Nonlinearfreevibrationofsingle-walledcarbonnanotubesusingnonlocal Timoshenkobeamtheory

8/9/2019 Nonlinearfreevibrationofsingle-walledcarbonnanotubesusingnonlocal Timoshenkobeamtheory

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Nonlinear free vibration of single-walled carbon nanotubes using nonlocalTimoshenko beam theory

J. Yang a,n, L.L. Ke b,c, S. Kitipornchai b

a School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO Box 71, Bundoora, Victoria 3083, Australiab Department of Building and Construction, City University of Hong Kong, Hong Kong, PR Chinac Institute of Engineering Mechanics, Beijing Jiaotong University, 100044 Beijing, PR China

a r t i c l e i n f o

Article history:Received 24 November 2009

Received in revised form

19 January 2010

Accepted 19 January 2010Available online 25 January 2010

Keywords:

SWCNTs

Nonlinear vibration

Timoshenko beam theory

Nonlocal elasticity

DQ method

a b s t r a c t

Nonlinear free vibration of single-walled carbon nanotubes (SWCNTs) is studied in this paper based onvon Karman geometric nonlinearity and Eringens nonlocal elasticity theory. The SWCNTs are modeled

as nanobeams where the effects of transverse shear deformation and rotary inertia are considered

within the framework of Timoshenko beam theory. The governing equations and boundary conditions

are derived by using the Hamiltons principle. The differential quadrature (DQ) method is employed to

discretize the nonlinear governing equations which are then solved by a direct iterative method to

obtain the nonlinear vibration frequencies of SWCNTs with different boundary conditions. Zigzag (5, 0),

(8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical calculations and the elastic modulus is

obtained through molecular mechanics (MM) simulation. A detailed parametric study is conducted to

study the influences of nonlocal parameter, length and radius of the SWCNTs and end supports on the

nonlinear free vibration characteristics of SWCNTs.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Nonlocal elasticity theory was proposed by Eringen [13] to

consider the scale effect in elasticity by assuming the stress at a

reference point to be a function of strain field at every point in the

body. It has been extensively applied to analyze the bending,

buckling, vibration and wave propagation of beam-like elements

in micro- or nanoelectromechanical devices [413]. Sudak [14]

studied infinitesimal column buckling of carbon nanotubes

(CNTs), incorporating the van der Waals (vdW) forces and small

scale effect, and showed that the critical axial strain decreases

compared with the results of classical beams. Wang [15]

discussed the molecular dispersion relationships for CNTs by

taking into account the small scale effect. Wang and Hu [16]

studied flexural wave propagation in a SWCNT by using thecontinuum mechanics and dynamic simulation. Lu et al. [17]

investigated the wave propagation and vibration properties of

single- or multi-walled CNTs based on nonlocal beam model.

Wang et al. [18] presented analytical solutions for the free

vibration of nonlocal Timoshenko beams. Reddy[19] developed

nonlocal theories for EulerBernoulli, Timoshenko, Reddy, and

Levinson beams. Analytical bending, vibration and buckling

solutions are obtained which bring out the nonlocal effect on

bending deformation, buckling load, and natural frequencies.More recently, Tounsi and his co-workers [2023] investigated

the sound wave propagation in single- and double-walled CNTs

taking into account the nonlocal effect, temperature and initial

axial stress. Furthermore, they [24,25] derived the consistent

governing equation of motion for the free vibration of fluid-

conveying CNTs with nonlocal effect, which is an important

application of nonlocal elastic theory in CNTs. Yang et al. [26]

investigated the pull-in instability of nano-switches subjected to

combined electrostatic and intermolecular forces within the

framework of nonlocal elasticity theory. Aydogdu[27]presented

a generalized nonlocal beam theory to study bending, buckling

and free vibration of nanobeams.

Previous theoretic and experimental investigations [28,29]

showed that the deformation of nanostructures, such as CNTs, isnonlinear in nature when subjected to large external loads.

