Heat Source in Infinite Plane with Elliptic Rigid Inclusion and Hole

Abstract

The plane thermoelastic problems of a stationary heat source in an infinite plane with an elliptic rigid inclusion and an elliptic hole are analyzed under thermally adiabatic and isothermal boundary conditions. The problems of an elliptic rigid inclusion are derived for the following cases: (1) the case that there are rigid-body displacement and rotation; and (2) the case that there is no rigid-body displacement or rotation. To analyze these problems, the following three fundamental solutions are derived: Problem A, in which a point heat source exists within an infinite domain; Problem B, in which the inclusion has a small amount of rotation; and Problem C, in which the inclusion is subjected to concentrated loads. Two cases can be obtained by superimposing these fundamental solutions. For the hole problem, the fundamental solution (Green's function) is also derived. In analysis, the complex stress functions, the mapping function, and the thermal dislocation method are used. The complex stress functions are obtained as a closed form. For analytic examples, the stress distributions are shown under thermally adiabatic and isothermal boundary conditions. For the crack problem, the stress intensity factors are shown for the location of the heat source.