Axisymmetric Indentation of a Thin Incompressible Elastic Layer

Axisymmetric indentation of an incompressible elastic layer by a frictionless rigid sphere is considered. Estimates of the contact radius and the force required to produce a given indentation are given when the thickness of the layer is small compared to the contact radius. The elastic layer is either bonded to or slips along a rigid substrate. The estimates are obtained by asymptotically matching a lubrication-type expansion valid in the contact region to an edge layer expansion studied using the Wiener--Hopf technique. A substantially larger force is required to equally indent a bonded layer compared to a slipping layer. In the former case the force is inversely proportional to the cube of the layer thickness, rather than to the layer thickness itself, if the layer is free to slip.