Nonlinear Vibrations of a Single-walled Carbon Nanotube for Delivering of Nanoparticles

7/21/2019 Nonlinear Vibrations of a Single-walled Carbon Nanotube for Delivering of Nanoparticles

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Nonlinear Dyn

DOI 10.1007/s11071-014-1255-y

ORIGINAL PAPER

Nonlinear vibrations of a single-walled carbon nanotube

for delivering of nanoparticles

Keivan Kiani

Received: 9 April 2013 / Accepted: 13 January 2014

Springer Science+Business Media Dordrecht 2014

Abstract Thecapability of carbonnanotubes(CNTs)

in efficient transporting of drug molecules into the bio-

logical cells has been the focus of attention of vari-

ous scientific disciplines during the past decade. From

applied mechanics points of view, translocation of a

nanoparticle inside the pore of a CNT would result

in vibrations. The true understanding of the interac-

tive forces between the moving nanoparticle and the

inner surface of the CNT is a vital step in factual

realization of such vibrations. Herein, by employing

the nonlocal Rayleigh beam theory, nonlinear vibra-tions of single-walled carbon nanotubes (SWCNTs)

as nanoparticle delivery nanodevices are studied. The

existing van der Waals interactional forces between

the constitutive atoms of the nanoparticle and those

of the SWCNT, frictional force, and both longitudinal

and transverse inertial effects of the moving nanoparti-

cle are taken into account in the proposed model. The

nonlinear-nonlocal governing equations are explicitly

obtained and then numerically solved using Galerkin

method and a finite difference scheme in the space

and time domains, respectively. The roles of the veloc-

ity and mass weight of the nanoparticle, small-scale

effect, slenderness ratio, and vdW force on the maxi-

mumlongitudinal and transverse displacements as well

as the maximum nonlocal axial force and bending

K. Kiani (B)

Department of Civil Engineering, K.N. Toosi University

of Technology, Valiasr Ave., P.O. Box 19967-15433,

Tehran, Iran

e-mail: [email protected]; [email protected]

moment within the SWCNT are examined. In general,

the obtained results reveal that the nonlinear analysis

should be performed when the nanotube structure is

traversed by a moving nanoparticle with high levels of

the mass weight and velocity.

Keywords Single-walled carbon nanotube

(SWCNT) Nanoparticle delivery NonlinearvibrationNonlocal Rayleigh beam theory

1 Introduction

Due to the brilliant mechanical properties of car-

bon nanotubes (CNTs) [15] as well as frictionless

nature of their inner surface for conveying fluids flow

[68], they are recognized as superior nanodevices for

nanoparticle delivery [913]. Among various forms

of CNTs, single-walled carbon nanotubes (SWCNTs)

have been broadly investigated because direct compar-

ison between thepredictedproperties by the theoretical

worksandthoseof experimentallyobserveddata would

be possible[14,15]. In order to control vibrations of

SWCNTs for transporting of nanoparticles, their vibra-

tion behaviors dueto anindividualmovingnanoparticle

should be rationally investigated.

Atomic simulationsof nanostructuresgenerally take

alotoftimeandlaborcosts.Ontheotherhand,ifatleast

onedimension of thenanostructure under study is large

enough in compare to other ones, such costs consider-

ably increase. As a result, exploiting alternative effi-

cient techniques in analyzing of such nanostructures

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K. Kiani

has been the focus of attentions of the nanotechnol-

ogy community during the past two decades. Nonlocal

continuum field theory of Eringen [16,17] is among

the successful ones, which has been frequently used

for modeling of SWCNTs. To this end, the equivalent

continuum structure (ECS) pertinent to the SWCNT

is considered. The ECS for a SWCNT is an isotropichollow cylinder solid whose most of its frequencies

are identical to those of the SWCNT under study. In

nonlocal modeling of the nanostructure, by using a so-

called small-scale effect parameter, the existing inter-

atomic bonds between the constitutive atoms of CNTs

are appropriately incorporated into the equations of

motion. For each problem, the small-scale effect para-

meter is commonly adjusted by comparing the pre-

dicted results by the nonlocal model with those of an

atomic model[1821]. In this research work,sinceonly

the longitudinal and transverse vibrations of SWCNTsdue to translocation of nanoparticles are of concern, a

nonlocal beam model is employed. Certainly, if cap-

turing the propagated circumferential waves within the

SWCNT due to the passage of a moving nanoparticle

would be also of interest, nonlocal shell models should

be replaced and then appropriately analyzed.

In the context of nonlocal continuum theory, the

investigations on the effects of small-scale as well as

mass weight and velocity of the moving nanoparticle

on the linear transverse vibrations of SWCNTs were

initiated by Kiani and Mehri [22]. Such studies werealso carried out for double-walled CNTs, and analyt-

ical expressions of elastic deformation fields for the

innermost and outermost tubes were obtained[23,24].

In other complementary works, through using nonlo-

cal beam theories, the inertial effects of the moving

nanoparticle were also taken into account in the mod-

eling of the problem[25,26]. Such studies explained

that under what situations the effects of inertial terms

of the moving nanoparticle due to the vibrations of the

hosted nanotube are not negligible at all, and should be

appropriately included in the modeling of the problem.

Simsek[27] examined transversely forced vibrations

of a SWCNT subjected to a moving harmonic force

in the context of nonlocal Euler-Bernoulli beam the-

ory for small deflections. In another work, Simsek [28]

studied laterally small vibrations of microbeams under

action of a moving microparticle on the basis of mod-

ified couple stress theory. In the latter two works, the

inertial effects of the moving micro-/nanoparticle were

not considered in the proposed models. There are also

some works on the influence of nanoparticle transloca-

tion on thesmall in-planeandout-of-planevibrations of

nanoplates [2932]. As it is seen, the undertaken works

for the problems of moving nanoparticle-SWCNT

interaction were restricted to small deflections. In some

cases, as itwas exploredin some details formacro-scale

structures subjected to a moving mass [33], the hypoth-esis of small displacements may be not reasonable. In

following up this matter, this work is mainly devoted

to answer this question that under what circumstances

the linear analysis (LA) of the problem would be no

longer satisfactory.

In the present scrutinization, nonlinear vibrations

of a SWCNT for transporting a nanoparticle with a

constant velocity are investigated in the framework of

nonlocal continuum theory of Eringen. The interac-

tional forces between the moving nanoparticle and the

vibrating SWCNT are taken into account. To this end,the mass weight of the nanoparticle, the vdW forces

between the atoms of the moving nanoparticle and

those of the SWCNT, and both the longitudinal and

transverse inertial effects of the moving nanoparticle

are incorporated into the above-mentioned interactive

forces. In the context of large displacements, the equa-

tions of motion of the SWCNT are obtained on the

basis of the nonlocal Rayleigh beam theory. Due to

the appearance of the inertial effects in the governing

equations, finding an analytical solution is a very prob-

lematic task. Thereby, the Galerkin method and a finitedifference scheme are implemented for discretization

of the nonlinear-nonlocal governing equations in the

space and time domains, respectively. The dynamic

axial andtransverse displacements as well as thenonlo-

cal axialandbending momentwithintheSWCNTacted

upon by a moving nanoparticle are numerically calcu-

lated.Theeffects of thecrucial factors on themaximum

values of the elastic field of the SWCNT are inspected

in some detail.

2 Definition and assumptions of the physical

problem

An ECS with simply supported ends acted upon by a

moving nanoparticle is considered as shown in Fig.1.

