Flexural waves in multi-walled carbon nanotubes using gradient elasticity beam theory

Computational Materials Science 67 (2013) 188–195

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Flexural waves in multi-walled carbon nanotubes using gradient elasticity beam theory J.X. Wu a, X.F. Li a,b,⇑, W.D. Cao c a

School of Civil Engineering, Central South University, Changsha 410075, China State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China c RIOH Transport Consultants Ltd., Beijing 100191, China b

a r t i c l e

i n f o

Article history: Received 14 June 2012 Received in revised form 7 August 2012 Accepted 10 August 2012 Available online 5 October 2012 Keywords: Carbon nanotube Wave propagation Two-parameter elastic matrix Gradient elasticity

a b s t r a c t The behavior of transverse waves propagating in carbon nanotubes (CNTs) in a free space and in an elastic matrix is investigated. The CNTs are modeled as Timoshenko beams of hybrid gradient elasticity theory and the elastic matrix is modeled as a bi-parameter Pasternak foundation. A governing equation with two scale factors is derived for Timoshenko beams where shear deformation and rotary inertia are taken into account. The dispersion relation of flexural waves in CNTs is given and confirmed by molecular dynamics simulations. A comparison of the phase velocity of single-walled CNTs is made when neglecting shear deformation and/or rotary inertia. The wave speed of acoustic branch is especially focused for multi-walled CNTs and the wave speed is dependent on van der Waals interaction. The effects of the surrounding medium and scale parameters on the velocity of bending waves are discussed, in particular for acoustic mode. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Since the discovery of carbon nanotubes (CNTs) in 1991, great interest has been aroused in theoretical and experimental researches on CNTs. A large number of researches show that CNTs possess many unique properties, in particular mechanical, physical, electrical and thermal properties. Such extraordinary performances make it possible that CNTs are promising candidates for a variety of nanometer-sized devices such as sensors, actuators, etc. An ideal CNT may be understood as a hexagonal network of carbon atoms that are rolled up to make a seamless hollow cylinder. Since the diameter of CNTs falls in the order of nanometer, controlled experiments at such small scale are still of difficulty. Although molecular dynamics simulation is accurate and feasible for a small number of molecules or atoms, it still consumes a large amount of time and remains formidable for large scale systems. Thus, for large-scale systems, continuum mechanics approach is recognized as a simple and effective method for treating nanoscale structures. Due to high aspect ratio, the classical Euler–Bernoulli and Timoshenko theories of beams have been used in CNTs [1,2]. A large amount of experimental evidence shows that the mechanical properties of CNTs have a strong size-dependent effect ⇑ Corresponding author at: School of Civil Engineering, Central South University, Changsha 410075, China. Tel.: +86 731 8816 7070; fax: +86 731 8557 1736. E-mail address: xfl[email protected] (X.F. Li). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.08.035

[3,4]. The size effect cannot be described by the classical elasticity theory since the classical theory is free of scale parameters. To account for the size effect, the nonlocal elasticity theory formulated by Eringen [5] has been widely employed to characterize the size-dependent effect of materials and structures at small scale. Along this line, many studies including bending, vibration, and stability analysis related to CNTs have been conducted by employing the nonlocal Euler–Bernoulli beam model [6–9] and the nonlocal Timoshenko beam model [10–12]. Due to CNTs’ extraordinary bending stiffness, they are often applied as reinforced fiber embedded in composites. Under such circumstances, the matrix surrounding the CNTs can be taken as an elastic medium. Usually, the surrounding medium affects wave characteristics of the CNTs. Natsuki et al. [13] analyzed wave propagation of CNTs embedded in an elastic medium using an elastic shell model. Dong and Wang [14] also employed a shell model to investigate wave characteristics of CNTs in an elastic matrix. Li et al. [15] studied transverse waves traveling in CNTs surrounded by an elastic medium using the nonlocal Timoshenko beam model. Shen et al. [16] formulated a generalized nonlocal beam theory to deal with flexural waves propagating in CNTs embedded in an elastic matrix. In the above-mentioned researches, the surrounding elastic matrix is assumed to be Winkler-type elastic foundation. Mustapha and Zhong [17] tackled free vibration of an axially loaded CNT in a bi-parameter elastic matrix. Wu et al. [18] analyzed the characteristics of bending waves running in CNTs in

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a bi-parameter elastic matrix and found that the scale parameter and elastic medium stiffness have different influences on wave speed depending on the frequency range. In this paper, the behavior of flexural waves propagating in CNTs in a free space and surrounded with an elastic medium is studied. The CNTs are modeled as Timoshenko beams using hybrid strain gradient elasticity theory, whereas the surrounding medium is modeled as a bi-parameter elastic medium. For single-walled and multi-walled CNTs, the phase velocity of acoustic branch is especially determined and the effects of two stiffness parameters of the surrounding elastic medium, scale factors, shear deformation and rotary inertia on transverse waves are discussed.

