International Journal of Engineering Science 87 (2015) 13–22

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International Journal of Engineering Science

journal homepage: www.elsevier .com/locate / i jengsci

On nonlinear behavior and buckling of fluid-transportingnanotubes

http://dx.doi.org/10.1016/j.ijengsci.2014.11.0050020-7225/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR China.E-mail address: wanglindds@hust.edu.cn (L. Wang).

H.L. Dai a,b,c, L. Wang a,b,⇑, A. Abdelkefi d, Q. Ni a,b

a Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR Chinab Hubei Key Laboratory for Engineering Structural Analysis and Assessment, Wuhan 430074, PR Chinac School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798, Singapored Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA

a r t i c l e i n f o

Article history:Received 21 August 2014Received in revised form 22 October 2014Accepted 1 November 2014

Keywords:Nanotube conveying fluidNonlinear modelNonlocal elasticity theoryPost-bucklingNatural frequency

a b s t r a c t

A general nonlinear nonlocal model for supported nanotubes conveying fluid is developed.Considering the geometric nonlinearity associated with the mid-plane stretching of thenanotube, the extended Hamilton’s principle is used to derive this general model basedon Eringen’s nonlocal elasticity theory. Analytical solutions for the nonlinear responsesof the nanotube are obtained from the constructed nonlinear equation. It is shown thatthe presence of the nonlocal effect tends to decrease the critical flow velocity and increasethe buckled static displacement of the nanotube. It is also demonstrated that the nonlocaleffect has a significant impact on the pre- and post-buckling natural frequencies of thenanotube while the mass ratio mainly influences the post-buckling frequencies and thegeometric nonlinearity term has no effect on these frequencies of the nanotube.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In the past two decades, hollow nanobeams, also known as nanotubes or nanopipes, have been reported to be a class ofnew nano-fluidic devices and systems. One of the attractive features of such structures is the ability of transporting liquid-like materials (e.g., water, cells, and nanoparticles). For this reason, hollow nanobeams are also expected to have the poten-tial for nanopipettes, fluid filtration devices, targeted drug delivery devices, biomimetic selective transport of ions, fountainpen nanochemistry for chromium etching, etc. (Longhurst & Quirke, 2007; Wang, 2005, 2009).

In the past years, a number of theoretical studies have been focused on understanding the responses of fluid-loaded nano-tubes/microtubes (Dai, Wang, & Ni, 2014; Ke & Wang, 2010; Kuang, He, Chen, & Li, 2009; Lee & Chang, 2008; Liang & Su,2013; Tang, Ni, Wang, Luo, & Wang, 2014; Wang, Guo, & Hu, 2009; Wang, Liu, Ni, & Wu, 2013; Wang & Ni, 2008; Yan,Wang, & Zhang, 2010; Yang, Jia, Yang, & Fang, 2014; Yoon, Ru, & Mioduchowski, 2006). The earliest research studies ofthe vibrations of fluid-conveying nanotubes were based on linear analyses without considering possible nonlinear sources.To determine the linear governing equations of these fluid-loaded nanotubes, both classical continuum models (Yoon et al.,2006) and non-classical continuum models (Ke & Wang, 2010; Kuang et al., 2009; Lee & Chang, 2008) have been developed. Itwas demonstrated that a cantilevered nanotube conveying fluid may lose stability via flutter at very high flow velocities

14 H.L. Dai et al. / International Journal of Engineering Science 87 (2015) 13–22

(Yoon et al., 2006). On the other hand, for supported nanotubes conveying fluid, it was found that the possible form of insta-bility is buckling (Lee & Chang, 2008), i.e. the original equilibrium point becomes statically unstable.

However, some key questions associated with the dynamical behaviors of supported nanotubes conveying fluid cannot beanswered except by nonlinear theory. For example, using the linear equation of motion, it is impossible to determine theresponses of the system when the flow velocity is beyond the critical value. Also, the existence of post-divergence flutterpredicted by linear theory (Wang, 2009) is questionable and has to be reassessed by nonlinear theory. Because the originalequilibrium has become unstable after buckling, motions would actually take place about a new equilibrium position. Theseand other questions are some of the aspects on this topic and should be tackled by performing nonlinear analyses.

