Exact Solution of Thermoelastic Problem for a One-Dimensional Bar without Energy Dissipation

We consider a homogeneous isotropic thermoelastic half-space in the context of the theory of thermoelasticity without energy dissipation. There are no body forces or heat source acting on the half-space. The surface of the half-space is affected by a time dependent thermal shock and is traction free. The Laplace transform with respect to time is used. The inverse transforms are obtained in an exact manner for the temperature, thermal stress, and displacement distributions. These solutions are represented graphically and discussed for several cases of the applied heating. Comparison is made between the predictions here and those of the theory of thermoelasticity with one relaxation time. 1. Introduction In 1967, Lord and Shulman [1] introduced the theory of generalized thermoelasticity with one relaxation time for an isotropic body. This theory corrects the unrealistic conclusions of the older theories (the uncoupled and the coupled theories of thermoelasticity) that heat waves travel with infinite speeds. The Heat conduction law of this theory is the Cattaneo law which is different from Fourier’s law utilized in both the coupled and the uncoupled theories. Among the contribution to the subject are the works in [2–6]. In 1972 Green and Lindsay [7] developed the theory of generalized thermoelasticity with two relaxation times, based on a generalized inequality of thermodynamics. In this theory both the equations of motion and of heat conduction are hyperbolic. The heat conduction law is the same as Fourier’s law when the system has a centre of symmetry. Among the contributions to this theory are the works in [8, 9]. Green and Naghdi [10–12] have formulated three new models of thermoelasticity. In one of these models Green and Naghdi [12] predict that the internal rate of production of entropy is identically zero; that is, there is no dissipation of thermal energy. This theory (GN theory) is known as thermoelasticity without energy dissipation theory. In the development of this theory the thermal displacement gradient is considered as a constitutive variable, whereas in the conventional development of a thermoelasticity theory, the temperature gradient is taken as a constitutive variable [12]. A couple of uniqueness theorems have been proved in [13, 14], and one-dimensional waves in a half-space and in an unbounded body have been studied in [15, 16]. In view of some experimental evidence available in favour of finiteness of heat propagation speed, generalized thermoelasticity theories are considered to be more realistic than the conventional theory in