vibrational analysis of double-walled carbon nanotubes based on the nonlocal donnell shell theory ...

8/12/2019 VIBRATIONAL ANALYSIS OF DOUBLE-WALLED CARBON NANOTUBES BASED ON THE NONLOCAL DONNELL SHELL THE

1/15

IJST, Transactions of Mechanical Engineering, Vol. 37, No. M2, pp 91-105

Printed in The Islamic Republic of Iran, 2013 Shiraz University

VIBRATIONAL ANALYSIS OF DOUBLE-WALLED CARBON NANOTUBESBASED ON THE NONLOCAL DONNELL SHELL THEORY

VIA A NEW NUMERICAL APPROACH*

R. ANSARI1, H. ROUHI

2**AND B. ARASH

3

1, 2Dept. of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, I. R. of IranEmail: [email protected]

3Dept. of Mechanical and Manufacturing Engineering, University of Manitoba,

Winnipeg, Manitoba, Canada, R3T 5V6

AbstractThis article describes an investigation into the free vibration of double-walled carbon

nanotubes (DWCNTs) using a nonlocal elastic shell model. Eringens nonlocal elasticity isimplemented to incorporate the scale effect into the Donnell shell model. Also, the van der Waals

interaction between the inner and outer nanotubes is taken into account. A new numerical solution

method from incorporating the radial point interpolation approximation within the framework of

the generalized differential quadrature (GDQ) method is developed to solve the problem.

DWCNTs with arbitrary layerwise boundary conditions are considered in this paper. It is shown

that applying the local Donnell shell model leads to overestimated results and one must recourse to

the nonlocal version to reduce the relative error. Also, this work reveals that in contrast to the

beam model, the present nonlocal elastic shell model is capable of predicting some new non-

coaxial inter-tube resonances in studying the vibrational response of DWCNTs.

Keywords Double-walled carbon nanotube, radial point interpolation method, differential quadrature method, layerwiseboundary conditions

1. INTRODUCTION

Ever since carbon nanotubes (CNTs) were discovered by Iijima at the NEC laboratory in Tsukuba, Japan

[1], extensive theoretical and experimental studies have been conducted on these novel materials [2].

DWCNTs as the special cases of multi-walled CNTs can be made in quantitative amounts from the chains

of fullerenes generated inside single-walled CNTs [3]. In recent years, DWCNTs have drawn a great deal

of attention from the scientific community due to their amazing mechanical, optical and chemical

properties.

It is generally accepted that atomistic modeling of nanostructures is very time consuming.

Accordingly, it is advantageous to develop continuum models for the analysis of nanostructures due to

their computational efficiency. Since the conventional continuum mechanics is scale free, great attemptshave recently been devoted to the enhancement of classical continuum models in order to better

accommodate the results from molecular dynamics (MD) simulations. In the most commonly used size-

dependant continuum theory, nonlocal continuum theory, developed by Eringen [4, 5], the scale effect is

simply introduced into the constitutive equations as a material parameter.

Recently, the vibration analysis of CNTs has been the subject of numerous studies based on both

local and nonlocal models using beam and shell theories [6-17]. For example, Ansari et al. [9] studied the

free vibration of DWCNTs based on the nonlocal Donnell shell model using an analytical solution

method. They also employed the MD simulations in order to calibrate the nonlocal parameter used in their

nonlocal model.

Received by the editors July 29, 2012; Accepted December 2, 2012.Corresponding author