The literature on nonlinear behaviors of nanotubes conveying fluid is relatively limited. Kuang et al. (2009) studied theeffects of geometric nonlinearity and another nonlinearity arising from van der Waals (vdW) forces on the transverse vibra-tion of double-walled carbon nanotubes (DWCNTs) conveying fluid. Rasekh and Khadem (2009) developed a nonlinear vibra-tion model for a fluid-conveying single-walled carbon nanotube (SWCNT) embedded in a Winkler-type elastic foundation.However, in these two mentioned studies, the nonlinear governing equations of motion were derived based on the classicalcontinuum theory. Using Newton’s law, Soltani and Farshidianfar (2012) derived the nonlinear equation of lateral motion ofa SWCNT conveying fluid based on Eringen’s nonlocal elasticity theory. It was shown that the effect of nonlocal parameter onthe nonlinear frequency is not pronounced. Based on the Hamilton’s principle, Farshidianfar and Soltani (2012) also derivedthe equation of motion for this problem. However, the nonlinear responses when the CNTs are buckled were not analyzed.Ali-Asgari, Mirdamadi, and Ghayour (2013) studied the natural frequency and responses of CNTs conveying fluid based onthe coupling of nonlocal theory and von Karman’s stretching. Arani and his co-authors (Arani & Kolahchi, 2014; Arani,Kolahchi, Haghighi, & Barzoki, 2013; Arani, Bagheri, Kolahchi, & Maraghi, 2013; Arani, Hashemiana, & Kolahchia, 2013) havedone a lot of work on the nonlinear vibrations and instability of fluid-conveying boron nitride nanotubes (BNNTs) and carbonnanocones (CNCs). The nonlinear frequencies and critical flow velocity of the BNNTs and CNCs have been investigated. Arani,Shajaria, Amira, and Loghmana (2012) and Arani, Shajaria, Atabakhshiana, Amira, and Loghmana (2013) also studied thenonlinear frequency and stability of smart composite microtubes made of Poly-vinylidene fluoride (PVDF) reinforced byBoron-Nitride nanotubes (BNNTs) with consideration of the effects of internal fluid flow, imposed electric potential, smallscale, volume percent, and orientation angle of the BNNTs. The smart microtube was modeled as either a thin shell (Araniet al., 2012) based on the nonlinear Donnell’s shell theory or an Euler–Bernoulli beam (Arani et al., 2013).

It is noted that all the research studies mentioned in the foregoing have not paid attention to the post-buckling responsesof nanotubes conveying fluid. Recently, Ghasemi, Dardel, Ghasemi, and Barzegari (2013) discussed the buckling and post-buckling of fluid-conveying multi-walled carbon nanotubes (MWCNTs). The nonlinear governing equations and boundaryconditions were derived based on Eringen’s nonlocal elasticity theory. However, their nonlinear governing equations haveno time-dependent terms and hence are only suitable for static analysis of nanotubes conveying fluid.

From the literature discussed above, it is found that most investigations on the nonlinear behaviors of nanotubesconveying fluid were limited to ‘‘specific’’ nanotubes, such as CNTs, BNNTs and CNCs. There is still lack of unified or generaltheoretical models for predicting the nonlinear responses of ‘‘generic’’ nanotubes conveying fluid. Consequently, the post-buckling dynamics of fluid-conveying nanotubes has not been well understood yet. This motivates the current work.

In the present work, we aim to develop a general nonlocal nonlinear Euler–Bernoulli beam model for dynamic analysis ofsimply-supported nanotubes conveying fluid. In particular, the nonlinear equation of motion, from which the dynamicalbehaviors of the system can be predicted, is derived based on the extended Hamilton’s principle and the nonlocal elasticitytheory which is presented in Section 2. Since a nonlocal nanoscale parameter has been introduced in this nonlinear model,the nonlinear equation of motion enables us to analyze the size effect on the transverse motion. In Section 3, the post-buckling configurations of the nanotubes conveying fluid are obtained analytically, showing the essential details of dynamicresponses of the system when the flow velocity is beyond the critical value. Furthermore, the size effect on the first two nat-ural frequencies of the nanotube under pre- and post-buckling conditions has been investigated in Section 4. Summary andconclusions are presented in Section 5.

