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We study the limit of large volume equilibrium Gibbs measures for a rathergeneral Hamiltonians. In particular we study Hamiltonians which arise innaturally in Nonlinear Elasticity and Hamiltonians (containing surface terms)which arises naturally in Fracture Mechanics. In both of these settings we showthat an integral representation holds for the limit. Moreover, we also show ahomogenization result for the Nonlinear Elasticity setting. This extends arecent result of R. Koteck\'y and S. Luckhaus \cite{LK-comm-math-phys}.

Sobolev and SBV Representation Theorems for large volume limit Gibbs measures