The gauge theory of dislocations: conservation and balance laws

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.

28 pages

Cited by in corpus (4)

• The gauge theory of dislocations: static solutions of screw and edge dislocations
• Distributional and regularized radiation fields of non-uniformly moving straight dislocations, and elastodynamic Tamm problem
• On the non-uniform motion of dislocations: The retarded elastic fields, the retarded dislocation tensor potentials and the Liénard-Wiechert tensor potentials
• On the Higgs mechanism and stress functions in the translational gauge theory of dislocations