2/15

H. Rouhi et al.

IJST, Transactions of Mechanical Engineering, Volume 37, Number M2 October 2013

92

For a DWCNT, different combinations of layerwise boundary conditions can be considered. In this

respect, Xu et al. [7] stated that: The relevance of the existing model in which both tubes have the same

boundary conditions for the vibration of double-walled CNTs is questionable. Therefore, developing

powerful numerical solution methods capable of treating layerwise boundary conditions in a DWCNT can

play an important role in the advancement of computational nanomechanics. Hence, the main aim of thecurrent work is to extend the study reported in [9] on DWCNTs with the same boundary conditions for the

inner and outer tubes to DWCNTs with layerwise boundary conditions. To this end, a novel numerical

method termed as RPIDQ is developed within the framework of hybrid radial point interpolation [18] and

differential quadrature method [19]. The effectiveness of the present model is assessed by MD simulation

as a benchmark of good accuracy. In addition, this study provides a comparison between the beam and

shell models in predicting the frequencies of DWCNTs. To accomplish this goal, an explicit formula is

also derived for the nonlocal frequencies of DWCNTs based on the beam model. Some new inter-tube

resonant frequencies and the related non-coaxial vibrational modes are identified in this work as a result of

incorporating circumferential modes into the shell model.

2. MODELING

In the theory of nonlocal elasticity, unlike the conventional continuum mechanics, the stress at a point is

considered to be a functional of the strain field at all points in the body. To bring the nonlocality into

formulation, the Eringen nonlocal constitutive equation is employed as [5]1 (1)here stands for the nonlocal parameter which leads to consideration of the scale effect and is theLaplacian operator; is the macroscopic stress tensor at a point. The stress tensor is related to strain bygeneralized Hookes law as

: (2)where is the fourth order elasticity tensor and : denotes the double dot product. Hookes law for thestress and strain relation is given by

0 0 0 0 0 00 0 0 00 0 0 00 0 0 0

(3)

where , and are Youngs modulus, shear modulus and Poissons ratio of the material, respectively.Consider an elastic cylindrical shell with radius , length and thickness for each tube of a DWCNT(see Fig. 1). There are different theories for cylindrical shells such as Flugges theory [20, 21] or

Donnells theory [22]. In Donnells shell theory, the shear and rotary inertia effects are taken into account.

Thus, it seems to be suitable for the vibration analysis of cylindrical shells. Also, it is frequently used for

the analysis of CNTs due to the relatively accurate results in spite of its theoretical simplicity. Based on

the Donnell shell theory, the three-dimensional displacement components, and in the, and directions respectively, as shown in Fig. 1, are assumed to be

, ,, ,, , , (4a)

, ,, ,, , , (4b)


3/15

Vibrational analysis of double-walled carbon nanotubes

October2013 IJST, Transactions of Mechanical Engineering, Volume 37, Number M2

93

, ,, ,,, (4c)where ,,are mid-surface displacements and , are mid-surface rotations. The mid-surface strainsand curvature changes are given by

, , , , 1 , , 1 (5)The strains at any point in the shell thickness can then be written in terms of mid-surface strains and

curvature changes as (6a) (6b)

(6c)

(6d) (6e)

Using Eqs. (3) to (6) the stress and moment resultants can be given as follows

// . . 1 1 1 (7a)

/

/ . . 1 1 1

(7b)

// . . 1 (7c) // . . (7d)

// . . 1 (7e)

// . . 12 1 1 (7f) // . . (7g)

// . . 1 (7h)where is the bending rigidity. The governing equations are

(8a)


4/15

IJST, Transa

94

in which tube throug

of the resul

Eqs. (8) ca

where the d

tions of Mech

, , h the vdW i

ing equatio

be stated in

ifferential o

anical Engine

1, ,nteraction f

s is given b

terms of th

erators are

Fig. 1.

ring, Volume

1

are the ine

rces [9]. Eq

y Eqs. (7).

five field v

1

iven in Ap

Schematic of

. Rouhi et al.