Fu et al. [28] investigated the nonlinear free vibration of

embedded multiwall CNTs considering inter-tube radial displace-

ment and the related internal degrees of freedom. Shen and Zhang

[30,31] considered the buckling and postbuckling behavior of

single- and double-walled CNTs in thermal environments. Yan

et al. [32]analyzed the nonlinear vibration characteristics of the

fluid-filled DWNTs. To the best of authors knowledge, however,

no previous work has been done concerning the small scale effect

on the nonlinear vibration behavior of nanostructures.

This paper makes the first attempt to study the nonlinear

free vibration of SWCNTs based on von Karman geometric

ARTICLE IN PRESS

Contents lists available atScienceDirect

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

Physica E

1386-9477/$- see front matter& 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.physe.2010.01.035

n Corresponding author. Tel.: +61 3 99256169; fax: +61 3 99256108.

E-mail address: [email protected] (J. Yang).

Physica E 42 (2010) 17271735
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ARTICLE IN PRESS

nonlinearity, Timoshenko beam theory and Eringens nonlocal

elasticity theory. The Hamiltons principle is employed to derive

the governing equations and boundary conditions which are

solved by using the differential quadrature (DQ) method. A direct

iterative technique is then used to obtain the nonlinear vibration

frequencies of nonlocal SWCNTs with different end supports. In

numerical calculations, zigzag (5, 0), (8, 0), (9, 0) and (11, 0)

SWCNTs are considered and the elastic modulus is obtained by

using MM simulation. The influences of nonlocal parameter,length and radius of the SWCNTs and end supports on the

nonlinear free vibration characteristics of the SWCNTs are

discussed in detail.

2. Nonlocal nanobeam model

Unlike the constitutive equation in classical elasticity,

Eringens nonlocal elasticity theory [13] states that the stress

at a pointx in a body depends not only on the strain at that point

but also on those at all points of the body. This observation is in

accordance with atomic theory of the lattice dynamics and

experimental observation of the phonon dispersion [2]. Thus,

the nonlocal stress tensor rat point x is expressed as

r

ZV

a9x0x9; tTx0 dx0; 1

where T(x) is the classical, macroscopic stress tensor at point x.

a(9x0x9,t) is the nonlocal modulus or attenuation functionintroducing into the constitutive equation the nonlocal effect at

the reference pointx produced by local strain at the source x0.9x0-x9 is the Euclidean distance, and t e0a=l is defined as the scalecoefficient that incorporates the small scale factor, where e0 is a

material constant determined experimentally or approximated by

matching the dispersion curves of plane waves with those of

atomic lattice dynamics, and a andl are the internal and external

characteristic lengths (e.g. crack length, wavelength), respectively.

The stress tensorT(x) at pointx in a Hookean solid is related to

the strain tensor e(x) at the point by the generalized Hookes law[19]

Tx Cx : ex; 2

whereC(x) is the fourth-order elasticity tensor and : denotes the

double-dot product.

From Eqs. (1) and (2), the integral nonlocal constitutive

relations can be represented in an equivalent differential form

as[2]

1tl2r2r T: 3

3. Nonlinear vibration analysis of nonlocal SWCNTs

Fig. 1 shows a SWCNT modeled as a Timoshenko nanobeamwith length L, radius r, and effective tube thickness h. I t is

assumed that the SWCNTs vibrate only in the xzplane. Based on

Timoshenko beam theory, the displacements of an arbitrary point

in the beam along the x- and z-axes, denoted by ~Ux;z; t and~Wx;z; t, respectively, are

~Ux;z; t Ux; t zCx; t; ~Wx;z; t Wx; t; 4

where U(x,t) and W(x,t) are displacement components in the

midplane,C is the rotation of beam cross-section and tis time.

The von Karman type nonlinear strain-displacement relations give

exx@U

@x

1

2

@W

@x

2z

@C

@x ; gxz

@W

@x C; 5

where,exx is the axial strain, and gxz is the shear strain.

