IX. Plane stress and plane strain in bipolar co-ordinates

Abstract

The problem of the equilibrium of an elastic solid under given applied forces is one of great difficulty and one which has attracted the attention of most of the great applied Mathematicians since the time of Euler. Unlike the kindred problems of hydrodynamics and electrostatics, it seems to be a branch of mathematical physics in which knowledge comes by the patient accumulation of special solutions rather than by the establishment of great general propositions. Nevertheless, the many and varied applications of this subject to practical affairs make it very desirable that these special solutions should be investigated, not only because of their intrinsic importance but also for the light which they often throw on the general problem. One of the most powerful methods of the mathematical physicist in the face of recalcitrant differential equations is to simplify his problem by reducing it to two dimensions. This simplification can only imperfectly be reproduced in the Nature of our three-dimensional world, but, in default of more general methods, it provides an invaluable weapon. It was shown by Airy that in the two-dimensional case the, stresses may be derived by partial differentiations from a single stress function, and it was shown later that, in the absence of body forces, this stress function satisfies the linear partial differential equation of the fourth order ∇ 4 X = 0, where ∇ 4 = ∇ 2 . ∇ 2 , and ∇ 2 is the two-dimensional Laplacian ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 .