Keywords: AFEM, buckling and post-buckling, SWNT.

Large deformation and buckling of carbon nanotubes (CNT) were observed in experiments [1,2]. Extensive theoretical research has been carried out to investigate this buckling behavior. The widely used theoretical methods can generally be divided into atomistic based methods [3,4,5,6] and continuum mechanics [7,8]. Huang's research group [9,10] proposed an atomic-scale finite element method (AFEM). Employing inter-atomic potential to consider the multi-body interactions, AFEM is as accurate as molecular dynamics (MD) simulations and is much faster than the widely used conjugate gradient method. This paper employs AFEM to study the buckling and post-buckling behavior of single-walled CNT (SWNT). Brenner's "second generation" empirical Potential (BSGEP) [11] is used here. All calculation is performed by ABAQUS via its UEL subroutine. The curve of the average strain energy per atom via strain for (8, 0) SWNT is presented, for comparison, and the strain energy curves of Srivastava et al [4], Liew et al [5], Xiao et al [6] are compared. It can be easily found that our energy curve approaches theirs closely, especially the energy curve of Xiao et al and ours almost coincide with each other. In the strain energy curve for (7, 7) SWNT of Yakobson et al [3], there are four abrupt releases of energy, while in ours, there are only two obvious jumps, where its morphologies change abruptly. Its morphology can also be obtained in detail. At small strains, it deforms linearly and keeps straight, and the strain energy grows as a quadratic function, until the critical strain of 0.0492, which represents less than 2% relative error compared with that of Yakobson et al. After that, the strain energy drops to about 22%, and it enters into the post-buckling stage. In the beginning of this stage, it displays three flattening "fins" perpendicular to each other. With increasing strain, the central fin becomes flatter. The strain energy increases approximately linearly until the second critical strain of 0.101. Then the strain energy drops to about 30.5% to another straight line and increases again. The slope of the second straight line is smaller than that of the first line. The flattening serves as a hinge, similar to that of Yakobson et al. In the stage of post-buckling, the energy increases approximately linearly with strain, which agrees with Yakobson et al and Liew et al. In Yakobson et al, the SWNT is squashed entirely at a strain of 0.13, while in our simulation, it can deformed steadily, with only its central part being squashed, until the strain of 0.176. The Brenner potential [12] was used in the MD simulation of Yakobson et al, which leads to the smaller energy releases. While the BSGEP is employed in our simulation, there are obvious drops in our energy curve, which is consistent with recent research. Due to the two clear energy drops in our curve, our maximum strain energy is much smaller than that of Yakobson et al. Judging from the above analysis, our results on post-buckling of (7, 7) SWNT are reasonable. AFEM uses much less computing time than MD simulations, thus it is an efficient way to study buckling and post-buckling of SWNT.

1

M.R. Falvo, G.J. Clary, R.M. Taylor II, V. Chi, F.P. Brooks Jr, S. Washburn, and R. Superfine, "Bending and buckling of carbon nanotubes under large strain", Nature, 389, 582-584, 1997. doi:10.1038/39282

2

O. Lourie, D.M. Cox, and H.D. Wagner, "Buckling and Collapse of Embedded Carbon Nanotubes", Phys. Rev. Lett., 81, 1638-1641, 1998. doi:10.1103/PhysRevLett.81.1638

3

B.I. Yakobson, C.J. Brabec, and J. Bernholc, "Nanomechanics of Carbon Tubes: Instabilities beyond Linear Response", Phys. Rev. Lett., 76, 2511-2514, 1996. doi:10.1103/PhysRevLett.76.2511

4

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5

K.M. Liew, C.H. Wong, X.Q. He, M.J. Tan, and S.A. Meguid, "Nanomechanics of single and multiwalled carbon nanotubes", Phys. Rev. B, 69, 115429, 2004. doi:10.1103/PhysRevB.69.115429

6

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7

C.Q. Ru, "Column buckling of multiwalled carbon nanotubes with interlayer radial displacements", Phys. Rev. B, 62, 16962-169967, 2000. doi:10.1103/PhysRevB.62.16962

8

X.Q. He, S. Kitipornchai, and K.M. Liew, "Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction", J. Mech. Phys. Solids, 53, 303-326, 2005. doi:10.1016/j.jmps.2004.08.003

9

B. Liu, Y. Huang, H. Jiang, S. Qu, and K.C. Hwang, "The atomic-scale finite element method", Comput. Methods Appl. Mech. Engrg., 193, 1849-1864, 2004. doi:10.1016/j.cma.2003.12.037

10

B. Liu, H. Jiang, Y. Huang, S. Qu, M.F, Yu, and K.C. Hwang, "Atomic-scale finite element method in multiscale computation with applications to carbon nanotubes", to appear in Phys. Rev. B, 2005. doi:10.1103/PhysRevB.72.035435

11

D.W. Brenner, O.A. Shenderova, J.A. Harrison, S.J. Stuart, B. Ni, and S.B. Sinnott, "A second-generation reactive empirical bond order (rebo) potential energy expression for hydrocarbons", J. Phys.: Condens. Matter, 14, 783-802, 2002. doi:10.1088/0953-8984/14/4/312

12

D.W. Brenner, "Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films", Phys. Rev. B, 42, 9458-9471, 1990. doi:10.1103/PhysRevB.42.9458

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