The strain energy Vis given by

V1

2

Z L0

ZA

sxxexxsxzgxzdAdx; 6

where A is the cross-sectional area of the beam, sxx and txz arenormal and shear stresses, respectively. By submitting Eq. (5) into

Eq. (6), the strain energy Vcan be represented as

V1

2

Z L0

ZA

sxx@U

@x

1

2

@W

@x

2" #sxxz

@C

@x sxz

@W

@x C

( )dAdx

1

2

Z L0

Nx@U

@x

1

2

@W

@x

2" # Mx

@C

@x Qx

@W

@x C

( )dx;

7

where the normal resultant force Nx, bending moment Mx, and

transverse shear force Qxare calculated from

Nx

ZA

sxx dA; Mx

ZA

sxxzdA; Qx

ZA

sxzdA: 8

The kinetic energy Kcan be calculated from

K12

Z L

0rA

@U

@t

2

rA @W

@t

2

rI @C

@t

2

" #dx; 9

whereIis the second moment of area and r is the mass density ofbeam material.

For a beam type structure, the thickness and width are much

smaller than its length. Therefore, for beams with transverse

motion in thexzplane, the nonlocal constitutive relations (3) can

be approximated to one-dimensional form as[19]

sxxe0a2d

2sxxdx2

Eexx; sxze0a2d

2sxzdx2

Ggxz; 10

where E and G are Youngs modulus and shear modulus,

respectively. The constitutive relations in classical elasticity

theories can be recovered by setting the nonlocal parametere0a =0.

Using the Hamiltons principleZ t0

dKdVdt 0; 11

substituting Eqs. (7) and (9) into Eq. (11), integrating by parts and

setting the coefficients of dU, dW and dC to zero leads to the

equations of motion as [3335]

@Nx@x

rA@2U

@t2 ; 12a

@Qx

@x

@

@x

Nx@W

@x rA @

2W

@t2 ; 12b

r

L

x

z

h

Fig. 1. A single wall carbon nanotube (SWCNT) modeled as a nonlocal Timoshenko

nanobeam.

J. Yang et al. / Physica E 42 (2010) 1727 17351728


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4. Solution method

The differential quadrature (DQ) method [37,38] is used to

solve nonlinear Eq. (20) and the associated boundary conditions

to determine the nonlinear free vibration frequencies of nonlocal

SWCNTs. The fundamental idea of the DQ method is to

approximate the derivative of a function at a sample point as a

linear weighted sum of the function values at all of the sample

points in the problem domain. Hence, u, w and c and their kthderivatives with respect to x can be approximated by

fu; w;cg XN

m 1

lmx umxm; t; wmxm; t;cmxm; t

; 25

@k

@xk u; w;c

9x xi XN

m 1

Ck im um xm; t ; wm xm; t ; cm xm; t

;

26

whereNis the total number of nodes distributed along the x-axis,

lm(x) is the Lagrange interpolation polynomials, and Cim(k) is the

weighting coefficients whose recursive formula can be found in

[3740]. The cosine pattern is used to generate the DQ pointsystem

xi1

2 1cos

pi1N1

; i 1; 2;. . .N: 27

Applying Eqs (25) and (26) to Eq. (20), one obtains a set of

nonlinear ordinary differential equations

a11XN

m 1

C2im

um1

Z

XNm 1

C1im

wmXN

m 1

C2im

wm

! I1 uim

2XN

m 1

C2im

um

!;

28a

a55 XN

m 1

C2im wmZXN

m 1

C1imcm !S1im2S2i I1 wim

2XN

m 1

C2im wmm2XN

m 1

C2im umm4XN

m 1

C4im um

!;

28b

d11XN

m 1

C2imcma55Z

XNm 1

C1im

wmZci

! I3

cim2XN

m 1

C2im

cm

" #;

28c

where the dot represents the derivative with respect to the

dimensionless timei,

S1ia11

ZX

N

m 1

C2

im

um XN

m 1

C1

im

wm 3

2ZX

N

m 1

C1

im

wm !2

24XN

m 1

C2im wmXN

m 1

C1im umXN

m 1

C2im wm

#; 29

S2ia11Z2

3XN

m 1

C2im wm

!39

XNm 1

C1im wmXN

m 1

C2im wmXN

m 1

C3im wm

24

3

2

XNm 1

C1im wm

!2 XNm 1

C4im wm

35a11

Z

XNm 1

C4im umXN

m 1

C1im wm

"

