Modeling of van der Waals force for infinitesimal deformation of multi-walled carbon nanotubes treated as cylindrical shells

This paper models van der Waals (vdW) force for axially compressed multi-walled carbon nanotubes (CNTs), whereby each tube is treated as a cylindrical shell continuum. Explicit formulas are derived for predicting the critical axial strain of a triple-walled CNT using a more refined vdW model. The analysis of a cylindrical shell continuum model of multi-walled CNTs using this refined vdW force model is carried out to study the influence of the effect of vdW interaction between different layers of a CNT and the size effect of a CNT on the vdW interaction. It is shown herein that the greatest contribution to the vdW interaction comes from the adjacent layers and the contribution from a remote layer may be neglected. The vdW interaction is found to be strongly dependent on the radius of the tube, especially when the radius is small enough (<7 nm). When the radius is large enough (>40 nm), the vdW interaction coefficient cij can be taken as a constant value (i.e. independent of radius). However, these constant values are different for the vdW interaction between two different layers of a multi-walled CNT. The effect of the vdW interaction on the critical axial strain of a triple-walled CNT for the cases of before and after buckling is also examined for various innermost radii.