Analytical evaluation of transformation matrix between functional spaces and its application in obtaining exact solutions of thermo-mechanical stresses in disks

Abstract

Thermo-mechanical stresses developed in a disk, inner boundary of which is subject to the nonstationary thermo-mechanical loadings due to variation in temperature and pressure inside the disk is considered in the presented paper. Problems of theoretical analysis are divided and described as follows: The first problem implies analytical solution of the heat conduction equation with nonhomogeneous boundary conditions of Robin type, and the second problem deals with obtaining analytical solution of dynamic equation of displacement for nonisothermal elastic body (Navier equation) with traction boundary conditions in the plane strain case of uncoupled thermo-elasticity. Finite Hankel transform is used to solve the problems. Two sets of orthogonal basis functions necessary for integral transform are obtained. The first one corresponds to the boundary conditions written for the heat conduction equation, and the second fits the boundary conditions for Navier equation. Integrals defining elements of transformation matrix between the two sets of basis functions are evaluated analytically. In general, convenient method of obtaining analytical solution of linear hyperbolic equation with nonhomogeneous term, which itself is solution of nonhomogeneous parabolic equation, is elaborated within the presented work. Using finite difference method, discrete counterparts of the described problems are constructed and, via the built-in ordinary differential equation (ODE) solvers in MATLAB, numerical solutions are obtained, which exhibit excellent accordance with analytical ones.