A general integral representation result for continuum limits of discrete energies with superlinear growth

We study the asymptotic behavior, as the mesh size tends to zero, of a general class of discrete energies defined as the superposition of 'inter-atomic' potentials accounting for pairwise interactions and depending on the positions of the 'atoms' in the reference lattice and in the deformed configuration. We highlight the dependence of such potentials on different quotients and show that, under superlinear growth conditions and decay assumptions with respect to long range interactions, all the possible variational limits are defined on Sobolev space and of integral type. We show that, in general, the energy density of the limit functional may be a quasi-convex nonconvex function even if very simple interactions are considered. We also treat the case of homogenization, giving a general asymptotic formula that can be simplified in many situations (e.g., in the case of nearest neighbor interactions or under convexity hypotheses).