6.6.1 Single Dispersive Wave Equation.

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Preface; Contents; 1 Instead of Introduction; 1.1 One-Dimensional Elastic Waves in Heterogeneous Solids; 1.1.1 Single Inclusion; 1.1.2 Periodic Laminate; 1.1.3 Functionally Graded Material; 1.1.4 Remarks; 1.2 Models for One-Dimensional Dispersive Waves in Solids with Microstructure; 1.2.1 Classical Wave Equation; 1.2.2 Strain Gradient Model; 1.2.3 Linear Version of the Boussinesq Equation; 1.2.4 Love-Rayleigh Equation for Rods Accounting for Lateral Inertia; 1.2.5 Refined Models; 1.2.6 Models with Higher-Order Time Derivatives; 1.2.7 Remarks; 1.3 Conclusions; References.

3.2.4 Elimination of the Internal Variable3.2.5 Rheology and Thermodynamics; 3.3 Material Thermomechanics; 3.3.1 Material and Spatial Time Derivatives; 3.3.2 Balance Laws; 3.3.3 Material Form of the Energy Conservation; 3.3.4 Material (Canonical) Momentum Conservation; 3.4 Single Internal Variable Theory; 3.4.1 Dissipation Inequality; 3.4.2 Simple Evolution Equation for Internal Variable; 3.5 Example: Phase Field Theory; 3.6 Conclusions; References; 4 Dual Internal Variables; 4.1 Introduction; 4.2 Dual Internal Variables; 4.2.1 Non-zero Extra Entropy Flux.

4.2.2 Evolution Equations for Internal Variables4.2.3 Fully Dissipative Case; 4.2.4 Non-dissipative Case; 4.3 Example: Cosserat Media; 4.3.1 Linear Micropolar Media; 4.3.2 Microrotation as an Internal Variable; 4.4 Example: Micromorphic Linear Elasticity; 4.4.1 The Mindlin Microelasticity; 4.4.2 Rearrangement; 4.4.3 Microdeformation Tensor as an Internal Variable; 4.4.4 Remark on Second Gradient Elasticity; 4.5 Conclusions; References; Part II Dispersive Elastic Waves in One Dimension; 5 Internal Variables and Microinertia; 5.1 Introduction.

5.2 Single Internal Variable: One-Dimensional Example5.2.1 Evolution Equation for a Single Internal Variable; 5.3 Dual Internal Variables in One Dimension; 5.3.1 Example: Linear Elasticity; 5.4 Summary and Discussion; References; 6 Dispersive Elastic Waves; 6.1 One-Dimensional Thermoelasticity in Solids with Microstructure; 6.2 Description with Single Internal Variable ; 6.3 Dispersive Wave Equation with Direct Coupling; 6.4 Dispersive Wave Equation with Gradient Coupling; 6.5 Description with Dual Internal Variables; 6.6 Microstructure Model with Direct Coupling.

Part I Internal Variables in Thermomechanics2 Introduction; 2.1 Micro versus Macro; 2.2 Internal Variables and Dynamic Degrees of Freedom; 2.2.1 Internal Variables of State; 2.2.2 Internal Dynamic Degrees of Freedom; 2.2.3 Similarity and Differences; 2.3 Generalization: Dual Internal Variables; 2.4 Historical Remarks; References; 3 Thermomechanical Single Internal Variable Theory; 3.1 Introduction; 3.2 Thermodynamic Rheology; 3.2.1 Balance Laws; 3.2.2 The Second Law of Thermodynamics; 3.2.3 Linear Solution of Dissipation Inequality for Isotropic Materials.

This book describes an effective method for modeling advanced materials like polymers, composite materials and biomaterials, which are, as a rule, inhomogeneous. The thermoelastic theory with internal variables presented here provides a general framework for predicting a material's reaction to external loading. The basic physical principles provide the primary theoretical information, including the evolution equations of the internal variables. The cornerstones of this framework are the material representation of continuum mechanics, a weak nonlocality, a non-zero extra entropy flux, and a consecutive employment of the dissipation inequality. Examples of thermoelastic phenomena are provided, accompanied by detailed procedures demonstrating how to simulate them.