Sufficient conditions for the validity of the Cauchy-Born rule close to \(\mathrm{SO}(n)\)
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.
scientific article; zbMATH DE number 5141333