The dispersive effects due to the presence of microstructure in solids are studied. The basic mathematical model is derived following Mindlin's theory. In the onedimensional case the governing equations of a linear system are presented. An approximation using the slaving principle indicates a hierarchy of waves. The corresponding dispersion relations are compared with each other. The choice between the models can be made on the basis of physical effects described by dispersion relations.

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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.Keywords Generalized continua · Canonical thermomechanics · Internal variables · Microstructure · Cosserat medium · Micromorphic medium MotivationGeneralized continuum theories extend conventional continuum mechanics by incorporating intrinsic microstructural effects in the mechanical behavior of materials [1][2][3][4][5]. Internal variable approach was always an alternative framework for the continuum modeling of such effects in materials [6][7][8][9][10][11][12]. However, the wellestablished theory of internal variables of state [13,14] cannot completely describe a generalized medium because an internal variable of state has no inertia, but it dissipates. If inertia is introduced, the internal variable must be treated as an actual degree of freedom [15]. Accordingly, the variable is not "internal" any more but can be controlled for instance at the boundary of a body.A more general thermodynamic framework of the internal variable theory presented recently [16] is based on a duality between internal variables, which make possible to derive evolution equations both for internal variables of state and internal degrees of freedom. A natural question relates to the ability of this duality concept to comprise inertial effects. To answer this question, we show how the dual internal variables can be introduced into continuum mechanics and how certain generalized continuum theories can be interpreted in terms of the dual internal variables.The most suitable framework for the generalization of continuum theory by weakly non-local dual internal variables enriched by an extra entropy flux is the material formulation of continuum thermomechanics [17,18]. Therefore, basic definitions of the canonical thermomechanics [17] are recalled in Sect. 2 of the paper. Then dual variables are introduced in Sect. 3 and evolution equations for both dissipative and non-dissipative processes are derived in Sect. 3.2. Linear Cosserat, micromorphic, and second gradient elasticity theories are

Results of heat pulse experiments in various artificial and natural materials are reported in this paper. The experiments are performed at room temperature with macroscopic samples. It is shown that temperature evolution does not follow Fourier's law but is well explained by the Guyer-Krumhansl equation. The observations confirm the ability of non-equilibrium thermodynamics to formulate universal constitutive relations for thermomechanical processes.