Symmetries of 2-Lattices and Second Order Accuracy of the Cauchy--Born Model

We show that the Cauchy--Born model of a single-species 2-lattice is second order if the atomistic and continuum kinematics are connected in a novel way. Our proof uses a generalization to 2-lattices of the point symmetry of Bravais lattices. Moreover, by identifying similar symmetries in multispeci...

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Bibliographic Details
Published in:Multiscale modeling & simulation 2013-01, Vol.11 (2), p.615-634
Main Authors: Van Koten, Brian, Ortner, Christoph
Format: Article
Language:English
Subjects:
Accuracy
Applied mathematics
Continuums
Density
Energy
Estimates
Graphene
HCP metals
Internal energy
Kinematics
Numerical analysis
Point defects
Strain
Symmetry
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Summary:We show that the Cauchy--Born model of a single-species 2-lattice is second order if the atomistic and continuum kinematics are connected in a novel way. Our proof uses a generalization to 2-lattices of the point symmetry of Bravais lattices. Moreover, by identifying similar symmetries in multispecies pair interaction models, we construct a new stored energy density, using shift gradients but not strain gradients, that is also second order accurate. These results can be used to develop highly accurate continuum models and atomistic/continuum coupling methods for materials such as graphene, hcp metals, and shape memory alloys. [PUBLICATION ABSTRACT]
ISSN:1540-3459
1540-3467
DOI:10.1137/120870220