Currents and dislocations at the continuum scale

A striking geometric property of elastic bodies with dislocations is their non-Riemannian nature in the sense that the deformation tensor cannot be written as the gradient of a one-to-one immersion, since its curl must not be zero, but equals to the density of dislocations, a concentrated Radon measure 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 with 1 ≤ p < 2, as required for dislocation-induced strain singularities. In its first part, this paper addresses the problem of mathematical modeling of dislocations. It is a key purpose of the paper to first build a framework where dislocations are described in terms of integer rectifiable 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 rectifiable 1-currents, with modeling purposes. In the second part of the paper, two variational problems are solved by the direct method of the calculus of variations, 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 assumption of the defect part of the energy must also be provided, and discussed.