Asymptotic estimates of buckling radial pressure for multi-walled carbon nanotubes at different variants of boundary conditions

Buckling of short multi-walled carbon nanotubes (MWCNTs) under external radial pressure is studied on the base of a multiple-shell model. The modified Mushtari-Donell-Vlasov type equations taking into account the van der Waals (vdW) interaction forces between adjacent tubes are used as the governing ones. In contrast to a majority of available studies on buckling of MWCNTs, which consider only the simply supported boundary conditions, this paper based on the asymptotic approach allows for the study of the buckling behavior of MWCNTs with different variants of the boundary conditions at the tube edges. At first, the pre-buckling membrane hoop stress-resultants induced by radial pressure are determined for each wall. Then, introducing a small parameter defined as a thickness-to-radius ratio, the asymptotic solutions of the boundary value problem are constructed for different cases which depend on the outermost radius of a MWCNT. The relevance of the present approach is confirmed by good agreement between asymptotic estimates and exact values of the buckling radial pressure for simply supported double- and triple-walled nanotubes determined on the base of the accepted shell model. In addition, the validity of the asymptotic estimates is justified by comparing theirs with existing data obtained on the base of the available multiple-shell model taking into account the pressure dependence of the interlayer vdW forces. The influence of the outermost radius, aspect ratio and boundary conditions as well on the buckling radial pressure is analyzed in this study.