Anisotropic Elasticity of a Prismatic Dislocation Loop in a Hexagonal Crystal

Abstract

Elasticity solutions for the stress and displacement fields around a finite circular prismatic dislocation loop in the basal plane of a hexagonal crystal have been obtained by taking into account crystal anisotropy. Following the method proposed by Elliott for the problems of axial symmetry in hexagonal crystals, the Hankel transforms of two stress functions were determined from the boundary conditions appropriate for a circular prismatic loop. From these stress functions, the components of stress and displacement have been derived in terms of a series of definite integrals which could in turn be expressed in terms of elliptic integrals. The expressions of anisotropic elasticity approach those of isotropic theory derived by Kroupa when the conditions of elastic isotropy are imposed on the elastic constants. Numerical calculations have been carried out specifically for the case of graphite. The over-all effect of crystal anisotropy is to reduce the magnitude of the stresses and the displacements around the loop, except for the component of displacement normal to the loop plane which is increased by about a factor of 2.