Non-Conventional Thermodynamics, Cattaneo’s Heat Conduction Law, Thermo-Diffusion Coupling and Variational Formulations

Conventional continuum thermodynamics is characterized by a classical form of the energy law and the second law of thermodynamics in the form of the Clausius–Duhem inequality. This thermodynamic framework fails to capture certain features in material response as, e.g., length scale effects, temperature waves in rigid heat conductors and diffusion phenomena. Typically, such material characteristics are associated with pronounced non-localities in time and space. To tackle these issues, non-conventional thermodynamic approaches might be appropriate. The employment of non-conventional thermodynamics is very attractive, as it enables the extension of the applicability of conventional thermodynamics in a simple way. A specific non-conventional thermodynamic framework has previously been proposed as a generalization of irreversible thermodynamics. Energy supply effects were neglected in this work. However, energy supply terms may become important when discussing thermodynamical consistency of many physical models. The present paper extends the applicability of the proposed non-conventional thermodynamic framework by accounting for energy supply densities and demonstrates its capabilities by addressing Cattaneo’s heat conduction law—known for predicting temperature waves—and thermo-diffusion coupling theories. It is shown that, within the adopted thermodynamics, the considered physical models are thermodynamically consistent and that the resulting field theories admit formulations within a variational framework for rate problems; in this sense, the models are properly formulated.