Homogenization of line tension energies

Abstract

We prove an homogenization result, in terms of $\Gamma$-convergence, for energies concentrated on rectifiable lines in $R^3$ without boundary. The main application of our result is in the context of dislocation lines in dimension 3. The result presented here shows that the line tension energy of unions of single line defects converge to the energy associated to macroscopic densities of dislocations carrying plastic deformation. As a byproduct of our construction for the upper bound for the $\Gamma$-Limit, we obtain an alternative proof of the density of rectifiable 1-currents without boundary in the space of divergence free fields.