Analytical and numerical methods for the solution to the rigid punch contact integral equations

In this article, the problem of a rigid punch pressed onto the surface of an elastic half-plane is studied. In a first instance, it is reminded that the standard frictionless hard contact situation, where the contact pressure is unbounded at contact ends, exhibits an analytical solution to the governing singular integral equation with Cauchy kernel. Thereafter, it is shown that the situation of contact regularization results in a singular integro-differential equation with Cauchy kernel. This latter case leads to bounded contact pressures at both contact ends, even in the frame of linear elasticity, which is of great interest in the presence of "peaking" phenomenon. However, this contact regularization requires a numerical treatment as opposed to the former. To that end, a novel simple but efficient numerical procedure, based on numerical integration in conjunction with a centered finite differences scheme, is presented and numerically illustrated through two examples at the end of the paper.