Inserted: 30 jul 2019
Last Updated: 30 jul 2019
We study the atomistic-to-continuum limit of a class of energy functionals for crystalline materials via $\Gamma$-convergence. We consider energy densities that may depend on interactions between all points of the lattice and we give conditions that ensure compactness and integral-representation of the continuum limit on the space of special functions of bounded variation. This abstract result is complemented by a homogenization theorem, where we provide sufficient conditions on the energy densities under which bulk- and surface contributions decouple in the limit. The results are applied to long-range and multi-body interactions in the setting of weak-membrane energies.
Keywords: Homogenization, $\Gamma$-convergence, discrete-to-continuum, free-discontinuity functionals, multi-body interactions