Analytical Solution for Two-Dimensional Coupled Thermoelastodynamics in a Cylinder

An infinitely long hollow cylinder containing isotropic linear elastic materials is considered to be under the effect of arbitrary boundary stress and thermal condition. The two-dimensional coupled thermoelastodynamic PDEs are specified based on motion and energy equations, which are uncoupled using Deresiewicz-Zorski potential functions. The Laplace integral transform and Bessel-Fourier series are used to derive solutions for the potential functions, then the displacements-, stresses- and temperature-potential relationships are used to determine displacements, stresses and temperature fields. It is shown that the formulation presented here is collapsed on the solution existed in the literature for a simpler case of axis-symmetric configuration. To solve the equation used and evaluate the displacements, stresses and temperature at any point and time, a numerical procedure is needed. In this case, the numerical inversion method proposed by Durbin is applied to evaluate the inverse Laplace transforms of different functions involved in this paper. For numerical inversion, there exist many difficulties such as singular points in the integrand functions, infinite limit of the integral and the time step of integration. Finally, the desired functions are numerically evaluated and the results show that the boundary conditions are accurately satisfied. The numerical evaluations are presented graphically to make engineering sense for the problem involved in this paper for different cases of boundary conditions. The results also indicate that although the thermal induced wave propagates with an infinite velocity, the time lag of receiving stress waves with significant amplitude is not zero. The effect of thermal boundary conditions are shown to be somehow oscillatory, which is due to reflective boundary conditions and may be used in designing of such an element.