Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics

In this paper, the small scaling parameter e₀ of the nonlocal Timoshenko beam theory is calibrated for the free vibration problem of single-walled carbon nanotubes (SWCNTs). The calibration exercise is performed by using vibration frequencies generated from molecular dynamics simulations at room temperature. It was found that the calibrated values of e₀ are rather different from published values of e₀. Instead of a constant value, the calibrated e₀ values vary with respect to length-to-diameter ratios, mode shapes, and boundary conditions of the SWCNTs. In addition, the physical meaning of the scaling parameter is explored. The results show that scaling parameter assists in converting the kinetic energy to the strain energy, thus enabling the kinetic energy to be equal to the strain energy. The calibrated e₀ presented herein should be useful for researchers who are using the nonlocal beam theories for analysis of micro and nano beams/rods/tubes.