2. Mathematical modeling

The system under consideration is composed of a uniform nanotube with internal fluid flow, as shown in Fig. 1. The nano-tube is simply-supported at both ends with x, t and w(x, t) showing the axial, time coordinates, and the transverse displace-ment, respectively. The length, elasticity modulus, mass per unit length, and cross-section area of the nanotube are denoted,respectively, by L, E, m, and A.

Ux

z

Fig. 1. Schematic of a nanotube containing internal fluid flow.

H.L. Dai et al. / International Journal of Engineering Science 87 (2015) 13–22 15

To derive the governing equation of motion of the present system, several assumptions are considered. First, we use theEuler–Bernoulli beam assumption to model the nanotube, i.e. the shear deformation and rotary inertia are neglected. Second,we consider the geometric nonlinearity which is only due to the stretching effect of the mid-plane of the nanotube. Third, theinternal fluid flow is assumed to be a steady plug-like flow. It should be mentioned that at nanoscales, the flow velocity pro-file may be nonuniform over the cross-section. However, the effect of internal fluid flow may be viewed as an ‘equivalent’uniform flow, according to the exploration discussed by Guo, Zhang, and Païdoussis (2010). Thus, we define an ‘equivalent’mass of internal fluid per unit length as M, flowing with an ‘equivalent’ steady flow velocity U.

The strain–displacement equation for a geometrically nonlinear Euler–Bernoulli beam accounting for the mid-planestretching can be written as (Pupov & Balan, 1998; Soltani, Kassaei, & Taherian, 2014)

exx ¼ u0 þ 12

w02

� �� zw00 ¼ e0 þ zj ð1Þ

where u is the axial displacement of a material point on the mid-plane (i.e. z = 0); the prime (0) denotes partial differentiation

with respect to x; exx is the normal strain at an arbitrary point of the nanotube at a distance z from the mid-plane; e0 and jare, respectively, the nonlinear longitudinal strain and curvature.

To derive the equations of motion of the nanotube, we use the extended Hamilton’s principle which can be written as

Z t2

t1

ðdTb þ dTf � dVÞdt ¼ 0 ð2Þ

where Tb and Tf are, respectively, the kinetic energy of the nanotube and the fluid, and V is the total potential energy of thenanotube.

The variation of the total potential energy of the system can be expressed as follows

dV ¼Z L

0

ZArxxdexx ¼

Z L

0Nxd u0 þ 1

2w0

2� �

�Mxdw00� �

dx ð3Þ

where rxx is the normal stress. Nx and Mx are the axial force and bending moment per unit length, respectively, which areexpressed as

Nx ¼Z

ArxxdA; Mx ¼

ZA

zrxxdA ð4Þ

The velocity of the nanotube element can be written in the form of

~Vb ¼@~r@t¼ _rx

~iþ _rz~k ð5Þ

where~r is the vector position to a point measured from the origin, and the over-dot ð�Þ denotes partial differentiation withrespect to time t. The velocity of the center of the fluid element is (Paidoussis, 1998)

~Vf ¼ ~Vb þ U~s ð6Þ

where ~s is the unite vector tangential to the nanotube and is given as

~s ¼ @rx

@s~iþ @rz

@s~k ð7Þ

where s means the length along the nanotube. Substituting Eq. (7) into Eq. (6), the flow velocity is written as

~Vf ¼@

@tþ U

@

@s

� �rx~iþ rz

~k� �

ð8Þ

Recalling that rz = w, @rx@s ’ 1 and @rx

@t � Oðe2Þ ’ 0, thus, the velocities of the nanotube and fluid flow can be, respectively,expressed as:

~Vb ¼ _w~k ð9Þ

~Vf ¼ U~iþ @w@tþ @w@x

U� �

~k ð10Þ

Therefore, variations of the kinetic energy of the nanotube and the fluid can be respectively stated as:

dTb ¼Z L

0

ZAqb~Vbd~VbdAdx ¼

Z L

0m _wd _wdx ð11Þ

dTf ¼Z L

0

ZAqf~Vf d~Vf dAdx ¼

Z L

0Mð _wþ Uw0Þdð _wþ Uw0Þdx ð12Þ

16 H.L. Dai et al. / International Journal of Engineering Science 87 (2015) 13–22

Substituting Eqs. (3), (11), (12) into Eq. (2) and using the standard variation techniques, the following correspondingvariation results are obtained

