Nonlocal Analysis of Natural Vibrations of Carbon Nanotubes

When the atomic level is considered, the dynamic properties have a major influence on nanostructures behavior because of their ultrahigh or very high natural frequencies. Vibrations are essential in the analysis of resonators, oscillators, and sensors for carbon nanotube-based devices. The present paper is dedicated to the eigenfrequencies analysis of single-walled carbon nanotubes. In the analysis, the small-scale coefficient is introduced, and the nonlocal elasticity theory is applied in the modal analysis of the carbon nanotubes. The correlation between nonlocal small-scale parameter and the vibrational behavior of the carbon nanotubes is studied. The determination of nonlocal parameter is based on the structural finite element model of the nanotube. A detailed parametric study is realized to investigate the effects of the length-to-diameter ratio L/D, nonlocal coefficient, the influence of boundary conditions on eigenfrequencies of carbon nanotubes. The main contribution of this work is the evaluation of the nonlocal parameter for single-walled carbon nanotubes based on the finite element approximation in modeling the dynamic behavior instead of commonly used MD simulations. It was clearly displayed that the application of the nonlocal beam models involves analysis of the small-scale parameter, whose value determines the final eigenfrequencies. The highest values of eigenfrequencies were observed for the local beam model. It was presented that the eigenfrequencies for carbon nanotubes decrease when the nanotube length increases for all analyzed modes. On the other hand, when the diameter of carbon nanotubes increases, the eigenfrequency values rise. The numerical validation of the nonlocal parameter revealed the high dependence on the boundary conditions and the mode number. However, the influence of the mode number and the boundary conditions is vanishing for long single-walled carbon nanotubes.