Circular prismatic dislocation loops in elastic bodies with spherical free surfaces

We present a solution to the boundary-value problem in the classical theory of elasticity for a circular prismatic dislocation loop (CPDL) in an elastic body with one or two spherical free surfaces such as a spherical particle, an infinite body with a spherical cavity, and a spherical shell of finite thickness. The axisymmetric position of the CPDL with respect to the spherical surfaces is assumed. Elastic stresses, dilatation and strain energy of the loop are presented in a concise and transparent form of the series with the Legendre polynomials, numerically calculated and illustrated by corresponding maps, and discussed in detail. It is shown that the stress and dilatation fields are strongly screened and distorted by the internal and external free surfaces of the systems under consideration as compared with the case of an infinite medium. The dilatation can change its sign in the subsurface layers of the particle, near the cavity and in the shell. Features of CPDL strain energy are discussed in detail for the cases when the loop experiences the influence of spherical free surfaces.