Z t2

t1

dVdt ¼Z t2

t1

Z L

0�N0xdu� ðNxw0Þ0dw�M00

xdw�

dxdt

¼Z t2

t1

NxdujL0 þ Nxw0dwjL0 �Mxdw0jL0 þM0xdwjL0

� �dt

þZ t2

t1

�Z L

0N0xdudx�

Z L

0ðNxw0Þ0dwdx�

Z L

0M00

xdwdx� �

dt ð13Þ

Z t2

t1

dTbdt ¼Z L

0

Z t2

t1

_wd _wdtdx ¼Z L

0m _wdwjt2

t1

� �dxþ

Z L

0�Z t2

t1

m €wdwdt� �

dx ð14Þ

Z t2

t1

dTf dt ¼Z L

0Mð _wþ Uw0Þdwjt2

t1

h idxþ

Z t2

t1

Mð _wþ Uw0ÞUdwjL0h i

dt þZ L

0�Z t2

t1

Mð €wþ U _w0Þdwdt� �

dx

þZ t2

t1

�Z L

0Mð _w0 þ Uw00ÞUdwdx

� �dt ð15Þ

Using the boundary conditions of a simply-supported nanotube, the equations of motion are obtained as

N0x ¼ 0 ð16Þ

ðmþMÞ€wþ 2MU _w0 þMU2w00 ¼ ðNxw0Þ0 þM00x ð17Þ

According to Eringen’s nonlocal elasticity theory (Eringen, 2002), the stresses at a point in the body not only depend onthe strain at that point, but also on the strains at all other points of the body. Thus, the nonlocal constitutive relation for thenormal stress is given by

rxx � ðe0aÞ2 @2rxx

@x2 ¼ Eexx ð18Þ

where e0 is a constant appropriate to each material and a is an internal characteristic length, such as the lattice parameter ormolecular diameter.

The axial force and bending moment per unit length Nx and Mx can be obtained by integrating both sides of Eq. (18) overthe cross-section area, respectively, i.e.

Z

ArxxdA� ðe0aÞ2 @2

@x2

ZArxxdA ¼ E

ZAexxdA ð19Þ

ZA

zrxxdA� ðe0aÞ2 @2

@x2

ZA

zrxxdA ¼ EZ

AzexxdA ð20Þ

Therefore,

Nx � ðe0aÞ2N00x ¼ EAe0 ð21Þ

Mx � ðe0aÞ2M00x ¼ �EIw00 ð22Þ

In the light of Eqs. (16) and (21), one obtains

Nx ¼ EAe0 ð23Þ

and integrate Eq. (23) once with respect to x, we get

EAuðxÞ þ EA2

Z x

0½w0ðnÞ�2dn ¼ Nxxþ c ð24Þ

where n is a dummy variable and c is a constant, since the nanotube ends do not move, i.e. u(0) = u(L) = 0, the stress resultantNx can be expressed as follows:

Nx ¼EA2L

Z L

0w0

2dx ð25Þ

H.L. Dai et al. / International Journal of Engineering Science 87 (2015) 13–22 17

From Eqs. (17) and (22), we obtain

Mx ¼ ðe0aÞ2 ðmþMÞ €wþ 2MU _w0 þMU2w00 � ðNxw0Þ0h i

� EIw00 ð26Þ

Substituting Eq. (26) into Eq. (22), the equation of the transverse motion of the nanotube can be written as

EIw0000 � ðe0aÞ2 ðmþMÞ €wþ 2MU _w0 þMU2w00 � ðNxw0Þ0h i00

� ðNxw0Þ0 þ ðmþMÞ €wþ 2MU _w0 þMU2w00 ¼ 0 ð27Þ

Substituting Eqs. (16) and (25) into Eq. (27), we get

EIw0000 �ðe0aÞ2 ðmþMÞ €w00 þ2MU _w000 þ ðMU2�EA2L

Z L

0w0

2dxÞw0000

� �þ MU2�EA

2L

Z L

0w0

2dx

� �w00 þ2MU _w0 þðmþMÞ €w¼0

ð28Þ

Using the following dimensionless quantities,

g ¼ wL; n ¼ x

L; s ¼ EI

M þm

� �1=2 t

L2 ; b ¼ MM þm

;u� ¼ MEI

� �1=2

UL; c ¼ AL2

I; en ¼

e0aL

ð29Þ

Eq. (28) can be rewritten in the dimensionless form as:

