Materials with Internal Variables and Relaxation to Conservation Laws

The theory of materials with internal state variables of Coleman and Gurtin CG] provides a natural framework to investigate the structure of relaxation approximations of conservation laws from the viewpoint of continuum thermomechanics. After reviewing the requirements imposed on constitutive theories by the principle of consistency with the Clausius-Duhem inequality, we pursue two speciic theories pertaining to stress relaxation and relaxation of internal energy, respectively. They each lead to a relaxation framework towards the theory of thermoelastic non-conductors of heat, equipped with globally deened "entropy" functions for the associated relaxation process. Next, we consider a semilinear model problem of stress relaxation. We discuss uniform stability and com-pactness for solutions of the relaxation system, in the zero-relaxation limit, and establish convergence to the system of isothermal elastodynamics, by using compensated compactness. Finally, we prove a strong dissipation estimate for the relaxation approximations proposed in Jin-Xin JX], when the limit system is equipped with a strictly convex entropy.