On the steady-state thermoelastic problem for the half-space

Abstract

This paper deals with the determination of the steady-state thermal stresses and displacements in a semi-infinite elastic medium which is bounded by a plane. The problem is treated within the classical theory of elasticity and is approached by the method of Green. It is shown that the stress field induced by an arbitrary distribution of surface temperatures is plane and parallel to the boundary. If the surface temperature is prescribed arbitrarily over a bounded “region of exposure” and is otherwise constant, the problem reduces to the determination of Boussinesq’s three-dimensional logarithmic potential for a disk in the shape of the region of exposure, whose mass density is equal to the given temperature. Moreover, it is found that there exists a useful connection between the solutions to Boussinesq’s and to the present problem for the half-space. An exact closed solution, in terms of complete and incomplete elliptic integrals of the first and second kind, is given for a circular region of exposure at uniform temperature. Exact solutions in terms of elementary functions are presented for a hemispherical distribution of temperature over a circular region, as well as for a rectangle at constant temperature.