analytical analysis of buckling and post-buckling of fluid conveying multi-walled carbon nanotubes

Applied Mathematical Modelling 37 (2013) 4972–4992

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Applied Mathematical Modelling

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Analytical analysis of buckling and post-buckling of fluid conveyingmulti-walled carbon nanotubes

Arman Ghasemi, Morteza Dardel ⇑, Mohammad Hassan Ghasemi, Mohammad Mehdi BarzegariDepartment of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran

a r t i c l e i n f o

Article history:Received 24 December 2011Received in revised form 21 August 2012Accepted 25 September 2012Available online 12 October 2012

Keywords:Carbon nanotubeNonlinear Timoshenko beamPost-bucklingZero structural stiffnessVan der Waals force

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.09.061

⇑ Corresponding author. Address: P.O. Box 484, DE-mail address: [email protected] (M. Dardel).

a b s t r a c t

In this work, buckling and post-buckling analysis of fluid conveying multi-walled carbonnanotubes are investigated analytically. The nonlinear governing equations of motionand boundary conditions are derived based on Eringen nonlocal elasticity theory. Thenanotube is modeled based on Euler–Bernoulli and Timoshenko beam theories. The VonKarman strain–displacement equation is used to model the structural nonlinearities. Fur-thermore, the Van der Waals interaction between adjacent layers is taken into account.An analytical approach is employed to determine the critical (buckling) fluid flow velocitiesand post-buckling deflection. The effects of the small-scale parameter, Van der Waals force,ends support, shear deformation and aspect ratio are carefully examined on the criticalfluid velocities and post-buckling behavior.

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1. Introduction

In the recent years, carbon nanotubes (CNTs) have been used as nanotubes for conveying fluids due to their excellentmechanical properties and hollow cylindrical geometry. So, they are broadly used in fabricating nano-fluidic devices suchas conveying aqueous fluids, resonance sensing, delivering drugs to a specific organ or syringe needles for interrogationof cells [1,2].

Investigating mechanical properties of the CNTs is important to characterize their behavior. Experimental techniques forexamining the mechanical characteristics in nano-scales are difficult and expensive; hence, in the last decade, the compu-tational and theoretical simulation models have been developed in many works. Most researches of nano-scale system havebeen carried out based on two methods, molecular dynamic (MD) [3–7] and size depended continuum theories [8–24]. Dueto the fact that the surface to volume ratio is considerable in nano-scale systems, the size effects are not negligible anymore.Classic continuum models are unable to take into account the size effects; therefore, some modified continuum models, suchas Eringen nonlocal elasticity theory have been proposed to deal with this problem [8–10]. In the Eringen nonlocal theory,size-dependent equations are achieved by incorporating an internal characteristic length (such as lattice spacing betweenindividual atoms) and a constant which is determined from experiments or by matching dispersion curves of atomic-latticedynamics [8–11]. This model has a good agreement with the corresponding results of molecular dynamics approach [25,26].

In the framework of complex structures, MD analysis is very complicated [11,14]; consequently, systems with small num-ber of atoms are proper to be fully analyzed based on MD solution [3,4,12,14]. Furthermore, some researchers used the ap-proach of mixing MD method with the nonlocal elasticity theory, in which at first the MD method is used to determinenonlocal parameter of a fully physically defined nano-structure then, the nonlocal elasticity is used to investigate the sys-tem’s behavior [5–7].

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A. Ghasemi et al. / Applied Mathematical Modelling 37 (2013) 4972–4992 4973

Also there are many works that their main purpose is to study the nonlocal parameter’s influence on the behavior of thesystem at some typical nonlocal parameter values by applying nonlocal theory [12–24]. These works did not relate to specificcases as these works have not fully defined physical properties.

In actual cases, calibrating of the nonlocal parameter depends on different parameters, such as diameter, aspect ratio,mode number, boundary conditions, Van der Waals effect, mass distribution of system, room temperature and the nanotubestructural geometric (having zigzag or armchair shape) [4–7,27]. Accordingly, in many researches, a typical value is chosenfor nonlocal parameter [12–24].

Wang and Liew [22] presented the application of nonlocal continuum mechanics of static analysis of micro and nano-structures. Adhikari and Chowdhury [27] derived the calibration constants necessary for employing single-walled carbonnanotubes. Chowdhury et al. [28] studied the vibration of multi-walled carbon nanotubes (MWCNTs) with an emphasison the effect of the mixed boundary conditions. There are some works such as [29–33] that investigate the vibration andinstability of fluid conveying CNTs These works used classic elasticity and linear Euler–Bernoulli theory in their analysis.Yao and Han [34] investigated axially compressed buckling of a MWCNT under temperature field. They used classic elasticityand Donnel shell theory. Chang [13] investigated thermal–mechanical vibration and instability of a fluid-conveying SWCNTembedded in an elastic medium based on nonlocal elasticity theory. Chang and Liu [21] studied small scale effect on flow-induced instability of double-walled carbon nanotubes (DWCNTs). In [13,21], the system is analyzed based on linear non-local Euler–Bernoulli beam model. Hao et al. [16] studied the small-scale effect on torsional buckling of MWCNTs. Sunet al. [23] obtained torsional buckling toque of MWCNTs embedded in an elastic medium using Donnel shell theory. Moham-madimehr et al. [24] studied the torsional buckling of a DWCNT embedded in Winkler and Pasternak foundations using non-local theory. Amara et al. [12] carried out nonlocal elasticity effect on column buckling of MWCNTs under temperature field.Murmu and Pradhan [19] proposed buckling solution of a SWCNT embedded in an elastic medium based on nonlocal Tim-oshenko beam theory. Narendar and Gopalakrishnan [20] calculated critical buckling temperature of SWCNTs embedded in aone-parameter elastic medium based on nonlocal Timoshenko beam. Yan et al. [15] investigated the nonlocal effect on axi-ally compressed buckling of triple-walled carbon nanotubes (TWCNTs) under temperature field.

In the many of the aforesaid works, just the buckling analysis of CNTs is investigated, and the post-buckling analysis hasbeen neglected. Shen and Zhang [35] presented buckling and post-buckling analysis for DWCNTs subjected to combined ax-ial and radial loads in thermal environments. Yao and Han [36] investigated Torsional buckling and post-buckling equilib-rium path of DWCNT. In [35,36], the classic continuum is used to model the systems. Setoodeh et al. [18] proposed exactsolution for post-buckling of SWCNTs based on nonlocal Euler–Bernoulli beam theory.

Since CNTs under compressive loading, such as fluid conveying nanotubes with both fixed end are exposed to the buck-ling kind of instability, investigation of buckling and post-buckling behavior of them is important [18,37].

In the present paper, both Euler–Bernoulli and Timoshenko beam theories are used to study the buckling and post-buckling behavior of SWCNTs and MWCNTs. The equations of motion and boundary conditions are derived based onthe Von Karman strain–displacement equation and nonlocal Eringen stress field. The Van der Waals interaction betweenadjacent layers is modeled as a distributed spring. The work done by centrifugal force of fluid flow is considered in bothtransverse and in-plane direction. It must be noted that for a pipe and beam with moderate or small aspect ratio, sheardeformation effects is not negligible; consequently, Timoshenko beam theory is more suitable than Euler–Bernoulli[7,11,19,20,38,39].

