Research Articles

Nonlocal Flügge Shell Model for Vibrations of Double-Walled Carbon Nanotubes With Different Boundary Conditions

[+] Author and Article Information
R. Ansari

Department of Mechanical Engineering,
University of Guilan,
P.O. Box 3756,
Rasht, Iran
e-mail: r_ansari@guilan.ac.ir

B. Arash

Department of Mechanical and
Manufacturing Engineering,
University of Manitoba,
Winnipeg, Manitoba, Canada, R3T 5V6

1Corresponding author.

Manuscript received January 8, 2012; final manuscript received July 23, 2012; accepted manuscript posted August 23, 2012; published online January 22, 2013. Assoc. Editor: George Kardomateas.

J. Appl. Mech 80(2), 021006 (Jan 22, 2013) (12 pages) Paper No: JAM-12-1007; doi: 10.1115/1.4007432 History: Received January 08, 2012; Revised July 23, 2012; Accepted August 23, 2012

In this paper, the vibrational behavior of double-walled carbon nanotubes (DWCNTs) is studied by a nonlocal elastic shell model. The nonlocal continuum model accounting for the small scale effects encompasses its classical continuum counterpart as a particular case. Based upon the constitutive equations of nonlocal elasticity, the displacement field equations coupled by van der Waals forces are derived. The set of governing equations of motion are then numerically solved by a novel method emerged from incorporating the radial point interpolation approximation within the framework of the generalized differential quadrature method. The present analysis provides the possibility of considering different combinations of layerwise boundary conditions. The influences of small scale factor, layerwise boundary conditions and geometrical parameters on the mechanical behavior of DWCNTs are fully investigated. Explicit expressions for the nonlocal frequencies of DWCNTs with all edges simply supported are also analytically obtained by a nonlocal elastic beam model. Some new intertube resonant frequencies and the corresponding noncoaxial vibrational modes are identified due to incorporating circumferential modes into the shell model. A shift in noncoaxial mode numbers, not predictable by the beam model, is also observed when the radius of DWCNTs is varied. The results generated also provide valuable information concerning the applicability of the beam model and new noncoaxial modes affecting the physical properties of nested nanotubes.

Copyright © 2013 by ASME
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Fig. 1

Schematic of a DWCNT treated as nested cylindrical shells

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Fig. 5

Mode shapes of a simply supported DWCNT predicted by the shell model: (a) with R1 = 1 nm and L/R1 = 10 (b) with R1 = 2 nm L/R1 = 10

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Fig. 6

Three dimensional mode shape associated with ω41(4) for a DWCNT with a free inner tube and a clamped outer tube (FF/CC) (R1 = 8.5 nm, L/R1 = 5)

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Fig. 2

First resonant frequencies from nonlocal continuum shell model and MD simulations for (5,5) @ (10,10) armchair DWCNT with CF/CF boundary conditions

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Fig. 3

First three resonant frequencies of a DWCNT with simply supported end conditions versus aspect ratio (R1 = 0.35 nm): (a) ω11 (b) ω21 c) ω31

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Fig. 4

Mode shapes of a simply-supported DWCNT predicted by the beam model: (a) with R1 = 1 nm and L/R1 = 10 (b) with R1 = 2 nm and L/R1 = 10




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