Receding contact between an inclusion of generalized shape and a remotely-stressed plate

Abstract

This paper examines the problem of frictionless contact and separation of a rigid inclusion from an infinite elastic plate. The inclusion initially fits perfectly into a hole in the plate, and separates from it upon remote stressing. In a significant generalization of the classical work of Wilson and Goree on receding contacts of elliptical inclusions, the present formulation allows the inclusion to be of quite general shape. In particular, superellipses provide a convenient way to describe a family of such generalized inclusion shapes by varying two parameters. A highly accurate conformal mapping scheme based on rational functions and the Adaptive Antoulas–Anderson (AAA) algorithm is used to map the interior of a unit circle to the exterior of these generalized inclusions. Notably, a rational function representation is far more efficient than a polynomial representation. The governing Singular Integro-Differential Equation (SIDE) of the contact is then derived using the conformal map and analytic continuation, and solved using standard orthogonal polynomial techniques. The contact size and transmitted pressure traction are obtained for a wide range of shapes, including superelliptical, curvilinear hexagonal, and hypotrochoidal square inclusions. The role of the remote stress ratio on the contact response is studied, and a phase diagram demarcating various contact regimes is obtained. A comparison of the pressures for selected inclusion shapes using finite element analysis (FEA) shows good agreement with the present (semi-analytical) SIDE results. For shapes like hypotrochoidal squares, subtle changes in the inclusion geometry can cause significantly different peak contact pressures, pointing to the importance of accurate modeling of geometry in conforming–receding contact problems.