The Involution Correspondence in Plane Elastostatics for Regions Bounded by a Circle

Abstract

It is shown that the elastic field induced by prescribed displacements or surface tractions acting on a circular disk (inner region) can be expressed in terms of the elastic field induced by the same quantities acting on the circular boundary (hole) of an infinite plane (outer region), and vice versa. It is shown further that this correspondence is an involution. This novel representation permits one to express the elastic field in a disk with either vanishing displacements or tractions along the boundary in terms of the elastic field of an infinite domain, provided all singularities are in the inner region. Similarly, the elastic field in the outer region can be expressed in terms of the elastic field of the infinite domain, provided all singularities reside in the outer region. The expressions so-derived possess simple algebraic character and are universal in the sense of being independent of the applied loads (singularities) in the two problems.