Applied Mathematics, 2011, 2, 619-624 doi:10.4236/am.2011.25082 Published Online May 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM A Problem of a Semi-Infinite Medium Subjected to Exponential Heating Using a Dual-Phase-Lag Thermoelastic Model Ahmed Elsayed Abouelregal Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt E-mail: ahabogal@mans.edu.eg Received January 2, 2011; revised March 39, 2011; accepted April 2, 2011 Abstract The problem of a semi-infinite medium subjected to thermal shock on its plane boundary is solved in the context of the dual-phase-lag thermoelastic model. The expressions for temperature, displacement and stress are presented. The governing equations are expressed in Laplace transform domain and solved in that do- main. The solution of the problem in the physical domain is obtained by using a numerical method for the inversion of the Laplace transforms based on Fourier series expansions. The numerical estimates of the dis- placement, temperature, stress and strain are obtained for a hypothetical material. The results obtained are presented graphically to show the effect phase-lag of the heat flux q  and a phase-lag of temperature gra- dient   on displacement, temperature, stress. Keywords: Generalized Thermoelasticity, Dual-Phase-Lag Model, Semi-Infinite Medium, Laplace Transform 1. Introduction Biot [1] (1956) introduced the theory of coupled ther- moelasticity to overcome the first shortcoming in the classical uncoupled theory of thermoelasticity where it predicts two phenomena not compatible with physical observations. First, the equation of heat conduction of this theory does not contain any elastic terms. Second, the heat equation is of a parabolic type, predicting infi- nite speeds of propagation for heat waves. The governing equations for Biot theory are coupled, eliminating the first paradox of the classical theory. However, both theo- ries share the second shortcoming since the heat equation for the coupled theory is also parabolic. Thermoelasticity theories that predict a finite speed for the propagation of thermal signals have aroused much interest in the last three decades. These theories are known as generalized therrnoelasticity theories. The first generalizations of the thermoelasticity theory is due to Lord and Shulman [2] who introduced the theory of gen- eralized thermoelasticity with one relaxation time by po- stulating a new law of heat conduction to replace the classical Fourier’ law. This law contains the heat flux vector as well as its time derivative. It contains also a new constant that acts as a relaxation time. The heat equ- ation of this theory is of the wave-type, ensuring finite speeds of propagation for heat and elastic waves. The re- maining governing equations for this theory, namely, the equations of motion and the constitutive relations remain the same as those for the coupled and the uncoupled theories. This theory was extended by Dhaliwal and She- rief [3] to general anisotropic media in the presence of heat sources. A generalization of this inequality was proposed by Green and Laws [4] Green and Lindsay obtained another version of the constitutive equations in [5]. The theory of thermoelasticity without energy dissipation is another ge- neralized theory and was formulated by Green and Na- ghdi [6]. It includes the thermal displacement gradient among its independent constitutive variables, and differs from the previous theories in that it does not accommo- date dissipation of thermal energy. Tzou [7,8] proposed the dual-phase-lag (DPL) model, which describes the interactions between phonons and electrons on the microscopic level as retarding sources causing a delayed response on the macroscopic scale. For macroscopic formulation, it would be convenient to use the DPL mode for investigation of the micro-structural