Relating Entropy Flux With Heat Flux in Two-Temperature Thermodynamic Model for Metal Thermoviscoplasticity

A rigorous derivation of the relation between the entropy flux and the heat flux in a recently developed two-temperature thermodynamic model of metal thermoviscoplasticity is presented. The two-temperature model exploits the internal variable theory of thermodynamics, wherein thermodynamic restrictions on the constitutive functions are based on the second law written in a form similar to the classical Clausius–Duhem (CD) inequality. Here, the weakly interacting thermodynamic subsystems, e.g., configurational and kinetic vibrational subsystems, enable defining their own temperatures, heat fluxes, and entropy fluxes. The CD-type inequality is then constructed with the assumption, as in rational thermodynamics, that entropy fluxes equal heat fluxes divided by respective absolute temperatures. Validity or otherwise of this restrictive assumption is however an open question in the context of two-temperature thermomechanics, and there are, indeed, known materials for which this assumption fails to hold. To settle this important point, we start with a detailed analysis based on a general entropy inequality, whose thermodynamic consequences are extracted using Müller–Liu procedure of Lagrange multipliers, and subsequently, appeal to material frame-indifference, material symmetry groups for additional constitutive restrictions. We conclude that, for isotropic–viscoplastic materials, subsystem entropy fluxes are indeed given by the respective heat fluxes divided by their own temperatures.