Wave propagation in fluid-filled single-walled carbon.pdf

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Wave propagation in fluid-filled single-walled carbon nanotube onanalytically nonlocal EulerBernoulli beam model

Yang Yang a, Lixiang Zhang a,n, C.W. Lim b

a Department of Engineering Mechanics, Kunming University of Science and Technology, 50 East Ring Road, Kunming 650051, Yunnan, Chinab Department of Building and Construction, City University of Hong Kong, Hong Kong, China

a r t i c l e i n f o

Article history:

Received 21 September 2011

Accepted 19 November 2011

Handling Editor: L.G. Tham

a b s t r a c t

An analytically nonlocal EulerBernoulli beam model for the wave propagation in fluid-filled

single-walled carbon nanotube (SWCNT) is established. The governing equations with the

nonlocal effects are derived on the variational principle, and used in the wave propagation

analysis of the SWCNT beam. Compared with the partially nonlocal EulerBernoulli beam

models used previously, the analytically nonlocal model presented in the present study

predicts well the effects of the stiffness enhancement and the wave damping at the high

wavenumber or the strong nonlocal effects area for the fluid-filled SWCNT beam. Though the

analytical model is less sensitive than the partially nonlocal model when the moving velocity

of the internal fluid is high enough, it simulates more of the high-order nonlocal effecting

information than the partially nonlocal model does in many cases.

&2011 Elsevier Ltd. All rights reserved.

1. Introduction

Since the discovery of carbon nanotubes (CNTs) in the early 1990s [1], the mechanical behaviors of CNTs have attracted

many research interests. Especially, the wave propagation behavior of a fluid-filled CNTs beam is an interesting and

challenging topic since CNTs are quite acceptable in nanobiological and nanomechanical applications such as nano fluid

conveyance and drug delivery [2,3]. The research approaches about mechanical behaviors of the fluid-conveyed CNTs

beam include experimental and theoretical analyses. However, the experimental results about the CNTs properties are not

accurate enough since the nano scale experiment is quite difficult to manipulate or control [4]. Therefore, theoretical

analyses are more acceptable for the research of CNTs.

Theoretical methods of studying on the mechanics behaviors of CNTs have molecular dynamic (MD) simulation and

mechanics modeling. MD simulation is the most common and accurate computational approach for analyzing CNTs, because the

behavior of every molecule of CNTs and fluid are simulated in code. However, MD approach is inefficient because of its complicate

calculation, time consuming and instability, especially for a large scale system[5,6]. Thus the elastic continuum models of CNTs

are developed. Ru et al. for the first time applied the classical continuum model on the dynamical analysis for a fluid-conveyed

CNTs beam[7,8]. They discussed the influences of the internal moving fluid on the free vibration of the CNTs beam and flow

induced structural instability of a SWCNT beam which was modeled as a classical EulerBernoulli beam. Zhangs group

investigated the dynamical behaviors for a fluid-conveyed multiple-walled carbon nanotubes (MWCNTs) beam based on the

EulerBernoulli beam models[9,10]. Similar dynamical analyses were also completed based on Donnells cylindrical shell model

for the MWCNTs beam[11,12]. However, applying directly this on the classical beam or shell models possibly leads to inaccurate

results, since the influences of the nanoscale effects on the mechanical properties of the CNTs beams cannot be simulated by the

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/jsvi

Journal of Sound and Vibration

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.jsv.2011.11.018

n Corresponding author. Tel./fax: 86 871 3303561.E-mail addresses: [email protected] (Y. Yang),[email protected],[email protected] (L. Zhang), [email protected] (C.W. Lim).

Journal of Sound and Vibration ] (]]]]) ]]]]]]

Please cite this article as: Y. Yang, et al., Wave propagation in fluid-filled single-walled carbon nanotube on analyticallynonlocal EulerBernoulli beam model, Journal of Sound and Vibration (2011), doi:10.1016/j.jsv.2011.11.018

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classical models[13]. Therefore, some advanced continuum elastic models are established to study the mechanical behaviors of

the CNTs beams including strain gradient models[13,14],couple stress models[1517], and nonlocal stress models[1825].

Nonlocal stress is a kind of effective model which is sensitive to the nano scale (also referred to as nonlocal) effects of CNTs.

Thus great number of articles about analyses for static and dynamical behaviors of the CNTs beams based on the nonlocal models

are published[1825]. In these studies, the dynamical analysis of the fluid-filled CNTs beam is one of the most attractive topics.

