A multiscale framework for computational nanomechanics: Application to the modeling of carbon nanotubes

A multiscale computational framework is presented that provides a coupled self-consistent system of equations involving molecular mechanics at small scales and quasi-continuum mechanics at large scales. The proposed method permits simultaneous resolution of quasi-continuum and atomistic length scales and the associated displacement fields in a unified manner. Interatomic interactions are incorporated into the method through a set of analytical equations that contain nanoscale-based material moduli. These material moduli are defined via internal variables that are functions of the local atomic configuration parameters. Point defects like vacancy defects in nanomaterials perturb the atomic structure locally and generate localized force fields. Formation energy of vacancy is evaluated via interatomic potentials and minimization of this energy leads to nanoscale force fields around defects. These nanoscale force fields are then employed in the multiscale method to solve for the localized displacement fields in the vicinity of vacancies and defects. The finite element method that is developed based on the hierarchical multiscale framework furnishes a two-level statement of the problem. It concurrently feeds information at the molecular scale, formulated in terms of the nanoscale material moduli, into the quasi-continuum equations. Representative numerical examples are shown to validate the model and demonstrate its range of applicability.