eccentric compression stability of multi-walled carbon nanotubes

COMPOSITES

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Composites Science and Technology 67 (2007) 1406–1414

SCIENCE ANDTECHNOLOGY

Eccentric compression stability of multi-walled carbonnanotubes embedded in an elastic matrix

X.Y. Wang, X. Wang *, X.H. Xia

Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University,

Shanghai 200240, PR China

Received 12 March 2006; received in revised form 3 September 2006; accepted 14 September 2006Available online 27 October 2006

Abstract

This paper reports the results of an investigation on the eccentric compression stability of multi-walled carbon nanotubes embedded inan elastic matrix. Based on continuum modeling, a multilayer shell model is presented for the eccentric compression buckling of multi-walled carbon nanotubes embedded in an elastic matrix, in which the effect of van der Waals forces between two adjacent tubes is takeninto account. The critical bending moment and the eccentric compression mode for three types of multi-walled carbon nanotubes withdifferent layer numbers and ratios of radius to thickness are calculated. Results obtained show that the eccentric compression bucklingmode corresponding the critical bending moment is unique, and is different from the purely axial compression buckling of an individualmulti-walled carbon nanotube. For different types of multi-walled carbon nanotubes, the effect of matrix stiffness on the critical bendingmoment of multi-walled carbon nanotubes under eccentric compression loading is obviously different, and is dependent on the innermostradius and layer numbers of the multi-walled carbon nanotubes. The critical bending stress exerted on the center tubes of nearly solidmulti-walled carbon nanotubes does not change as the ratio of the axial compression loading to the bending membrane force increases.The new features and meaningful numerical results in this paper are helpful for the application and the design of nanostructures in whichmulti-walled carbon nanotubes act as basic elements.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Carbon nanotubes; B. Modeling; B. Mechanical properties

1. Introduction

Carbon nanotubes discovered by Iijima [1] at the begin-ning of the last decade, has attracted worldwide attentionrelated to the use of the nanotubes in material reinforce-ment, field emission panel display, parts of nano-devices,hydrogen storage, (high frequency) micromechanical oscil-lators and sensors [2–4]. Lau and Hui [5] presented recentdevelopments on the nanotubes and investigations in theapplications on nanotube composites, and gave muchattention on the examination of the mechanical propertiessuch as tensile strength of an individual nanotube or a bun-dle of nanotube-rope, and the buckling properties of nano-

0266-3538/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compscitech.2006.09.006

* Corresponding author.E-mail address: [email protected] (X. Wang).

tube-composite structures due to shrinkage of matrixesafter curing. Multi-walled carbon nanotubes are composedof concentric layers of single-walled carbon nanotube(SWNT). When multi-walled carbon nanotubes (MWNTs)occur in large strain deformation, they bend at large anglesand may start to elastically ripple, buckle, and form kinks,which induces abrupt changes of physical properties of theMWNTs. The investigations on tension, compression,bending or torsional deformations of single- or multi-walled carbon nanotubes have been the subject of numer-ous experiments and theoretical approaches [6–15].

Generally, there are two theoretical approaches tounderstand the mechanical behavior of nano-structures:atomistic molecular-dynamics simulations and continuummechanics. The molecular dynamic (MD) simulationshave been conducted by several investigators [7,16].

mailto:[email protected]

X.Y. Wang et al. / Composites Science and Technology 67 (2007) 1406–1414 1407

However, the computational problem here is that the timesteps involved in the MD simulations are limited to sys-tems with a maximum atom number of about 109 bythe scale and cost of computation [17]. Yakobson et al.[7] introduced an atomistic model for axially compressedbuckling of single-walled nanotube and also compared itwith a simple continuum shell model. They found thatall changes of buckling patterns displayed by the molecu-lar-dynamics simulations could be predicted by the con-tinuum shell model. Based on a continuum mechanicsmodel, Wang et al. [18–20] constructed different three-dimensional finite element models to obtain an effectivebending modulus of CNTs with various ripplingdeformations.

