Concentrated forces and couples in an elastic half-plane with a hole.

Abstract

The plane problem of the theory of elasticity for the linearly elastic semiinfinite plane with a free-form hole bounded by a smooth contour is investigated. The half-plane is considered to be subjected to the tension at infinity, external load at the rectilineal boundary and at the surface of the cavity. Concentrated forces and moments are also supposed to be applied at points within the body involved. The problem has been formulated using the Kolosov-Muskhelishvili complex stress potential technique. Results have been obtained by the superposition of two auxiliary problems. The first of them is the problem of the intact half-plane (without a hole) under given outside load at the straight boundary, at infinity and under known concentrated forces and couples. The second one is the problem of the intact half-plane under unknown inside load (applied at points within the body) to be defined. Complex potentials for single forces and moments and for distributed forces at points within a semiplane being used, the solution found thoroughly satisfies boundary conditions at the straight-line border of the half-plane and at infinity. For the surface of the cavity resolving Fredholm integral equations of the first kind in unknown load are derived. Further the problem has been reduced to the system of linear algebraic equations. Moreover the formulas for periodic load at the rectilinear edge and periodic forces and couples at points within the elastic semiplane with a hole has been written. The results obtained can be easily adapted to the problems of different types of singularities at any cavity within the elastic half-plane. A worked out example for the semiinfinite plane with a circular cut is presented, the single force being applied near the cut in the direction normal to its surface and to the straight-line boundary of the half-plane. Some details of the numerical implementation of the method proposed is discussed. It is found that increase of the radius of curvature of the boundary leads to the stress growth in that area.