Shell Element Based Model for Wave Propagation in

Carbon Nanotube

A. Chakraborty∗

India Science Lab, GM R&D, Bangalore, India

A spectrally formulated finite element is developed to study very high frequency elasticwaves in carbon nanotubes (CNTs). A multi-walled nanotube (MWNT) is modelled asan assemblage of shell elements connected throughout their length by distributed springs,whose stiffness is governed by the van der Waals force acting between the nanotubes. Thespectral element is formulated using the recently developed strategy based on singularvalue decomposition (SVD) and polynomial eigenvalue problem (PEP). The element canmodel a MWNT with any number of walls. Studies are carried out to investigate theeffect of the number of walls on the spectrum and dispersion relation. It is shown thatthe shell model captures first anti-symmetric Lamb wave modes. It is also shown throughnumerical examples that the developed element efficiently captures the response of MWNTfor Tera-hertz level frequency loading.

I. Introduction

Since their discovery in 1991, CNTs are currently occupying the center-stage of theoretical and experi-mental attention.1 An extensive account of the theoretical modelling aspects of CNTs, both atomistic

and continuum approaches can be found in,2 where Euler-Bernoulli (EB) beam theory based Spectral FiniteElement (SFE) was developed for MWNTs. In the present study, a new spectral element based on shelltheory is developed. There are few earlier works on the shell-theory based model of CNTs. Yakobson etal.3, 4 noticed the unique features of fullerenes and developed a continuum shell model, where the analyticalexpressions for the energy of a shell in terms of local stresses and deformations were provided. Ru5, 6 followedthis continuum shell model to investigate buckling of CNTs subjected to axial compression. This kind ofcontinuum shell models can be used to analyze the static or dynamic mechanical properties of nanotubes.However, these models neglect the detailed characteristics of nanotube chirality, and are unable to accountfor forces acting on the individual atoms.

Spectral Finite Element Method (SFEM)7 is arguably the most suitable technique for studying wavepropagation in structural waveguides and especially in MWNT due to high frequency content loading (THzlevel). The basic SFEM is further improved by Chakraborty and Gopalakrishnan where the problem is castas PEP8 and SVD method of wave amplitude extraction is utilized. This approach puts SFEM in a solidground and gives it the most generalized form to handle any kind of structure. This method is so far utilizedin plate element formulation9 and beam approximation of MWNT.2

In the present work, membrane shell model based SFE is considered for modelling MWNT. Wavenumbersand group speeds are determined as functions of the number of walls, which facilitate to get deep insightin the wave propagation response of MWNTs. The present element formulation exploits the PEP methodextensively, which is most suitable in the present case as number of walls has been taken as a parameter inelement formulation. The element is used to capture the behavior of CNT for high frequency loading.

II. Element Formulation

According to the theory of cylindrical membrane shell, the governing equations are

δu : A11uxx + A12R−1wx = I0u (1)

∗Researcher, Vehicle Structures & Safety, Professional Member, AIAA

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δw : −R−1(A12ux + A22R−1w) = I0w (2)

δv : A55vxx = I0v (3)

where, u, v and w are the displacement components of the shell in the X , Y and Z direction, respectivelyand the Z is measured downward positive (see Fig. 1).

X, u

Y, v

Z, w

Figure 1. The spectral element and the associated dof

To solve this problem, the solutions for the unknown variables are sought in the form

ξ(x, t) =N∑

n=1

Nk∑p=1

ξe−jkpxe−jωnt , ξ = {u, v, w}, j2 = −1 , (4)

where ωn and kp are the circular frequency at the n − th sampling point and the wavenumber in the Xdirection, respectively. The N is the index corresponding to the Nyquist frequency in fast Fourier transform(FFT), which is used for computer implementation of the Fourier transform. The Nk is the number of wavemodes in the X direction, which in this case is 2. The A11, A12, A55 and I0 are the integrated materialproperties over the cross-section of the tube (see9).

The frequency domain transformed equations can be further simplified by eliminating w from Eq. (2),which gives:

(A11 + A12βR−1)uxx + I0ω2u = 0 , β = RA12/(I0R

2ω2 − A22) , w = βux (5)

Henceforth, we are considering only this axial degree of freedom, u for the MWNT model since w can beobtained from u and v can be found separately. The equations for u for double-wall nanotube become:

A(1)11 u1,xx + I

(1)0 ω2u1 + A

(1)12 R−1

1 (β11u1,x + β12)u2,x) = a1(u2 − u1) , (6)

A(2)11 u2,xx + I

(2)0 ω2u2 + A

(2)12 R−1

2 (β21u1,x + β22)u2,x) = −a1(u2 − u1) , (7)

where the superscript indicates the wall numbers. The βij are the coefficients that establish the relationbetween u and w, i.e.,

w1 = β11u1,x + β12u2,x w2 = β21u1,x + β22u2,x , (8)

which can be obtained from the relation:[I(1)0 ω2 − A

(1)22 R−1

1 − c1 c1

c1 I(2)0 ω2 − A

(2)22 R−1

2 − c1

]{w1

w2

}=

{A

(1)12 u1,x

A(2)12 u2,x

}, (9)

where c1 and a1 are the interaction coefficients between the nanotubes which can be computed by the relationgiven in.2

