Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction

Explicit formulas are derived for the van der Waals (vdW) interaction between any two layers of a multi-walled carbon nanotube (CNT). Based on the derived formulas, an efficient algorithm is established for the buckling analysis of multi-walled CNTs, in which individual tubes are modeled as a continuum cylindrical shell. The explicit expressions are also derived for the buckling of double-walled CNTs. In previous studies by Ru (J. Appl. Phys. 87 (2000b) 7227) and Wang et al. (Int. J. Solids Struct. 40 (2003) 3893), only the vdW interaction between adjacent two layers was considered and the vdW interaction between the other two layers was neglected. The vdW interaction coefficient was treated as a constant that was not dependent on the radii of the tubes. However, the formulas derived herein reveal that the vdW interaction coefficients are dependent on the change of interlayer spacing and the radii of the tubes. With the increase of radii, the coefficients approach constants, and the constants between two adjacent layers are about 10% higher than those reported by Wang et al. (Int. J. Solids. Struct. 40 (2003) 3893). In addition, the numerical results show that the vdW interaction will lead to a higher critical buckling load in multi-walled CNTs. The effect of the tube radius on the critical buckling load of a multi-walled CNT is also examined.