3 XN

m 1

C3im um XN

m 1

C2im wm3 XN

m 1

C2im um XN

m 1

C3im wm

XN

m 1

C1im umXN

m 1

C4im wm

#: 30

The associated boundary conditions can be handled in the

same way. For example, the boundary conditions of a hinged

hinged SWCNT are written as

d11XN

m 1

C11mcma11Z

XNm 1

C21mumXN

m 1

C11mwm

"

3

2Z

XNm 1

C11mwm

!2 XNm 1

C21mwmXN

m 1

C11mumXN

m 1

C21mwm

35

m2 I3XN

m 1C21m cmI1Z w1m2I1Z

XNm 1

C21mum !

0;

u1 w1 0; at z 0; 31a

d11XN

m 1

C1Nmcma11Z

XNm 1

C2NmumXN

m 1

C1Nmwm

"

3

2Z

XNm 1

C1Nmwm

!2 XNm 1

C2NmwmXN

m 1

C1NmumXN

m 1

C1Nmwm

35

m2 I3XN

m 1

C2NmcmI1Z wNm

2I1ZXN

m 1

C2Nmum

! 0;

uN wN 0; at z 1: 31b

Denoting the unknown dynamic displacement vector

d uif gT; wif g

T; ci Tn oT

; i 1; 2;. . .N; 32Table 2

Dimensionless linear and nonlinear fundamental frequencies of nonlocal (8, 0)

SWCNTs: results with varying total number of nodes N.

N HH CH CC

ol onl ol onl ol onl

5 0.45690 0.66723 0.81629

6 0.41943 0.43755 0.60107 0.60978 0.81256

7 0.42263 0.44016 0.60342 0.61924 0.80536 0.83042

8 0.42333 0.44055 0.60523 0.61963 0.80550 0.82071

10 0.42333 0.44055 0.60526 0.61975 0.80551 0.81888

16 0.42333 0.44055 0.60526 0.61975 0.80551 0.81888

20 0.42333 0.44055 0.60526 0.61975 0.80551 0.81888

Table 3

Dimensionless linear fundamental frequency of a hingedhinged nonlocal

Timoshenko nanobeam.

L/h (e0a)2=0.5 (e0a)

2=1.5 (e0a)2=2.5

Ref.[19] Present Ref.[19] Present Ref .[19] Present

10 9.6335 9.6331 9.2101 9.2097 8.8380 8.8377

20 9.6040 9.5942 9.1819 9.1726 8.8110 8.8020

100 9.5135 9.4765 9.0953 9.0600 8.7279 8.6940

Table 1

The elastic modulus of zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs obtained by

using molecular mechanics simulation.

(n, m) Number of

atoms

Length

(nm)

Diameter

(nm)

Elastic modulus

(TPa)

(5, 0) 240 4.7971 0.391 1.1468

(8, 0) 384 4.8659 0.626 1.1556

(9, 0) 432 4.8749 0.705 1.1572

(11, 0) 528 4.8857 0.861 1.1621



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Eqs. (28) and (31) can be expressed in matrix form as

KL KNLd M d 0; 33

where M is the mass matrix, KL is the linear stiffness matrix and

KNL is nonlinear stiffness matrix that is the functions in d. M, KLandKNLare 3N 3Nmatrices.

Expanding the dynamic displacement vector d in the form of

d deioi; 34

whereo OLffiffiffiffiffiffiffiffiffir=E

p represents the dimensionless frequency,O is

the nonlinear vibration frequency of the SWCNT,

d ui T

; wi T

; ci Tn oT

is the vibration mode shape

vector. Substituting Eq. (34) into Eq. (33) yields the nonlinear

eigenvalue equations as follows:

KL KNLd

o2Md 0; 35

This nonlinear equation can be solved through a direct

iterative process below

Step 1: By neglecting the nonlinear matrix KNL, a linear

eigenvalue (ol) and the associated eigenvector are obtained fromEq. (35). The eigenvector is then appropriately scaled up such that

the maximum transverse displacement is equal to a given

vibration amplitude wmax. Note that wmax=w(0.5)for clamped

clamped and hingedhinged SWCNTs while wmax=w(0.57) for a

clampedhinged SWCNT.Step 2: Using the eigenvector to calculate KNL, a new

eigenvalue and eigenvector are obtained from the updated

eigensystem (35).