The ECS is a hollow isotropic elastic solid of length lb,

inner/outer radiusri/ro, elasticity modulusEb, cross-

sectional area Ab, and moment inertia Ib. A mov-

ing mass with mass weight mg and constant veloc-

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Nonlinear vibrations of SWCNTs for nanoparticle delivery

Fig. 1 aA SWCNT for

delivering of a nanoparticle.

bA simply supported ECS

pertinent to the SWCNT

acted upon by a moving

nanoparticle

(a)

(b)

Mg v

xM

lb

xM

lb

Directi

onofm

ovemen

t

ofthen

anopar

ticle

ityv enters the hollow space of the SWCNT from the

left-hand side. The location of the moving nanopar-

ticle on the inner surface of the ECS is denoted by

(xM= vt,zM= ri )wheretis the time parameter(see Fig.1). Due to the fairly strong attraction forces

between the constitutive atoms of the moving nanopar-

ticle and those of the SWCNT, in which thez direc-

tion component of the resultant force is represented

by FvdW, the moving nanoparticle would be in con-tact with the inner surface of the SWCNT through the

course of vibration (i.e., 0t lbv). The longitudinal

friction force between the outer surface of the mov-

ing nanoparticle and the inner surface of the SWCNT,

Ff, is assumed to obey the hypothesis of the Coulomb

friction theory. Thereby,

Ff=kMg+FvdWMD

2uzDt2

(xM,zM), (1)

where k is the kinetic friction coefficient, D2

Dt2 is

the second material derivative with respect to time,

uz= uz(x, t)is the transverse displacement field ofthe ECS, and its longitudinal one is represented by

ux= ux(x, t). By taking into account of both lon-gitudinal and transverse inertial effects of the moving

nanoparticle, the longitudinal and transverse interac-

tional forces at the contact point, which are, respec-

tively, denoted byFcxandFcz , are stated by

Fcx=

FfMD2ux

Dt2

(xM,zM)

=M

k

D2uzDt2 D2uxDt2

(xM,zM)

H(lbxM), (2a)

Fcz= M

g D2uz

Dt2

(xM,zM)

H(lbxM), (2b)

where= g+ FvdWM

, andHis the Heaviside func-

tion. Based on theRayleigh beam theory, the longitudi-

nal and the transverse components of the displacement

field would beux(x,z, t)=u(x, t)zw,x(x, t)anduz(x,z, t)=w(x, t)whereu(x, t)andw(x, t)denotethe longitudinal and the transverse dynamic displace-

ments of the neutral axisof the ECS,and [.],xrepresentsthe first derivative of[.]with respect tox. By introduc-ing such displacements to Eqs. (2a) and (2b), one can

arrive at

Fcx=Mk D2wDt2

D2u

Dt2riD

2w,x

Dt2

(xxM)H(lbxM), (3a)

Fcz=M

g D2w

Dt2

(xxM)H(lbxM), (3b)

whererepresents the Dirac delta function. There are

some notes on using Eqs. (3a) and(3b) that should be

paid attention to by the interested readers:

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K. Kiani

1. The moving nanoparticle is traveling on a straight

line. Therefore, only the longitudinal and the trans-

verse components of displacements of the SWCNT

are taken into account. For an arbitrary path of

motion of the nanoparticle on the inner surface of

the SWCNT, special treatments should be consid-

ered in both modeling and analyzing of the problemunder study.

2. Duringthecourse ofexcitation,the moving nanopar-

ticle would be entirely in contact with the SWCNT.

Therefore, the full inertial effects of the moving

nanoparticle due to the motion of the SWCNT have

been considered. Now, if someone is interested in

studying the separation of the moving nanoparticle

from the inner surface of the SWCNT, the initi-

ation of such a phenomenon could be monitored

by checking the sign of the contact force in the

z direction. When such a positive contact forcebecomes negative at a special time, the separation

definitely occurs. During the course of separation,

the particle moves within the gaseous continuum

of the pore of the SWCNT, and we have Fcz= 0.At such a time interval, the motion of the mov-

ing nanoparticle could be readily investigated via

Newtons second law. Due to the vdW interactional

forces as well as the gravitational force, surely, the

moving nanoparticle would touch again the inner

surface of the SWCNT (i.e., reattachment). In this

study, the possibility of separation of the movingnanoparticle from the inner surface of the SWCNT

will be studied. Furthermore, the role of nonlin-

earity of the strains, for the case of large deflec-

tion of the SWCNT, on such an interesting phe-

nomenon is addressed. However, the phenomenon

of attachmentreattachment is not captured by the

proposed model, since it is assumed that the fairly

strong attraction forces between the constitutive

atoms of the nanoparticle and those of the SWCNT

would prevent the moving nanoparticle from sepa-

ration.

3. The cause of motion of the nanoparticle has not

been considered, since only the effects of the parti-

cle translocation on the vibrations of the nanostruc-

ture are of particular interest. Surely, any cause of

movement of the nanoparticle and its interactional

effects with the dynamic displacement field of the

SWCNT would result in more complicated govern-

ing equations as well as more difficulties in solving

the governing equations of the problem.

In thefollowing part, thederivation of thenonlinear-

nonlocal governing equations for slender SWCNTs

subjected to a moving nanoparticle will be explained.

For the sake of generality in studying the problem, the

equations of motion are presented in the dimensionless

form. The initial and boundary conditions are imposed

to the equations of motion. For solving the resultingboundary value problem, the Galerkin method is pro-

posed in the continuing. By application of such a pow-

erful method and using appropriate mode shapes, the

nonlinear governing equations are deduced to the non-

linear ordinary differential equations (ODEs). Using a

finite difference scheme, the resulting ODEs are solved

in the time domain, and the generated dynamic dis-

placements and forces of the SWCNT due to a moving

nanoparticle are determined.

3 Nonlocal modeling of the problem under study

According to the von-Karman beam theory for the

SWCNT which has been modeled based on the NRBT,

the nonlinear axial strain and stress accounting for

largedeflections-small rotationsof theECSare approx-

imated as

xx=ux,x+1

2

u2x,x+w2x,x

u,x

+1

2 u2,x+w2,xzw,xx, (4a)xx= EbxxEb

u,x+

1

2

u2,x+w2,x

zw,xx

.

(4b)

In the context of nonlocal continuum theory of Erin-

gen, the only nonlocal stress component of the present

model is expressed by [17,3436]

nlxx(e0a)2nlxx,xx= xx, (5)

wheree0ais called small-scale parameter. The mag-nitude of this parameter could be determined by com-

paring the predicted dispersion curves by the proposed

nonlocalmodelwith thoseofanatomistic-basedmodel.

By multiplying both sides of Eq.(5) byzandz2, and

then integrating the resulting statements over the cross-

sectional area of the ECS, the nonlocal axial force,

Nnlb , and the bending moment, Mnlb , within the ECS

are related to their classical (i.e., local) counterparts as

follows:

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Nnlb (e0a)2Nnlb,xx=Nb=

Ab

xxdA=EbAb

u,x+1

2

u2,x+w2,x

,

(6a)

Mnlb (e0a)2Mnlb,xx=Mb=

Ab

zxxdA= EbIbw,xx. (6b)

In order to derive the governing equations of the ECS

fordelivering a nanoparticleon thebasisof thenonlocal

continuum theory, the kinetic energy of the ECS,T(t),

its elastic strain energy,U(t), and the work done by

the exerted forces of the moving nanoparticle on the

ECS,W(t), should be appropriately evaluated. These

parameters are stated as

T(t)=12

lb0

b Ab u2 + w2+Ibw2,xdx, (7a)U(t)=1

2

lb0

u,x+

1

2

u2,x+w2,x

Nnlb

w,xxMnlb

dx, (7b)

W(t)=

(Fcxux+Fczuz)(xxM)

