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Vibration of Double-Walled Carbon Nanotubes Coupledby Temperature-Dependent Medium under a MovingNanoparticle with Multi Physical FieldsAli Ghorbanpour Araniab, Mir Abbas Roudbaria & Keivan Kianica Faculty of Mechanical Engineeringb Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Iranc Department of Civil Engineering, K.N. Toosi University of Technology, Tehran, IranAccepted author version posted online: 28 Oct 2014.

To cite this article: Ali Ghorbanpour Arani, Mir Abbas Roudbari & Keivan Kiani (2014): Vibration of Double-Walled CarbonNanotubes Coupled by Temperature-Dependent Medium under a Moving Nanoparticle with Multi Physical Fields, Mechanics ofAdvanced Materials and Structures, DOI: 10.1080/15376494.2014.952853

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ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 1

Vibration of Double-Walled Carbon Nanotubes Coupled by Temperature-Dependent

Medium under a Moving Nanoparticle with Multi Physical Fields

Ali Ghorbanpour Arani 1, 2, Mir Abbas Roudbari 1 and Keivan Kiani3

1Faculty of Mechanical Engineering, 2Institute of Nanoscience & Nanotechnology, University of

Kashan, Kashan, Iran, 3Department of Civil Engineering, K.N. Toosi University of

Technology, Tehran, Iran

Address correspondence to Ali Ghorbanpour Arani, Faculty of Mechanical Engineering,

University of Kashan, Kashan, Iran. E-mail addresses: aghorban@ kashanu.ac.ir;

[email protected]; [email protected]; [email protected]

ABSTRACT

A numerical model on nonlinear vibration of double-walled carbon nanotubes (DWCNTs)

subjected to a moving nanoparticle and multi physical fields is proposed. DWCNTs are

considered with the kinematic assumption of Euler-Bernoulli beam theory. The surrounding

elastic substrate is simulated as Pasternak foundation which is assumed to be temperature-

dependent. Hamilton's principle, incremental harmonic balanced method, Galerkin and time

integration method with direct iteration are employed to establish the equations of motion of

zigzag DWCNTs. The study reveals that for the weak van der Waals forces, DWCNTs has the

positive and the negative deflections as if it vibrates under a moving nanoparticle.

Keywords: DWCNTs, vibration, nanoparticle, temperature-dependent medium, physical fields.

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1. INTRODUCTION

The particular nature of carbon atoms and molecular perfection of carbon nanotubes (CNTs)

provide them with remarkably high levels of material properties such as physical, chemical,

stiffness and strength. Because of their novel properties, CNT holds substantial promise as

building blocks for nanoelectronics, nanodevices, solar cells, space elevators, nano opto

mechanical systems (NOMS) and nanocomposites [1-6]. Based on the number of walls, CNTs

are designated as single-walled, double-walled or multi-walled. The study of the novel properties

of single-walled (SWCNTs) and double-walled carbon nanotubes (DWCNTs) is important for

future materials development and one of the important aspects is their behavior in the presence

of magnetic and thermal fields. The DWCNTs or multi-walled carbon nanotubes (MWCNTs) are

made up of two or multiple layered cylinders along the thick direction, which interact with the

van der Waals (vdW) forces. Such forces play a critical role in vibration patterns of DWCNTs

[7-10].

Magnetic field effects in nanotubes could be important for exciting potential applications in

the area of nano-magneto-mechanical-systems (NMMS), micro-magneto-mechanical-systems

(MMMS), nanosensors, spintronics and nanocomposites.

Reddy et al. [11] worked on the magnetic properties of metal-filled multi-walled nanotubes by

using vibrating sample magnetometry. Heremans et al. [12] carried out measurements of the

magnetic moment and susceptibility of carbon nanotubes. They experimentally illustrated that

magnetic moment and susceptibility behavior is a function of magnetic field strength and

temperature. Vibration response of DWCNTs subjected to an externally applied longitudinal

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magnetic field was reported by Mumu et al. [9]. An analytical approach to study the effect of a

longitudinal magnetic field on the transverse vibration of a magnetically sensitive DWCNT was

proposed. Kiani [13] studied elastic wave propagation in magnetically affected double-walled

carbon nanotubes. Nonlocal Rayleigh, Timoshenko, and higher-order beam theories were

adopted. The flexural and shear frequencies as well as their corresponding phase and group

velocities of the sound waves for the proposed models were obtained. The role of strength of

longitudinal magnetic field, geometry properties of the DWCNT, size dependency, wave

number, and elastic properties of the surrounding matrix on the characteristics of the sound

waves were explained. Also, he proposed characterization of free vibration of elastically

supported DWCNTs subjected to a longitudinally varying magnetic field [14]. Using nonlocal

