From Volterra dislocations to strain-gradient plasticity

TL;DR

This paper rigorously derives a strain-gradient plasticity model from a continuum dislocation framework, introducing a novel geometric approach that unifies dislocation models beyond previous admissible strain methods.

Contribution

It presents a new continuum-based derivation of strain-gradient plasticity from dislocation theory using a geometric, Lagrangian approach with circulation conditions.

Findings

Established a Γ-limit for bodies with finitely-many dislocations

Developed new geometric rigidity estimates for dislocated bodies

Unified strain-gradient models with dislocation theory in a continuum framework

Abstract

We rigorously derive a strain-gradient model of plasticity as a $\Gamma$-limit of continuum bodies containing finitely-many edge-dislocations (in two dimensions). The key difference from previous such derivations is the elemental notion of a dislocation: we work in a continuum framework in which the lattice structure is represented by a smooth frame field, and the presence of a dislocation manifests in a circulation condition on that frame field; the resulting model is a Lagrangian approach with a multiplicative strain decomposition. The multiplicative nature of the geometric incompatibility generates many technical challenges, which require a systematic study of the geometry of bodies containing multiple dislocations, the definition of new notions of convergence, and the derivation of new geometric rigidity estimates pertinent to dislocated bodies. Our approach places the strain-gradient limit in a unified framework with other models of dislocations, which cannot be addressed within the "admissible strain" approach used in previous works.