Pressure of a die on an elastic layer of finite thickness

Abstract

This article considers the problem of the pressure of an axisymmetric die on an elastic layer of finite thickness. Solution to this problem is obtained in terms of asymptotic series in powers of h −1, where h is the dimensionless thickness of the layer. The cases of plane and parabolic dies are studied in detail. The solution obtained adequately described the state of stress when the thickness of layer is of the order of the diameter of the surface of contact. ∗ ∗ After this article was submitted for printing, it came to the attention of the authors that another paper, written by Lebedev and Ufliand [1] considers the axially-symmetric problem of a plane circular die on a layer of finite thickness. The solution given in [1] will be applicable, generally speaking, for any dimensionless thickness h. However, its use involves the numerical evaluation of a certain Fredholm integral with subsequent numerical finding of quadratures. In this respect the asymptotic formulas presented in this work are important, since in numerous cases they give directly all the fundamental characteristics of the problem.