Internal Variables in Thermoelasticity (eBook)

Preface 6
Contents 8
1 Instead of Introduction 10
1.1 One-Dimensional Elastic Waves in Heterogeneous Solids 10
1.1.1 Single Inclusion 12
1.1.2 Periodic Laminate 12
1.1.3 Functionally Graded Material 13
1.1.4 Remarks 14
1.2 Models for One-Dimensional Dispersive Waves in Solids with Microstructure 15
1.2.1 Classical Wave Equation 15
1.2.2 Strain Gradient Model 16
1.2.3 Linear Version of the Boussinesq Equation 18
1.2.4 Love-Rayleigh Equation for Rods Accounting for Lateral Inertia 19
1.2.5 Refined Models 20
1.2.6 Models with Higher-Order Time Derivatives 21
1.2.7 Remarks 23
1.3 Conclusions 25
References 26
Part I Internal Variables in Thermomechanics 28
2 Introduction 29
2.1 Micro versus Macro 29
2.2 Internal Variables and Dynamic Degrees of Freedom 31
2.2.1 Internal Variables of State 33
2.2.2 Internal Dynamic Degrees of Freedom 35
2.2.3 Similarity and Differences 36
2.3 Generalization: Dual Internal Variables 36
2.4 Historical Remarks 37
References 38
3 Thermomechanical Single Internal Variable Theory 42
3.1 Introduction 42
3.2 Thermodynamic Rheology 43
3.2.1 Balance Laws 43
3.2.2 The Second Law of Thermodynamics 44
3.2.3 Linear Solution of Dissipation Inequality for Isotropic Materials 45
3.2.4 Elimination of the Internal Variable 46
3.2.5 Rheology and Thermodynamics 47
3.3 Material Thermomechanics 49
3.3.1 Material and Spatial Time Derivatives 50
3.3.2 Balance Laws 52
3.3.3 Material Form of the Energy Conservation 54
3.3.4 Material (Canonical) Momentum Conservation 56
3.4 Single Internal Variable Theory 58
3.4.1 Dissipation Inequality 60
3.4.2 Simple Evolution Equation for Internal Variable 61
3.5 Example: Phase Field Theory 61
3.6 Conclusions 63
References 63
4 Dual Internal Variables 66
4.1 Introduction 66
4.2 Dual Internal Variables 67
4.2.1 Non-zero Extra Entropy Flux 69
4.2.2 Evolution Equations for Internal Variables 69
4.2.3 Fully Dissipative Case 70
4.2.4 Non-dissipative Case 71
4.3 Example: Cosserat Media 72
4.3.1 Linear Micropolar Media 72
4.3.2 Microrotation as an Internal Variable 72
4.4 Example: Micromorphic Linear Elasticity 74
4.4.1 The Mindlin Microelasticity 74
4.4.2 Rearrangement 75
4.4.3 Microdeformation Tensor as an Internal Variable 76
4.4.4 Remark on Second Gradient Elasticity 77
4.5 Conclusions 78
References 78
Part II Dispersive Elastic Waves in One Dimension 80
5 Internal Variables and Microinertia 81
5.1 Introduction 81
5.2 Single Internal Variable: One-Dimensional Example 82
5.2.1 Evolution Equation for a Single Internal Variable 84
5.3 Dual Internal Variables in One Dimension 85
5.3.1 Example: Linear Elasticity 87
5.4 Summary and Discussion 88
References 89
6 Dispersive Elastic Waves 91
6.1 One-Dimensional Thermoelasticity in Solids with Microstructure 91
6.2 Description with Single Internal Variable 92
6.3 Dispersive Wave Equation with Direct Coupling 94
6.4 Dispersive Wave Equation with Gradient Coupling 95
6.5 Description with Dual Internal Variables 96
6.6 Microstructure Model with Direct Coupling 98
6.6.1 Single Dispersive Wave Equation 99
6.7 Microstructure Model with Gradient Coupling 100
6.7.1 Single Dispersive Wave Equation 101
6.8 Unified Dispersive Wave Equation 101
6.9 Conclusions 103
References 104
7 One-Dimensional Microelasticity 105
7.1 Introduction 105
7.2 Mindlin's Microstructure Theory 106
7.3 Unidirectional Case 107
7.3.1 Longitudinal Motion 107
7.3.2 One-Dimensional Approximation 108
7.4 Dimensionless Variables 109
