Analysis of wave propagation in carbon nanotubes via elastic shell theories

International Journal of Engineering Science 45 (2007) 227–241

www.elsevier.com/locate/ijengsci

Analysis of wave propagation in carbon nanotubesvia elastic shell theories

K.M. Liew a,*, Q. Wang b

a Department of Building and Construction, City University of Hong Kong, Hong Kong, Chinab Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6

Received 2 April 2007; accepted 2 April 2007Available online 24 May 2007

Abstract

This paper investigates wave propagation in both single-walled carbon nanotubes (SWNTs) and double-walled carbonnanotubes (DWNTs) via two developed elastic shell theories: Love’s thin cylindrical shell theory and the Cooper–Naghdithick cylindrical shell theory. In studying DWNTs, the van der Waals effect is accounted for and modeled with the twotheories. The elastic thick shell theory, in which the shear and inertia effects are taken into account, is developed firstto investigate the wave propagations of CNTs to provide more accurate wave dispersions for higher modes. The materialproperties of the CNTs that are used in the two shell theories are proposed, and the expression of the inertia moment of thecross area in the thick shell theory is recommended. The dispersion results that are derived via the two theories are com-pared to show the feasibility of those theories in studying CNTs. Radius-dependent wave propagation results in SWNTsand DWNTs are also studied via the two theories.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Carbon nanotubes; Bending rigidity; Wave propagation; Continuum models; Phase velocity; Thin shell bending theory;Cooper–Naghdi shell theory; Phonon dispersion of carbon nanotubes

1. Introduction

Eversince carbon nanotubes (CNTs) were discovered by Iijima [1] they have attracted worldwide attention[2,3]. CNTs have a wide range of applications, including in atomic-force microscopes, field emitters, friction-less nano-actuators, nano-motors, nano-bearings, nano-springs, nano-fillers for composite materials, andnanoscale electronic devices [4]. In numerical and analytical investigations of CNTs, continuum models, suchas elastic shell and beam models, have been applied because molecular simulations are expensive and almostinapplicable for large-scale problems [5–9]. Ru [10,11] used the elastic shell model to conduct buckling anal-yses of CNTs. Yakobson et al. [12] noticed the unique features of fullerenes and developed a continuum shellmodel to study different instability patterns of CNTs under different compressive loads. Krishman et al. [13]

0020-7225/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijengsci.2007.04.001

* Corresponding author.E-mail address: [email protected] (K.M. Liew).

mailto:[email protected]

228 K.M. Liew, Q. Wang / International Journal of Engineering Science 45 (2007) 227–241

estimated Young’s modulus of SWNTs using classical elastic beam theory. Wang [14] measured the effectivein-plane stiffness and bending rigidity of armchair and zigzag CNTs through the analysis of a representativevolume element of the graphene layer via continuous elastic models. Yoon et al. [15] studied resonant frequen-cies and the associated vibration modes of an individual multi-walled carbon nanotube (MWNT) that wasembedded in an elastic medium. Zhang et al. [16,17] applied local and non-local elastic shell theories to studythe elastic buckling of a long DWNT that was embedded in an elastic medium using the energy method, andthe elastic buckling of MWNTs under uniform external radial pressure. To summarize, elastic thin shell theoryhas been proposed and applied to efficiently investigate the mechanical behavior of CNTs. Wang et al. [18]used the Tersoff–Brenner potential and ab initio calculations to show that the elasticity of the thin shell theoryevolves from isotropic to square symmetric with the decreasing tube diameter, which leads to significant diam-eter dependence for all of the elastic moduli and the representative wall thickness. He et al. [19–22] investigatedthe buckling behavior of CNTs using elastic shell model having the van der Waals effect modeled by the Len-nard–Jones potential.

In the analysis of shell structures, Love [23] put forth the simplest membrane shell model, in which thetransverse forces and bending and twisting moments are negligible. This model is appropriate for thin shellstructures in which only normal and shear forces that act at the mid-surface of the shell are considered. Someslightly modified theories based on this simple model were presented by Vlaso [24] and Sanders [25]. Love[23,26] and Reissner [27] proposed the first approximation of thin shell theory based on two key postulations:(a) that the transverse normal stress is negligible; and (b) that normals to the reference surface of the shellremain normal to it and do not change in length during deformation. The models that have been developedbased on the above postulations are usually referred to collectively as Love’s bending theory or lower-ordershell theory. Many studies have been carried out on shell structures based on Love’s bending theory. Forcylindrical shells of moderate thickness, Mirsky and Herrmann [28] incorporated the shear effects in boththe axial and circumferential directions, and the rotary-inertia effects, in studying axially symmetric waves.Cooper and Naghdi [29] presented a theory that included both shear and rotary-inertia effects for the non-axi-ally symmetric motion of shell structures, which is referred as thick shell theory. To show the applicability ofthin shell theory in predicting wave solutions for higher modes, Mirsky [30] investigated the approximate the-ory for the vibration of orthotropic, thick, cylindrical shells in which the effect of transverse normal stress wasretained. The results showed that the frequencies for the fourth and fifth modes at infinite wavelength are veryclose or nearly equal for thin shells, but can be differed by approximately 20% for thick shells. Greenspon [31]conducted another study of wave propagation in cylindrical shells using different shell theories, and showedthat the Cooper–Naghdi theory [29] – which states that only the transverse shear effect and rotary inertiashould be considered – can be applied to all shells (with an error within 20%) where the ratio of radius andthickness, and wave mode, are both less than 5. Used in this way, shell theory is sufficient when dealing withwave propagation in thick cylindrical shells.

