Analytical and numerical analyses of neo-Hookean thin-wall composite spheres

Abstract

Thin-wall composite spheres (TWCSs) are very common in both natural and artificial structures. Their response to mechanical loading was investigated in the past almost solely in the limit of infinitesimal deformations. We examine, within the framework of finite deformation elasticity, the mechanics of incompressible TWCSs with neo-Hookean core and shell phases subjected to general homogeneous displacement and traction boundary conditions. We derive explicit general forms for the displacement and the pressure fields in both phases in terms of a power series about the shear and the tension magnitudes and the shell volume fraction. The predictions of the analytical solutions are analyzed and compared with corresponding results of finite element simulations for TWCSs with different ratios between the phases shear moduli. In addition to an extension of the work of Weil and deBotton [22] from simple shear to general homogeneous boundary conditions, we modify the power series solution and provide a reliable solution for any combination of phases shear modulus. We demonstrate that a relatively small number of terms in the series is required for a good agreement with the numerical simulations up to a stretch ratio of 1.5 when considering the local fields, and up to a stretch ratio of 2 when considering the average fields. The analysis emphasizes the interaction between the shell and the core and reveals the different roles of the coating under different boundary conditions. We highlight interesting similarities and dissimilarities between the spatial distributions of the local stresses and the variations of the average stresses developing in TWCSs with stiff and soft shells, under displacement and traction boundary conditions.