Continuum model for long-wavelength phonons in two-dimensional graphite and carbon nanotubes

A continuum model yielding dispersion relations for long-wavelength acoustic and optical phonons in two-dimensional graphite and achiral single-walled carbon nanotubes is developed. Close relations with the theory of thin elastic plates and cylindrical shells are established. The effective thickness of a nanotube determining the limits of validity of the continuum model is estimated. A qualitative analysis of vibrations of chiral nanotubes is presented.