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Mathematical structure of generalized heat equations in rigid solids is analyzed from the point of view of internal variables theory. Internal variables are used for accounting for the influence of inner microstructure on heat conduction. It is shown that all the known extensions of the classical heat equation can be recovered in the framework of the internal variables theory. The introduction of a single internal variable does not change the parabolic type of the heat conduction equations. Hyperbolic generalized heat conduction equations can be derived only by means of dual internal variables.

Application of internal variables to the description of the influence of a microstructure on the overall behavior of solids is presented and discussed. It is demonstrated on the one-dimensional example that capabilities of strain gradient models are much less than those provided by internal variables.

In this paper, elastic wave propagation in a one-dimensional micromorphic medium characterized by two internal variables is investigated. The evolution equations are deduced following two different approaches, namely using: (i) the balance of linear momentum and the Clausius–Duhem inequality, and (ii) an assumed Lagrangian functional (including a gyroscopic coupling) together with a variational principle. The dispersion relation is obtained and the possibility of the emerging band gaps is shown in such microstructured materials. Some numerical simulations are also performed in order to highlight the dispersive nature of the material under study.

The description of heat conduction in microstructured solids is presented in the framework of the dual internal variable approach. One of the internal variables is identified with microtemperature, i.e. the fluctuation of macroscopic temperature due to the inhomogeneity of the body. It is shown that the microstructure influence may result in a hyperbolic heat propagation for the microtemperature. The macroscale heat conduction is described by a parabolic equation which is coupled with the hyperbolic equation for the microtemperature.

It is demonstrated that the dual internal variable approach is able to predict a hyperbolic character of heat conduction at the microscale. One of the internal variables is identified with microtemperature, i.e., the fluctuation of macroscopic temperature due to the inhomogeneity of the body. The macroscopic heat conduction equation remains parabolic, but coupled with the hyperbolic evolution equation for the microtemperature.

The overall description of thermomechanical processes in microstructured solids includes both direct and indirect couplings of equations of motion and heat conduction at the macrolevel. In addition to the conventional direct coupling, there exists the coupling between macromotion and microtemperature evolution. This means that the macrodeformation can induce microtemperature perturbations due to the heterogeneity in the presence of a microstructure. These perturbations, propagating with finite speed, can induce, in turn, corresponding changes in macrotemperature.

The introduction of double dual internal variables provides the complete extension of the classical thermoelasticity theory onto the case of microstructured solids. This extension keeps the structure of canonical balances of momentum and energy and provides the thermodynamically consistent evolution equations for microdeformation and microtemperature. Evolution equations in the case of dual internal variables are hyperbolic and coupled with the equations of macromotion.

The theory of a single internal variable of state is well established including the weak nonlocality and the enrichment by extra entropy flux. The theory is based on the consideration of the internal variable of state as a tool for taking into account the internal dissipation. Inertial effects are absent in this theory by definition, which leads to parabolic evolution equations for the internal variables of state. Due to the simplicity, the single internal variable theory is the necessary first step in the construction of more sophisticated material models.

The introduction of nonlinear terms at both macro- and micro-levels does not change the formalism of the internal variable approach. In the non-dissipative case, the nonlinear terms can be balanced with dispersion providing the well-known models of solitonic behavior. The interplay between micro-and macro- nonlinearities allows to achieve more sophisticated models of nonlinear dispersive wave propagation than those in the homogeneous solids.

Several well-known dispersive wave propagation models together with generalization of the Mindlin-type models are derived by using internal variables. The adopted phenomenological approach is based on the material formulation of continuum mechanics and provides the full thermodynamic consistency due to the dual internal variables concept.

It is shown that inertial terms appear naturally in the thermodynamic theory with dual internal variables and the conditions of their appearance is well understandable in terms of mechanical notions. This demonstrates the difference between the standard single internal variable theory and the dual internal variables concept.

It is shown how dual weakly non-local internal variables and extra entropy fluxes can be introduced in the framework of canonical thermomechanics on the material manifold. This extension of the single internal variable formalism allows one to derive a hyperbolic evolution equation for internal variables in the non-dissipative case. Since the dissipation inequality is the basis of the derivation, it ensures the thermodynamic consistency of the obtained evolution equations.

Prediction of the thermoelastic behavior of microstructured materials suggests a more general description of thermal processes in addition to the generalized continuum description extending the conventional continuum mechanics for incorporating intrinsic microstructural effects. Double dual internal variables are introduced in order to couple inertial microstructural effects like microdeformation and diffusive microstructural effects like microtemperature. The full coupled system of governing equations provides a complete extension of the classical thermoelasticity theory onto the case of microstructured solids.

International audience; Reflection and transmission of elastic waves at the interface between two distinct micromorphic media are considered in the one-dimensional setting. Dual internal variable approach is used for the description of the microstructure influence on the global motion. It is shown that reflection and transition coefficients for plane waves depend on the coupling between macro-and micro-motions as well as on the choice of the microstructural interaction at the interface. Numerical simulations exhibiting results with promising technological implications are shown.

