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On the stability of Bravais lattices and their Cauchy–Born approximations*

Published online by Cambridge University Press:  26 July 2011

Thomas Hudson  and
Christoph Ortner
Thomas Hudson
Affiliation:
Mathematical Institute, Oxford, OX1 3LB, UK. [email protected]; [email protected]
Christoph Ortner
Affiliation:
Mathematical Institute, Oxford, OX1 3LB, UK. [email protected]; [email protected]
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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.

Keywords

Bravais latticeCauchy–Born modelstability
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 1 , January 2012 , pp. 81 - 110
Copyright
© EDP Sciences, SMAI, 2011

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