Type II thermoelasticity. A new aspect

Abstract

In this paper we investigate the thermomechanical theory of thermoelasticity without energy dissipation. We consider the case where the assumptions on the constitutive tensor are different from the usual ones in the current literature. These assumptions are suitable because they correspond to situations where the material is prestressed and has initial entropy flux. When we restrict our attention to the one-dimensional problem, we first establish a condition on the coefficients to guarantee the hyperbolicity of the system. Then we prove that the problem is well posed and stable. When the hyperbolicity condition is not satisfied we prove that the problem is ill-posed. We also prove a similar result when the dimension is greater than one whenever the domain satisfies a certain condition. Though we cannot expect in general the stability of solutions in dimension greater than one, we prove the uniqueness and stability of radial solutions in the general case. To obtain the results, we need the use of a conservation law which has not previously been considered in the literature. Similar results are not known for other thermoelastic theories.