Axisymmetric and beamlike vibrations of multiwall carbon nanotubes

Several vibration problems of multiwall carbon nanotubes (MWNTs) are studied in detail based on a multiple-elastic shell model. According to recent data available in the literature, an updated value of bending stiffness for single-wall carbon nanotubes (SWNTs) is suggested, which is in a much better agreement with atomistic model for phonon-dispersion relation of SWNTs. For axisymmetric vibrations (with circumferential wave number n=0), it is found that longitudinal (L) modes of individual tubes of a MWNT have almost identical frequencies and are usually coupled with each other through Poisson-ratio effect-induced radial (R) vibrations and interlayer van der Waals interaction. Especially in the transition zone of R- and L modes, the significant Poisson-ratio effect leads to mixed R-L modes with comparable longitudinal and radial displacements. On the other hand, for beamlike vibrations (with n=1), the present multiple-shell model is found to be in good agreement with the multiple-beam model for almost coaxial bending (B) modes of large- and small-radius MWNTs and noncoaxial B modes of small-radius MWNTs (e.g., of the outermost radius less than 2 nm), with relative errors less than 10%. However, for high-order noncoaxial modes of large-radius MWNTs, the relative errors between the two models increase up to 50% in extreme cases due to larger non-beamlike deformation of the cross section while both models give similar overall vibration modes through the entire length of MWNTs. In particular, for lower circumferential wave numbers (n=0-10), the lowest frequency always corresponds to the minimum half-axial wave number m=1 for simply supported end conditions. When the wave vector decreases, the lowest frequency decreases and the associated mode shifts from an R mode with larger n to a coaxial B mode with n=1.