Lee and Chang[24]applied the nonlocal EulerBernoulli beam model to analyze the nano scale effects on the natural frequency

and mode shape for free vibration of the fluid-filled SWCNT beam. Their results show that the frequency and mode shape are

obviously influent by the nano scale effects. Narendar[25]investigated the terahertz wave characteristics of a fluid-filled SWCNT

beam based on the nonlocal Timoshenko beam model. The influences of the nano scale effects and fluid density on the flexural

and shear wave propagations were studied when the fluid velocity is at 1000 m/s.

Though the nonlocal models can simulate the nano scale effects on the mechanical behaviors of a CNTs beam, some

contradictory predictions and surprising conclusions based on the nonlocal models are induced. For example, the bendingbehaviors of a cantilever CNTs beam with a point loading at the free end are not affected by the nano scale effects[19]and on the

nonlocal beam models. This means the bending behaviors of the cantilever CNTs beam are completely same to a classical beam

when the point loading is added. Another inaccurate factor is that all the nonlocal models predict stiffness decreasing due to the

nano scale effects, which is contradict with the results of experiment and MD simulation [20]. Reasons for these unsatisfied

results are that the nonlocal models used are formulated by directly extending the classical beam models without rigorous

verification, and hence certain very important higher-order nonlocal terms have been inadvertently neglected[26]. For instance,

the nonlocal bending moment of a CNTs beam based on nonlocally constitutive relation is directly used in the classical beam

models without any modification. Thus, the nonlocal effects only on the bending moment are considered, instead of the governing

equations and boundary conditions being neglected [26]. Generally, these nonlocal models which are obtained by directly

extending the classical models are called as partially nonlocal models (PN).

Lim and his colleagues established an analytically nonlocal model (AN) according to the variation principle [2632]. The

new nonlocal governing equations and boundary conditions containing high-order nonlocal terms were obtained in their

work. The mechanical behaviors of CNTs beams including bending, buckling, vibration and wave propagation based on theAN models were confirmed more reasonable than the PN models. The unsatisfied problems in the PN models mentioned

above were solved with the AN models according to Lims studies[2632]. However, the applications of the AN models are

limited because of the complicated calculation.

In this paper, an analytically nonlocal EulerBernoulli beam model (ANE) is derived and employed to analyze the

characteristics of the wave propagation in a fluid-filled SWCNT beam. Using the novel ANE model for a CNTs beam, the

influences of the fluid and the nano scale effects on the wave propagating characters are studied in details.

2. ANE model for fluid-filled SWCNT beam

In the nonlocal continuum elastic theory, the stress at a material point is considered to not only depend on the strain at

this point, but also on all other points in a domain near to this point [18]. So, the constitutive relation is stated as

rijr ZOa9r0r9,tr0ijr0dO (1a)

Nomenclature

a CC bond length

Ac cross-section area of CNTs

Af fluid filled area

e0 nonlocal effect constant of CNTs

E Youngs modulusi

ffiffiffiffiffiffiffi1p

I second-order moment

k wavenumber

Kc kinetic energy of SWCNT

Kf kinetic energy of fluid

L length of CNTs

Mx nonlocal bending moment

Mef effective bending moment

r displacement vector

t time variable

T temporal period

u strain energy density

V strain energy

w lateral deflection

W amplitude

x axial coordinates

y vertical coordinate

a nonlocal modulus

eij strain tensorrc mass density of SWCNTrf mass density of fluidsij stress tensorO whole volume of SWCNT

energy function

t material parameter of SWCNTl Lameconstants

m Lameconstantsdij Kronecker delta

o angular frequencyr2 Laplace operatory rotary angle of CNTs beam section


Y. Yang et al. / Journal of Sound and Vibration ] (]]]]) ]]]]]]2

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s0ijr0 lekkr0dij2meijr0 (1b)

where rij(r) is the stress at pointr, r0ijr0is the stress at all other points else r0 in the domain O,a is the nonlocal module

which depends on the distance 9r0r9 and the nonlocal material parametert,l andm are the Lameconstants and eij(r0) isthe strain at point r0. In Eq. (1a), it is obviously that the stress r ij(r) at pointrdepends on all the points r0. Thus the volumeintegration is necessary to calculate rij(r). In Eq. (1b), the stress r

0ijr0at point r0 meets the classical constitutive equation

(Hookes law). However, it is mathematically difficult to obtain the analytical solution because of the volume integration in

Eq. (1a), Eringen simplified Eq. (1a) to a second-order differential equation in a two-dimension coordinate space when

Greens function is used as

1e0a2r2rijr0ij (2)wheree0is the nonlocal parameter that depends on the material properties, a is the internal length of the bond relation or

the lattice parameter,r2 @2=@x2 @2=@y2 is the Laplace operator, rij and r0ij are the abbreviations of rij(r) and r0ijr0,respectively.