Because electronic and transport properties of carbonnanotubes could be extremely sensitive to even very smalldistortion of their otherwise perfect cylindrical geometry[21,22], recently considerable attention has turned to buck-ling behavior of single- or multi-walled carbon nanotubes[23–26]. Ru [27] presented an elastic double-shell modelfor infinitesimal buckling of a double-walled carbon nano-tubes embedded in an elastic medium under axial compres-sion. Ru [28] also studied the effect of van der Waalsinteraction between the inner and outer nanotubes on axialbuckling of double-walled carbon nanotubes, and derived asimple formula for the axially critical strain. Wang andYang [29] presented the effect of thermal environment onaxially critical load of multi-walled CNTs by utilizing acontinuum mechanics model. Wang et al. [30] has recentlystudied elastic buckling of individual MWNTs subjected toradial external pressure based on the multilayer shellsmodel from Ru [26,31] and their results showed that thepredicted critical pressure using continuum mechanicsmodel is in a reasonably good agreement with some exper-iment results from Tang et al. [32], which further offers anevidence that the elastic shell model can be used to studybuckling behavior of single- or multi-walled carbonnanotubes.

Most of the previous works on buckling behavior of sin-gle- or multi-walled carbon nanotubes have mainly focusedon the purely axial compression buckling behavior or thepure bending behavior of carbon nanotubes. From Arroyoet al. [33] and Williams et al. [34], a number of experimentshave suggested that multi-walled carbon nanotubes can beused as basic elements of nano-electromechanical systems,such as AFM tip, CNT-nanomechanical switches, and elec-tric, optical and mechanical nano-sensors and nanoprobesin metrology, biological, chemical and physical investiga-tions, which may be subjected to eccentric compressionloading.

To our knowledge, few detailed report on the eccentriccompression buckling of multi-walled carbon nanotubesis available in literatures. This is apparently due to the factthat experimental research and molecular dynamics simula-tion for eccentric compression buckling of MWNTsembedded in an elastic matrix remain a formidable task.Therefore, eccentric compression stability of MWNTs

embedded in an elastic matrix remains an open topic inthe literature.

Based on a multiple-elastic shell model considering vander Waals interaction force between the inner and outerlayers of nanotubes, the present work reports the resultsof an investigation on the eccentric compression stabilityproblems of MWNTs embedded in an elastic matrix.According to the ratio of radius to thickness, theMWNTs discussed here are classified into three cases:thin, thick, and nearly solid. The critical loading andthe eccentric compression stability mode are calculatedfor various ratios of radius to thickness. Results carriedout show that the eccentric compression stability modecorresponding the critical loading of MWNTs embeddedin an elastic matrix is unique, and is different from theaxially compressed buckling of MWNTs [26]. It is seenfrom calculation examples that the distribution of maxi-mum critical bending stresses exerted on each tube ofMWNTs under eccentric compression loading is depen-dent on the ratio of radius to thickness and the ratio ofeccentric compression loading. For different types ofMWNTs, the effect of matrix stiffness on the critical bend-ing moment of MWNTs under eccentric compressionloading is different, and is dependent on the innermostradius and layer numbers of the MWNTs.

Compared to all known previous studies, the new fea-tures, and some meaningful and interesting numericalresults in the present work will stimulate further interestin this topic and may be used as a useful reference forthe application and the design of various nanostructuresin which multi-walled carbon nanotubes act as basicelements.

2. Basic equation

The mechanics properties of single- or multi-walled car-bon nanotubes have been effectively studied based on elas-tic-shell models from Falvo et al. [6], Yakobson et al. [7],and Ru [26,27]. Motivated by these ideas, MWNTs canbe taken as a set of concentric cylindrical shells with vander Waals interaction between adjacent layers, which isshown in Fig. 1. In the present model, because the inter-laminar shear forces between layers of multi-walled carbonnanotubes is negligible from Charlier and Michenaud [35],for infinitesimal buckling of a cylindrical shell with radiusR, thickness h, Young’s modulus E and Poisson’s ratio l,the stability equation of a cylindrical shell subjected toeccentric compression loading can be given in terms ofw(x,y) as follows [24,36]

Dr8w ¼ r4pðx; yÞ þ N 0xðx; yÞ

o2

ox2r4wþ N 0

yðx; yÞo2

oy2r4w

� Eh

R2

o4wox4

ð1Þ

where x and y express the axial and circumferential coordi-nates of shell, w(x,y) denote the additional inward normal

xF

xFxF

xFh

Rk+1

Rk

x

The van der Waals forces

The van der Waals forces

L

The kth tube

The (k+1)th tube

MM

elastic matrix

elastic matric

Fig. 1. A multiwall carbon nanotube embedded in an elastic matrix, undereccentric compression composed of an uniform compressive loading and apure bending moment.