!

2

1  l1

!

2 @2 @/ 2 @ M ¼ EI 1  l ; 2 @x2 @x2 @x

ð7Þ

where I is the moment of inertia of cross-sectional area A, and M is the bending moment,

I¼

Z

z2 dA; M ¼

A

Z

zrxx dA:

ð8Þ

A

Performing a straightforward integration for both sides of Eq. (6) yields 2

1  l1

! !  2 @2 @w 2 @ ; A 1  l Q ¼ G / þ s 2 @x @x2 @x2

ð9Þ

where Q is the shearing force, 2. Gradient elasticity beam model

Q¼

Z

rxz dA:

ð10Þ

A

Here we employ a more general version of elasticity theory incorporating both stress and strain gradients. The constitutive equation of this theory reads [19]

    2 2 1  l1 O2 rij ¼ 1  l2 O2 ðkdij ekk þ 2leij Þ;

ð1Þ

where rij and eij denote the stress and strain tensors, k and l are the Lamé constants, l1 = e1a and l2 = e2a are scale coefficients, where e1 and e2 are nondimensional material constants that can be determined by experiment or molecular dynamics simulation, and O2 is the Laplacian operator. Note that the strain gradient elasticity theory is recovered from l1 = 0, and the nonlocal elasticity theory is recovered from l2 = 0. So they are two special cases of the above hybrid gradient elasticity theory. Recently, the elasticity theory including stress gradient and strain gradient have been extended to deal with the size effect of CNTs by some researchers such as [20–23]. Here, we apply the above relation to analyze the behavior of flexural waves in CNTs where Timoshenko beam assumption is adopted. Now, a theoretical derivation of governing equations is carried out for one-dimensional (1D) Timoshenko beams. Since 1D nanostructures are concerned, it is reasonable to set ryy = rzz = ryz = rxy = 0, and the constitutive equations reduce to

1

2 l1

@2 @x2

2 2 @ 1  l1 2 @x

!

rxx ¼ !

rxz ¼

! @2 Eexx ; 1 @x2 ! 2 2 @ 1  l2 2 Gcxz ; @x 2 l2

ð2Þ

ð3Þ

where E and G represent Young’s modulus and shear modulus, respectively. Here, the longitudinal coordinate is denoted as x, and the coordinate perpendicular to the neutral axis of the undeformed beam as z. Denote the transverse deflection and rotation angle of the cross section as w(x, t) and /(x, t), respectively. For Timoshenko beams, the normal strain exx and shear strain cxz can be expressed as

exx ¼ z

@/ @w ;c ¼ / þ : @x xz @x

ð4Þ

2

2

1  l1

@2 @x2 @2 @x2

!

! @2 @/ z ; @x @x2 !  2 @w 2 @ /þ ¼ G 1  l2 2 : @x @x

rxx ¼ E 1  l22

ð5Þ

rxz

ð6Þ

!

@Q @2w þ q ¼ qA 2 ; @x @t @M @2/  Q ¼ qI 2 ; @x @t

Multiplying both sides of Eq. (5) by z and then performing integration over any cross section, we obtain

ð11Þ ð12Þ

where q is distributed loads on a lateral surface, and q is the mass density of the CNTs. From Eqs. (11) and (12), we have

@2M @3/ @2w ¼ qI 2 þ qA 2  q; 2 @x @t @x @t @2Q @3w @q ¼ qA 2  : @x2 @t @x @x

ð13Þ ð14Þ

Substituting Eqs. (13) and (14) into Eqs. (7) and (9) results in 2

! ! 2 @3/ @2w @/ 2 @ q A  q þ EI 1  l ; þ 2 @x2 @x @t 2 @x @t 2 ! !  2 @3w @q @w 2 @ þ Gs A 1  l2 2 /þ : qA 2  @x @x @t @x @x

M ¼ l1 qI 2

Q ¼ l1

ð15Þ ð16Þ

Insertion of M and Q into Eqs. (11) and (12), we obtain a set of governing equations of Timoshenko beams

! ! !  2 2 @ @w @2w 2 @ 2 @ /þ 1  l2 2 ¼ 1  l1 2 Gs A qA 2  q ; ð17Þ @x @x @x @x @t ! !  2 2 2 @ / @w 2 @ 2 @ EI 1  l2 2 /þ ¼ Gs A 1  l2 2 @x @x @x2 @x ! 2 2 @ / 2 @ þ qI 1  l1 2 : ð18Þ @x @t 2 It is worth pointing out that the above equations are coupled with respect to / and w. To further simplify them, we introduce a new function F to satisfy the following equation

