The study of nonlinear deformation of a plane with an elliptical inclusion for harmonic materials

Abstract

Analytical methods are used to study nonlinear deformation of a plane with an elliptical inclusion. The elastic properties of a material of the plane and inclusion are described with a semi-linear material. The external load is constant nominal (Piola) stresses at infinity. At the inclusion boundary, the conditions of the continuity for stresses and displacements are satisfied. Semi-linear material belongs to the class of harmonic, the methods of the theory of functions of a complex variable are applicable to solving nonlinear plane problems. Stresses and displacements are expressed in terms of two analytical functions of a complex variable, determined by the boundary conditions on the inclusion contour. It is assumed that the stress state of an inclusion is uniform (the tensor of nominal stresses is constant). This hypothesis made it possible to reduce the difficult nonlinear problem of conjugation of two elastic bodies to the solution of two more simpler problems for a plane with an elliptical hole. The validity of this hypothesis is justified by the fact that the constructed solution exactly satisfies all the equations and boundary conditions of the problem. The same hypothesis was used earlier by other authors to solve linear and nonlinear problems of an elliptical inclusion. In the article, a comparative analysis of the stresses and strains is carried out for two models of harmonic materials — semi-linear and John’s. Various variants of values of elasticity parameters of the inclusion and matrix have been considered.