IUTAM Symposium on Progress in the Theory and Numerics of Configurational Mechanics (eBook)

Table of Contents 6
Preface 9
Acknowledgements 11
On Discontinuities of Material Momentum and Eshelby Stress in Hyperelasticity and Thermoelasticity 12
1 Motivation 12
2 Notations 14
3 Balances of Physical Momentum and Energy 14
4 Material Momentum and Eshelby Stress 15
5 Hyperelastic Material 17
6 Thermoelastic Material 18
7 Conclusion 20
References 21
On a Constraint-Based Regularization Technique for Configurational r- Adaptivity and 3D Shape Optimization 22
1 Introduction 22
2 Mechanical Problem 24
3 Configurational r-Adaptivity 25
4 Configurational Shape Optimization 27
5 Fictitious Energies 29
6 Numerical Experiments 31
6.1 Cracked Specimen 31
6.2 Sheet Metal Part 33
7 Conclusions 34
Acknowledgement 35
References 35
Some New Properties of the Eshelby Stress Tensor 37
1 Introduction 37
2 Evolution of Microstructural Defects by Considering the Eshelby Stress Tensor 38
3 Example 42
4 Discussion 44
References 45
On Configurational Aspects of Finite Deformation Inelasticity: A Variational Approach Versus the Transformation of Balance of Momentum 46
1 Introduction 46
2 Basic Considerations 48
3 Variational Approach 50
4 Two Configurational Field Formulations 51
4.1 Variational Approach 51
4.2 Transformation-Based Approach 53
5 Comparison and Discussion 54
References 54
Configurational Forces Derived from the Total Variation of the Rate of Global Dissipation 56
1 Introduction 56
2 Representation of Configurational and Deformational Motions 57
2.1 Preliminaries 57
2.2 Absolute and Material Time Differentiation 58
2.3 Global and Localized Dissipation Inequality 59
3 Dissipation Functional for Changing Material Configuration 60
3.1 Preliminaries 60
3.2 Basic Format of the Dissipation Functional – Configurational and Material Parts 61
3.3 Total Variation of the Rate of Global Dissipation Due to Configurational Changes 61
4 Model Problem: Bar with Interface Separating Parts with Dissimilar Material Properties 62
4.1 Problem Formulation – Preliminaries 61
4.2 Elastic-Plastic Model – Linearized Format 62
4.3 Explicit (Classical) Configurational Forces at the Material Interface 64
4.4 Implicit (Configurational-Induced) Configurational Forces at the Material Interface – Tangent Problem 64
5 Numerical Example 65
6 Discussion and Conclusions 66
References 67
On Crack Analysis of Functionally Graded Materials with Material Forces 69
1 Introduction 69
2 Kinematics and Balance Equations 70
2.1 Parameterization of Geometry and Displacement 70
2.2 Time Dependent Migrating Control Volume 72
2.3 Dissipation Inequality 72
2.4 Evolution Equation for Crack Propagation 74
2.5 Summary of Coupled Initial-Boundary Value Problem 74
2.6 Solution of Coupled Initial-Boundary Value Problem 75
3 Compact Tension Specimen with Graded Material 76
Acknowledgment 78
References 78
Momentum and Material Momentum in Superconductors 80
1 Introduction 80
2 Electric Currents 82
3 Maxwellian Fields and London’s Conjectures 83
4 Time Dependent Ginzburg–Landau Equation: Variational Approach 84
5 Euler Equations for Compressible Fluids: Momentum of the Supercurrent 86
6 Material (or Configurational) Momentum of the Supercurrent 87
7 Concluding Remarks 88
References 89
Dislocations, Microforce and Micromomentum in Second Order Finite Elasto- Plasticity 90
1 Introduction 90
2 Plastic Connection 92
3 Balance Equations 93
4 Free Energy Imbalance. Thermodynamic Restrictions 95
5 Conclusions 99
Acknowledgement 99
References 99
A Variational Framework for Dual Solutions in the Physical and Material Space 101
1 Introduction 101
2 A General Optimal Control Approach 102
3 Variational Balance Laws in the Physical and Material Spaces 104
4 Duality Techniques for the Physical Problem 105
4.1 The Primal Physical Problem 105
4.2 The Dual Physical Problem 105
4.3 Sensitivity Relation Using the Dual Solution 106
4.4 The Material Residual of the Dual Problem 107
5 Duality Techniques for the Material Problem 108
5.1 The Primal Material Problem 108
5.2 The Dual Material Problem 108
5.3 Duality Relation for the Coupled Problem 109
6 Conclusions 109
References 110
On the Nonlocal Symmetries, Group Invariant Solutions and Conservation Laws of the Equations of Nonlinear Dynamical Compressible Elasticity 112
1 Introduction 112
2 Nonlinear Elasticity: Boundary Value Problems in 1D 113
3 Nonlocally Related Systems of 1D Nonlinear Elasticity 114
4 Point and Nonlocal Symmetry Classification of the Lagrange System EW{ x, t v, s, ., w} . L{ y, s
5 Calculation of Group Invariant Solutions Arising from the Lagrange System EW{ x, t v, s, ., w} ( 5)
6 Conservation Laws of Dynamical Nonlinear Elasticity 122
7 Conclusions 124
References 125
Configurational Forces in the Theory of Two- Phase Plates 126
1 Introduction 126
2 Basic Equations of the Direct Theory of Plates 127
3 Variation Principle of Total Energy 130
4 Kinetic Equation 132
5 Tension of Two-Phase Rectangular Plate 132
6 Conclusion 134
Acknowledgements 134
References 134
On Configurational Formulations in the Director Theory of Rods 136
