Derivations of the stress-strain relations for viscoanelastic media and the heat equation in irreversible thermodynamic with internal variables

By using a procedure of classical irreversible thermodynamics with internal variable (CIT-IV), some possible interactions among heat conduction and viscous-anelastic flows for rheological media are studied. In particular, we introduce as internal variables a second rank tensor εαβ(1) \varepsilon _{\alpha \beta }^{(1)} that is contribution to inelastic strain and a vector ξ_α that influences the thermal transport phenomena and we derive the phenomenological equations for these variables in the anisotropic and isotropic cases. The stress-strain equations, the general flows and the temperature equation in visco-anelastic processes are obtained and when the medium is isotropic, we obtain that the total heat flux J^(q) can be split in two parts: a first contribution J⁽⁰⁾, governed by Fourier law, and a second contribution J⁽¹⁾, obeying Maxwell-Cattaneo-Vernotte (M-C-V) equation.