2. Formulation of the problem

Fig. 1 shows a multi walled MWCNT which conveys fluid. The ith layer has the flexural rigidity EIpi, the cross section area

Api, the equal thickness h and the length L. In addition, there is Van der Waals radius between adjacent layers, in which the

Van der Waals force exists. The fluid which flows in the inner layer, has density per unit length mf and steady axial flowvelocity U.

(a) (b)

nr

1r

h

2r

x

zVan der Waals

radius

U

Fig. 1. (a) The schematic of fluid-conveying MWCNT and (b) the cross section of MWCNT.

4974 A. Ghasemi et al. / Applied Mathematical Modelling 37 (2013) 4972–4992

2.1. Equation of motion in accordance with the Euler–Bernoulli beam theory

The Euler–Bernoulli beam theory displacement field is given by [40]:

uiðx; zÞ ¼ u0iðxÞ � zw00i

; wiðx; zÞ ¼ w0iðxÞ; ð1Þ

where the prime (0) denotes partial differentiation with respect to x. ui(x,z) and wi(x,z) are total displacement along x and zaxis of layer i, respectively. Subscript ‘0’ denotes displacement of the mid-plane axis. The nonlinear strain–displacement rela-tionship is given by [40]:

�xxi¼ �zw000i

þ 0:5w020iþ u00i

: ð2Þ

The variation of the potential energy ðdVÞ is:

dV ¼XN

i¼1

Z L

0

ZZApi

rxxid�xxi

dApidx: ð3Þ

Fig. 2 shows an element of the inner layer of MWCNT before and after deformation. For enough small value of the rotation ofmid-plane axis – h –, it is admissible to assume that, tan h ¼ �w001

� h, sin h ¼ h, cos h ¼ 1 and dw01 � dx. As a result, the cur-vature radius of the element – R – can be written as [40]:

1R¼ dh

ds� �

dw001

ds; ds ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidx2 þ dw2

01

q� dx! 1

R� �w0001

: ð4Þ

The centrifugal force of fluid flow per unit length is:

Fc ¼ mfU2

R¼ �mf U2w0001

: ð5Þ

Therefore, according to Fig. 2, the variation of the work is done by centrifugal force of fluid flow is:

dWC ¼Z L

0ðFC sin hdu01 þ FC cos hdw01 Þdx ¼

Z L

0ðmf U2w0001

w001du01 �mf U

2w0001dw01 Þdx: ð6Þ

MWCNTs can be considered as a set of concentric SWNTs with the Van der Waals interaction between nearby layers [38]. TheVan der Waals pressure per unit length, that exerts on the ith layer of the nanotube by the (i � 1) th and the (i + 1) th layersare [38]:

FVi�1!i¼ �ci�1ðw0i

�w0i�1Þð1�m0Þ; FViþ1!i

¼ ciðw0iþ1�w0i�1

Þð1� n0Þ; i ¼ 1; . . . ;N; ð7Þ

where m0 = 1 for i = 1, m0 = 0 for i – 1, n0 = 1 for i = N and n0 = 0 for i – N. Moreover, ci – the Van der Waals interaction coef-ficient between ith and (i + 1) th layer – is given by [38]:

ci ¼320� 2ri ðerg=cm2Þ

0:16D2 ; D ¼ 0:142 ðlength of carbon — carbon boundÞ: ð8Þ

The variation of the work done by Van der Waals force is:

dWV ¼XN

i¼1

Z L

0ðFVi�1!i

þ FViþ1!iÞdw0i

dx: ð9Þ

By applying extended Hamilton principle to Eqs. (3), (6), (9), the system’s differential equations of motions can be expressedas follows:

Fig. 2. An element of fluid-conveying of inner layer MWCNT.


� N0i �m0mf U2w0001w001¼ 0;

�M00i � ðNiw0i0 Þ

0 þm0mf U2w0001þ ð1�m0Þci�1ðw0i

�w0i�1Þ � ð1� n0Þciðw0iþ1

�w0iÞ ¼ 0; i ¼ 1; . . . ;N

ð10Þ

and the boundary conditions are:

Nidu0ijL0 ¼ 0; �Midw00i

jL0 ¼ 0; ðM0i þ Niw00i

Þdw0ijL0 ¼ 0; i ¼ 1; . . . ;N; ð11Þ

Ni and Mi are resultant of axial stress and area moment of axial stress on the cross section of layer i, respectively, which aredefined as follows:

Ni ¼ZZ

Api

rxxidApi

; Mi ¼ZZ

Api

zrxxidApi

; i ¼ 1; . . . ;N: ð12Þ

In Eq. (12), rxxiis the nonlocal axial stress. According to the Eringen nonlocal theory the differential type of relationship be-

tween nonlocal stress tensor (r) and Hookean stress tensor (t) for slow varying field is approximated, by [8–10]:

r� l2L2$2r ¼ t; l ¼ e0aL; ð13Þ

where, e0 is a constant that depends on the system properties; a is internal characteristic length (i.e. the distance betweenindividual atoms), and L is external characteristic length (i.e. macroscopic length of system). l is nonlocal parameter anddenotes the ratio of internal to the external characteristic length of the system [8,9].

By considering Eq. (13), nonlocal axial stress can be written as:

rxxi� ðe0aÞ2r00xxi

¼ E½u00i� zw000i

þ 0:5ðw00iÞ2�; i ¼ 1; . . . ;N: ð14Þ

Using nonlocal stress field, Eqs. (10) and (12), gives M and N as follows:

Ni ¼ EApi½u00iþ 0:5ðw00i

Þ2� �m0mf U2ðe0aÞ2½w001w00001þ ðw0001

Þ2�; ð15Þ

Mi ¼ �EIpiw000iþ ðe0aÞ2½m0mf U2w0001

þ ð1�m0Þci�1ðw0i�w0i�1

Þ � ð1� n0Þciðw0iþ1�w0i

Þ � EApið1:5w000i

ðw00iÞ2 þw000i

u00i

þw00iu000iÞ� þm0mf U

2ðe0aÞ4½ðw001Þ2wð4Þ01

þ 4w00001w0001

w001þ ðw0001

Þ3�:

These following parameters are the dimensionless quantities used in the present work, and are introduced as follows.

�c ¼ffiffiffiffiffiffiffiffiffiffiffiffi

Ip1

L2Ap1

s; gi ¼

w0i

L�c; #i ¼

u0i

L�c2 ; /i ¼wi

�c; n ¼ x

L; m ¼

ffiffiffiffiffiffiffiffiffiffiffimf L

2

EIp1

sU;

l ¼ e0aL; ai ¼

Ipi

Ip1

; �ci ¼ciL

4

EIp1

; Ki ¼GL2Api

ksi

EIp1

; li ¼Api

A1:

ð16Þ

Using Eqs. (10), (11) and (15), (16) gives the dimensionless equations of motion and boundary conditions in the terms ofdisplacements in Eqs. (17) and (18). In these equations the prime (0) denotes partial differentiation with respect to n.