Rayleigh beam theory, the explicit expressions of the governing equations were obtained and

then numerically solved via an efficient numerical scheme. For magnetically affected DWCNTs

with simply supported, fully clamped, simple-clamped, and clamped-free ends, the flexural

frequencies as well as the corresponding vibration modes were evaluated for different varying

magnetic fields. The influences of the small-scale parameter and the magnetic field strength on

the dominant flexural frequencies of the DWCNTs were explained and discussed. Another paper

with the title of longitudinally varying magnetic field influenced transverse vibration of

embedded DWCNTs was reported by Kiani [15]. Using nonlocal Rayleigh, Timoshenko, and

higher-order beam theories, the free dynamic deflection of elastically supported DWCNTs

subjected to a longitudinally varying magnetic field was examined. By employing reproducing

kernel particle method (RKPM), the equations off motion of the magnetically affected DWCNT

(MADWCNT) for each model were reduced to a set of algebraic equations. For four common

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boundary conditions, namely fully simple, fully clamped, simple-clamped, cantilevered supports,

the dominant frequencies of the nanostructure were calculated. In particular cases, the predicted

results by the RKPM were compared with those of the exact solution.

In recent years, so many analysis of nonlocal vibration of nanotubes with or without moving

nanoparticle with consideration of thermal and surface effects have been reported. Fu et al. [16]

worked on analysis of nonlinear vibration for embedded carbon nanotubes. By using the

incremental harmonic balanced method, the iterative relationship of nonlinear amplitude and

frequency for the SWCNTs and DWCNTs were expressed. In the numerical calculation, the

amplitude frequency response curves of the nonlinear free vibration for the SWCNTs and

DWCNTs were obtained.

Dynamic response of a SWCNT subjected to a moving nanoparticle was studied in the

framework of the nonlocal continuum theory by Kiani [17], who considered the inertial effects of

the moving nanoparticle and the existing friction between the nanoparticle surface and the inner

surface of the SWCNT. Analytical solutions of the problem were provided for the

aforementioned nonlocal beam models with simply supported boundary conditions. Simsek [18]

presented an analytical method for the forced vibration of an elastically connected double-carbon

nanotube system (DCNTS) carrying a moving nanoparticle based on the nonlocal elasticity

theory. He showed that the velocity of the nanoparticle and the stiffness of the elastic layer have

significant effects on the dynamic behaviour of DCNTS. An analytical method of the small scale

parameter on the vibration of single-walled boron nitride nanotube (SWBNNT) under a moving

nanoparticle was presented by Ghorbanpour Arani et al. [19]. The effects of electric field, elastic

medium, slenderness ratio and small scale parameter were investigated on the vibration

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behaviour of SWBNNT under a moving nanoparticle. Results indicated the importance of using

surrounding elastic medium in decrease of normalized dynamic deflection (NDD). Also,

Ghorbanpour Arani and Roudbari [20] examined the effect of the nonlocal piezoelastic surface

on the vibration of visco-Pasternak coupled boron nitride nanotube (BNNT) system acted upon

by a moving nanoparticle. BNNTs were coupled by visco-Pasternak medium and single-walled

zigzag structure BNNT was selected in that study. Hamilton's principle was employed to derive

the corresponding higher-order equations of motion for both nanotubes. The influence of the

smart controller was proved on the fundamental longitudinal frequency.

Wang et al. [21] studied the thermal effect on vibration and instability of CNTs conveying

fluid. Results were demonstrated for the dependence of natural frequencies on the flow velocity

as well as temperature change. The influence of temperature change on the critical flow velocity

at which buckling instability occurs was investigated. It was concluded that the effect of

temperature change on the instability of SWCNTs conveying fluid is significant. A nonlocal

beam model for nonlinear analysis of CNTs on elastomeric substrates was proposed by Shen et

al. [22]. The SWCNT was modeled as a nonlocal nanobeam with a small scale effects. The

elastomeric substrate with finite depth was modeled as a two-parameter elastic foundation. The

thermal effects were included and the material properties of both SWCNTs and the substrate

were assumed to be temperature-dependent.

The natural frequency of nanotubes with consideration of surface effects was studied by Lee

and Chang [23]. They showed that the effect of surface stresses on the natural frequency of the

nanotube is very significant. Surface effects on the vibrational frequency of DWCNTs using the

nonlocal Timoshenko beam model were proposed by Lei et al. [24]. The influence of the surface

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elasticity modulus, residual surface stress, nonlocal parameter, axial half-wave number and

aspect ratio were investigated in detail. Wang [25] worked on the vibration analysis of fluid-

conveying nanotubes with consideration of surface effects. In a different model, the effects of

both inner and outer surface layers on the nanotubes were taken into consideration. He showed

that the surface effects with positive elastic constant or positive residual surface tension tend to

increase the natural frequency and critical flow velocity. For small tube thickness or large aspect

ratio, the stability of the nanotubes will be greatly enhanced due to the surface effect. Surface

effects on the transverse vibration and axial buckling of double nanobeam system (DNBS) were

examined by Wang and Wang [26] based on a refined Euler-Bernoulli beam model. It is found

that surface effects get quite important when the cross-sectional size of beams shrinks to nano-

meters.

