A Thick Hollow Sphere Compressed by Equal and Opposite Concentrated Axial Loads: An Asymptotic Solution

Abstract

We consider an elastic hollow sphere with midsurface radius R and thickness 2h which is subjected to two equal and opposite concentrated loads acting at the ends of a diameter. The three-dimensional linear elasticity solution to this problem consists of (i) a narrow Saint Venant component extending a distance of order O(h) from each load point, (ii) a wider edge bending component extending a distance of order $O(\sqrt{Rh})$ from each load point, and (iii) a membrane component which permeates the whole sphere. Because of the stress singularities at the load points, the Saint Venant component is extremely complex and difficult to calculate. We determine the other two (outer) components of the solution without any explicit knowledge of the Saint Venant (or inner) component. This is achieved by a rigorously valid method which does not depend on any physical heuristic, such as Saint Venant's principle. Our method depends on the fact, established here for the first time, that the eigenfunctions for the sphe...