Buckling characteristics of embedded multi-walled carbon nanotubes

An analytical algorithm is proposed to describe the buckling behaviour of multi-walled carbon nanotubes (CNTs) that are embedded in a matrix with consideration of the van der Waals (vdW) interaction. The individual tube is treated as a cylindrical shell, but the tube deflections are coupled with each other due to the vdW interaction. The interaction between the matrix and the outermost tube is modelled as a Pasternak foundation. Based on the proposed model, an accurate expression and a simple approximate expression are derived for the buckling load of double-walled CNTs that are embedded in a matrix. The approximate expression clearly indicates that the vdW force is coupled with the matrix parameters. A numerical simulation is carried out, and the results reveal that the increase in the number of layers leads to a decrease in the critical buckling load for multi-walled CNTs with a fixed innermost radius. In contrast, when the outermost radius is fixed, the critical buckling load increases with the increase in the number of layers for multi-walled CNTs without a matrix. However, for multi-walled CNTs that are embedded in a matrix, the critical buckling load decreases first and then increases with the increase in the number of layers. This implies that there is a given number of layers for a multi-walled CNT at which the critical buckling load is the lowest, and that this number depends on the matrix parameters. © 2005 The Royal Society.