On the stability of Bravais lattices and their Cauchy–Born approximations*
Mathematical Institute, Oxford, OX1 3LB, UK. firstname.lastname@example.org; email@example.com
Revised: 24 March 2011
We investigate the stability of Bravais lattices and their Cauchy–Born approximations under periodic perturbations. We formulate a general interaction law and derive its Cauchy–Born continuum limit. We then analyze the atomistic and Cauchy–Born stability regions, that is, the sets of all matrices that describe a stable Bravais lattice in the atomistic and Cauchy–Born models respectively. Motivated by recent results in one dimension on the stability of atomistic/continuum coupling methods, we analyze the relationship between atomistic and Cauchy–Born stability regions, and the convergence of atomistic stability regions as the cell size tends to infinity.
Mathematics Subject Classification: 35Q74 / 49K40 / 65N25 / 70J25 / 70C20
Key words: Bravais lattice / Cauchy–Born model / stability
© EDP Sciences, SMAI, 2011