On Some Finite Deformations of Inhomogeneous Compressible Elastic Solids

We study several simple boundary value problems for a special class of inhomogeneous, isotropie, compressible elastic solids: inflation of a spherical shell; bending, stretching and shearing of a rectangular block; and straightening, stretching and shearing of a sector of a hollow cylinder. The purpose of the study is twofold: to establish new exact solutions to boundary value problems concerning inhomogeneous, isotropie, compressible elastic solids that have technological relevance (to the deformation of layered composites); and to reaffirm a thesis concerning an inherent difficulty with regard to the homogenization techniques that are in vogue that appeal to the stored energy in the homogenized body being the same as that for the inhomogeneous body. In addition to establishing new exact solutions for a class of inhomogeneous elastic solids, the study reinforces the thesis of Saravanan and Rajagopal that great care ought to be exercised in approximating even a very mildly inhomogeneous body by an equivalent homogeneous body when large deformations are involved.