Step 3: The eigenvector is scaled up again and step 2 is

repeated until the relative error between the given vibration

amplitude and the maximum transverse displacement wmax iswithin 0.1%.

5. Molecular mechanics simulation for the elastic modulus of

SWCNTs

It was assumed in many previous studies [9,11,13,17,28]

dealing with vibration behavior of SWCNTs that the elastic

modulus of the SWCNT is about 1 TPa. In fact, the elastic modulus

is different for different diameters and chirality of the SWCNT and

can be determined through molecular mechanics (MM) simula-

tion. The present paper employs the MM simulation to obtain the

elastic modulus of zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs

shown inFig. 2. The interatomic interactions in the SWCNTs are

described by the COMPASS force field (condensed-phased

optimized molecular potential for atomistic simulation studies)

[41]. This is the first and only ab initio force field to enable

accurate simulation and simultaneous prediction of structural,

conformational, vibrational, and thermophysical properties for a

broad range of molecules both in isolation and in the condensed-

phase. The MM simulations are carried out at a temperature of 0 K

to avoid the thermal effect [42]. The thickness of the SWCNT is

selected as 0.34 nm. It is assumed that the two ends of the SWCNT

are fixed boundaries. The simulations of the SWCNT under

compression can be identified through a minimizer processor,

which enables the atoms in CNTs to rotate and move relative to

each other following a certain minimization algorithm tominimize the strain energy so that an equilibrium state can be

identified. In the present analysis, energy minimization is

conducted using the smart minimizer that switches from the

steepest-descent to conjugated gradient and then to the Newton

method. The strain energy of the SWCNT is collected at every

compression deformation with the incremental displacement step

of 0.01 nm. Once the strain energy at every compression step is

available, the second derivative of the strain energy with respect

to the compression can easily be obtained through a simple finite

difference method. The modulus and radius of the zigzag (5, 0), (8,

0), (9, 0) and (11, 0) SWCNTs are listed inTable 1and will be used

in the next section for nonlinear vibration analysis of SWCNTs.

These values agree well with the results obtained from the

molecular dynamic simulation [43], molecular structural

mechanics [44]and experiment[45].

It should be pointed out that for CNTs modeled as a 1-D

isotropic solid such as the Timoshenko beam model used in the

present paper, elastic modulus is the most important elastic

parameter that influences the vibration frequencies of the

SWCNTs. As in many previous studies, see, for example,

Refs. [9,18], our focus is placed on the effect of diameter and

chirality on the elastic modulus of the SWCNTs by using MM

simulation while the shear modulus is approximately determined

fromG =0.5E/(1+n). The Poissons ratio is taken as the value of thegraphite, i.e. v=0.19.

6. Numerical results

Table 2 lists the dimensionless fundamental frequencies

(oOLffiffiffiffiffiffiffiffiffir=E

p ) of (8, 0) SWNTs (m=0.15) with varying total

numbers of nodes N. o l and onl denote the linear and nonlinearfrequencies (wmax=0.4), respectively. The SWNTs are modeled as

nonlocal Timoshenko nanobeams with hingedhinged (HH),

clampedhinged (CH) and clampedclamped (CC) boundary

conditions. Consider the SWNTs with radius r=0.313 nm, length

L=5 nm, Youngs modulus E=1.1556 TPa, Poissons ratio v=0.19,

effective tube thickness h=0.34 nm and shear correction factor

Ks=0.563 [18]. It is seen that the accuracy of the results is

improved with an increasing number of nodes Nand convergent

results are obtained whenNZ10. Hence,N=10 is used in all of the

following numerical calculations.

Table 4

Dimensionless linear fundamental frequency of SWCNTs with different boundary conditions.

