Analytical study of cavitation in elastic solids with affinely varied mechanical property

A tiny spherical cavity will expand unboundedly in a homogenous incompressible neo-Hookean solid when the applied hydrostatic tension reaches five times of the shear modulus of the solid. Such phenomenon is usually referred as cavitation in a soft solid. In the previous studies, the soft solid is often assumed to have homogeneous mechanical properties. In this paper, the effects of inhomogeneity have been investigated by assuming the shear modulus in the solid varies affinely with the radial coordinate. A bifurcation problem is considered for a graded neo-Hookean solid subjected to uniform hydrostatic tension on the external boundary. We find that the cavitation solution with a traction-free internal cavity can be analytically expressed using hypergeometric function. For large hydrostatic tension, the equilibrium configuration can bifurcate from the trivial solution (no cavity) to the cavitation solution (cavity occurs). Both the trivial solution and cavitation solution can be stable and unstable determined by minima of their potential energy. When the external hydrostatic tension is greater than a critical value, the internal cavity can expand both continuously or discontinuously inside the solid. We hope the results obtained in this article will be helpful to understand the cavitation phenomenon in complex soft materials.