37, Number

tia terms.

s. (8) are m

hus, for the

ariables

1

endix A.

a CNT treate

2

3

lso, denoltiplied byth tube of, , ,

1 1

1

1

as an elastic

es the press1 DWCNT,

, ,

1 1

1 1

shell

O

ure exerted. The lefty the use o1,2as

1

tober 2013

(8b)

(8c)

(8d)

(8e)

on the thhand side

Eqs. (7),

(9a)

(9b)

(9c)

(9d)

(9e)


5/15



95

For any tube of a DWCNT, different boundary conditions may be considered by the combination of

simply-supported (S), clamped (C) and free (F) edges. For example:

Simply-supported-Simply-supported (SS)

0, 0 , (10)Clamped-Clamped (CC) 0, 0 , (11)

Clamped-Free (CF)

0, 0 0, (12)

3. SOLUTION

a) Radial point interpolation method

In the radial point interpolation method (RPIM), the trial function is given by [18]

, (13a)

, , , (13b)

, , , (13c)

where is the number of nodes in the neighborhood of a given point , are radial basis functions inthe space coordinates and is the coefficient corresponding to the given point . The vector ofcoefficients can be determined by enforcing Eq. (13) to pass through all the nodes within the supportdomain of point 1,2, , (14)or in matrix form

(15)in which

, , , and

is called the moment matrix given by

(16)

From Eq. (15), one can have

(17)Substituting Eq. (17) into Eq. (13) gives


6/15

H. Rouhi et al.


96

(18)in which

(19)

The th derivative of the RPIM shape functions is readily obtained by (20)b) RPIDQ analog of field equations

The modal displacement functions for the th tube are taken as, cos (21a)

,

sin (21b)

, cos (21c) , cos (21d) , sin (21e)

Substituting these modal functions into the field equations of the DWCNT and discretizing them at

the th given point using the RPIDQ approximation give

1 (22a)

1

(22b)

1 1

(22c)

1

(22d)


7/15



97

1

(22e)

where the algebraic operators are given in Appendix A.

c) RPIDQ analog of boundary conditions

By the RPIDQ approximation, the discrete counterparts of the equations governed by the boundary

conditions become

Simply-supported-simply-supported (SS)

V W

0 ; 1 0 ; 0 0,

(23)

Clamped-Clamped (CC)

0 0, (24)Clamped-Free (CF)

0 0, 1

0 ;

0 ;

0 ;

0 ;

0 0 1, (25)

d) Derivation of eigenvalue problem

Rearranging the quadrature analogs of field equations and the boundary conditions within the

framework of a generalized eigenvalue problem leads to

0 (26)

where the subscripts and refer to the boundary and domain grid points, respectively. The displacementvectors and are defined by

, 1,2 (27)

and


8/15

H. Rouhi et al.


98

, 1,2 (28)Using the condensation technique [23], Eq. (26) can be transformed into the standard form of

0 (29)from which the eigenvalues () can be extracted. The smallest value of is the fundamental frequency.4. RESULTS AND DISCUSSION

The constant values needed for numerical evaluations are E 1 TPa, 0.34 , 2.3g cm and 0.3. The configuration of layerwise boundary conditions, for example, will be indicated by (SS/CC),where the pair of SS corresponds to the inner tube and the pair of CC corresponds to the outer tube. Also,

for a given inter-tube mode number , for convenience, the frequency associated with the th axial andth circumferential modes will be denoted by . For all the calculations performed in this work, thefollowing radial basis function which is one of the commonest forms with adjustable parameters is

employed [18]

exp , (30)To define the support domain at a given point, the dimension of the support domain can be

determined by , where is the dimensionless size of the support domain and is acharacteristic length that relates to the nodal spacing near the point at . For uniformly distributed nodes,is the distance between two neighboring nodes. For non-uniformly distributed nodes, alternatively, can be characterized as an average nodal spacing in the support domain of . The physical meaning ofthe dimensionless size of the support domain

is the factor of the average nodal spacing. In one-

dimensional cases, an average value of can be computed by 1 , in which is anestimated support domain, , at the point . It is worth mentioning that should be a reasonably goodestimate of and is the number of nodes that are covered by a known domain with the dimension of. The first three resonant frequencies of a SS/SS DWCNT for up to 19regular and irregular gridsampling points are listed in Table 1. This table indicates quite obviously the converging trend of the

present numerical solution with increasing number of sampling points for both given radii of support

domain.

