From Volterra dislocations to strain-gradient plasticity
Raz Kupferman, Cy Maor
TL;DR
The paper addresses deriving a strain-gradient plasticity model as a $\Gamma$-limit for continuum bodies containing finitely-many edge-dislocations in $2$D. It adopts a Lagrangian framework with a multiplicative decomposition of the strain gradient and represents the lattice by a smooth frame field, with dislocations appearing as circulation constraints on that frame field. The authors develop new geometric notions of convergence and rigidity estimates for dislocated bodies and place the limit in a unified framework with other dislocation models, extending beyond the admissible-strain paradigm. This establishes a rigorous bridge between discrete dislocations and continuum strain-gradient theories, enabling analysis of low-energy defects and providing a foundation for future extensions to mixed defects and slender geometries such as graphene.
Abstract
We rigorously derive a strain-gradient model of plasticity as a $Γ$-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.