Exploring the Effect of Dihedral Energy on the Nonlinear Mechanics of the Carbon Nanotubes Using a Multiscale Modeling

Abstract

A hierarchical multiscale finite element model is employed to investigate the effect of dihedral energy term on the numerical simulation of two-dimensional materials. The numerical examples of the carbon nanotubes and graphene sheets are studied employing a refined constitutive model in conjunction with a multiscale finite element method. The constitutive law refined with the greater accuracy on the bending modulus using second generation reactive empirical bond order potential with dihedral energy term is employed to investigate the linear and nonlinear response of the carbon nanotubes incorporating material and Green–Lagrange geometric nonlinearities. The inclusion of the dihedral energy term predicts bending modulus close to those of through first principle calculations. The deformations at the nanoscale and macroscopic scales are related through the Cauchy–Born rule. The effect of the dihedral energy term on the response of the carbon nanotubes is studied in detail. The governing equation of motion for the carbon nanotubes is formulated through Hamilton’s energy principle. The spatial approximation of the carbon nanotubes at the continuum scale is attained through the finite element method. The membrane locking in the circumferential strain is eliminated through the membrane consistent interpolation functions obtained through the least-square method.