The T(3) gauge theory of static dislocations in continuous solids was investigated. The most general linear constitutive relationships were used, in terms of the elastic distortion tensor and dislocation density tensor for the force and pseudo-moment stresses of an isotropic solid. The constitutive relationships contained six material parameters. In this theory, both the force and pseudo-moment stresses were asymmetrical. The theory involved four characteristic lengths, ℓ1, ℓ2, ℓ3 and ℓ4, which were explicitly given. Firstly, the three-dimensional Green’s tensor of the master-equation for the force stresses in the translational gauge theory of dislocations was derived. An investigation was then made of the generalized plane strain (anti-plane strain and plane strain). Using the stress-function method, modified stress functions were found for screw and edge dislocations. The solution for the screw dislocation was given in terms of one independent length, ℓ1 = ℓ4. For the edge dislocation, only two characteristic lengths, ℓ2 and ℓ3, arose; with one of them being the same (ℓ2 = ℓ1) as for the screw dislocation. Thus, the present theory involved only two independent lengths for the generalized plane strain. If the two lengths, ℓ2 and ℓ3, of an edge dislocation were equal, an edge dislocation was obtained which was the gauge theoretical version of a modified Volterra edge dislocation. In the case of symmetrical stresses, well-known results were recovered.

The Gauge Theory of Dislocations - Static Solutions of Screw and Edge Dislocations. M.Lazar, C.Anastassiadis: Philosophical Magazine, 2009, 89[3], 199-231