The elastic fields of closed dislocation loops in isotropic crystals were developed for differential geometric parametric segments in covariant-contravariant vector forms. The displacement vector field, strain and stress tensor fields, as well as the self-energy and mutual interaction energies, were all expressed in terms of 3 covariant basis vectors: the unit tangent, the unit radius and the Burgers vector, and their contravariant reciprocals. Differential affine transformations were shown to map the scalar unit interval [0,1] directly onto the vector displacement, and second-rank tensor strain and stress fields of a dislocation segment. The resultant affine differential mappings were independent of any coordinate system and could be easily integrated, using analytical or numerical methods, to obtain the total field of closed dislocation loops. The method was applied to a simplified geometry for which analytical expressions could be obtained, and was illustrated by numerical simulations of mesoscopic plastic deformation.

Affine Covariant-Contravariant Vector Forms for the Elastic Field of Parametric Dislocations in Isotropic Crystals. N.M.Ghoniem, J.Huang, Z.Wang: Philosophical Magazine Letters, 2002, 82[2], 55-63