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

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