This book presents a theoretical treatment of nonlinear behavior of solids and structures in such a way that it is suitable for numerical computation, typically using the Finite Element Method. Starting out from elementary concepts, the author systematically uses the principle of virtual work, initially illustrated by truss structures, to give a self-contained and rigorous account of the basic methods. The author illustrates the combination of translations and rotations by finite deformation beam theories in absolute and co-rotation format, and describes the deformation of a three-dimensional continuum in material form. A concise introduction to finite elasticity is followed by an extension to elasto-plastic materials via internal variables and the maximum dissipation principle. Finally, the author presents numerical techniques for solution of the nonlinear global equations and summarizes recent results on momentum and energy conserving integration of time-dependent problems. Exercises, examples and algorithms are included throughout.
Author(s): Steen Krenk
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 361
Cover......Page 1
Title......Page 3
Copyright......Page 4
Dedication......Page 5
Contents......Page 7
Preface......Page 11
1 Introduction......Page 13
1.1 A simple non-linear problem......Page 14
1.1.1 Equilibrium......Page 15
1.1.2 Virtual work and potential energy......Page 18
1.2 Simple non-linear solution methods......Page 19
1.2.1 Explicit incremental method......Page 20
1.2.2 Newton–Raphson method......Page 21
1.2.3 Modified Newton–Raphson method......Page 25
1.3 Summary and outlook......Page 26
1.4 Exercises......Page 27
2 Non-linear bar elements......Page 29
2.1 Deformation and strain......Page 30
2.2 Equilibrium and virtual work......Page 32
2.3 Tangent stiffness matrix......Page 36
2.4 Use of shape functions......Page 38
2.5 Assembly of global stiffness and forces......Page 43
2.6 Total or updated Lagrangian formulation......Page 48
2.7 Summing up the principles......Page 51
2.8 Exercises......Page 55
3 Finite rotations......Page 59
3.1 The rotation tensor......Page 61
3.2 Rotation of a vector into a specified direction......Page 65
3.3 The increment of the rotation variation......Page 67
3.4 Parameter representation of an incremental rotation......Page 72
3.5 Quaternion parameter representation......Page 75
3.5.1 Representation of the rotation tensor......Page 76
3.5.2 Addition of two rotations......Page 77
3.5.3 Incremental rotation from quaternion parameters......Page 79
3.5.4 Mean and difference of two rotations......Page 80
3.6 Alternative representation of the rotation tensor......Page 81
3.7 Summary of rotations and their virtual work......Page 84
3.8 Exercises......Page 85
4 Finite rotation beam theory......Page 88
4.1 Equilibrium equations......Page 89
4.2 Virtual work, strain and curvature......Page 90
4.3 Increment of the virtual work equation......Page 93
4.3.1 Constitutive stiffness......Page 94
4.3.2 Geometric stiffness......Page 95
4.3.3 The load increments......Page 97
4.4 Finite element implementation......Page 98
4.4.1 Element stiffness matrix......Page 99
4.4.2 Loads and internal forces......Page 101
4.4.3 Shear locking......Page 103
4.5 Summary of ‘elastica’ beam theory......Page 110
4.6 Exercises......Page 111
5 Co-rotating beam elements......Page 112
5.1 Co-rotating beams in two dimensions......Page 113
5.1.1 Co-rotation form of the tangent sti.ness......Page 116
5.1.2 Element deformation stiffness......Page 119
5.1.3 Total tangent stiffness......Page 122
5.1.4 Finite element implementation......Page 124
5.2 Co-rotating beams in three dimensions......Page 129
5.2.1 Co-rotation form of the tangent stiffness......Page 132
Initial non-symmetric form of Kr......Page 136
Symmetric form of Kr......Page 137
5.2.2 Element deformation stiffness......Page 139
Local geometric stiffness......Page 140
5.2.3 Total tangent stiffness......Page 142
5.2.4 Finite element implementation......Page 145
Incremental formulation......Page 146
Total formulation......Page 149
5.3 Summary and extensions......Page 151
5.4 Exercises......Page 153
6 Deformation and equilibrium of solids......Page 157
6.1 Deformation and strain......Page 158
6.1.1 Non-linear strain......Page 160
6.1.2 Decomposition into deformation and rigid body motion......Page 163
6.2 Virtual work and stresses......Page 166
6.2.1 Piola–Kirchhoff stress......Page 167
6.2.2 Cauchy and Kirchhoff stresses......Page 170
6.2.3 Stress rates......Page 172
6.3 Total Lagrangian formulation......Page 177
6.3.1 Equilibrium and residual forces......Page 178
6.3.2 Tangent stiffness......Page 179
6.3.3 Finite element implementation......Page 182
6.4.1 Transformation from total to updated format......Page 186
6.4.2 Virtual work in the current configuration......Page 188
6.4.3 Finite element implementation......Page 192
6.5 Summary of non-linear motion of solids......Page 197
6.6 Exercises......Page 198
7 Elasto-plastic solids......Page 201
7.1 Elastic solids......Page 202
7.1.1 Stress invariants......Page 204
7.1.2 Strain invariants and small strain elasticity......Page 210
7.1.3 Isotropic elasticity at finite strain......Page 212
7.2 General plasticity theory......Page 215
7.2.1 Reversible deformation......Page 216
7.2.2 Maximum plastic dissipation rate......Page 219
7.2.3 Evolution equations......Page 224
7.2.4 Isotropic and kinematic hardening......Page 228
7.3 Von Mises plasticity models......Page 230
7.3.1 Yield surface and flow potential......Page 231
7.3.2 Explicit integration......Page 234
7.3.3 Radial return algorithm......Page 237
7.4 General aspects of plasticity models......Page 241
7.4.1 Combined isotropic and kinematic hardening......Page 242
7.4.2 Internal variables and non-associated flow......Page 246
7.4.3 General computational procedure......Page 249
7.5 Models for granular materials......Page 253
7.5.1 Flow potential and yield surface......Page 254
7.5.2 Elasticity and hardening......Page 259
7.6 Finite strain plasticity......Page 261
7.7 Summary......Page 264
7.8 Exercises......Page 265
8 Numerical solution techniques......Page 268
8.1 Iterative solution of equilibrium equations......Page 269
8.1.1 Non-linear iteration strategies......Page 271
8.1.2 Direction and step-size control......Page 272
8.2 Orthogonal residual method......Page 275
8.3 Arc-length methods......Page 282
8.3.1 General constraint formulation......Page 284
8.3.2 Hyperplane constraints......Page 286
8.3.3 Hypersphere constraint......Page 290
8.4 Quasi-Newton methods......Page 295
8.5 Summary......Page 299
8.6 Exercises......Page 300
9 Dynamic effects and time integration......Page 302
9.1 Newmark algorithm for linear systems......Page 304
9.1.1 Energy balance and stability......Page 307
9.1.2 Numerical accuracy and damping......Page 312
9.2 Non-linear Newmark algorithm......Page 316
9.3 Energy-conserving integration......Page 321
9.3.1 State-space formulation......Page 322
9.3.2 Non-linear kinematics for Green strain......Page 323
Special properties of Green strain......Page 324
Geometric and constitutive stiffness matrices......Page 325
9.3.3 Energy-conserving algorithm......Page 327
The iteration process......Page 328
9.4.1 Spectral analysis of linear systems......Page 335
9.4.2 Linear algorithm with energy dissipation......Page 337
9.4.3 Non-linear algorithm with energy dissipation......Page 339
9.5 Summary and outlook......Page 343
9.6 Exercises......Page 345
References......Page 348
Index......Page 357
