Remarks on Circular Edges and Open Questions

Abstract

This last chapter is devoted to our latest results on circular edges, and some open questions that are the aim and scope of future research. In daily practice, in reality, most edges are curved in three-dimensional domains and therefore these are of utmost engineering interest. Here we concentrate on circular edges (a “penny-shaped crack” being a special renowned case) in a 3-D domain, and derive explicitly a singular series expansion in the vicinity of such an edge for the simplest scalar elliptic operator, the Laplace operator. The displacements and stress fields associated with the elasticity system are provided in a recent paper [208].