1. Introduction ................................................. 1
1.1. Texture formation during the plastic flow process ....... 1
1.2. Properties of materials with texture .................... 2
1.3. Mathematical and physical theory of plasticity .......... 2
2. General principles ........................................... 7
2.1. Motion and deformation .................................. 7
2.1.1. Motion description in material and spatial
coordinates ...................................... 7
2.1.2. Deformation, strain-rate and spin measures ....... 9
2.1.3. Finite elastic-plastic deformations ............. 13
2.2. Stress state ........................................... 18
2.2.1. Stress and stress-rate measures ................. 18
2.2.2. Conjugate stress and strain-rate measures ....... 21
2.2.3. Equilibrium equations and boundary conditions ... 22
2.2.4. Principle of virtual velocities ................. 23
2.3. Constitutive relations ................................. 24
2.3.1. Physical admissibility .......................... 24
2.3.2. Constitutive principles ......................... 27
2.3.3. Material symmetry ............................... 28
2.3.4. Plastic flow rule and plastic spin rule ......... 29
3. Anisotropic plasticity ...................................... 34
3.1. Anisotropic yield conditions and hardening rules ....... 34
3.1.1. General form of anisotropic yield condition ..... 34
3.1.2. The Hill orthotropic criterion .................. 40
3.1.3. Hardening evolution laws ........................ 43
3.2. Generalized relations of anisotropic plasticity ........ 48
3.2.1. Generalized plastic potential ................... 48
3.2.2. Conjugate flow rule and conjugate plastic spin
rule ............................................ 49
3.2.3. Generalized Prandtl-Reuss equations ............. 50
3.3. Models of plastically anisotropic materials ............ 56
3.3.1. Rigid-plastic materials ......................... 56
3.3.2. Elastic-plastic materials ....................... 59
3.3.3. Finite element formulation ...................... 60
3.4. Advanced plastic deformations .......................... 64
3.4.1. Application to sheet metal forming processes .... 64
3.4.2. Differences between theoretical predictions
and experimental results ........................ 69
3.4.3. More physical approach - theories with plastic
corner effect ................................... 74
4. Sources of induced plastic anisotropy ....................... 80
4.1. Physical approach ...................................... 80
4.1.1. Background of considerations .................... 80
4.1.2. Plastic behaviour of single crystals ............ 84
4.1.3. Plastic behaviour of crystalline aggregates ..... 93
4.2. Structure and kinematics of single crystals ........... 101
4.2.1. Crystallographic lattice frame and system of
slip systems ................................... 101
4.2.2. Stereographic projection of crystalline
lattice ........................................ 106
4.2.3. Motion of the lattice .......................... 110
4.2.4. Motion of the material ......................... 113
4.2.5. Coupling of the lattice motion and the
material motion ................................ 118
4.2.6. Uniform deformations of single crystals ........ 118
4.2.7. Finite elastic-plastic deformations of
single crystals ................................ 121
5. Plasticity of crystals ..................................... 125
5.1. Rigid-plastic crystals behaviour ...................... 125
5.1.1. General form of yield criterion ................ 125
5.1.2. The Schmid law - independent slip systems ...... 127
5.1.3. Crystals with interacting slip systems ......... 131
5.1.4. Generalized plastic potential for strain rate
and plastic spin ............................... 135
5.1.5. Regularized Schmid law - interacting slip
systems ........................................ 137
5.1.6. Complete system of equations ................... 142
5.2. Elastic-plastic crystals behaviour .................... 145
5.2.1. General form of constitutive relations ......... 145
5.2.2. The Schmid model ............................... 148
5.2.3. Rate-dependent model ........................... 150
5.2.4. Regularized Schmid model ....................... 150
5.2.5. Lattice reorientations ......................... 155
5.2.6. Latent hardening ............................... 157
5.2.7. Complete system of equations ................... 159
6. From textures to plastic anisotropy ........................ 161
6.1. Elements of texture analysis .......................... 161
6.1.1. The Orientation Distribution Function .......... 161
6.1.2. Direct and inverse pole figures ................ 166
6.1.3. Methods of texture analysis .................... 171
6.2. Continuous model of textured materials ................ 177
6.2.1. Preliminary remarks ............................ 177
6.2.2. Extended physical space ........................ 180
6.2.3. Textured material body ......................... 181
6.2.4. The equation of texture evolution .............. 183
6.2.5. Global virtual power principle ................. 186
6.2.6. Finite element analysis of crystal aggregate ... 188
6.2.7. Finite element analysis of textured
continuum ...................................... 190
6.3. Deformation textures development ...................... 194
6.3.1. Fibre and rolling textures ..................... 194
6.3.2. Single crystal behaviour ....................... 200
6.3.3. Polycrystal behaviour .......................... 203
7. Refined models of anisotropic plasticity ................... 208
7.1. Non-quadratic plastic potentials ...................... 208
7.1.1. Direct generalizations ......................... 208
7.1.2. Tricomponent plane stress yield surface ........ 212
7.1.3. Six-component yield surface .................... 217
7.2. Dual plastic potentials ............................... 220
7.2.1. Quadratic strain-rate potential ................ 220
7.2.2. Texture-adjusted strain-rate potential ......... 222
7.2.3. Convex non-quadratic strain-rate potential ..... 225
References .................................................... 229
Index ......................................................... 237
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