Cavity Formation at the Center of a Composite Incompressible Nonlinearly Elastic Sphere

Abstract

In this paper, the effect of material inhomogeneity on void formation and growth in incompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid composite sphere composed of two arbitrary homogeneous isotropic incompressible elastic materials perfectly bonded across a spherical interface. Under a uniform radial tensile dead load, a branch of radially symmetric configurations involving a traction-free internal cavity centered at the origin bifurcates from the undeformed configuration. In contrast to the situation for a homogeneous neo-Hookean sphere, bifurcation here may occur either locally to the right or to the left. In the latter case, the cavity has finite radius on first appearance. This discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon observed in certain structural mechanics problems. Explicit conditions determining the type of bifurcation are established for the general composite sphere. An analysis of the stress distribution is carried out and the effect of cavitation at the center on possible interface debonding is explored for the special case when the constituent materials are both neo-Hookean. It is shown that, in a quasi-static loading process, cavitation has the effect of preventing debonding at the interface.