Remarks on cavity formation in fiber-reinforced incompressible non-linearly elastic solids

In this paper we focus on cavity formation in fiber-reinforced incompressible non-linearly elastic solids. In particular, the material under consideration is a base neo-Hookean augmented by a function that accounts for the existence of a unidirectional reinforcing. This function characterizes the anisotropy of the material and is referred to as reinforcing model. Previous works has dealt with the analysis of a specific reinforcing model, the so called standard reinforcing model, that is a quadratic function that depends only on the fiber reinforcement stretch. Here, two different reinforcing models are examined: a power law that depends only on the fiber stretch and a quadratic function that depends simultaneously on the fiber stretch and fiber shearing. Closed form analytic solutions are found for the classical problem of cavity formation in a sphere under uniform radial tensile dead-load with the fiber in the radial direction. It is shown that for some of the new reinforcing models under study the cavitation instabilities obtained in previous works for the standard reinforcing model are not possible.