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doi: 10.1098/rspa.2005.1526, 3785-38054612005Proc. R. Soc. A

K.M Liew, X.Q He and S Kitipornchaicarbon nanotubesBuckling characteristics of embedded multi-walled

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Buckling characteristics of embeddedmulti-walled carbon nanotubes

BY K. M. LIE W, X. Q. HE A N D S. KITIPORNCHAI

Department of Building and Construction, City University of Hong Kong,Tat Chee Avenue, Kowloon, Hong Kong

([email protected])

An analytical algorithm is proposed to describe the buckling behaviour of multi-walledcarbon nanotubes (CNTs) that are embedded in a matrix with consideration of the van

der Waals (vdW) interaction. The individual tube is treated as a cylindrical shell, butthe tube deflections are coupled with each other due to the vdW interaction. Theinteraction between the matrix and the outermost tube is modelled as a Pasternakfoundation. Based on the proposed model, an accurate expression and a simpleapproximate expression are derived for the buckling load of double-walled CNTs that areembedded in a matrix. The approximate expression clearly indicates that the vdW forceis coupled with the matrix parameters. A numerical simulation is carried out, and theresults reveal that the increase in the number of layers leads to a decrease in the criticalbuckling load for multi-walled CNTs with a fixed innermost radius. In contrast, when theoutermost radius is fixed, the critical buckling load increases with the increase in the

number of layers for multi-walled CNTs without a matrix. However, for multi-walledCNTs that are embedded in a matrix, the critical buckling load decreases first and thenincreases with the increase in the number of layers. This implies that there is a givennumber of layers for a multi-walled CNT at which the critical buckling load is the lowest,and that this number depends on the matrix parameters.

Keywords: van der Waals interaction; multi-walled carbon nanotube;

critical buckling load; cylindrical shell model; Pasternak foundation

1. Introduction

There has been much research activity on carbon nanotubes (CNTs) since theirdiscovery in 1991 by Iijima of the NEC Laboratory in Tsukuba, Japan (Iijima1991). In such research, atomistic-based methods (Yakobson et al. 1996;Hernandez et al. 1998; Sanchez-Portal et al. 1999) and continuum mechanics(Govindjee & Sackman 1999; Harik 2002; Lau et al. 2004) are the two maintheoretical methods that are used to study the mechanical behaviour of CNTs.However, the atomistic-based methods are currently limited by computingcapability. For example, in our molecular dynamics (MD) simulation (Liew et al.2004a) of buckling behaviour, the calculation for a single-walled (10,10) CNT

with 2000 atoms required 36 h on a single CPU SGI origin 2000 system. Thecomputational time increases sharply with the increasing number of atoms, and

Proc. R. Soc. A (2005) 461, 37853805

doi:10.1098/rspa.2005.1526

Published online 23 September 2005

Received 21 July 2004Accepted 9 June 2005 3785 q 2005 The Royal Society

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thus explodes exponentially with multi-walled CNTs. For example, a four-walled(5,5), (10,10), (15,15) and (20,20) CNT with a length-to-diameter ratio L/DZ9.1contains 15 097 atoms, and the calculation of its elasto-plastic deformation up tofailure takes over two months (Liew et al. 2004b).

Several continuum models (such as elastic beam, truss, and shell) have been

proposed for the analysis of CNTs. Harik (2002) studied the validity of thecontinuum-beam models to analyse the constitutive behaviour of CNTs, andgave the applicability criterions of the Euler beam model in the study of CNTs.Li & Chou (2003a,b) developed a space truss/frame model to investigate themechanical properties of single-walled CNTs, and examined their Youngsmodulus and shear modulus. A truss model and an equivalent-plate model wereproposed by Odegard et al. (2002) by linking computational chemistry and solidmechanics. By introducing additional rods, the energy that is associated withbond-angle variation was found to be equal to the strain energy of the rods,which allowed the authors to determine the Youngs modulus. Yakobson et al.