1408 X.Y. Wang et al. / Composites Science and Technology 67 (2007) 1406–1414

displacements of the middle surface due to buckling, p(x,y)is the total inward normal pressure at the point (x,y), D isthe effective bending stiffness of the shell andN 0

xðx; yÞ and N 0yðx; yÞ express the initial membrane forces

before buckling which are not uniform prior to bucklingunder an eccentric compression [24,36].

Because the van der Waals interaction pressures (perunit area) between the kth tube and the (k + 1)th tubeare equal and opposite, the pressures pk(k+1) and p(k+1)k,exerted on the corresponding point on the tubes k andk + 1, respectively, should be related by

pðkþ1Þk ¼ �Rk

Rkþ1

pkðkþ1Þ; ðk ¼ 1; 2; . . . N � 1Þ ð2Þ

where Rk expresses the radius of the kth tube and all thelayers of MWNTs have the same Young’s modulus E, Pois-son’s ratio l.

Applying Eqs. (1) and (2) to each layer of all concentrictubes of MWNTs, eccentric compression stability ofMWNTs embedded in an elastic matrix is governed bythe N coupled equations as follows:

D1r8w1 ¼ r4p12 þ N 0x1

o2

ox2r4w1 þ N 0y1

o2

oy2r4w1 � Eh1

R21

o4w1

ox4

DN�1r8wN�1 ¼ r4 pðN�1ÞN � RN�2

RN�1pðN�2ÞðN�1Þ

� �

þN 0xðN�1Þ

o2

ox2r4wN�1 þ N 0yðN�1Þ

o2

oy2r4wN�1

� EhN�1

R2N�1

o4wN�1

ox4

DNr8wN ¼ r4pMN þr4pNðN�1Þ þ N 0

xNo2

ox2r4wN

þN 0yN

o2

oy2r4wN � EhNR2

N

o4wNox4

ð3Þwhere wk(x,y)(k = 1,2, . . .,N) is the inward deflection ofthe kth tube, the subscripts 1, 2, . . .N are used to denotethe quantities associated with each layer from the inner-most tube to the outermost tube, and pM

N expresses a inter-action pressure between the outermost layer of MWNTsand the surrounding elastic matrix, which is given by

pMN ¼ �dwN ð4Þ

In the above formula, d expresses a spring constant deter-mined by the surrounding elastic matrix and the outermostdiameter of MWNTs and wN expresses the normal dis-placement of the outermost tube of MWNTs.

Based on Winkler linear elastic foundation, Ru [27]introduced a spring constant, d, to describe the effect ofan infinite matrix surrounding carbon nanotube on axiallycompressed buckling of a double-walled carbon nanotube.But, the physical relation among the spring constant,d, thecharacteristic of surrounding elastic matrix and the outer-most diameter of MWNTs has been not clearly given intheir paper. The Whitney–Riley model in this paper maybe used to determine a spring constant d in Eq. (4) for afinite elastic matrix or an infinite elastic matrix [37]. Thepressure, p, between the outermost tube and the surround-ing elastic matrix prior to buckling is uniform. Here, thesurrounding elastic matrix is considered as a hollow cylin-der with inner-radius RN and outer-radius b. The radial dis-placement of the hollow cylinder under the uniforminternal pressure can be expressed as

ur ¼ ðAr þ Br�1Þp ð5Þwhere

A ¼ ð1þ lMÞð1� 2lMÞEM

1

b2=R2N � 1

;

B ¼ 1þ lM

EM

b2

b2=R2N � 1

ð6Þ

EM and lM express the Young’s module and Poisson ratioof matrix, respectively.

If a multi-walled carbon nanotube is embedded in aninfinite elastic matrix, we have b!1, so that Eq. (6) issimplified to

A ¼ 0; B ¼ 1þ lM

EMR2

N ð7Þ

Neglecting the dependence of the spring constant d on thecurvature of buckling mode as a second-order effect, thensubstituting Eqs. (6) and (7) into Eq. (5), the spring con-stants obtained from Whitney–Riley model for a finite elas-tic matrix and an infinite elastic matrix can be, respectively,determined as

d ¼ pMN ður¼RN ¼ 1Þ ¼ Emðb2 � R2

NÞRN ð1þ lmÞ½ð1� 2lmÞR2

N þ b2Þð8aÞ

d ¼ pMN ður¼RN ¼ 1Þ ¼ Em

ð1þ lmÞRNð8bÞ

3. Analysis for eccentric compressive stability

The van der Waals force between two carbon atoms canbe described by the Lennard–Jones model from Girifalco[38]. Since the innermost radius of MWNTs is usuallymuch larger than 0.5 nm from Iijima et al. [39], the inter-