(zzM) d H(lbxM), (7c)

where the over-dot sign denotes the derivative with

respect to the time, and represents the inner surface

region of the ECS. In order to derive the equations of

motion for the problem at hand, the Hamiltons princi-

ple is exploited:t2

t1(TU+W) dt=0,in which

t1andt2are two arbitrary times, andis the variation

symbol. Thereby, the nonlocal governing equations are

obtained as

bAbuNnlb,x=Fcx(xxM)H(lbxM), (8a)bAbwIbw,xx

Nnlb w,x,x

Mnlb,xx= Fczri Fcx,x (xxM)H(lbxM), (8b)

by combining Eqs.(6a) and (6b) with Eqs. (8a) and

(8b), the nonlocal axial force and bending moment

within the ECS in terms of ECSs displacements are

derived as,

Nnlb =Nb+(e0a)2 [bAbuFcx(xxM)H(lbxM)],x, (9a)

Mnlb=Mb+(e0a)2

bAbwIbw,xx

Nnlb w,x,x

Fcz

ri Fcx,x(xxM)H(lbxM)

,

(9b)

by substituting Eqs. (9a) and (9b) into Eqs. (8a) and

(8b), one can arrive at

bAbu(e0a)2u,xx

Nb,x= Fcx(xxM)(e0a)2 (Fcx(xxM)),xx

H(lbxM), (10a)

bAb

w(e0a)2w,xxbIb w,xx(e0a)2w,xx xx

Nnlb w,x

,x

(e0a)2Nnlb w,x

,xx x

Mb,xx

= Fczri Fcx,x (xxM)(e0a)2 ((Fczri Fcx,x (xxM),xxH(lbxM). (10b)Inorder toexpress Eqs. (10a)and(10b) intermsof only

displacements, from Eqs. (6a) and (6b),NbandMbas

a function of displacement components are substituted

into these equations. On the other hand, in the context

of small rotation,Nnlb w,x

,x

(e0a)2Nnlb w,x

,xxx

EbAb

u2,x+ 12

u2,x+w2,x

w,x

,xAs a result, from

Eqs. (10a) and (10b), the nonlocal equations of motion

of aSWCNTtransportingan individualmovingnanopar-

ticle in terms of displacements accounting for largedeflections are obtained as

bAbu(e0a)2u,xx

EbAb u2,x+12 u2,x+w2,x

,x

= Fcx(xxM)(e0a)2 (Fcx(xxM)),x xH(lbxM),(11a)

bAb

w(e0a)2w,x xbIb w,x x(e0a)2w,x xxx

EbAb

u2,x+

1

2

u2,x+w2,x

w,x

,x

+EbIbw,x xxx=

Fczri Fcx,x

(xxM) (e0a)2 FczriFcx,x (xxM),x xH(lbxM).

(11b)

Since only the influence of the exerted forces by the

moving nanoparticle on the deformation field of the

SWCNT is of interest, the initial deflection of the

SWCNT due to its own weight is neglected. Hence,

the following initial conditions are considered,

u(x, t=0)=0, w(x, t=0)=0. (12)

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K. Kiani

In the case of simply supported SWCNT with fixed

movable ends, the following conditions should be

satisfied:

u(x=0, t)=0, w(x=0, t)=w(x=lb, t)=0,Nnlb (x=lb, t)=0, Mnlb (x=0, t)

=Mnlb (x=lb, t)=0. (13)Formoregenerality, the followingdimensionless quan-

tities are introduced:

= xlb

, M=xMlb , u =ulb

, w = wlb

, = 1l2b

EbIbbAb

t,

=e0alb

, =lbrb

, M= MbAblb

, = vCL

, =

glbCL

,

= g

, =lbri

,[.]=[.], +2[.],+()2[.], ,(14)

whereis a dimensionless operator,rbis the gyra-tion radius of theECSs cross-section (i.e.,rb

= IbAb ),and CLis the speed of the longitudinal wave within theECS (i.e.,CL =

Ebb

). By introducing the dimen-

sionless parameters in Eq. (14) to Eqs. (11), (12),

and (13), the dimensionless nonlinear governing equa-

tions of a SWCNT subjected to a moving nanoparticle

based on the nonlocal continuum theory of Eringen are

expressed by:

u,

2 u2,+12 u2,+w2,

,

=M k ()2 w

u1

w

(xxM)

H(1M), (15a)

w, 2w,

2

u2,+1

2

u2,+w2,

w,

,

+w,

=M

()2 w1

k

()2 w

u1

w

,

(M)H(1M),(15b)

with the following initial and boundary conditions,

u( , =0)=0, w(, =0)=0,u(=0, )=0, w(=0, )=w(=1, )=0,N

nl

b (=1, )=0, Mnl

b (=0, )=Mnl

b (=1, )=0,(16)

where the dimensionless operator is defined by

[.] = [.] 2[.], . Using Eqs. (9a), (9b), and (14),the dimensionless nonlocal axial force and bending

moment within the SWCNT subjected to a moving

nanoparticle are calculated in terms of dimensionless

displacements as follows:

Nnl

b =2

u,+1

2

u2,+w2,

+2

u, M

k

()2 wu1

w,

(M)H(1M)

,

,

(17a)

Mnl

b = w, +2

w, 2w,

Nnl

b w,, M()2

w

1

k ()2 w

u1

w,

,

(M)H(1M)

,

(17b)

where Nnl

b = Nnlb l

2b

EbIband M

nl

b = Mnlb lb

EbIb. Due to the

appearance of both longitudinal and transverse iner-

tial effects of the moving nanoparticle as well as the

existence of the nonlinear terms in the formulations of

the nonlocal equations of motion, seeking an analyti-cal solution to Eqs. (17a) and(17b) is a very problem-

atic task. Thereby, suggestion of an efficient numerical

scheme in solving such equations would be of great

beneficial. In the following part, Galerkin method plus

to a special finite difference scheme is employed for

fulfilling such a crucial job.

4 Solving the coupled nonlinear partial differential

equations of motion of a SWCNT

for nanoparticle delivery

The only unknown dimensionless displacements of the

ECSare discretized in terms of mode shapesas follows:

u( , )=N Mi=1

ui ( )ui (),

w(,)=N Mi=1

wi ()wi (), (18)

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whereui ( )andwi ( )are thei th mode shapes asso-

ciated with the longitudinal and transverse displace-

ments of the SWCNT, respectively.ui ( )andwi ( )in

order are the unknown parameters pertinent to thei th

mode shapes of longitudinal and transverse displace-

ments that should be determined at the required times,

N Mis the number of vibrational modes which is con-sidered in the analysis of the problem under study. For

the considered boundary conditions of the SWCNT,

the following mode shape functions are taken into

account [25,26]:

ui ( )=sin ((i0.5) ) , wi ( )=sin (i ) .(19)

Now both sides of Eqs. (15a) and (15b) are, respec-

tively, multiplied by uand w where denotes thevariational sign. Through integrating of the resulting

relations over the dimensionless space interval [0,1],

and taking the necessary integration by parts, one can

arrive at the following set of ODEs:

Mbx, = fb, (20)

or

M

uu

b Muw

b

M

wu

b M

ww

b u,

w, = f

u

b

f

w

b , (21)where

u=T, (22a)w=< w1, w2, . . . , wN M >T, (22b)

Muu

b

i j

=1

0

ui

uj+2ui,uj,

d

+M

ui (M)2ui,(M)

uj (M)H(1M),(22c)

Muwb i j =Mui (M)2ui,(M) sgn w

()2

k

wj (M)

1

wj,(M)

H(1M),

(22d)M

ww

b

i j

=1

0

wi

wj +2wi,wj,

+ 2

wi,wj,+2wi,wj,

d

+M

wi (M)2wi,(M)

wj (M)

