Acoustic modes of finite length homogeneous and layered cylindrical shells: Single and multiwall carbon nanotubes

We present a numerical study of the normal modes of vibration of both homogeneous and heterogeneous finite length cylindrical shells of arbitrary wall thickness with applications toward single and multiwall carbon nanotubes in the continuum limit. The method is checked by comparison of computed and measured resonance spectra for a machined aluminum cylindrical shell. The dependence of the natural frequencies of various radial modes with the length and radius of single wall tubes is investigated and compared to atomistic models and Raman spectroscopy data. The radial dependence for the radial breathing mode and four harmonics of the squash mode are found to be well fitted by power laws and agree with analytical solutions in the thin wall limit. A general model for an elastically heterogeneous layered cylindrical shell is applied to multiwall tubes with graphene sheets and gaps between the sheets represented by two different materials. The frequency dependence on length and diameter is investigated for tubes composed of two to four concentric shells.