Fig. 1indicates a fluid-filled simply supported SWCNT beam in the Cartesian coordinate system with the vertical deflection w,

the fluid velocityU, and the beam length L, wherexandydenote the axial and vertical coordinates, respectively. If only the stress

and strain inx direction are considered, Eq. (2) is simplified into one-dimensional formulation as

sxe0a2d

2sxdx2

Eex (3)

wheres(x) ande(x) is the normal stress and strain inxdirection of the SWCNT beam and Eis Youngs modulus. The second term

at the left hand of Eq. (3) is the high-order nonlocal term which denotes the nonlocal effects on the stress/strain relation.Obviously, the classical constitutive equation (Hookes law) is recovered withe0a0 if the nonlocal effects are too insignificant tobe considered.

The bending moment at the cross section of the SWCNT beam is defined as

MxZ ysdAc (4)

whereAcis the cross-section area of the SWCNT beam and s is the abbreviation ofs(x) in Eq. (3). The normal strain for theEulerBernoulli beam is written as

e y d2w

dx2 (5)

Substituting Eq. (3) into Eq. (4), one arrives

Mxe0a2@2Mx@x2

EI@2w@x2

(6)

IZ y2dAc (7)

Eq. (7) denotes the second-order moment of the area over the cross section. Hence, the solutions of Eqs. (3) and (6) are

shown as

s EX1n1

e0a2n1e/2n1S (8)

Mx EIX1n1

e0a2n1w/2nS (9)

w

y

x

o

U

U

L

Fig. 1. Simply supported fluid-filled SWCNT beam.


Y. Yang et al. / Journal of Sound and Vibration ] (]]]]) ]]]]]] 3



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where/nS denotes the nth derivatives with respect to x. Eqs. (6) and (9) define the nonlocal bending moments of theSWCNT beam. If the nonlocal bending moments in Eqs. (6) and (9) are directly substituted into the classical Euler

Bernoulli beam model with a fluid filling [33], the governing equation of the partially nonlocal EulerBernoulli beam (PNE)

with fluid filling is obtained as

M/2Sx rfAfrcAc@2w

@t2rfAfU2

@2w

@x22rLAfU

@2w

@x@t 0 (10)

where Af is the fluid filling area of the SWCNT beam, rc and rf are the mass densities of the SWCNT beam and fluid,respectively. Based on Eq. (10), the PNE model for the fluid-filled SWCNT beam is established and applied to analyze thevibration behaviors. However, as discussed above, the PN models are not accurately enough to predict the nonlocal effects

of the SWCNT beam[1825]. Thus the ANE model is necessary to analyze the fluid-filled SWCNT beam. According to the

idea of the AN model[26], the normal strain energy density of the SWCNT beam is

unZ e

0sede (11)

where

unu1 u2 u3 (12)and

u112Ee2

u212EX1n1

1n1e0a2ne/nS2

u3EX1n1

e0a2/n 1SXnm1

1m1e/mSe/2n1mS( )

(13)

Thus the strain energy in the whole SWCNT beam is

VZ T

0

ZOc

undOcdt (14)

whereOcis the volume of the SWCNT beam except the space of the fluid filling and Tis the vibration temporal period. The

kinetic energy of the SWCNT beam is

KcrcAc

2Z T

0Z L

0

dw

dt 2" #

dxdt (15)

Furthermore, the kinetic energy of the fluid is considered as

KfrfAf

2

Z T0

Z L0

Ucosy2 _wUsiny2dxdt (16)

where y is the rotary angle of the SWCNT beam section, and _wdw=dt. Since y is quite small, approximations ofsiny y tanyw 1h i and cosyE1 are assumed. Thus, Eq. (16) is reduced to

KfrfAf

2

Z L0

U2 _wUw/1S2dx (17)

