Large deformations of a plate with an elastic elliptic inclusion for John's harmonic material

Abstract

Exact analytical solution of a non-linear plane-strain problem is obtained for a plate with an elastic elliptic inclusion subjected to uniform remote nominal (Piola) stresses. The conditions of continuity are performed for the nominal stresses and displacements at a contour of inclusion. Mechanical properties of a plate and an inclusion are described by model of a John's harmonic material. This model has allowed to use complex-variable methods for a solution of non-linear plane-strain problems. It is supposed that a state of stress inside inclusion is uniform (tensor of nominal stresses is constant). By this assumption the complicated non-linear problem of conjugation of two bodies of different materials reduce to the solution of two more simple problems for a plate with an elliptic hole. The validity of this hypothesis is proved by that obtained solution satisfies precisely to all equations and boundary conditions of problem. Similar hypothesis was used at a solution of linear and non-linear problems about elliptic inclusion.