Solution for the elastic field in a layered medium under axisymmetric contact loading

Abstract

This study addresses the problem of the calculation of the elastic stress and displacement field within isotropic layered media in frictionless contact with rigid axisymmetric indenters. For a prescribed surface stress distribution, the integral transform approach is recalled using a matrix formulation which lends itself to generalizations to multilayered systems. It leads to an analytical solution for the Hankel transform of the elastic field which can readily be numerically inverted in the real space using available discrete Hankel transform algorithms. As an example, the shear stresses induced by the sphere indentation of a coated substrate are calculated as a function of the geometrical confinement of the contact and of the compressibility of the layer. The calculation was carried out using the surface pressure distribution provided by an exact solution to the coated contact problem. In addition, the elastic fields were also determined using an elliptic approximation of the contact pressure distribution. It is shown that the interface shear stress is strongly dependent on the details of the applied pressure profile close to the edge of the contact. In confined layers close to incompressibility, the elliptic approximation is found to result in a systematic overestimate of the interface shear stresses.