Taking the variation with respect to w for Kf, Kcand V, one yields

dKf rfAf Z T

0Z L

0

w2U _

w

/1S

U2

w

/2S

dwdxdt

rfAfZ L

0 _wdwT0dxU

Z L0

w/1SdwT0dxUZ T

0 _wdwL0dtU2

Z T0

w/1SdwL0dt

(18)

dKc rcAcZ T

0

Z L0

wdwdxdtrcAcZ L

0

_wdw T

0dx (19)

where

wd2w=dt2

dVZ T

0

ZOdundOdt

Z T0

ZOdu1 du2 du3dOdt EI

Z T0

Z L0

X1n1

2n3e0a2n1w/2n1S" #

dwdxdt

Z T

0 EIX1n1

2n3e0a2n

1w/

2n

1Sdw X

1

n12n3e0a

2n

1w/

2nSdw/

1S

"(





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X1n1

2n1e0a2nw/2n1Sdw/2SX1n1

2ne0a2n1w/2n 1Sdw/3S

X1n1

2n1e0a2n1w/2n1Sdw/4SX1n1

2n1e0a2n2w/2n 1Sdw/5S

X1

n1

2n1e0a2n 3w/2n3Sdw/6S #L

0

9=;dt (20)

Anther form of this equation with Mx is stated as

dV EIZ T

0

Z L0

M/2Sef

dwdxdtEIZ T

0M/1Sx 2

X1n1

e0a2nM/2n1Sx !

dw Mx2X1n1

e0a2nM/2nSx !

dw/1S

"

e0a2M/1Sx 2X1n1

t2n 1 M2n 1h ix

!dw/2S 2e0a4

X1n1

e0a2n1M/2nSx !

dw/3S

e0a4M/1Sx 2X1n1

e0a2n2M/2n1Sx !

dw/4S e0a6M/2Sx 2X1n1

e0a2n3 M/2n1Sx !

dw/5S

e0a8M/3Sx 2 X1

n1

e0a2n4M/2n3Sx !dw/6S #

L

0

dt (21)

whereMef is the effectively nonlocal bending moment, namely

MefMx2X1n1

e0a2nM/2nSx EIX1n1

2n3e0a2n1w/2nS (22)

According to the variational principle, the stationary condition is written as

d dVdKcdKf 0 (23)

Substituting Eqs. (18)(22) into Eq. (23) and considering all the boundary conditions withL0and initial conditions withT0 (seeAppendix A), the governing equation of the fluid-filled SWCNT beam based on the ANE model is obtained as

M/2Sef

rfAfrcAc@2w

@t2rfAfU2

@2w

@x22rLAfU

@2w

@x@t 0 (24)

By comparing Eq. (24) with Eq. (10), it is obvious that the bending moment Mxin the PNE model is replaced with the

effective moment Mef. Moreover it is worthy to note that for the ANE models without fluid filling, the replacing relation

fails to directly apply to the nonlocal field [29]. In other words, the nonlocal bending moment Mx does not meet the

nonlocal replacing relation. Eqs. (24) and (10) are returned to the classical beam models when e0a-0[33].

Substituting Eq. (22) into Eq. (24) and omitting the higher-order terms O((e0a)6), the governing equation of the fluid-

filled SWCNT beam is obtained as

3EIe0a4w/8Se0a2EIw/6SEIw/4SrfAfU2w/2SrfAfrcAc@2w

@t22rfAfU

@2w

@t@x 0 (25)

The solutions for Eq. (25) is assumed as

wWeikxot (26)whereWis the amplitude of the wave mode, k is the wavenumber ando is the angular frequency. Substituting Eq. (26)into Eq. (25), one yields

o rfAfUk7ffiffiffiffiffiffiDA

prfAfrcAc

(27)

where

DAr2fA2fU2k2 rfAfrcAc3EIe0a4k8 EIe0a2k6 EIk4rfAfU2k2 (28)

Eq. (27) reflects a spectrum relation of the wave propagation in the fluid-filled SWCNT beam based on the ANE model. It

is confirmed by Eq. (27) that the angular frequency depends on the wavenumber k, the fluid velocityU, and the nanoscale

parametere0a. It is clear that the value of the angular frequency in Eq. (27) may be a complex number when DAo0, and

the real and imaginary parts are represented as

Reo rfAfUk

rfAfrcAc (29)





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Imo 7 iffiffiffiffiffiffiDA

prfAfrcAc

(30)

