> 0 is a constant. This equation, which is parabolic, has a very unpleasant feature: a thermal disturbance at any point in the body is felt instantly at every other point; or in terms more suggestive than precise, the speed of propagation of disturbances is infinite. In this paper we develop a general theory of heat conduction for nonlinear materials with memory, a theory which has associated with it finite propagation speeds. In Section 3 we determine the restrictions that thermodynamics places on our constitutive relations. We show that our theory differs f rom other theories of heat conduction in that the heat-flux, like the entropy, is determined by the functional for the free-energy. In Section 6 we study the propagation of certain types of weak discontinuities. We show that in certain circumstances waves travelling in the direction of the heat-flux vector propagate faster than waves travelling in the opposite direction. In Section 7 we deduce the linearized theory appropriate to infinitesimal temperature gradients. We show that the linearized constitutive equation for the heat-flux q has the form: 1

A general theory of heat conduction with finite wave speeds

The equations of generalized thermoelasticity for an anisotropic medium are derived. Also, a uniqueness theorem for these equations is proved. A variational principle for the equations of motion is obtained.

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Generalized thermoelasticity for anisotropic media

A quasi-static uncoupled theory of thermoelasticity based on the heat conduction equation with a time-fractional derivative of order α is proposed. Because the heat conduction equation in the case 1≤α≤2 interpolates the parabolic equation (α = 1) and the wave equation (α = 2), the proposed theory interpolates a classical thermoelasticity and a thermoelasticity without energy dissipation introduced by Green and Naghdi. The Caputo fractional derivative is used. The stresses corresponding to the fundamental solutions of a Cauchy problem for the fractional heat conduction equation are found in one-dimensional and two-dimensional cases.

Fractional heat conduction equation and associated thermal stress

A three-phase-lag model of the linearized theory of coupled thermoelasticity is formulated by considering the heat condition law that includes temperature gradient and the thermal displacement gradient among the constitutive variables. The Fourier law is replaced by an approximation to a modification of the Fourier law with three different translations for the heat flux vector, the temperature gradient and also for the thermal displacement gradient. The model formulated is an extension of the thermoelastic models proposed by Lord–Shulman, Green–Naghdi and Tzou.

On A Thermoelastic Three-Phase-Lag Model

Fractional order theory of thermoelasticity

In this work a generalized dynamical theory of thermoelasticity is formulated using a form of the heat transport equation which includes the time needed for acceleration of the heat flow. The theory takes into account the coupling effect between temperature and strain rate, but the resulting coupled equations are both hyperbolic. Thus, the paradox of an infinite velocity of propagation, inherent in the existing coupled theory of thermoelasticity, is eliminated. A solution is obtained using the generalized theory which compares favourably with a known solution obtained using the conventional coupled theory.

Y. Shulman

Papers

A generalized dynamical theory of thermoelasticity