目錄 |
1. | Numerical Algorithms for the Simulation of Finite Plasticity with Microstructures / Boris Kramer | 1 |
1.1. | Introduction | 1 |
1.2. | Preliminaries and Notation | 3 |
1.3. | Convergent Adaptive Finite Element Method for the Two-Well Problem in Elasticity | 4 |
1.3.1. | Review of the Model Problem | 5 |
1.3.2. | Adaptive Algorithm | 6 |
1.3.3. | Convergence for the Deformation Gradient | 9 |
1.4. | Guaranteed Lower Energy Bounds for the Two-Well Problem | 10 |
1.4.1. | Nonconforming FEM and Discrete Energy Functional | 10 |
1.4.2. | Lower Energy Bounds | 12 |
1.4.3. | Guaranteed Error Control for the Pseudo-stress | 15 |
1.4.4. | Numerical Experiments | 16 |
1.5. | Discontinuous Galerkin Method for Degenerate Convex Minimization Problems | 17 |
1.5.1. | Optimal Design Benchmark | 18 |
1.5.2. | Discontinuous Galerkin Methods | 20 |
1.5.3. | Lifting Operator R | 20 |
1.5.4. | Connection with the Nonconforming Method | 21 |
1.5.5. | Adaptive Finite Element Method | 22 |
1.5.6. | Computational Experiments | 22 |
1.5.7. | ₀أ-shaped Domain | 23 |
1.5.8. | Slit Domain | 24 |
1.6. | Conclusions and Outlook | 25 |
2. | Variational Modeling of Slip: From Crystal Plasticity to Geological Strata / Carolin Kreisbeck | 31 |
2.1. | Introduction | 31 |
2.2. | Experimental Observation of Slip Microstructures in Nature | 33 |
2.2.1. | Chevron Folds in Rocks | 34 |
2.2.2. | Kink Bands in Stacks of Paper under Compression | 34 |
2.2.3. | Simple Laminates in Shear Experiments in Crystal Plasticity | 36 |
2.3. | The Hunt-Peletier-Wadee Model for Kink Bands | 37 |
2.4. | Variational Modeling of Microstructure | 38 |
2.5. | Models in Crystal Plasticity with One Active Slip System | 41 |
2.5.1. | Variational Formulation of Crystal Plasticity | 42 |
2.5.2. | Relaxation Results in Crystal Plasticity with One Slip System | 44 |
2.5.3. | Heuristic Origin of the Laminates | 46 |
2.5.4. | Relation to Kink Bands in Rocks | 49 |
2.5.5. | Elastic Approximation | 51 |
2.5.6. | Higher-Order Regularizations | 52 |
2.6. | Beyond One Slip-System | 53 |
2.6.1. | Two Slip Systems in a Plane | 53 |
2.6.2. | Three Slip Systems in a Plane | 54 |
3. | Rate-Independent versus Viscous Evolution of Laminate Microstructures in Finite Crystal Plasticity / Klaus Hackl | 63 |
3.1. | Introduction | 63 |
3.2. | Variational Modeling of Microstructures | 64 |
3.3. | Single Slip Crystal Plasticity | 67 |
3.4. | Partial Analytical Relaxation via Lamination | 67 |
3.5. | Rate-Independent Evolution | 70 |
3.5.1. | Evolution Equations | 70 |
3.5.2. | Laminate Rotation | 71 |
3.5.3. | Laminate Initiation | 72 |
3.5.4. | Numerical Scheme | 72 |
3.6. | Simulation of Rotating Laminates | 73 |
3.7. | Viscous Evolution | 75 |
3.7.1. | Evolution Equations | 76 |
3.7.2. | Laminate Rotation | 77 |
3.7.3. | Laminate Initiation | 77 |
3.8. | Comparison of the Laminate Evolution for the Rate-Independent Case and the Viscosity Limit | 78 |
3.9. | Conclusion and Discussion | 85 |
4. | Variational Gradient Plasticity: Local-Global Updates, Regularization and Laminate Microstructures in Single Crystals / Christian Miehe | 89 |
4.1. | Introduction | 90 |
4.2. | A Multifield Formulation of Gradient Crystal Plasticity | 93 |
4.2.1. | Introduction of Long-Range Field Variables | 93 |
4.2.2. | Introduction of Short-Range Field Variables | 96 |
4.2.3. | Energy Storage, Dissipation Potential and Load Functionals | 99 |
4.2.4. | Rate-Type Variational Principle and Euler Equations | 102 |
4.2.5. | Explicit Form of the Micro-force Balance Equations | 103 |
4.3. | Algorithmic Formulation of Gradient Crystal Plasticity | 103 |
4.3.1. | Time-Discrete Field Variables in Incremental Setting | 103 |
4.3.2. | Update Algorithms for the Short-Range Field Variables | 104 |
4.3.3. | Time-Discrete Incremental Variational Principle | 105 |