The spectrum relation of the PNE model is also obtained by substituting Eq. (26) into Eq. (10) as

o rfAfUk7ffiffiffiffiffiffiDP

prfAfrcAc

(31)

where

DPr2fA2fU2k2 rfAfrcAcEIe0a4k8e0a2EIk6 EIk4rfAfU2k2 (32)

The real and imaginary parts of the angular frequency based on the PNE models are as follows:

Reo rfAfUkrfAfrcAc (33)

Imo 7 iffiffiffiffiffiffiDP

prfAfrcAc

(34)

By comparing Eqs. (27)(30) with (31)(34), we know that the terms with (e0a)4 and (e0a)

2 in DA and DP take

an opposite sign, which leads clearly to the different results of the wave propagation behaviors as discussed in next

section.

3. Results and discussions

The material and geometry parameters of the SWCNT beam and the fluid used are taken as E1 TPa,I1.78 1038 m4, Af3.0 1019 m2, rf1 103 kg m3, Ac3.63 1019 m2, rc2.27 103 kg m3, a0.142 nm(CC bond length).

According to Eqs. (27) and (31), the spectrum relation of Re(o) via the wavenumber k for the fluid-filled SWCNT beamis shown inFig. 2when the fluid velocity is taken as U300 m/s and the nanoscale parameter of the beam material ase0a0.05, 0.1, and 0.2 nm. The letter Ein Fig. 2denotes the solutions based on the classical EulerBernoulli beam modelwithout nonlocal effects. As illustrated inFig. 2, the solutions of the angular frequencies on the ANE model are higher than

those on the PNE model at the low wavenumber area. Therefore, the ANE model predicts higher stiffness for the fluid-filled

SWCNT beam than the PNE model does. It is first time to confirm this stiffness enhancement effects for the fluid-filledSWCNT beam on the ANE model, even though the similar results haves already obtained for the SWCNT beam without

fluid filling [29]. However, the angular frequency predictions on the ANE model indicated a sharp decrease when the

wavenumber is over a critical value. For the parameter e0a0.05, 0.1, and 0.2 nm, the critical wavenumbers are 15.5, 7.5,and 3.4 109 m1, respectively. The physical reason for the angular frequency decreasing is that the nonlocal effectscontribute an additional damping of the wave propagation. Obviously, the wave length becomes smaller as the

wavenumber increases. Thus, if the nonlocal effects domain in Eq. (1a) is larger than the size of one wave length, the

characteristics of the wave propagation have a significant decaying change. In this case, the kinetic energy of the wave

propagation is exhausted due to the nonlocal effects and the propagating frequency decreases rapidly. A similar

phenomenon is confirmed for the wave propagation in a SWCNT beam without fluid filling [29]. However, the nonlocal

effects based on the PNE model is not to induce the rapid decaying for the wave propagation when the propagating

frequency is at the high wavenumber area as shown inFig. 2. Another case is that the classical EulerBernoulli beam model

does not contribute any damping to the wave propagation because the nonlocal effects are exclusive. In fact, the imaginary

parts based on the PNE and the classical EulerBernoulli beam models are zero for any value of wavenumber, which meansthere is no a damping supply in these two models. Therefore, the ANE model predicts stronger nonlocal effects on the wave

propagation behaviors of the fluid-filled SWCNT beam.

Figs. 3 and 4illustrate, respectively, the real and imaginary parts as functions of the motion velocity of the fluid within

the SWCNT beam for wavenumber k 1 109 m1. FromFig. 3,the angular frequency decreases with the increase of thefluid velocity on the PNE model when the fluid velocity Uo1100 m/s. In contrary, the angular frequency on the ANE model

increases first, and then decreases sharply to zero when the fluid velocity Uo1100 m/s. The results show that these two

models contribute the different influences on the stiffness of the SWCNT beam. From previous analysis, the nonlocal

effects on the ANE model lead to the stiffness enhancement of the SWCNT beam, which increases the value of the angular

frequency. However, the nonlocal effects on the PNE model decrease the stiffness of the SWCNT beam. Thus, as the

nanoscale parameter e0a varies from 0.05 to 0.2 nm, the angular frequency on the ANE model increases, while the PNE

model shows a reversed case. When the fluid velocity U41100 m/s, the angular frequency becomes a conjugate complex

number. The real and imaginary parts are presented in Eqs. (29) and (30) for the ANE model and in Eqs. (33) and (34) for

the PNE model. The comparisons of the real parts are indicated in Fig. 3for the ANE, PNE, and E models. The imaginaryparts are shown inFig. 4, in which the decaying effects of the wave propagation are clearly represented when the fluid





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velocityU41100 m/s. FromFig. 4, it is obvious that the propagation decayings on the ANE and PNE models increase with

the fluid velocity rising. However, the ANE model is stronger since the value of the imaginary part increases much fasterthan the PNE model. The solutions on the classical beam model are placed between those on the ANE and PNE models.

