Analysis of radial expansion, eversion, and cavitation of soft functionally graded material spheres

Abstract

We study radial expansion, cavitation, and eversion of spherical shells made of incompressible, isotropic, and functionally graded (i.e. inhomogeneous) soft (or rubber-like) materials that are increasingly being used in prosthetics, seals, tires, flexible electronics, soft robots, and many other applications. We consider all geometric and material nonlinearities and assume the sphere material to be Mooney–Rivlin material whose two parameters, C1( R) and C2( R), are smooth functions of the radial coordinate, R, in the stress-free undeformed configuration. The shell’s inversion illustrates non-uniqueness of solutions in finite elasticity since sphere’s bounding surfaces are traction free in the reference and the deformed configurations but stresses/strains in the interior are different. Assuming that a shell under a dead tensile pressure on the outer surface cavitates when the radial stretch at the inner surface equals four, we delineate effects of functions C1( R) and C2( R) on the cavitation pressure. It is found that for power-law variations with indices m and n, respectively, for C1( R) and C2( R) the cavitation pressure can be controlled by suitably choosing m and n. Large positive and negative values of m and n are deleterious for a sphere loaded only by a pressure on the inner surface since they produce high hoop stresses within the sphere. Other results given in the paper will enable one to tailor functions C1( R) and C2( R) to either mitigate cavitation or initiate it at a desired pressure or have prescribed through-the-thickness variations of stresses to optimize sphere’s performance.