There is a growing interest in the terahertz physics of nanoscale materials and devices [32–35], which is open-ing a new topic on phonon dispersion of CNTs, especially on the terahertz frequency range. Some studies onthis topic have been reported in the literature, but mainly for SWNTs [36–39]. Yoon et al. [39] studied the wavepropagation of DWNTs and multiple walled carbon nanotubes (MWNTs), in which they modeled the van derWaals force via their multiple-beam theory. Yoon et al. [40] studied the rotary inertia and shear deformation oftransverse wave propagation in CNTs with elastic beam theory, and listed fundamental characteristic waveresults for MWNTs. Wang and Varadan [41] developed classical thin beam theory and Timoshenko beam the-ory to study wave propagation in both SWNTs and DWNTs. Chakraborty et al. [42] developed a spectrallyformulated finite element to study elastic waves in CNTs, where the frequency content of the exciting signalwas at the terahertz level. Wang et al. [43] employed a multi-elastic shell theory to symmetrically study the freevibration of MWNTs. Their theory was based on Flugge [44] equations of elastic shells. As can be seen from thisreview, only elastic thin shell theory and thin and thick beam theories have been employed in the analysis ofwave propagation in CNTs. The applicability of thin shell theory has not been investigated, and neither hasthe feasibility of thin shell theory in the analysis of the mechanical behavior of CNTs.

This study will fill that gap and develop elastic thick shell theory in the analysis of the wave propagation ofCNTs. The derivation of the CNT material properties that are used in thick shell theory is proposed, with theaim of using numerical simulations to investigate the applicability of the two shell theories to CNT wave

K.M. Liew, Q. Wang / International Journal of Engineering Science 45 (2007) 227–241 229

dispersions. The feasibility of thin shell theory in particular is discussed based on a comparison of wave solu-tions via Love’s thin shell theory and the Cooper–Naghdi thick shell theory for SWNTs and DWNTs. Thickshell theory is applied to study the wave propagation of DWNTs by using a developed double shell theory thataccounts for van der Waals force. Radius-dependent wave propagation results for SWNTs and DWNTs arealso studied via the two theories.

2. Wave propagation in CNTs via Love’s thin shell theory

Elastic thin shell theory is first employed to study wave propagation of SWNTs. The theory operates underLove’s shear-rigidity assumption, which shows that a certain plane that is perpendicular to the mid-plane willremain so after deformation. The coordinate x is the direction along the shell, h is the direction of the polarangle, and r is the radial direction. The corresponding strains in the current coordinate system are expressed as[23]

ex ¼ouox� z

o2wox2

; ð1Þ

eh ¼1

Rwþ ov

oh

� �� z

o2w

R2oh2; ð2Þ

cxh ¼ovoxþ ou

Roh� 2z

o2wRoxoh

; ð3Þ

where ex, eh, and cxh are the normal and shear strains, and u; v;w are displacement variables in x; h; and rdirections.

Incorporating equilibrium state analysis and constitutive relations into the cylindrical shell model [45,46],wave propagation equations for SWNTs via Love’s thin shell theory are given as follows in terms of displace-ment variables:

Eh1� t2

o2u

ox2þ Ehtð1� t2ÞR

owoxþ o

2voxoh

� �þ Eh

2ð1þ tÞo

2vRohox

þ o2u

R2oh2

� �¼ qh

o2u

ot2; ð4Þ

Eh

ð1� t2ÞR2

owohþ o2v

oh2

� �þ Ehtð1� t2ÞR

o2uoxoh

þ Eh2ð1þ tÞ

o2vox2þ o2u

Roxoh

� �

þ Dð1� tÞ2R2

o3wox2oh

þ D

R4

o3w

oh3þ Dt

R2

o3wohox2

¼ qho2vot2

; ð5Þ

Do4wox4þ Dð1þ tÞ

R2

o4w

ox2oh2þ D

R4

o4w

oh4þ Eh

ð1� t2ÞR2wþ ov

oh


ouoxþ qh

o2wot2¼ 0; ð6Þ

where E is Young’s modulus, h is the thickness, R is the radius of the mid-plane, q is the mass density, t isPoisson’s ratio, and D ¼ Eh3

