# Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems

created by braidesa on 16 Mar 2017
modified on 06 Apr 2017

[BibTeX]

published on line

Inserted: 16 mar 2017
Last Updated: 6 apr 2017

Journal: ESAIM: M2AN
Year: 2017
Doi: doi.org/10.1051/m2an/2017011

Abstract:

In this paper we provide rigorous statements and proofs for the asymptotic analysis of discrete energies defined on a two-dimensional triangular lattice allowing for fracture in presence of a microscopic impenetrability constraint. As the lattice parameter goes to $0$, we prove that any limit deformation with finite energy is piecewise rigid and we prove a general lower bound with a suitable Griffith-fracture energy density which reflects the anisotropies of the underlying triangular lattice. For such a continuum energy we also provide a class of (piecewise rigid) deformations satisfying opening-crack" conditions on which the lower bound is sharp. Relying on these results, some consequences have been already presented in a companion paper to validate models in Computational Mechanics in the small-deformation regime. (Braides and Gelli, JMPS 2016 http:/cvgmt.sns.itpaper2697)