� lið#00i þ g0ig00i Þ �m0�c2m2½g001g01 � l2ðgð4Þ1 g01 þ 3g0001 g001Þ� ¼ 0;

aigð4Þi þm0v2g001 þ ð1�m0Þ�ci�1ðgi � gi�1Þ � ð1� n0Þ�ciðgiþ1 � giÞ

� lið1:5g00i ðg0iÞ2 þ g00i #

0i þ g0i#

00i Þ

� l2½m0m2gð4Þ1 þ ð1�m0Þðg00i � g00i�1Þ � ð1� n0Þ�ciðg00iþ1 � g00i Þ

� lið1:5gð4Þi ðg0iÞ

2 þ 9g000i g00i g0i þ 3ðg00i Þ

3 þ gð4Þi #0i þ 3g000i #00i þ 3g00i #

000i þ g0i#

ð4Þi Þ

�m0�c2m2ðgð4Þ1 ðg01Þ2 þ 4g0001 g001g

01 þ ðg001Þ

3Þ�

�m0�c2l4m2½gð6Þ1 ðg01Þ2 þ 8gð5Þ1 g001g

01 þ 14gð4Þ1 g0001 g01 þ 13gð4Þ1 ðg001Þ

2 þ 18ðg0001 Þ2g001� ¼ 0;

ð17Þ

½lið0:5ðg0iÞ2 þ #0iÞ �m0l2�c2m2ðg0001 g01 þ ðg001Þ

2Þ�d#ij10 ¼ 0; ð18Þ

½aig00i � l2ðm0m2g001 þ ð1�m0Þ�ci�1ðgi � gi�1Þ � ð1� n0Þ�ciðgiþ1 � giÞ � lið1:5g00i ðg0iÞ2 þ g00i #

0i þ g0i#

00i ÞÞ

�m0l4�c2m2ðgð4Þ1 ðg01Þ2 þ 4g0001 g001g

01 þ ðg001Þ

3Þ�dg0ij10 ¼ 0;

½�aig000i þ lið0:5ðg0iÞ3 þ #0ig0iÞ þ l2ðm0m2ðg0001 � �c2ðg0001 ðg01Þ

2 þ ðg001Þ2g01ÞÞ þ ð1�m0Þ�ci�1ðg0i � g0i�1Þ � ð1� n0Þ�ciðg0iþ1 � g0iÞ

� lið1:5g000i ðg0iÞ2 þ 3ðg00i Þ

2g0i þ g000i #0i þ 2g00i #

00i þ g0i#

000i ÞÞ þm0l4�c2m2ðgð5Þ1 ðg01Þ

2 þ 6gð4Þ1 g001g01 þ 4ðg0001 Þ

2g01 þ 7g0001 g001Þ�dgij10 ¼ 0:


2.2. Equation of motion in accordance with the Timoshenko beam theory

The Timoshenko beam theory displacement field is [40]:

uiðx; zÞ ¼ u0iðxÞ þ zwiðxÞ; wiðx; zÞ ¼ w0i

ðxÞ; ð19Þ

where wi(x) denotes the rotation of cross section of ith layer of the nanotube. The nonlinear strain–displacement relationshipis [40]:

�xxi¼ u00i

þ zw0i þ 0:5ðw00iÞ2; cxzi

¼ w00iþ wi: ð20Þ

Similar to the procedure described in the previous section, the equations of motion are as follows:

� N0i �m0mf U2w0001

w001¼ 0;

� Q 0i � ðw00iNiÞ0 þm0mf U

2w0001þ ð1�m0Þci�1ðw0i


�w0iÞ ¼ 0;

�M0i þ Q i ¼ 0; i ¼ 1; . . . ;N:

ð21Þ

And the boundary conditions are:

Nidw0ijL0 ¼ 0; Midwij

L0 ¼ 0;

ðNiw00iþ Q iÞdw0i

jL0 ¼ 0; i ¼ 1; . . . ;N:ð22Þ

Qi is the resultant of the shear stress on the cross section of the ith layer, and is defined as:

Qi ¼ ksi

ZZApi

rxzidApi

; i ¼ 1; . . . ;N; ð23Þ

where ks is the shear correction coefficient for circular tube and it is given by [39]:

ksi¼ 6ð1þ mÞð1þ siÞ2

ð7þ 6mÞð1þ siÞ2 þ ð20þ 12mÞs2i

; i ¼ 1; . . . ;N; ð24Þ

where m is Poisson’s ratio and si is the ratio of inner radius to the outer radius of ith layer. Using Eq. (13), gives the resultantnonlocal axial and nonlocal shear stresses as follows:

rxxi� ðe0aÞ2r00xxi

¼ Eðu00iþ zw0i þ 0:5ðw00i

Þ2Þ;

rxzi� ðe0aÞ2r00xzi

¼ Gðwi þw00iÞ; i ¼ 1; . . . ;N:

ð25Þ

By using Eqs. (13) and (31), Ni, Mi and Qi: will be acquired as the next equation:

Ni ¼ EApiðu00iþ 0:5ðw00i

Þ2Þ � ðe0aÞ2m0mf U2ðw0000iw00iþ ðw000i

Þ2Þ;

Qi ¼ GApiksiðw00iþwiÞ þ ðe0aÞ2½m0mf U

2w0000iþ ð1�m0Þci�1ðw00i


�w00iÞ� EApi

ð3ðw000iÞ2w00i

þ1:5ðw00iÞ2w0000i

þ u0000iw00iþ2u000i

w000iþ u00i

w0000iÞ� þ ðe0aÞ4m0mf U2½wð5Þ01

ðw001Þ2 þ 6wð4Þ01

w0001w001þ4ðw00001

Þ2w001þ7w00001

ðw0001Þ2�;ð26Þ

Mi ¼ EIpiw0i þ ðe0aÞ2m0mf U

2½w0001þ ð1�m0Þci�1ðw0i


�w0iÞ � EApi

ð1:5w000iðw0i0 Þ

2 þw000iu00i

þ u000iw00iÞ� þ ðe0aÞ4m0mf U2½wð4Þ01

ðw001Þ2 þ 4w00001

w0001w001þ ðw0001

Þ2�:

Considering Eqs. (21), (22), (26) and (16) gives dimensionless equations of motion and boundary conditions in terms of dis-placement as follows:

� lið#00i þ g0ig00i Þ �m0�c2m2½g001g01 � l2ðgð4Þ1 g01 þ 3g0001 g001Þ� ¼ 0;

�Kiðg00i þ /0iÞ þm0v2g001 þ ð1�m0Þ�ci�1ðgi � gi�1Þ � ð1� n0Þ�ciðgiþ1 � giÞ

� lið1:5g00i ðg0iÞ2 þ g00i #

0i þ g0i#

00i Þ

� l2½m0m2gð4Þ1 þ ð1�m0Þðg00i � g00i�1Þ � ð1� n0Þ�ciðg00iþ1 � g00i Þ

� lið1:5gð4Þi ðg0iÞ

2 þ 9g000i g00i g0i þ 3ðg00i Þ

3 þ gð4Þi #0i þ 3g000i #00i þ 3g00i #

000i þ g0i#

ð4Þi Þ

�m0�c2m2ðgð4Þ1 ðg01Þ2 þ 4g0001 g001g

01 þ ðg001Þ

3Þ�

�m0�c2l4m2½gð6Þ1 ðg01Þ2 þ 8gð5Þ1 g001g

01 þ 14gð4Þ1 g0001 g01 þ 13gð4Þ1 ðg001Þ

2 þ 18ðg0001 Þ2g001� ¼ 0;

� ai/00i þKiðg0i þ /iÞ ¼ 0:

