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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 , 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 the companion paper [A. Braides et al., J. Mech. Phys. Solids 96 (2016) 235–251] to validate models in Computational Mechanics in the small-deformation regime.
Keywords: Variational theory of fracture, discrete-to-continuum analysis, Γ-convergence, Lennard−Jones potentials
Braides, Andrea 1 ; Gelli, Maria Stella 2
Braides, Andrea; Gelli, Maria Stella. Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1903-1929. doi: 10.1051/m2an/2017011
@article{M2AN_2017__51_5_1903_0,
author = {Braides, Andrea and Gelli, Maria Stella},
title = {Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1903--1929},
year = {2017},
publisher = {EDP-Sciences},
volume = {51},
number = {5},
doi = {10.1051/m2an/2017011},
zbl = {1381.49011},
mrnumber = {3731554},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017011/}
}
TY - JOUR AU - Braides, Andrea AU - Gelli, Maria Stella TI - Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1903 EP - 1929 VL - 51 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017011/ DO - 10.1051/m2an/2017011 LA - en ID - M2AN_2017__51_5_1903_0 ER -
%0 Journal Article %A Braides, Andrea %A Gelli, Maria Stella %T Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1903-1929 %V 51 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017011/ %R 10.1051/m2an/2017011 %G en %F M2AN_2017__51_5_1903_0
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