Nonlinear continuum mechanics for finite element analysis

283 Pages·1997·1.924 MB·English

by  Javier Bonet, Richard D. Wood

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 finite element
techniques in the same book, Bonet and Wood provide a complete, clear,
and unified 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 finite 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
i
ii
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
ThePittBuilding,TrumpingtonStreet,CambridgeCB21RP,UnitedKingdom
CAMBRIDGE UNIVERSITY PRESS
TheEdinburghBuilding,CambridgeCB22RU,UnitedKingdom
40West20thStreet,NewYork,NY10011-4211,USA
10StamfordRoad,Oakleigh,Melbourne3166,Australia
°c CambridgeUniversityPress1997
Thisbookisincopyright. Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished1997
PrintedintheUnitedStatesofAmerica
TypesetinTimesandUnivers
Library of Congress Cataloging-in-Publication Data
Bonet,Javier,1961–
Nonlinearcontinuummechanicsforfiniteelementanalysis/Javier
Bonet,RichardD.Wood.
p. cm.
ISBN0-521-57272-X
1. Materials–Mathematicalmodels. 2. Continuummechanics.
3. Nonlinearmechanics. 4. Finiteelementmethod. I.Wood.
RichardD. II.Title.
TA405.B645 1997
620.1010015118–dc21 97-11366
CIP
A catalog record for this book is available from
the British Library.
ISBN052157272Xhardback
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To Catherine, Doreen and our children
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CONTENTS
Preface xiii
1 INTRODUCTION 1
1.1 NONLINEAR COMPUTATIONAL MECHANICS 1
1.2 SIMPLEEXAMPLESOFNONLINEARSTRUCTURALBEHAVIOR 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