The pressurized hollow sphere problem in finite elastostatics for a class of compressible materials

Abstract

This investigation is concerned with the problem of a hollow sphere subjected to uniform internal and external pressure within the equilibrium theory of finite elasticity. The sphere is composed of homogeneous, isotropic, compressible materials of special type, namely harmonic materials. Explicit closed-form solutions for the deformation and stress fields are obtained. The true stress distribution, expressed as a function of the undeformed coordinates, is shown to be essentially independent of material properties. The two cases of internal pressure only, and external pressure only, are examined in detail. In the former case, there is a critical value of the applied pressure at which the maximum hoop stress in the sphere, occurring at the inner surface, becomes unbounded. Results appropriate for thin shells are also obtained. For the case of external pressure only, a critical value of the applied pressure exists for which the cavity closes. The maximum hoop stress does not always occur at the cavity wall. For nearly solid spheres, or equivalently, for a cavity in an unbounded medium, explicit results are provided for the corresponding stress concentration factor. For sufficiently small values of applied pressures, all the foregoing results coincide with those of classical linear isotropic elastostatics.