Boundary condition m=0.1 m=0.3 m=0.5

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

HH 3.0243 3.0210(0.11%) 2.6538 2.6385(0.58%) 2.2867 2.2665(0.89%)

CH 3.6939 3.6849(0.24%) 3.2115 3.1724(1.23%) 2.7471 2.6982(1.81%)

CC 4.3471 4.3269(0.47%) 3.7895 3.7032(2.33%) 3.2420 3.1372(3.34%)

Table 5

Comparisons of nonlinear frequency ratio onl/ol for isotropic homogeneoushingedhinged beam withL/h=100, h =0.3 in.

Wmax=Y Present FEM[46]

1.0 1.11920 1.1181

2.0 1.41801 1.4178

3.0 1.80919 1.8094

4.0 2.24511 2.2455

5.0 2.70429 2.7052

J. Yang et al. / Physica E 42 (2010) 1727 1735 1731


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ARTICLE IN PRESS

Table 3 gives the dimensionless linear fundamental

frequencies (oOL2ffiffiffiffiffiffiffiffiffiffiffiffiffirA=EI

p ) of a hingedhinged nonlocal

Timoshenko nanobeam with various slenderness ratio L/h. The

parameters used in this example are[19]:L =10,E=30 106,r=1,

v=0.3, Ks=5/6. Our results are in good agreement with the

analytical results given by Reddy[19]using nonlocal Timoshenko

beam theory as well.

Table 4 presents the dimensionless linear fundamental

frequencies (oOL2 ffiffiffiffiffiffiffiffiffiffiffiffiffirA=EIp ) of the SWNTs based on nonlocalTimoshenko beam model. The analytical solutions given by

0.0 0.2 0.4 0.6 0.81.00

1.05

1.10

1.15

1.20

wmax

H-H:

= 0.00 (l= 0.4680)

= 0.10 (l= 0.4465)

= 0.15 (l= 0.4233)

= 0.20 (l= 0.3963)

0.0 0.2 0.4 0.6 0.81.00

1.02

1.04

1.06

1.08

1.10

C-H:

= 0.00 (l= 0.6765)

= 0.10 (l= 0.6420)

= 0.15 (l= 0.6053)

= 0.20 (l= 0.5628)

wmax

0.01.00

1.02

1.04

1.06

1.08

wmax

nl/

l

nl/

l

nl/

l

C-C:

= 0.00 (l= 0.9036)

= 0.10 (l= 0.8560)

= 0.15 (l= 0.8055)

= 0.20 (l= 0.7473)

0.2 0.4 0.6 0.8

Fig. 3. The effect of nonlocal parameter m on nonlinear frequency ratio versusamplitude curves of (8, 0) SWCNTs with L =5 nm: (a) hingedhinged; (b) clamped

hinged; and (c) clampedclamped.

0.01.00

1.04

1.08

1.12

1.16

wmax

nl/

l

nl/

l

nl/

l

H-H:

L = 5 nm (l= 0.4233)

L = 8 nm (l= 0.2742)

L = 12 nm (l= 0.1853)

L = 16 nm (l= 0.1397)

1.00

1.02

1.04

1.06

1.08

1.10

C-H:

L = 5 nm (l= 0.6053)L = 8 nm (

l= 0.4085)

L = 12 nm (l= 0.2808)

L = 16 nm (l= 0.2130)

wmax

1.00

1.01

1.02

1.03

1.04

1.05

1.06

C-C:

L = 5 nm (l= 0.8055)

L = 8 nm (l= 0.5666)

L = 12 nm (l= 0.3972)

L = 16 nm (l= 0.3036)

wmax

0.2 0.4 0.6 0.8

0.0 0.2 0.4 0.6 0.8

0.0 0.2 0.4 0.6 0.8

Fig. 4. The effect of lengthLon nonlinear frequency ratio versus amplitude curves

of (8, 0) SWCNTs withm=0.15: (a) hingedhinged; (b) clampedhinged; and (c)clampedclamped.



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ARTICLE IN PRESS

Wang et al. [18] are also provided for a direct comparison.

The parameters used in this example are taken as [18]:

radius r=0.339 nm, Youngs modulus E=5.5 TPa, Poissons ratio

v=0.19, effective tube thicknessh =0.066 nm and shear correction

factor Ks=0.563. The figures in the brackets are the relative

errors between the present and analytical solutions.

