Preface to the Second Edition ................................. VII
Preface to the First Edition ................................... IX
Part I Continuum Mechanics and Elastoplasticity Theory
1 Preliminaries ................................................ 3
1.1 Introduction ............................................ 3
1.2 Some Historical Remarks ................................. 5
1.3 Notation ................................................ 9
2 Continuum Mechanics and Linearized Elasticity ............... 15
2.1 Kinematics ............................................. 16
2.2 Balance of Momentum; Stress ............................ 22
2.3 Linearly Elastic Materials ............................. 27
2.4 Isotropic Elasticity ................................... 29
2.5 A Thermodynamic Framework for Elasticity ............... 32
2.6 Initial-Boundary and Boundary Value Problems for
Linearized Elasticity .................................. 36
2.7 Thermodynamics with Internal Variables ................. 37
3 Elastoplastic Media ......................................... 41
3.1 Physical Background and Motivation ..................... 41
3.2 Three-Dimensional Elastoplastic Behavior ............... 47
3.3 Examples of Yield Criteria ............................. 61
3.4 Yield Criteria for Dilatant Materials .................. 66
3.4.1 Examples ........................................ 66
3.4.2 A further note on non-smooth yield surfaces ..... 69
3.5 Hardening Laws ......................................... 70
3.6 Single-crystal Plasticity .............................. 74
3.7 Strain-gradient Plasticity ............................. 82
3.7.1 Polycrystalline plasticity ...................... 82
3.7.2 Gradient single-crystal plasticity .............. 87
3.8 Bibliographical Remarks ................................ 94
4 The Plastic Flow Law in a Convex-Analytic Setting ........... 95
4.1 Some Results from Convex Analysis ...................... 96
4.2 Basic Plastic Flow Relations of Elastoplasticity ...... 106
4.3 Strain-gradient Plasticity ............................ 117
4.3.1 The Aifantis model ............................. 118
4.3.2 Polycrystalline strain-gradient plasticity ..... 119
4.3.3 Strain-gradient single-crystal plasticity ...... 121
Part II The Variational Problems of Elastoplasticity
5 Basics of Functional Analysis and Function Spaces .......... 125
5.1 Results from Functional Analysis ...................... 125
5.2 Function Spaces ....................................... 135
5.2.1 The Spaces Cm(Ω), Сm(Ω), and Lp(Ω) ............. 135
5.2.2 Sobolev Spaces ................................. 139
5.2.3 Spaces of Vector-Valued Functions .............. 147
6 Variational Equations and Inequalities ..................... 151
6.1 Variational Formulation of Elliptic Boundary Value
Problems .............................................. 151
6.2 Elliptic Variational Inequalities ..................... 163
6.3 Parabolic Variational Inequalities .................... 171
6.4 Qualitative Analysis of an Abstract Problem ........... 174
7 The Primal Variational Problem of Elastoplasticity ......... 187
7.1 Classical Elastoplasticity with Hardening ............. 189
7.1.1 Variational formulation ........................ 189
7.1.2 Analysis of the problem ........................ 195
7.2 Classical Single-crystal Plasticity ................... 201
7.3 Strain-gradient Plasticity ............................ 203
7.3.1 The Aifantis model ............................. 203
7.3.2 The Gurtin model of strain-gradient
plasticity ..................................... 204
7.4 Strain-gradient Single-crystal Plasticity ............. 213
7.4.1 Weak formulation of the problem ................ 213
7.4.2 Well-posedness ................................. 215
7.5 Stability Analysis .................................... 219
8 The Dual Variational Problem of Classical
Elastoplasticity ........................................... 225
8.1 The Dual Variational Problem .......................... 226
8.2 Analysis of the Stress Problem ........................ 230
8.3 Analysis of the Dual Problem .......................... 242
8.4 Rate Form of Stress-Strain Relation ................... 246
Part III Numerical Analysis of the Variational Problems
9 Introduction to Finite Element Analysis .................... 251
9.1 Basics of the Finite Element Method ................... 253
9.2 Affine Families of Finite Elements .................... 255
9.3 Local Interpolation Error Estimates ................... 259
9.4 Global Interpolation Error Estimates .................. 265
10 Approximation of Variational Problems ...................... 269
10.1 Approximation of Elliptic Variational Equations ....... 269
10.2 Numerical Approximation of Elliptic Variational
Inequalities .......................................... 273
10.3 Approximation of Parabolic Variational Inequalities ... 282
11 Approximations of the Abstract Problem ..................... 285
11.1 Spatially Discrete Approximations ..................... 286
11.2 Time-Discrete Approximations .......................... 288
11.3 Fully Discrete Approximations ......................... 295
11.4 Convergence Under Minimal Regularity .................. 301
12 Numerical Analysis of the Primal Problem ................... 319
12.1 Error Analysis of Discrete Approximations of the
Primal Problem ........................................ 320
12.1.1 Problems of classical elastoplasticity with
hardening ...................................... 320
12.1.2 Problems of strain-gradient plasticity ......... 329
12.2 Solution Algorithms ................................... 337
12.3 Convergence Analysis of the Solution Algorithms ....... 348
12.4 Regularization Technique and A Posteriori Error
Analysis .............................................. 355
12.5 Fully Discrete Schemes with Numerical Integration ..... 363
13 Numerical Analysis of the Dual Problem ..................... 371
13.1 Time-Discrete Approximations of the Stress Problem .... 373
13.2 Time-Discrete Approximations of the Dual Problem ...... 379
13.3 Fully Discrete Approximations of the Dual Problem ..... 383
13.4 Predictor-Corrector Algorithms ........................ 393
13.5 Computation of the Closest-Point Projections .......... 401
References .................................................... 405
Index ......................................................... 415
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