Nonlocal elastic beam models for flexural wave propagation in double-walled carbon nanotubes

Flexural wave in a double-walled carbon nanotube is studied. The scale effect of the carbon nanotube on the wave dispersion is investigated through the nonlocal elastic beam theories. The flexural wave dispersion predicted by the nonlocal elastic Timoshenko beam theory has a good agreement with that by molecular dynamics simulations in a wide frequency range up to the terahertz region. The results show that only the nonlocal elastic Timoshenko beam model can predict the small-scale effect on the dispersion of flexural wave in double-walled carbon nanotube when the wave number is large. Moreover, an estimation of the scale coefficient e0 for the nonlocal elastic double Timoshenko beam model is suggested by validations from the molecular dynamics simulations. The noncoaxial flexural vibration of the double-walled carbon nanotube can be observed from molecular dynamics simulations at high frequency range. The van der Waals interaction is found to have little effect on the noncoaxial flexural vibration of the carbon nanotube, and the nonlocal elastic Timoshenko beam theory is found to be inapplicable in modeling the noncoaxial wave propagation in carbon nanotubes.