ð27Þ


And boundary conditions are:

½lið0:5ðg0iÞ2 þ #0iÞ �m0l2�c2m2ðg0001 g01 þ ðg001Þ

2Þ�d#ij10 ¼ 0;

½ai/0i þ l2ðm0m2g001 þ ð1�m0Þ�ci�1ðgi � gi�1Þ � ð1� n0Þ�ciðgiþ1 � giÞ � lið1:5g00i ðg0iÞ

2 þ g00i #0i þ g0i#

00i ÞÞ

þm0l4�c2m2ðgð4Þ1 ðg01Þ2 þ 4g0001 g001g

01 þ ðg001Þ

3Þ�d/ijL0 ¼ 0;

½Kiðg0i þ /iÞ þ lið0:5ðg0iÞ3 þ #0ig0iÞ

þ l2ðm0m2ðg0001 � �c2ðg0001 ðg01Þ2 þ ðg001Þ

2g01ÞÞ þ ð1�m0Þ�ci�1ðg0i � g0i�1Þ� ð1� n0Þ�ciðg0iþ1 � g0iÞ � lið1:5g000i ðg0iÞ

2 þ 3ðg00i Þ2g0i þ g000i #

0i þ 2g00i #

00i þ g0i#

000i ÞÞ

þm0l4�c2m2ðgð5Þ1 ðg01Þ2 þ 6gð4Þ1 g001g

01 þ 4ðg0001 Þ

2g01 þ 7g0001 g001Þ�dgijL0 ¼ 0:

ð28Þ

3. Buckling and post-buckling solution methodology

The derived differential equations of motion and boundary conditions are highly nonlinear. In this paper, a methodologyis presented to obtain buckling mode shapes and post-buckling configurations analytically. For this purpose, at first, the lin-ear part of equations is solved analytically to obtain critical flow velocities and exact mode shapes. In the next stage, thepost-buckled configuration is considered as a multiplication of the exact normalized mode shapes by the amplitude of buck-ling mode. Then, substituting the assumed post-buckled configuration in nonlinear equations of motion, the post-buckledamplitude is obtained at each desired flow velocity.

3.1. Buckling mode shapes in accordance with the Euler–Bernoulli beam model

In this approach by assuming an exponential solution as gi(n) = Hiekn; and substituting it in the linear part of Eq. (21), the

following expressions can be obtained for a DWCNTs.

½ða1 � l2m2Þk4 þ ðm2 � l2�c1Þk2 þ �c1�H1 þ ðl2�c1k2 � �c1ÞH2 ¼ 0;

½a2k4 � l2�c1k

2 þ �c1�H2 þ ðl2�c1k2 � �c1ÞH1 ¼ 0:

ð29Þ

By writing above equation in matrix form and setting its determinant to zero, the characteristic equation can be written asfollows:

½a1a2 � l2a2m2�k8 þ ½a2 � l2ða1 þ a2Þ�c1 þ l4�c1m2�k6 þ ½ða1 þ a2Þ�c1 � 2l2�c1m2�k4 þ �c1m2k2 ¼ 0: ð30Þ

Solving Eq. (30) gives eight roots ki in the terms of given velocity. Accordingly the general solutions of the characteristicequation are:

g1ðnÞ ¼ C1 þ C2nþ C3ek3n þ C4e�k3n þ C5ek5n þ C6e�k5n þ C7ek7n þ C8e�k7n;

g2ðnÞ ¼ C1 þ C2nþH3C3ek3n þH4C4e�k3n þH5C5ek5n þH6C6e�k5n þH7C7ek7n þH8C8e�k7n;ð31Þ

where

Hj ¼ �l2�c1k

2j � �c1

a2k4j � l2�c1k

2j þ �c1

;

kj used in Eq. (31) is a function of fluid velocity, and is unknown. Substituting g1(n) and g2(n) obtained from Eq. (31) into thelinear part of Euler–Bernoulli boundary conditions (Eq. (18)) gives eigenvalue problem, which contains eight equations withthe nine undetermined parameters. These nine parameters are C1, . . . ,C8 in addition to the fluid velocity. The eigenvalueproblem can be written in the matrix form as follows:

½BðvÞ�8�8fC1 � � � C8 gT ¼ 0; ð32Þ

B(m) is varied according to different end supports. Eigenvalues (velocities at buckling modes mc) and ortho-normalized set ofeigenvector of fC1 � � � C8 gT , will be obtained from Eq. (32). By obtaining the value of mc, the characteristic equation’s roots,k1, . . . ,k8, can be calculated. Therefore, buckled mode shapes of first and second layers of the nanotubes will be calculatedfrom Eq. (31). From now, the first and second layers’ buckled mode shapes are denoted by g1 and g2 respectively.

3.2. Post-buckling configuration in accordance with the Euler–Bernoulli beam model

Buckling is the verge of instability at which the configurations of the system change. After buckling, the amplitude of con-figuration gradually increases with increasing the load. From mathematical point of view, the buckling problem is an eigen-value problem; so the buckled mode shape has one unknown parameter which is considered as undetermined coefficientamplitude. This coefficient is obtained from solving post-buckling problem. The post-buckled mode shapes are introducedas follows:


g1ðnÞ ¼ bg1g1ðnÞ; g2ðnÞ ¼ bg2

g2ðnÞ; ð33Þ

where bg1and bg2

are unknown post-bucked configuration’s amplitude coefficients which should be calculated. For post-buckling analysis, the nonlinear parts should be considered. If all the nonlinear terms are considered, the boundary condi-tions will get very complicated. A simple tactic to overcome this difficulty is to transfer all nonlinear terms from boundaryconditions into the differential equations, and retaining the linear parts; as a result, the linear buckled mode shape can beused directly for post-buckling analysis.

The buckled mode shape of in-plane displacement (#i) exists in nonlinear constitutive equations; however, it can besolved in terms of buckled mode shape of transverse displacement. By using in-plane equation of motion and its correspond-ing boundary conditions, in-plane displacement can be rewritten as follows:

#1ðnÞ ¼ b2g1#1ðnÞ; #2ðnÞ ¼ b2

g2#2ðnÞ; ð34Þ

#1 ¼ZZ½�ð1þ �c2m2Þg001g01þl2�c2m2ðgð4Þ1 g01þ3g0001 g001Þ�dndn

þZZ½�ð1þ �c2m2Þg001g01þl2�c2m2ðgð4Þ1 g01þ3g0001 g001Þ�dndn

��n¼0�ZZ½�ð1þ �c2m2Þg001g01þl2�c2m2ðgð4Þ1 g01þ3g0001 g001Þ�dndn

��n¼1

!n

�ZZ½�ð1þ �c2m2Þg001g01þl2�c2m2ðgð4Þ1 g01þ3g0001 g001Þ�dndn

��n¼0;

#2 ¼ �ZZ

g001g01dndnþ �

ZZg001g

01dndn

��n¼0þZZ

g001g01dndn

��n¼1

!nþ

ZZg001g

01dndn

��n¼0:

