Thermodynamics of Rate-independent Processes in Viscous Solids at Small Strains

Abstract

So-called generalized standard solids (of the Halphen–Nguyen type) involving also activated rate-independent processes such as plasticity, damage, or phase transformations are described as a system of a momentum equilibrium equation and a variational inequality for inelastic evolution of internal-parameter variables. The stored energy is considered as temperature dependent and then the thermodynamically consistent system is completed with the heat-transfer equation. Existence of a suitably defined “energetic” solution is proved by a nontrivial combination of theory of rate-independent processes by Mielke et al. [Handbook of Differential Equations, Elsevier, Amsterdam, 2005, pp. 461–559; Models of Continuum Mechanics in Analysis and Engineering, H.-D. Alber, R. Balean, and R. Farwig, eds., Shaker Ver., Aachen, 1999, pp. 117–129; Nonlinear Differ. Equ. Appl., 11 (2004), pp. 151–189; Arch. Ration. Mech. Anal., 162 (2002), pp. 137–177] adapted for coupling with viscous/inertial effects and of sophisticated estimates by Boccardo and Gallouët of the temperature gradient of the heat equation with $L^1$-data. Illustrative examples are presented, too.