Stresses analysis of a plane with elliptical inclusion for the model of the semi-linear material

Abstract

The exact analytical solution of nonlinear problem of elasticity (plane stress and plane strain) for a plane with elastic elliptic inclusion is obtained. The external constant nominal (Piola) stresses are given at infinity of a plane. The continuity conditions for the stress and displacement are satisfied on the boundary of the inclusion. The mechanical properties of the plane and inclusion are described by a model of semi-linear material. For this material methods of the theory of functions of a complex variable are using for solving nonlinear plane problems. The stresses and displacements are expressed through two analytic functions of a complex variable, which are defined from the nonlinear boundary conditions on a boundary of inclusion. The assumption is made that the stress state of the inclusion is homogeneous (the tensor of nominal stresses is constant). The hypothesis has allowed to reduce a difficult nonlinear problem of elliptic inclusion to the solution of two simpler problems (first and second) for a plane with an elliptic hole. The validity of this hypothesis has proven to that obtained solution precisely satisfies to all equations and boundary conditions of a problem. The similar hypothesis was used earlier for solving linear and nonlinear problems of elliptic inclusion. Using general solution, the solutions of some particular nonlinear problems are obtained: a plane with a free elliptic hole and a plane with a rigid inclusion. The stresses are calculated on the contour of inclusion for different material parameters.