Asymptotic analysis of Lennard-Jones systems beyond the nearest-neighbour setting: A one-dimensional prototypical case

Abstract

We consider a one-dimensional system of Lennard-Jones nearest- and next-to-nearest-neighbour interactions. It is known that if a monotone parameterization is assumed then the limit of such a system can be interpreted as a Griffith fracture energy with an increasing condition on the jumps. In view of possible applications to a higher-dimensional setting, where an analogous parameterization does not always seem reasonable, we remove the monotonicity assumption and describe the limit as a Griffith fracture energy where the increasing condition on the jumps is removed and is substituted by an energy that accounts for changes in orientation (‘creases’). In addition, fracture may be generated by ‘macroscopic’ or ‘microscopic’ cracks.