A computationally efficient multiscale finite element formulation for dynamic and postbuckling analyses of carbon nanotubes

The efficient multiscale membrane locking free shell elements are developed to study the dynamic and postbuckling characteristics of carbon nanotubes incorporating material and geometric nonlinearities. The constitutive relation at continuum level is derived through the Cauchy-Born rule incorporating the effect of curvature tensor on bond lengths and using the Tersoff-Brenner atomic interaction potential per unit area of a unit cell. The membrane locking is eliminated by using the smoothed shape functions derived through the least square strain smoothing technique for the interpolation of the transverse displacement in the circumferential strain. The performance of the four/eight noded inconsistent/consistent Kirchhoff rectangular and improved discrete Kirchhoff quadrilateral (IDKQ) shell elements is investigated. It is found that the four noded elements with smoothed interpolation of transverse displacement in the circumferential strain yield accurate results and are computationally efficient. The multiscale modelling results are found to be in close agreement with the molecular mechanics simulations. The significant effect of material nonlinearity on the nonlinear dynamic and postbuckling responses is predicted.