An atomistic-based finite deformation membrane for single layer crystalline films
M. Arroyo and T. Belytschko
Journal of the Mechanics and Physics of Solids, 50, 1941-1977 (2002).

ABSTRACT
A general methodology to develop hyper-elastic membrane models applicable to
crystalline films one-atom thick is presented. In this method, an extension of
the Born rule based on the exponential map is proposed. The exponential map
accounts for the fact that the lattice vectors of the crystal lie along the
chords of the curved membrane, and consequently a tangent map like the standard
Born rule is inadequate. In order to obtain practical methods, the exponential
map is locally approximated. The effectiveness of our approach is demonstrated
by numerical studies of carbon nanotubes. Deformed configurations as well as
equilibrium energies of atomistic simulations are compared with those provided
by the continuum membrane resulting from this method discretized by finite
elements.