The torsional buckling of an individual multi-walled carbon nanotube under two different loading conditions is studied in this article. The multiple shell model is adopted and the effects of van der Waals forces between adjacent nanotubes are taken into account. An examination with an individual double-walled carbon nanotube shows that the effect of the change of interlayer spacing on the torsional buckling force can be neglected if only the innermost radius is larger than a certain value. Under this condition, single buckling equations are derived and explicit formulas for the critical torsional loads in terms of the buckling modes are obtained. It is found that the critical torsional load of a multi-walled carbon nanotube with torque exerted on the outermost tube is higher than that of the same multi-walled carbon nanotubes under the torques being proportionally applied to each individual layer of the multi-walled carbon nanotubes. For thin multi-walled carbon nanotubes with large radii, the critical torque linearly scales with its thickness, but the critical shear force (per unit length) of the multi-walled carbon nanotubes uniformly twisted along the cross section does not increase as its layer number (thickness) increases, which is due to the interlayer slips between adjacent nanotubes.
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