1

sgn

w

()2

k

wj,(M)

1

wj,(M)

H(1M), (22e)

Mwu

b

i j

= M

wi (M)2wi,(M)

uj,H(1M),

(22f)

f

u

b

i=

10

2

u,+12

u2,+w2,

1+u,

ui,d

+M

ui (M)2ui,(M)

1sgn

w

()2

2w,(M, )

+ ()2 w,(M, )

2u, (M, )+()2 u,(M, )H(1M),

(22g)f

w

b

i=

10

2w,wi,

u,+

1

2

u2,

+ w2,+w,wi, d+Mwi (M)2wi,(M)

()2 + 2w, (M, )+()2 w,(M, ),

1 1

2+ k sgn

w()2

+ 1

2u, (M, )+()2 u,(M, )

,

H(1M). (22h)

In order to evaluate the unknown parameters of

Eq. (21)at each time, let y=x,. Therefore, Eq.(20)could be rewritten as,

M z,=f,

z=

y

x

, M=

Mb 0

0 I

,f=

fby

, (23)

whereIis the identity matrix. For discretizing Eq.(23)

in the time domain, the finite difference method is

exploited. For this purpose, z, is approximated by

z,= zi+1zi , where= i+1 i , zi = z(i ),andzi+1=z(i+1). By substituting such a discretizedform ofz,into Eq.(23),

Mzi+1Mzi f= 0,M=(1 )Mi+Mi+1 ,f= (1 )fi+fi+1 , (24)whereis the weight parameter of time, and its value

is commonly considered in the range of 0 1. In all

calculations, the value of this parameter is set equal

to 0.7. By employing Newtons method for calculating

zi+1in Eq. (24)at each time step,

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K. Kiani

Kzi+1=f,K=M f,zz=zoldi+1 ,f= M zoldi+1+M zi+ f,zi+1=z newi+1 zoldi+1 ,

(25)

wherez

old

i+1is the previous value ofzi+1through theiteration process. The elements off,zhave been givenin Appendix. The unknown parameters in Eq. (25)

are determined by performing iteration process until

achievingthe accurateresults for znewi+1at each time step.

5 Results and discussion

To show the capabilities of the proposed model in

predicting dynamic response of SWCNTs due to the

nanoparticle delivery, a fairly comprehensive paramet-ric study is carried out in accordance with the NA

explained in Sect.4.The limitations of the LA in pre-

dicting the elastic field of the SWCNT due to a moving

nanoparticle are also of particular interest. To this end,

consider a SWCNT acted upon by a moving nanopar-

ticle with the following data[37]:ri = 1 nm, ro =1.34 nm, b = 2500 kg/m3,Eb = 1 TPa, =2, andk=0.3. In the following parts, the influencesof mass weight and velocity of the moving nanoparti-

cle, vdW interactional force, and small-scale parame-

ter on the maximum axial displacement and nonlocalaxial force as well as the maximum deflection and non-

local bending moment of the SWCNT subjected to a

moving nanoparticle are examined in some details. In

order to study the problem in a more reasonable frame-

work, the following normalized fields are introduced:

uN= uumax,st , wN= w

wmax,st, NnlbN=

Nnlb

Nmax,st,

andMnlbN= M

nlb

Mmax,st. In these relations,umax,stand

Nmax,stdenote themaximum dimensionless values of

axial displacement and local axial force due to the sta-

tically applied frictional force at the midspan point of

the SWCNT, respectively. Additionally, wmax,stand

Mmax,strepresentthemaximum dimensionless valuesof deflection and local bending moment due to the stat-

icallyapplied weight of thenanoparticle at themidspan

point of the SWCNT, respectively. Since the effects of

theinterested parameters on themaximum elastic fields

of the SWCNT are of particular concern, the maximum

values of the above-mentioned normalized fields are,

respectively, denoted byuN,max,wN,max,NnlbN,max,

andMnlbN,max.

To show the efficiency of the proposed method-

ology, a convergence study is performed. For this

purpose, the relative errors of the maximum normal-

ized transverse displacement and normalized nonlocal

bending moment are defined by erel,w =wN,maxwN,max(N M=15)wN,max(N M=15) and erel,M =MnlbN,maxMnlbN,max(N M=15)MnlbN,max(N M=15)

. In Fig. 2, the plottedresults oferel,w anderel,Mas a function ofN Mare

provided. As it is seen in Fig.2,the predicted relative

errors of both maximum transverse displacement and

nonlocal bending moment of the SWCNT acted upon

bya movingnanoparticlewoulddecreaseas thenumberof mode shapes increases. Furthermore, for fairly high

levels ofN M, the effect of the vibration mode number

on the above-mentioned relative errors would reduce.

Such a scrutiny reveals that the proposed numerical

scheme is an effective one in nonlinear dynamic analy-

sisofnanotubestructures subjectedto moving nanopar-

ticles.

Fig. 2 Convergence check

of the proposed numerical

model; (e0a=1 nm, =20,M=0.1, VN=0.3)

3 5 7 10 150

0.01

0.02

0.03

0.04

0.05

0.06

NM

erel,w

3 5 7 10 150

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

NM

erel,M

1 3


9/19


Fig. 3 Comparison of the

predicted normalized

transverse displacement

under the moving object by

the proposed model with

those of Lee [38]:

aVN=0.11,bVN

=0.5; (M

=0.2,

rb= 0.03lb , ( ) LA,() NA, () Lee[38])

0 0.25 0.5 0.75 10

0.3

0.6

0.9

1.2

wN

(M

,)

0 0.25 0.5 0.75 10

0.45

0.9

1.35

1.8

wN

(M

,)

M

(a)

(b)

For the sake of verification, the predicted results

by the proposed model are compared with those of

Lee [38] in some special cases. Lee [38] studied lin-

early dynamical responses of simply supported Timo-

shenko beams which are acted upon by a moving mass.

In order to compare the predicted results by the NRBT

with those of the Timoshenko model, a slender beam

with lbrb

= 1003 is considered. The predicted normal-

ized deflections of the beam under the moving object

versus its dimensionless position have been plotted in

Fig.3a, b for two levels of the velocity of the movingobject. The predicted results by the proposed model

based on the LA and nonlinear analysis (NA) as well

as those of Lee [38] have been provided in these fig-

ures. Asit can beseenin Fig. 3a, b, there is a reasonably

good agreement between the linearly predicted results

of the present model and those of Lees model[38] for

most of the positions of the moving object. A more

detailed scrutiny of the plotted results reveals that the

maximum relative error between the predicted normal-

ized deflection by the LA and those of the model of

Lee [38] are limited to 1.5 and 3 percent forVN=0.11and 0.5, respectively. For the lower level of thevelocity

of the moving body, the predicted normalized deflec-

tions under the moving body on the basis of the NA

are in line and very close to the predicted values by the

LA. However, for the greater levels of the velocity, the

discrepancies between the results of the LA and those

of the NA are more obvious. As it is seen in Fig. 3b,

the linear model generally overestimates the results of

thenonlinear model. In the following, a comprehensive

parametric studywill bepresentedto determinethelim-

itations of the linear model in predicting the maximum

elastic fields of the SWCNTs subjected to a moving

nano-object.

