Chin. Phys. B Vol. 21, No. 1 (2012) 014601 Generalized thermoelasticity of the thermal shock problem in an isotropic hollow cylinder and temperature dependent elastic moduli Ibrahim A. Abbas a)b) and Mohamed I. A. Othman c) † a) Department of Mathematics, Faculty of Science and Arts - Khulais, King AbdulAziz University, Jeddah, Saudi Arabia b) Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt c) Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt (Received 28 March 2011; revised manuscript received 21 July 2011) In this paper, we construct the equations of generalized thermoelasicity for a non-homogeneous isotropic hollow cylider with a variable modulus of elasticity and thermal conductivity based on the Lord and Shulman theory. The problem has been solved numerically using the finite element method. Numerical results for the displacement, the temperature, the radial stress, and the hoop stress distributions are illustrated graphically. Comparisons are made between the results predicted by the coupled theory and by the theory of generalized thermoelasticity with one relaxation time in the cases of temperature dependent and independent modulus of elasticity. Keywords: generalized thermoelsticity, thermal shock, temperature dependent elastic moduli, finite element method PACS: 46.25.Hf DOI: 10.1088/1674-1056/21/1/014601 1. Introduction In the postwar years, we have seen a rapid devel- opment of themoelasticity stimulated by various engi- neering sciences. [1] Most investigations were done un- der the assumption of the temperature-independent material properties, which limited the applicability of the solutions obtained to certain ranges of tempera- ture. At high temperature, the material characteris- tics, such as the modulus of elasticity, the Poisson’s ratio, and the coefficient of thermal conductivity, are no longer constants. [2] In recent years, due to progress in various fields of science and technology, taking into consideration the real behaviour of the material char- acteristics becomes an actual necessity. In some in- vestigations, they have been taken as functions of coordinates. [3,4] The classical uncoupled theory of thermoelastic- ity predicts two phenomena that are not compatible with the physical observations. First, the equation of heat conduction of this theory does not contain any elastic terms contrary to the fact that elastic changes produce heat effects. Second, the heat equation is of parabolic type, predicting infinite speeds of propaga- tion for heat waves. Biot [5] introduced the theory of coupled (CD) thermoelasticity to overcome the first shortcoming. The governing equations for this theory are coupled, eliminating the first paradox of the classical theory. However, his theory still has the second shortcoming, since the heat equation for the coupled theory is also parabolic. Lord and Shulman [6] (LS) introduced the the- ory of generalized thermoelasticity with one relaxation time. In this theory, a modified law of heat conduc- tion including both the heat flux and its time deriva- tive replaces the conventional Fourier’s law. The heat equation associated with this theory is a hyperbolic one, and hence automatically eliminates the para- dox of infinite speeds of propagation inherent from both the uncoupled and the coupled theories of ther- moelasticity. Most often, the solutions obtained us- ing this theory differ little qualitatively from those obtained using either the coupled or the uncoupled theories, though, the solutions differ quantitatively. However, for many problems involving steep heat gra- dients and when short time effects are sought, this theory gives markedly different results compared to those predicted by any of the other theories. This is the case encountered in many problems in indus- try, especially inside nuclear reactors where very high † Corresponding author. E-mail: othman@yahoo.com c  2012 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn 014601-1