Small amplitude radial oscillations of an incompressible, isotropic elastic spherical shell

Abstract

The small amplitude radial oscillation and infinitesimal stability about an equilibrium configuration of an arbitrary incompressible, isotropic and homogeneous elastic spherical shell under constant inflation pressure loading is studied for both thick- and thin-walled shells and for a spherical cavity within an unbounded continuum. The classical criterion of infinitesimal stability yields a general stability theorem relating the frequency and the pressure response. It follows that points at which the pressure is stationary are unstable or neutrally stable. All results are expressed in terms of the shear response function for a general incompressible, isotropic elastic material, and specific results are illustrated for the Mooney—Rivlin and Gent material models, the latter having limited extensibility. The classical neo-Hookean material exhibits results that are lower bounds for both models. A criterion obtained by others to characterize the possible bifurcation from a spherical to an aspherical shape is cast in terms of the general shear response function, and the physical nature of all elastic materials for which a shape bifurcation may occur is characterized. The result is illustrated in several examples.