The time history plots of the displacements and non-

local forces of the midspan point of the SWCNT used

for a nanoparticle delivery are demonstrated in Figs.4

and 5 for different levels of the velocity of the nanopar-

ticle. The predicted results have been provided on the

basis of both LAand NA.In these figures, frepresents

the dimensionless time of leaving the SWCNT by themoving nanoparticle. For a low level of the velocity of

the moving nanoparticle (i.e.,VN=0.1), the predictedresults by the LA and those of the NA are coincident

with a good accuracy (see Fig.4a,5a). For a fairly low

level of the velocity (i.e.,VN=0.2), the predicted lon-gitudinal displacement and nonlocal axial force of the

midspan point of the SWCNT by the LA and those of

the NA are fairly coincident. The predicted deflection

and nonlocal bending moment of the midspan point of

the SWCNT based on the LA and those of the NA are

roughly close to each other. Thediscrepancies betweenthe predicted results by the LA and those of the NA are

more obvious at the locally minimum and maximum

points of the plotted results. As the velocity of the mov-

ing nanoparticle increases, the discrepancies between

the predicted results by the LA and those of the NA

magnify. For the normalized axial displacement and

nonlocalaxial force,thismatteris more apparentduring

the course of free vibration as well as at the end of the

course of forced vibration.However, for thenormalized

1 3


10/19

K. Kiani

0 1 25

0

5

uN

(0.5,

)

0 1 25

0

5

uN

(0.5,)

0 1 25

0

5

uN

(0.5,

)

0 1 210

0

10

uN

(0.5,

)

0 1 215

0

15

uN

(0.5,

)

/f

0 1 25

0

5

Nbn

l (0.5,

)

0 1 25

0

5

Nbn

l (0.5,

)

0 1 25

0

5

Nbn

l (0.5,

)

0 1 210

0

10

Nbn

l (0.5

,)

0 1 215

0

15

Nbn

l (0.5,

)

/f

(a)

(b)

(c)

(d)

(e)

Fig. 4 Time history plots of the normalized axial displacement

and nonlocal axial force of the midspan point of the SWCNTtraversed by a moving nanoparticle for different levels of the

velocity:a VN

=0.1, b VN

=0.3,c VN

=0.5, d VN

=0.7,

eVN= 0.9; (= 50,M= 0.3,e0a= 1 nm; ( ) LA,() NA)

deflectionandnonlocal bending moment, such discrep-

ancies are more obvious in both courses of forced and

free vibrations. As it is observed in Figs. 4and5, the

peak points of both dynamic displacements and nonlo-

cal forces of the SWCNT move from the first phase to

the second oneas the velocity of the moving nanoparti-

cle grows. In the following parts, the influences of both

the velocity and mass weight of the moving nanoparti-

cle on the maximum values of both displacements and

nonlocal forces within the SWCNT acted upon by a

moving nanoparticle are explained.

An interesting study has been conducted to exam-

ine the influence of the existing interactional vdW

forces between the constitutive atoms of the nanopar-

ticle and those of the SWCNT on the nonlinear

dynamic response of the SWCNT subjected to a mov-

ing nanoparticle. For this purpose, the plots of the pre-

dicted both linear and nonlinear results of the displace-

ments as well as nonlocal forces of the SWCNT are

provided in Fig.6ac for different levels of the mov-

ing nanoparticle velocity. For low levels of the moving

nanoparticle velocity (i.e.,VN


11/19


0 1 22

0

2

wN

(0.5,

)

0 1 22

0

2

wN

(0.5,

)

0 1 22

0

2

wN

(0.5,

)

0 1 23

0

3

wN

(0.5,

)

0 1 22

0

2

wN

(0.5,

)

/f

0 1 21

0

1

Mbn

l (0.5,

)

0 1 22

0

2

Mbn

l (0.5,

)

0 1 22

0

2

Mbn

l (0.5,

)

0 1 22

0

2

Mbn

l (0.5,

)

0 1 22

0

2

Mbn

l (0.5,

)

/f

(a)

(b)

(c)

(d)

(e)

Fig. 5 Time history plots of the normalized deflection and non-

local bending moment of the midspan point of the SWCNT tra-

versed by a moving nanoparticle for different levels of the veloc-

ity: a VN= 0.1, bVN= 0.3, c VN= 0.5, dVN= 0.7, eVN= 0.9; (= 50, M= 0.3,e0a= 1 nm; ( ) LA,() NA)

moment by the LA and those of the NA would increase

as the magnitude of the vdW force increases. For mod-

erate levels of the moving nanoparticle velocity (i.e.,

VN=0.3), the predicted values of both axial displace-ment and nonlocal axial force within the SWCNT by

both the LA and NA linearly magnify with the vdWforce; however, the predicted values of both the deflec-

tion and nonlocal bending moment of the SWCNT

slightly vary with the vdW interactional force. For

>3, the LA can predict both the predicted axial dis-

placements and nonlocal axial forces of the SWCNT

by the NA with relativeerror lower than 3 %. Neverthe-

less, the differences between the predicted deflections

as well as nonlocal bending moments by the LA and

those of the NA are in the range of 78.5 %. For a fairly

high level of the moving nanoparticle velocity (i.e.,

VN= 0.5), the discrepancies between the predictedresults by the LA and those of the NA would com-

monly increase with the magnitude of the vdW force.

Almost for all values of , the NA underestimates the

predicted results by the LA. Based on the NA, the pre-dicted normalized maximum deflections and nonlocal

bending moment reduce with the magnitude of vdW

force.

The effect of mass weight of the nanoparticle on the

vibrations of SWCNTs for nanoparticle delivery is of

interest in this part. The plots of normalized maximum

displacements and nonlocal forces in terms of dimen-

sionless mass of the nanoparticle for different levels

of the nanoparticles velocity have been provided in

1 3


12/19

K. Kiani

1 3 51

6

11

uN,max

1 3 51

1.05

1.1

1.15

wN,max

1 3 51

5

9

NbN,max

nl

1 3 50.9

1

MbN,max

nl

1 3 51

6

11

uN,max

1 3 51.35

1.45

1.55

wN,max

1 3 51

5

9

NbN,max

nl

1 3 51.25

1.4

MbN,max

nl

1 3 51

6

11

uN,max

1 3 51.6

1.8

2

wN,max

1 3 51

5

9

NbN,max

nl

1 3 51.4

1.7

MbN,max

nl

(a)

(b)

(c)

Fig. 6 Effect of the vdW interactional force on the maximum

values of normalized displacements and nonlocal forces of the

SWCNT for different levels of the nanoparticles velocity: a

VN=0.1, b VN=0.3,c VN=0.5; [( ) LA, () NA;=50,M=0.3,e0a=1 nm]

Figs. 7ae. Regarding the maximum values of dynami-

calaxial displacementas well as nonlocal axial force of

the SWCNT in which acted upon by a moving nanopar-

ticle, for VN0.3 and all considered levels of the massweight of the nanoparticle, the discrepancies between

the predicted results by the LA and those of the NA

are lesser than 5 %. In the case ofVN= 0.5, suchdiscrepancies are lower than 5 % for M 0.25. ForM >0.25, the discrepancies between the results of the

LA and those of the NA increase with the mass weight

of themoving nanoparticle. ForM=0.5, the NA over-estimates the result of the LA with relative error about

