Uniform in-plane stresses and strains inside an incompressible nonlinear elastic elliptical inhomogeneity

Abstract

We establish the uniformity of the internal in-plane elastic field of stresses and strains inside an incompressible nonlinear elastic elliptical inhomogeneity embedded in an infinite linear isotropic elastic matrix subjected to uniform remote in-plane stresses. The nonlinear elastic material occupying the inhomogeneity can incorporate a power-law hardening material. The original boundary value problem is ultimately reduced to a single non-linear equation. It is proved rigorously that the non-linear equation has a unique solution. Once the non-linear equation is solved numerically, the uniform elastic field within the nonlinear elastic elliptical inhomogeneity and the non-uniform elastic field in the linear elastic matrix can be determined.