Towards uniformly $\Gamma$-equivalent theories for nonconvex discrete systems

Abstract

In this paper we consider a one-dimensional chain of atoms whichinteract with their nearest and next-to-nearest neighboursvia a Lennard-Jones type potential. We prescribe the positionsin the deformed configuration of the first two and the last two atomsof the chain.  &nbsp We are interested in a good approximation of the discrete energy of thissystem for a large number of atoms, i.e., in the continuum limit.  &nbsp We show that the canonical expansion by $\Gamma$-convergence does not provide an accurate approximationof the discrete energy if the boundary conditions for the deformation are close to the threshold betweenelastic and fracture regimes.This suggests that a uniformly $\Gamma$-equivalent approximation of the energy should be made, asintroduced by Braides and Truskinovsky, to overcome the drawback of the lack of accuracy of the standard$\Gamma$-expansion.  &nbsp In this spirit we provide a uniformly $\Gamma$-equivalent approximation of the discrete energy at first order, which arisesas the $\Gamma$-limit of a suitably scaled functional.