Plasticity : mathematical theory and numerical analysis

Cover......Page 1
Plasticity......Page 4
Preface to the Second Edition......Page 8
Preface to the First Edition......Page 10
Contents......Page 14
1.1 Introduction......Page 18
1.2 Some Historical Remarks......Page 20
1.3 Notation......Page 24
2 Continuum Mechanics and Linearized Elasticity......Page 29
2.1 Kinematics......Page 30
2.2 Balance of Momentum; Stress......Page 36
2.3 Linearly Elastic Materials......Page 41
2.4 Isotropic Elasticity......Page 43
2.5 A Thermodynamic Framework for Elasticity......Page 46
2.6 Initial–Boundary and Boundary Value Problems for Linearized Elasticity......Page 50
2.7 Thermodynamics with Internal Variables......Page 51
3.1 Physical Background and Motivation......Page 54
3.2 Three-Dimensional Elastoplastic Behavior......Page 60
3.3 Examples of Yield Criteria......Page 74
3.4.1 Examples......Page 79
3.4.2 A further note on non-smooth yield surfaces......Page 82
3.5 Hardening Laws......Page 83
3.6 Single-crystal Plasticity......Page 87
3.7.1 Polycrystalline plasticity......Page 95
3.7.2 Gradient single-crystal plasticity......Page 100
3.8 Bibliographical Remarks......Page 107
4 The Plastic Flow Law in a Convex-Analytic Setting......Page 108
4.1 Some Results from Convex Analysis......Page 109
4.2 Basic Plastic Flow Relations of Elastoplasticity......Page 119
4.3 Strain-gradient Plasticity......Page 130
4.3.1 The Aifantis model......Page 131
4.3.2 Polycrystalline strain-gradient plasticity......Page 132
4.3.3 Strain-gradient single-crystal plasticity......Page 134
5.1 Results from Functional Analysis......Page 136
5.2.1 The Spaces Cm(Ω), Cm(Ω), and Lp(Ω)......Page 146
5.2.2 Sobolev Spaces......Page 150
5.2.3 Spaces of Vector-Valued Functions......Page 158
6.1 Variational Formulation of Elliptic Boundary Value Problems......Page 162
6.2 Elliptic Variational Inequalities......Page 174
6.3 Parabolic Variational Inequalities......Page 182
6.4 Qualitative Analysis of an Abstract Problem......Page 185
7 The Primal Variational Problem of Elastoplasticity......Page 197
7.1.1 Variational formulation......Page 199
7.1.2 Analysis of the problem......Page 205
7.2 Classical Single-crystal Plasticity......Page 211
7.3.1 The Aifantis model......Page 213
7.3.2 The Gurtin model of strain-gradient plasticity......Page 214
7.4.1 Weak formulation of the problem......Page 223
7.4.2 Well-posedness......Page 225
7.5 Stability Analysis......Page 229
8 The Dual Variational Problem of Classical Elastoplasticity......Page 234
8.1 The Dual Variational Problem......Page 235
8.2 Analysis of the Stress Problem......Page 239
8.3 Analysis of the Dual Problem......Page 251
8.4 Rate Form of Stress–Strain Relation......Page 255
9 Introduction to Finite Element Analysis......Page 258
9.1 Basics of the Finite Element Method......Page 260
9.2 Affine Families of Finite Elements......Page 262
9.3 Local Interpolation Error Estimates......Page 266
9.4 Global Interpolation Error Estimates......Page 272
10.1 Approximation of Elliptic Variational Equations......Page 275
10.2 Numerical Approximation of Elliptic Variational Inequalities......Page 279
10.3 Approximation of Parabolic Variational Inequalities......Page 288
11 Approximations of the Abstract Problem......Page 290
11.1 Spatially Discrete Approximations......Page 291
11.2 Time-Discrete Approximations......Page 293
11.3 Fully Discrete Approximations......Page 300
11.4 Convergence Under Minimal Regularity......Page 306
12 Numerical Analysis of the Primal Problem......Page 323
12.1.1 Problems of classical elastoplasticity with hardening......Page 324
12.1.2 Problems of strain-gradient plasticity......Page 333
12.2 Solution Algorithms......Page 341
12.3 Convergence Analysis of the Solution Algorithms......Page 352
12.4 Regularization Technique and A Posteriori Error Analysis......Page 359
12.5 Fully Discrete Schemes with Numerical Integration......Page 367
13 Numerical Analysis of the Dual Problem......Page 375
13.1 Time-Discrete Approximations of the Stress Problem......Page 377
13.2 Time-Discrete Approximations of the Dual Problem......Page 383
13.3 Fully Discrete Approximations of the Dual Problem......Page 387
13.4 Predictor–Corrector Algorithms......Page 397
13.5 Computation of the Closest-Point Projections......Page 405
References......Page 409
Index......Page 419