Solution of the dynamic problem of thermoelasticity in stresses for a strip

The method for solving a two-dimensional dynamic thermoelasticity problem in stresses for a strip with a rectangular cross-section is proposed. The initial system of equations in stresses, which describes the plane-stress state of the strip, is selected. A solution method is developed based on approximating the distributions of all components of the dynamic stress tensor using cubic polynomials with respect to the thickness coordinate of the strip. This reduces the original system of two-dimensional unsteady equations to a system of one-dimensional unsteady equations involving integral characteristics along the thickness. To solve the resulting system of equations, a finite integral transform is applied with respect to the transverse coordinate of the strip, and the Laplace transform is used for the time variable. General solutions to the considered dynamic thermoelasticity problem under unsteady thermal and force actions on the strip are presented. A criterion for assessing the bearing capacity of the strip is proposed. The analysis reveals the existence of four types of resonant frequencies under the specified unsteady conditions.