The stress analysis of thin contact layers: a singular perturbation method for integral equations

Abstract

In this paper, a contact problem is considered with a circular indenter pressed normally against a semi-infinite elastic composite that consists of a contact layer with uniform thickness welded together with another dissimilar medium. The indentation is modelled by means of a real continuous function g ( t ), and an integral equation representation for the displacement is derived in terms of g ( t ) on the contact boundary. The integral equation is evaluated numerically for g ( t ) when the contact layer is finite. In the case when the thickness of the contact layer becomes small compared with the radius of the contact region, a singular perturbation technique is used to derive an asymptotic expansion for g ( t ). This leads to a Wiener–Hopf equation formulation for the semi-infinite geometry in the Fourier-transformed domain of the inner coordinates. Subsequently, Van Dyke's matching principle is used to match the inner with the outer solution of g ( t ). The results are illustrated and verified with a simple limit solution derived using an integral argument.