layer interaction potential between two adjacent tubes canbe simply approximated by the potential obtained for twoflat graphite monolayers, denoted by g(d), where d is theinterlayer spacing [40]. This model has been previousshown to provide good agreement with experimentalresults carried out by Lu [41]. For the present infinitesimalbuckling analysis, the van der Waals pressure, pV (x,h), atany point between two adjacent tubes should be a linearfunction of the deflection jump at that point, so we have[26]

pV ¼dgðdÞ

dd

� �d¼t

þ cðDwÞ; ð9aÞ

where

c ¼ d2g

dd2

��d¼t

ð9bÞ

In the above formula, t is the initial interlayer spacing priorto buckling, which is equal or very close to the representa-tive thickness of single-walled carbon nanotube, (Dw) ex-presses the deflection jump due to buckling, and c

expresses the van der Waals interaction coefficient. Becausethe initial interlayer spacing is very close to the equilibriuminterlayer spacing at which dg(d)/dd = 0 described by Girif-alco [38], the first term of Eq. (9a) is negligible, reflectingthe fact that all initial interlayer pressures vanish. Saitoet al. [42] gave a van der Waals interaction coefficient as

c ¼ 320 erg=cm2

0:16 g2ðg ¼ 1:42� 10�8 cmÞ ð9cÞ

Here, because the present analysis is limited to infinitesimalbuckling, the interaction coefficient c is calculated at theinitial interlayer spacing (about 0.34 nm). The curvature-dependency of the interaction coefficient c is neglected herebecause it is very small when the innermost radii are muchlarger than 0.6 nm [40,43]. Thus, the van der Waals forcesbetween two adjacent layers due to buckling are expressedas

p12 ¼ c½w2 � w1�; p23 ¼ c½w3 � w2�; . . . ;

pðN�1ÞN ¼ c½wN � wN�1� ð10Þ

Before buckling, the initial membrane force of the kth tubeof MWNTs embedded in an elastic matrix, under eccentriccompressive loading composed of a uniform compressionand a pure bend, is expressed as

N 0xk ¼ F xk þ N xk cos

yRk

ð11Þ

where F xk ¼ F x ¼ F 0x=N (k = 1, 2, . . . N) expresses the axial

uniform membrane force of the kth tube induced by axiallyuniform compressive loading, F 0

x , and Nxk expresses themaximum bending membrane force in the kth tube inducedby the bending moment Mk(k = 1, 2, . . . N) of the kthlayer.

Thus the total bending moment applied on MWNTs iswritten as

M ¼ M1 þM2 þ � � � þMN ð12aÞFurthermore, the bending moments, M1,M2, . . . ,MN, ap-plied on each tubes of MWNTs are dependent on their ra-dii, which is expressed as

M1 : M2 : � � �MN ¼ R31 : R3

2 : � � �R3N ð12bÞ

Combining the Eqs. (12a) and (12b), the bending momentsMk applied on the kth tube of MWNTs can be expressedas

Mk ¼R3

kPNk¼1R3

k

M ; ðk ¼ 1; 2; . . . ;NÞ ð13Þ

The distribution of the bending stress applied on the kthtube is expressed as

rbxk ¼ rbk cos

yRk; ð0 6 y 6 pÞ ð14Þ

where rbk expresses the maximum bending stress appliedon the kth tube.

Integrating Eq. (14) over the cross section of the kthtube, the bending moment applied on the kth tube is rewrit-ten as

Mk ¼Z 2pRk

0

rbxkh � dy � R cos

yRk

ð15Þ

Combining the Eqs. (14) and (15), the maximum bendingmembrane force in the kth tube can be expressed as

