Surface energies in nonconvex discrete systems

We analyze the variatlonal limit of one-dimensional next-to-nearest neighbours (NNN) discrete systems as the lattice size tends to zero when the energy densities are of multiwell or Lennard-Jones type. Properly scaling the energies, we study several phenomena as the formation of boundary layers and phase transitions. We also study the presence of local patterns and of anti-phase transitions in the asymptotic behaviour of the ground states of NNN model subject to Dirichlet boundary conditions. We use this information to prove a localization of fracture result in the case of Lennard-Jones type potentials.