Nonlinear Continuum Mechanics for Finite Element Analysis

by Javier Bonet , Richard D. Wood

• ISBN13: 9780521838702
• ISBN10: 0521838703
• eBook ISBN(s): 9781139637176
• Additional ISBN(s): 9780511389139
• Edition: 2nd
• Format: Hardcover
• Copyright: 2008-03-24
• Publisher: Cambridge University Press

Summary

The first edition of this successful text considered nonlinear geometrical behavior and nonlinear hyperelastic materials, and the numerics needed to model such phenomena. By presenting both nonlinear continuum analysis and associated finite element techniques in one, Bonet and Wood provide, in the new edition of this successful text, a complete, clear, and unified treatment of these important subjects. New chapters dealing with hyperelastic plastic behavior are included, and the authors have thoroughly updated the FLagSHyP program, freely accessible at www.flagshyp.com.

Table of Contents

Preface	p. xv
Introduction	p. 1
Nonlinear Computational Mechanics	p. 1
Simple Examples of Nonlinear Structural Behavior	p. 2
Cantilever	p. 2
Column	p. 3
Nonlinear Strain Measures	p. 4
One-Dimensional Strain Measures	p. 5
Nonlinear Truss Example	p. 6
Continuum Strain Measures	p. 10
Directional Derivative, Linearization and Equation Solution	p. 13
Directional Derivative	p. 14
Linearization and Solution of Nonlinear Algebraic Equations	p. 16
Mathematical Preliminaries	p. 22
Introduction	p. 22
Vector and Tensor Algebra	p. 22
Vectors	p. 23
Second-Order Tensors	p. 28
Vector and Tensor Invariants	p. 37
Higher-Order Tensors	p. 41
Linearization and the Directional Derivative	p. 47
One Degree of Freedom	p. 48
General Solution to a Nonlinear Problem	p. 49
Properties of the Directional Derivative	p. 52
Examples of Linearization	p. 53
Tensor Analysis	p. 57
The Gradient and Divergence Operators	p. 58
Integration Theorems	p. 60
Analysis of Three-Dimensional Truss Structures	p. 63
Introduction	p. 63
Kinematics	p. 65
Linearization of Geometrical Descriptors	p. 67
Internal Forces and Hyperelastic Constitutive Equations	p. 68
Nonlinear Equilibrium Equations and the Newton-Raphson Solution	p. 70
Equilibrium Equations	p. 70
Newton-Raphson Procedure	p. 71
Tangent Elastic Stiffness Matrix	p. 72
Elasto-Plastic Behavior	p. 74
Multiplicative Decomposition of the Stretch	p. 74
Rate-independent Plasticity	p. 76
Incremental Kinematics	p. 80
Time Integration	p. 83
Stress Update and Return Mapping	p. 83
Algorithmic Tangent Modulus	p. 86
Revised Newton-Raphson Procedure	p. 88
Examples	p. 89
Inclined Axial Rod	p. 89
Trussed Frame	p. 89
Kinematics	p. 94
Introduction	p. 94
The Motion	p. 94
Material and Spatial Descriptions	p. 95
Deformation Gradient	p. 97
Strain	p. 101
Polar Decomposition	p. 105
Volume Change	p. 110
Distortional Component of the Deformation Gradient	p. 112
Area Change	p. 115
Linearized Kinematics	p. 116
Linearized Deformation Gradient	p. 116
Linearized Strain	p. 117
Linearized Volume Change	p. 118
Velocity and Material Time Derivatives	p. 118
Velocity	p. 118
Material Time Derivative	p. 119
Directional Derivative and Time Rates	p. 120
Velocity Gradient	p. 122
Rate of Deformation	p. 122
Spin Tensor	p. 125
Rate of Change of Volume	p. 128
Superimposed Rigid Body Motions and Objectivity	p. 130
Stress and Equilibrium	p. 134
Introduction	p. 134
Cauchy Stress Tensor	p. 134
Definition	p. 134
Stress Objectivity	p. 138
Equilibrium	p. 139
Translational Equilibrium	p. 139
