Calculus of Variations and Geometric Measure Theory

A. Braides - M. Solci

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

created by braidesa on 04 Jul 2014
modified on 28 Mar 2020


Published Paper

Inserted: 4 jul 2014
Last Updated: 28 mar 2020

Journal: Math. Mech. Solids
Volume: 21
Pages: 915-930
Year: 2016
Doi: 10.1177/1081286514544780


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 seems not always 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.

Keywords: Gamma-convergence, fracture mechanics, Lennard-Jones potentials, discrete-to-continuum, atomistic systems