(1996) applied a traditional continuum shell model to predict the buckling of asingle-walled CNT and compared the model with an MD simulation. Theirresults show that the continuum shell model can obtain the buckling pattern.Based on the traditional shell model, Ru (2000, 2001a,b) proposed a continuumshell model with consideration of the van der Waals (vdW) interaction to studythe buckling of double-walled CNTs. He proposed a linear proportionalrelationship between the variation of the vdW force and the normal deflectionjump to model the vdW forces, but his model can only be applied to a double-walled CNT. Wang et al. (2003) extended the shell model to the bucklinganalysis of multi-walled CNTs, but Rus model is not suitable for multi-walled

CNTs as the effects of all of the other layers except for the adjacent layers on thevdW interaction are neglected.

It is well known that CNTs have extremely good mechanical properties.Hence, many researchers have explored the possibility of increasing the strengthof various composites by using CNTs as fibres (Jin et al. 1998; Schadler et al.1998; Bower et al. 1999). Currently, most of the research on CNT-reinforcedcomposites is focused on the atomistic-based method (Frankland & Brenner2000), and limited literature can be found on the experimental and continuumtheory for the analysis of CNT-reinforced composites. Lourie et al. (1998)reported experimental observations on the buckling and collapse of CNTs that

are embedded in epoxy resin. Their experimental observations of the buckling ofCNTs are strikingly similar to the theoretical predictions of Yakobson et al.(1996). In addition, Srivastava et al. (1999) studied the nanoplasticity of single-walled CNTs under uniaxial compression by using tight-binding MD. Theircomputed critical stress is also in good agreement with the experimentallyestimated range of values that was reported by Lourie et al. (1998).

To address the lack of a continuum theory for the analysis of CNT-reinforcedcomposites, a continuum model is proposed in this paper for the bucklinganalysis of multi-walled CNTs that are embedded in a matrix with vdWinteraction taken into consideration. The interaction between the outermost tube

and the matrix is modelled as a Pasternak foundation. The validity of theproposed model is demonstrated by comparing it to the existing MD simulationresults.

K. M. Liew and others3786

Proc. R. Soc. A (2005)


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2. Model development

Consider an axially compressed multi-walled CNT that is embedded in an elasticmatrix, as shown in figure 1, in which the individual tube is treated as acylindrical shell of radius Ri, thickness h, and Youngs modulus E. Each tuberefers to a coordinate system (x, q), where x is the axial coordinate and q thecircumferential angular coordinate. The multi-walled CNT is empty inside, andthe outermost tube is bonded to the matrix. The ends of all of the tubes areassumed to be simply supported.

(a) Matrix model

Previous studies by Wagner et al. (1998) on the stress transfer between aCNT and a polymer matrix show that CNTpolymer adhesion is quite strong,

and that not only the normal stress but also the shear stress transfers from theCNT to the polymer matrix. Thus, in the analysis of infinitesimal buckling, weassume that the relation between the pressure and the deflection of theoutermost tube surface can be described by the Pasternak foundation model(Pasternak 1954) i.e.

pNx; qZKKWwx; qCGbV2wx; q; 2:1

where the first parameter KW is the Winkler foundation modulus (Winkler1867), the second parameter Gb is the stiffness of the shearing layer, N is thenumber of layers of the outermost tube and V2 is the Laplace operator, whichis defined as

V2Z

v2

vx2C

1

RN

v2

vq2 : 2:2

RI

ROvan der

Waals forces

h

Nx

Nx

L

Figure 1. A continuum cylindrical shell model of a multi-walled CNT embedded in an elasticmatrix under axial compression and van der Waals interaction.

3787Buckling characteristics



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This model assumes that the elastic foundation consists of a closed spaced andindependent springs, where the top ends of the springs are connected to anincompressible layer that resists only transverse shear deformation. This modelcan describe the interaction between the pressure and deflection of theoutermost tube and a shear interaction between the springs. When setting the

second parameter GbZ0, the Pasternak model is reduced to the Winklermodel, i.e.

pNx; qZKKWwx; q: 2:3

(b ) Basic formulas

Based on the classical thin shell theory (Timoshenko & Gere 1961), the basicequations for the elastic buckling of a multi-walled CNT that is embedded in amatrix can be derived as the N coupled equations, i.e.

