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# Methods and Applications of Analysis

## Volume 23 (2016)

### Number 1

### Currents and dislocations at the continuum scale

Pages: 1 – 34

DOI: http://dx.doi.org/10.4310/MAA.2016.v23.n1.a1

#### Authors

#### Abstract

A striking geometric property of elastic bodies with dislocations is that the deformation tensor cannot be written as the gradient of a one-to-one immersion, its curl being nonzero and equal to the density of the dislocations, a measure concentrated in the dislocation lines. In this work, we discuss the mathematical properties of such constrained deformations and study a variational problem in finite-strain elasticity, where *Cartesian maps* allow us to consider deformations in $L^p$ with $1 \leq p \lt 2$, as required for dislocation-induced strain singularities. Firstly, we address the problem of mathematical modeling of dislocations. It is a key purpose of the paper to build a framework where dislocations are described in terms of *integral* 1-*currents* and to extract from this theoretical setting a series of notions having a mechanical meaning in the theory of dislocations. In particular, the paper aims at classifying integral 1-currents, with modeling purposes. In the second part of the paper, two variational problems are solved for two classes of dislocations, at the mesoscopic and at the continuum scale. By *continuum* it is here meant that a countable family of dislocations is considered, allowing for branching and cluster formation, with possible complex geometric patterns. Therefore, modeling assumptions of the defect part of the energy must also be provided, and discussed.

#### Keywords

Cartesian maps, integer-multiplicity currents, dislocations, finite elasticity, modeling, variational problem

#### 2010 Mathematics Subject Classification

32C30, 46E40, 49J45, 49Q15, 53C65, 58A25, 74B20, 74G65, 74N05