Extended theory of elastica for free torsional, longitudinal, and radial breathing vibrations of single-walled carbon nanotubes

Single-walled carbon nanotube (SWCNT) can be viewed as a two-dimensional elastica obtained from rolling up graphene into a cylinder. By extending the formulation of elastica to dynamic problems, the present study provides an analysis for free torsional, longitudinal, and radial breathing vibrations of SWCNTs. With incorporating the effects of both geometrical and material nonlinearities, the exact closed-form expressions are obtained for the deformation field and the natural angular frequency of the elastica corresponding to the freely vibrating SWCNT. It is shown that torsional, longitudinal, and radial breathing modes of deformation are coupled to each other in free vibration of SWCNTs. However, for the special cases of zigzag and armchair SWCNTs as well as the limiting case of large-radius SWCNTs, torsional mode of deformation is decoupled from the other two modes.