Cauchy-Born rule and crystallization in 2D mass-spring lattices

The Cauchy-Born (CB) rule postulates that when a monatomic crystal is subjected to a small linear displacement of its boundary, then all atoms will follow this displacement. In the absence of previous mathematical results, we study the validity of this rule in the model case of a 2D cubic lattice interacting via harmonic springs between nearest and diagonal neighbours.
Establishing validity of the CB rule is a variant of the celebrated ``crystallization problem''. Another variant, obtained by replacing linear displacement boundary conditions by zero applied forces, will also be discussed in my lecture.
We find that for favourable values of the spring constants and spring equilibrium lengths the CB rule is a theorem, and that for unfavourable values the rule is incorrect. Moreover in the latter case the overestimation of the lattice energy per unit volume by the CB rule cannot be cured by quasiconvexification (and not even by convexification) of the CB energy.
Roughly speaking, validity of the CB rule can be viewed as a lattice analogon of quasiconvexity, and our proof of CB goes via verifying an appropriate analogon of polyconvexity, which we call lattice polyconvexity.
(In case of zero applied forces, the tool of lattice polyconvexity has to be replaced by a recent geometric rigidity theorem due to Friesecke, James and Müller.)
Reference: G.Friesecke, F.Theil, J.Nonl.Sci. 12 No. 5 2002 445-478