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

  • Stefan Müller

    Universität Bonn, Germany
  • Sergio Conti

    Universität Bonn, Germany
  • Georg Dolzmann

    University of Maryland, College Park, United States
  • Bernd Kirchheim

    Universität Leipzig, Germany


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.

Cite this article

Stefan Müller, Sergio Conti, Georg Dolzmann, Bernd Kirchheim, Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n). J. Eur. Math. Soc. 8 (2006), no. 3, pp. 515–539

DOI 10.4171/JEMS/65