4.3.4. | Space-Time-Discrete Incremental Variational Principle | 106 |
4.4. | Example 1: Analysis of an F.C.C. Crystal Grain Aggregate | 108 |
4.4.1. | Slip Systems and Euler Angles | 108 |
4.4.2. | Voronoi-Tessellated Unit Cell under Shear | 109 |
4.5. | Example 2: Laminate Microstructure in Single Crystals | 110 |
4.5.1. | Double Slip Systems | 111 |
4.5.2. | Implications of Same Plane Double Slip | 112 |
4.5.3. | Laminate Deformation Microstructure in Single Crystal Copper | 114 |
4.6. | Conclusion | 118 |
5. | Variational Approaches and Methods for Dissipative Material Models with Multiple Scales / Alexander Mielke | 125 |
5.1. | Introduction | 125 |
5.2. | Variational Formulations for Evolution | 127 |
5.2.1. | Generalized Gradient Systems and the Energy-Dissipation Principle | 128 |
5.2.2. | Rate-Independent Systems and Energetic Solutions | 132 |
5.3. | Evolutionary ₀أ-Convergence | 134 |
5.3.1. | pE-convergence for Generalized Gradient Systems | 134 |
5.3.2. | pE-convergence for Rate-Independent Systems | 137 |
5.4. | Justification of Rate-Independent Models | 138 |
5.4.1. | Wiggly Energies Give Rise to Rate-Independent Friction | 139 |
5.4.2. | 1D Elastoplasticity as Limit of a Chain of Bistable Springs | 141 |
5.4.3. | Balanced-Viscosity Solutions as Vanishing-Viscosity Limits | 143 |
5.5. | Rate-Independent Evolution of Microstructures | 147 |
5.5.1. | Laminate Evolution in Finite-Strain Plasticity | 148 |
5.5.2. | A Two-Phase Shape-Memory Model for Small Strains | 149 |
6. | Energy Estimates, Relaxation, and Existence for Strain-Gradient Plasticity with Cross-Hardening / Patrick W. Dondl | 157 |
6.1. | Introduction | 158 |
6.2. | A Continuum Model for Strain-Gradient Plasticity with Cross Hardening | 159 |
6.2.1. | Plastic Shear | 160 |
6.2.2. | Locks and Cross-Hardening | 161 |
6.2.3. | Geometrically Necessary Dislocations | 162 |
6.2.4. | The Model | 163 |
6.3. | Relaxation of the Single-Slip Condition | 164 |
6.4. | Some Remarks about Existence of Minimizers | 168 |
6.5. | Energy Estimates for a Shear Experiment | 168 |
6.6. | Conclusions | 171 |
7. | Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations / Caterina Ida Zeppieri | 175 |
7.1. | Introduction | 175 |
7.2. | Key Mathematical Challenges | 183 |
7.3. | Heuristics for Scaling Regimes | 184 |
7.3.1. | The Core Energy of a Single Dislocation | 184 |
7.3.2. | The Core Energy of Many Dislocations | 186 |
7.3.3. | The Interaction Energy | 187 |
7.4. | Main Result | 189 |
7.4.1. | Set-Up | 189 |
7.4.2. | Results | 191 |
7.5. | Ideas of Proof | 193 |
8. | Microstructure in Plasticity, a Comparison between Theory and Experiment / Patrick W. Dondl | 205 |
8.1. | Introduction | 205 |
8.2. | Modeling Continuum Plasticity | 207 |
8.3. | A Single-Pass Shear Deformation Experiment and the Resulting Microstructure | 208 |
8.3.1. | Sample Preparation and Shear Deformation Experiments | 208 |
8.3.2. | Digital Image Correlation for Strain Mapping and EBSD for Texture Mapping | 209 |
8.3.3. | Outcome of the Single Crystal Shear Deformation Experiments | 210 |
8.3.4. | Energy Minimizing Microstructure | 212 |
8.3.5. | An Analysis of the Substructure Within the Lamination Bands | 215 |
8.4. | Conclusions | 216 |
9. | Construction of Statistically Similar RVEs / Jorg Schroder | 219 |
9.1. | Introduction | 220 |
9.2. | Statistically Similar RVEs | 222 |
9.2.1. | Method | 223 |
9.2.2. | Lower and Upper Bounds of RVEs | 224 |
9.2.3. | Statistical Measures | 225 |
9.3. | Construction and Analysis of SSRVEs | 233 |
9.3.1. | Objective Functions | 235 |
9.3.2. | Coupled Micro-macro Simulations | 240 |
9.3.3. | SSRVEs Based on Different Sets of Statistical Measures | 241 |
9.3.4. | Comparison of Stress on Microscale | 244 |
9.3.5. | Analysis of Bounds | 248 |
9.4. | Conclusion | 250 |
| Author Index | 257 |