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*Preprint*

**Inserted:** 6 may 2015

**Last Updated:** 6 may 2015

**Year:** 2015

**Abstract:**

This paper deals with a variational problem for dislocations in which the curl of the deformation tensor is constrained by a concentrated measure in a set of lines, called the dislocation density, while the energy density involves the deformation tensor and its gradient, specifically, the curl and the divergence in two distinct terms. To solve this constrained variational problem in finite elasticity, the notion of integral $1$-current is used in the spirit of previous work by the same authors. No assumptions on the lines are made except the classical requirement to be closed loops or end at the crystal boundary. Since the displacement field is by essence multiple valued, it is chosen to work with torus-valued maps. Moreover, graphs of harmonics maps are at the heart of such a problem, and therefore our theory is grounded in an analysis of their properties with a view to dislocation modelling. Our main result shows that dislocation density and displacement graph boundary are bound notions. Generalizations of distributional determinants and cofactors appropriate for our purposes are also discussed. Indeed, it is shown that they are Radon measures whose singular parts are expressed in terms of the displacement graph boundary.

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