Fig. 2. (a) Spectrum relation on ANE and PNE models and (b) zoom of (a).

Fig. 3. Real part of angular frequency via fluid velocity.





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Fig. 5 presents the Argand diagrams of the SWCNT beam when wavenumber k1 109 m1 and the fluid velocityvaries from 1000 m/s to 3000 m/s. The plots based on the ANE and PNE models with the different values ofe0acoincide

nearly together. Thus it is clear that the Argand diagrams of the two models are insensitive to the nonlocal effects. Thedivergence points of the Argand diagrams are point A for the ANE model and point P for the PNE and E models,

Fig. 4. Imaginary part of angular frequency via fluid velocity.

Fig. 5. (a) Argand diagrams with different value ofe0a and (b) zoom of (a).





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respectively, when the real part takes 2.2 THz and 1.6 THz. Moreover, the values of the imaginary parts for all the three

models increase with the real parts rising, which confirms that the damping of the wave propagation goes up with the fluid

velocity increasing for two models.

Figs. 6 and 7 indicate the real and imaginary parts as functions of the nanocsale parameter e0a with wavenumber

k1 109 m1. First, when e0ao0.45 nm, the angular frequency takes a pure real number. In Fig. 6, the real part on theANE model increases when the fluid velocity varies from 100 m/s to 500 m/s, while the PNE solution decreases in the same

condition. The reason is that the stiffness enhancement for the ANE model and the stiffness decline for the PNE model. It is

similar with the case inFig. 3when the fluid velocity Uo1100 m/s. The real parts on the ANE and PNE models indicatedifferent tendencies with the nanoscale parameters. For the ANE model, the frequency increases first and then decreases

sharply whene0a40.25 nm. From previous analysis, the stiffness enhancement of the nonlocal effects on the ANE model

increases the frequency of the propagation. However, the nonlocal domain in Eq. (1a) becomes larger as the nanoscale

parameter e0a increases. If the nonlocal domain is larger than the size of one wave length, the wave cannot propagate

through and a sharp decaying occurs. Thus, the angular frequency decreases rapidly when e0a40.25 nm for the ANE

model. This phenomenon fails to be predicted with the PNE model.

InFig. 7, the imaginary parts increase with the nanoscale parameter increasing when e0a40.42 nm for the ANE model

and e0a40.5 for the PNE model. Thus e0a0.42 nm and e0a0.5 nm are the divergence points for the two models,respectively. According to the information, these two models have the same predictions of the wave decaying when the

nanosclae parameter is high enough. However, the decaying on the ANE model is much stronger than that on the PNE

model, which is similar with the case in Fig. 4. Therefore, it is confirmed that the ANE model predicts stronger nonlocal

effects on the wave propagation behaviors of the fluid-filled SWCNT beam.

Fig. 8shows the Argand diagrams with the different values of the fluid velocity when wavenumber k1 109

m1

andthe nanosclae parameter e0a varies from 0 to 1 nm. From Fig. 8, it is obvious that the plots on the PNE model with the

Fig. 6. Real part of angular frequency via e0a.

Fig. 7. Imaginary part of angular frequency via e0a.





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different values of the fluid velocity are close to each other than the plots on the ANE model, because the ANE model is

more sensitive to the varying fluid velocity. When the nanoscale parameter increases from 0 to 0.42 nm, only the real partexists and the value decreases to 0 from 7 for the ANE model. As the nanoscale parameter increases from 0.42 nm to 1 nm,

the value of the angular frequency becomes a complex number and the divergence occurs on the ANE model as shown in

Fig. 8(b). Thus the divergence point are e0a0.42 nm for the ANE model. Reason for the divergence is that the highnonlocal effects contribute a big damping when e0a40.42 nm. The PNE model predicts similar solutions when the

divergence point is e0a0.5 nm. Therefore, it is different fromFig. 5that the divergence point on the ANE model occursbefore. It is similar with Fig. 5 that the imaginary part increases with the real part, which means the damping of the

propagation is enhanced with increasing nanoscale parameter. It confirms once again that the ANE model predicts more

nonlocal effects, and it implies that more damping is used in the ANE model.

