A non-linear continuum theory of material bodies with continuously distributed dislocations was presented, based upon a gauge theoretical approach. Firstly, the canonical conservation laws that corresponded to the group of translations and rotations in material space were derived using Noether’s theorem. These equations gave the canonical Eshelby stress tensor as well as the total canonical angular momentum tensor. The canonical Eshelby stress tensor was neither symmetric nor gauge-invariant. Based upon the Belinfante-Rosenfeld procedure, the gauge-invariant Eshelby stress tensor was obtained, which could be symmetrical relative to the reference configuration only for isotropic materials. The gauge-invariant angular momentum tensor was also obtained. Decomposition of the gauge-invariant Eshelby stress tensor, into an elastic and a dislocation part, led to derivation of the famous Peach-Koehler force.

On the Nonlinear Continuum Theory of Dislocations: a Gauge Field Theoretical Approach. E.Agiasofitou, M.Lazar: Journal of Elasticity, 2010, 99[2], 163-78