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Research Papers

Buckling Analyses of Double-Wall Carbon Nanotubes: A Shell Theory Based on the Interatomic Potential

[+] Author and Article Information
W. B. Lu, J. Wu, K. C. Hwang

Department of Engineering Mechanics, AML, Tsinghua University, Beijing 10084, China

X. Feng1

Department of Engineering Mechanics, AML, Tsinghua University, Beijing 10084, Chinafengxue@northwestern.edu

Y. Huang1

Department of Mechanical Engineering and Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL 60208y-huang@northwestern.edu

The complete expansion of the velocity $vθ$ includes the terms $sin nθ sin(mπZ/L)$, $sin nθ cos(mπZ/L)$, $cos nθ sin(mπZ/L)$, and $cos nθ cos(mπZ/L)$$(n=0,1,2,…, m=1,2,3,…)$, which are equivalent to $cos((mπZ/L)−nθ)$, $cos((mπZ/L)+nθ)$, $sin((mπZ/L)−nθ)$, and $sin((mπZ/L)+nθ)$. It can be shown that these four terms lead to the same bifurcation condition as the first one $cos((mπZ/L)−nθ)$ in $vθ$ in Eq. 46.

1

Corresponding authors.

J. Appl. Mech 77(6), 061016 (Sep 01, 2010) (6 pages) doi:10.1115/1.4001286 History: Received September 05, 2009; Revised January 20, 2010; Published September 01, 2010; Online September 01, 2010

Abstract

Based on the finite-deformation shell theory for carbon nanotubes established from the interatomic potential and the continuum model for van der Waals (vdW) interactions, we have studied the buckling of double-walled carbon nanotubes subjected to compression or torsion. Prior to buckling, the vdW interactions have essentially no effect on the deformation of the double-walled carbon nanotube. The critical buckling strain of the double-wall carbon nanotubes is always between those for the inner wall and for the outer wall, which means that the vdW interaction decelerates buckling of one wall at the expenses of accelerating the buckle of the other wall.

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Figures

Figure 1

The (absolute value of) axial force, normalized by the mean circumference π(Rin+Rout) of double-wall CNT, versus the (absolute value of) engineering axial strain εaxial for four double-wall CNTs in compression, namely, two zigzag ones (14,0)@(23,0) and (26,0)@(35,0), and two armchair ones (8,8) @(13,13) and (15,15) @ (20,20); results with and without van der Waals interactions are presented

Figure 2

The buckling strain εcrcompression versus L/m for the double-wall CNT (8,8)@(13,13) in compression, where L is the CNT length, and m(=1,2,3,…) is the eigen mode number; n=0 and 1,2,3,… correspond to axisymmetric and nonaxisymmetric buckling, respectively

Figure 3

The buckling strain εcrcompression versus L/m for nonaxisymmetric buckling n=1 of double-wall CNT (8,8)@(13,13), where L is the CNT length and m(=1,2,3,…) is the eigen mode number. The buckling strain for single-wall CNTs (8,8) and (13,13) are also presented.

Figure 4

The torque, normalized by π(Rin+Rout)2/2, versus the normalized twist κ(Rin+Rout)/2 for four double-wall CNTs in torsion, namely, two zigzag ones (14,0)@(23,0) and (26,0)@(35,0), and two armchair ones (8,8)@(13,13) and (15,15)@(20,20); results with and without van der Waals interactions are presented

Figure 5

The buckling twist κcr versus L/m for the double-wall CNT (8,8)@(13,13) in torsion, where L is the CNT length and m(=1,2,3,…) is the eigen mode number

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