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Research Papers

Analysis of Carbon Nanotubes and Graphene Nanoribbons With Folded Racket Shapes

[+] Author and Article Information
R. H. Plaut1

Department of Civil and Environmental Engineering, Virginia Tech,Blacksburg, VA 24061rplaut@vt.edu

A. D. Borum, D. A. Dillard

Department of Engineering Science and Mechanics,Virginia Tech,Blacksburg, VA 24061

1

Corresponding author.

J. Eng. Mater. Technol 134(2), 021009 (Mar 27, 2012) (10 pages) doi:10.1115/1.4006178 History: Received August 02, 2011; Revised February 18, 2012; Published March 26, 2012; Online March 27, 2012

Folding of carbon nanotubes and graphene nanoribbons into a shape that looks like a tennis racket is considered. An elastic continuum model is utilized in two types of analysis. The first is called an “adhesion model,” in which the adjacent sides of the racket handle are assumed to be straight and bonded together with constant or no separation. The nanotube or nanoribbon is represented as an elastica. This model has been treated in the literature, but new analytical results are derived here, involving the geometry, work of adhesion, and bending and adhesion energies. Expressions are determined for (i) the length for which the total energy is the same as for the straight unstrained equilibrium configuration and (ii) for the minimum length for existence of a stable racket equilibrium shape. The second type of analysis uses the Lennard-Jones potential to model the attractive (van der Waals) and repulsive forces between the two sides of the racket. A nanoribbon is investigated, and the derivative of the interatomic potential is integrated along the length and across the width. Numerical solutions of the integro-differential equations are obtained with a new technique utilizing the finite difference method and minimization of the squares of the resulting algebraic equations. The results are presented for two cases with different flexural rigidities. The separation between the two sides of the handle decreases in the direction of the racket head (loop), and the handle experiences internal compression under the external attractive and repulsive forces. For the adhesion model, the dimensions of the head are proportional to the square root of the flexural rigidity, and this relationship is approximately satisfied in the numerical results based on the Lennard-Jones model.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Perspective of graphene nanoribbon in racket shape

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Figure 2

Centerline of carbon nanotube or graphene nanoribbon in racket shape

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Figure 3

Element of loop from S to S + dS

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Figure 4

Nondimensional equilibrium shapes of top half of racket for a = 1 (no handle) and do  = 0, 0.05, and 0.10

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Figure 5

Nondimensional bending energy, ubend , adhesion energy, uadh , and half the work of adhesion, wadh /2, as functions of arc length a of half loop

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Figure 6

Nondimensional work of adhesion wadh as function of separation do for a = 1

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Figure 7

Coordinate system for Lennard-Jones model

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Figure 8

Shapes of top half of racket for cases I and II

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Figure 9

Vertical component FY acting on top half of racket for case I, and shape

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Figure 10

Horizontal component FX and vertical component FY acting on top half of racket for case I

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Figure 11

Resultant force per unit length acting on top half of racket for case I

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Figure 12

Vertical internal force P and horizontal internal force Q in top half of racket for case I

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Figure 13

Bending moment M in top half of racket for case I

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Figure 14

Effect of width Wo on extreme values for case I

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