The analytical solution of plane problem for a plate with an elastic elliptic inclusion

Abstract

The exact analytical solution of the plane problem of elasticity (plane strain and plane stress) for the infinite elastic plate containing an elliptic elastic inclusion of different material is obtained. At infinity of the plate the constant stresses are given. Stresses and displacements are continuous at the interface. Methods of the theory of functions of a complex variable and the conformal transformation are applied for solving the problem. The basic assumption is that the state of stress in the elliptic inclusion is homogeneous under constant stresses at infinity. Acceptance of this hypothesis allows us to reduce the solution of the difficult problem for a plate with elastic inclusion to the solution of two boundary value problems (the first and the second) for the plate with an elliptic hole. The validity of this hypothesis in our work is proved by the fact that the obtained solution precisely satisfies to all boundary conditions. The equations of equilibrium and compatibility are carried out identically by introducing Kolosov — Muskhelisvili complex potentials. The calculation of the stresses is performed in MATLAB environment, and the graphs of stresses for various kinds of loadings at infinity and different materials of the inclusion and the plate are presented.