An analytical solution for a plane thermal stress problem expressed in elliptical coordinates. (3rd Report. Steady-state thermal stresses in a confocal hollow elliptical plate).

Abstract

The analytical solutions are given for both a steady-state heat conduction problem and an associated plane thermal stress problem expressed in elliptical coordinates in a confocal hollow elliptical plate subjected to nonaxisymmetric heatings on the elliptical boundaries, and with heat loss from the upper and lower surfaces into the surrounding media. The temperature function can be expressed in the form of an infinite series of Mathieu and modified, Mathieu functions. The associated plane thermal stress problem can be formulated in terms of Airy's stress function. In this formulation, a single-valuedness of the rotation component is assured by using Michell's condition in elliptical coordinates derived already by one of the present authors. The numerical calculations of the distributions of the temperature and circumferential thermal stress in the confocal hollow elliptical plates are carried out for the cases of various aspect ratios of the elliptical hole and various Biot's numbers on the upper and lower surfaces.