The gauge theory of dislocations: conservation and balance laws

TL;DR

This paper develops a gauge theory framework for dislocations, deriving conservation and balance laws using Noether's theorem, and explores implications for isotropic, linear elastic materials with dislocations.

Contribution

It introduces an improved translational gauge theory of dislocations including density and current tensors, and derives associated conservation and balance laws.

Findings

Conserved translational and rotational currents for the total Lagrangian.

Balance laws with configurational forces, moments, and powers.

Reciprocity theorem for the gauge theory of dislocations.

Abstract

We derive conservation and balance laws for the translational gauge theory of dislocations by applying Noether's theorem. We present an improved translational gauge theory of dislocations including the dislocation density tensor and the dislocation current tensor. The invariance of the variational principle under the continuous group of transformations is studied. Through Lie's-infinitesimal invariance criterion we obtain conserved translational and rotational currents for the total Lagrangian made up of an elastic and dislocation part. We calculate the broken scaling current. Looking only on one part of the whole system, the conservation laws are changed into balance laws. Because of the lack of translational, rotational and dilatation invariance for each part, a configurational force, moment and power appears. The corresponding J, L and M integrals are obtained. Only isotropic and homogeneous materials are considered and we restrict ourselves to a linear theory. We choose constitutive laws for the most general linear form of material isotropy. Also we give the conservation and balance laws corresponding to the gauge symmetry and the addition of solutions. From the addition of solutions we derive a reciprocity theorem for the gauge theory of dislocations. Also, we derive the conservation laws for stress-free states of dislocations.