The difference is very small at m=0.1 but tends to increaseas m increases. This is because the nonlocal effect is notincluded in the shear constitutive relationship in their

work[18].

Table 5 gives nonlinear frequency ratio onl/ol at different

maximum vibration amplitudes Wmax=Y (=1.0, 2.0, 3.0, 4.0, 5.0)

0.01.00

1.02

1.04

1.06

1.08C-C:

(5,0): r = 0.1955 nm (l= 0.6402)

(8,0): r = 0.3130 nm (l= 0.8055)

(9,0): r = 0.3525 nm (l= 0.8551)

(11,0): r = 0.4305 nm (l= 0.9415)

wmax

nl/

l

1.00

1.02

1.04

1.06

1.08

1.10C-H:

(5,0): r = 0.1955 nm (l= 0.4666)

(8,0): r = 0.3130 nm (l= 0.6053)(9,0): r = 0.3525 nm (

l= 0.6496)

(11,0): r = 0.4305 nm (l= 0.7304)

wmax

nl/

l

1.00

1.04

1.08

1.12

1.16

1.20H-H:

(5,0): r = 0.1955 nm (l= 0.3164)

(8,0): r = 0.3130 nm (l= 0.4233)

(9,0): r = 0.3525 nm (l= 0.4598)

(11,0): r = 0.4305 nm (l= 0.5302)

wmax

nl/

l

0.2 0.4 0.6 0.8

0.0 0.2 0.4 0.6 0.8

0.0 0.2 0.4 0.6 0.8

Fig.5. The effect of radius ron nonlinear frequency ratio versus amplitude curves

of the SWCNTs with m=0.15 and L=5 nm: (a) hingedhinged; (b) clampedhinged; and (c) clampedclamped.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

C-C:

= 0.00

= 0.10

= 0.15

= 0.20

x/L

w

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

C-H:

= 0.00

= 0.10

= 0.15

= 0.20

x/L

w

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

H-H:= 0.00

= 0.10

= 0.15

= 0.20

x/L

w

Fig. 6. The effect of nonlocal parameter on the nonlinear mode shapes (deflection

w) of (8, 0) SWCNTs with wmax=0.4 andL =5 nm: (a) hingedhinged; (b) clamped




8/9

ARTICLE IN PRESS

for isotropic homogeneous hingedhinged withL/h=100,h =0.3 in.

Here, Yffiffiffiffiffiffiffi

I=Ap

is the radius of the gyration of the beam withIand

A as the cross-section area and area moment of inertia, onlandolare the dimensionless nonlinear and linear frequencies,

respectively. The results obtained by the present direct iterative

method and finite element method[46]are listed inTable 5. Good

agreement was observed between the results obtained by the

direct iterative method and finite element method.

We now investigate the nonlinear free vibration of hinged

hinged (HH), clampedhinged (CH) and clampedclamped (C

C) nonlocal SWCNTs. Zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs

are considered and their elastic modulus and radius are listed in

Table 1. The frequency is normalized aso OL ffiffiffiffiffiffiffiffiffir=Ep inFigs. 37.InFigs. 35, the linear fundamental frequenciesolare also given.Unless otherwise stated, it is assumed that the length of theSWCNTs L =5 nm, Poissons ratio v=0.19, effective tube thickness

h =0.34 nm and shear correction factor Ks=0.563[18].

Fig. 3shows the nonlocal effect on the nonlinear frequency ratioonl/olversus amplitude curves for the (8, 0) SWCNTs withL =5 nm.Note that the nonlocal parameter m=0 corresponds to classicalSWCNTs without nonlocal effect. The SWCNTs exhibit a typical hard-

spring behavior, i.e., the nonlinear frequency ratio increases as the

vibration amplitude is increased for all boundary conditions. The

nonlocal parameter has a significant effect on the nonlinear vibration

behavior. At a given vibration amplitude, an increase in the nonlocal

parameter leads to both smaller linear and nonlinear frequencies but

a higher nonlinear frequency ratio. The clampedclamped SWCNT has

the highest while the hingedhinged one has the lowest linear

frequency, nonlinear frequency and nonlinear frequency ratio since

the end support is the strongest for the clampedclamped SWCNT

and the weakest for the hingedhinged SWCNT.