!2 2 @ 4 F q2 @4F 2 @ 1 þ 1  l 1 @x4 Gs E @x2 @t4 ! ! 2 2 qA @2F 2 @ 2 @ 1  l2 2 1  l1 2 þ EI @x @x @t2 ! !   2 2 q q @4F 2 @ 2 @ 1  l2 2 1  l1 2  þ E Gs @x @x @x2 @t2 ! 2 1 2 @ 1  l1 2 q: ¼ EI @x 2 l2

Eqs. (2) and (3) can be written as

1  l1

and Gs is the effective shear modulus, Gs = jG, j being a shear correction factor depending on the shape of the cross-section. The fundamental equations of motion for 1D nanostructures read

@2 @x2

!2

ð19Þ

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J.X. Wu et al. / Computational Materials Science 67 (2013) 188–195

Thus if the transverse deflection w and rotation / are chosen to have the following form in terms of F:

! @ 2 @F /¼ 1 ; ð20Þ @x2 @x ! ! ! 2 2 2 EI @2F qI @2F 2 @ 2 @ 2 @ þ ; w ¼ 1  l2 2 F  1  l2 2 1  l 1 Gs A @x @x @x2 Gs A @x2 @t2 2 l2

ð21Þ

C–C bond length a in (27) for the purpose of maintaining the dimension of b0 and b1 to be same. For transverse waves propagating in CNTs, a proper solution can be expressed as

Fðx; tÞ ¼ FeiðkxxtÞ ;

ð28Þ

pffiffiffiffiffiffiffi where i ¼ 1; F is the amplitude, k is the wavenumber, and x is the circular frequency. Substituting Eqs. (27) and (28) into the governing Eq. (19), and noting that

we find that Eqs. (17) and (18) are automatically satisfied. In other words, we have derived a single governing equation to describe dynamic behavior of Timoshenko beams. If making the shear modulus be high enough, meaning Gs ? 1, we find that

  2 2 2 q ¼ ðb0 þ b1 a2 k Þ 1 þ l2 k FeiðkxxtÞ ;

w¼

where

! 2 2 @ 1  l2 2 F; @x

ð22Þ

in place of (21) and the governing equation for Rayleigh beams can be derived from Eq. (19), namely,

! ! " # 2 @2 @4w @ 2 qA q @2w 2 @ w 1 þ 1  l1 2 @x2 @x4 @x @t 2 EI E @x2 ! 2 1 2 @ ¼ 1  l1 2 q: EI @x 2 l2

ð23Þ

If one further neglects the effect of rotary inertia, meaning q I = 0, an Euler–Bernoulli beam is recovered [16,21,23]. In addition, if neglecting rotary inertia, that is qI = 0, while shear deformation is taken into consideration, the corresponding beams are sometime called shear beams. In this case, Eqs. (17) and (18) become

!

 @w ¼ /þ @x

!

!

2 2 @ @2w 2 @ 2 @ qA 2  q ; ð24Þ 1  l2 2 1  l1 2 @x @x @x @t ! !  2 2 2 @ / @w 2 @ 2 @ /þ EI 1  l2 2 : ð25Þ ¼ Gs A 1  l2 2 @x @x @x2 @x

Gs A

1

@2 @x2

!2

2

þ 1  l1

@4F @x4 !

@2 @x2

2

1  l2

q2 

Thus by solving the quadratic Eq. (29) in x2, we obtain dispersion relation as follows

x1;2 ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u tb1  b1  b0 b2 b0

:

ð30Þ

Transverse waves are seen to have two branches. The lower velocity is acoustic mode, and higher velocity is optical mode [25]. With the above frequencies, recalling the phase velocity c = x/k, one immediately gets two phase velocities c1 and c2 in a SWCNT. From the above, the phase speeds for several special cases are derived as follows:

ð31Þ

ð26Þ

3. Flexural waves of SWCNTs In general, in order to reinforce the strength of composites, CNTs are commonly embedded in an elastic medium, and the surrounding elastic medium has a strong effect on mechanical behaviors of CNTs such as stability and dynamic behavior. To analyze transverse wave characteristics of CNTs surrounded with an elastic medium, Pasternak-type elastic foundation model is employed to simulate the interaction of CNTs with the surrounding elastic medium. Pasternak-type foundation model can be stated as

@2w q ¼ b0 w þ b1 a ; @x2

2 2 2 1 þ l1 k Gs E       1 qA q q 2  2 2 2 2 k 1 þ l1 k þ 1 þ l2 k ; b1 ¼ þ 2 EI E Gs  2    1 2 2 4 2 2 2 2 2 1 þ l1 k 1 þ l2 k : b0 þ b1 a2 k b 2 ¼ 1 þ l2 k k þ EI b0 ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 2 u b þ b a2 k2 EIk 1 þ l2 k 1 : c¼t 0 þ 2 2 2 2 2 ðA þ Ik Þqk ðA þ Ik Þq 1 þ l1 k

In the absence of the scale parameters, the governing equation for classical shear beams [24] is reduced from the above.

