tailieunhanh - Buckling of double - walled carbon nanotubes

In this note we deal with the approximate solution of the buckling problem of a clamped-free double-walled carbon nanotube. First the finite difference method is utilized to solve this case. Then this approach is verified by solving the buckling problem of a double-walled carbon nanotube that is simply supported at both ends for which the exact solution is available. | Vietnam Journal of Mechanics, VAST, Vol. 34, No. 4 (2012), pp. 217 – 224 BUCKLING OF DOUBLE-WALLED CARBON NANOTUBES Isaac Elishakoff1 , Kévin Dujat2 , Maurice Lemaire2 of Mechanical Engineering, Florida Atlantic University, USA 2 French Institute for Advanced Mechanics, Aubière, France 1 Department Abstract. In this note we deal with the approximate solution of the buckling problem of a clamped-free double-walled carbon nanotube. First the finite difference method is utilized to solve this case. Then this approach is verified by solving the buckling problem of a double-walled carbon nanotube that is simply supported at both ends for which the exact solution is available. Keyword: Buckling, nanotube, finite difference method, clamped-free. 1. INTRODUCTION The studies on buckling of carbon nanotubes (CNTs) include those of Yakobson et al. [1], Cornwell and Wille [2], Yao and Lordi [3], Garg et al. [4], Lu et al. [5], Wang et al. [6], Falvo et al. [7], Guo et al. [8], Sudak [9], He et al. [10] and Wang [11,12]. However, buckling of carbon nanotubes is not yet studied sufficiently. For example, we are not able to conclude yet if this kind of a structure exhibits the same buckling behavior as uniform beams. In their work on the buckling of double-walled carbon nanotubes (DWCNTs) Elishakoff and Pentaras [13] found, with Galerkin method, that the ratio between the critical loads of a clamped-clamped DWCNT and one simply supported at both ends is about four whereas the analogous ratio for the uniform beam equals four exactly. They did not investigate the case of clamped-free DWCNTs because of the necessity to satisfy different boundary conditions for inner and outer nanotubes. This study fills the existing gap. Hereinafter, we use the finite difference method that has been widely used in the past for buckling analysis of various structures (see for example, works by Salvadori [14], Iremonger [15], Chajes [16], and Mikhailov [17]). 2. BUCKLING DIFFERENTIAL EQUATIONS IN