Shell element based model for wave propagation analysis in multi-wall carbon nanotubes

A spectrally formulated finite element is developed to study very high frequency elastic waves in carbon nanotubes (CNTs). A multi-walled nanotube (MWNT) is modelled as an assemblage of shell elements connected throughout their length by distributed springs, whose stiffness is governed by the van der Waals force acting between the nanotubes. The spectral element is formulated using the recently developed strategy based on singular value decomposition (SVD) and polynomial eigenvalue problem (PEP). The element can model a MWNT with any number of walls. Studies are carried out to investigate the effect of the number of walls on the spectrum and dispersion relation. The importance of shell element based model over the beam model is established. The zone of validity of the previously developed beam model is also investigated. It is shown that the shell model is required to capture the symmetric Lamb wave modes. It is also shown through numerical examples that the developed element efficiently captures the response of MWNT for Tera-hertz level frequency loading.