Theory for highly stressed thick-walled elastic spheres and their vibration without small strain approximations

Abstract

This paper considers how the radii and vibration of an elastic thick-walled sphere is affected by high stresses imposed by high external pressures or by high internal pressures. For simplicity, the sphere consists of idealized Hookean material allowing unlimited compression or expansion. The sphere is filled by material with an isotropic pressure that generally differs from the pressure on the outer surface of the sphere. The analysis includes large strain nonlinear curvature effects where, unlike linear elasticity theory, the difference in radial and tangential strains is not small compared to those strains. Despite the sphere’s robustness, destructive physical consequences are predicted for nonzero Poisson coefficients where, for instance, a high-pressure tank inner stresses make the inner radius approach the less stressed outer radius. This would be avoided if the Poisson coefficient changed with large strains and approached zero. To model the effect of high stresses and strains on breathing mode vibration, an effective spring constant is derived from quadratic radius deviations of the system potential energy from equilibrium. Insights from analysis avoiding linear elasticity approximations may be applicable to improved understanding of deep-sea marine creature survival, improved underwater vessel design for large depths, and safer containers of fluids at high pressures.