Uniform elastic field inside an elliptical inhomogeneity with Neuber’s nonlinear stress–strain law under generalized plane strain deformations

We prove the uniformity of the in-plane and anti-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 and anti-plane stresses. The elastic material occupying the elliptical inhomogeneity obeys Neuber's special nonlinear stress-strain law. The original boundary value problem is finally reduced to a single non-linear equation for the constant effective strain within the inhomogeneity, which is rigorously proved to have a unique solution. Once the non-linear equation is solved numerically, we establish the uniform elastic field within the elliptical inhomogeneity and the non-uniform elastic field in the linear elastic matrix.