Plane contact problem for a half-space with a poisson's ratio that varies with depth

Abstract

This paper contains the general solution to the problem of equilibrium of an inhomogeneous half-space or layer for which the shear modulus is constant and Poisson's ratio is an arbitrary function of depth. It was assumed that the applied external normal and tangential loads are such that, in the inhomogeneous half-space or layer, plane deformation conditions are observed. In particular, the variation in Poisson's ratio under conditions when it varies continuously from one finite value on the surface to another finite value at an infinitely great depth was considered in detail. This dependence on depth provides an opportunity to vary both the boundary values of Poisson's ratio and the rate of variation with increase in distance from the surface. With the aid of this solution the distribution of stresses in the inhomogeneous halfspace when the external load degenerates into a single concentrated normal load was investigated. This analysis made it possible to reach a number of conclusions which are of interest for friction and wear theory. The solution obtained for the concentrated force is based on the integral equation which was set up to solve plane contact problems for an inhomogeneous half-space with a Poisson's ratio that varies with depth. An approximate ( i.e. close enough to be exact) solution of the contact problem for the pressure of a round cylinder-shaped indenter on the surface of an inhomogeneous halfspace was obtained. This solution enables the stress-strain state in the deep layers of the contacting area to be investigated relatively easily. The analysis of the solution of this contact problem was carried out.