Small Scale Effect on the Buckling Analysis of a Double-Walled Carbon Nanotube under External Radial Pressure Using Energy Method

Document Type : Research Article



In this paper, using energy method, small scale effects on the buckling analysis of a double-walled carbon nanotube (DWCNT) under external radial pressure is studied. The constitutive equations derived for a DWCNT using the nonlocal theory of elasticity which Eringen are presented for the first time. By minimizing the second variation of the total energy for a DWCNT, hence, the value of the nonlocal critical buckling load is obtained. It is seen from the results that the nonlocal critical buckling load increases with increasing the circumferential wave number. Moreover, it is seen that the nonlocal critical buckling load is lower than that of the local one. It is shown from the results that the ratio of the critical buckling load decreases with increasing the length of nanotubes while it increases with decreasing the radius of the outer tube.


[1]     P. Ball, "Roll up for the revolution", Nature, vol. 414, pp. 142-144, 2001.
[2]     C. Q. Ru, “Axially compressed buckling of a double walled carbon Nanotube embedded in an elastic medium", Journal of the Mechanics and Physics of Solids, vol. 49, pp. 1265-1279, 2001.
[3]     C. Y. Wnag, C. Q. Ru ,A. Mioduchowski, “Axially compressed buckling of pressured multiwall carbon Nanotube”, International Journal of Solids and Structures, vol. 40, pp. 3893-3911, 2003.
[4]     X. Guo, A. Y. T. Leung, X. Q. Jiang, Y. Huang, “”, Composite, vol. 39 (part B), pp. 202-208, 2008.
[5]     H. L. Ball, W. J. Chang, "A closed-form solution for critical buckling temperature of a single-walled carbon Nanotube", Physica E, vol. 41, pp. 1492-1494, 2009.
[6]     A. Ghorbanpour Arani, R. Rahmani, A. Arefmanesh, and S. Golabi, ”Buckling analysis of multi-walled carbon nanotubes under combined loading considering the effect of small length scale”, Journal of Mechanical Science and Technology, vol.  22, pp. 429-439, 2008.
[7]     A.R. Ranjbartoreh, A. Ghorbanpour, and B. Soltani, ”Double-walled carbon nanotube with surrounding elastic medium under axial pressure”. Physica E, vol. 39, pp. 230-239, 2007.
[8]     A. C. Eringen,”On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves”, Journal of Applied Physics, vol. 54, pp.4703-4710, 1983.