Rotational Equilibrium	p. 141
Principle of Virtual Work	p. 142
Work Conjugacy and Alternative Stress Representations	p. 144
The Kirchhoff Stress Tensor	p. 144
The First Piola-Kirchhoff Stress Tensor	p. 145
The Second Piola-Kirchhoff Stress Tensor	p. 148
Deviatoric and Pressure Components	p. 151
Stress Rates	p. 152
Hyperelasticity	p. 155
Introduction	p. 155
Hyperelasticity	p. 155
Elasticity Tensor	p. 157
The Material or Lagrangian Elasticity Tensor	p. 157
The Spatial or Eulerian Elasticity Tensor	p. 158
Isotropic Hyperelasticity	p. 160
Material Description	p. 160
Spatial Description	p. 161
Compressible Neo-Hookean Material	p. 162
Incompressible and Nearly Incompressible Materials	p. 166
Incompressible Elasticity	p. 166
Incompressible Neo-Hookean Material	p. 169
Nearly Incompressible Hyperelastic Materials	p. 171
Isotropic Elasticity in Principal Directions	p. 174
Material Description	p. 174
Spatial Description	p. 175
Material Elasticity Tensor	p. 176
Spatial Elasticity Tensor	p. 178
A Simple Stretch-based Hyperelastic Material	p. 179
Nearly Incompressible Material in Principal Directions	p. 180
Plane Strain and Plane Stress Cases	p. 183
Uniaxial Rod Case	p. 184
Large Elasto-Plastic Deformations	p. 188
Introduction	p. 188
The Multiplicative Decomposition	p. 189
Rate Kinematics	p. 193
Rate-Independent Plasticity	p. 197
Principal Directions	p. 200
Incremental Kinematics	p. 204
The Radial Return Mapping	p. 207
Algorithmic Tangent Modulus	p. 209
Two-Dimensional Cases	p. 211
Linearized Equilibrium Equations	p. 216
Introduction	p. 216
Linearization and Newton-Raphson Process	p. 216
Lagrangian Linearized Internal Virtual Work	p. 218
Eulerian Linearized Internal Virtual Work	p. 219
Linearized External Virtual Work	p. 221
Body Forces	p. 221
Surface Forces	p. 222
Variational Methods and Incompressibility	p. 224
Total Potential Energy and Equilibrium	p. 225
Lagrange Multiplier Approach to Incompressibility	p. 225
Penalty Methods for Incompressibility	p. 228
Hu-Washizu Variational Principle for Incompressibility	p. 229
Mean Dilatation Procedure	p. 231
Discretization and Solution	p. 237
Introduction	p. 237
Discretized Kinematics	p. 237
Discretized Equilibrium Equations	p. 242
General Derivation	p. 242
Derivation in Matrix Notation	p. 245
Discretization of the Linearized Equilibrium Equations	p. 247
Constitutive Component: Indicial Form	p. 248
Constitutive Component: Matrix Form	p. 249
Initial Stress Component	p. 251
External Force Component	p. 252
Tangent Matrix	p. 254
Mean Dilatation Method for Incompressibility	p. 256
Implementation of the Mean Dilatation Method	p. 256
Newton-Raphson Iteration and Solution Procedure	p. 258
Newton-Raphson Solution Algorithm	p. 258
Line Search Method	p. 259
Arc-Length Method	p. 261
Computer Implementation	p. 266
Introduction	p. 266
User Instructions	p. 267
Output File Description	p. 273
Element Types	p. 276
Solver Details	p. 277
Constitutive Equation Summary	p. 277
Program Structure	p. 284
Main Routine flagshyp	p. 284
Routine elemtk	p. 292
Routine radialrtn	p. 298
Routine ksigma	p. 299
Routine bpress	p. 301
Examples	p. 302
Simple Patch Test	p. 302
Nonlinear Truss	p. 303
Strip With a Hole	p. 304
Plane Strain Nearly Incompressible Strip	p. 305
Elasto-plastic Cantilever	p. 306
Appendix: Dictionary of Main Variables	p. 308
Bibliography	p. 312
Index	p. 314
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