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

This paper investigates the combined torsional buckling of multi-walled carbon nanotubes (MWNTs) coupling with radial pressures. The analysis is based on the continuum mechanics model, and the effect of the van der Waals interaction between adjacent tubes is taken into account. A buckling condition is derived for determining the critical shear membrane force under combined torsional buckling, which clearly indicates the role of radial pressures. The critical shear membrane force and the buckling mode are worked out for three typical MWNTs subjected to various internal pressure or external pressure. It is shown that the effect of internal pressure or external pressure on the critical shear membrane force for combined torsional buckling of MWNTs is related to the types of MWNTs. This effect is strong for thin MWNTs, moderate for thick MWNTs and small for solid MWNTs. Numerical results also indicate that the buckling mode corresponding to the critical shear membrane force of MWNTs is unique and only dependent on the structure of MWNTs. In particular, for combined torsional buckling of MWNTs with very small internal pressure or external pressure, the buckling mode is just that for the corresponding pure torsional buckling.