Screw dislocations in periodic media: Variational coarse graining of the discrete elastic energy

We study the asymptotic behavior, as the lattice spacing ? tends to zero, of the discrete elastic energy induced by topological singularities in an inhomogeneous ? periodic medium within a two-dimensional model for screw dislocations in the square lattice. We focus on the |log?| regime which, as ?->0 allows the emergence of a finite number of limiting topological singularities. We prove that the ?-limit of the |log?| scaled functionals as ?->0 is equal to the total variation of the so-called "limiting vorticity measure" times a factor depending on the homogenized energy density of the unscaled functionals.