIVITY 92

4 STRESS AND EQUILIBRIUM 96
4.1 INTRODUCTION 96
4.2 CAUCHY STRESS TENSOR 96
4.2.1 De¬nition 96
4.2.2 Stress Objectivity 101
4.3 EQUILIBRIUM 101
4.3.1 Translational Equilibrium 101
4.3.2 Rotational Equilibrium 103
4.4 PRINCIPLE OF VIRTUAL WORK 104
4.5 WORK CONJUGACY AND STRESS REPRESENTATIONS 106
4.5.1 The Kirchho¬ Stress Tensor 106
4.5.2 The First Piola“Kirchho¬ Stress Tensor 107
4.5.3 The Second Piola“Kirchho¬ Stress Tensor 109
4.5.4 Deviatoric and Pressure Components 112
4.6 STRESS RATES 113

5 HYPERELASTICITY 117
5.1 INTRODUCTION 117
5.2 HYPERELASTICITY 117
5.3 ELASTICITY TENSOR 119
5.3.1 The Material or Lagrangian Elasticity Tensor 119
5.3.2 The Spatial or Eulerian Elasticity Tensor 120
5.4 ISOTROPIC HYPERELASTICITY 121
5.4.1 Material Description 121
5.4.2 Spatial Description 122
5.4.3 Compressible Neo-Hookean Material 124
5.5 INCOMPRESSIBLE AND NEARLY
INCOMPRESSIBLE MATERIALS 126
5.5.1 Incompressible Elasticity 126
5.5.2 Incompressible Neo-Hookean Material 129
5.5.3 Nearly Incompressible Hyperelastic Materials 131
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5.6 ISOTROPIC ELASTICITY IN PRINCIPAL DIRECTIONS 134
5.6.1 Material Description 134
5.6.2 Spatial Description 135
5.6.3 Material Elasticity Tensor 136
5.6.4 Spatial Elasticity Tensor 137
5.6.5 A Simple Stretch-Based Hyperelastic Material 138
5.6.6 Nearly Incompressible Material in Principal Directions 139
5.6.7 Plane Strain and Plane Stress Cases 142
5.6.8 Uniaxial Rod Case 143

6 LINEARIZED EQUILIBRIUM EQUATIONS 146
6.1 INTRODUCTION 146
6.2 LINEARIZATION AND NEWTON“RAPHSON PROCESS 146
6.3 LAGRANGIAN LINEARIZED INTERNAL VIRTUAL WORK 148
6.4 EULERIAN LINEARIZED INTERNAL VIRTUAL WORK 149
6.5 LINEARIZED EXTERNAL VIRTUAL WORK 150
6.5.1 Body Forces 151
6.5.2 Surface Forces 151
6.6 VARIATIONAL METHODS AND INCOMPRESSIBILITY 153
6.6.1 Total Potential Energy and Equilibrium 154
6.6.2 Lagrange Multiplier Approach to Incompressibility 154
6.6.3 Penalty Methods for Incompressibility 157
6.6.4 Hu-Washizu Variational Principle for Incompressibility 158
6.6.5 Mean Dilatation Procedure 160

7 DISCRETIZATION AND SOLUTION 165
7.1 INTRODUCTION 165
7.2 DISCRETIZED KINEMATICS 165
7.3 DISCRETIZED EQUILIBRIUM EQUATIONS 170
7.3.1 General Derivation 170
7.3.2 Derivation in Matrix Notation 172
7.4 DISCRETIZATION OF THE LINEARIZED
EQUILIBRIUM EQUATIONS 173
7.4.1 Constitutive Component “ Indicial Form 174
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7.4.2 Constitutive Component “ Matrix Form 176
7.4.3 Initial Stress Component 177
7.4.4 External Force Component 178
7.4.5 Tangent Matrix 180
7.5 MEAN DILATATION METHOD FOR INCOMPRESSIBILITY 182
7.5.1 Implementation of the Mean Dilatation Method 182
7.6 NEWTON“RAPHSON ITERATION AND SOLUTION
PROCEDURE 184
7.6.1 Newton“Raphson Solution Algorithm 184
7.6.2 Line Search Method 185
7.6.3 Arc Length Method 187


8 COMPUTER IMPLEMENTATION 191
8.1 INTRODUCTION 191
8.2 USER INSTRUCTIONS 192
8.3 OUTPUT FILE DESCRIPTION 196

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