Fig. 4 shows the effect of beam length L on the nonlinear

frequency ratio versus amplitude curves for (8, 0) SWCNTs with

m=0.15. Both linear frequency and nonlinear frequency ratiodecrease as the lengthL increases. As L changes from 5 to 16 nm,

the linear frequency drops remarkably while the nonlinear

frequency ratio decreases slightly. The effect of beam length L

on the nonlinear frequency ratio is seen to be very small and is

negligible for long SWCNT (LZ16 nm).

Fig. 5 shows the effect of the radius r on the nonlinear

frequency ratio versus amplitude curves for SWCNTs with m=0.15andL =5 nm. Again, zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs

are considered. The radius changes from 0.1955 nm of the (5, 0)SWCNT to 0.4305 nm of the (11, 0) SWCNT. Results show that an

increase in the radius significantly raises the linear fundamental

frequency but slightly increases the nonlinear frequency ratio.

The nonlinear fundamental mode shapes for the displacement w

and rotation c of (8, 0) SWCNTs are plotted in Figs. 6 and 7with

various nonlocal parameter (m=0.0, 0.1, 0.15, 0.2) at wmax=0.4 andL=5 nm. The maximum displacementw occurs at the midpoint of

the hingedhinged and clampedclamped SWCNTs buts slightly

deviates from the center of the clampedhinged SWCNT. The

nonlocal parameter nearly has no effect on the nonlinear mode

shape (w andc) for the hingedhinged SWCNT, but it is relatively

large for the clampedhinged and clampedclamped SWCNTs. The

similar phenomenon is also found by Wang et al. [18] for linear

vibration modes of the nonlocal Timoshenko beams. Wang et al. [18]proved that the linear vibration modes of the hingedhinged beam

do not include any nonlocal parameter, which is included in the

linear vibration modes of the clampedhinged and clamped

clamped beams. It is should be pointed out that though the nonlocal

parameter has not effect on both the linear and nonlinear modes of

the hingedhinged SWCNT, it has significant effect to both the linear

and nonlinear frequencies of the hingedhinged SWCNT, as can be

seen from all of the results inTables 3, 4andFig. 3.

7. Conclusions

This paper investigates the nonlinear free vibration of SWCNTs

based on von Karman geometric nonlinearity, Timoshenko beam

0.0 0.2 0.4 0.6 0.8 1.0-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

H-H:

= 0.00

= 0.10

= 0.15

= 0.20

x/L

0.0 0.2 0.4 0.6 0.8 1.0-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

C-H:

= 0.00

= 0.10

= 0.15

= 0.20

x/L

0.0 0.2 0.4 0.6 0.8 1.0-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

C-C:

= 0.00

= 0.10

= 0.15

= 0.20

x/L

Fig. 7. The effect of nonlocal parameter on the nonlinear mode shapes (rotation

c) of (8, 0) SWCNTs withwmax=0.4 andL =5 nm: (a) hingedhinged; (b) clamped




9/9

ARTICLE IN PRESS

theory and Eringens nonlocal elasticity theory. Theoretical

formulations include the small scale effect and the influences of

transverse shear deformation and rotary inertia. The differential

quadrature (DQ) method and a direct iterative approach are

employed to obtain the nonlinear vibration frequencies and mode

shapes of nonlocal nanobeams with different end supports. Zigzag

(5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical

calculation and their elastic modulus is obtained by using MM

simulation. Numerical results show that: (1) at a given vibrationamplitude, an increase in nonlocal parameter leads to smaller

linear and nonlinear frequencies but a higher nonlinear frequency

ratio; (2) both linear frequency and nonlinear frequency ratio

become lower as the length of SWCNT increases and the radius

decreases; (3) the nonlocal parameter has an insignificant effect

on the nonlinear mode shape but can considerably change the

linear and nonlinear frequencies.

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Nonlinearfreevibrationofsingle-walledcarbonnanotubesusingnonlocal Timoshenkobeamtheory