The pressure of a punch in the form of an elliptic paraboloid on an elastic layer of finite thickness

Abstract

The problem of the indentation (without friction) of an absolutely solid body into an elastic layer is investigated. It is assumed that the diameter of the contact area which is unknown in advance, is small compared with the layer thickness. A model unilateral contact problem of the pressure on the elastic half-space of a punch with a surface which is close to an elliptic paraboloid is derived using the method of matched asymptotic expansions. The asymptotic solution of the model problem and the asymptotic form of the boundary of the contact area are constructed using Nazarov's method. A uniformly valid asymptotic representation is found for the density of the contact pressures. The asymptotic solution of the axisymmetric problem is written out in explicit form.