Homogenization of line tension energies

Abstract

We prove an homogenization result, in terms of $\Gamma$-convergence, for energies concentrated on rectifiable lines in ${\mathbb{R}}^3$ 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.