2

ð29Þ

The phase speed of a hybrid gradient Rayleigh beam can be directly obtained from Eq. (29) only if setting Gs ? 1. Under such a circumstance, b0 vanishes and the phase speed reduces to

! " # @ 2 @ 2 qA q @2F F  @x2 @t2 EI Gs @x2

! 2 1 2 @ 1  l1 2 q: ¼ EI @x

b0 x4  2b1 x2 þ b2 ¼ 0;

3.1. Hybrid Rayleigh beam theory (HRBT)

Thus, Eq. (19) simplifies to 2 l2

we obtain an algebraic equation

ð27Þ

where b0, b1 are, respectively, spring stiffnesses describing the normal and tangential constraint. Here, we have introduced the

Since the wave speed c depends on the wavenumber, transverse waves are still dispersive. Moreover, the optical mode branch disappears owing to neglect of rotary inertia. In other words, waves with higher frequencies are not taken into account, and only acoustic branch is retained. 3.2. Hybrid shear beam theory (HSBT) In this case, contrary to Rayleigh beams, shear deformation is taken into account. Rotary inertia, however, is removed. By making use of Eqs. (26) and (29), the wave speed in this case can be derived below

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 2 u Gs ðb0 þ b1 a2 k2 Þ Gs EIk 1 þ l2 k : c¼t þ 2 2 2 qðGs A þ EIk Þ 1 þ l21 k2 ðGs A þ EIk Þqk

ð32Þ

Clearly, in this case there is only flexural waves of acoustic mode and the optical mode also disappears. However, it will be viewed that the HSBT is closer to the HTBT than the HRBT.

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J.X. Wu et al. / Computational Materials Science 67 (2013) 188–195

3.3. Nonlocal Timoshenko beam theory (NTBT) The phase speed of transverse waves based on the nonlocal Timoshenko beam theory can be reduced from Eq. (29) only if setting l2 = 0, i.e.

c1;2

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u h   i3 u ! u !2 4 2 2 2 u 2 u 4Gs EIk þ b0 þ b1 a k 1 þ l1 k 7 u Gs A Gs þ E 6 Gs A Gs þ E 1 u 7: 6 t ¼u  þ þ  5 2 2 4 2 2 4 t 2 q q qIk qIk q Ik 2 1 þ l1 k

As an example, considering a SWCNT in a free space, using the nonlocal Timoshenko beam theory [15,26], the wave speed is found to be

c1;2

For classical Rayleigh beams and shear beams, the phase speeds can be determined from the above as two special cases. In particular, the well-known phase speed for Euler–Bernoulli beams can pffiffiffiffiffiffiffiffiffiffiffiffiffi be further recovered as c ¼ k EI=qA.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u !2 u u u u 1 Gs A Gs þ E t Gs A Gs þ E 4Gs E7 6 ¼u  þ  2 5: 4 2þ t  2 2 2 q q q q Ik q Ik 2 1 þ l1 k ð34Þ

If further neglecting the effect of shear deformation, the phase speed of Rayleigh beams [18]

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 u b þ b1 a2 k EIk   ; c¼t 0 þ 2 2 2 2 2 ðA þ Ik Þqk 1 þ l1 k ðqA þ qIk Þ

ð35Þ

can be recovered. When the surrounding medium is absent, the wave speed reduces to the one given in [7]. 3.4. Strain gradient Timoshenko beam theory (SGTBT) In Eq. (29) if the scale factor l1 is set to 0, one finds that the phase speed of transverse waves based on the gradient beam theory can be obtained. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 3ffi " # u !2 u 2 u1 þ l2 k2 Gs A Gs þ E u 4Gs 4 4 b0 þ b1 a2 k 5 u Gs A Gs þ E 2 u  : t c1;2 ¼t þ þ  EIk þ  2 2 2 q q qIk2 qIk2 q2 Ik4 1þl k 2

ð33Þ

To show the effects of the scale coefficients l1 and l2 on the wave velocity, the phase velocity of transverse waves propagating in a (5, 5) armchair SWCNT in a free space is displayed in Fig. 1. In numerical calculation, we take the mass density q = 2.237 g/cm3, Young’s modulus E = 0.39 TPa, Poisson’s ratio v = 0.28, shear modulus G = 0.5E/(1 + v) and thickness t = 0.342 nm [27]. For a CNT with hollow circular cross section, the shear correction factor j = 0.8 [28]. From Fig. 1, transverse waves have two branches, acoustic and optical branches. For small wavenumbers, the phase speeds are insensitive to the scale coefficients regardless of acoustic and optical branches, whereas for large wavenumbers, the phase speeds are very sensitive for both acoustic and optical branches. This indicates that the scale parameters l1 = e1a and l2 = e2a play a crucial role only for large wavenumbers and have little effect on phase speeds for small wavenumbers. This can be explained as follows. The size effects are apparent only for high frequency of the excitation. According to the fact that elastic waves are understood as the long-wave limits of lattice dynamics, the classical wave speed may be taken as an upper bound. For this purpose, it is reasonable to require e2 < e1. On the other hand, of much importance is the acoustic mode since it is easily excited. Based on the above two points, in what follows more attention is paid to the phase speed of acoustic mode in the case of e2 < e1 unless otherwise stated. The effects of the scale coefficients e1 and e2 on the phase speed of acoustic mode are examined in Figs. 2 and 3, respectively. It is observed that the wave speed curves are indistinguishable for very small wavenumbers. With wavenumbers rising, the wave speed