12ð1�t2Þ is the bending rigidity of CNTs in shell theories.A key issue in applying continuum shell theories to the analysis of CNTs is the determination of material

properties, such as bending rigidity. As the atomic structure of CNTs is of discrete nature, unlike what hasbeen proposed for beam structures, the derivation of all material properties in beam models from continuousstructural mechanics cannot be directly applied to the study of CNTs. Ru [10] proposed that the effectivebending rigidity of CNTs in elastic thin shell theory should be regarded as an independent material parameterthat is not related to the equilibrium thickness by the elastic bending stiffness formula. Actually, in all of thelower-order models for beams, plates, and shells, the common assumption used is the ‘‘straight normal pos-tulate,’’ which states that the longitudinal deformation at any point in the flexural direction is proportional tothe distance between that point and the mid-plane of the mid-surface of the structure. However, the atomiclayer in CNTs cannot be divided into different layers, and the flexural strain or deformation is actually con-centrated on a narrow region around the centerline of the atomic layer rather than distributed linearly over thethickness direction [10]. Therefore, the material parameters of CNTs are proposed by [43] in-plane stiffnessEh = 360 J/m2, mass density qh = (2.27 g/cm3) · 0.34 nm, Poisson’s ratio t = 0.2, and bending rigidityD = 2 eV. A new value of bending rigidity, D = 2 eV, rather than D = 0.85 eV [12], was recently proposed,


and the results of phonon dispersion relations for a CNT were compared with those based on local elasticitytheory for a cylindrical medium [47].

Wave propagation is studied by considering the following solutions:

uðx; h; tÞ ¼ Ueiðnxþmh�xtÞ; ð7Þvðx; h; tÞ ¼ V eiðnxþmh�xtÞ; ð8Þwðx; h; tÞ ¼ W eiðnxþmh�xtÞ; ð9Þ

where n and m are the wavenumbers in the longitudinal and radial directions of the CNT, respectively, x is thefrequency, and U ; V ; and W are the magnitudes of the wave propagation.

Substituting Eqs. (7)–(9) into Eqs. (4)–(6) yields a set of homogeneous equations,

a11 a12 a13

a21 a22 a23

a31 a32 a33

264

375

U

V

W

8><>:

9>=>; ¼ f0g; ð10Þ

where the elements in the matrix aij ði; j ¼ 1; . . . ; 3Þ are given in Appendix A.The relationship between wave number n and wave phase velocity c ¼ x

n can thus be determined by search-ing the condition for non-trivial solutions of U, V, and W, i.e.,

det

a11 a12 a13

a21 a22 a23

a31 a32 a33

264

375 ¼ 0; ð11Þ

where det denotes the determinant of a matrix.We now discuss the application of thin shell theory to the wave propagation of DWNTs. In the analysis of a

DWNT, the van der Waals interaction effect at the interface of the inner and outer tubes has to be taken intoaccount in modeling. In linear analysis, the van der Waals interaction pressure at any point between two adja-cent tubes has been modeled by a linear function of the deflection jump at that point [48]. Wang et al. [49]characterized the interaction between atoms of two constituent tubes by the van der Waals interaction poten-tial. The van der Waals effect was estimated using Lennard–Jones pair potential. In particular, the followingestimate for the increase of the intertube interaction energy due to bending was proposed:

U ¼ aCd2

Z L

0

ðw1ðxÞ � w2ðxÞÞ2 dx; ð12Þ

where the energy constant C = 1.0 · 1020 J/m4, d is the average diameter of the two tubes, and a is a non-dimensional parameter as a function d, or curvature [18].

In terms of the foregoing model of the van der Waals interaction, the governing equations for the inner andouter tubes can be written as follows via thin shell theory:

Eh1� t2

o2ui

ox2þ Ehtð1� t2ÞRi

owi

oxþ o

2vi

oxoh

� �þ Eh

2ð1þ tÞo

2vi

Riohoxþ o

2ui

R2i oh2

!¼ qh

o2ui

ot2; ð13a–bÞ

Eh

ð1� t2ÞR2i

owi

ohþ o2vi

oh2

� �þ Ehtð1� t2ÞRi

o2ui

oxohþ Eh

2ð1þ tÞo2vi

ox2þ o2ui

R1oxoh

� �

þ Dð1� tÞ2R2

i

o3wi

ox2ohþ D

R4i

o3wi

oh3þ Dt

R2i

o3wi

ohox2¼ qh

o2vi

ot2; ð14a–bÞ

Do4wi

ox4þ D

R2i

o4wi

ox2oh2þ D

R4i

o4wi

oh4þ Dt

R2i

o4wi

ox2oh2

þ Eh

ð1� t2ÞR2i

wi þovi

oh


oui

oxþ qh

o2wi

ot2þ aCdðwj � wiÞ ¼ 0; ð15a–bÞ


where i; j ¼ 1; 2; i 6¼ j. ui; vi;wi;Ri are the displacement variables in the x; h; and r directions and the radii ofthe inner and outer tubes, respectively, and d = R1 + R2.

