Non-Linear Elastic Deformations

Cover

Non-Linear Elastic Deformations

Copyright © 1984 by R. W. Ogden/Ellis Harwood Ltd.
ISBN 0-486-69648-0
TA 17.6.034 1997 620.1' 1232-dc21
LCCN 97-16162 CIP

Table of Contents

Acknowledgements

Preface

CHAPTER 1 Tensor Theory
1.1 EUCLIDEAN VECTOR SPACE
1.1.1 Orthonormal bases and components
1.1.2 Change of basis
1.1.3 Euclidean point space : Cartesian coordinates
1.2 CARTESIAN TENSORS
1.2.1 Motivation : stress in a continuum
1.2.2 Definition of a Cartesian tensor
1.2.3 The tensor product
1.2.4 Contraction
1.2.5 Isotropic tensors
1.3 TENSOR ALGEBRA
1.3.1 Second-order tensors
1.3.2 Eigenvalues and eigenvectors of a second-order tensor
1.3.3 Symmetric second-order tensors
1.3.4 Antisymmetric second-order tensors
1.3.5 Orthogonal second-order tensors
1.3.6 Higher-order tensors
1.4 CONTRAVARIANT AND COVARIANT TENSORS
1.4.1 Reciprocal basis. Contravariant and covariant components
1.4.2 Change of basis
1.4.3 Dual space. General tensors
1.5 TENSOR FIELDS
1.5.1 The gradient of a tensor field
1.5.2 Symbolic notation for differential operators
1.5.3 Diffeientiation in Cartesian coordinates
1.5.4 Differentiation in curvilinear coordinates
1.5.5 Curves and surfaces
1.5.6 Integration of tensor fields
REFERENCES

CHAPTER 2 Analysis of Deformation and Motion
2.1 KINEMATICS
2.1.1 Observers and frames of reference
2.1.2 Configurations and motions
2.1.3 Reference configurations and deformations
2.1.4 Rigid-body motions
2.2 ANALYSIS OF DEFORMATION AND STRAIN
2.2.1 The deformation gradient
2.2.2 Deformation of volume and surface
2.2.3 Strain, stretch, extension and shear
2.2.4 Polar decomposition of the deformation gradient
2.2.5 Geometrical interpretations of the deformation
2.2.6 Examples of deformations
2.2.7 Strain tensors
2.2.8 Change of reference configuration or observer
2.3 ANALYSIS OF MOTION
2.3.1 Deformation and strain rates
2.3.2 Spins of the Lagrangean and Eulerian axes
2.4 OBJECTIVITY OF TENSOR FIELDS
2.4.1 Eulerian and Lagrangean objectivity
2.4.2 Embedded components of tensors
REFERENCES

CHAPTER 3 Balance Laws, Stress and Field Equations
3.1 MASS CONSERVATION
3.2 MOMENTUM BALANCE EQUATIONS
3.3 THE CAUCHY STRESS TENSOR
3.3.1 Linear dependence of the stress vector on the surface normal
3.3.2 Cauchy's laws of motion
3.4 THE NOMINAL STRESS TENSOR
3.4.1 Definition of nominal stress
3.4.2. The Lagrangean field equations
3.5 CONJUGATE STRESS ANALYSIS
3.5.1 Work rate and energy balance
3.5.2 Conjugate stress tensors
3.5.3 Stress rates
REFERENCES

CHAPTER 4 Elasticity
4.1 CONSTITUTIVE LAWS FOR SIMPLE MATERIALS
4.1.1 General remarks on constitutive laws
4.1.2 Simple materials
4.1.3 Material uniformity and homogeneity
4.2 CAUCHY ELASTIC MATERIALS
4.2.1 The constitutive equation for a Cauchy elastic material
4.2.2 Alternative forms of the constitutive equation
4.2.3 Material symmetry
4.2.4 Undistorted configurations and isotropy
4.2.5 Anisotropic elastic solids
4.2.6 Isotropic elastic solids
4.2.7 Internal constraints
4.2.8 Differentiation of a scalar function of a tensor
4.3 GREEN ELASTIC MATERIALS
4.3.1 The strain-energy function
4.3.2 Symmetry groups for hyperelastic materials
4.3.3 Stress-deformation relations for constrained hyperelastic materials
4.3.4 Stress-deformation relations for isotropic hyperelastic materials
4.3.5 Strain-energy functions for isotropic elastic materials
4.4 APPLICATION TO SIMPLE HOMOGENEOUS DEFORMATIONS
REFERENCES