Table 1. Convergence of resonant frequencies (THz) of a SS/SS DWCNT

( 8.5 , 5, 1.34g cm , 0)Number ofnodes Regular Irregular Regular Irregular Regular Irregular

9 0.1335 0.1338 0.2644 0.2665 0.4096 0.4462

11 0.1333 0.1334 0.2640 0.2642 0.4017 0.396213 0.1330 0.1331 0.2637 0.2646 0.4002 0.4011

15 0.1328 0.1330 0.2635 0.2639 0.3998 0.400517 0.1328 0.1328 0.2634 0.2635 0.3997 0.399919 0.1328 0.1328 0.2634 0.2634 0.3997 0.3997

9 0.1334 0.1335 0.2641 0.2652 0.4050 0.4420

11 0.1332 0.1333 0.2639 0.2640 0.4012 0.4026

13 0.1329 0.1330 0.2636 0.2638 0.3999 0.400915 0.1328 0.1329 0.2634 0.2637 0.3997 0.4001

17 0.1328 0.1328 0.2634 0.2634 0.3997 0.399819 0.1328 0.1328 0.2634 0.2634 0.3997 0.3997


9/15

October2013

To val

those of M

(10,10) D

calibrated s

calibratedobtained fr

To inv

variation o

Fig. 3. This

shell modeeffects in t

more pron

ratio is also

between th

parameter i

for stiffer b

idate the so

D simulatio

CNT again

uch that the

alue for m the nonl

Fig. 2. Fund

estigate the

nonlocal to

figure sho

becomes se nonlocal

unced for l

studied in

e boundary

s affected b

oundary con

Fig. 3. V

Vibratio

lution proce

reported i

t nanotube

nonlocal s

is 1.15 ncal shell mo

mental frequ(5,5) @ (1

effects of n

local freque

s that as the

aller thanodel make

wer values

ig. 4. As s

conditions

the type of

ditions.

riation of fre

with

al analysis of

IJS

dure presen

[9]. The v

spect ratio i

ell model is

. It is obsedel with its

encies from t0,10) DWCN

nlocal para

ncy ratio ag

nonlocal pa

that of its lDWCNT

of aspect r

own in this

ncreases. A

boundary c

uency ratio

ifferent value

double-walled

T, Transaction

ed in this

ariation of

s shown in

capable of

rved that tdjusted no

e present shT under SS/S

eter and as

inst forrameter inc

cal counterore flexibl

tio. The ef

figure, as th

ccording to

nditions, so

ersus nonloc

s of aspect ra

carbon nanot

s of Mechanic

ork, the res

undamental

ig. 2. The n

predicting t

ere is a golocal param

ll model andS boundary c

ect ratio o

different va

eases, the fr

part. It phy. Also, it is

ect of boun

e nonlocal p

this figure,

that the no

l parameter f

tio ( 10

bes

l Engineerin

lts generat

frequencies

onlocal para

e results o

d agreemeter and the

MD simulatinditions

the frequen

ues of aspe

equency obt

ically meanobserved th

dary conditi

arameter in

the signifi

local influe

or a SS/CC D), Volume 37,

ed are comp

for a SS/S

meter nMD simul

t between tones reporte

n [9] for a

cies of DW

t ratio is ill

ained for th

s that the sat the effec

ons on the

reases, the

ance of the

ce is more

WCNT

umber M2

99

ared with

(5,5) @

eds to be

tion. The

he resultsd in [9].