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Substituting Eq. 4 in Eqs. 6 and 7 the differential operators give way to an algebraic equation, which canbe cast as a quadratic eigenvalue problem (QEP) as

Find (v �= 0, k) such that, Ψ(k)v = (k2A2 + kA1 + A0)v = 0 , (10)

where k is the wavenumber kp at nth frequency and v is the vector of unknown constants. The QEP issolved by linearization (casting it as a generalized eigenvalue problem) and the detail is omitted here. Oncethe eigenvalues and the eigenvectors are known, the total solution can be constructed by taking the linearcombination of all the individual solutions.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Frequency, [THz]

Wav

enum

ber ×

thic

knes

s

Imaginary part

Real part

Figure 2. Spectrum relation: DWNT

III. Numerical Examples

First, the variation of the wavenumbers kp is studied. It is assumed that the nanotubes are having athickness of 0.35 nm with internal radius of the first tube at 5 nm and inter-tube spacing of 0.142 nm.Further, the elastic modulus, poisson’s ratio and density of each tube are taken as 1 TPa, 0.3 and 1300kg/m3, respectively. Figure 2 shows the wavenumber variation for a double wall nanotube. It is interestingto find that the waves are inhomogeneous, i.e., they simultaneously allow real (propagating) and imaginary(evanescent) parts, which persist for all frequencies. Figure 3 shows the variation of the phase and groupspeeds. It is clear that the present shell model captures the first two anti-symmetric modes for the two walls.The coupling between the nanotubes shifts the modes with non-zero cut-off frequency (around 0.015 and0.15 THz).

Subsequently, the formulated element is verified by comparing its response with 3D FE model analyzedin LS-DYNA. To avoid modelling difficulties in LS-DYNA, nominal steel material is considered for a tubewith length to radius ratio of 10. Even for this large thickness, it is shown that the present model worksperfectly. The tube with cantilever boundary condition is subjected to a high frequency loading shown inFig. 4, applied axially at the free end. The axial velocity of the tube at the free end is shown in Fig. 5, whichshows excellent comparison with LS-DYNA simulation.

IV. Conclusion

A novel spectral element is developed to model multi-wall nanotube with minimal cost of computation.The element can accommodate any number of walls and successfully captures all the wave propagationcharacteristics. However, limited by the displacement field assumptions, the element can predict only theanti-symmetric Lamb wave modes. Further, the coupling introduces cut-off frequencies in these modes, i.e.,initial frequency zone where no bending mode exists.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50

Frequency, (THz)

Pha

se S

peed

& G

roup

Spe

ed, [

km/s

]

Group speeds

Phase speeds

Figure 3. Dispersion relation: DWNT

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5x 10

−5

Frequency (KHz)

Fre

quen

cy A

mpl

itude

0 100 200 300

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (µ sec)

Load

(N

)

Figure 4. Broadband pulse loading, inset shows time domain data

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−3

−3

−2

−1

0

1

2

3x 10

−5

Axi

al v

eloc

ity, [

m/s

]

Time, [s]

SFEMLS−DYNA

Figure 5. Axial velocity due to axial load

Although the element is an improvement over the existing beam model, it is not enough to correctlyrepresent the dynamics of MWNTs, which require shell element based modeling.

References

1Tomanek, D. and Enbody, R., Science and Applications of Nanotubes, Kluwer/Plenum, NY, 2000.2Chakraborty, A., Sivakumar, M., and Gopalakrishnan, S., “Spectral element based model for wave propagation analysis

in multi-wall carbon nanotubes,” Int. J. Sols. Str., Vol. 43(2), 2006, pp. 279–294.3Yakobson, B., Brabec, C., and Bernholc, J., “Nanomechanics of carbon tubes: instabilities beyond linear range,” Phys.

Rev. Lett., Vol. 76, 1996, pp. 2511–2514.4Yakobson, B., Brabec, C., and Bernholc, J., “Structural mechanics of carbon nanotubes: from continuum elasticity to

atomistic fracture,” J. Computer-Aid. Mat. Design, Vol. 3, 1996, pp. 173–182.5Ru, C. Q., “Effective bending stiffness of carbon nanotubes,” Physical Review B , Vol. 62, 2000, pp. 9973–9976.6Ru, C. Q., “Elastic buckling of single-walled carbon nanotube ropes under high pressure,” Physical Review B , Vol. 62,

2000, pp. 10405–10408.7Doyle, J. F., Wave Propagation in Structures, Springer, 1997.8Lancaster, P., Theory of Matrices, Academic Press, 1969.9Chakraborty, A. and Gopalakrishnan, S., “A Spectrally Formulated Plate Element for Wave Propagation Analysis in

Anisotropic Material,” Comput. Meth. Appl. Mech. Engg., Vol. 194 (42-44), 2005, pp. 4425–4446.

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