From discrete to continuum models: The Cauchy-Born rule

The Cauchy-Born rule postulates that when a monatomic crystal is subjected to a small linear displacement of its bounday, then all atoms will follow this displacements. In absence of previous mathematical result, we study the validity of this rule in the model case of a 2D cubic lattice interacting via harmonic springs.

Our main result is that for favourable values of the spring constants and equilibrium spring lengths, the CB rule is actually a theorem.

Simple counterexamples show that for unfavourable spring parameters or large displacements the CB rule fails. Moreover the resulting overestimation of the lattice energy per unit volume by the CB rule can not be cured by convexificaton (let alone quasiconvexification) of the CB energy.

The main tool in the proof is a novel notion of lattice polyconvexity which allows to overcome the difficulty that the elastic energy as a function of atomic positions can never be convex, due to frame-indifference.