An isotropic elastic medium containing a cylindrical borehole with a rigid plug

Abstract

An analytical and numerical analysis is made of an isotropic elastic medium containing a cylindrical borehole of infinite length in which is located a tightly fitting rigid plug of finite length. Both the pulling of the plug and the occurrence of a radial misfit are considered. The boundary conditions are mixed, with zero radial and shear stresses at the bore surface outside the plug region and displacements given across the plug surface. Using integral representations for a Love auxiliary function, the crucial step is the analytical incorporation of the square root singularity at boundary condition junctions. This is done by using Neumann Bessel function series representations of the integrand kernels of boundary condition stresses such that discontinuous Weber–Schafheitlin integrals can be used to satisfy these conditions exactly. Displacement conditions are solved in terms of integrals of products of Bessel functions. The solutions provide expressions for the far field behaviour of a Kelvin point load solution for the plug pull case and a combined centre of expansion plus double force for radial misfit. Numerical results show good convergence of the method and the correct singular behaviours of borehole surface stresses.