Thermoelasticity with Finite Wave Speeds — A Survey

Abstract

At present there are at least two different generalizations of the classical linear thermoelasticity. The first one proposed by Green and Lindsay (1972) (G-L Theory) involves two relaxation times of a thermoelastic process, while the second theory due to Lord and Shulman (1967) (L-S Theory) admits only one relaxation time. Both theories have been developed in an attempt to eliminate the paradox of an infinite velocity of thermoelastic propagation inherent in the classical case. In this paper a number of general results concerning these theories for a homogeneous isotropic solid are presented. They include (G-L Theory): 1. Domain of Influence Theorem, 2. Decomposition Theorem, 3. Uniqueness Theorem, and 4. Variational Principle. Theorem 1 asserts that in the new theory thermoelastic disturbances produced by the data of bounded support propagate with a finite velocity only. Theorem 2 which is quite similar to a Boggio result in linear elastodynamics shows that a thermoelastic disturbance can be split into two fields each of which propagates with a different finite velocity. Theorem 3 covers uniqueness for a stress-temperature boundary value problem with arbitrary initial tensorial data. Finally, the variational principle gives an alternative description of the theory in terms of a thermoelastic process (S,q), where S and q represent the stress tensor and the heat flux vector, respectively. In the L-S theory uniqueness of a stress-flux initial-boundary value problem is discussed. A number of suggestions concerning those areas of the theory that are critically in need of further investigation are also given.