Free transverse vibrations of a double-walled carbon nanotube: gradient and internal inertia effects

This is a modest contribution on higher-order continuum theory for predicting size effects in small-scale objects. It relates to a preceding article of the journal by the same authors (AMSS, 2013, 26: 9–20) which considered the longitudinal dynamical analysis of a gradient elastic fiber but, in addition to an internal length, an internal time parameter is also introduced to model delay/acceleration effects associated with the underlying microstructure. In particular, the free transverse vibration of a double-walled carbon nanotube (DWNT) is studied by employing gradient elasticity with internal inertia. The inner and outer carbon nanotubes are modeled as two individual elastic beams interacting with each other through van der Waals (vdW) forces. General explicit expressions are derived for the natural frequencies and the associated inner-to-outer tube amplitude ratios for the case of simply supported DWNTs. The effects of internal length (or scale) and internal time (or inertia) on the vibration behavior are evaluated. The results indicate that the internal length and time parameters of the adopted strain gradient-internal inertia generalized elasticity model have little influence on the lower order coaxial and noncoaxial vibration modes, but a significant one on the higher order modes.