Nonlinear vibrations of a single-walled carbon nanotube for delivering of nanoparticles were

investigated by Kiani [27]. The vdW interaction forces between the constitutive atoms of

nanoparticle and those of the SWCNT, frictional force, and both longitudinal and transverse

inertial effects of the moving nanoparticle were taking into account in the proposed model. The

nonlinear-nonlocal governing equations were explicitly obtained and then numerically solved

using Galerkin method and a finite deference scheme in the space and time domains,

respectively. The roles of the velocity and mass weight of the nanoparticle, small-scale effect,

slenderness ratio and vdW force on the maximum longitudinal and transverse displacements as

well as the maximum nonlocal axial force and bending moment within the SWCNT were

examined. Also, in another study, he proposed nanoparticle delivery via stocky single-walled

carbon nanotubes (A nonlinear-nonlocal continuum-based scrutiny) [28]. Based on the

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Rayleigh, Timoshenko, and higher-order beam theories, the dimensionless equations of motion

were constructed and then numerically solved. The effects of the mass weight and velocity of the

nanoparticle, radius and length of the SWCNT, the above mentioned vdW forces, and the small-

scale parameter on the maximum displacements and forces within the SWCNT were exclusively

explored. Moreover, another paper with the title of on the interaction of a SWCNT with a

moving nanoparticle using nonlocal Rayleigh, Timoshenko, and higher-order beam theories were

studied [29]. The moving nanoparticle was modeled by a moving point load by considering its

full inertial effects and Coulomb friction with the inner surface of the equivalent continuum

structure (ECS). The ECS under the moving nanoparticle was modeled based on the Rayleigh,

Timoshenko, and higher-order beam theories in the context of the nonlocal continuum theory of

Eringen. The dimensionless discrete equations of motion associated with the nonlocal beam

models were then obtained by using Galerkin method. The effects of slenderness ratio of the

ECS, ratio of mean radius to thickness of the ECS, mass weight and velocity of the moving

nanoparticle, and small scale parameter on the dynamic response of the SWCNT were explored.

However, to date, no report has been found in the literature on the magneto-thermo in- phase

and out-of-phase vibration behavior of DWCNTs under a moving nanoparticle with surface

effect. Herein, it is aimed to provide a comprehensive model for nonlinear in-phase and out-of-

phase vibrations of DWCNTs under a moving nanoparticle with multi physical fields using

elasticity theory. The surrounding elastic substrate is simulated as Pasternak foundation and is

assumed to be temperature-dependent. Hamilton's principle, incremental harmonic balanced

method (IHBM), Galerkin and time integration method of Newmark [30] in conjunction with

direct iteration method (DIM) [31] are employed to establish the corresponding higher order

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equations of motion, nonlinear in-phase and out-of-phase frequencies for nanotubes and

transverse displacement of coupled DWCNTs. It is important to note that the mentioned effects

play an important role in the nonlinear dynamic and vibration responses of the nanodevices

which are of interest in the area of nano-mechanic structures such as MMMs and NMMs.

2. NONLOCAL ELASTICITY THEORY

Based on Eringen nonlocal elasticity model [32], the stress at a reference point x in a body is

considered as a function of strains of all points in the near region. This assumption is agreement

with experimental observations of atomic theory and lattice dynamics in phonon scattering in

which for a homogeneous and isotropic elastic solid the partial nonlocal form is:

),(),(:)( xdVxxExV

(1)

where symbols ‘ :’ is the inner product with double contraction, E is young’s modulus and

)(x denotes the nonlocal stress tensor at x . The kernel function ),( xx is the nonlocal

modulus, xx is the Euclidean distance and lae0 , where 0e is constant suitable to each

material, a is an interior features size and l is an external characteristic size. The volume

integral in Eq. (1) is over the region V occupied by the body. The kernel function is given as [32]

),.()2(),( 012

lxxKlx , (2)

where 0K is the modified Bessel function. Combining Eqs. (1) and (2), the following equation

can be written:

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220 )(1 ae , (3)

where 2 is the Laplacian operator. The above nonlocal constitutive Eq. (3) has been recently

used widely for the study of micro and nanostructure elements. Figure 1 depicts DWCNTs which

is coupled by Pasternak foundation with length l , thickness h , inner and outer radiuses 1R and

2R , respectively. CNTs are simulated with Euler-Bernoulli beam model.

).,(),,(~,0),,(~

,),(),(),,(~

txWtzxWtzxV

xtxWztxUtzxU

(4)

where ( , )U x t , ( , )W x t and ( , )V x t are displacement components of middle surface. Using Eq. (4),

strain displacement relation can be written:

22

2

)(21

xw

xwzxx

. (5)

Therefore, the only nonzero nonlocal stress within the nanotube structures are expressed as [19]

),()( 0*2

0* TEae xxxxxx (6)

where xx* , 0 and T are the stress, thermal expansion coefficient and temperature, respectively.