7.5 Numerical Scheme 111
7.6 Numerical Simulation 112
7.6.1 Reference Solution 112
7.6.2 Prediction by the Mindlin Microelasticity 114
7.7 Conclusions 116
References 116
8 Influence of Nonlinearity 118
8.1 Introduction 118
8.2 Constitutive Model 119
8.2.1 Single Dispersive Wave Equation 120
8.3 Examples of Nonlinear Dispersive Wave Equations 121
8.3.1 The Boussinesq Equation 121
8.3.2 The Korteveg--de Vries Equation 121
8.3.3 The Benjamin--Bona--Mahoney Equation 122
8.3.4 The Camassa--Holm Equation 123
8.4 Conclusions 124
References 124
Part III Thermal Effects 126
9 The Role of Heterogeneity in Heat Pulse Propagation in a Solid with Inner Structure 127
9.1 Geometry and Material Properties 127
9.2 The Fourier Law in Two-Dimensional Case 127
9.3 Numerical Scheme 128
9.4 Numerical Details 129
9.4.1 Initial and Boundary Conditions 129
9.5 Results of Numerical Simulations 131
9.5.1 Homogeneous Case 131
9.5.2 Inhomogeneous Case 132
9.6 Conclusions 133
References 134
10 Heat Conduction in Microstructured Solids 135
10.1 Introduction 135
10.2 Heat Conduction in Microstructured Solids. Single Internal Variable Explication 138
10.2.1 Evolution Equation for the Single Internal Variable 140
10.2.2 Quadratic Free Energy 141
10.3 Heat Conduction in Microstructured Solids with Dual Internal Variables 142
10.3.1 Quadratic Free Energy 143
10.3.2 Hyperbolicity of Evolution Equations for Internal Variables 144
10.3.3 Parabolicity of Heat Conduction Equation 146
10.4 Summary and Discussion 146
References 147
11 One-Dimensional Thermoelasticity with Dual Internal Variables 150
11.1 Introduction 150
11.1.1 Classical Thermoelasticity in Homogeneous Solids in One Dimension 152
11.1.2 Single Internal Variable Theory 153
11.1.3 Dual Internal Variables 156
11.1.4 Interpretation of Internal Variables 158
11.2 Conclusions 162
References 164
12 Influence of Microstructure on Thermoelastic Wave Propagation 166
12.1 Introduction 166
12.2 One-Dimensional Thermoelastic Wave Propagation in Solids with Microstructure 167
12.3 Numerical Results 169
12.4 Conclusions 173
References 175
Part IV Weakly Nonlocal Thermoelasticity for Microstructured Solids 176
13 Microdeformation and Microtemperature 177
13.1 Introduction 177
13.2 Governing Equations 179
13.2.1 Piola--Kirchhoff Formulation 179
13.2.2 Material Formulation 180
13.3 Double Dual Internal Variables 181
13.3.1 Microdeformation 184
13.3.2 Microtemperature 185
13.4 One-Dimensional Example 185
13.4.1 Microdeformation in One Dimension 186
13.4.2 Microtemperature in One Dimension 187
13.4.3 Boundary Conditions 188
13.5 Conclusions 188
References 190
Summary 193
Appendix A Sketch of Thermostatics 195
A.1 Fluids 196
A.1.1 Ideal Gas 197
A.2 Elastic Solids 197
A.2.1 Free Energy 199
A.3 Internal Variables 200
A.3.1 Weakly Nonlocal Internal Variables -- Thermostatics for Gradients 201
Appendix B Finite-Volume Numerical Algorithm 203
B.1 Introduction 203
B.2 Examples of Conservation Laws 203
B.2.1 Euler Equations of Gas Dynamics 204
B.2.2 Shallow Water Equations 204
B.2.3 Heat Conduction Equation 205
B.2.4 Linear Elasticity 205
B.2.5 Local Equilibrium Approximation 206
B.2.6 Excess Quantities and Numerical Fluxes 206
B.2.7 Riemann Invariants 208
B.2.8 Riemann Problem 209
B.2.9 Excess Quantities at the Boundaries Between Cells 210
B.3 Numerical Scheme for Thermoelastic Waves 211
B.3.1 Local Equilibrium Approximation 212
B.3.2 Excess Quantities 213
B.3.3 Conclusions 216
B.4 Figure Captions 217
Index 221