ð36Þ

In particular, in a free space, we have wave speed

ð37Þ 3.5. Classical Timoshenko beam theory (CTBT) In the absence of the surrounding elastic medium and letting l1 = l2 = 0, Eq. (29) becomes

q2 Gs E

x4 



qA EI

 þ

q E

þ

q Gs

  2 4 k x2 þ k ¼ 0;

c1;2

15

e =0.9 1

TO

TA e =0.36 2

e =0.9 2

10

e =1.25 2

5

ð38Þ

and hence, the phase speed of the classical Timoshenko beams is

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi3 u 2 !2 u u u1 6 Gs A Gs E u t Gs A þ Gs þ E  4Gs E7: ¼u 5 t2 4 2 þ q þ q  q2 qIk qIk2 q q

20

Phase Speed , c (km/s)

c1;2

25

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u !2 u u u1 þ l2 k2 6 Gs A Gs þ E u G A G þ E 4G E s s s 7 2 ¼u t þ  2 5: 4 2þ t 2 q q q qIk qIk2

0 −2 10

−1

10

0

10

1

10

10

2

Nondimensional Wavenumber , ka

ð39Þ

Fig. 1. The effects of gradient coefficients l1 and l2 on phase speed of transverse waves propagating in a (5, 5) armchair SWCNT without surrounding elastic medium.

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J.X. Wu et al. / Computational Materials Science 67 (2013) 188–195

6

6 e =0.18

e1=0.9

5

e =0.36

2

e =1.5

2

Phase Speed , c (km/s)

1

Phase Speed , c (km/s)

1

e2=0.54

e =2

4

3

2

4

3

2

1

1

0 −2 10

e =0.9

e2=0.36

5

1

−1

10

0

10

1

10

0 −2 10

2

10

10

Nondimensional Wavenumber , ka

curves for different values of e1 and e2 are remarkably different. In particular, in Fig. 2 an increase in e1 gives rise to a decrease of the corresponding wave speed and the position of wavenumber at which the wave speed reaches a peak value is shifted smaller. By contrast, in Fig. 3 an increase in e2 gives rise to an increase of the corresponding wave speed and the position of wavenumber at which the wave speed reaches a peak value seems not to be altered. Fig. 4 gives a comparison of the phase velocity based on several representative hybrid gradient beam models for a (5, 5) armchair SWCNT. If using hybrid Timoshenko beam theory (HTBT), two branches of flexural waves are viewed, which are labeled with TA and TO for acoustic mode and optical mode, respectively. However, only one wave speed is given if the hybrid Rayleigh, shear, and Euler–Bernoulli beam theories are used, denoted as HRBT, HSBT, and HEBT, respectively. For comparison, we also display the results using the molecular dynamics simulation [27]. In our model, two scale coefficients l1(=e1a) and l2(=e2a) are required to be determined. To this end, e1 and e2 are chosen to fit the results of molecular dynamic simulation. Then, l1 = 0.9a and l2 = 0.36a are identified here, a being the C–C bond length. From Fig. 4, it is found that all of the phase speeds are very close for acoustic mode for small wavenumbers, and they agree with molecular dynamics simulations well. However, with the wavenumber rising, the phase speeds given from the HEBT and HRBT are greatly deviated from the results of molecular dynamics simulation. On contrary, the phase speeds given from HSBT and HTBT for acoustic mode are still in good agreement with molecular dynamics simulations when the wave number becomes larger. This suggests that HEBT and HRBT are accurate only for lower wavenumbers or frequencies and they may overestimate wave speed for higher wavenumbers or frequencies. Nonetheless, it is noted that for large wavenumbers or frequencies, the wave speed using HRBT tends to the wave speed of HTBT for optical mode. On the other hand, HSBT and HTBT for acoustic mode are more adequate in a wider wavenumber range. For any wavenumber, the wave speed of HSBT is very close to that of HTBT for acoustic mode. Moreover, the governing equation or HSBT is simpler than that for the HTBT. Due to this reason, for simplicity, in the following we may approximate HTBT for acoustic mode by HSBT. This also infers that shear deformation plays a more dominant role for the lower wave speed of acoustic mode, where rotary inertia plays a more dominant role for the higher wave speed of optical mode. Their effects are completely different.

