Currents and dislocations at the continuum scale

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A striking geometric property of elastic bodies with dislocations is their non-Riemannian nature in the sense that the deformation cannot be written as the gradient of a one-to-one immersion, and hence that no displacement field can properly be defined as model variable. In fact, the deformation curl equals to the density of dislocations which is a concentrated Radon measure in the dislocation lines. In this paper we consider a countable family of dislocations, discuss the mathematical properties of such constraint deformations and study a variational problem in finite-strain elasticity. It turns out that both the deformation and the dislocation lines may be modelled by means of the mathematical theory of currents. In particular, Cartesian maps allow one to consider deformations in L^p with 1\leq p<2, which are appropriate for dislocation-induced strain singularities. Moreover, integer-multiplicity currents are perfectly suited to describe either the static or the dynamics of families of dislocations which mutually interact, and possibly form complex structures such as clusters. Though the evolution of dislocations is not considered in this paper, it is the main motivation of our approach. As a matter of fact, this work describes a conservative ground state where dislocations are assumed to obey energy minimization principles, and over which any relevant effect resorting to thermodynamics outside equilibrium might be added in a subsequent step within the proposed mathematical formalism.