Group velocity of wave propagation in carbon nanotubes

The group velocities of longitudinal and flexural wave propagations in single- and multi-walled carbon nanotubes are studied in the frame of continuum mechanics. The dispersion relations between the group velocity and the wavenumber for flexural and longitudinal waves, described by a beam model and a cylindrical shell model, are established for both single- and multi-walled carbon nanotubes. The effect of microstructures in carbon nanotubes on the wave dispersion is revealed through the non-local elastic models of a beam and a cylindrical shell, including the second-order gradient of strain and a parameter of microstructure. It is shown that the microstructures in the carbon nanotubes play an important role in the dispersion of both longitudinal and flexural waves. In addition, the non-local elastic models predict that the cut-off wavenumber of the dispersion relation between the group velocity and the wavenumber is approximately 2×10¹⁰m^{- 1} for the longitudinal and flexural wave propagations in both single- and multi-walled carbon nanotubes. This may explain why the direct molecular dynamics simulation cannot give a proper dispersion relation between the phase velocity and the wavenumber when the wavenumber approaches approximately 2×10¹⁰m^-1, much lower than the cut-off wavenumber for the dispersion relation between the phase velocity and the wavenumber predicted by continuum mechanics.