Static and dynamic characteristics of a compound sphere using Initial Stress Reference Independence

Abstract

We study the influence of residual stress on the static and dynamic characteristics of a spherical structure. We consider a shrink-fitted compound sphere, where residual stress is developed due to the interference between outer and inner spherical layers. We calculate this residual stress and model the residually stressed material with suitable hyperelastic strain energy satisfying Initial Stress Reference Independence (ISRI). Under the action of inhomogeneous residual stress, the compound sphere behaves as an anisotropic, functionally graded composite sphere, which is reflected by the model. We apply this model to obtain stress distributions and inflation characteristics of the compound sphere. Our results demonstrate the hypothesis that residual stress moderates the stress concentration and reduces the stress gradient in biological/engineering structures. We obtain critical stretch and pressure for spherically symmetric deformations. An increase in interference/residual stress increases the critical stretch/pressure and improves the structural stability. We then apply an analytical approach to calculate the natural frequency of the compound sphere for small amplitude radial vibration. We observe that an increase in residual stress increases the natural frequency and widens the range of stable free vibration. We numerically investigate the free and forced vibration of the compound sphere. Forced vibration amplitude reduces as interference/residual stress increases, which is reflected in the frequency–amplitude plots. The frequency response shows a small hardening non-linearity. We investigate the subharmonic and superharmonic responses for small and large amplitude vibration.