25 %. Commonly, the discrepancies between the pre-

dicted both axial displacement and nonlocal force by

the NA and those of the LA magnify with the velocityof the moving nanoparticle. Concerning lateral vibra-

tion of theexploited SWCNT for nanoparticledelivery,

fora low level of themovingnanoparticlevelocity (i.e.,

VN=0.1) and the considered range ofM, the discrep-ancies between both the deflection and nonlocal bend-

ing moment of the SWCNT by the NA and those of

the LA are lower than 3 %. In such a circumstance, the

predicted results by the LA would be trustable with a

good accuracy. Generally, such discrepancies increase

with the velocity of the nanoparticle, particularly for

those nanoparticles with high values of mass weight

(see Fig.7d, e). According to the LA, for velocity of

the moving nanoparticle up toVN=0.5, there exists aroughly linear relationship between mass weight of the

moving nanoparticle and both values of the normalized

maximum deflection and nonlocal bending moment

within the SWCNT (see Fig. 7be). On the basis of

the NA, the nonlinear variations of such parameters as

a function of the mass weight of the moving nanopar-

ticle are so obvious forVN0.5.Another instructive study is carried out to determine

theinfluenceof themoving nanoparticlevelocityon the

vibrational behavior of carryingnanoparticle SWCNTs

based on both LA and NA. In the case ofM=0.3 ande0a= 1 nm, the predicted displacements and nonlo-cal forces within the SWCNT in terms of the normal-

ized velocity of the moving nanoparticle are demon-

strated in Fig.8.Irrespective of the initial fluctuations

of the normalized maximum deflection and nonlocal

bending moment (forVN


13/19


0 0.25 0.53.6

3.8

4

uN,max

0 0.25 0.51

1.1

1.2

wN,max

0 0.25 0.52.2

2.4

2.6

NbN,max

nl

0 0.25 0.50.9

0.95

1

MbN,max

nl

0 0.25 0.53

3.5

4

uN,max

0 0.25 0.51.2

1.4

1.6

wN,max

0 0.25 0.52

2.5

3

NbN,max

nl

0 0.25 0.51.2

1.4

1.6

MbN,max

nl

0 0.25 0.50

5

10

uN,max

0 0.25 0.51.5

2

2.5

wN,max

0 0.25 0.52

4

6

NbN,max

nl

0 0.25 0.51

1.5

2

MbN,max

nl

0 0.25 0.50

20

40

uN,max

0 0.25 0.51.5

2

2.5

wN,max

0 0.25 0.50

10

20

NbN,max

nl

0 0.25 0.51

2

3

MbN

,max

nl

0 0.25 0.50

50

M

uN,max

0 0.25 0.51.5

2

2.5

M

wN,max

0 0.25 0.50

20

40

M

NbN,max

nl

0 0.25 0.50

2

4

M

MbN,max

nl

(a)

(b)

(c)

(d)

(e)

Fig. 7 Effect of the nanoparticles weight on the maximum



VN = 0.1, b VN = 0.3, c VN = 0.5, d VN = 0.7, eVN=0.9; [( ) LA, () NA; =60,e0a=1 nm]

tically lessen with the velocity of the moving nanopar-

ticle up toVN= 1. A close scrutiny shows that theLA overestimates the predicted wN,maxby the NA

with relative error lower than 10 %. ForVN 0.68, the LA underesti-

mates the predictedMnlbN,maxby the NA. Excluding the

initial fluctuations of the normalized maximum axial

displacement and nonlocal force forVN


14/19

K. Kiani

0 0.5 12

6

10

14

18

22

VN

uN,max

0 0.5 10.8

1

1.2

1.4

1.6

1.8

2

2.2

VN

wN,max

0 0.5 12

4

6

8

10

12

14

VN

NbN,max

nl

0 0.5 10.8

1.2

1.6

2

2.4

VN

MbN,max

nl

Fig. 8 Effect of the nanoparticle velocity on the maximum values of normalized displacements and nonlocal forces of the SWCNT;

[( ) LA, () NA;=50,M=0.3,e0a=1 nm]

in capturing the predicted results of interest by the NA,

theLAcould reproduce thenormalized maximum elas-

tic fields of the SWCNT on the basis of the NA with

relative error lower than 2 and 10 % forVN=0.1 and0.3, respectively. It is worth mentioning that no regu-

lar pattern is observed for discrepancies of the results

of the LA and those of the NA in terms of the veloc-

ity of the moving nanoparticle. As the magnitude of

the small-scale parameter increases, the discrepancies

between the predictednormalizedmaximum deflection

by the LA and that of the NA would generally mag-

nify. Except the caseVN=0.9, this fact is also true forthe maximum normalized values of the nonlocal bend-

ing moment. ForVN=0.9, the discrepancies betweenthe predicted normalized maximum nonlocal axial and

bending moment of the SWCNT by the LA and those

of the NA would lessen with the small-scale parameter.

Generally, no regular pattern for the discrepancies of

the results of the LA and those of the NA as a function

of the small-scale parameter is detectable for various

levels of the velocity of the moving nanoparticle.

The influence of the slenderness ratio on the gener-

ated displacements and nonlocal forces within the car-

rying nanoparticle SWCNTs is of interest. The plots of

the longitudinal and transverse displacements as well

as nonlocal axial force and bending moment in terms

of the slenderness ratio are provided in Fig. 10ac.

Such plots are provided for three levels of the velocity

of the moving nanoparticle (i.e., VN= 0.1, 0.3, and0.5). In the case ofVN=0.1 (see Fig.10a), the nor-malized displacements and nonlocal forces decrease

as the slenderness ratio of the nanostructure increases.

Further investigations display that the discrepancies

between the predicted results by the NA and those of

the LA would generally lessen with the slenderness

ratio. Regarding the caseVN=0.3 (see Fig.10b), thepredicted normalized axial force and bending moment

would reduce as the slenderness ratio of the SWCNT

increases. However, such a fact is not generally true for

the longitudinaland transverse displacements. Interest-

ingly, the NA predicts that the maximum normalized

transverse displacement would slightly magnify as the

slenderness ratio increases. Nevertheless, the obtained

results based on the LA showthat this parameter would

lessen with the slenderness ratio. Such a fact is more

obvious forVN= 0.5 (see Fig.10c). Excluding thenormalized maximum longitudinal displacement, the

discrepancies between the predicted results by the LA

and those of the NA would generally reduce as the

slenderness ratio increases. In the case ofVN= 0.5(see Fig.10c), except the predicted normalized maxi-

mumtransverse displacement, both LAandNA display

1 3


15/19


0 1 23.7

3.75

3.8

uN,max

0 1 21

1.1

1.2

wN,max

0 1 22.2

2.4

2.6

NbN,max

nl

0 1 20.9

0.95

1

MbN,max

nl

0 1 23

3.5

4

uN,max

0 1 21.2

1.4

1.6

wN,max

0 1 22.4

2.6

2.8

NbN,ma

x

nl

0 1 2

1.4

1.6

MbN,max

nl

0 1 25

5.5

6

uN,max

0 1 21.6

1.8

2

wN,max

0 1 23.4

3.6

3.8

NbN,max

nl

0 1 21.5

1.6

1.7

MbN,max

nl

0 1 25

10

15

uN,max

0 1 21.9

2

2.1

wN,max

0 1 24

6

8

N

bN,max

nl

0 1 21.9

2.1

2.3

M

bN,max

nl

0 1 210

20

30

e0a(nm)

uN,max

0 1 21.5

2

2.5

e0a(nm)

wN,max

0 1 25

10

15

e0a(nm)

NbN,max

nl

0 1 21.8

2

2.2

e0a(nm)

MbN,max

nl

(a)

(b)

(c)

(d)

(e)

Fig. 9 Effect of the small-scale parameter on the maximum



VN = 0.1, b VN = 0.3, c VN = 0.5, d VN = 0.7, eVN=0.9; [( ) LA, () NA; =50,M=0.3]

that thenormalized maximum dynamic responsewould

decrease with the slenderness ratio. Further, the dis-

crepancies between the results of the NA and those of

the LA would reduce as the slenderness ratio increases.

Among all studied cases, in the case ofVN=0.5, vari-ation of the slenderness ratio has the most influence on

the variation of the maximum dynamic response.