Similarly, wave propagation in the DWNT can be studied by considering the solutions

ujðx; h; tÞ ¼ U jeiðnxþmh�xtÞ; ð16a–bÞ

vjðx; h; tÞ ¼ V jeiðnxþmh�xtÞ; ð17a–bÞ

wjðx; h; tÞ ¼ W jeiðnxþmh�xtÞ; ð18a–bÞ

where subscript j ¼ 1; 2 stands for variables at the inner and outer tubes.Substituting Eqs. (16)–(19) into Eqs. (13)–(15) yields the following set of homogeneous equations:

�a11 �a12 �a13 �a14 �a15 �a16

�a21 �a22 �a23 �a24 �a25 �a26

�a31 �a32 �a33 �a34 �a35 �a36

�a41 �a42 �a43 �a44 �a45 �a46

�a51 �a52 �a53 �a54 �a55 �a56

�a61 �a62 �a63 �a64 �a65 �a66

2666666664

3777777775

U 1

V 1

W 1

U 2

V 2

W 2

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;¼ f0g; ð19Þ

where the elements in the matrix �aij ði; j ¼ 1; . . . ; 6Þ are given in Appendix A.Therefore, a wave solution for the DWNT can be derived from an eigenvalue problem to achieve non-triv-

ial solutions of Ui; V i;W i ði ¼ 1; 2Þ as follows:

det

�a11 �a12 �a13 �a14 �a15 �a16

�a21 �a22 �a23 �a24 �a25 �a26

�a31 �a32 �a33 �a34 �a35 �a36

�a41 �a42 �a43 �a44 �a45 �a46

�a51 �a52 �a53 �a54 �a55 �a56

�a61 �a62 �a63 �a64 �a65 �a66

2666666664

3777777775¼ 0: ð20Þ

3. Wave propagations of CNTs via the Cooper–Naghdi thick shell theory

This section we investigate the wave propagations of SWNTs and DWNTs via the Cooper–Naghdi thickshell theory.

Cooper and Naghdi [29] proposed a shell theory that takes shear and rotary inertia into consideration. Thekinematic relations in the theory, from which the strain expressions can be easily derived, are given as follows:

Uxðx; h; z; tÞ ¼ uðx; h; tÞ þ zbxðx; h; tÞ; ð21ÞU hðx; h; z; tÞ ¼ vðx; h; tÞ þ zbhðx; h; tÞ; ð22ÞUzðx; h; z; tÞ ¼ wðx; h; tÞ; ð23Þ

where bx and bh represent the axial and circumferential shear effects, Ux, Uh, and Uz are the global displace-ment fields, and their corresponding components irrelevant of co-ordinate z are u, v, and w, respectively.

The constitutive relations in thick shell theory are given as [45]

rxx ¼E

1� t2ðexx þ tehhÞ; ð24Þ

rhh ¼E

1� t2ðehh þ texxÞ; ð25Þ

sxh ¼ cxhE

2ð1þ tÞ ; ð26Þ


rxz ¼ jE

2ð1þ tÞ cxz; ð27Þ

rhz ¼ jE

2ð1þ tÞ chz; ð28Þ

where j is the shear coefficient. According to Mirsky [30], the shear coefficients are determined to be p2/12 bycomparing the cut-off frequencies of the thickness shear modes in the axial and circumferential directions tothose of the exact theory. In addition, Mirsky pointed out that the chosen coefficients only slightly affect thenumerical results of wave propagation in the useful range. Even for thick shells, the values of shear coefficientsdo not differ by more than 10% from the true values. In addition, Cooper and Naghdi [29] adopted the coef-ficients 5/6 from a natural sequence of consistent assumptions for the stresses and displacements that areemployed in Reissner’s variational theorem. In this study, we choose j to be 0.8333 throughout oursimulations.

The equilibrium equations of the cylindrical shell can be derived based on the variational theorem of Reiss-ner [27], from which wave propagation equations for SWNTs via the Cooper–Naghdi thick shell theory areobtained as follows:

Eh1� t2

o2uox2þ Ehtð1� t2ÞR

owoxþ o2v

oxoh

� �þ Eh

2ð1þ tÞRo2vohox

þ o2u

Roh2

� �¼ qh

o2uot2þ qI

Ro2bx

ot2; ð29Þ

Eh

ð1� t2ÞR2

owohþ o

2v

oh2


o2u

oxohþ Eh

2ð1þ tÞo

2vox2þ o

2uRoxoh

� �

þ jEh2ð1þ tÞR

owRohþ bh �

vR

� �¼ qh

o2vot2þ qI

Ro2bh

ot2; ð30Þ

� jEh

2ð1þ tÞo

2wox2þ obx

oxþ o

2w

R2oh2þ obh

Roh� ov

R2oh

� �þ Eh

ð1� t2ÞR2wþ ov

oh

� �

þ Ehtð1� t2ÞR

ouox¼ �qh

o2wot2

; ð31Þ

Do

2bx

ox2þ Dt

o2bh

Roxohþ Dð1� tÞ

4

o2bh

Roxohþ Dð1� tÞ

4

o2bx

R2oh2

� jEh

2ð1þ tÞowoxþ bx

� �¼ qI

o2bx

ot2þ o2u

Rot2

� �; ð32Þ

Dð1� tÞ4

o2bh

ox2þ Dð1� tÞ

4

o2bx

R2oxohþ D

R2

o2bh

oh2þ Dt

Ro2bx

oxoh

� jEh

2ð1þ tÞowRohþ bh �

vR

� �¼ qI

o2bh

ot2þ o

2vRot2

� �; ð33Þ

where qI is inertia effect of the cross area in shell structures. As discussed in the previous section, bendingrigidity D should be set as an independent parameter in CNT mechanical analysis, so parameter qI shouldbe defined according to the particular structure of the CNT and should not be calculated from classicalmechanics concepts by assuming that the cross-section of the shell structure is uniformly distributed. Thisparameter is defined hereinafter to not only reveal the unusual atomic structure of CNTs, but also to accom-modate its classical mechanics definition as follows:

