On the solution of a contact problem for a rhombus weakened with a full-strength hole

Abstract

This paper addresses a problem of plane elasticity theory for a doubly connected body whose external boundary is a rhombus with its diagonals lying at the coordinate axes OX and OY . The internal boundary is the required full-strength hole and the symmetric axes are the rhombus diagonals. Absolutely smooth stamps with rectilinear bases are applied to the linear parts of the boundary, and the middle points of these stamps are under the action of concentrated forces, so there are no friction forces between the stamps and the elastic body. The hole boundary is free from external load and the tangential stresses are zero along the entire boundary of the rhombus. Using the methods of complex analysis, the analytical image of Kolosov-Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body), and the equation of an unknown part of the boundary are determined under the condition that the tangential normal stress arising at it takes the constant value. Such holes are called full-strength holes. Numerical analysis are performed and the corresponding graphs are constructed.