Combined torsional buckling of multi-walled carbon nanotubes coupling with axial loading and radial pressures

This paper reports the results of an investigation on combined torsional buckling of multi-walled carbon nanotubes (MWNTs) under combined torque, axial loading and radial pressures based on the continuum mechanics model, which takes into account the effect of the van der Waals interaction between adjacent tubes. A buckling condition is derived for determining the critical buckling torque and associated buckling mode. In particular, for combined torsional buckling of double-walled carbon nanotubes, an explicit expression is obtained and some detailed results are demonstrated. According to the innermost radius-to-thickness ratio, MWNTs are classified into three types: thin, thick, and (almost) solid. Numerical results are worked out for the critical buckling torque and associated buckling mode for all the three types of MWNTs subjected to various axial stresses (axial tensile stresses or axial compressive stresses), internal pressures, and external pressures. It is shown that, the axial tensile stress or the internal pressure will make the MWNTs resist higher critical buckling torque, while the axial compressive stress or external pressure will lead to a lower critical buckling torque. The effect of axial stress (axial tensile stress or axial compressive stress) on the critical buckling torque of MWNTs is very small for all the three types of MWNTs, while the effect of the internal pressure or external pressure is related to the types of MWNTs, which is strong for the thin MWNTs, moderate for the thick MWNTs, and small for the solid MWNTs. Numerical results also indicate that, the associated buckling mode is unique and dependent on the structure of MWNTs. Especially, for combined torsional buckling of MWNTs with very small axial stress and radial pressures, the buckling mode is just the one for the corresponding pure torsional buckling.