qI ¼ Dð1� t2Þ qhEh¼ 2 eV� ð1� 0:2Þ � 2:27 g=cm3 � 0:34 nm

360 J=m2¼ 5:495 kg: ð34Þ

Wave propagation is studied for SWNTs by considering the following solutions:


uðx; h; tÞ ¼ Ueiðnxþmh�xtÞ; ð35Þ

vðx; h; tÞ ¼ V eiðnxþmh�xtÞ; ð36Þ

wðx; h; tÞ ¼ W eiðnxþmh�xtÞ; ð37Þ

bxðx; h; tÞ ¼ Bxeiðnxþmh�xtÞ; ð38Þ

bhðx; h; tÞ ¼ Bheiðnxþmh�xtÞ: ð39Þ

Substituting Eqs. (35)–(39) into Eqs. (29)–(33) yields the following set of homogeneous equations:

bij

� �U

V

W

Bx

Bh

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼ f0g; ð40Þ

where the elements in the above matrix bij ði; j ¼ 1; . . . ; 5Þ are given in Appendix B. Hence, the relationshipbetween wave number n and wave phase velocity c ¼ x

n can be determined by searching the condition fornon-trivial solutions of U, V, W, Bx, and Bh through the solution of an eigenvalue problem as discussed inthe previous section.

To save space, the governing equations for DWNTs via the Cooper–Naghdi shell theory are listed asfollows by adopting the proposed van der Waals model:

Eh1� t2

o2ui

ox2þ Ehtð1� t2ÞRi

owi

oxþ o

2vi

oxoh

� �þ Eh

2ð1þ tÞRi

o2vi

ohoxþ o

2ui

Rioh2

� �¼ qh

o2ui

ot2þ qI

Ri

o2bxi

ot2; ð41a–bÞ

Eh

ð1� t2ÞR2i

owi

ohþ o2vi

oh2


o2ui

oxohþ Eh

2ð1þ tÞo2vi

ox2þ o2ui

Rioxoh

� �

þ jEh2ð1þ tÞRi

owi

Riohþ bhi �

vi

Ri

� �¼ qh

o2vi

ot2þ qI

Ri

o2bhi

ot2; ð42a–bÞ

� jEh

2ð1þ tÞo2wi

ox2þ obxi

oxþ o2wi

R2i oh2þ obhi

Rioh� ovi

R2i oh

!þ Eh

ð1� t2ÞR2i

wi þovi

oh

� �

þ Ehtð1� t2ÞRi

oui

ox¼ �qh

o2wi

ot2� aCd wj � wi

� �; ð43a–bÞ

Do

2bxi

ox2þ Dt

o2bhi

Rioxohþ Dð1� tÞ

4

o2bhi

Rioxohþ Dð1� tÞ

4

o2bxi

R2i oh2

� jEh

2ð1þ tÞowi

oxþ bxi

� �¼ qI

o2bxi

ot2þ o

2ui

Riot2

� �; ð44a–bÞ

Dð1� tÞ4

o2bhi

ox2þ Dð1� tÞ

4

o2bxi

R2i oxoh

þ D

R2i

o2bhi

oh2þ Dt

Ri

o2bxi

oxoh

� jEh

2ð1þ tÞowi

Riohþ bhi �

vi

Ri

� �¼ qI

o2bhi

ot2þ o2vi

Riot2

� �; ð45a–bÞ

where subscripts i; j ¼ 1; 2 and i 5 j. Subscript i ¼ 1; 2 indicates the variables in the inner and outer tubes sep-arately and Ri ði ¼ 1; 2Þ are radii for the inner and outer tubes. Wave propagation solutions for the DWNTare assumed to be in the following forms:


ujðx; h; tÞ ¼ U jeiðnxþmh�xtÞ; ð46a–bÞ

vjðx; h; tÞ ¼ V jeiðnxþmh�xtÞ; ð47a–bÞ

wjðx; h; tÞ ¼ W jeiðnxþmh�xtÞ; ð48a–bÞ

bxjðx; h; tÞ ¼ Bxjeiðnxþmh�xtÞ; ð49a–bÞ

bhjðx; h; tÞ ¼ Bhjeiðnxþmh�xtÞ; ð50a–bÞ

where subscript j ¼ 1; 2 stands for variables at the inner and outer tubes. Similarly, substituting Eqs. (46)–(50)into Eqs. (41)–(45) yields a set of homogeneous equations as

�bij

� �Xh iT ¼ f0g; ð51Þ

where hX i ¼ hU 1; V 1;W 1Bx1;Bh1;U 2; V 2;W 2;Bx2;Bh2i. The elements in the above matrix �bij ði; j ¼ 1; . . . ; 10Þare given in Appendix B. Wave solutions for the DWNT can thus be derived from det½�bij� ¼ 0.

