Nonlinear continuum mechanics for finite element analysis

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010178735	TA405 B66 2008	Open Access Book	Book	Searching... Unknown
Searching... PSZ JB	30000010200070	TA405 B66 2008	Open Access Book	Book	Searching... Unknown

Designing engineering components that make optimal use of materials requires consideration of the nonlinear characteristics associated with both manufacturing and working environments. The modeling of these characteristics can only be done through numerical formulation and simulation, and this requires an understanding of both the theoretical background and associated computer solution techniques. By presenting both nonlinear continuum analysis and associated finite element techniques under one roof, Bonet and Wood provide, in this 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. Worked examples and exercises complete each chapter, making the text an essential resource for postgraduates studying nonlinear continuum mechanics. It is also ideal for those in industry requiring an appreciation of the way in which their computer simulation programs work.

Preface	p. xv
1 Introduction	p. 1
1.1 Nonlinear Computational Mechanics	p. 1
1.2 Simple Examples of Nonlinear Structural Behavior	p. 2
1.2.1 Cantilever	p. 2
1.2.2 Column	p. 3
1.3 Nonlinear Strain Measures	p. 4
1.3.1 One-Dimensional Strain Measures	p. 5
1.3.2 Nonlinear Truss Example	p. 6
1.3.3 Continuum Strain Measures	p. 10
1.4 Directional Derivative, Linearization and Equation Solution	p. 13
1.4.1 Directional Derivative	p. 14
1.4.2 Linearization and Solution of Nonlinear Algebraic Equations	p. 16
2 Mathematical Preliminaries	p. 22
2.1 Introduction	p. 22
2.2 Vector and Tensor Algebra	p. 22
2.2.1 Vectors	p. 23
2.2.2 Second-Order Tensors	p. 28
2.2.3 Vector and Tensor Invariants	p. 37
2.2.4 Higher-Order Tensors	p. 41
2.3 Linearization and the Directional Derivative	p. 47
2.3.1 One Degree of Freedom	p. 48
2.3.2 General Solution to a Nonlinear Problem	p. 49
2.3.3 Properties of the Directional Derivative	p. 52
2.3.4 Examples of Linearization	p. 53
2.4 Tensor Analysis	p. 57
2.4.1 The Gradient and Divergence Operators	p. 58
2.4.2 Integration Theorems	p. 60
3 Analysis of Three-Dimensional Truss Structures	p. 63
3.1 Introduction	p. 63
3.2 Kinematics	p. 65
3.2.1 Linearization of Geometrical Descriptors	p. 67
3.3 Internal Forces and Hyperelastic Constitutive Equations	p. 68
3.4 Nonlinear Equilibrium Equations and the Newton-Raphson Solution	p. 70
3.4.1 Equilibrium Equations	p. 70
3.4.2 Newton-Raphson Procedure	p. 71
3.4.3 Tangent Elastic Stiffness Matrix	p. 72
3.5 Elasto-Plastic Behavior	p. 74
3.5.1 Multiplicative Decomposition of the Stretch	p. 74
3.5.2 Rate-independent Plasticity	p. 76
3.5.3 Incremental Kinematics	p. 80
3.5.4 Time Integration	p. 83
3.5.5 Stress Update and Return Mapping	p. 83
3.5.6 Algorithmic Tangent Modulus	p. 86
3.5.7 Revised Newton-Raphson Procedure	p. 88
3.6 Examples	p. 89
3.6.1 Inclined Axial Rod	p. 89
3.6.2 Trussed Frame	p. 89
4 Kinematics	p. 94
4.1 Introduction	p. 94
4.2 The Motion	p. 94
4.3 Material and Spatial Descriptions	p. 95
4.4 Deformation Gradient	p. 97
4.5 Strain	p. 101
4.6 Polar Decomposition	p. 105
4.7 Volume Change	p. 110
4.8 Distortional Component of the Deformation Gradient	p. 112
4.9 Area Change	p. 115
4.10 Linearized Kinematics	p. 116
4.10.1 Linearized Deformation Gradient	p. 116
4.10.2 Linearized Strain	p. 117
4.10.3 Linearized Volume Change	p. 118
4.11 Velocity and Material Time Derivatives	p. 118
4.11.1 Velocity	p. 118
4.11.2 Material Time Derivative	p. 119
4.11.3 Directional Derivative and Time Rates	p. 120
4.11.4 Velocity Gradient	p. 122
4.12 Rate of Deformation	p. 122
4.13 Spin Tensor	p. 125
4.14 Rate of Change of Volume	p. 128
4.15 Superimposed Rigid Body Motions and Objectivity	p. 130
5 Stress and Equilibrium	p. 134
5.1 Introduction	p. 134
5.2 Cauchy Stress Tensor	p. 134
5.2.1 Definition	p. 134
5.2.2 Stress Objectivity	p. 138
