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

Conti, 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 -