4. Results and discussion

The first derivation from Love’s thin shell theory for SWNTs is the decoupled pure torsional motion (T)from Eq. (5) for axi-symmetric wave motion, if o

oh ¼ 0 is set, which is expressed as

cT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Eh2qhð1þ tÞ

s¼ 13941 m=s: ð52Þ

Another asymptotic phase velocity for the coupled radius–longitudinal (R–L) modes can be obtained as

cðR–LÞasym ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ehqhð1� t2Þ

s¼ 22042 m=s: ð53Þ

We also consider the large wavelength and small wavenumber limits for the axi-symmetric motion via thethin shell theory. As n! 0 is set, the radius-dependent cut-off frequency is derived as

xcut ¼1

R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEh

qhð1� t2Þ

s¼ 22042

RHz: ð54Þ

The cut-off frequency is in the THz range. For example, for a SWNT with R = 2 nm, the cut-off frequencyxcut = 11.02 THz is derived.

The second derivation from Cooper–Naghdi thick shell theory for SWNTs is the two decoupled pure tor-sional motion about v and bh for the axi-symmetric wave motion from Eqs. (30) and (33). Two asymptoticvalues for the torsional coupled modes at small and large wavenumber limits are simply listed as followsfor SWNTs:

cT 1 ¼

Eh2ð1þ tÞ þ

Dð1� tÞ4R2

qhþ qI=R2

0BB@

1CCA

1=2

when n! 0; ð55Þ

cT 2 ¼

Eh2 1þ tð Þ �

Dð1� tÞ4R2

qhD 1� tð Þ

4R2þ Eh

2ð1þ tÞqI

R2

0BB@

1CCA

1=2

when n!1: ð56Þ

In the following numerical simulations, the wave dispersion relations are provided in terms of the non-

dimensional phase velocity, �c ¼ c=ffiffiffiffiffiffiffiffiffiffiffiffiffi

Eh2qhð1þtÞ

q, versus the non-dimensional wave number in the longitudinal

direction of the shell structure, �n ¼ nh=2p, where h = 0.34 nm is the thickness of the carbon graphite sheet.


Fig. 1 compares the non-dimensional phase velocity and the non-dimensional wavenumber via Love’s thinshell theory and the Copper–Naghdi thick shell theory for axi-symmetric wave propagation of a SWNT withR = 2 nm. The two shell theories provide identical solutions for the third mode and virtually alike variationsfor the second mode at larger wavenumbers. However, for the first mode, the thin shell theory provides anover-estimated phase velocity at larger wavenumbers even though both theories present undistinguishableresults at smaller wavenumbers. A further comparison of the two shell theories on wave dispersions for theSWNT at wavenumbers in the circumferential direction m = 1 is given in Fig. 2, from which it can be noticedthat the difference between the phase velocities for the second and third modes based on the two theories isindiscernible at larger wavenumbers, but obvious at very small wavenumbers. For the first mode, differentia-tion on the phase velocities via the two theories can even be envisaged at smaller wavenumbers for this asym-metric wave propagation. This becomes more evident at higher wavenumbers in the circumferential direction,m = 5, as shown in Fig. 3. Nevertheless, there are no differences in the wave solutions for the second and thirdmodes in Fig. 3.

In the wave propagation of DWNTs dual modes are found, as was also the case in a noncoaxial resonancestudy of MWNTs [48]. Fig. 4a shows the phase velocities for a DWNT with r1 = 2 nm for the first two dual

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.2 0.4 0.6 0.8 1.0Non-dimensional wavenumber

No

n-d

imen

sio

nal

ph

ase

velo

city

Love's theory 1stmode

Love's theory 2ndmodeLove's theory 3rdmode

Higher-ordertheory 1st mode

Higher-ordertheory 2nd mode

Higher-ordertheory 3rd mode

Fig. 1. Comparison of the two shell theories for SWNTs at m = 0.

Non-dimensional wavenumber0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

No

n-d

imen

sio

nal

ph

ase

velo

city


Love's theory 2ndmode

Love's theory 3rdmode





0.0

2.0

4.0

6.0

8.0

10.0

12.0


No

n-d

imen

sio

nal

ph

ase

velo

city



Love's theory 3rdmode


Higher-order 2ndmode



0.0

0.5

1.0

1.5

2.0


No

n-d

imen

sio

nal

ph

ase

velo

city

Love's theory1st mode

Love's theorydual 1st mode

Higher-ordertheory 1stmodeHigher-orderdual 1st mode

Fig. 4a. Comparison of the two shell theories for DWNTs for the first two dual modes at m = 0.


modes at m = 0. Our first observation is that phase velocities for the two dual modes converge only at largerwavenumbers based on both shell theories. This indicates that there will be coaxial wave propagation resultsonly at larger wavenumbers in the longitudinal direction for DWNTs because the higher curvature at shorterwavelength, or large wavenumbers, makes the van der Waals effect much stronger, which leads to virtuallycoaxial wave motions. Our second observation is similar to that for Fig. 1 – there is an obvious differencein the phase velocities derived based on the two shell theories only at larger wavenumbers. The phase velocitiesfor the second dual wave modes for DWNTs are shown in Fig. 4b. Only minor differences for the dual modesare found via the thick shell theory at very small wavenumbers in the longitudinal direction. Similar to theobservation for the SWNT, the differences of the phase velocities converge at larger wavenumbers based onthe two shell theories.

