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

We study the asymptotic behavior, as the lattice spacing ɛ tends to zero, of the discrete elastic energy induced by topological singularities in an inhomogeneous ɛ periodic medium within a two-dimensional model for screw dislocations in the square lattice. We focus on the |logɛ| regime which, as ɛ→0allows the emergence of a finite number of limiting topological singularities. We prove that the Γ-limit of the |logɛ| scaled functionals as ɛ→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.