NTs, the

strated in

nonlocal

all scaleof is

frequency

ifference

nonlocal

rominent


10/15

IJST, Transa

100

Now,

obtained us

of local be

Appendix

layerwise e

in Figs. 5a

against the

used in the

Furthermor

discrepanc

values. Alsbeam mode

boundary c

present loc

beam and

Figs. 5a an

inaccurate

circumfere

frequency

sufficientlyFigure

basis of be

via the bea

model in p

mode shap

coaxial vib

predictable.

modes, the

the associat

ones relate

tions of Mech

Fig. 4.

a compariso

ing beam m

m model re

for SS/S

d conditio

and b are th

length of na

se figures. I

e, it is foun

between th

o, it is obsel. Figs. 5c a

onditions, n

l Donnell s

hell models

b. Thus, o

esults for v

tial mode i

urves corre

long DWC6 depicts th

m model de

m model c

edicting ne

s predicted

ational mod

Hence, unl

application

ed non-coax

to higher ci

anical Engine

Variation of

ith different

is made b

del. To this

ported in [7

DWCNTs

s such as F

e local and

notube, res

t can be se

that the be

e beam and

ved that thend d show t

amely FF/C

hell model

for shorter

e may concl

brations of

nto accoun

sponding to

Ts behave le vibrationa

veloped in

rresponding

inter-tube

y the shell

es such as

ke the bea

f the prese

ial vibration

rcumferenti

ring, Volume

frequency rat

types of layer

etween the

aim, the res

] and also t

are presen

/CC and SS

onlocal (ectively. Fo

n that the

m model te

shell model

nonlocal efe fundame

C and SS/C

and the loc

DWCNTs c

ude that dep

WCNTs.

. It should

the beam a

ike a beam.l mode shap

ppendix B.

to coaxial

resonant fr

odel are al

and fomodel whi

t shell mod

al modes. F

l mode nu

. Rouhi et al.

37, Number

io versus non

wise boundar

esults gene

ults of the p

e ones obt

ted in Fig.

/CC are con 0.5 )r the beam

undamental

ds to overe

s becomes

fect is moretal frequen

C. These fi

l beam mo

an also be

ending on t

he reason

be noted t

nd shell m

es of a SS/S

This figure

and non-co

equencies a

so shown in

r radius ch is capabl

l leads to p

rthermore,

bers as the i

2

ocal paramet

y conditions

ated by the

resent nonlo

ined from t

5. Differe

sidered in t

fundamenta

odel, the f

frequency

timate the f

ore pronou

prominenties of DW

ures also p

el develop

een in thes

e length of

ere is that,

at as the l

dels tend t

S DWCNT

shows two

xial modes.

d the relat

Fig. 7. As r1 ande of predicti

edict furthe

on-coaxial

nner radius

er for a DWC/ 10)present she

cal Donnell

e nonlocal

t boundary

ese figures.

frequencie

equencies

ecreases as

equencies

nced for D

n the shellNTs with t

ovide a co

d in [7]. T

figures si

anotube, th

he beam m

ngth of na

converge.

or two diffe

ibrational

To show t

d non-coax

vealed in t

and fog only two

inter-tube

ibrational

f DWCNT

O

NTs

ll model an

shell model

beam mode

conditions

Presented g

of a SS/SS

iven by Eq.

the length

f DWCNT

CNTs of l

odel thano different

parison be

e differenc

ilar to that

e beam mod

del does n

notube incr

This reveal

rent inner ra

ode shapes

e ability o

ial vibration

is figure, fu

r radius inter-tube v

ibrational

odes may s

is increased.

tober 2013

the ones

and those

l given in

including

aphically

DWCNT

(B.8) are

increases.

o that the

w length

hat in thelayerwise

ween the

between

shown in

l leads to

t take the

ases, the

s that the

dii on the

predicted

the shell

al modes,

ther non-2 areibrational

odes and

hift to the


11/15

October2013

Fig. 6. M

Fig.

ode shapes of

Vibratio

(a)

(c). Compariso

mo

a SS/SS DW1 n

al analysis of

IJS

between the

els for a SS/

CNT predicteand

double-walled

T, Transaction

fundamental

S DWCNT (

d by the bea10b) with

carbon nanot

s of Mechanic

frequencies o

0.35

model devel

2 nm an

bes

l Engineerin

(b)