U , W and TK are strain energy, work done by applied load and kinetic energy, respectively.

The strain energy for DWCNTs can be written as [19]

,}21{

0

* i

l

iiA

xxxx dxdAUi

(7)

where i shows the number of nanotubes layers such as inner and outer tubes ( 2,1i ).

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The component of Lorentz forces along the x , y and z directions are [9]:

).()(

),(

,0

2

22

2

2

2

2

22

2

2

2

2

22

xWH

yzV

yW

xWHf

yzW

yV

xVHf

f

xxz

xy

x

(8)

So the Lorentz forces work can be written as:

,)(21

02211

l

zzLF dxwAfwAfW (9)

For the DWNT with two layers, the pressure at any point between any two adjacent tubes

depends on the difference of their deflections at that point. Thus the vdW forces can be

expressed as:

),( 12 wwCFvdW (10)

where vdWF and C are the vdW forces and coefficient between two tubes. The vdW forces work

is as follows:

,][21

021

l

vdWvdWvdW dxwFwFW (11)

The lateral and shear forces of the elastic medium which is simulated by Pasternak foundation

can be considered as applied forces which may be written as [19]

),()(2

22

2 xwGwKF pwem

(12)

where wK and pG are spring and shear modulus of elastic medium, respectively. Since the

substrate is relatively soft, the foundation stiffness wK may be expressed by [22]

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)],2exp()562(5[)2)(1(4 11

2122

0

0

l

Ekw (13)

In which

.)1(

,)1(

,),exp()2( 020111s

s

s

s EEl

H

(14)

E , s and sH are the Young’s modulus, Poisson’s ratio and depth of the foundation,

respectively. It is assumed that the Young’s modulus of the foundation is temperature-dependent,

whereas Poisson’s ratio depends weakly on temperature change and is assumed to be a constant.

pG is taken to be 1–10th of the value of wK for the same tube in thermal environmental

conditions [22]. Furthermore, in order to introduce a more complete model, Jiang et al. [33]

established a cohesive law for CNTs and polymer interfaces. The cohesive law, cohesive energy

and cohesive strength are obtained directly from van der Waals interaction which was proposed

by Lennard-Jones potential model [34]. So, as a future outlook of this study, the Pasternak model

can be replaced by Jiang cohesive law due to improve the mechanical behaviour of composite.

Therefore, the work done by elastic medium and transverse load is

,])([21

0 12 l

ememdxpwwFW (15)

where P is the distributed transverse load along x axis and TF is the elastic medium forces. The

kinetic energy can be written as

2,1,])([21 2

0

idxdAtwK

iii

li

AT

i

(16)

The governing equations of motion for embedded DWCNTs can be derived by Hamilton’s

principles as follows:

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0)(10

t

TT dtWKU . (17)

After taking the variation of strain energy, work done by applied forces and kinetic energy

and setting the coefficients of 1w and 2w

equal to zero lead to

,)(21

,)(:

,)(:

2

2222

22

2

2

1111

21

2

1

seTix

zvdWemxx

zvdWxx

NNxwEAN

wAAfFFx

wN

xxMw

pAfFwAxw

Nxx

Mw

(18)

From Eqs. (6) and (18) and ixix dAzM , the governing equation for DWCNTs are given by:

,0

))(2

)(1()()()(

,

))(2

)(1()()()(

22

2222

2

2

22222*

2

11

1211

2

2

11111*

1

AfAfFF

FFwNNx

wEAx

wAwAwEI

ppAfAf

FFwNNxwEA

xwAwAwEI

ZZvdWvdW

ememseT

ZZ

vdWvdWseT

(19)

1w and 2w are transverse displacements of the inner and outer nanotubes, respectively. Also

20 )( ae is the nonlocal parameter used in the nonlocal continuum model. The cross sectional

area iA and cross-sectional area moment of inertial iI in which the sub-index i=1 and 2 shows

that the indexed parameter is associated with the inner and outer tubes, respectively.

Superscripts and ' denote the first derivatives with respect to t and x , respectively.

The effect of the moving nanoparticle on the nanotube can be presented by load ),( txP as

follows [18]

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)(δ),( npxxPtxP . (20)

In this simulation, P denotes the magnitude of the moving load, indicates the Dirac-delta

function and npx is the position of nanoparticle in the inner nanotube.

On the basis of the theory of thermal elasticity mechanics, the axial force TN can be written as

[21]

TEAN T 0 , (21)

The effective flexural rigidity and surface effect parameter can be expressed as [23, 24]

where sE is the surface elasticity modulus and 0 is the residual surface stress under unstrained

conditions.