Substituting Eqs. (33) and (34) in nonlinear Euler–Bernoulli equations of motions and transferring nonlinear part of theboundary conditions into the equations of motion in addition to applying the Galerkin method, results in an algebraic equa-tion in the term of unknown post-buckling amplitude coefficients (bg1

and bg2) as follows:

bg1

Z 1

0fða1 � l2m2Þgð4Þ1 þ ðm2 � l2�c1Þg001 þ �c1g1gg1dnþ bg2

Z 1

0fl2�c1g002 � �c1g2gg1dn

þ b3g1

Z 1

0f�l1ð1:5g001g

021 þ g001#1 þ g01#

001Þ þ l2l1ð1:5gð4Þ1 g0021 þ 9g0001 g001g

01 þ 3g0031 þ gð4Þ1 #01 þ 3g0001 #

001 þ 3g001#

0001 þ g1#

ð4Þ1 Þ

þ l2�c2m2ðgð4Þ1 g0021 þ 4g0001 g001g01 þ g0031 Þ � l4�c2m2ðgð6Þ1 g021 þ 8gð5Þ1 g001g

01 þ 14gð4Þ1 g0001 g01 þ 13gð4Þ1 g021 þ 18g00021 g001Þ

� ½l2l1ð1:5g001ðg01Þ2 þ g001#

01 þ g01#

001Þ þ l4�c2m2ðgð4Þ1 ðg01Þ

2 þ 4g0001 g001g01 þ ðg001Þ

3Þ�ðd0ðn� 1Þ � d0ðnÞÞgg1dn ¼ 0; ð35Þ

bg1

Z 1

0fl2�c1g001 � �c1g1gg2dnþ bg2

Z 1

0fa2gð4Þ2 � l2�c1g002 þ �c1g2gg2dnþ b3

g2

Z 1

0f�l2ð1:5g002g

022 þ g002#2 þ g02#

002Þ

þ l2l2ð1:5gð4Þ2 g0022 þ 9g0002 g002g02 þ 3g0032 þ gð4Þ2 #02 þ 3g0002 #

002 þ 3g002#

0002 þ g2#

ð4Þ2 Þ � l2l2½1:5g002ðg02Þ

2 þ g002#02 þ g02#

002�ðd

0ðn� 1Þ� d0ðnÞÞgg2dn ¼ 0;

where, d is the Dirac delta function with the following properties:
Z 1
0d0ðn� aÞuðnÞdn ¼ �

Z 1

0dðn� aÞu0ðnÞdn;

Z 1

0dðn� aÞuðnÞdn ¼ uðaÞ; 0 6 a 6 1: ð36Þ

By solving Eq. (35), the post-buckling amplitude coefficient bg1and bg2

and consequently, the post-buckling configurationwill be obtained.

3.3. Buckling mode shapes in accordance with the Timoshenko beam model

The solution procedure in Timoshenko theory is similar to the solution procedure presented for Euler–Bernoulli theory.By assuming an exponential solution as gi(n) = Hie

kn and /i(n) = Uiekn; and substituting them in Eq. (27), the following expres-

sions will be resulted for a DWCNT:

ð�m2m2k4 þ ðv2 �K1 � l2�c1Þk2 þ �c1ÞH1 �K1kU1 þ ðl2�c1k2 � �c1ÞH2 ¼ 0;

K1kH1 þ ðK1 � a1k2ÞU1 ¼ 0;

ðl2�c1k2 � �c1ÞH1 þ ð�K2 � l2�c1k

2 þ �c1ÞH2 �K2kU2 ¼ 0;

K2kH2 þ ðK2 � a2k2ÞU2 ¼ 0:

ð37Þ

Using Eq. (37) gives the characteristic equation of the system as it is written in the next equation:

� l2a1a2v2ðK2 þ l2�c1Þk10 þ ½a1a2K2ðv2 �K1Þ þ l2ð2a1a2�c1v2 þK1K2a2v2 � a1a2�c1ðK1 þK2ÞÞþ l4�c1v2ða1K2 þ a2K1Þ�k8 þ ½a1a2�c1ðK1 þK2 � v2Þ �K1K2a2v2 �K1K2a1v2 þ l2�c1ð�2K1a2v2 þK1K2ða1 þ a2Þ� 2K2a1v2Þ � l4K1K2�c1v2�k6 þ ½�K1K2�c1ða1 þ a2Þ þ ðK2a1 þK1a2Þ�c1v2 þ 2l2K1K2�c1v2�k4 �K1K2�c1v2k2 ¼ 0:


An important point which should be considered is that, the above characteristic equation is of tenth order, but the systemhas eight boundary equations; consequently, the problem might be indeterminate. Because the value of the coefficient ofterm k10 is very smaller than the other terms’ coefficients, two roots of Eq. (38) are much larger than the other roots. Theterm k10 is created as a result of interaction between the nonlocal stress of Timoshenko displacement and the axial load.Solving the above characteristic equation with neglecting these two larger roots give g1(n) and g2(n) same as Eq. (31) and/1(n) and /2(n) as follows:

/1ðnÞ ¼ �C2 þ C3U13ek3n þ C4U

14e�k3n þ C5U

15ek5n þ C6U

16e�k5n þ C7U

17ek7n þ C8U

18e�k7n;

/2ðnÞ ¼ �C2 þ C3U23ek3n þ C4U

24e�k3n þ C5U

25ek5n þ C6U

26e�k5n þ C7U

27ek7n þ C8U

28e�k7n;

ð39Þ

where, Hj is used in Eq. (31) and U1j and U2

j are used in Eq. (39) are:

U1j ¼ �

Kkj

K1 � a1k2j

;

Hj ¼l2a1v2k6

j � ½a1ðK1 � v2Þ þ l2ða1�c1 �K1v2Þ�k4j � ½K1ðv2 � l2�c1Þ � a1�c1�k2

j �K1�c1

�l2a1�c1k4j þ ðl2K1�c1 þ a1�c1Þk2

j �K1�c1;

U2j ¼ Hj

K2kj

K2 � a2k2j

:

ð40Þ

Substituting g1(n), g2(n), /1(n), and /2(n) into the linear part of Timoshenko boundary conditions (Eq. (28)) and then, solvingthe outcome eigenvalue problem, gives the critical (buckling) fluid velocities and normalized buckled mode shapes. Normal-ized transverse deflection and shear deformation mode shapes of first and second layers of the system is noted by g1; /1; g2

and /2, respectively.

3.4. Post-buckling configuration in accordance with the Timoshenko beam model

The post-buckled configurations can be considered as follows:

g1ðnÞ ¼ bg1g1ðnÞ; g2ðnÞ ¼ bg2

g2ðnÞ; /1ðnÞ ¼ bu1/1ðnÞ; /2ðnÞ ¼ b/2 /2ðnÞ; ð41Þ

where bg1, b/1 , bg2

and b/2 are post-bucked configuration’s amplitude coefficient.The nonlinear post-buckling equations of the Timoshenko theory with the presence of the nonlinear part of boundary

conditions in the equations of motion are:

bg1

Z 1

0f�l2v2gð4Þ1 þ ðv2 �K1 � l2�c1Þg0021 þ �c1g1gg1dnþ b/1

Z 1

0f�K1/

01gg1dnþ bg2

Z 1

0fl2�c1g002 � �c1g2gg1dn

þ b3g1

Z 1

0f�l1ð1:5g001g

021 þ g001#1 þ g01#

001Þ þ l2l1ð1:5gð4Þ1 g0021 þ 9g0001 g001g

01 þ 3g0031 þ gð4Þ1 #01 þ 3g0001 #

001 þ 3g001#

0001 þ g1#

ð4Þ1 Þ

þ l2�c2m2ðgð4Þ1 g0021 þ 4g0001 g001g01 þ g0031 Þ � l4�c2m2ðgð6Þ1 g021 þ 8gð5Þ1 g001g

