A boundary perturbation analysis for elastic inclusions and interfaces

Abstract

A first-order boundary perturbation method based on Muskhelishvilli's complex variable representations is formulated for the two-dimensional elasticity problem of a nearly-circular inclusion embedded in an infinite dissimilar material. Universal relations which are independent of loading conditions are established among the solutions for a homogeneous infinite plane, a perfectly-circular inclusion and a slightly-perturbed non-circular inclusion. It is shown that the solution to a circular inclusion can be constructed algebraically from that of an infinite homogeneous plane while the stress distribution along the interface circle leads to the perturbation solutions for a nearly-circular inclusion. Explicit results arc given for inclusions with smooth polygon shapes by considering cosine wavy perturbations along a reference circle. A similar analysts is carried out for a bimaterial interface whose shape deviates slightly from a straight line. Our perturbation results can be used to study elastically-induced morphological perturbations of surfaces, interfaces, voids, precipitates and inclusions in a stressed solid. As an example, we demonstrate that, under sufficiently large stresses, material surfaces become unstable against a range of diffusional perturbations bounded by two critical wavelengths. Also, as suggested from our perturbation analysis and verified by a finite element calculation, even slight surface undulation caused by an unstable morphological perturbation can result in substantial stress concentration along the surface, which may cause plastic deformation or brittle fracture before the bulk stress reaches a critical level.