General case of contact problems for a regular polygon weakened with full-strength hole

Abstract

The paper addresses a problem of plane elasticity theory for a doubly connected body whose external boundary is a regular polygon and the internal boundary is the required full-strength hole including the origin of coordinates. The full-strength hole is cycle symmetric. It is assumed that to every link of the broken line conforming the outer boundary of the given body are applied absolutely smooth rigid punches with rectilinear bases, which are under the action of the force P that applies to their middle points. There is no friction between the surface of the given elastic body and the punches. The uniformly distributed normal stress Q is applied to the hole boundary. Using the methods of complex analysis, the analytical image of Kolosov–Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body) and the shape of the hole’s contour are determined under the condition that the tangential normal stress arising at it takes a constant value. A similar problem is considered for a square and an equilateral triangle, which are weakened with full-strength holes. Using the method developed here, the partially unknown boundary value problems under consideration is reduced to known boundary value problems of the theory of analytic functions. The solutions are presented in quadratures, and full-strength contours are constructed.