Torsional vibration of carbon nanotubes: Comparison of two nonlocal models and a semi-continuum model

• Develop two kinds of nonlocal torsional vibration model for carbon nanotubes.
• Present a semi-continuum model to explain the inconformity of two nonlocal models.
• Link the nonlocal theory with surface properties of nanomaterials.

In recent years, people were puzzled about two reverse nonlocal models in studying transverse bending of nanobeams. Following the ideologies of both nonlocal models, two kinds of torsional models were constructed to investigate the nonlocal torsional vibration of carbon nanotubes, respectively. Just like the transverse bending of nanobeams, it is strange to observe two opposite size-dependent performances. The first nonlocal continuum model (weakened model) was based on equilibrium equations and nonlocal torsional shear stress relation. Natural frequency decreases with an increase in nonlocal nanoscale parameter, or it increases with increasing length of the carbon nanotube. Thus the torsional stiffness of carbon nanotubes is weakened. On the other hand, the second nonlocal model (enhanced model) was developed from the strain energy variational principle. Natural frequency increases (or decreases) with increasing nonlocal nanoscale parameter (or length of the carbon nanotube), or the nanostructural stiffness is strengthened. For judgment, a torsional semi-continuum model with discrete atomic layers in the cross section of a carbon nanotube was proposed. The relaxation effects on surface atoms were considered in the torsional semi-continuum model. It is concluded that the relaxation type (attractive or repulsive) of surface atoms results in two different nonlocal results. Consequently, both the existing reverse models are proved to be valid.