(d)f shell and be

oped in Appe

10

, Volume 37,

am

ndix B: a) wi

umber M2

101

th


12/15

IJST, Transa

102

The free v

work. To t

set of gove

combined

the nonloca

leads to ov

Through c

beam mod

DWCNTs

model. By

predicted.

observed to

1. Iijima, S2. Elishak

Wiley,3. Bandow

Wall Ca

Lett., V

4. Eringen,surface

5. Eringen,6. Li, R.

elastic

7. Xu, K.boundar

tions of Mech

Fig. 7. Mode

bration of

is end, a no

ning equati

ith the GD

l shell mod

restimated

mparison b

l, it was c

s compared

he present

dditionally

likely happ

. (1991). Heli

ff, I. et al. (2

ondon., S., Takizaw

rbon Nanotu

l. 337, pp. 48

A. C. (1983

aves.J. App

A. C. (2002)

Kardomate

edia by a no

., Aifantis,

y conditions

anical Engine

(a)

shapes of a

and

WCNTs w

local Donn

ns were th

Q method.

l and MD s

esults and

tween the

ncluded tha

to the shell

onnell shell

, a shift in

n when the

cal microtub

012). Carbon

a, M., Hiraha

es Derived f

-54.

). On differe

l. Phys., Vol.

.Nonlocal co

s, G. A. (20

local elastic

. C., and Ya

etween inner

ring, Volume

S/SS DWCN

10 b

5. CONCL

ith differen

ell shell mo

n numerical

ood agree

mulation. It

ne must rec

esults gene

t the beam

model, due

model som

non-coaxial

radius of D

RE

s of graphitic

nanotubes a

a, K., Yudas

om the Chai

tial equation

54, pp. 4703-

ntinuum field

7). Vibratio

shell model.

, Y. H., 200

and outer tub

. Rouhi et al.

37, Number

T predicted b

) with

2UDING R

layerwise

el was deve

ly solved vi

ent was ob

was indicat

ourse to the

ated by the

model tend

o not incor

new non-c

modes, w

CNTs is v

FERENCE

carbon.Nat

d nanosenso

aka, M. & Iiji

s of Fulleren

s of nonloca

4710.

theories. Ne

characteristi

SME J. Appl

, Vibration

es.ASME J.

2

y the shell m

nm

MARKS

oundary c

loped whic

a the radial

served betw

ed that appl

nonlocal v

shell mode

to overesti

orating circ

axial inter-t

ich is unpr

ried.

S

re, Vol. 8, pp

rs: Vibration,

ma, S. (2001

es in Single-

elasticity an

York, Sprin

cs of multiw

Mech., Vol.

of double-w

ppl. Mech.,

(b)

del: a) with

10

nditions wa

accounts fo

point interp

een the calc

ing the loca

rsion to red

l and the o

mate the re

mferential

ube resonan

edictable b

. 354-356.

buckling an

. Raman Sca

all Carbon

d solutions o

ger-Verlag.

alled carbon

74, pp. 1087-

alled carbon

ol. 75, p. 02

O

1 nm

s investigat

r the scale e

olation appr

ulated frequ

l Donnell s

uce the rela

es obtained

sonant freq

mode numb

ces of DWC

the beam

ballistic im

ttering Study

anotubes. C

f screw dislo

nanotubes e

1094.

anotubes wi

1013.

tober 2013

ed in this

ffect. The

ximation

encies by

ell model

ive error.

from the

encies of

r into the

NTs were

model, is

act. ISTE-

of Double-

em. Phys.

cation and

bedded in

h different


13/15



103

8. Arash, B. & Ansari, R. (2010). Evaluation of nonlocal parameter in the vibrations of single-walled carbon

nanotubes with initial strain. Physica E, Vol. 42, pp. 2058-2064.

9. Ansari, R., Rouhi, H. & Sahmani, S. (2011). Calibration of the analytical nonlocal shell model for vibrations of

double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics. Int. J. Mech.

