The contact problem between an elliptical punch and an elastic half-space with friction. The case of the punch subjected to shearing.

Abstract

This paper is concerned with a contact problem between an elastic half-space and a rigid elliptical punch subjected to translation parallel to the surface of the half-space. In such a problem, the normal contact stress within the contact region is much smaller than the tangential stress and is usually assumed to be zero for simplicity of the analysis. In this study, we treat the problem as a mixed boundary value problem in which the surface displacements are specified inside the contact region and the surface stresses are zero outside. Therefore, the normal contact stress is not zero within the contact region. The problem is formulated in the form of simultaneous integral equations with kernels involving Bessel functions. It is reduced to the inhomogeneous Hilbert problem with infinite unknown functions by using Abel transformation and Plemelj formulae. A general solution for the contact stress and the surface displacement is given. The distributions of stress and displacement are shown for the case of a flat-bottom punch and compared with those of the result when the normal contact stress is assumed to be zero.