Materials with internal variables and relaxation to conservation laws (Q1300698)

The author considers the early theory of materials with internal variables of state as introduced by \textit{B. D. Coleman} and \textit{M. E. Gurtin} [Phys. Fluids 10, 1454-1458 (1967; Zbl 0173.27701)], and its application to a class of viscoelastic materials as described by \textit{I. Suliciu} and more particularly by \textit{C. Faciu} and \textit{M. Mihailescu-Suliciu} [Int. J. Solids Struct. 23, 1505-1520 (1987; Zbl 0653.73018)]. First, the requirements imposed on the constitutive theory of such materials by the thermodynamical admissibility are reviewed. Both stress relaxation and energy relaxation are considered as specific examples. These two yield a relaxation framework for the theory of thermoelastic non-conductors of heat, where the associated relaxation process is related to a globally defined entropy function. By way of application, a semilinear model problem of stress relaxation is envisaged. This allows for a thorough discussion on the uniform stability, on the compactness of relaxation system in the zero-relaxation limit, and on the convergence of the system to that of isothermal elastodynamics by considerations of compensated compactness (L. Tartar). Finally, a strong-dissipation estimate of the relaxation approximation proposed earlier by other authors [\textit{S. Jin} and \textit{Z. Xin}, Commun. Pure Appl. Math. 48, No. 3, 235-276 (1995; Zbl 0826.65078)] for relaxing schemes of systems of conservation laws, is proved when the limit system possesses a strictly convex entropy.