An analytical solution for a plane thermal stress problem expressed in elliptical co-ordinates. 1st report. Transient thermal stresses in an elliptical plate.

Abstract

As an example of transient thermal stress problems expressed in rectangular curvilinear coordinates for which no analytical solutions have been reported, analytical solutions are presented for a transient two-dimensional temperature field and an associated thermal stress problem in an elliptical plate expressed in elliptical co-ordinates. The transient temperature function of the elliptical plate subjected to an abrupt change in temperature on the elliptic boundary can be expressed in the form of an infinite series, including Mathieu and modified Mathieu functions of the first kind of even integral order. The associated thermal stress problem can be formulated in terms of thermoelastic displacement potential and Papkovich-Neuber's stress functions. Numerical calculations are carried out for the distributions of circumferential thermal stress on the elliptic boundary and the major and minor axes in the elliptical plate for three cases of eccentricity.