10

0

10

1

10

2

Nondimensional Wavenumber , ka Fig. 3. Effects of scale coefficient l2 on the phase speed of acoustic branches when l1 > l2.

12

10

e =0.9 1

Phase Speed , c (km/s)

Fig. 2. Effects of scale coefficient l1 on the phase speed of acoustic branches when l1 > l2.

−1

HTBT−TO HTBT−TA HRBT HEBT HSBT MD

e =0.36 2

8

6

4

2

0 −2 10

−1

10

0

10

1

10

2

10

Nondimensional Wavenumber , ka Fig. 4. Phase speed of transverse waves propagating in a (5, 5) armchair SWCNT using different hybrid strain gradient nonlocal beam model and molecular dynamics simulation.

Fig. 5 gives a comparison of the phase velocity of transverse waves propagating in a (5, 5) armchair SWCNT using various theories under the assumption of Timoshenko beams including classical, nonlocal, strain gradient and hybrid theories, labeled CTBT, NTBT, SGTBT, and HTBT, respectively. As mentioned before, it is observed in Fig. 4 that the phase speed predicted by HTBT is in good agreement with molecular dynamics simulations when l1 = 0.9a and l2 = 0.36a. For other theories, the prominent difference between their predictions and molecular dynamics simulations is present. In addition, for the NTBT, the phase velocity (34) is found to tend to vanishing as wavenumber is sufficiently large, and this is what we do not expect. For the SGTBT, the phase velocity (37) is larger than its classical counterpart, and this is not inconsistency with the fact that elastic waves as the long-wave limits of lattice dynamics. As a consequence, with the size effect included, only the HTBT can give a reasonable interpretation. To illustrate the effects of the elastic foundation on the wave speed, the variation of the phase velocity of transverse waves propagating in a (5, 5) armchair SWCNT embedded in an elastic

193

J.X. Wu et al. / Computational Materials Science 67 (2013) 188–195 10

10

6

e =0.9 1

1

e =0.36 2

4

β / E=0 0

β / E=0.01 0

β / E=0.02 0

6

4

2

2

0 −2 10

2

8

Phase Speed , c (km/s)

Phase Speed , c (km/s)

8

β1=0

e =0.9 e =0.36

HTBT CTBT NTBT SGTBT

10

−1

10

0

10

1

10

0

2

10

−2

10

Nondimensional Wavenumber , ka

−1

10

0

10

1

10

2

Nondimensional Wavenumber , ka

Fig. 5. Phase speed of transverse waves propagating in a (5, 5) armchair SWCNT based on different Timoshenko beam theories.

Fig. 7. The effects of parameter b0 on the phase speed of transverse waves propagating in a SWCNT embedded in an elastic medium.

for the inner tubes, and 10 1

0

2

β / E=0

8

Phase Speed , c (km/s)

qN ¼ aN1 ðwN  wN1 Þ  b0 wN þ a2 b1

β =0

e =0.9 e =0.36

ð41Þ

for the outmost tube, where aj is van der Waals interaction coefficient, taking the form

1

β / E=10 1

(

β1/ E=20

aj ¼

6

0;

j¼0

320ð2Rj Þerg=cm2 0:16a2

ð42Þ

; j ¼ 1; 2; . . . ; N  1

where Rj measured in nm is the radius of the relative inner tube. Here for the outmost tube, besides van der Waals interaction, the contribution of the surrounding medium has been considered. Now we take the deflection for each tube in the form

4

2

0 −2 10

@ 2 wN ; @x2

F j ðx; tÞ ¼ F j eiðkxxtÞ ;

−1

10

0

10

1

10

2

10

Nondimensional Wavenumber , ka Fig. 6. The effects of parameter b1 on the phase speed of transverse waves propagating in a SWCNT embedded in an elastic medium.

medium using HTBT is showed in Figs. 6 and 7 for variable values of b1 or b0, respectively. The parameters b0 and b1 have different influences on the phase velocity of acoustic mode. On one hand, they only affect the phase velocity for small wavenumbers ka < 1. The effects almost completely disappear with wavenumbers rising to ka > 10. On the other hand, the influence range of b1 is wider than that of b0. Considering the fact that b0 describes the effect of normal stress of the surrounding medium, while b1 describes the effect of tangential stress of the surrounding medium, we come to a conclusion that the effect of tangential stress on the phase speed has a wider range of wavenumbers.

where the quantity with subscript j = 1, 2, . . . , N denotes the one corresponding to the j-th tube, F j is the amplitude. Substituting Eq. (43) into Eqs. (40) and (41) yields, respectively,

  2 2 q1 ¼ a1 ðF 1  F 2 Þ 1 þ l2 k eiðkxxtÞ ;

j ¼ 1; 2; . . . ; N  1;

