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

Abstract

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.

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“…Later we found the solutions of the problem of a generalized thermoelastic infinite medium with a cylindrical cavity subjected to a ramptype heating and loading [2006a] and the two-dimensional generalized thermoelasticity problem for a half-space subjected to ramp-type heating [2006c]. In [Youssef and Al-Lehaibi 2007] we used the statespace approach in the problem of two-temperature generalized thermoelasticity while in [Bassiouny and Youssef 2008] we solved the two-temperature generalized thermopiezoelasticity problem of a finite rod subjected to different types of thermal loading. In [Youssef 2008] solved the two-dimensional problem of a two-temperature generalized thermoelastic half-space subjected to ramp-type heating.…”

“…The stresses obtained from the stress functions are also plotted in Figure 2. For comparison the analogous solution obtained in [Hutchinson 1968] using the Mises yield condition is also shown. The results obtained here are qualitatively similar to those obtained in [Unger 2008] using an approximate singular solution.…”

“…are the second and third invariants [Chakrabarty 1987] of the deviatoric stress tensor, σ 1 and σ 2 being the principal stresses and σ 0 the yield stress in tension. A representation of this yield condition is shown in Figure 1 in the principal stress plane (σ 1 , σ 2 ) along with a comparison with the Mises and Tresca yield conditions.…”

Effect of Hall Current in Thermoelastic Materials with Double Porosity Structure

“…Also Ezzat et al [27], Maghraby et al [28], Youssef and Al-Lehaibi [29], Othman [30], Elisbai and Youseff [31] and Sherief and El-Sayed [32] investigated different types of problems in different media by using the state space approach.…”

Vibration of gold nano beam in context of two-temperature generalized thermoelasticity subjected to laser pulse

“…Youssef and Elsibai (2010) investigated the vibration of gold nano-beam induced by many types of thermal loading. Youssef (2006) investigated two-temperature generalized thermoelasticity theory together with a general uniqueness theorem and solved many applications in the context of this theory with Al-Lehaibi (2007), Harby (2007) and Bassiouny (2008). In this work, the model of vibration of gold nano-beam induced by laser pulse heating will be developed in the context of two-temperature generalized thermoelasticity and non-Fourier heat conduction.…”