Such comparisons of the two dual modes for the DWNT based on the two shell theories are further inves-tigated in Figs. 5a and 5b at wavenumbers in the circumferential direction m = 3. For the first dual modes,there is again a difference between the phase velocities at smaller wavenumbers. In addition, the indispensabledifferences of phase velocities for the second dual modes based on the two shell theories are similar to that in

Non-dimensional wavenumber

No

n-d

imen

sio

nal

ph

ase

velo

city

0.0

1.4

2.8

4.2

5.6

7.0

0.0 0.2 0.4 0.6 0.8 1.0

Love's theory1st mode

Love's theorydual 1st mode


Higher-ordertheory dual 1stmode

Fig. 5a. Comparison of the two shell theories for DWNTs for the first dual modes at m = 3.

0.0

1.5

3.0

4.5

6.0

7.5

9.0

0.0 0.2 0.4 0.6 0.8 1.0


No

n-d

imen

sio

nal

ph

ase

velo

city

Love's theory2nd mode

Love's theorydual 2nd mode

Higher-ordertheory 2ndmode

Higher-ordertheory dual 2ndmode

Fig. 5b. Comparison of the two shell theories for DWNTs for the second dual modes at m = 3.


No

n-d

imen

sio

nal

ph

ase

velo

city

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.2 0.4 0.6 0.8 1.0


Love's theory dual2nd mode


Higher-ordertheory dual 2ndmode

Fig. 4b. Comparison of the two shell theories for DWNTs for the second dual modes at m = 0.



asymmetric wave propagation for SWNTs as shown in Fig. 2. For the second dual modes, there are noobvious differences in the phased velocities based on either of the shell theories. In addition, both shell theoriesprovide similar wave solutions for the dual modes.

The radius-dependent wave dispersions for the first mode at m = 0 are shown in Fig. 6 for three SWNTswith r = 1 nm, r = 2 nm, and r = 5 nm, respectively. We can see from the figure that phase velocities which areinsensitive to SWNT radius are found at larger wavenumbers for both shell theories. Meanwhile, a lowerphase velocity is observed for SWNTs with larger radii. For the wave dispersions, Fig. 7 displays the variationof �c versus �n at m = 0 for the first dual modes. An important conclusion from that can be drawn from thisfigure is that coaxial motions are available for DWNTs with larger radii because there is less differencebetween the phase velocities for the first dual modes for DWNTs with larger radii. This indicates that coaxialmotions subsist for DWNTs with larger radii. Ru [50] studied the degraded axial buckling of MWNTs andfound that coaxial buckling also occurs for CNTs with larger radii.

0.0

0.5

1.0

1.5

2.0


No

n-d

imen

sio

nal

ph

ase

velo

city

Love's theory 1stmode r = 1 nm

Higher-ordertheory 1st mode r = 1 nm





Fig. 6. Radius effect on wave propagation for SWNTs at m = 0.

0.0

0.4

0.8

1.2

1.6

2.0


No

n-d

imen

sio

nal

ph

ase

velo

city Higher-order

theory 1st mode r = 1 nm

Higher-ordertheort dual 1stmode r = 1 nm


Higher-ordertheory dual 1stmode r = 2 nm


Higher-ordertheory dual 1stmode r = 5 nm

Fig. 7. Radius effect on wave propagation for DWNTs at m = 0.


5. Conclusions

This paper has developed both Love’s thin shell theory and the Cooper–Naghdi thick shell theory to inves-tigate the wave propagations of SWNTs and DWNTs. The difference of wave dispersions based on the twotheories has been studied, and the feasibility of the widely used thin shell theory has been investigated. Wecan now draw conclusions based on the numerical simulations provided. For SWNTs, there are obvious dif-ferences of phase velocities for the first mode at higher wavenumbers in the longitudinal direction for axi-sym-metric motions, but only minor differences in velocity for the second mode at very small wavenumbers, and nodifference in the third mode. However, the differences in phase velocity for the first mode become much moreobvious at smaller wavenumbers for asymmetric wave motions. For the first dual modes in the wave propa-gation of DWNTs, the phase velocities for the two dual modes converge only at larger wavenumbers based onboth shell theories, whereas for the second dual modes, they exhibit converged values for all wavenumbersbased on the two shell theories. In addition, velocities that are insensitive to the CNT radius can be foundat larger wavenumbers based on both shell theories for SWNTs, but there are coaxial motions for DWNTswith larger radii.

Acknowledgements

The work described in this paper was supported by the City University of Hong Kong Strategic ResearchGrant [Project No. 7002080]. The second author (QW) is grateful for the support from the Canada ResearchChair (CRC) program from the Canadian Government and the University of Manitoba.

