Axisymmetric indentation of a rigid cylinder on a layered compressible and incompressible halfspace

We propose a solution for an elastic, axisymmetric, indentation problem. The indenter is a rigid cylinder on an elastic layer in contact with an elastic substrate. The goal is to provide a contact law between the applied force and the displacement of the coating in two cases: frictionless interaction and perfect binding between the coating and the substrate. As examples we have considered situations in which the substrate is either softer, similarly stiff, or stiffer than the coating, both for compressible or incompressible materials.