Some analytical results for plane contact problems

Abstract

Traditionally, two-dimensional elasticity problems have been solved using an Airy stress function or a Muskhelishvili complex potential approach. The principal advantage of the Muskhelishvili approach is that the interior stress field may be found directly from the boundary tractions, unlike the Airy approach where a degree of guesswork is necessary to obtain a suitable potential function. This paper demonstrates the equivalence of these two approaches to contact problems. The paper then extends the Muskhelishvili method to out-of-plane contact problems. To demonstrate the usefulness of this method, it is applied to two important problems motivated by fretting fatigue studies. The stick/slip behaviour of a Hertzian indenter, when the substrate is subjected to a bulk tensile stress field, is determined and the complete stress fields for both a flat-ended punch and a parabolic punch, when the punch is sliding in an out-of-plane direction, are found.