1� e2n u�2 � 1

2cZ 1

0

@g@n

� �2

dn

" #( )@4g@n4 þ u�2 � 1

2cZ 1

0

@g@n

� �2

dn

" #@2g@n2 þ

@2g@s2

þ 2ffiffiffib

pu�

@2g@n@s

� e2n

@4g@n2@s2

þ 2ffiffiffib

pu�

@4g@n3@s

!¼ 0 ð30Þ

3. Buckling instability and post-buckling configuration

It is noted that as the flow velocity increases to a critical value (u�cr), the nanotube can display instabilities, i.e. buckling forsupported systems, or flutter for cantilevered systems (Paidoussis, 1998). To determine the buckled configuration of thenanotube, a static analysis is performed in which all time-dependent terms in Eq. (30) are dropped. Therefore, the governingequation of the buckled configuration is given by:

1� e2n u�2 � 1

2cZ 1

0

dgdn

� �2

dn

" #( )d4gdn4 þ u�2 � 1

2cZ 1

0

dgdn

� �2

dn

" #d2gdn2 ¼ 0 ð31Þ

Following Nayfeh and Emam (2008), the exact solution to Eq. (31) can be determined. As mentioned in the foregoing, theintegral in Eq. (25) (and hence in Eq. (31)) is a constant for a given g(n). Therefore, one can assume that integral to be

C ¼ 12

Z 1

0

dgðnÞdn

� �2

dn ð32Þ

Then, Eq. (31) can be rewritten as:

d4gðnÞdn4 þ k2 d2gðnÞ

dn2 ¼ 0 ð33Þ

where

k2 ¼ u�2 � cC1� e2

n½u�2 � cC� ð34Þ

For simply-supported boundary conditions, the following displacement field is assumed

gðnÞ ¼ c0 sin kn ð35Þ

where c0 is unknown to be determined. Using Eq. (35) and the corresponding boundary conditions yields

sin k ¼ 0 ð36Þ

Solving this characteristic equation, it is easy to find that the first two eigenvalues are p and 2p, which corresponds to thefirst and second static buckled configurations, respectively. Substituting Eq. (32) into Eq. (34) and using Eq. (35), one obtains

k2 ¼u�2 � 1

4 cc20k

2

1� e2n u�2 � 1

4 cc20k

2� ð37Þ

18 H.L. Dai et al. / International Journal of Engineering Science 87 (2015) 13–22

or

Fig.

c0 ¼ �2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu�2 � k2 1� e2

nu�2� �

ck2 þ e2nck4

sð38Þ

Thus, using Eq. (38), the constant c0 corresponding to any eigenvalue k may be calculated analytically, and hence the sta-tic buckled configurations of the nanotube can be determined. For other types of boundary conditions, similar procedurecould be performed for predicting the buckled configurations of the nanotube. In this paper, however, the first buckled con-figuration of simply-supported nanotubes conveying fluid will be given. The plotted curves in Fig. 2 show the positive buck-led displacement (g(0.5)) of the nanotube as a function of the flow velocity for three different values of nonlinear termcoefficient c and when en = 0.1. It is noted that, for low flow velocities (e.g., u⁄ = 2.5), the nanotube is stable and hencethe static transverse deflection is zero, indicating that the nanotube remains in its undeformed static equilibrium state(monostable configuration). For higher flow velocities beyond a certain critical value (u�cr = 2.997 when en = 0.1), however,the system becomes unstable, the form of instability being buckling (bistable configuration). Clearly, it follows from the plot-ted curves in Fig. 2 that an increase in the nonlinear term coefficient c is accompanied by a decrease in the dimensionlessbuckled displacement. This result is expected because the nonlinear term coefficient is only present in the denominator ofthe amplitude of the static buckled displacement c0, as shown in Eq. (38).

