Geodesic
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

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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 : 10.1051/m2an/2011014
Classification : 35Q74, 49K40, 65N25, 70J25, 70C20
Keywords: Bravais lattice, Cauchy-Born model, stability 
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     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},
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     doi = {10.1051/m2an/2011014},
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     zbl = {1291.35388},
     language = {en},
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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

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