ON THE STABILITY OF BRAVAIS LATTICES AND THEIR CAUCHY-BORN APPROXIMATIONS

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.