On the stability of Bravais lattices and their Cauchy-Born approximations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 1, p. 81-110

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.

DOI : https://doi.org/10.1051/m2an/2011014
Classification:  35Q74,  49K40,  65N25,  70J25,  70C20
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {1},
     year = {2012},
     pages = {81-110},
     doi = {10.1051/m2an/2011014},
     zbl = {1291.35388},
     mrnumber = {2846368},
     language = {en},
     url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 1, pp. 81-110. doi : 10.1051/m2an/2011014. http://www.numdam.org/item/M2AN_2012__46_1_81_0/

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