Fig. 3 shows the variations of the static buckled displacement of the nanotube as a function of the flow velocity for threedifferent values of en when c = 500. We note that an increase in the nonlocal parameter results in a significant decrease in thecritical flow velocity. It is also demonstrated that higher values of the static displacement are obtained when increasing thenonlocal parameter.

The evolution of the critical flow velocity as a function of the nonlocal parameter is presented in Fig. 4. Clearly, the criticalflow velocity is not linearly dependent on the nonlocal parameter. In fact, the variation of the critical flow velocity is neg-ligible for small values of the nonlocal parameter. For higher values of the nonlocal parameter (en > 0.05), a fast decreasein the critical flow velocity is obtained when increasing the value of the nonlocal parameter. We can conclude that the non-local effect strongly affects the onset of buckling and tends to destabilize the system’s response.

4. Post-buckling natural frequencies

It can be obtained from the previous section that the first positive buckled equilibrium can be written as

~gðnÞ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu�2�p2ð1�e2

nu�2Þcp2þe2

ncp4

rsinpn. In order to investigate the dynamics of the nanotube under the post-buckling condition, we

introduce

gðn; sÞ ¼ ~gðnÞ þ gðn; sÞ ð39Þ

where gðn; sÞ is a small dynamic disturbance around the first buckled configuration gðn; sÞ. Substituting Eq. (39) into Eq. (30),one can obtain

1� e2n u�2 � 1

2cZ 1

0

d~gdnþ @g@n

� �2

dn

" #( )d4 ~gdn4 þ

@4g@n4

!� e2

n@4g

@n2@s2þ 2

ffiffiffib

pu�

@4g@n3@s

!

þ u�2 � 12cZ 1

0

d~gdnþ @g@n

� �2

dn

" #d2 ~gdn2 þ

@2g@n2

!þ @

2g@s2 þ 2

ffiffiffib

pu�

@2g@n@s

¼ 0 ð40Þ

Then, we use the equation of motion of the static buckled configuration which is given by

2 3 4 50

0.05

0.1

0.15

Flow velocity

Dis

plac

emen

t

γ = 250γ = 500γ = 1000

2. Variations of the buckled displacement of the nanotube as a function of the flow velocity at x = 0.5L for different values of c when en = 0.1.

2 3 4 50

0.05

0.1

0.15

Flow velocityD

ispl

acem

ent

en= 0.00

en= 0.15

en= 0.30

Fig. 3. Variations of the buckled displacement of the nanotube as a function of the flow velocity at x = 0.5L for different values of en when c = 500.

0 0.1 0.2 0.32

2.3

2.6

2.9

3.2

Nonlocal parameter

Crit

ical

flow

vel

ocity

Fig. 4. Variations of the critical flow velocity (u�cr) as a function of the nonlocal parameter en when c = 500.

H.L. Dai et al. / International Journal of Engineering Science 87 (2015) 13–22 19

1� e2n u�2 � 1

2cZ 1

0

d~gdn

� �2

dn

" #( )d4 ~gdn4 þ u�2 � 1

2cZ 1

0

d~gdn

� �2

dn

" #d2 ~gdn2 ¼ 0 ð41Þ

Consequently, the equation of motion around the first buckled equilibrium for the simply-supported nanotube conveyingfluid based on the nonlocal theory can be obtained (we remove the hat for g):

@4g@n4 � e2

n u�2 � 12cZ 1

0

d~gdn

� �2

dn

" #@4g@n4 þ e2

n12cZ 1

0

@g@n

� �2

dnþ cZ 1

0

d~gdn

@g@n

dn

" #d4 ~gdn4 þ

@4g@n4

!