01 þ 14gð4Þ1 g0001 g01 þ 13gð4Þ1 g021 þ 18g00021 g001Þ

� ½l2l1ð1:5g001ðg01Þ2 þ g001#

01 þ g01#

001Þ þ l4�c2m2ðgð4Þ1 ðg01Þ

2 þ 4g0001 g001g01 þ ðg001Þ

3Þðd0ðn� 1Þ � d0ðnÞÞgg1dn ¼ 0; ð42Þ

bg1

Z 1

0fl2�c1g001 � �c1g1gg2dnþ bg2

Z 1

0fð�K2 � l2�c1Þg002 þ �c1g2gg2dnþ b/2

Z 1

0f�K2/

02gg2dn

þ b3g2

Z l

0f�l2ð1:5g002g

022 þ g002#2 þ g02#

002Þ þ l2l2ð1:5gð4Þ2 g0022 þ 9g0002 g002g

02 þ 3g0032 þ gð4Þ2 #02 þ 3g0002 #

002 þ 3g002#

0002 þ g2#

ð4Þ2 Þ

� l2l2½1:5g002ðg02Þ2 þ g002#

02 þ g02#

002�ðd

0ðn� 1Þ � d0ðnÞÞgg2dn ¼ 0:

Solving Eq. (42) gives buckled configuration for each layer in each desired flow velocity (v).

4. Results

4.1. Validation of the presented equations and solutions

The validity of the presented equation of motions, boundary conditions, buckling and post-buckling solution is examinedby comparing the buckling fluid velocities and static post-buckled configuration’s bifurcation diagram of the present workwith the results presented in [18] for a SWCNT with diameter d = 1 nm and length to diameter ratio of L/r = 10. It should benoted that the square of dimensionless fluid velocity ðv2

c Þ in present work is equal to dimensionless axial load (pc) in [18].Table 1 contains the values of v2

c=p2 of present work and pc/p2 of [18] for the different values of small-scale parameter andtwo boundary conditions according to Euler–Bernoulli beam theory (EBT). Table 1 shows that there are an excellent agree-ment between the present analytical solution and those in [18]. Fig. 3 shows a supercritical bifurcation diagram of the firstthree dimensionless buckled configurations of the present work and corresponding results of [18] with the small-scale

Table 1First three dimensionless critical buckling flow velocity m2

c =p2 of SWCNT for various values of small-scale parameter and two boundary conditions.

e0a

0 nm 0.5 nm 1 nm 1.5 nm 2 nm

Present Ref. [18] Present Ref. [18] Present Ref. [18] Present Ref. [18] Present Ref. [18]

S–SFirst mode 1 1 0.97592 0.9759 0.91017 0.9102 0.81829 0.8183 0.71696 0.7170Second mode 4 4 3.64068 3.6407 2.86783 2.8678 2.11835 2.1183 1.55090 1.5509Third mode 9 9 7.36458 7.3646 4.76628 4.7663 3.00140 3.0014 1.97669 1.9767

H–CFirst mode 2.04574 2.0457 1.94745 1.9474 1.70208 1.7021 1.40670 1.4067 1.13173 1.1317Second mode 6.04680 6.0468 5.26175 5.2618 3.78683 3.7868 2.58102 2.5810 1.78520 1.7852Third mode 12.0471 12.0471 9.28663 9.2866 5.50346 5.5035 3.27790 3.2779 2.0930 2.0936

Fig. 3. Nonlocal static supercritical bifurcation diagrams for the first three buckled configurations of a SWCNTs (a) simple–simple and (b) simple–clamped.

Table 2The first three exact dimensionless critical flow velocity mc of SWCNT for various values of small-scale parameter e0a and aspect ratios (r1 = 1 nm).

L/r1 e0a

0 nm 1 nm 1.5 nm 2 nm

EBT TBT EBT TBT EBT TBT EBT TBT

S–S 10 3.14159 2.93221 2.99716 2.79740 2.84185 2.65244 2.6609 2.482796.28318 4.98024 5.32013 4.21693 4.57244 3.62426 3.91239 3.10150⁄

9.42477 6.17265 6.85866 4.49590 5.44267 3.67074⁄ 4.41692 –C–S 4.49303 3.89288 4.09864 3.55951 3.72606 3.24273 3.34211 2.91429

7.72525 5.56929 6.11347 4.43122⁄ 5.04714 3.67074⁄ 4.19752 3.10150⁄

10.90412 6.50307 7.37000 – 5.68784 – 4.54496 –

S–S 15 3.14159 3.04315 3.07487 2.97852 2.99416 2.90325 2.89705 2.806856.28318 5.59091 5.79530 5.15678 5.32018 4.73401 4.81637 4.285718.42477 7.47036 7.98027 6.32540 6.85866 5.50611 5.86859 4.65224

C–S 4.49331 4.19419 4.30442 4.01906 4.09864 3.82993 3.85456 3.604127.72525 6.50398 6.86792 5.79014 6.11347 5.15964 5.38117 4.54521

10.90412 8.11558 8.81994 6.57929 7.37000 5.50618 6.17941 4.65225

S–S 20 3.14159 3.08566 3.10353 3.04769 3.05785 3.00283 8.99716 2.943246.28318 5.86439 5.99433 5.59480 5.68370 5.30488 5.32018 4.965588.42477 8.16363 8.52557 7.38477 7.69620 6.66637 6.85586 5.94093

C–S 4.49331 4.31752 4.30442 4.21327 4.09864 4.09298 3.85456 3.940677.72525 6.96253 6.86792 6.49799 6.11347 6.03002 5.38117 5.51757

10.90412 9.04607 8.81994 7.94910 7.37000 7.01299 6.17941 6.20300


Table 3Applying Routh–Hurwitz criterion for calculation of buckling flow velocity limit and structural negative stiffness.

EBT TBT

Characteristic equation (1 � l2v2)k4 + v2k2 = 0 a1l2v2k6 + [K1(a1 � l2v2) � a1v2]k4 + K1v2k2 = 0a1 = 1 � l2v2, a2 = v2 a1 = a1l2v2k6, a2 = K1(a1 � l2v2) � a1v2, a3 = K1v2

The Routh–Hurwitz schemak2 a1 a2 k4 a1 a2 a3

k3 4a1 2a2 0

k1 2a1 0 k2 a2/2 a3 0k1 2a2 � 8a1a3/a2 0 0

k0 a2 0 k0 a3 0 0

Table 4The first three exact dimensionless critical flow velocity mc of DWCNT for various values of small-scale parameter e0a and aspect ratios (r1 = 1 nm,r2 = 1.675 nm).

