Wave stability in isotropic temperature-rate-dependent thermoelasticity

Temperature-rate-dependent thermoelasticity is a theory of thermoelasticity in which the stress, entropy and heat flux are permitted to depend on the rate of change of temperature and the temperature gradient, as well as the usual variables of temperature and deformation gradient. This has the effect of introducing two relaxation times into the equations of thermoelasticity. Another important effect is that heat now travels at a finite speed rather than the infinite speed implied by the diffusion equation. In an isotropic temperature-rate-dependent thermoelastic material, it is found that four plane harmonic waves may propagate: two purely elastic transverse waves and two longitudinal waves that are dispersive and attenuated. All four waves are stable in the sense that their amplitude remains bounded. An alternative theory that forces heat to travel at finite speed is that of generalized thermoelasticity, in which the rate of change of heat flux also appears in the heat conduction equation, thereby introducing a relaxation time. Two different methods of combining the effects of temperature-rate-dependent thermoelasticity and generalized thermoelasticity are discussed and it is found that the transverse waves are unaltered but that one or both of the longitudinal waves become unstable.