Nxk ¼ rbkh ¼ Mk

pR2k

¼ RkM

pPN

k¼1R3k

; ðk ¼ 1; 2; � � � ;NÞ ð16Þ

For the bending of an elastic cylindrical shell, prior tobuckling, the Brazier effect causes the circular cross-sec-tion of the shell to become more ovalized uniformly overthe whole shell length as the bending curvature increases[44]. Considering that the MWNTs is so long that theconstraints at two ends have no considerable effect onthe magnitude of the critical internal force so that theBrazier effect of the pre-buckling shell does not affectthe buckling additional deformation in the stability gov-erning Eq. (1), the eccentric compressive buckling modesof the kth layer of multi-walled carbon nanotubes canbe taken as [36]

wk ¼ sinnpxL

X3

n¼1

fkn coskyRk

¼ sinnpxL

fk1 cosy

Rkþ fk2 cos

2yRkþ fk3 cos

3yRk

� �;

k ¼ 1; 2; . . . ;N ð17Þ

where fk1, fk2, fk3 (k = 1,2, . . .. . .N) are real constants,Rk(k = 1,2, . . .. . .N) are the radius of the kth tube, L isthe length of MWNTs and ( n = 1,2,. . .. . .) are the axialhalf wavenumbers of the MWNTs.

Table 1The geometrical data for examples of MWNTs

Examples: Thin MWNTs(1,2)

ThickMWNTs (3)

Nearlysolids (4)

Example numbers: 1 2 3 4

R1 (nm) (Innermostradius)

8.5 18 2.7 0.65

R1/Nh 5.00 6.62 0.99 0.24(h = 0.34 nm, the effective

thickness of a SWNTs)L (nm) (length) 118.32 244.56 60.96 36.36N (number of layers) 5 8 8 8


Based on Galerkin’s method [36], we haveR L0

R 2pR1

0 Z1 sin npxL cos y

R1dydx ¼ 0R L

0

R 2pR1

0Z1 sin npx

L cos 2yR1

dydx ¼ 0R L0

R 2pR1

0Z1 sin npx

L cos 3yR1

dydx ¼ 0

8>>><>>>:. . . . . . . . . . . . . . . . . .R L

0

R 2pRN

0ZN sin npx

L cos yRN

dydx ¼ 0R L0

R 2pRN

0ZN sin npx

L cos 2yRN

dydx ¼ 0R L0

R 2pRN

0ZN sin npx

L cos 3yRN

dydx ¼ 0

8>><>>:

ð18Þ

where

Z1 ¼ Dr8w1� cr4½w2�w1� �r4 N 0x1

o2w1

ox2þN 0

y1

o2w1

oy2

� �

þEh

R21

o4w1

ox4

. . . . . . . . . . . . . . . . . . . . .

ZN�1 ¼ Dr8wN�1� cr4½wN �wN�1�RN�2

RN�1

ðwN�1�wN�2Þ�

�r4 N 0xðN�1Þ

o2wN�1

ox2þN 0

yðN�1Þo2wN�1

oy2

� �þ Eh

R2N�1

o4wN�1

ox4

ZN ¼ Dr8wN þ dr4wN þr4½c RN�1

RNðwN �wN�1Þ�

�r4 N 0xN

o2wN

ox2þN 0

yN

o2wN

oy2

� �þ Eh

R2N

o4wN

ox4ð19Þ

Substitution of Eq. (19) into Eq. (18) yields a set of alge-braic equations for fk1, fk2, fk3 (k = 1,2, . . .. . .N), whichis expressed as

A11f11 þ A12f12 þ A14f21 ¼ 0

A21f11 þ A22f12 þ A23f13 þ A25f22 ¼ 0

A32f12 þ A33f13 þ A36f23 ¼ 0

8><>:. . . . . . . . . . . . . . . . . .

A½3ðN�1Þþ1�½3ðN�2Þþ1�fðN�1Þ1 þ A½3ðN�1Þþ1�½3ðN�1Þþ1�fN1

þA½3ðN�1Þþ1�½3ðN�1Þþ2�fN2 ¼ 0


þA½3ðN�1Þþ2�½3ðN�1Þþ2�fN2 þ A½3ðN�1Þþ2�½3ðN�1Þþ3�fN3 ¼ 0


þA½3ðN�1Þþ3�½3ðN�1Þþ3�fN3 ¼ 0

8>>>>>>>><>>>>>>>>:

ð20ÞThe detailed expression of coefficientsAij in Eq. (20) is toolengthy and is not put in this paper.

Eq. (25) can be rewritten as

½AðM ; F 0x ; nÞ�3N�3N

f11

f12

f13

..

.

fN3

266666664

377777775¼ 0 ð21Þ

where M, F 0x and n express the total bending moment and

the axially uniform compressive force applied on MWNTs,

and the axial half wavenumbers of the MWNTs,respectively.

