State-space approach of two-temperature generalized thermoelasticity of one-dimensional problem

In this paper, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory [Youssef, H., 2005a. The dependence of the modulus of elasticity and the thermal conductivity on the reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity, J. Appl. Math. Mech., 26(4), 4827; Youssef, H., 2005b. Theory of two-temperature generalized thermoelasticity, IMA J. Appl. Math., 1–8]. The medium is assumed initially quiescent. Laplace transform and state space techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to thermal shock and traction free. The inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the two-temperature parameter.