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. 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 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