On the Inflation of Residually Stressed Spherical Shells

Abstract

A well known result from the non-linear theory of elasticity applied to spherical shells is that the classical Mooney-Rivlin constitutive law may give either a monotonic or a non-monotonic pressure-inflation response for finite deformation. Specifically, this is determined by two factors: the relative shell thickness, and the relative \(I_{1}\) to \(I_{2}\) contribution in the M-R constitutive law. Here we consider how a residual stress field may affect this behavior. Using a constitutive framework for hyperelastic materials with residual stress that has been especially applied to tubes, this paper focuses on finite thickness spherical shells while using a similar prototypical energy response. In this context we examine different residual stress states, and show how certain of these lead to more workable analytical results than others. All of them enable Taylor and asymptotic expansions in the small and large inflation limit. On this basis it is shown how particular residual stress fields can cause a monotonic inflation graph to become non-monotonic, and vice versa.