Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n)

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. {\bf 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.