We investigate the stability of Bravais lattices and their Cauchy-Born approximations under periodic perturbations. We formulate a general interaction law and derive its Cauchy-Born continuum limit. We then analyze the atomistic and Cauchy-Born stability regions, that is, the sets of all matrices that describe a stable Bravais lattice in the atomistic and Cauchy-Born models respectively. Motivated by recent results in one dimension on the stability of atomistic/continuum coupling methods, we analyze the relationship between atomistic and Cauchy-Born stability regions, and the convergence of atomistic stability regions as the cell size tends to infinity.
Mots-clés : Bravais lattice, Cauchy-Born model, stability
@article{M2AN_2012__46_1_81_0,
author = {Hudson, Thomas and Ortner, Christoph},
title = {On the stability of {Bravais} lattices and their {Cauchy-Born} approximations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {81--110},
publisher = {EDP-Sciences},
volume = {46},
number = {1},
year = {2012},
doi = {10.1051/m2an/2011014},
mrnumber = {2846368},
zbl = {1291.35388},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2011014/}
}
TY - JOUR AU - Hudson, Thomas AU - Ortner, Christoph TI - On the stability of Bravais lattices and their Cauchy-Born approximations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 81 EP - 110 VL - 46 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011014/ DO - 10.1051/m2an/2011014 LA - en ID - M2AN_2012__46_1_81_0 ER -
%0 Journal Article %A Hudson, Thomas %A Ortner, Christoph %T On the stability of Bravais lattices and their Cauchy-Born approximations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 81-110 %V 46 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011014/ %R 10.1051/m2an/2011014 %G en %F M2AN_2012__46_1_81_0
Hudson, Thomas; Ortner, Christoph. On the stability of Bravais lattices and their Cauchy-Born approximations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 1, pp. 81-110. doi : 10.1051/m2an/2011014. http://www.numdam.org/articles/10.1051/m2an/2011014/
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