Sci., Vol. 53, pp. 786-792.10. Ansari, R. & Rouhi, H. (2012). Analytical treatment of the free vibration of single-walled carbon nanotubes

based on the nonlocal flugge shell theory.ASME J. Eng. Mater. Technol., Vol. 134, p. 011008.

11. Ghavanloo, E. & Fazelzadeh, S. A. (2012). Vibration characteristics of single-walled carbon nanotubes based on

an anisotropic elastic shell model including chirality effect.Appl. Math. Model, Vol. 36, pp. 4988-5000.

12. Yan, Y., Wang, W. & Zhang, L. (2012). Free vibration of the fluid-filled single-walled carbon nanotube based

on a double shell-potential flow model.Appl. Math. Model, Vol. 36, pp. 6146-6153.

13. Murmu, T., McCarthy, M. A. & Adhikari, S. (2012). Vibration response of double-walled carbon nanotubes

subjected to an externally applied longitudinal magnetic field: A nonlocal elasticity approach. J. Sound Vib.,

Vol. 331, pp. 5069-5086.

14. Hu, Y. G., Liew, K. M. & Wang, Q. (2012). Modeling of vibrations of carbon nanotubes. Procedia Eng., Vol.31, pp. 343-347.

15. Khosrozadeh, A. & Hajabasi, M. A. (2012) Free vibration of embedded double-walled carbon nanotubes

considering nonlinear interlayer van der waals forces.Appl. Math. Model., Vol. 36, pp. 997-1007.

16. Ambrosini, D. & de Borbn, F. (2012). On the influence of the shear deformation and boundary conditions on

the transverse vibration of multi-walled carbon nanotubes. Comput. Mater. Sci., Vol. 53, pp. 214-219.

17. Lim, C. W., Li, C. & Yu, J. L. (2012). Free torsional vibration of nanotubes based on nonlocal stress theory. J.

Sound Vib., Vol. 331, pp. 2798-2808.

18. Liu, G. R. (2003).Mesh free methods moving beyond the finite element method. CRC PRESS.

19. Bellman, R. E., Kashef, B. G. & Casti, J. (1972). Differential quadrature: a technique for rapid solution of

nonlinear partial differential equations.J. Comput. Phys., Vol. 10, pp. 40-52.

20. Flugge, W. (1960). Stresses in shells. Springer, Berlin.

21. Darvizeh, M., Ansari, R., Darvizeh, A. & Sharma, C. B. (2003). Buckling of fibrous composite cylindrical shells

with non-constant radius subjected to different types of loading.Iran. J. Sci. Technol. Trans. B Eng., Vol. 27, pp.

535-550.

22. Donnell, L. H. (1976).Beam, plates and shells. New York: McGraw-Hill.

23. Bathe, K. J. (1996). Finite element procedures. Englewood Cliffs, NJ Prentice Hall.

APPENDIX A

Differential operators:

1 12 21 1 1 12 21 1 1 1 0 12 21 1 1 1 1 0

1

1 1

1

0


14/15

H. Rouhi et al.


104

12 1 2 1 12 1 2 1 0

12 1 2

1

Algebraic operators:

1 12 21 1 12 21 1 1 0 12 21 1 1 0

1 1 0 12 1 2 12 1 2 0 1

21

2

APPENDIX B: EXACT SOLUTION FOR THE NONLOCAL BEAM MODEL

sin , 1,2 (B.1) 1 1 0 1 1 0 (B.2)Substituting Eq. (B.1) into Eq. (B.2) gives the following algebraic equations 1 1 1

0

1 1 1 0(B.3) 0 (B.4)

where

1 1 1 1

(B.5)

1

00 1 (B.6)


15/15



105

(B.7)The nontrivial solution of (B.4) gives 0. The following explicit formula for the resonantfrequencies of the DWCNT can be obtained as

22 4 (B.8)

where 1 1

1 1 1 1

vibrational analysis of double-walled carbon nanotubes based on the nonlocal donnell shell theory ...

Documents