Analysis of a prismatic elastic contact of finite length

This paper is concerned with a contact problem which is geometrically two dimensional, but of finite extent in a third dimension. Two different contact models (common edge contact and incomplete contact) are analyzed, using a finite element model to investigate the 3D end effects. The object is to take the 2D plane strain solution in each model as a reference, and to show how it must be modified to allow for the 3D finite extent contact problem with free end faces. It is shown that, for a sufficiently long prismatic contact, the in-plane stress distribution at the mid-plane matches the solution to the 2D plane strain problem. Additionally, the end effect is evaluated using the finite element results to show how it decays with distance from the free end. The decay is exponential and governed by a dominant length-scale of the problem. For a common edge contact, this length-scale is the contact width. However, for a Hertzian contact, the contact width varies in the third dimension and the governing length scale is the radius of curvature, typically much larger than the contact width.