Using model of strain gradient membrane shell to characterize longitudinal wave dispersion in multi-walled carbon nanotubes

Although many studies have been conducted on the vibration and wave propagation in multi-walled carbon nanotubes, our molecular dynamics results on such nanotubes with small diameters show that the van der Waals force has little effect on the longitudinal wave dispersion in most wave number cases, but limited by the capability of MD simulations, the situation in large multi-walled carbon nanotubes remains elusive. Here we develop a model of multi-strain gradient membrane shell is developed for the analysis the longitudinal wave propagation and the dispersion in multi-walled carbon nanotubes. Wave number dependent solutions are derived from the model of multi-membrane shell with the strain gradient taken into consideration. The effect of the van der Waals force on the dispersion of longitudinal waves is studied. When the diameter of the nanotube is small, the continuum model of strain gradient shell shows that the intertube van der Waals force has weak effects. When the diameter is large, the effects of the van der Waals force on the wave dispersion become obvious. With a decrease inter-layer distance, the van der Waals interaction becomes stronger, and its effects on the dispersion relation increase too. The van der Waals force can increase the critical frequency, especially in multi-walled carbon nanotubes of large diameter. This study also reveals the significance of the small-scale effect on the longitudinal wave propagation of multi-walled carbon nanotubes. When the wave number is lower, the strain gradient has little effect on the longitudinal wave dispersion. When the wave number becomes very high, the strain gradient effect becomes significant, and the difference of the dispersion relation given by the two models is obvious. The consideration of the van der Waals force can weaken the small size effect.