Thermoelastic wave propagation suggests a coupling between elastic deformation and heat conduction in a body. Microstructure of the body influences the both processes. Since energy is conserved in elastic deformation and heat conduction is always dissipative, the generalization of classical elasticity theory and classical heat conduction is performed differently. It is shown in the paper that a hyperbolic evolution equation for microtemperature can be obtained in the framework of the dual internal variables approach keeping the parabolic equation for the macrotemperature. The microtemperature is considered as a macrotemperature fluctuation. Numerical simulations demonstrate the formation and propagation of thermoelastic waves in microstructured solids under thermal loading.

The extension of the thermoelasticity theory by weakly non-local dual internal variables enriched by an extra entropy flux for the thermomechanical description of the behavior of microstructured solids is presented. The internal variables take into account the distributed effect of microdeformations or microtemperatures (and their gradients) on the overall macroscopic behavior. The evolution equations for microtemperatures can be hyperbolic, which can induce wave-like propagation for macrotemperature due to the coupling of equations.

The basic ideas for describing the dispersive wave motion in microstructured solids are discussed in the one-dimensional setting because then the differences between various microstructure models are clearly visible. An overview of models demonstrates a variety of approaches, but the consistent structure of the theory is best considered from the unified viewpoint of internal variables. It is shown that the unification of microstructure models can be achieved using the concept of dual internal variables.

The formal structure of generalized continuum theories is recovered by means of the extension of canonical thermomechanics with dual weakly non-local internal variables. The canonical thermomechanics provides the best framework for such generalization. The Cosserat, micromorphic, and second gradient elasticity theory are considered as examples of the obtained formalization.

The origin of microinertia of micromorphic theories is investigated from the point of view of non-equilibrium thermodynamics. In the framework of dual internal variables microinertia stems from a thermodynamic equation of state related to the internal variable with the properties of mechanical momentum., 14 pages, no figures

International audience; This paper describes the mathematical models derived for wave propagation in solids with internal structure. The focus of the overview is on one-dimensional models which enlarge the classical wave equation by higher-order terms. The crucial parameter in models is the ratio of characteristic lengths of the excitation and the internal structure. Novel approaches based on the concept of internal variables permit one to take the thermodynamical conditions into account directly. Examples of generalisations include frequency-dependent multiscale models, nonlinear models and thermoelasticity. The substructural complexity within the framework of elasticity gives rise to dispersion of waves. Dispersion analysis shows that acoustic and optical branches of dispersion curves together describe properly wave phenomena in microstructured solids. In the case of nonlinear models, the governing equations are of the Boussinesq type. It is argued that such models of waves in solids with microstructure display properties that can be analysed as phenomena of complexity.

The dispersive wave motion in solids with microstructure is considered in the one-dimensional setting in order to understand better the mechanism of dispersion. It is shown that the variety of dispersive wave propagation models derived by homogenization, continualisation, and generalization of continuum mechanics can be unified in the framework of dual internal variables theory.

Internal variables are introduced in the framework of canonical thermomechanics on the material manifold. The canonical equations for energy and pseudomomentum cannot be separated by means of the scale separation because these equations should concern all fields together and, therefore, all scales together. However, the intrinsic interaction force requires a kinetic relation for internal variables. This kinetic relation depends on representation of the internal variable as “internal variable of state” or “internal degree of freedom”.

Dispersive wave propagation in solids with microstructure is discussed in the small-strain approximation and in the one-dimensional setting. It is shown that the generalizations of wave equation based on continualizations of discrete systems as well as on homogenization methods can be recovered in the framework of the internal variable theory in the case of non-dissipative processes.

The canonical thermomechanics on the material manifold is enriched by the introduction of dual weakly non-local internal variables and extra entropy fluxes. In addition to the dissipative reaction-diffusion equation for a single internal variable of state, a hyperbolic evolution equation for the internal degree of freedom can be also recovered in the non-dissipative case. It is demonstrated that the Mindlin mi-cromorphic theory can be represented in terms of dual internal variables in a natural way in the framework of the canonical thermomechanics.

Dynamic degrees of freedom and internal variables are treated in a uniform way. The unification is achieved by means of the introduction of a dual internal variable. This duality provides the corresponding evolution equations depending on whether the Onsager-Casimir reciprocity relations are satisfied or not., 16 pages, revised

Results of numerical experiments are presented in order to compare direct numerical calculations of wave propagation in a laminate with prescribed properties and corresponding results obtained for an effective medium with the microstructure modelling. These numerical experiments allowed us to analyse the advantages and weaknesses of the microstructure model.

The wave motion in micromorphic microstructured solids is studied. The mathematical model is based on ideas of Mindlin and governing equations are derived by making use of the Euler–Lagrange formalism. The same result is obtained by means of the internal variables approach. Actually such a model describes internal fields in microstructured solids under external loading and the interaction of these fields results in various physical effects. The emphasis of the paper is on dispersion analysis and wave profiles generated by initial or boundary conditions in a one-dimensional case.