Deformation of a plate with elliptic elastic inclusion

Abstract

The exact analytical solution of a plane problem of elasticity (plane strain or plane stress) for an infinite elastic plate containing elliptic elastic inclusion of different material is received. The constant normal and shear stresses are given at infinity of a plate. The stresses and displacements are continuous on the interface of inclusion and plate. The methods of the theory functions of a complex variable and of conformal transformation are applied to the solution of this plane problem. The basic assumption which is used for construction of the solution consists that the stress state in the elliptic inclusion is homogeneous at constant external stresses at infinity of a plate. The acceptance of this hypothesis has allowed to reduce the solution of the complicated interface problem for a plate with elastic inclusion to the solution of two simple boundary value problems (the first and the second) for a plate with an elliptic hole, their exact solutions are known. The validity of the signed hypothesis in our work is proved to that the received solution precisely satisfies to all boundary conditions of a boundary value problem. Thus the equations of equilibrium and compatibility conditions are carried out identically by introduction of Kolosov-Muskhelisvili complex potentials. The calculations of stresses in package Matlab have been executed and graphics are constructed for various kinds loadings of plates at infinity and different materials of inclusion and plate. Refs 26. Figs 4.

Keywords:

plane problems, elliptic inclusion, complex functions method, conformal transformation