Dwuwymiarowe Zagadnienie Teorii Naprężeni Cieplnych

The general (complex) solution of the equations of the plane theory of thermal stress is obtained assuming that the temperature is an analytic function. By considering the solution for the infinite region, the potentials ϕ (z) and ψ (z) involved in the general solution are given an interpretation, enabling the application of the theory of analytic functions to reduce a given boundary-value problem to boundary-value problems for some analytic functions. The equations for the stress and the displacement are obtained in curvilinear coordinates connected with the conformal mapping of a circle or a semi-plane into the given region S.
In addition, following the argument of Muschelishvili, the stresses and the displacements are expressed for the circle and the elastic semi-plane in terms of the temperature and an additional analytic function defined outside the region considered. A method is proposed for using the above relations in the case of a concentrated source of heat.
The last section of the paper is concerned with the determination of the thermal stress in a circular panel acted on by a connected source of heat of constant intensity W. The temperature T0 at the edge of the panel is also constant. The planes bounding the panel are thermally insulated.
Part of the periphery of the panel is free, the remaining part being fixed. A closed-form solution is obtained. By passing to the limit we obtain from it the solutions for panels with free and fixed edge. Diagrams of σrr and σrψ are given for a portion of the fixed edge.