An analytical solution for a plane thermal stress problem expressed in elliptical coordinates. (2nd Report. Steady-state thermal stresses in an elliptical plate with heat transfer on the upper and lower surfaces).

Abstract

As another example of heat conduction and plane thermal stress problems in which the solutions are expressed in Mathieu and modified Mathieu functions, analytical solutions in elliptical coordinates are given for both a steady-state temperature field and an associated plane thermal stress problem in an elliptical plate subjected to unaxisymmetric heating on the elliptic boundary with heat transfer on the upper and lower surfaces. When heat loss from the upper and lower surfaces into the surrounding media exists, the temperature function must be expressed in Mathieu and modified Mathieu functions and thermal stresses occur even in a steady-state temperature field. The associated plane thermal stress problem can be formulated in terms of Airy's stress function. Numerical calculations are carried out for the distributions of temperature and circumferential thermal stress in the elliptical plate subjected to unaxisymmetric heating expressed in the form of a fourth-order equation and Heviside step function of the η coordinate on the elliptic boundary.