Calculus of Variations and Geometric Measure Theory

A. Braides - M. S. Gelli

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

created by braidesa on 16 Mar 2017
modified on 18 Nov 2017


Published Paper

Inserted: 16 mar 2017
Last Updated: 18 nov 2017

Journal: ESAIM: M2AN
Volume: 51
Pages: 1903-1929
Year: 2017
Doi: 10.1051/m2an/2017011


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)