Towards uniformly Gamma-equivalent theories for nonconvex discrete systems

In this paper we consider a one-dimensional chain of atoms which interact with their nearest and next-to-nearest neighbours via a Lennard-Jones type potential. We prescribe the positions in the deformed configuration of the first two and the last two atoms of the chain. We are interested in a good approximation of the discrete energy of this system for a large number of atoms, i.e., in the continuum limit. We show that the canonical expansion by Γ-convergence does not provide an accurate approximation of the discrete energy if the boundary conditions for the deformation are close to the threshold between elastic and fracture regimes. This suggests that a uniformly Γ-equivalent approximation of the energy should be made, as introduced by Braides and Truskinovsky, to overcome the drawback of the lack of accuracy of the standard Γ-expansion. In this spirit we provide a uniformly Γ-equivalent approximation of the discrete energy at first order, which arises as the Γ-limit of a suitably scaled functional.