Nonlinear vibration and bending instability of a single-walled carbon nanotube using nonlocal elastic beam theory

Due to the nonlocal Euler–Bernoulli elastic beam theory, the effects of rippling deformation on the bending modulus and the structural bending instability of a single-walled carbon nanotube (SWCNT) are investigated. The nonlinear vibrational model of a cantilevered SWCNT is solved using the perturbation method of multiscales. The nonlinear resonant frequency and the associated effective bending modulus of the carbon nanotube (CNT) are derived analytically. The effects of the nonlocal parameter, the external harmonic force, and the diameter-to-length ratio on the effective bending modulus are discussed widely. Moreover, the model can predict special kind of structural instability due to the rippling deformation called rippling instability. The results show that the nonlocal theory forecasts larger values for the effective bending modulus compared with the classical beam theory, especially for the stubby CNTs. Meanwhile, the rippling instability threshold will move to the higher values of the diameter-to-length ratio based on the nonlocal beam theory comparing with the local ones.