Realizing the accuracy levels of the linear analysis

for the problem under study would be of great impor-

tance in practical applications, for instance, SWCNTs

for drug delivery. To this end, a relative error para-

meter is defined by erel = |[.]L A [.]N A|/|[.]L A|,where[.] is the parameter under study (i.e., [.] =uN,max, orwN,max, orN

nlbN,max, orM

nlbN,max). In

the plane ofVN M, the contour plots pertinent totheerel =0.05, 0.1, 0.2, and 0.30 are presented forthe normalized maximum displacements and nonlocal

forces in Figs.11 and12.The demonstrated results

have been given for three levels of the small-scale para-

meter (i.e.,e0a= 0, 1, and 2 nm). According to theplotted results in these figures, higher values of both

mass weight and velocity of the moving nanoparticle

would result in more discrepancies between the pre-

dictedresults by the LAand those ofthe NA.The region

between two arbitrary contours specifies the zone in

which the LA could estimate the predicted results by

the NA with the relative error in the range of those spe-

cific values pertinent to the aforementioned contours.

As it is seen in Figs.11and12,for nearly a half area

of the consideredVN Mplane, the LA could pre-dict the results of the NA with accuracy lower than 5

percent. The contour lines of deflections and nonlo-

cal bending moments of the SWCNT subjected to a

moving nanoparticle for various small-scale parame-

ters generally pursue the same trend (see Fig.12a, b).

However, this matter is not exactly true for axial dis-

placement and nonlocal axial force (see Fig.11a, b).

Higher density of the contour lines implies more sen-

1 3


16/19

K. Kiani

20 40 603.5

4

4.5

uN,max

20 40 601

1.05

1.1

1.15

wN,max

20 40 602

3

4

5

NbN,max

nl

20 40 601

1.2

1.4

MbN,max

nl

20 40 603

4

5

6

uN,max

20 40 601.4

1.6

1.8

wN,max

20 40 602

4

6

8

NbN,max

nl

20 40 601

1.5

2

MbN,max

nl

20 40 600

5

10

15

uN,max

20 40 601.5

2

2.5

wN,max

20 40 600

10

20

30

NbN,max

nl

20 40 601.5

2

2.5

MbN,max

nl

(a)

(c)

(b)

Fig. 10 Effect of the slenderness ratio on the maximum values of normalized displacements and nonlocal forces of the SWCNT for

different levels of the nanoparticles velocity: aVN=0.1,bVN=0.3,cVN=0.5; [( ) LA, () NA;M=0.3,e0a=1 nm]

Fig. 11 Contour plots of

erelfor:aNormalizedmaximum longitudinal

displacement,bNormalized

maximum nonlocal axial

force; ((...)e0a=0,( )e0a=1 nm, ()e0a=2 nm;=50)

0 0.25 0.50

0.25

0.5

0.75

1

M

VN

0.05

0.05

0.05

0.05

0.05

0.1

0.1

0.2

0.2

0.3

0.3

0 0.25 0.50

0.25

0.5

0.75

1

M

VN

0.05

0.05

0.05

0.05

0.1

0.1

0.2

0.2

(a) (b)0.1

0.3

0.3

sitivity of the accuracy level to the variation of both

mass weight and velocity of the moving nanoparticle.

According to Fig.11a, b, for a moving nanoparticle

with M= 0.5, the variation of the velocity of the

moving nanoparticle in the range of 0.50.6 has the

most influence on the variation of the normalized max-

imum axial displacement as well as the nonlocal axial

force.

1 3


17/19


Fig. 12 Contour plots of

erelfor:aNormalized

maximum deflection,

bNormalized maximum

nonlocal bending

moment; ((...)e0a=0,( )e0a=1 nm, ()e0a

=2 nm;

=50)

0 0.25 0.50

0.25

0.5

0.75

1

M

VN

0.05

0.0

5

0.05

0.05

0.1

0 0.25 0.50

0.25

0.5

0.75

1

M

VN

0.05

0.

05

0.05

0.1

0.1

0.2

0.3

0.1 0.05

(a) (b)

6 Conclusions

Nonlinear longitudinal and transverse vibrations of

SWCNTs fornanoparticledelivery areexploredvia the

nonlocal Rayleigh beam theory. Without considering

the cause of motion of the nanoparticle, it is assumed

that the nanoparticle slips on a straightpath on the inner

surface of the SWCNT. By considering the interac-

tionalvdWforcesbetween theconstitutive atoms of the

nanoparticle and those of the SWCNT, a simple fric-

tional model is employed. Both longitudinal and trans-

verse inertial effects of the moving nanoparticle are

incorporated into the interactional forces. By making

some reasonable assumptions, the nonlocal governing

equations of the model are constructed. The resulting

nonlinear-coupled equations of motion are solved via

Galerkin approach. The influences of the velocity and

the mass weight of the nanoparticle, the vdW interac-

tional force, the small-scale parameter, and the slen-

derness ratio on the maximum elasto-dynamic fields of

the SWCNT are addressed in some detail. The major

obtained results are as

1. As the velocity and the mass weight of the moving

nanoparticle magnify, the discrepancies between

the predicted results by the LA and those of the

NA would increase.

2. The maximum longitudinal displacement as well

as nonlocal axial force within the SWCNT would

increase as the vdW force between the mov-

ing nanoparticle and the nanotube intensifies. For

lower levels of the velocity, both maximum trans-

verse displacement and nonlocal bending moment

would increase as the magnitude of the vdW force

increases. However, for higher velocities, these

parameters would decrease with the vdW force.

Further, the discrepancies between the results of

the LA and those of the NA would magnify with

the vdW force.

3. Generally, the discrepancies between the results of

the LA and those of the NA would decrease as the

slenderness ratio of the SWCNT increases.

4. For moderate levels of the velocity of the moving

nanoparticle, the maximum elasto-dynamic fields

of the SWCNT would generally increase with the

small-scale parameter. However, for high levels of

the moving nanoparticle velocity, variation of the

small-scale parameter has a trivial influence on the

variation of the elasto-dynamic fields of the carrier

SWCNT.

Appendix

Theconstitutive submatrices of thematrixf,zwith theirelements are as,

f,z=

fy

fx

I 0

; f

y=

fu

b

u,

fu

b

w,

fwb

u,

fwb

w,

,f

x= fubu fubw

fwb

u

fwb

w

, (26)1 3


18/19

K. Kiani

wheredenotes the sign of partial derivative. The ele-

ments of the matrices fx

and fy

are calculated as fol-

lows

f

u

b

u i j=

1

02

1+3u,+0.5w2,

ui,uj,d

M()2 ui (M) 2ui,(M)

uj,(M)H(1M), (27a)

fu

b

w

i j

= 1

0

2 ui,

w,

wj,

+32w,wj,

1+u,d

+M()2

ui (M)2ui,(M)

wj, (M)

1

sgn

w

( )

2H(1M), (27b)f

w

b

u

i j

= 1

0

uj,

w,

wj,

+ 2w,wj,

1+u,d

+M()2

wi (M)

2wi,(M)

uj,(M)H(1M), (27c)f

w

b

w

i j

=

1

0

2 u,+0.5u2,+1.5w2,+0.5w,

wi,wj,

+1+u,+0.5

u2,+w2,

wi,

wj,

d+M()2

wi (M)2wi,(M)

12

k

sgn

w

()2

wj,(M)

wj,(M)H(1M), (27d)

f

u

b

u,i j = 2M

ui (M)

2ui,(M)

uj,(M)H(1M), (27e) f

u

b

w,

i j

=2M

1

sgn (

w()2

ui (M)2ui,(M)

wj,(M)H(1M),

(27f)f

w

b

u,

i j

= 2M

wi (M)