CHAPTER 5 Boundary-Value Problems
5.1 FORMULATION OF BOUNDARY-VALUE PROBLEMS
5.1.1 Equations of motion and equilibrium
5.1.2 Boundary conditions
5.1.3 Restrictions on the deformation
5.2 PROBLEMS FOR UNCONSTRAINED MATERIALS
5.2.1 Ericksen's theorem
5.2.2 Spherically symmetric deformation of a spherical shell
5.2.3 Extension and inflation of a circular cylindrical tube
5.2.4 Bending of a rectangular block into a sector of a circular tube
5.2.5 Combined extension and torsion of a solid circular cylinder
5.2.6 Plane strain problems : complex variable methods
5.2.7 Growth conditions
5.3 PROBLEMS FOR MATERIALS WITH INTERNAL CONSTRAINTS
5.3.1 Preliminaries
5.3.2 Spherically symmetric deformation of a spherical shell
5.3.3 Combined extension and inflation of a circular cylindrical tube
5.3.4 Flexure of a rectangular block
5.3.5 Extension and torsion of a circular cylinder
5.3.6 Shear of a circular cylindrical tube
5.3.7 Rotation of a solid circular cylinder about its axis
5.4 VARIATIONAL PRINCIPLES AND CONSERVATION LAWS
5.4.1 Virtual work and related principles
5.4.2 The principle of stationary potential energy
5.4.3 Complementary and mixed variational principles
5.4.4 Variational principles with constraints
5.4.5 Conservation laws and the energy momentum tensor
REFERENCES

CHAPTER 6 Incremental Elastic Deformations
6.1 INCREMENTAL CONSTITUTIVE RELATIONS
6.1.1 Deformation increments
6.1.2 Stress increments and elastic moduli
6.1.3 Instantaneous moduli
6.1.4 Elastic moduli for isotropic materials
6.1.5. Elastic moduli for incompressible isotropic materials
6.1.6 Linear and second-order elasticity
6.2 STRUCTURE AND PROPERTIES OF THE INCREMENTAL EQUATIONS
6.2.1 Incremental boundary-value problems
6.2.2 Uniqueness : global considerations
6.2.3 Incremental uniqueness and stability
6.2.4 Variational aspects of incremental problems
6.2.5 Bifurcation analysis : dead-load tractions
6.2.6 Bifurcation analysis: non-adjoint and self-adjoint data
6.2.7 The strong ellipticity condition
6.2.8 Constitutive branching and constitutive inequalities
6.3 SOLUTION OF INCREMENTAL BOUNDARY-VALUE PROBLEMS
6.3.1 Bifurcation of a pre-strained rectangular block
6.3.2 Global aspects of the plane-strain bifurcation of a rectangular block
6.3.3 Other problems with underlying homogeneous deformation
6.3.4 Bifurcation of a pressurized spherical shell
6.4 WAVES AND VIBRATIONS

REFERENCES

CHAPTER 7 Elastic Properties of Solid Materials
7.1 PHENOMENOLOGICAL THEORY
7.2 ISOTROPIC MATERIALS
7.2.1 Homogeneous pure strain of an incompressible material
7.2.2 Application to rubberlike materials
7.2.3 Homogeneous pure strain of a compressible material
7.3 THE EFFECT OF SMALL CHANGES IN MATERIAL PROPERTIES
7.4 NEARLY INCOMPRESSIBLE MATERIALS
7.4.1 Compressible materials and the incompressible limit
7.4.2 Nearly incompressible materials
7.4.3 Pure homogeneous strain of a nearly incompressible isotropic material
7.4.4 Application to rubberlike materials
REFERENCES

APPENDIX 1 Convex Functions
REFERENCES

APPENDIX 2 Glossary of Symbols

Index

Back Cover