The Somigliana ring dislocation

A basic elasticity solution applicable to an important class of internal stress problems related, for example, to fiber-matrix composites and spalling of cylindrical coatings is obtained. The basic problem that has been solved is that of the singular stress-displacement field resulting from the introduction of a Somigliana ring dislocation in an isotropic linear elastic solid. The Burgers vector of this dislocation has two components, one being normal to the plane of the circular ring dislocation (Volterra type) and the other being in the radial direction of the ring dislocation everywhere (Somigliana type). The analytical solution, in terms of complete elliptic integrals of the first, second and third kinds, is obtained using the Love stress function and Fourier transform. The results are verified numerically and by examining various limiting cases, including the straight edge dislocation as the radius of the dislocation loop tends to infinity, the orthogonal pair of dipoles as the radius tends to zero, and the Lamé solution of a cylindrical bar and a cylindrical hole in an infinite medium as the axial location of the dislocation tends to minus infinity. The resulting analytical solution is considered as a step towards evaluating both the extended stress field around and interactions among various three-dimensional defects such as cylindrical cracks, fiber-tips and fiber-matrix debonding.