Sufficient conditions for the validity of the Cauchy-Born rule close to \(\mathrm{SO}(n)\)
type: Item
instance of: scholarly article
title: Sufficient conditions for the validity of the Cauchy-Born rule close to \(\mathrm{SO}(n)\)
zbMATH Open document ID: 1146.74006
DOI: 10.4171/JEMS/65
author: Sergio Conti
author: Georg Dolzmann
author: Bernd Kirchheim
author: Stefan Müller
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2007-04-10T00:00:00Z
review text: Summary: 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 \textit{G. Friesecke} and \textit{F. Theil} [J. Nonlinear Sci. 12, No. 5, 445--478 (2002; Zbl 1084.74501)] for a two-dimensional 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 \(\text{SO}(n)\) and use interpolation to convert ordinary null Lagrangians into discrete ones.
Mathematics Subject Classification ID: 74A25
Mathematics Subject Classification ID: 74E15
Mathematics Subject Classification ID: 74A60
Mathematics Subject Classification ID: 74B99
zbMATH DE Number: 5141333
zbMATH Keywords: affine displacement
zbMATH Keywords: monatomic crystal
zbMATH Keywords: polyconvexity
zbMATH Keywords: discrete null Lagrangian
MaRDI profile type: Publication
cites work: Convexity conditions and existence theorems in nonlinear elasticity
cites work: Null Lagrangians, weak continuity, and variational problems of arbitrary order
cites work: Regularity of quasiconvex envelopes
cites work: Fine phase mixtures as minimizers of energy
cites work: Q3433838
cites work: Equilibrium configurations of crystals
cites work: A variational model for reconstructive phase transformations in crystals, and their relation to dislocations and plasticity
cites work: Direct methods in the calculus of variations
cites work: Q5335594
cites work: Variational methods for crystalline microstructure. Analysis and computation
cites work: The null set of the Euler-Lagrange operator
cites work: Nilpotent energies in liquid crystal theory
cites work: Q3683687
cites work: Equilibrium theory for X-ray observations of crystals
cites work: Variational methods for elastic crystals
cites work: The lower quasiconvex envelope of the stored energy function for an elastic crystal
full work available at URL: https://doi.org/10.4171/jems/65
OpenAlex ID: W2120721991
description: scientific article; zbMATH DE number 5141333