# Sufficient conditions for the validity of the Cauchy-Born rule close to $\mathrm{SO}\left(n\right)$

Sergio Conti; Georg Dolzmann; Bernd Kirchheim; Stefan Müller

Journal of the European Mathematical Society (2006)

- Volume: 008, Issue: 3, page 515-539
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topConti, Sergio, et al. "Sufficient conditions for the validity of the Cauchy-Born rule close to $\mathrm {SO}(n)$." Journal of the European Mathematical Society 008.3 (2006): 515-539. <http://eudml.org/doc/277330>.

@article{Conti2006,

abstract = {The Cauchy–Born rule provides a crucial link between continuum theories of elasticity and the atomistic nature of matter. In its strongest form it says that application of affine displacement
boundary conditions to a monatomic crystal will lead to an affine deformation of the whole crystal lattice. We give a general condition in arbitrary dimensions which ensures the validity of the Cauchy–Born rule for boundary deformations which are close to rigid motions.
This generalizes results of Friesecke and Theil [J. Nonlin. Sci. 12 (2002), 445–478] for a twodimensional model. As in their work, the key idea is to use a discrete version of polyconvexity
(ordinary convexity of the elastic energy as a function of the atomic positions is ruled out by frame indifference). The main point is the construction of a suitable discrete null Lagrangian which allows one to separate rigid motions. To do so we observe a simple identity for the determinant function on $\mathrm \{SO\}(n)$ and use interpolation to convert ordinary null Lagrangians into discrete ones.},

author = {Conti, Sergio, Dolzmann, Georg, Kirchheim, Bernd, Müller, Stefan},

journal = {Journal of the European Mathematical Society},

keywords = {Cauchy-Born rule; atomistic models; null Lagrangian; affine displacement; monatomic crystal; polyconvexity; discrete null Lagrangian},

language = {eng},

number = {3},

pages = {515-539},

publisher = {European Mathematical Society Publishing House},

title = {Sufficient conditions for the validity of the Cauchy-Born rule close to $\mathrm \{SO\}(n)$},

url = {http://eudml.org/doc/277330},

volume = {008},

year = {2006},

}

TY - JOUR

AU - Conti, Sergio

AU - Dolzmann, Georg

AU - Kirchheim, Bernd

AU - Müller, Stefan

TI - Sufficient conditions for the validity of the Cauchy-Born rule close to $\mathrm {SO}(n)$

JO - Journal of the European Mathematical Society

PY - 2006

PB - European Mathematical Society Publishing House

VL - 008

IS - 3

SP - 515

EP - 539

AB - The Cauchy–Born rule provides a crucial link between continuum theories of elasticity and the atomistic nature of matter. In its strongest form it says that application of affine displacement
boundary conditions to a monatomic crystal will lead to an affine deformation of the whole crystal lattice. We give a general condition in arbitrary dimensions which ensures the validity of the Cauchy–Born rule for boundary deformations which are close to rigid motions.
This generalizes results of Friesecke and Theil [J. Nonlin. Sci. 12 (2002), 445–478] for a twodimensional model. As in their work, the key idea is to use a discrete version of polyconvexity
(ordinary convexity of the elastic energy as a function of the atomic positions is ruled out by frame indifference). The main point is the construction of a suitable discrete null Lagrangian which allows one to separate rigid motions. To do so we observe a simple identity for the determinant function on $\mathrm {SO}(n)$ and use interpolation to convert ordinary null Lagrangians into discrete ones.

LA - eng

KW - Cauchy-Born rule; atomistic models; null Lagrangian; affine displacement; monatomic crystal; polyconvexity; discrete null Lagrangian

UR - http://eudml.org/doc/277330

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.