L/r2 e0a

0 nm 1 nm 1.5 nm 2 nm


S–S 10 7.349557 6.93348 7.22667 6.81716 7.08141 6.67968 6.69201 6.5012412.66603 10.94918 12.05091 10.36609 11.16666⁄ 9.64113⁄ 8.37555⁄ 8.35784⁄

14.27872 12.01713 13.15794 11.19983⁄ – – – –C–S 10.20797 9.08880 9.88308 8.79445 9.51522 8.46868 8.37555⁄ 8.35784⁄

13.98484 11.70425 12.97813 10.84091 11.16666⁄ 9.64113⁄ – –14.64780 12.19073 13.19743 11.19983⁄ – – – –

S–S 15 7.42292 7.22324 7.36786 7.16769 7.29832 7.10002 7.20419 7.0082814.32101 13.04692 13.92331 12.67883 13.46992 12.25977 12.5625⁄ 11.7373318.99905 16.42377 18.07637 15.54914 16.75000⁄ 14.63262 – 12.5367⁄

C–S 10.55027 9.94367 10.38909 9.791166 10.19740 9.61062 9.93127 9.3743216.98568 14.91488 16.32024 14.30934 15.58092 13.65601 12.5625⁄ 12.5367⁄

20.42662 17.28519 19.15821 16.12362 16.75000⁄ 14.99023 – 12.5367⁄

S–S 20 7.43792 7.32152 7.40548 7.28958 7.36552 7.25025 7.31065 7.1963714.69911 13.86697 14.45335 13.63433 14.16282 13.35936 13.78402 13.0009321.02687 18.84303 20.32363 18.19483 19.53677 17.47167 16.7500⁄ 16.59054

C–S 10.61506 10.25348 10.52168 10.16323 10.40829 10.05383 10.25105 9.9065617.83861 16.36523 17.40476 15.96153 16.90362 15.49853 16.23332 14.9146823.51226 20.53570 22.52008 19.62769 21.42543 18.65704 16.7500⁄ 16.71569

Table 5The first three exact dimensionless critical flow velocity mc of TWCNT for various values of small-scale parameter e0a and aspect ratios (r1 = 1 nm, r2 = 1.675 nm,r3 = 2.35 nm).

L/r3 e0a

0 nm 1 nm 1.5 nm 2 nm


S–S 10 13.18814 12.85264 13.07605 12.74318 12.93987 12.61020 11.7500⁄ 11.3839⁄

20.66536 17.60313 18.81132 15.08191⁄ 15.6666⁄ 12.9106⁄ – –20.98783 17.78376 19.05059 – – – – –

C–S 17.98253 17.03628 17.67487 15.08191⁄ 15.6666⁄ 12.9106⁄ 11.7500⁄ 11.3839⁄

20.78810 18.32152 18.83804 – – – – –23.00516 18.46935 21.50424 – – – – –

S–S 15 13.38659 13.22678 13.33414 13.17494 13.26943 13.11098 13.18039 13.0230025.39797 24.36644 25.04471 22.58728⁄ 23.5000⁄ 19.8660⁄ 17.6250⁄ 17.0759⁄

31.71942 26.61329 28.74673 – – – – –C–S 18.95342 18.15720 18.80588 18.31102 18.62548 19.8660⁄ 17.6250⁄ 17.0759⁄

29.45194 26.41390 28.45608 22.58728⁄ 23.5000⁄ – – –31.85640 27.34179 28.64103 – – – – –

S–S 20 13.42098 13.32960 13.39117 13.29999 13.35418 13.26325 13.30291 13.2123326.37628 25.70529 26.15210 25.48639 25.87975 25.22021 23.5000 22.7678⁄

36.94538 35.32723 36.33515 34.59759 31.33333⁄ 25.8213⁄ – –C–S 19.13197 18.84240 19.04632 18.75733 18.94076 18.65273 18.79573 18.50939

31.77854 30.56203 31.38420 30.17760 30.89120 25.8213⁄ 23.5000 22.7678⁄

40.43911 35.40207 37.59682 37.39782 31.33333⁄ – – –


Fig. 4. First three normalized mode shapes of DWCNT with L/r1 = 15 and e0a = 1 nm of TBT for (a) S–S buckled configuration, (b) S–S buckled axis’ rotation,(c) S–C buckled configuration and (d) S–C buckled axis’ rotation.


parameter e0a = 1 nm for two end supports according to EBT. These comparisons show that in lower flow velocities there isan exact agreement between the present work and results given in [18]. But, as the work done by centrifugal force at in-plane direction has noticeable effects in high velocities – which is considered in the present work and ignored by Setoodehet al. [18], the post-buckled configuration values in the present work are little bit greater than those are given in [18].

4.2. Results for single and MWCNTs

In the following, MWCNTs are considered as concentric tubes with Young’s modulus E = 1 TPa, the Poisson’s ratio v = 0.3,inner diameter of 2 nm, thickness of hp = 0.34 nm and Van der Waals radius between each layer rvd = 0.335 nm [41]. The sim-ple–simple ends (S–S) and the simple–clamped ends (S–C) are chosen as the MWCNT boundary conditions.

Table 2 lists the exact first three dimensionless critical flow velocities of a SWCNT at various values of small-scale param-eters, aspect ratios and two end supports according to both EBT and Timoshenko beam theory (TBT). Table 2 demonstratesthat increase in the small-scale parameter values, decreases critical flow velocities. Therefore, the classic continuum theoryin which, e0a = 0 nm is overestimated. According to the nonlocal stress–strain field (Eqs. (13), (30) and (38)), increasing the

Fig. 5. Nonlocal supercritical post-buckling bifurcation diagrams of different buckled configurations of SWCNT with L/r1 = 10 for (a) S–S ends and EBT, (b)S–C ends and EBT, (c) S–S ends and TBT and (d) S–C ends and TBT.


nonlocal parameter, reduces the overall structural stiffness. Consequently, a system with higher nonlocal parameter bucklesat a lower fluid velocity.

The results of both EBT and TBT indicate that the value of buckling fluid velocities in S–C ends is greater than S–S ends,because in physical point of view, the structural stiffness of CNT in S–C ends is greater than S–C ends. The effect of ends sup-port in stiffness of CNT is studied extensively in [29]. There is a good agreement between the conclusion of Ghavanloo et al.[29] and those of the present work.

The comparison between the results of EBT and TBT in Table 2 shows that the shear deformation’s effects decreases crit-ical flow velocities, since shear deformation reduces total structural stiffness of the system. In addition, the shear deforma-tion’s effect is more significant in the low aspect ratios of nanotubes. In EBT the cross section remains perpendicular to theneutral axis of the beam, but in TBT theory the cross section rotates with respect to the neutral axis of the beam which addssome flexibility to the beam [40]. Moreover, in higher modes the stiffness is greater, so the shear deformation effect is more

Fig. 6. Nonlocal supercritical post-buckling bifurcation diagrams of DWCNT with L/r1 = 10 for different modes of (a) S–S ends and EBT; (b) S–C ends andEBT, (c) S–S ends TBT and (d) S–C ends and TBT.


considerable; and also, since a system with S–C ends is stiffer than S–S ends, the effect of shear deformation is higher in S–Cends cases.