þ u�2 � 12cZ 1

0

d~gdn

� �2

dn

" #@2g@n2 �

12cZ 1

0

@g@n

� �2

dnþ cZ 1

0

d~gdn

@g@n

dn

" #d2 ~gdn2 þ

@2g@n2

!þ @

2g@s2

þ 2ffiffiffib

pu�

@2g@n@s� e2

n@4g

@n2@s2þ 2

ffiffiffib

pu�

@4g@n3@s

!¼ 0 ð42Þ

By dropping the nonlinear and damped terms in Eq. (42), the linear free vibration problem can be obtained, the result is

@2g@s2 � e2

n@4g

@n2@s2þ @

4g@n4 � e2

n u�2 � 12cZ 1

0

d~gdn

� �2

dn

" #@4g@n4 þ u�2 � 1

2cZ 1

0

d~gdn

� �2

dn

" #@2g@n2

þ e2ncZ 1

0

d~gdn

@g@n

dnd4 ~gdn4 � c

Z 1

0

d~gdn

@g@n

dnd2 ~gdn2 ¼ 0 ð43Þ

The first two natural frequencies are discussed in this paper; therefore, a second-order modal truncation is adopted totransform the partial differential equation (PDE) into the ordinary differential equation (ODE). To this end, we introduce

gðn; sÞ ¼X2

i¼1

/iðnÞriðsÞ ð44Þ

20 H.L. Dai et al. / International Journal of Engineering Science 87 (2015) 13–22

where /i(n) and ri(s) are the mode shape functions and the corresponding generalized modal coordinates. Following Nayfehand Emam (2008), the post-buckling mode shape functions are the same as the pre-buckling mode functions for the simply-supported beam systems which are given by

Fig. 5.b = 0.2

/iðnÞ ¼ffiffiffi2p

sinðipnÞ ð45Þ

that simultaneously satisfy the orthogonality condition

Z 1

0/iðxÞ/jðxÞdx ¼

0; i – j1; i ¼ j

ð46Þ

and the following boundary conditions

gðn ¼ 0; sÞ ¼ gðn ¼ 1; sÞ ¼ 0 ð47Þ

g00ðn ¼ 0; sÞ ¼ g00ðn ¼ 1; sÞ ¼ 0 ð48Þ

Substituting Eqs. (44), (45) into Eq. (43) and applying the orthogonality conditions, we can write the Eq. (43) in the fol-lowing form

½M�f€rg þ ½C�f _rg þ ½K�frg ¼ 0 ð49Þ

where M, C, and K are 2-order matrices, then the natural frequencies of the nanotube can be obtained by solving theimaginary parts of the eigenvalues of the following matrix E

E ¼0 I

�M�1K �M�1C

� �ð50Þ

In which I represents the 2-order unit matrix.The plotted curves in Fig. 5 show the variations of the first two natural frequencies of the nanotube as function of the flow

velocity for different values of nonlocal parameter (en) when b = 0.2 and c = 500. In the pre-buckled region (the flow velocityis smaller than its critical value), it is clear that an increase in the flow velocity is accompanied by a decrease in the first twonatural frequencies of the nanotube. At buckling, the first natural frequency becomes equal to zero and the system loses itsmonostable equilibrium. In the post-buckling region, the system becomes bistable (two non-zero stable solutions and oneunstable solution) and the first two natural frequencies increase as the flow velocity is increased. Inspecting Fig. 5, in thecase when en is set equal to zero, the first two frequencies decrease when the flow speed varies from 0 to p (the critical flowvelocity u�cr), then it increases when the flow speed is beyond p. In the pre-buckling region, the results show that an increasein the nonlocal parameter (en) is followed by a rapid decrease in the first two natural frequencies of the nanotube. For exam-ple, the critical flow velocity u�cr decreases from p to 2.286 when en increases from 0 to 0.3. This result is expected from thebuckling problem analysis, as shown in Fig. 4. In the post-buckling region, the first natural frequency increases when thenonlocal parameter en is increased. On the other hand, the second natural frequency is smaller when the nonlocal parameteren is increased for nondimensional flow velocity values smaller than 5. Therefore, the existence of nonlocal parameterstrongly affects the critical flow velocity as well as the natural frequencies of the nanotube.

The effects of the mass ratio b and the nonlinear term coefficient c on the first two natural frequencies are presented,respectively, in Figs. 6 and 7. It is noted that the mass ratio does not affect the nondimensional critical flow velocity. Thisis expected because the critical flow speed does not depend on the mass ratio, as shown in Eqs. (37) and (38). Furthermore,it follows from the plotted curves in Fig. 6 that an increase in the mass ratio b results in a quick variation in the first twofrequencies when the nanotube behaves in the pre-buckling region. In the post-buckling region, the first natural frequencydecreases as increasing the mass ratio. However, the second natural frequency of the nanotube increases when increasingthe mass ratio in the whole flow velocity range.