3. SOLUTION METHODOLOGY

3.1. First approach; Galerkin and IHBM

In this section, the problem will be solved numerically by using the Galerkin and IHBM. The

unknown deflection functions are approximated as [18]

)()(),(1

111 txtxwi

ii

,

)()(),(1

222 txtxwi

ii

(24)

3*)( REEIEI s (22)

RN Se 04 (23)

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where i1 )(t and i2 )(t are the unknown time-dependent generalized coordinates and )(1 xi and

)(2 xi are the eigenmodes of undamped simply-supported nanotubes which are express as [18]

),(sin)()( 21 Lxixx ii

,...3,2,1i (25)

Substituting Eq. (25) into Eq. (24) and the result equation into Eq. (19) multiplying both sides of

the resulting equation with )(1 xj and )(2 xj , respectively, integrating it over the domain (0, L)

yields

,][

)()(

)(21)(

21)(

)()(

0 10 1112

11

0 1112

111101

120 111

12

11 0

123

1111 0

123

111''''

101

1

1101

11101

1111

011

0 11*

1

L

j

L

jixi

i

L

jixi

iji

L

ii

i

L

jiii

i

ji

L

iiiji

L

iiiji

L

seTi

i

jise

L

Ti

iji

L

iiji

i

L

ii

L

jii

dxPPAH

dxAHdxCdxC

dxEAdxEAdxNN

dxNNdxAdxAdxEI

(26)

.0

)()(

)(21)(

21)(

)(

)()(

0 2222

12

0 2222

122201

120 221

12

21 0

22

23

2221 0

22

23

222''''

201

2

2201

22201

2221

021 0

222

1 022222

10

1 0222

*2

L

jixi

i

L

jixi

iji

L

ii

i

L

jiii

i

ji

L

iiiji

L

iiiji

L

seTi

i

jise

L

Ti

iji

L

iiji

i

L

ii

L

jiwi

i

L

jipwijii

L

i

L

pijii

AH

dxAHdxCdxC

dxEAdxEAdxNN

dxNNdxAdxAdxk

dxGkdxGdxEI

(27)

Substituting the derivative of Eq. (25) into Eqs. (26) and (27)

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),)/(1())/(1()/(

))/(1()/)(())/(1()/(4

))/(1())()/(1()/()(

21

2221

122

1221

12

1212

14*

1

LpLLHA

LLNNLLEALALCLEI

x

seT

(28)

.0))/(1()/(

))/(1()/)(())/(1()/(4

))/(1(

))()/(1())/(1()/)(()/()(

2222

2

222

2222

22

2

122

222

24*

2

LLHA

LLNNLLEALA

LCLLGkLEI

x

seT

pw

(29)

The in-phase and out-of-phase nonlinear nondimensional fundamental frequencies (NFF) of

the coupled nanotubes can be obtained from IHBM [10, 16]. The following dimensionless

parameters are now introduced:

,)()(

,)(,)(,)(

,)(),1(,

,25.0,)()(,

,,)(,

,,,

1

*12

111

1

22

2

*2

*1

2

1

111

22

11

1

1

tA

EIL

APL

LAN

LAN

L

AH

Le

Le

EIEI

AA

AC

AG

LAk

ra

ra

AIr

L

PSe

SeT

T

xH

cp

Rw

K

(30)

By substituting these into Eqs. (28) and (29), the dimensionless nonlinear vibration governing

equations are obtained as follows:

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,)()(})()()()(1{)( 2223

112222

21

22

L

P

L

C

L

Se

L

T

L

H

L

C

L

aaad

ad

(31)

.0)(

})()()()()()(1{)(

12

322

22222222

22

a

aad

ad

L

C

L

Se

L

T

L

H

L

C

L

K

L

R

L

(32)

So, the nonlinear amplitude frequency response curves can be obtained from this method.

3.2. Second approach; nonlinear Newmark and IDM

The equations of motion (28) and (29) can be written in a compact matrix form as follows

)},({)}(]{[)}()})]{(({[)}(]{[)}(]{[ tFtKttKtCtM LNL (33)

where ][M , ][C , ][ LK and ][ NLK are mass, damping, linear and nonlinear stiffness matrices,

respectively, and )]([ tF is generalized load matrix generated by the moving nanoparticle.

Also .)}(),({)}({ 21Tttt

Based on time integration method of Newmark [30] in conjunction with the DIM [31], the

nonlinear equation of motion can be solved.

4. NUMERICAL SOLUTION OF THE PROBLEM

Herein, the roles of the surface effect, temperature change, slenderness ratio, and vdW force,

on both linear and nonlinear frequencies are going to be investigated. The properties of the

DWCNTs are considered as follows [16, 22]:

TPa1.1E , 3g/cm1300 , nm32.2iD , nm30 D , m001.0sH , )GPa(0034.022.3 TE

, 48.0s

The surface elasticity and surface stress values were obtained using atomistic calculations in

which sE is calculated from surface elastic constants 112 nickel [23].

