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

Abstract

SUMMARYA 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.

A heterogeneous multiscale modeling framework for hierarchical systems of partial differential equations

“…Heterogeneous methods are also used for material and spatial randomness [25], and to generate missing data in the macroscopic models via stochastic up-scaling of microscopic models [26]. The idea of merging heterogeneous models has been the basis for the development of bridging scale methods in micro and nanomechanics [27,28], and in coupling atomistic and continuum models [29][30][31]. Fracture mechanics is another area where models with different physics are coupled together to capture the localized effects in the process zones around the cracks [32,33].…”

“…Fracture mechanics is another area where models with different physics are coupled together to capture the localized effects in the process zones around the cracks [32,33]. First principles-based approaches employing inter-atomic potentials and the Cauchy-Born rule have also been used for quasi-continuum modeling of the mechano-electronic properties of materials [27,34], and to resolve physics around defects and in the shear bands in crystalline solids [28,32].…”

“…The solution of the two systems follows along the procedure presented in Masud et al [21,27]. For completeness, we outline the main steps here.…”

A heterogeneous modeling method for porous media flows

“…Hierarchical models are also used for material and spatial randomness (Ostoja‐Starzewski ) and to generate missing data in macroscopic models via stochastic upscaling of microscopic models (Ganapathysubrmanian ). The notion of linking hierarchical models has been the basis for the development of bridging scale methods in micromechanics and nanomechanics (Masud and Kannan , Shenoy ) and in the coupling of atomistic and continuum models (Curtin , Hou ). Fracture mechanics is another area where different models are employed at different scale levels to capture localized effects in the process zones around cracks (Shilkrot , Wagner ).…”

Concurrent coupling of atomistic and continuum models at finite temperature

“…The first concurrent atomistic-continuum multiscale models coupled molecular statics and continuum models, representing a thermal (0 K) mechanics [1][2][3][4][5][6][7]. The fully coupled problem of deformation and transport (at finite temperatures) was addressed more recently [8][9][10][11][12][13][14][15].…”