Appendix A

2

a11 ¼ �Eh

1� t2n2 � Eh

2ð1þ tÞm

R2þ qhx2; a12 ¼

Ehtð1� t2Þ

nmRþ Eh

2ð1þ tÞnmR

; a13 ¼Ehtnð1� t2ÞR ;

a21 ¼ a12; a22 ¼ �Eh

1� t2

m2

R2� Eh

2ð1þ tÞ n2 þ qhx2;

a23 ¼ �Eh

ð1� t2ÞR2mþ Dð1� tÞ

2

n2m

R2þ Dm3

R4þ Dtn2m

R2; a31 ¼ �a13; a32 ¼

Eh1� t2

m

R2;

a33 ¼ Dn4 þ Dð1þ tÞn2m2

R2þ Dm4

R4þ Eh

ð1� t2ÞR2� qhx2:

Replace R with R1 in all the following expressions about amn ðm; n ¼ 1; 2; 3Þ:
�aki ¼ aki ði ¼ 1; 2; 3Þ; �akj ¼ 0 ðj ¼ 4; 5; 6Þ; k ¼ 1; 2;
�a31 ¼ a31; �a32 ¼ a32; �a33 ¼ a33 � aCd; �a34 ¼ �a35 ¼ 0; �a36 ¼ aCd:

Replace R with R2 in all the following expressions about amn ðm; n ¼ 1; 2; 3Þ:
�aki ¼ aðk�3Þði�3Þ ði ¼ 4; 5; 6Þ; �akj ¼ 0 ðj ¼ 1; 2; 3Þ; k ¼ 4; 5;
�a64 ¼ a31; �a65 ¼ a32; �a66 ¼ a33 � aCd; �a61 ¼ �a62 ¼ 0; �a63 ¼ �a36:

Appendix B

2

b11 ¼ �Eh

1� t2n2 � Eh

2ð1þ tÞm

R2þ qhx2; b12 ¼

Ehtð1� t2Þ

nmRþ Eh

2 1þ tð ÞnmR

; b13 ¼Ehtnð1� t2ÞR ;

b14 ¼qIx2

R; b15 ¼ 0; b21 ¼ b12; b22 ¼ �

Eh1� t2

m2

R2� Eh

2ð1þ tÞ n2 � jEh

2ð1þ tÞRþ qhx2;


b23 ¼ �Eh

ð1� t2ÞR2m� jEh

2 1þ tð ÞR2; b24 ¼ 0; b25 ¼

qIx2

Rþ jEh

2ð1þ tÞR ; b31 ¼ �b13;

b32 ¼Eh

1� t2

m

R2; b33 ¼

jEh n2 þ m2=R2� �2ð1þ tÞ þ Eh

ð1� t2ÞR2� qhx2; b34 ¼

jEhn2 1þ tð Þ ;

b35 ¼ �jEhm

2ð1þ tÞR ;

b41 ¼ b14; b42 ¼ 0; b43 ¼ �b34; b44 ¼ qIx2 � Dn2 � Dð1� tÞm2

4R2� jEh

2ð1þ tÞ ;

b45 ¼Dtnm

Rþ Dð1� tÞnm

4R; b51 ¼ 0; b52 ¼ b25; b53 ¼ �b35; b54 ¼ b45;

b55 ¼ qIx2 � D

R2n2 � Dð1� tÞn2

4R2� jEh

2ð1þ tÞ :

Replace R with R1 in all the following expressions about bmn ðm; n ¼ 1; 2; 3; 4; 5Þ:
�bki ¼ bki ði ¼ 1;2;3;4;5Þ; �bkj ¼ 0 ðj ¼ 6;7;8;9;10Þ; k ¼ 1;2;�b31 ¼ b31; �b32 ¼ b32; �b33 ¼ b33 � aCd; �b34 ¼ b34; �b35 ¼ b35; �b36 ¼ �b37 ¼ �b39 ¼ �b3;10 ¼ 0; �b38 ¼ aCd:�bki ¼ bki ði ¼ 1;2;3;4;5Þ; �bkj ¼ 0 ðj ¼ 6;7;8;9;10Þ; k ¼ 4;5:
Replace R with R2 in all the following expressions about bmn ðm; n ¼ 1; 2; 3; 4; 5Þ:
�bki ¼ bðk�5Þði�5Þ ði ¼ 6; 7; 8; 9; 10Þ; �bkj ¼ 0 ðj ¼ 1; 2; 3; 4; 5Þ; k ¼ 6; 7;�b81 ¼ 0; �b82 ¼ 0; �b83 ¼ aCd; �b84 ¼ �b85 ¼ 0; �b86 ¼ b31; �b87 ¼ b32;
�b88 ¼ b33 � aCd; �b89 ¼ b34;

�b8;10 ¼ b35

�bki ¼ bðk�5Þði�5Þ ði ¼ 6; 7; 8; 9; 10Þ; �bkj ¼ 0 ðj ¼ 6; 7; 8; 9; 10Þ; k ¼ 9; 10:

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Analysis of wave propagation in carbon nanotubes via elastic shell theories