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