How does collective ``solid mechanics'' behaviour
emerge from (accurate quantum-mechanical or approximate) many-atom potential energy surfaces?
I will discuss basic open problems and recent mathematical results, including
(1) identification of the 3D fcc (face-centered cubic)
lattice as a local minimizer of the celebrated Lennard-Jones pair interaction model
(2) an analysis of the Cauchy-Born rule for elasticity tensor determination in a simple 2D atomistic model. We find that the CB rule
-- is actually a theorem when short-range interactions are dominant
-- but fails when long-range interactions are dominant or when sufficiently strong pressure is applied. In the failure regime, atomic-scale fluctuations of the atomic position field become favourable.
Joint work with Florian Theil (Warwick).
Copyright. All Rights Reserved.