Conservation and balance laws were derived, for the translational gauge theory of dislocations, by applying Noether's theorem. An improved translational gauge theory for dislocations was presented which included the dislocation density tensor and the dislocation current tensor. The invariance of the variational principle under a continuous group of transformations was studied. Using the Lie infinitesimal invariance criterion, conserved translational and rotational currents were obtained for the total Lagrangian; made up of an elastic and dislocation part. The broken scaling current was calculated. Looking only at one part of the whole system, the conservation laws were converted into balance laws. Because of the lack of translational, rotational and dilatational invariance for each part, a configurational force, moment and power appeared. The corresponding J, L and M integrals were obtained. Only isotropic and homogeneous materials were considered, and the theory was strictly linear. Constitutive laws were chosen for the most general linear form of material isotropy. Conservation and balance laws were also given which corresponded to gauge symmetry and the addition of solutions. From the latter, a reciprocity theorem was derived for the gauge theory of dislocations. Conservation laws for stress-free states of dislocations were also derived.

The Gauge Theory of Dislocations - Conservation and Balance Laws. M.Lazar, C.Anastassiadis: Philosophical Magazine, 2008, 88[11], 1673-99