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The dimensionless velocity parameter ( ) can be expressed as [18]

1cr

np

vv

, (34)

where Lvcr 11 and 1 is the main magnitude of the i when 1i . Also, the dimensionless

parameter *t represents dimensionless time as follows [18]

Ltv

Lx

t pp * . (35)

Figure 2 demonstrates the influence of the temperature change on the nondimensional

fundamental frequency (NFF) with nondimensional amplitude. NFF can be obtained as follows

.Linear

NonlinearNFF

(36)

The linear free vibration frequencies for DWNT are calculated as follows:

).)/(1(),)/(1(],/[}))/(1({],/[}))/(1({

],/[})/)((

)/)(()/)(()/()/()/()/()/(){(

],/[})/)(()/)(()/()/()/()/(){(

),(),(

,)4(21

222

211

22

212

1

24

224

2422

222

42

*2

142

2421

221

41

*1

212121

2

LASLASSCLbSCLb

SLNN

LNNkLGkLGCLCLHALHALEIa

SLNNLNNCLCLHALHALEIa

bbaaaa

SeT

SeTwpwp

xx

SeTSeT

xx

Linear

(37)

As it is illustrated, in presence of temperature-dependent spring and shear modulus, NFF has

smaller values than without elastic medium. Also, increase in the values of temperature lead to

decrease in the magnitudes of NFF. So the spring and shear modulus of surrounding

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temperature-dependent elastic medium has a pronounced effect on the NFF response curves of

DWCNTs.

Figure 3 shows the effect of the length on the NFF for nonlinear and linear out-of-phase

vibration for selected values of nonlocal parameters. It is interesting to note that at lower values

of length )20( nmL the difference between nonlinear and linear out-of-phase vibration is

remarkable. Also, frequency increases with decrease in the values on small scale effect. The

variation of NFF with longitudinal magnetic field for various values of vdW coefficients is

depicted in figure 4. It can be realized from this figure that curves of the DWCNTs are steep

when the vdW force is small which is called in-phase vibration, but with the increment of the

force, the curves tend towards a flat curve which is called out-of-phase vibration. It is important

to note that the increase in the values of longitudinal magnetic field lead to increase in

magnitudes of NFF.

Figure 5 demonstrates the influences of nondimensional amplitudes on the NFF with and

without surface stress effects. Because of the very large surface to volume ratio of nanotube,

surface effects play an important role in determining the NFF of these materials and should be

considered. It is clear from this figure that in presence of surface stress effect NFF has higher

values in a specific nondimensional amplitude value. The effect of the dimensionless velocity

parameter on the NDD of the inner and outer nanotubes for selected values of the small scale

parameters is given in figures 6 and 7. It can be seen that increase in the values of nonlocal

parameters lead to increase in the magnitude of NDD. Also, for the large values of the velocity

parameter, the NDD tend to zero. The influence of the temperature change on the NDD for

various values of the small scale effect of the inner and outer nanotubes is given in figures 8 and

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9. As it is depicted, in presence of temperature-dependent spring and shear modulus NDD for

inner nanotube has larger values in a similar case of outer nanotube. Also, increase in the values

of temperature lead to increase in the magnitudes of NDD for both nanotubes. In fact, in

presence of thermal effect, DWCNTs becomes softer. It is remarkable that the magnitudes of

NDD for selected values of temperature-dependent spring and shear modulus are not same which

is because of the stiffer DWCNTs caused by the surrounding elastic medium. Figures 10 and 11

show the influence of the vdW forces on the NDD of the inner and outer nanotubes for various

values of the longitudinal magnetic field. The most interesting feature is that the inner and outer

nanotubes behave reversely as the longitudinal magnetic field increases. Indeed, for inner

nanotube NDD decreases with increase in the values of longitudinal magnetic field and for outer

nanotube NDD increases with increase in the values of longitudinal magnetic field. Also, for the

case of in-phase vibration mode there is no sensible deflection for the outer nanotube. Likewise,

the variation of NDD for inner nanotube in a specific vdW forces is more than outer ones.

Figures 12 and 13 illustrate the in-phase and out-of-phase variations of the NDD with time

histories for selected values of vdW forces of inner and outer nanotubes. It is interesting to note

that DWCNTs has the positive and the negative deflections as if it vibrates under a moving

nanoparticle as a harmonic load. This condition occurred for the weak vdW forces. Also, the

most obvious observation from this figure is that for the weak vdW forces coupling between two

nanotubes, the variation of NDD for both inner and outer nanotubes are remarkable, whereas, as

the stiffness of vdW forces increases, the features of the NDD curves of the two nanotubes begin

to resemble each other. Likewise, for the strong vdW forces coupling between two nanotubes,

there is a similar NDD for inner and outer nanotubes which is called rigid coupling. The effect of

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the nanoparticle moving load on the NDD for linear and nonlinear cases is illustrated in figure

14. As expected, the nonlinear NDD is smaller than the linear one, and as the value of the

moving load increases, the difference between the linear and the nonlinear NDD increases. This

is because of the internal axial force in the beam contributes to the lateral stiffness resulting from

the geometrical nonlinearity.