5.3 Equilibrium	p. 139
5.3.1 Translational Equilibrium	p. 139
5.3.2 Rotational Equilibrium	p. 141
5.4 Principle of Virtual Work	p. 142
5.5 Work Conjugacy and Alternative Stress Representations	p. 144
5.5.1 The Kirchhoff Stress Tensor	p. 144
5.5.2 The First Piola-Kirchhoff Stress Tensor	p. 145
5.5.3 The Second Piola-Kirchhoff Stress Tensor	p. 148
5.5.4 Deviatoric and Pressure Components	p. 151
5.6 Stress Rates	p. 152
6 Hyperelasticity	p. 155
6.1 Introduction	p. 155
6.2 Hyperelasticity	p. 155
6.3 Elasticity Tensor	p. 157
6.3.1 The Material or Lagrangian Elasticity Tensor	p. 157
6.3.2 The Spatial or Eulerian Elasticity Tensor	p. 158
6.4 Isotropic Hyperelasticity	p. 160
6.4.1 Material Description	p. 160
6.4.2 Spatial Description	p. 161
6.4.3 Compressible Neo-Hookean Material	p. 162
6.5 Incompressible and Nearly Incompressible Materials	p. 166
6.5.1 Incompressible Elasticity	p. 166
6.5.2 Incompressible Neo-Hookean Material	p. 169
6.5.3 Nearly Incompressible Hyperelastic Materials	p. 171
6.6 Isotropic Elasticity in Principal Directions	p. 174
6.6.1 Material Description	p. 174
6.6.2 Spatial Description	p. 175
6.6.3 Material Elasticity Tensor	p. 176
6.6.4 Spatial Elasticity Tensor	p. 178
6.6.5 A Simple Stretch-based Hyperelastic Material	p. 179
6.6.6 Nearly Incompressible Material in Principal Directions	p. 180
6.6.7 Plane Strain and Plane Stress Cases	p. 183
6.6.8 Uniaxial Rod Case	p. 184
7 Large Elasto-Plastic Deformations	p. 188
7.1 Introduction	p. 188
7.2 The Multiplicative Decomposition	p. 189
7.3 Rate Kinematics	p. 193
7.4 Rate-Independent Plasticity	p. 197
7.5 Principal Directions	p. 200
7.6 Incremental Kinematics	p. 204
7.6.1 The Radial Return Mapping	p. 207
7.6.2 Algorithmic Tangent Modulus	p. 209
7.7 Two-Dimensional Cases	p. 211
8 Linearized Equilibrium Equations	p. 216
8.1 Introduction	p. 216
8.2 Linearization and Newton-Raphson Process	p. 216
8.3 Lagrangian Linearized Internal Virtual Work	p. 218
8.4 Eulerian Linearized Internal Virtual Work	p. 219
8.5 Linearized External Virtual Work	p. 221
8.5.1 Body Forces	p. 221
8.5.2 Surface Forces	p. 222
8.6 Variational Methods and Incompressibility	p. 224
8.6.1 Total Potential Energy and Equilibrium	p. 225
8.6.2 Lagrange Multiplier Approach to Incompressibility	p. 225
8.6.3 Penalty Methods for Incompressibility	p. 228
8.6.4 Hu-Washizu Variational Principle for Incompressibility	p. 229
8.6.5 Mean Dilatation Procedure	p. 231
9 Discretization and Solution	p. 237
9.1 Introduction	p. 237
9.2 Discretized Kinematics	p. 237
9.3 Discretized Equilibrium Equations	p. 242
9.3.1 General Derivation	p. 242
9.3.2 Derivation in Matrix Notation	p. 245
9.4 Discretization of the Linearized Equilibrium Equations	p. 247
9.4.1 Constitutive Component: Indicial Form	p. 248
9.4.2 Constitutive Component: Matrix Form	p. 249
9.4.3 Initial Stress Component	p. 251
9.4.4 External Force Component	p. 252
9.4.5 Tangent Matrix	p. 254
9.5 Mean Dilatation Method for Incompressibility	p. 256
9.5.1 Implementation of the Mean Dilatation Method	p. 256
9.6 Newton-Raphson Iteration and Solution Procedure	p. 258
9.6.1 Newton-Raphson Solution Algorithm	p. 258
9.6.2 Line Search Method	p. 259
9.6.3 Arc-Length Method	p. 261
10 Computer Implementation	p. 266
10.1 Introduction	p. 266
10.2 User Instructions	p. 267
10.3 Output File Description	p. 273
10.4 Element Types	p. 276
10.5 Solver Details	p. 277
10.6 Constitutive Equation Summary	p. 277
10.7 Program Structure	p. 284
10.8 Main Routine flagshyp	p. 284
10.9 Routine elemtk	p. 292
10.10 Routine radialrtn	p. 298
10.11 Routine ksigma	p. 299
10.12 Routine bpress	p. 301
10.13 Examples	p. 302
10.13.1 Simple Patch Test	p. 302
10.13.2 Nonlinear Truss	p. 303
10.13.3 Strip With a Hole	p. 304
10.13.4 Plane Strain Nearly Incompressible Strip	p. 305
10.13.5 Elasto-plastic Cantilever	p. 306
10.14 Appendix: Dictionary of Main Variables	p. 308
Bibliography	p. 312
Index	p. 314

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