Affine covariant-contravariant vector forms for the elastic field of parametric dislocations in isotropic crystals

The elastic field of closed dislocation loops in isotropic crystals is 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 are all expressed in terms of three covariant basis vectors: the unit tangent t, the unit radius e and the Burgers vector b, and their contravariant reciprocals. Differential affine transformations are shown to map directly the scalar unit interval (is an element of [0; 1]) on to vector displacement, and second-rank tensor strain and stress fields of a dislocation segment, described by the parameter omega. The resulting affine differential mappings are independent of coordinate systems and can be readily integrated by analytical or numerical methods to obtain the total field of closed dislocation loops. The method is applied to simplified geometry, where analytical expressions can be obtained and is illustrated in numerical simulations of mesoscopic plastic deformation.