Geodesic
Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n)
Journal of the European Mathematical Society, Tome 8 (2006) no. 3, pp. 515-539

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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 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 SO(n) and use interpolation to convert ordinary null Lagrangians into discrete ones.
DOI : 10.4171/jems/65
Classification : 60-XX, 00-XX
Keywords: Cauchy-Born rule, atomistic models, null Lagrangian 
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Sergio Conti; Georg Dolzmann; Bernd Kirchheim; Stefan Müller. Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n). Journal of the European Mathematical Society, Tome 8 (2006) no. 3, pp. 515-539. doi: 10.4171/jems/65

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