| Issue |
ESAIM: M2AN
Volume 46, Number 1, January-February 2012
|
|
|---|---|---|
| Page(s) | 81 - 110 | |
| DOI | https://doi.org/10.1051/m2an/2011014 | |
| Published online | 26 July 2011 | |
On the stability of Bravais lattices and their Cauchy–Born approximations*
Mathematical Institute, Oxford, OX1 3LB, UK. This email address is being protected from spambots. You need JavaScript enabled to view it.
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Received:
22
November
2010
Revised:
24
March
2011
Abstract
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.
Mathematics Subject Classification: 35Q74 / 49K40 / 65N25 / 70J25 / 70C20
Key words: Bravais lattice / Cauchy–Born model / stability
This work was supported by an undergraduate vacation bursary of the Oxford Centre for Nonlinear PDE (funded by EPSRC), and by the EPSRC grant “Analysis of atomistic/continuum coupling methods”.
© EDP Sciences, SMAI, 2011
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