Thus, the existence condition of a non-zero solution offNk (N = 1, 2, . . . N; k = 1, 2, 3) in Eq. (26) is given by

det½AðM ; F 0x ; nÞ�3N�3N ¼ 0 ð22Þ

This condition determines the critical bending momentM(n) exerted on MWNTs for different axially uniformcompressive force F 0

x . Thus the corresponding maximumbending stress, rbk(n), exerted on the kth tube can be ob-tained by utilizing Eq. (16).

4. Four calculation examples and discussion for eccentric

compression stability

In calculation examples, D1 = � � � = Dk = D = 0.85 eVand h1 = � � � = hk = h = 0.34 nm, Eh1 = � � � = Ehk = Eh

= 360 J/m2, l = 0.3 and L = 12RN (the outmost radius)are taken from [26], so that the critical bending momentM(n) can be obtained for a given ratio, Fx1/Nx1, betweencompressive membrane force, Fx1, and the maximum bend-ing membrane force, Nx1, applied on the first tube. Thusthe corresponding maximum bending critical stresses,rbk(n) = Nxk(n)/h exerted on the kth tube can be describedin Figures. According to the radius-to-thickness ratio ofMWNTs, all MWNTs embedded in an elastic matrix,under eccentric compressive loading, are classified intothe three typical cases, which are shown in Table 1. Here,we calculate four examples for all three types ((a) thinMWNTs, (b) thick MWNTs and (c) nearly solid) ofMWNTs embedded in an elastic matrix, under eccentriccompression combined bending and axial compressionloading, respectively. The relationship between the bendingmoment of MWNTs embedded in an elastic matrix, undereccentric compression loading, and the axial half wave-numbers n is shown in Fig. 2 for four examples as shownin Table 1.

An interesting result is that the axial half wavenumberscorresponding to the critical bending moment in MWNTsunder eccentric compression are unique for all three typesof MWNTs, and is different from the relationship betweenthe axially compressed buckling stress and the wavenum-bers (m,n) from Wang et al. [26]. As the result, there is onlyone combination of the wavenumbers, n, which corre-

00.0

0.5

1.0

1.5

2.0

2.5

3.0Critical bending moments Mc r (10 –15 Nm )

under eccentric compression (Fx1/Nx1= 0.5);

A: Mcr=2.413 for example 1

B: Mcr =6.456 for example 2

C: Mcr =11.81 for example 3

D: Mcr=0.512 for example 4

D

C

BA

01( M tne

mom gnidne

B41-

)m

N

Axial buckling wavenumber n50 100 150 200

Fig. 2. The relationships between the bending moment (M) and the axialbuckling wavenumber (n) for examples 1–4.

5.5

6.0

6.5

7.0d/c=0.02

5Layer

4Layer

3Layer

2Layer

1Layer

sserts gnidneb lacitirC

kb)a

PG(

Eccentric compressive loading Fx1

/ Nx1

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 3a. The critical bending stresses for each layer versus the ratio ofuniform compressive load to bending load for example 1, where d is thespring coefficient of matrix and c is van der Waals interaction coefficient.

0.02.4

2.6

2.8

3.0

3.2

3.4

d/c=0.02

1Layer


kb)a

PG(

8Layer7Layer6Layer5Layer4Layer3Layer2Layer

0.2 0.4 0.6 0.8 1.0Eccentric compressive loading F

x1/ N

x1

Fig. 3b. The critical bending stresses for each layer versus the ratio ofuniform compressive load to bending load for example 2, where d is thespring coefficient of matrix and c is van der Waals interaction coefficient.

8

9

10

11

12

13

14

15

16d/c=0.02


/ Nx1


kb)a

PG( 8

7

6

5

4

3

2

1=k

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 3c. The critical bending stresses for each layer versus the ratio ofuniform compressive load to bending load for example 3, where d is thespring coefficient of matrix and c is van der Waals interaction coefficient.


sponds to the critical bending moment under eccentriccompressive loading, so that the axial half wavenumbersof buckling modes of MWNTs under eccentric compres-sion loading can be determined uniquely.