0 1 2 3 4 5 60

5

10

15

20

25

Flow velocity

The

first

freq

uenc

y

en=0.0

en=0.1

en=0.2

en=0.3

0 1 2 3 4 5 625

30

35

40

Flow velocity

The

seco

nd fr

eque

ncy

en=0.0

en=0.1

en=0.2

en=0.3

Variations of the first two natural frequencies of the nanotube as function of the flow velocity for different values of the nonlocal parameter whenand c = 500.

0 1 2 3 4 5 60

5

10

15

20

25

Flow velocity

The

first

freq

uenc

y

β=0.2β=0.5β=0.8

0 1 2 3 4 5 632

36

40

44

48

Flow velocity

The

seco

nd fr

eque

ncy

β =0.2β =0.5β =0.8

Fig. 6. Variations of the first two natural frequencies of the nanotube as function of flow velocity for different values of b when en = 0.1, c = 500.

0 1 2 3 4 5 60

5

10

15

20

25

Flow velocity

The

first

freq

uenc

y

γ =500γ =2000γ =5000

0 1 2 3 4 5 632

34

36

38

40

Flow velocity

The

seco

nd fr

eque

ncy

γ =500γ =2000γ =5000

Fig. 7. The first two natural frequencies of the nanotube as function of the flow velocity for different values of c when en = 0.1, b = 0.2.

H.L. Dai et al. / International Journal of Engineering Science 87 (2015) 13–22 21

Clearly, the nonlinear term coefficient c does not affect the first two natural frequencies of the nanotube when increasingthe flow velocity, as shown in Fig. 7. This result is due to the fact that the mode shape of a simply-supported beam does notdepend on the buckling static position, as shown in Eq. (45). For a fixed-hinged or fixed–fixed nanotube, the mode shapes aredependent on the static buckling position as demonstrated in the work of Nayfeh and Emam (2008) and hence the nonlinearterm coefficient may affect the natural frequencies of the nanotube.

5. Conclusions

The nonlinear dynamic responses of simply-supported nanotubes conveying fluid have been investigated by developing ageneral nonlinear model based on nonlocal elasticity theory. The nonlinear equation of motion was derived by consideringthe geometric nonlinearity associated with mid-plane stretching of the nanotube. It was shown that the nanotube conveyingfluid remains in its undeformed static equilibrium state (monostable configuration) at low flow velocities while undergoes abuckling instability at a higher critical flow velocity for different values of the nonlocal parameter. In the post-bucklingregion (bistable configuration), the buckled displacement increases as the flow velocity is increased. It was demonstratedthat the nonlocal parameter decreases the critical flow velocity and hence tends to destabilize the nanotube’s response.The results showed that the buckled displacement of the nanotube would be increased when considering the nonlocal effect.More importantly, it was also shown that the first two (pre- and post-buckling) natural frequencies of the nanotube arestrongly dependent on the nonlocal parameter.

The buckling instability discussed in the foregoing occurs in the range u⁄ = 2–4 generally. The corresponding dimensionalflow velocities are of real concern for engineering applications (Yoon et al., 2006). Application of this work is related to thestatic stability and buckling configuration of targeted drug delivery devices that may be characterized as nanotubes contain-ing internal fluid flow. It should be noted that one of the basic requirements in the design of high-performance drug deliverydevices is to ensure the stability of the system. Yet, another potential application of this research study is the nanofluidicdevice, an instrument in which the nanotube/nanochannel is used to transport fluids to attain specific mechanical behaviorof interest, which may find application in micro- and nano-technology (Paidoussis, 2008; Rinaldi, Prabhakar, Vengallatore, &Paidoussis, 2010).

Acknowledgments

The authors gratefully acknowledge the support provided by the National Natural Science Foundation of China (11172107and 11172109), the Natural Science Foundation of Hubei Province (2013CFA130, 2014CFA124) and the FundamentalResearch Funds for the Central Universities, HUST (2014YQ007).

22 H.L. Dai et al. / International Journal of Engineering Science 87 (2015) 13–22

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