Nonlinear deformation of plane with elastic elliptic inclusion for model of semi-linear material

Abstract

The problems of elasticity for composite materials with inclusions have a great practical significance for physics, mechanics and other fields of science. In this paper the exact analytical solution to nonlinear problem of plane with elliptic inclusion is obtained. The constant nominal (Piola) stresses are given at infinity. Mechanical properties of plane and inclusion are modeling by harmonic semi-linear material. The stresses and displacements are expressed through two analytic functions of a complex variable, which are defined from nonlinear boundary conditions. Acceptance of a hypothesis, that nominal stress tensor is constant inside inclusion, has allowed to reduce a difficult problem of interface of two elastic bodies to the solution of two more simple problems for a plane with an elliptic hole. Validity of this hypothesis is proved to that obtained solution precisely satisfies to all equations and boundary conditions of a problem. The nominal hoop stresses were calculated on an interface of plane and inclusion for different material parameters.