Three-Dimensional Generalized Thermoelasticity with Variable Thermal Conductivity

Abstract—In this paper, a three-dimensional model of the generalized thermoelasticity with one relaxation time and variable thermal conductivity has been constructed. The resulting nondimensional governing equations together with the Laplace and double Fourier transforms techniques have been applied to a threedimensional half-space subjected to thermal loading with rectangular pulse and traction free in the directions of the principle co-ordinates. The inverses of double Fourier transforms, and Laplace transforms have been obtained numerically. Numerical results for the temperature increment, the invariant stress, the invariant strain, and the displacement are represented graphically. The variability of the thermal conductivity has significant effects on the thermal and the mechanical waves.


I. INTRODUCTION
HE uncoupled thermoelasticity which is called the classical thermoelasticity theory, predicts that the phenomena are incompatible with the behavior of the thermoelastic materials. The heat conduction equation is separated and does not contain any elastic effect while the fact that the elastic changes lead to heat changes and the heat conduction equation is of the parabolic type which generates infinite speeds of propagation for heat waves.
Biot introduced the theory of coupled thermoelasticity (CTE) to fix the shortcoming in the uncoupled theory of thermoelasticity [1]. In the context of the CTE, the equations of motion are hyperbolic type, and heat conduction equation is of diffusion type which generates infinite speeds of thermal changes which are not compatible with the physical behavior. To fix the second paradox, many modifications of dynamic thermoelasticity theories were introduced by [2]- [5] based on second sound phenomena. Many problems based on the above theories have been solved [6]- [12].
State-space methods are one of the essentials of the modern control theory and it is the foundation for many studies in stochastic systems, nonlinear systems, and optimal control. There is no limit to the order as it works for any number of independent first-order differential equations [13]- [15]. Moreover, this approach is useful because the linear systems based on time parameter could be analyzed as time-invariant linear systems and problems formulated could be programmed on a computer easily. High-order linear systems can be analyzed by state-space methods where multiple input and multiple output systems can be treated quickly. Solving thermoelastic problems by using the state-space approach in which the governing equations of the problem are rewritten in context of state-space variables, namely, the temperature increment, the displacement, or the strain and their gradients, has been developed by [13].
Godfrey has reported decreases of up to 45% in the thermal conductivity of various samples of silicon nitride between 1 and 400 °C. So, we have to know the effects of these variations on the stress and displacement distributions in metal components [16]. So, the temperature dependence of material properties must be taken into consideration in the thermal stress analysis of these elements. Many applications have been introduced assuming variable thermal conductivity [10], [17], [18].

II. THE GOVERNING EQUATIONS
The governing equation of an isotropic and homogeneous elastic medium in the context of the generalized thermoelasticity with one relaxation time without any heat sources or any external body forces in general co-ordinates take the following form: The equations of motion [7], [19]: where is the temperature increment and is the reference temperature such that. The heat equation [10], [18]: The constitutive relations in the form The strain-displacement relation in the form  ( 5 ) where is the thermal conductivity in the normal case. Differentiating (5) with respect to the coordinates, we get (6) Differentiating (6) again with respect to the coordinates, we obtain (7) Differentiating (5) with respect to time, we get (8) Hence, we obtain ( 9 ) We approximate the variation of the thermal diffusivity with temperature by the linear law [10], [18]: (10) where is a small constant is called parameter of the thermal conductivity change and it takes positive or negative values according to the material properties where the thermal conductivity of some materials increases and other decreases while heating.
Substituting from (10) into the mapping in (5), we get ( 1 1 ) which gives (12) Substitute in (1) and (3), we get (13) ( 1 4 ) For linearity, we will neglect the small nonlinear terms in (13) and (14). Hence, we obtain (15) and (16) III. FORMULATION OF THE PROBLEM Consider an isotropic, homogeneous and elastic body in three dimensional occupies the region which is defined by where the body is quiescent initially and has been shocked thermally moreover tractions free on the bounding of the surface in the directions of the principal axis. The governing equations will be taken in the context of the generalized thermoelasticity when the body has no heat sources or any external forces. By using the Cartesian coordinates and the components of the displacement , we can write them as follows: The equations of motion: The heat equation: The constitutive relations are in the forms: , , , and (27) We can rewrite (17) where , .
For simplicity, we dropped the primes. By summing (32)-(34) and using (27), we get (40) We will consider the invariant stress to be the mean value of the principal stresses as follows: (41) By using (36)-(38), we obtain (42) where .
Using Laplace transform defined for any function as: (43) Applying the above transform for both sides of the above equations, we obtain (44) Applying the above transform, we have the following system of ordinary differential equations (60) and (61) where and

IV. STATE SPACE FORMULATION
We shall take as state variables in the physical domain the quantities , and .
Regarding the Laplace transform of these four variables, the equations can be written as [13], [14]: The formal solution of (63) is given by:   where I is the identity matrix of order 4.
To determine the coefficient, we use the Cayley-Hamilton theorem again as follows: To complete the solution we have to know the vector matrix , so we have to apply certain boundary conditions, so we consider that the bounding plane to the surface has no traction on the principal axis and thermally shocked, which gives (80) and (81) where (constant) is the intensity of the thermal shock and is the Heaviside unit step function. Thus, from (11)