ð40Þ



iðkxxtÞ

e

;

In addition, as pointed out previously, we here use HSBT as an approximation of HTBT and in this case we obtain a system of linear algebraic equations for F j as follows:

" 2 2 1 þ l2 k

4

a1

k þ  2 2 EI1 1 þ l1 k

... " 2 2 1 þ l2 k 

qj ¼ aj ðwj  wjþ1 Þ  aj1 ðwj  wj1 Þ;

2 2 l2 k

j ¼ 2; . . . ; N  1; ð45Þ h   i  2 2 2 2 iðkxxtÞ qN ¼ aN1 F N1  aN1 þ b0 þ a b1 k F N 1 þ l2 k e : ð46Þ

4

k þ

2 2

Compared with SWCNTs, there is van der Waals interaction among the tubes of multi-walled carbon nanotubes (MWCNTs). It can be evaluated by the derivative of the Lennard–Jones potential. Here omitting details, we take van der Waals interaction as a linear spring form and then for the j-th tube it can be stated as

ð44Þ 

qj ¼ ½ðaj þ aj1 ÞF j  aj1 F j1  aj F jþ1  1 þ

1 þ l1 k

4. Flexural waves of MWCNTs

ð43Þ

aj1 EIj

2 2

1 þ l1 k 

aN1 EIN

4

k þ

q A1 EI1

aj þ aj1 EIj

F j1 

... " 2 2 1 þ l2 k



aj EIj

þ

 

q Gs

qAj EIj

k

2



þ

#

x2 F 1 

q Gs

2

k



a1 EI1

F 2 ¼ 0;

# 2

x Fj

F jþ1 ¼ 0;

aN1 þ b0 þ a2 b1 k2

F N1 ¼ 0:

EIN

ð47Þ

ð48Þ  

qAN EIN

þ

q Gs

k

2



# 2

x FN ð49Þ

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J.X. Wu et al. / Computational Materials Science 67 (2013) 188–195

In order for the above system in F j to have a nontrivial solution, the determinant of the coefficient matrix of the system has to vanish. That is,

det f ðx; kÞ ¼ 0;

k

; j ¼ 1; 2; . . . ; N:

ð51Þ

2



m1  m2 x F 1 þ m3 F 2 ¼ 0;

n3 F 1 þ n1  n2 x2 F 2 ¼ 0:

4 3

0

−2

10

−1

0

10

10

1

10

10

2

Nondimensional Wavenumber , ka

ð53Þ

a

q

q

a

A1 4 1 2 1 k þ ; m2 ¼ þ k ; m3 ¼  ; 2 2 EI EI Gs EI1 1 1 1 þ l1 k 2 2 2 2 1 þ l2 k 4 A2 1 þ b0 þ a b1 k 2 k þ ; n2 ¼ þ k ; n3 2 2 EI2 EI2 Gs 1 þ l1 k

a

q

q

¼

a1 EI2

:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 u 2  2 u 1 u1 4m1 n1 m1 n1 4ðm1 n1  n3 m3 Þ5 : ¼ t þ  þ  k 2 m2 n2 m2 n2 m2 n2

ð54Þ

It should be pointed out that the above phase velocity is given by an explicit expression. However, if using the accurate governing equation for HTBT, such a simple expression cannot be derived. Fig. 8 presents a comparison of the phase velocity of acoustic mode using HTBT and HSBT, respectively. Two speeds are very close for any wavenumber. This implies that when focusing the phase velocity of acoustic mode, HSBT in place of HTBT is suitable and does not give rise to significant error. Prior to the presentation of the phase velocity of transverse waves propagating in a MWCNT, let us expound the effects of van der Waals interaction. To achieve this, we give a comparison of the phase velocity for three different cases, i.e. van der Waals interaction is present, neglected, and imposed to be infinity. The three cases for a DWCNT correspond to a1 given by (42), a1 = 0, and a1 ? 1, respectively. For the case of a1 = 0, two tubes in fact are independent, while for the case of a1 ? 1, two tubes are equivalent to a tube composed of perfectly bonded tubes with A = A1 + A2,I = I1 + I2. A comparison of the phase velocity of the above-mentioned three cases is showed in Fig. 9 for a DWCNT in a free space. From Fig. 9, neglecting van der Waals interaction or imposing it to be sufficiently large, the obtained phase velocities have a remarkable error, as compared with its accurate values. In order to keep our evaluation from getting tedious computation and to gain insight into the characteristic of wave propagation in CNTs, the following analysis is carried out only for a DWCNT, triple-walled carbon nanotube (TWCNT), and quadruple-walled carbon nanotube (QWCNT) embedded in an elastic medium, respectively. Figs. 10 and 11 are devoted to the variation of the phase velocity of transverse waves of acoustic mode for MWCNTs in a free space and in an elastic matrix, respectively. We find that the wave speeds have no difference for large wavenumbers ka > 1, while for small wavenumbers ka < 1, the wave speed for SWCNT is smallest. With the number of walls rising, the lowest wave speed

