On the Formulation of Continuum Thermodynamic Models for Solids as General Equations for Non-equilibrium Reversible-Irreversible Coupling

Abstract

The purpose of this work is the comparison of some aspects of the formulation of material models in the context of continuum thermodynamics (e.g., Šilhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997) with their formulation in the form of a General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC: e.g., Grmela and Öttinger in Phys. Rev. E 56: 6620-6632, 1997; Öttinger and Grmela in Phys. Rev. E 56: 6633-6655, 1997; Öttinger in Beyond equilibrium thermodynamics, Wiley, New York, 2005; Grmela in J. Non-Newton. Fluid Mech. 165: 980-998, 2010). A GENERIC represents a generalization of the Ginzburg-Landau model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to formulate non-equilibrium thermodynamic models for complex fluids, it has recently been applied to anisotropic inelastic solids in a Eulerian setting (Hütter and Tervoort in J. NonNewton. Fluid Mech. 152: 45-52, 2008; 53-65, 2008; Adv. Appl. Mech. 42: 254-317, 2009) as well as to damage mechanics (Hütter and Tervoort in Acta Mech. 201: 297-312, 2008). In the current work, attention is focused for simplicity on the case of thermoelastic solids with heat conduction and viscosity in a Lagrangian setting (e.g., Šilhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997, Chaps. 9-12). In the process, the relation of the two formulations to each other is investigated in detail. A particular point in this regard is the concept of dissipation and its model representation in both contexts. This work is dedicated to the memory of our colleague and friend Don Carlson.