From Fig. 2, it is seen that the critical bending momentsof MWNTs embedded in an elastic matrix with (d/c =0.02), under eccentric compressive loading Fx1/Nx1 = 0.5,for these four examples are 2.413 · 10�15 Nm (n = 88),6.456 · 10�15 Nm (n = 127), 11.81 · 10�15 Nm (n = 67),and 0.512 · 10�15 Nm (n = 51), respectively. Comparingcurves in Fig. 2, the maximum critical bending momentappears in a thick MWNTs with smaller innermost radius(Example 3) under eccentric compressive loading Fx1/Nx1

= 0.5, which shows that this type of MWNTs can resisthigher bend loading, and the minimum critical bendingmoment occurs in a nearly solid MWNTs embedded inan elastic matrix (Example 4), which shows that this typeof MWNTs can only resist lower bend loading. FromTable 1 and curves in Fig. 2, it is seen that the critical bend-ing moment of MWNTs under eccentric compressive load-ing Fx1/Nx1 = 0.5, is dependent on the innermost radius tothickness ratio of MWNTs embedded in an elastic matrix.

Fig. 3a–3d show the maximum critical bending stressapplied on each tube versus eccentric loading ratioFx1/Nx1 for four examples of MWNTs shown in Table 1.From Fig. 3a and 3b, it is seen that the maximum criticalbending stresses applied on each tube of thin MWNTs(Examples 1 and 2) decrease as the ratio of axial compres-sive force to the bending membrane force, Fx1/Nx1,increases. Fig. 3a and 3b show that the variation tendencyof the maximum critical bending stresses exerted on eachtube of thick or nearly solid MWNTs (Examples 3 and 4)versus the ratio of axial compressive force to the bendingmembrane force, Fx1/Nx1, are related to the thickness ofMWNTs embedded in an elastic matrix. It is noted thatthe critical bending stresses exerted on the innermost tubesof thick or nearly solid MWNTs increases as the ratio ofaxial compressive force to the bending membrane force,

4

8

12

16

20

24

28

d/c=0.02


/ Nx1


kb)a

PG(

8

7

6

5

4

3

2

1=k

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 3d. The critical bending stresses for each layer versus the ratio ofuniform compressive load to bending load for example 4, where d is thespring coefficient of matrix and c is van der Waals interaction coefficient.


Fx1/Nx1, increases, the critical bending stresses exerted onthe outermost tubes of thick or nearly solid MWNTsdecrease as the ratio of axial compressive force to the bend-ing membrane force, Fx1/Nx1, increases, and the criticalbending stresses exerted on the center tubes of nearly solidMWNTs do not change as the ratio of axial compressiveforce to the maximum bending membrane force, Fx1/Nx1,increases.

Fig. 4 shows the effect of the stiffness of matrix on thecritical bending moment of MWNTs under eccentrix com-pressive loading (Fx1/Nx1 = 0.5). From Fig. 4, it is seen thatthe critical bending moment of MWNTs increases as thestiffness of matrix surrounding the MWNTs increases.For different types of MWNTs, the effect of matrix stiffnesson the critical bending moment is different, and is depen-dent on the innermost radius and layer numbers ofMWNTs. The effect of matrix stiffness on the critical bend-ing moment of a thick MWNT with smaller innermostradius (Example 3) is very obvious.

0.002

4

6

8

10

12

14

for example 1 for example 2 for example 3

M tnemo

m gnidneb lacitirC

cr01(

51 -)

mN

d/c0.02 0.04 0.06 0.08 0.10

Fig. 4. The critical bending moments versus the ratio of the springcoefficient of matrix to van der Waals interaction coefficient for examples1–3 in Table 1.

5. Simple validation of the present method

Because few detailed reports on the combined bendingand axial compression buckling behavior of multi-walledcarbon nanotubes under eccentric compression loading isavailable in previous literatures, an exact comparison ofthe present result with existed results is difficult. In orderto obtain minimum comparisons with existing results ofpure bending buckling of MWNTs, the combined stabilityproblem of MWNTs under eccentric compression loadingcan be simplified to a pure bending buckling problem byletting the axial compression loading F 0

x in Eq. (22) equalzero.

First, literatures [7,38] presented an analytical formulaof critical bending moment (stress or strain) for an infini-tesimal buckling of single-shell under pure bending as

rcr ¼ ecrE ¼M cr

pR2h¼ Eh

Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1� l2Þ

p ð23Þ

Here, the linear buckling of a single-walled carbon nano-tube under pure bending is considered. Corresponding cal-culating parameters are taken as

R ¼ 8:5 nm; h ¼ 0:34 nm;

E ¼ 1:04 TPa ðEh ¼ 360 J=m2Þ and l ¼ 0:3 ð24Þ

Substituting Eq. (24) into Eq. (23), yields

M�cr ¼ 1:975� 10�15 N m ð25Þ

Substituting Eq. (24) into Eq. (22) and letting the axialcompression loading F 0

x in Eq. (22) equal zero, gives

M cr ¼ 1:960� 10�15 N m ð26ÞComparing formula (25) and formula (26), it is seen thattwo critical moments of single-walled CNTs under purebending obtained from different methods are a goodagreement.

