Materials with internal variables and relaxation to conservation laws

• Published in: Archive for rational mechanics and analysis 1999-05, Vol.146 (2), p.129-155
• Main Author: TZAVARAS, A. E
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Mathematical methods in physics; Numerical approximation and analysis; Ordinary and partial differential equations, boundary value problems; Physics; Solid mechanics; Structural and continuum mechanics; Studies; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Vibrations and mechanical waves
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s002050050139

.The theory of materials with internal state variables of Coleman & 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 specific theories pertaining to stress relaxation and relaxation of internal energy. They each lead to a relaxation framework for the theory of thermoelastic non-conductors of heat, equipped with globally defined "entropy" functions for the associated relaxation process. Next, we consider a semilinear model problem of stress relaxation. We discuss uniform stability and compactness 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.[PUBLICATION ABSTRACT]