The present work is concerned with a very recently proposed heat conduction model—an exact heat conduction model with a delay term for an anisotropic and inhomogeneous material—and some important theorems within this theory. A generalized thermoelasticity theory was proposed based on the heat conduction law with three phase-lag effects for the purpose of considering the delayed responses in time due to the micro-structural interactions in the heat transport mechanism. However, the model defines an ill-posed problem in the Hadamard sense. Subsequently, a proposal was made to reformulate this constitutive equation of heat conduction theory with a single delay term and the spatial behavior of the solutions for this theory have been investigated. A Phragmen–Lindelof type alternative was obtained and it has been shown that the solutions either decay in an exponential way or blow-up at infinity in an exponential way. The obtained results are extended to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction to the delay term and a Phragmen–Lindelof type alternative was obtained for the forward and backward in time equations. In the present work, we consider the basic equations concerning this new theory of thermoelasticity for an anisotropic and inhomogeneous material and make an attempt to establish some important theorems in this context. A uniqueness theorem has been established for an anisotropic body. An alternative characterization of the mixed initial-boundary value problem is formulated and a variational principle as well as a reciprocity principle is established.

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This work is concerned with a recent thermoelastic model. We investigate the propagation of plane harmonic waves in the context of this very recently proposed heat conduction model, an exact heat conduction model with a single delay term, established by Quintanilla. This model attempted to reformulate the heat conduction model that takes into account microstructural effects in heat transport phenomena and provided an alternative heat conduction theory with a single delay term. We aim to study the harmonic plane waves propagating in a thermoelastic medium by employing this new model and derive the exact dispersion relation solution. We mainly focus on a longitudinal wave coupled to a thermal field and find two different modes of this wave. We derive asymptotic expressions for several important characterizations of the wave fields: phase velocity, specific loss, penetration depth, and amplitude ratio. Analytical expressions for these wave characteristics are obtained for different cases of very-high- and low-frequency regions for elastic- and thermal-mode longitudinal waves. To verify the analytical results, we also carry out computational work to obtain numerical results of the wave characterizations for intermediate values of frequency and illustrate the results graphically. We show that our analytical and numerical results are in perfect match. On the basis of the analytical and numerical results, a thorough analysis of the effects of the single delay parameter on various wave characteristics is presented. We highlight several characteristic features of the new thermoelastic model, as compared with other models.

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The present work is concerned with the thermoelasticity theory of Green and Naghdi of type I, II and III. By considering a mixed initial-boundary value problem for an isotropic medium in the context of all three models of type I, II and III in a unified way, we derive an identity in terms of the temperature and potential. On the basis of this identity, we establish the domain of influence theorem for the Green–Naghdi-II model. This theorem implies that for a given bounded support of thermomechanical loading, the thermoelastic disturbance generated by the pair of temperature and potential of the system vanishes outside a well-defined bounded domain. This domain is shown to depend on the support of the load, that is, on the initial and boundary data. It is also shown that under Green–Naghdi-II model, the thermoelastic disturbance propagates with a finite speed that is dependent on the thermoelastic parameters.