In order to validate this study, a simplified analysis suggested by Ansari et al. [10] on the

nonlocal beam theory for nonlinear vibrations of embedded multi-walled CNTs in thermal

environment is proposed. The values of NFF for DWCNTs with temperature-dependent medium,

multi physical fields and surface stress effects against nondimensional amplitude are compared

with the magnitudes of NFF without additional effects and reference [10], which are depicted in

figure 15. Based on this figure, there is no remarkable difference among the curves and

acceptable results can be achieved.

Figure 16 demonstrates the comparison of linear and nonlinear frequencies without additional

effects versus the frequency curves of (5, 5) @ (10, 10) DWCNT with simply supported

boundary conditions at two ends (SS/SS) based on the molecular dynamic (MD) simulation

method. As demonstrated in this figure, the results corresponding to vibration of DWCNT are

similar to those previously reported by Ansari et al. [35].

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5. CONCLUSION

In this paper, nonlinear in-phase and out-of-phase vibrations of double-walled carbon

nanotubes (DWCNTs) under a moving nanoparticle with multi physical fields using nonlocal

elasticity theory and surface stress effect is proposed. The main motivations of this study are

using DWCNTs under a moving nanoparticle in presence of longitudinal magnetic field, the

nonzero nonlocal stresses, such as bulk and surface and the effect of temperature-dependent

elastic medium. In this study, the surrounding elastic medium is simulated by Pasternak

foundation which acts as a polymer matrix. Based on the numerical approach the higher order

equations of motion of zigzag DWCNTs are solved. As it is showed, the variation of nonlinear

in-phase and out-of-phase NFF with respect to nonlocal parameters, nondimensional amplitude,

length, longitudinal magnetic field and temperature-dependent spring and shear modulus were

investigated. Also, the variation of NDD for coupled DWCNTs with the time history of

nanoparticle movement and dimensionless velocity parameter for various values of vdW forces

and temperature-dependent elastic medium stiffness were carried out. The following results may

be obtained from this study

For the case of weak vdW forces the positive and the negative deflections show that the

outer nanotube vibrates under a moving harmonic load. Likewise, for the strong vdW

forces coupling between two nanotubes, there is a similar NDD for inner and outer

nanotubes which is called rigid coupling.

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The spring and shear modulus of surrounding temperature-dependent elastic medium has

a pronounced effect on the NFF response curves of DWCNTs.

The curves of the DWCNTs are steep when the vdW force is small which is called in-

phase vibration, but with the increment of the forces, the curves tend towards a flat curve

which is called out-of-phase vibration.

Increase in the values of temperature lead to increase in the magnitudes of NDD for both

nanotubes. In fact, thermal effect makes DWCNTs becomes softer. It is remarkable that

the magnitudes of NDD for selected values of temperature-dependent spring and shear

modulus are not same which is because of the stiffer DWCNTs caused by the

surrounding elastic medium.

The inner and outer nanotubes for various values of vdW forces behave reversely as the

longitudinal magnetic field increases. Also, for the case of in-phase vibration mode there

is no sensible deflection for the outer nanotube. Likewise, the variation of NDD for inner

nanotube in a specific vdW forces is more than outer ones.

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[32] C. Eringen, On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation

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[33] L.Y. Jiang, Y. Huang, H. Jiang, G. Ravichandran, H. Gao, K.C. Hwang, A cohesive law for carbon

nanotube/polymer interfaces based on the van der Waals force, J. Mech. Phys. Solids., vol. 54, pp.

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[34] H. Tan, L.Y. Jiang, Y. Huang, B. Liu, K.C. Hwang, The effect of van der Waals-based interface

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[35] R. Ansari, S. Ajori, B. Arash, Vibrations of single- and double-walled carbon nanotubes with

layerwise boundary conditions: A molecular dynamics study, Curr. Appl Phys., vol. 12, pp.707-

711, 2012.

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FIG. 1. Temperature-dependent medium coupled by DWCNTs with longitudinal magnetic field

and surface effect under moving nanoparticle.

FIG. 2. The influence of the temperature change on the NFF with nondimensional amplitude.

FIG. 3. The effect of the length on the NFF for non-linear and linear out-of-phase vibration for

selected values of nonlocal parameters.

FIG. 4. The variation of NFF with longitudinal magnetic field for various values of vdW

coefficients.

FIG. 5. The influences of nondimensional amplitudes on the NFF with and without surface stress

effects.

