On the time differential dual-phase-lag thermoelastic model

Abstract

This paper studies the time differential dual-phase-lag model of a thermoelastic material, where the elastic deformation is accompanied by thermal effects governed by a time differential equation for the heat flux with dual phase lags. This coupling gives rise to a complex differential system requiring a special treatment. Uniqueness and continuous dependence results are established for the solutions of the mixed initial boundary value problems associated with the model of the linear theory of thermoelasticity with dual-phase-lag for an anisotropic and inhomogeneous material. Two methods are developed in this paper, both being based on an identity of Lagrange type and of a conservation law applied to appropriate initial boundary value problems associated with the model in concern. The uniqueness results are established under mild constitutive hypotheses (right like those in the classical linear thermoelasticity), without any restrictions upon the delay times (excepting the class of thermoelastic materials for which the delay time of phase lag of the conductive temperature gradient is vanishing and the delay time in the phase lag of heat flux vector is strictly positive, when an ill-posed model should be expected). The continuous dependence results are established by using a conservation law and a Gronwall inequality, under certain constitutive restrictions upon the thermoelastic coefficients and the delay times.