Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity

TL;DR

This paper develops a relaxation theory for energies concentrated on line defects called dislocations, modeled as 1-currents with lattice-valued multiplicities, including explicit examples and technical approximation results.

Contribution

It introduces a relaxation framework for dislocation energies modeled by lattice-valued 1-currents, with explicit calculations and a structure theorem.

Findings

Relaxation of dislocation energies with lattice multiplicities is explicitly characterized.

A structure theorem for 1-currents with lattice multiplicity is established.

An example shows nontrivial relaxation behavior for certain Burgers vectors.

Abstract

In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may be identified with divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In this paper we develop the theory of relaxation for these energies and provide one physically motivated example in which the relaxation for some Burgers vectors is nontrivial and can be determined explicitly. From a technical viewpoint the key ingredients are an approximation and a structure theorem for 1-currents with multiplicity in a lattice.