Second, a comparison of the present solution with someexisting results from an experimental test [45] and the mod-

00.00

0.01

0.02

0.03

0.04

0.05

0.06

Poncharal et al. (experiment) Pantano et al. (calculation) Present method

Innermost diameter di=0.678 nm

Layer numbers of carbon nanotubes

niarts gnidneb lacitirC

4 8 12 16

Fig. 5. Comparison of critical bending strains of an individual MWNTsunder pure bending loading based on different methods.

01.0

1.5

2.0

2.5

3.0

01( M

51- )

mN

n

Literature [24] Present solution

1 2 3 4 5 6 7

Fig. 6. Bending moment M on the axial half wave number n of CNTs withR1/L = 0.02333, under pure bending.


ified finite-element method [46] is shown in Fig. 5. It is seenfrom Fig. 5 that critical bending strains from differentmethods appear in a good agreement.

Finally, a further comparison of the bending moment onaxial half wavenumber n of double-walled CNTs underpure bending given by the present method with curve 3 inFig. 2c of literature [24] is shown in Fig. 6. Here, calcula-tion data are the same as those in Section 5.1 of the litera-ture [24]. The reasonable agreement of the bendingmoments on axial half wavenumber of double-walledCNTs under pure bending based on two different methodsvalidates the present work further.

6. Conclusions

This paper reports the results of an investigation on thecombined stability of MWNTs embedded in an elasticmatrix, under eccentric compression loading, based on amultilayer shell mode considering the van der Waals forcebetween two adjacent tubes described by Ru [27]. However,it is worth to note that the van der Waals interaction is notonly from the adjacent tube but also from other tubes, asshown in He et al.’s works [47,48]. It is also seen from lit-eratures [47,48] that the effect of van der Waals interactionmodel on the critical loading of MWNTs is dependent onthe innermost radius of MWNTs, and gradually decreasesas the innermost radius of MWNTs increases. Therefore, itshould be stressed that the present van der Waals interac-tion model is rigorous only for the buckling problem ofMWNTs with larger radius. Certainly, analysis for eccen-tric compression buckling of MWNTs based on a rigorousvan der Waals interaction between any two tubes ofMWNTs is a topic of greater interest for further work.Main consequences in the paper are given by

(1) The axial buckling half wavenumbers (n) correspond-ing to the critical bending moment of MWNTsembedded in an elastic matrix, under eccentric com-pression loading are unique for various types of

MWNTs with different ratios of radius to thickness,and is obviously different from the relationshipbetween the axially compressed buckling stress andthe axial half wavenumbers (n). As the result, thereis only one combination of (n) which correspondsto the critical load, so that the axial half wavenum-bers of the complex buckling mode of MWNTs undereccentrix compression loading can be determineduniquely.

(2) For different types of MWNTs, the effect of matrixstiffness on the critical bending moment of MWNTsunder eccentric compression loading is different,which is dependent on the innermost radius and layernumbers of the MWNTs. The critical bending stres-ses exerted on some center tubes of nearly solidMWNTs do not change as the ratio of axial compres-sion loading to the maximum bending membraneforce, Fx1/Nx1, changes.

(3) It is noted that in the paper, the van der Waal’s forcebetween neighboring concentric tubes is directly pro-portional to the difference in the deflection in the twolayers due to linearized characteristics of the presentinfinitesimal buckling. Although these linearized lim-itations cannot reflect significant difference betweenstrong van der Waals repulsive forces and relativelyweak attractive forces, infinitesimal complex bucklinganalysis has the potential to capture first singularityof combined deformation of MWNTs embedded inan elastic matrix, under eccentric compressionloading.

(4) Because electronic and transport properties of carbonnanotubes could be extremely sensitive to even verysmall distortion of their perfect cylindrical geometry,the new features, and interesting numerical results ofthe present work may stimulate further interest in thistopic, and may be used as a useful reference for theapplication and the design of various nanostructuresin which multi-walled carbon nanotubes act as basicelements.

Acknowledgment

The authors thank the referees for their valuablecomments.

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eccentric compression stability of multi-walled carbon nanotubes

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