On the Nonlinear Continuum Theory of Dislocations: A Gauge Field Theoretical Approach

Abstract

A nonlinear continuum theory of material bodies with continuously distributed dislocations is presented, based on a gauge theoretical approach. Firstly, we derive the canonical conservation laws that correspond to the group of translations and rotations in the material space using Noether's theorem. These equations give us the canonical Eshelby stress tensor as well as the total canonical angular momentum tensor. The canonical Eshelby stress tensor is neither symmetric nor gauge-invariant. Based on the Belinfante-Rosenfeld procedure, we obtain the gauge-invariant Eshelby stress tensor which can be symmetric relative to the reference configuration only for isotropic materials. The gauge-invariant angular momentum tensor is obtained as well. The decomposition of the gauge-invariant Eshelby stress tensor in an elastic and in a dislocation part gives rise to the derivation of the famous Peach-Koehler force.

The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations

“…Using an affine gauge approach [28] it turns out that the gauge potential P has the geometrical meaning of the translational part of the generalized affine connection and α is the translational part of the affine curvature (see also [43,57]). A systematic investigation of conservation and balance laws in dislocation gauge theory using Lie-point symmetries has been carried out by Lazar and Anastassiadis [54,56] and Agiasofitou and Lazar [2]. An important result was a straightforward definition and physical interpretation of the Peach-Koehler force analogous to the Lorentz force in electrodynamics since there is a lot of confusion about the physical nature of the Peach-Koehler force in the literature.…”

“…The relaxed formulation of micromorphic elasticity has some similarities to the gauge theory of dislocations given by Lazar [43,44,45], Lazar and Anastassiadis [54,55] and Agiasofitou and Lazar [2]. In fact, in [55,49] a simplified static version of the isotropic Eringen-Claus model for dislocation dynamics [9] has been investigated with H = 0 and µ c ≥ 0, with a focus on the gauge theory of dislocations.…”

“…33) describing "skew-symmetric" edge dislocations with the property α (2) = −(α(2) ) T , and the tentor possesses the form α 21 α 13 + α 31 α 21 + α 12 2 α 22 α 23 + α 32 α 31 + α 13 α 32 + α23 2 " edge dislocations with the property α (3) = (α (3) ) T and also single screw dislocations. For the three-dimensional elastoplastic dislocation problem, screw dislocations correspond to connected with displacement vector u = (u 1 , u 2 , u 3 ).…”

Micromechanics and dislocation theory in anisotropic elasticity

“…The Peach-Koehler force is defined as (see, e.g., Lazar and Kirchner [2013]; Agiasofitou and Lazar [2010])…”

Untitled

“…Among the properties of admissibility, we require that F and ᏸ be related by condition (1-1) and that F be the gradient of a Cartesian map away from L. Therefore, both F and ᏸ are represented by particular types of integral currents. In dislocation gauge theory, an energy like (1-4) was used in [Lazar and Anastassiadis 2008; Agiasofitou and Lazar 2010], where the decomposition in an elastic and a dislocation part is given. From a mathematical viewpoint, that is, with variational techniques in appropriate functional spaces, problem (1-3) has been discussed and was first solved in [Müller and Palombaro 2008] with a single fixed dislocation loop in the crystal bulk (thus implying a minimization in F only) and later extended in for an unfixed countable family of dislocation currents satisfying certain boundary conditions.…”