gradually increases when CNTs are in a free space. When CNTs are embedded in an elastic medium, the above trend is opposite. 5. Conclusions Flexural wave behavior in CNTs embedded in a free space and embedded in an elastic matrix was investigated based on HTBT. CNTs are modeled as Timoshenko beams of gradient elasticity theory, and a bi-parameter foundation model was adopted to model the interaction of CNTs with the surrounding elastic medium. Some conclusions are drawn as follows:  The dynamic governing equations of HTBT were derived and further simplified. When one or two scale coefficients vanishes, the nonlocal, strain gradient, and classical theories of Timoshenko beams are recovered.

10 TO−With vdW force TA−With vdW force inner tube−Without vdW force outer tube−Without vdW force two tubes bonded

9 8

Phase Speed , c (km/s)

1 þ l2 k

From the above system, we obtain the phase velocity as follows

c1;2

5

Fig. 8. A comparison of the phase speed of transverse waves propagating in a (5, 5)@(10, 10) armchair DWCNT using the HTBT and HSBT. 2 2

n1 ¼

6

1

ð52Þ

where

m1 ¼

7

2

As an example, for a double-walled carbon nanotube (DWCNT) embedded in an elastic medium, we have the system of linear algebraic Eqs. (47)–(49)



HTBT HSBT

8

Phase Speed , c (km/s)

xj

9

ð50Þ

where f(x,k) is given in Appendix A. Eq. (50) gives a dispersion relation of transverse waves traveling in a MWCNT surrounded by an elastic medium. Since Eq. (50) is a polynomial equation in x2, the phase velocity c can be found to have N distinct roots for a MWCNT composed of N nanotubes, namely

cj ¼

10

TO

7 6 TA

5 4 3 2 1 0 −2 10

−1

10

0

10

1

10

2

10

Nondimensional Wavenumber , ka Fig. 9. A comparison of the phase speed of a (5, 5)@(10, 10) armchair DWCNT in the absence of the surrounding elastic medium due to van der Waals interaction coefficient a1.

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J.X. Wu et al. / Computational Materials Science 67 (2013) 188–195

Appendix A

6

Phase Speed , c (km/s)

5

SWCNT DWCNT TWCNT QWCNT

e1=0.9 e2=0.36

2

D11

6 D21 6 6 6 6 f ðx; kÞ ¼ 6 6 6 6 4

4

3

D22

D23

D32

D33 .. .

D34 .. .

..

Djðj1Þ

Djj

.

DNðN1Þ

2

D11 ¼

1

0 −2 10

7 7 7 7 7 7: 7 7 7 Djðjþ1Þ 5

0

0

2 2

1 þ l2 k

0

10

1

10

10

a1

EIj

1 þ l2 k 2

1 þ l1 k

Djðjþ1Þ ¼  12

aj1

11

1

2

β / E=0.01 β / E=20 0

10

1

4

k þ 2

aj

EIj

DNðN1Þ ¼  SWCNT DWCNT TWCNT QWCNT

e =0.9 e =0.36

aj þ aj1

EI1

þ

q Gs

2



x2 ;

k

aN1 EIN

1 þ l2 k

EIj

 

qAj EIj

þ

q Gs

k

2



x2 ; j ¼ 2; . . . ; N  1;

; j ¼ 2; . . . ; N  1; ;

2 2

DNN ¼

qA1

; j ¼ 2; . . . ; N  1;

2 2

Djj ¼



DNN

;

EI1

Djðj1Þ ¼ 

Fig. 10. Phase speed against the nondimensional wavenumber for a MWCNT in a free space.

a1

4

k þ  2 2 EI1 1 þ l1 k

D12 ¼  −1

Nondimensional Wavenumber , ka

Phase Speed , c (km/s)

3

D12

4

k þ 2 2 1 þ l1 k

aN1 þ b0 þ a2 b1 k2 EIN

 

qAN EIN

þ

q Gs

k

2



x2 :

9

References

8 7 6 5 4 3 −2 10

−1

10

0

10

1

10

Nondimensional Wavenumber , ka Fig. 11. Phase speed against the nondimensional wavenumber for a MWCNT in an elastic matrix.

 For the phase velocity of acoustic mode, the effect of shear deformation is more dominant than that of rotary inertia, and HSTB can be taken an approximation of HTBT for any wavenumber.  The phase speeds predicted by HSBT and HTBT provide a good agreement with the results of molecular dynamics simulations in a wider range of wavenumbers.  The surrounding matrix has a prominent influence on the wave speed for small wavenumbers. Moreover, The stiffness coefficients b1 and b0 play different contributions for the phase speed and influence range.

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