Static analysis and vibration characteristics of some noncarbon nanotubes through atomistic continuum coupled modelling

A computationally efficient mixed atomistic-continuum coupled modelling is employed to investigate the vibrational characteristics and large deformation static response of some nitride and phosphide-based nanotubes. The atomic entities bond lengths/angles and continuum entities strains/curvature tensors are coupled through the kinematics of quadratic-type Cauchy–Born rule considering curvature effects on the atomic entities. The finite element model is formulated in a cylindrical coordinate system considering geometric nonlinearity through nonlinear strain–displacement relations and material nonlinearity through interatomic potential. The present study is carried out using a four nodded membrane consistent element wherein smoothened interpolation functions derived through least square minimization are used to interpolate the circumferential strain to avoid membrane locking. The atomic interactions are modelled using Tersoff–Brenner type interatomic potential with recently reported new empirical parameters. The effects of length and diameter, boundary conditions and length-to-diameter ratio on the natural frequency of the different nanotubes are also reported. The results obtained through multiscale modelling are also compared with those obtained through molecular dynamics (MD) simulation. The effect of material nonlinearity on the fundamental frequency of the nanotubes under applied pressure is also reported.