Wave Stability for Near-Incompressibility at Uniform Temperature or Entropy in Generalized Isotropic Thermoelasticity

Abstract

This work is an extension of a 1998 work by Leslie and Scott to the theory of generalized isotropic thermoelasticity in which Fourier's constitutive equation for heat conduction is replaced by one in which a relaxation time (ro) is associated with the heat flux vector. The effect is to alter the character of the governing field equations from parabolic to hyperbolic type so that all disturbances propagate with finite speed. The theory of longitudinal plane wave propagation in generalized isotropic thermoelasticity is developed here along the lines developed by the authors for classical thermoelasticity. It is found that if the two longitudinal wave modes intersect in the complex plane of squared wave speeds, regarded as functions of frequency, then they do so in two semicircles in the lower half-plane (which guarantees stability). The equivalence of the two types of near-incompressibility can be demonstrated for To > 0 just as in the classical case To = 0.