2wi,(M)

uj,(M)H(1M), (27g)

f

w

b

w,

i j

=2M

wi (M)2wi,(M) 1

2

k

sgn w()2wj,(M)

wj,(M)H(1M). (27h)

References

1. Wong, E.W., Sheehan, P.E., Lieber, C.M.: Nanobeam

mechanics: elasticity, strength, and toughness of nanorods

and nanotubes. Science277, 19711975 (1997)

2. Yu, M., Lourie, O., Dyer, M.J., Kelly, T.F., Ruoff, R.S.:

Strength and breaking mechanism of multiwalled carbon

nanotubes under tensile load. Science287, 637640 (2000)

3. Dresselhaus, M.S., Dresselhaus, G., Avouris, P.: CarbonNanotubes: Synthesis, Structure, Properties, and Applica-

tions. Springer, New York (2001)

4. Rao, C.N.R., Satishkumar, B.C., Govindaraj, A., Nath, M.:

Nanotubes. Chem. Phys. Chem.2(2), 78105 (2001)

5. Dai, H.J.: Carbon nanotubes: opportunities and challenges.

Surf. Sci.500, 218241 (2002)

6. Mao, Z., Sinnott, S.B.: A computational study of molecular

diffusion and dynamics flow through CNTs. J. Phys. Chem.

B104, 46184624 (2000)

7. Sokhan, V.P., Nicholson, D., Quirke, N.: Fluid flow in

nanopores: an examination of hydrodynamic boundary con-

ditions. J. Chem. Phys. 115, 38783887 (2001)

8. Skoulidas, A., Ackerman, D.M., Johnson, K.J., Sholl, D.S.:

Rapid transport of gases in carbon nanotubes. Phys. Rev.Lett.89, 185901 (2002)

9. Wang, Y., Yang, S.T., Wang, Y., Liu, Y., Wang, H.: Adsorp-

tion and desorption of doxorubicin on oxidized carbon nan-

otubes. Colloids Surf. B97(1), 6269 (2012)

10. Depan, D., Misra, R.D.K.: Hybrid nanostructured drug car-

rier with tunable and controlled drug release. Mater. Sci.

Eng. C32(6), 17041709 (2012)

11. Chen, B.D., Yang, C.L., Yang, J.S., Wang, M.S., Ma, X.G.:

The translocation of DNA oligonucleotide with the water

stream in carbon nanotubes. Comput. Theor. Chem.991(1),

9397 (2012)

12. Karchemski, F., Zucker, D., Barenholz, Y., Regev, O.: Car-

bon nanotubes-liposomes conjugate as a platform for drugdelivery into cells. J. Control. Release 160(2), 339345

(2012)

13. Wu, N., Wang, Q., Arash, B.: Ejection of DNA molecules

from carbon nanotubes. Carbon50(13), 49454952 (2012)

14. Iijima, S.: Helical microtubules of graphitic carbon. Nature

354, 5658 (1991)

15. Bethune, D.S., Klang, C.H., deVries, M.S., Gorman, G.,

Savoy, R., Vazquez, J., Beyers, R.: Cobalt-catalysed growth

of carbon nanotubes with single-atomic-layer walls. Nature

363, 605607 (1993)

16. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng.

Sci.10, 116 (1972)

1 3


19/19


17. Eringen, A.C.: Nonlocal Continuum Field Theories.

Springer, New York (2002)

18. Zhang, Y.Y., Wang, C.M., Tan, V.B.C.: Assessment of Tim-

oshenko beam models for vibrational behavior of single-

walled carbon nanotubes using molecular dynamics. Adv.

Appl. Math. Mech.1(1), 89106 (2009)

19. Khademolhosseini, F., Phani, A.S., Nojeh, A., Rajapakse,

N.: Nonlocal continuum modeling and molecular dynam-ics simulation of torsional vibration of carbon nanotubes.

Nanotechnol. IEEE Trans.11(1), 3443 (2012)

20. Duan, W.H., Wang, C.M., Zhang, Y.Y.: Calibration of non-

local scaling effect parameter for free vibration of carbon

nanotubes by molecular dynamics. J. Appl. Phys. 101(2),

024305 (2007)

21. Ansari, R., Rouhi, H., Sahmani, S.: Calibration of the ana-

lytical nonlocal shell model for vibrations of double-walled

carbon nanotubes with arbitrary boundary conditions using

molecular dynamics. Int. J. Mech. Sci. 53(9), 786792

(2011)

22. Kiani, K., Mehri, B.: Assessment of nanotube structures

under a moving nanoparticle using nonlocal beam theories.

J. Sound Vib.329(11), 22412264 (2010)23. Kiani, K.: Application of nonlocal beam models to double-

walled carbon nanotubes under a moving nanoparticle. Part

I: theoretical formulations.ActaMech.216, 165195 (2011)

24. Kiani, K.: Application of nonlocal beam models to double-

walled carbon nanotubes under a moving nanoparticle. Part

II: parametric study. Acta Mech.216, 197206 (2011)

25. Kiani, K.: Longitudinal and transverse vibration of a single-

walled carbon nanotube subjected to a moving nanoparticle

accounting for both nonlocal and inertial effects. Physica E

42(9), 23912401 (2010)

26. Kiani, K., Wang, Q.: On the interaction of a single-walled

carbon nanotube with a moving nanoparticle using nonlocal

Rayleigh,Timoshenko,and higher-orderbeam theories.Eur.

J. Mech. A31(1), 179202 (2012)27. Simsek, M.: Vibration analysis of a single-walled carbon

nanotube under action of a moving harmonic load based on

nonlocal elasticity theory. Physica E43(1), 182191 (2010)

28. Simsek, M.: Dynamic analysis of an embedded microbeam

carrying a moving microparticle based on the modified cou-

ple stress theory. Int. J. Eng. Sci.48(12), 17211732 (2010)

29. Kiani, K.: Small-scale effect on the vibration of thin

nanoplates subjected to a moving nanoparticle via nonlo-

cal continuum theory. J. Sound Vib. 330(20), 48964914

(2011)

30. Kiani, K.: Nonlocal continuum-based modeling of ananoplate subjected to a moving nanoparticle. Part I: the-

oretical formulations. Physica E 44(1), 229248 (2011)

31. Kiani, K.: Nonlocal continuum-based modeling of a

nanoplate subjected to a moving nanoparticle. Part II: para-

metric studies. Physica E 44(1), 249269 (2011)

32. Kiani, K.: Vibrations of biaxially tensioned-embedded

nanoplates for nanoparticle delivery. Ind. J. Sci. Tech. 6(7),

48944902 (2013)

33. Kiani, K., Nikkhoo, A.: On the limitations of linear beams

for the problems of moving mass-beam interaction using a

meshless method. Acta Mech. Sin. 28(1), 164179 (2011)

34. Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of

nonlocal continuum models to nanotechnology. Int. J. Eng.

Sci.41, 305312 (2003)35. Wang, Q., Wang, C.M.: The constitutive relation and

small scale parameter of nonlocal continuum mechanics for

modelling carbon nanotubes. Nanotechnology 18, 075702

(2007)

36. Wan, Q.: Wave propagation in carbon nanotubes via nonlo-

cal continuum mechanics. J. Appl. Phys.98, 124301 (2005)

37. Gupta, S.S., Batra, R.C.: Continuum structures equivalent in

normal mode vibrations to single-walled carbon nanotubes.

Comput. Mater. Sci.43, 715723 (2008)

38. Lee, H.P.: Dynamics response of a Timoshenko beam on

a Winkler foundation subjected to a moving mass. Appl.

Acoust.55(3), 203215 (1998)

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