1 Introduction 136
2 Fundamental Relations of Balance and Jump 137
3 Classical Derived Relations of Balance and Jump 139
4 Derivation of the Configurational Formulations of Balance and Jump 141
Acknowledgement 143
References 143
Macroscopic Elasticity of Nanoporous Silicon: Bulk and Surface Effects 144
1 Introduction 144
2 Elasticity of Nanoporous Silicon: Bulk Effects 145
2.1 Keating Model for Bulk Silicon 145
2.2 Extension to Porous Silicon 146
3 Oxidation of Porous Silicon: Bulk and Surface Effects 149
3.1 Experimental Evidence 149
3.2 Surface Effects 150
3.3 Numerical Results and Size Effects 151
Acknowledgements 152
References 153
Internal Variables and Generalized Continuum Theories 154
1 Motivation 154
2 Canonical Thermomechanics on the Material Manifold 156
3 Dual Internal Variables 157
4 Example: Micromorphic Linear Elasticity 160
5 Conclusions 163
References 163
Stratified Energies: Ground States with Cracks 164
1 Introduction 164
2 Curvature Varifolds with Boundary 165
3 Transplacement Fields and Bulk Energy 167
3.1 Sobolev Maps and Related Cartesian Currents 168
3.2 The Bulk Energy 169
4 A Skeletal Model Admitting Formation of Cracks 170
4.1 The Energy Functional 171
4.2 Ground States: Existence Theorems 171
Acknowledgement 173
References 173
Crack Curving Based on Configurational Forces and Their Gradients 174
1 Introduction 174
2 Crack Curving in LEFM 175
3 Maximum Dissipation for Regular Curved Cracks 176
3.1 Zeroth Order Approximation 177
3.2 First Order Approximation 177
3.3 Second Order Approximation 178
4 Transition to Configurational Forces 179
5 Finite Element Framework 180
6 Step by Step Propagation Scheme 182
7 Conclusions 183
References 183
Anisotropic Elasticity of Grade Three: Conservation and Balance Laws 184
1 Introduction 184
2 Anisotropic Elasticity of Grade Three 185
3 Conservation – Balance Laws in Elasticity of Grade Three 187
3.1 Structure of Conservation Laws 187
3.2 Translations in Space and Time 188
3.3 Rotations in Space 190
3.4 Scaling 190
3.5 Addition of Solutions 191
4 Balance Laws in Elasticity of Grade Three 192
4.1 Balance Laws and the J, L and M-Integrals 192
4.2 Dynamical Reciprocal Theorem 194
5 Conclusions 195
Acknowledgement 195
References 195
Evaluation of Crack-Driving Forces at Finite Viscoelasticity: Theory and Experiment 197
1 Introduction 197
2 Viscoelasticity of Elastomers 198
3 Fracture Mechanical Parameters 200
3.1 Classical Fracture Mechanical Approaches 201
3.2 Material Force Method 201
4 Material Forces of Viscoelastic Material 203
5 Example 205
6 Conclusions 205
Acknowledgements 206
References 206
On Configurational Forces within GreenÒ Naghdi Thermo- Hyperelasticity 207
1 Kinematics of Spatial and Material Motion Problem 207
2 Thermoelastic Model 209
2.1 Type I – Classical Theory 209
2.2 Type III – Non-Classical Theory 210
2.3 Spatial Motion Problem 211
2.4 Material Motion Problem 212
3 Discretization and Numerical Example 213
4 Conclusions 217
Acknowledgement 218
References 218
Translational Conservation and Balance Laws in the Gauge Theory of Dislocations 219
1 Introduction 219
2 Dislocation Gauge Theory 220
3 Conservation Laws 224
4 Canonical Currents of Translations in Space and Time 225
4.1 Elastic Subsystem 226
4.2 Dislocation Subsystem 227
5 Gauge-Invariant Currents 228
5.1 Elastic Subsystem 228
5.2 Dislocation Subsystem 229
6 Conclusion 230
Acknowledgement 230
References 231
Configurational Forces in Continuous Theories of Elastic Ferroelectrics 232
1 Introduction 232
2 Sharp Interfaces and Configurational Forces in Ferroelectrics 233
3 The Level-Set Method 234
4 Continuous Energy and Configurational Forces 235
4.1 The Energy Function and the Level Set 235
4.2 Two Level Set Functions 237
5 Computational Results 238
Acknowledgement 241
References 241
A Variationally Consistent Approach for Crack Propagation Based on Configurational Forces 242
1 Introduction 242
2 The Strong Discontinuity Approach Ò Fundamentals 244
2.1 Kinematics Associated with Mixed Mode Brittle Failure 244
2.2 Kinematics Associated with Slip Bands 245
3 A Non-Local Crack Initiation Criterion 246
3.1 Fundamentals 246
3.2 Mixed Mode Brittle Failure 248
3.3 Slip Bands 248
4 Conclusions 249
References 249
Computational Homogenization of Defect Driving Forces 251
1 Introduction 251
2 Governing Equations 253
2.1 Governing Equations for the Spatial Motion Problem 253
2.2 Governing Equations for the Material Motion Problem 254
3 Homogenization 254
3.1 Averaged Variables for the Spatial Motion Problem 255
3.2 Averaged Variables for the Material Motion Problem 255
4 Scale-Transition and Boundary Conditions 256
4.1 Spatial Motion Problem 256
4.2 Material Motion Problem 257
5 Examples 258
6 Conclusions 260
References 261
On the Computation of Configurational Forces in Anisotropic Hyperelastic Solids 262
1 Introduction 263
2 Continuum Mechanical Preliminaries. 263
3 Balance of Configurational Forces 265
4 Polyconvex Anisotropic Energy Functions 266
5 Numerical Example: Cracked SET Specimen 268
6 Conclusion 270
Acknowledgement 270
References 270
Authors Index 272