A Helical Cauchy-Born Rule for Special Cosserat Rod Modeling of Nano and Continuum Rods

摘要

We present a novel scheme to derive nonlinearly elastic constitutive laws for special Cosserat rod modeling of nano and continuum rods. We first construct a 6-parameter (corresponding to the six strains in the theory of special Cosserat rods) family of helical rod configurations subjected to uniform strain along their arc-length. The uniformity in strain then enables us to deduce the constitutive laws by just solving the warping of the helical rod's cross-section (smallest repeating cell for nanorods) but under certain constraints. The constraints are shown to be critical in the absence of which, the 6-parameter family reduces to a well known 2-parameter family of uniform helical equilibria. An explicit formula for the 6-parameter helical map is derived which maps atoms in the repeating cell of a nanorod to their images for the purpose of repeating cell energy minimization. A scheme for the passage from nano to continuum scale is also presented to derive the constitutive laws of a continuum rod via atomistic calculations of nanorods. The bending, twisting, stretching and shearing stiffnesses of diamond nanorods and carbon nanotubes are computed to demonstrate our theory. We show that our scheme is more general and accurate than existing schemes allowing us to deduce shearing stiffness and several coupling stiffnesses of a nanorod for the first time.