Plane elasticity solution for a half-space weakened by a circular hole and loaded by a concentrated force

We present a complete analytical solution for the plane elasticity problem of a concentrated force acting on a half-space weakened by a circular hole. The solution is based on the algorithm proposed by G.B. Jeffery in 1920 for elasticity problems, using bipolar coordinates to characterize the geometry. We found the biharmonic stress function describing stresses and strains in the half-space and the associated biharmonic function that allows determining the displacement field. Both functions are given in the form of Fourier series with quite compact coefficients. Particular attention is paid to the displacements at the plane boundary of the half-space. It is discussed how these displacements can be used in an inverse procedure for the identification of a circular hole diameter and position.