*accepted MAA*

**Inserted:** 8 oct 2013

**Last Updated:** 4 sep 2015

**Year:** 2014

**Abstract:**

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, since deformation curl is nonzero but equals to the density of dislocations which is a concentrated Radon measure in the dislocation lines. Considering a countable family of dislocations, we discuss the mathematical properties of such constraint deformations and study a variational problem in finite-strain elasticity. In particular we model dislocation lines by the mean of currents with coefficients in $\mathbb{Z}^3$, whereas Cartesian maps allow one to consider deformations in $L^p$ with $1\leq p<2$, which are appropriate for dislocation-induced strain singularities.

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