In following the value of the velocities are marked by (⁄) superscript, representing situations in which system gets zerostructural stiffness. To clear the difference between buckling and zero structural stiffness, first, it should be mentioned thatbuckling happens when the resultant of the structural stiffness and the stiffness term due to the external forces of the systembecome zero, but the structural stiffness remains positive. In the governing equations of the system which are derivedregarding to the nonlocal elasticity, the external forces are participated directly in the structural stiffness (for examplethe term of ða1 � l2v2Þgð4Þ1 in Eq. (17) which is a function of flow velocity). In this situation, increasing flow velocity can makestructural stiffness equal to zero before the occurrence of linear buckling, so the system cannot have the buckling kind ofinstability afterward. As a result, the velocities which are marked by (⁄) are the limits of linear buckling analysis. These

Fig. 7. Nonlocal supercritical post-buckling bifurcation diagrams for different modes of simple–simple supported SWCNT with (a) L/r1 = 10, (b) L/r1 = 15 and(c) L/r1 = 20.


velocities can be obtained by using Routh–Hurwitz stability criterion on the characteristic equation of system as is shown inTable 3 for a SWCNT.

The Routh–Hurwitz criterion predict buckling velocity limit of SWCNT according to EBT as (1 � l2v2) < 0. It means thestructural stiffness (1 � l2v2)k4 changes from positive to negative. For SWCNT according to TBT, with applying Routh–Hur-witz criterion in Table 3, buckling velocity limit is predicted by [K1(a1 � l2v2 � a1v2] � 4K1a1l2v4 < 0.

Tables 4 and 5 present the exact first three dimensionless critical flow velocities of DWCNT’s and triple-walled CNT’s(TWCNT) respectively. Comparing Tables 2, 4 and 5 indicates that the Van der Waals force increases dimensionless criticalflow velocities which its effect is more noticeable on the tubes with greater lengths. In the dimensional form of system,increasing the length, decreases structural stiffness but has no effect on the stiffness of Van der Waals forces. In this work,the system is transformed to dimensionless form. As a result, dimensionless structural stiffness does not change with chang-ing of the length of systems. In addition, the dimensionless Van der Waals force grows by increasing the length of the system.

Fig. 8. Nonlocal supercritical post-buckling bifurcation diagrams for different modes of simple–clamped supported SWCNT with (a) L/r1 = 10, (b) L/r1 = 15and (c) L/r1 = 20.


Tables 4 and 5 show that increase in the nonlocal parameter values, decreases dimensionless buckling velocities. This ef-fect is more noticeable in higher modes and S–C ends. There are some terms of nonlocal parameter effective in structuralstiffness which are a function of fluid velocity and shear deformation (Eqs. (30) and (38)). These terms have subtractive effecton structural stiffness. Because in greater value of nonlocal parameter these terms got amplified, structural stiffness reducesmore by increasing fluid velocity and shear deformation effects in high values of nonlocal parameter. The range of bucklingvelocities grows with number of layers and Van der Waals force effect. Therefore, the effect of the nonlocal parameter instructural stiffness is more pronounced in MWCNTs than SWCNTs.

The velocities with star superscript in Tables 4 and 5 for DWCNT and TWCNT can be obtained by the same method asexplained in Table 3. The results show that more zero structural kind of instability happens for the systems that have morenumber of layers.

Fig. 9. Nonlocal supercritical post-buckling bifurcation diagrams for different modes of simple–simple supported DWCNT with (a) L/r1 = 10, (b) L/r1 = 15and (c) L/r1 = 20.


Fig. 4 shows first three normalized mode shapes of both layers of a DWCNT according to TBT for S–S and simple–clampedS–C ends support. This figure shows the deflections and the rotations of the DWCNT along the nanotubes’s span at three dif-ferent buckled mode shapes.

Fig. 5 illustrates the supercritical bifurcation diagrams of post-buckling configuration of SWCNT For different values ofsmall scale parameters. Fig. 5 shows the diagrams for both EBT and TBT and two boundary conditions. This figure illustratesthat the increase in the small scale parameter, increases the post-buckling deflection and decreases the stability margin forall of the modes at the every boundary condition and for both EBT and TBT. The behaviors illustrated in Fig. 5 diagrams, arecaused by decreasing in overall structural stiffness of the system as a result of increasing in the small scale parameter. More-over, Fig. 5 shows that the effect of small-scale parameter is more significant in higher modes and S–C ends case.

Fig. 6 shows post-buckling supercritical bifurcation diagrams of first and second layers of DWCNT for different value ofthe small scale parameters, two boundary conditions according to EBT and TBT. A comparison between Figs. 5 and 6 showsthat increasing the number of layers and considering the Van der Waals force, increase the stability of system and decrease

Fig. 10. Nonlocal supercritical post-buckling bifurcation diagrams for different modes of simple–clamped supported DWCNT with (a) L/r2 = 10, (b) L/r2 = 15and (c) L/r2 = 20.


the post- buckled deflection of nanotubes, because of the increase in the total stiffness of the system. Increasing the nonlocalparameter and fluid velocity reduces structural stiffness, and consequently amplifies post-buckled deflection. Since the effectof the nonlocal parameter in structural stiffness is coupled with fluid velocity, the amounts of decrease in structural stiffnessfor higher modes are greater. As a result, the difference between post-buckling amplitudes of the different modes enlargeswith the increase in the fluid velocity.

Figs. 7–10 provide a compression of post-buckling bifurcation configuration of SWCNT and DWCNT between EBT and TBTfor different aspect ratios and boundary conditions. Comparing the results of S–S ends with S–C ends shows that in all casesthe S–C ends have more stability and less buckled deflection than S–S ends and it demonstrates that the S–C ends is stiffer.Comparison between TBT and EBT in these figures illustrate that shear deformation effects increase the bucked amplitudesand decreases buckling flow velocities. In addition, its effect is more noticeable in low aspect ratios, higher modes, S–C endsand DWCNT case, because in these situations the reduction of stiffness by the shear deformation effect is more significant.


5. Conclusions

In the current study, buckling and post-buckling analysis of fluid conveying MWCNT are investigated based on nonlocalnonlinear Euler–Bernoulli and Timoshenko beam theories. The results show that considering nonlocal parameter, decreasesbuckling flow velocities and increases post-buckled deflection; which demonstrates that classic elastic theory is inaccurate.Moreover, increasing numbers of layers and considering Van der Waals force effect, increases stability margin and decreasespost-buckled deflection. The comparison between EBT and TBT clarifies that shear deformation, decreases dimensionlesscritical flow velocities and increases the amplitude of post-bucked configurations. Furthermore, the shear deformation effectis more significant when the CNTs have a lower aspect ratio. In this work, the results are obtained for three first modes andS–S and S–C boundary conditions. The S–C ends cases have more instability and less buckled deflection than the S–S ends,because the S–C ends are stiffer than S–S ends. The results indicate the effect of small-scale parameter and shear deformationis more significant in the S–C cases and the higher modes.

Appendix A

List of derivatives’ symbols that are used in system’s equations of motion.

The notation symbol
Definition of the notation 0 First derivative respect to x and n in dimensional and dimensionless relations respectively 00 Second derivative respect to x and n in dimensional and dimensionless relations respectively 000 Third derivative respect to x and n in dimensional and dimensionless relations respectively ( )(4) Fourth derivative respect to x and n in dimensional and dimensionless relations respectively ( )(5) Fifth derivative respect to x and n in dimensional and dimensionless relations respectively ( )(6) Sixth derivative respect to x and n in dimensional and dimensionless relations respectively
Appendix B

Sample buckling and post-buckling code for S–S according to EBT.


Appendix C. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apm.2012.09.061.

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analytical analysis of buckling and post-buckling of fluid conveying multi-walled carbon nanotubes

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