FIG. 6. The effect of the dimensionless velocity parameter on the NDD of the inner nanotubes

for selected values of the small scale parameters

FIG. 7. The effect of the dimensionless velocity parameter on the NDD of the outer nanotubes

for selected values of the small scale parameters

FIG. 8. The influence of the temperature change on the NDD for various values of the small

scale effect of the inner nanotubes

FIG. 9. The influence of the temperature change on the NDD for various values of the small

scale effect of the outer nanotubes

FIG. 10. The influence of the vdW forces on the NDD of the inner nanotubes for various values

of the longitudinal magnetic field.

FIG. 11. The influence of the vdW forces on the NDD of the outer nanotubes for various values

of the longitudinal magnetic field.

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FIG. 12. The in-phase and out-of-phase variations of the NDD with time histories for selected

values of vdW forces of inner nanotubes.

FIG. 13. The in-phase and out-of-phase variations of the NDD with time histories for selected

values of vdW forces of outer nanotubes.

FIG. 14. The effect of the nanoparticle moving load on the NDD for linear and nonlinear cases.

FIG. 15. The comparison of the NFF of the nanotube with nondimensional amplitude between

present studies and [10].

FIG. 16. Comparison of linear and nonlinear frequencies without additional effects for

nmae 20 with literature.

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FIG. 1.

0 0.5 1 1.5 2 2.51

1.05

1.1

1.15

1.2

1.25

Nondimensional fundamental

frequencies (NFF)

Nondimensional amplitude

Without elastic mediumT=3000C, 0=3.4584e-6 (1/

0K)

T=5000C , 0=4.5361e-6(1/0K)

T=7000C , 0=4.6677e-6(1/0K)

FIG.2.

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1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-8

1

1.5

2

2.5

3

3.5

4

4.5x 10

12

Frequency (Hz)

Length (m)

e0a=0 nm

e0a=1 nm

e0a=2 nme0a=0 nm

e0a=1 nm

e0a=2 nmLinear frequency

Nonlinear frequency

FIG.3.

0 2000 4000 6000 8000 10000 12000 14000 160001

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8


frequencies (NFF)

Hx (A/m)

C~0C=0.3e7 PaC=0.3e8 PaC=0.3e9 Pa

Out-of-phase mode

In-phase mode

FIG.4.

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0 0.5 1 1.5 2 2.5 3 3.51

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08


frequencies (NFF)


Without surface effectWith surface effect

FIG.5.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Normalized dynamic deflection (NDD)

Dimensionless velocity parameter ()

e0a=0 nm

e0a=0.5 nm

e0a=1 nm

e0a=1.5 nm

e0a=2 nm

FIG.6.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Dimensionless velocity parameter ()

e0a=0 nm

e0a=0.5 nm

e0a=1 nm

e0a=1.5 nm

e0a=2 nm

FIG.7.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75


e0a(nm)

T=3000C, 0=3.4584e-6 (1/0K)

T=5000C , 0=4.5361e-6(1/0K)

T=7000C , 0=4.6677e-6(1/0K)

FIG.8.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75


e0a(nm)

T=3000C, 0=3.4584e-6 (1/0K)

T=5000C , 0=4.5361e-6(1/0K)

T=7000C , 0=4.6677e-6(1/0K)

FIG.9.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 105

1.785

1.79

1.795

1.8

1.805

1.81


Hx (A/m)

C~0 pa (In-phase)C= 0.3e5 paC= 0.3e7 pa

FIG.10.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 105

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Hx (A/m)

C~0 pa (In-phase)C= 0.3e5 paC= 0.3e7 pa

FIG.11.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Time history(t*)

C~0 (In-phase)C=0.3e7 PaC=0.3e9 PaC=0.3e12 Pa

FIG.12.

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2014

ACCEPTED MANUSCRIPT


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Time history(t*)

C~0 (In-phase)C=0.3e7 PaC=0.3e9 PaC=0.3e12 Pa

FIG.13.

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

P (N)


LinearNonlinear

FIG.14.

Dow

nloa

ded

by [

McM

aste

r U

nive

rsity

] at

04:

23 0

1 N

ovem

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2014

ACCEPTED MANUSCRIPT


0 0.5 1 1.5 2 2.5 3 3.51

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5


frequencies (NFF)


Present study(with additional effects)Present study (without additional effects)R. Ansari et al. [10]

FIG.15.

2 3 4 5 6 7 8 9 10

x 10-9

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Length (m)

Frequency (THz)

Present study, Nonlinear frequency, (e0a=2 nm),without additional effectPresent study, Linear frequency, (e0a=2 nm)without additional effectAnsari et al. [35], Flitted data of (5,5)@(10,10) (SS/SS)Ansari et al. [35], (5,5)@(10,10) (SS/SS)

FIG.16.

Dow

nloa

ded

by [

McM

aste

r U

nive

rsity

] at

04:

23 0

1 N

ovem

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2014

Documents

by Temperature-Dependent Medium under a Moving b Institute of …wp.kntu.ac.ir/k_kiani/papers-2015/Vibration of Double... · 2015. 11. 23. · Kashan, Kashan, Iran, 3Department of