4. Conclusions

Based on the nonlocal continuum elastic theory and variational principle, an analytically nonlocal EulerBernoullibeam model for the wave propagation in a fluid-filled SWCNT beam is established. Compared with the partially nonlocal

Fig. 8. (a) Argand diagrams with different fluid velocities and (b) zoom of (a).





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EulerBernoulli beam model used previously, the analytically nonlocal model suggested in the present study can predict

the effects of the stiffness enhancement and the wave propagation decaying at the high wavenumber region, while the

partially nonlocal model fails to. Furthermore, the analytically nonlocal model has much stronger nonlocal effects on the

propagation of the wave. When the fluid velocity is high enough, the fluid-filled SWCNT beam on the analytically nonlocal

model is less sensitive to the nonlocal effects.

Acknowledgments

The authors thank the National Natural Science Foundation of China (NSFC) (Grant no. 50839003), the Science

Foundation of Educational Ministry of China (Grant no. 200806740005), and the National Natural Science Foundation of

Yunnan Province (Grant no. 2008GA027) for financial supports of this research.

Appendix A. Analysis of other boundary conditions

The boundary condition terms in Eqs. (21) and (23) contain the high-order derivatives ofMx and w. Eq. (23) is usable

only when all these high-order boundary conditions in Eq. (21) are satisfied. The boundary conditions for simply

supported, clamped and free cases are discussed as follows.

(1) For simply supported boundary

Mx is replaced with Mefin Eq. (21), the simply supported boundary conditions are derived as

Mef9x0,L Mx2X1n1

e0a2nM/2nSx" #

x0,L 0, w9

x0,L 0 (A1)

Furthermore, the simply supported case yields

M/1Sx 2X1n1

e0a2nM/2n 1Sx" #

x0,LM/1Sef 9x0,La0, w/1Sa0 (A2)

Moreover, all the equations become the classical beam model when e0a0. Thus, it yieldsMef9x0,LEIw/2S9x0,L 0 (A3)

Eqs. (A1)(A3), and (23) are met in the case

M/2nSef

9x0,L 0, n41 (A4)

Therefore, the simply supported boundary conditions are expressed as

w9x0,Lw/2S9x0,Lw/4S9x0,L. . .w/2nS9x0,L 0 (A5)

The zero and first order terms have the same forms for the PNE and classical models whenMxis directly replaced with

Mef, but the similar boundary conditions are unavailable. So, the approach of directly replacing Mxis inappropriate in

the boundary conditions.

(2) For clamped boundary

Similar to the simply supported case, the natural boundary conditions for the clamped ends are written asM/1S

ef 9

x0,La0, w9x0,L 0 (A6)

Mef9x0,La0, w/1S9

x0,L 0 (A7)

From Eqs. (21)(23), we yield

M/nSef 9x0,La0 (A8)

w/nS9x0,L 0 (A9)

The boundary conditions of the ANE model for n

0, 1, 2, 3? are stated as

M/1Sef 9

x0,La0, Mef9x0,La0 (A10)





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[29] C.W. Lim, Y. Yang, Nonlocal elasticity for wave propagation in carbon nanotubes: the physics and new prediction of nanoscale in nonlocal stressfield, Journal of Computational and Theoretical Nanoscience 7 (2010) 988995.

[30] C.W. Lim, Y. Yang, Wave propagation in carbon nanotubes: nonlocal elasticity induced stiffness and velocity enhancement effects, Journal ofMechanics of Materials and Structures 5 (2010) 459476.

[31] C.W. Lim, J.C. Niu, Y.M. Yu, Nonlocal stress theory for buckling instability of nanobeams: new prediction on stiffness strengthening effects ofnanoscales, Journal of Computational and Theoritical Nanoscience 7 (2010) 21042111.

[32] Y. Yang, L.X. Zhang, C.W. Lim, Wave propagation in double-walled carbon nanotubes on a novel analytically nonlocal Timoshenko-beam model,Journal of Sound and Vibration 330 (2011) 17041717.

[33] M.P. Paidoussis, FluidStructure Interactions Slender Structures and Axial Flow , Vol.1, Academic Press, San Diego, 1998.

Please cite this article as: Y. Yang, et al., Wave propagation in fluid-filled single-walled carbon nanotube on analytically



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