L1w1ZV41p1;

LiwiZV4ipi;

LNwNZV4NpN;

9>>>>>>>>=>>>>>>>>;

2:4

where wi (iZ1, 2,., N) is the deflection of the ith tube, pi is the pressure that is

exerted on the tube i due to the vdW interaction between layers, and Li is thedifferential operator that is given by

LiZDiV8iKNx

v2

vx2V

4iK

NqR2i

v2

vq2V

4iC

Eh

R2i

v4

vx4; 2:5

in which xis the axial and q the circumferential coordinate, NxZsxhand NqZsqhare the uniform forces per unit length in the axial and circumferential directionsof the ith tube prior to buckling with sx being the axial and sq being thecircumferential stress, Di is the bending stiffness of the ith tube, and

V2iZ

v2

vx2C

1

R2i

v2

vq2 : 2:6

Due to the infinitesimal deflection between any two layers, the pressure at anypoint of the tubes can be expressed as

pix; yZXNjZ1

pijx; qCDpix; q; 2:7

where pijx; q is the initial vdW pressure contribution to the ith layer from thejth layer prior to buckling, N is the total number of layers of the multi-walledCNT, and Dpi(x, q) is the pressure increment (after buckling) that is due to the




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vdW interaction and the tube-matrix interaction, i.e.

Dp1ZXNjZ1

D p1j;

DpiZXNjZ1

D pij;

DpNZXNjZ1

D pNjKKWwNCGbV2wN;

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

2:8

where D pijx; q is the pressure increment contribution to the pressure incrementDpi that is exerted on the ith layer from the jth layer. As only the infinitesimal

buckling is considered, the pressure incrementD

pijx;q

that is due to the vdWinteraction is assumed to be linearly proportional to the buckling deflectionbetween two walls, i.e.

D pijZ cijwiKwj; 2:9

where cij is a coefficient and is determined by the derivation of the vdW forces inthe sequel. Finally, we obtain the following governing buckling equations of amulti-walled CNT

L1w1ZV41w1

XNjZ1

c1jKXNjZ1

c1jV41wj;

LiwiZV4iwi

XNjZ1

cijKXNjZ1

cijV4iwj;

LNwNZV4NwN

XNjZ1

cNjKXNjZ1

cNjV4NwjKKWV

4NwNCGbV

6NwN:

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

2:10

It can be observed that the equations are coupled due to the vdW interaction.

(c) vdW interaction

It can be observed from equation (2.10) that all of the governing equations arecoupled with each other due to the vdW interaction, which is characterized bythe initial pressure pij (before buckling) and the coefficient cij (after buckling).Thus, the key issue for the buckling analysis is to develop an efficient approachfor the description of the vdW interaction. In our previous work (He et al. 2005),we derived explicit formulas to describe the vdW interaction between any twotubes of a multi-walled CNT. The vdW interaction can be characterized by

pijZ20483s12

9a4

X5

kZ0

K1k2kC1

5k

E12ij K

10243s6

9a4

X2

kZ0

K1k2kC1

2k

E6ij

" #Rj; 2:11




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and

cijZK1001p3s12

9a4E13ij K

1120p3s6

9a4E7ij

!Rj; 2:12

where aZ1.42 A is the CC bond length, Rj is the radius of the jth layer, the

subscripts iand jdenote the ith and jth layers, respectively, and E6ij, E7ij, E12ij andE13ij are the elliptical integrals.

3. Solution to the buckling analysis

The boundary conditions for the simply supported tubes are as follows.

wkZv

2wkvx2Z 0; at xZ 0 and xZL: 3:1

The deflection function that satisfies the boundary conditions equation (3.1) canbe approximated by

wkZAk sinmpx

Lsin nq; 3:2

where Ak (kZ1, 2,., N) are Nunknown coefficients, L is the length of the multi-walled CNT, and m and n are the axial half wavenumber and circumferentialwavenumber, respectively.

The substitution of equation (3.2) into equation (2.10) gives us

mp

L

2C

n

Rk

2

!2KXNjZ1

ckjDCNx

mp

L2

2K

pkRkD

n

Rk

2

8>:C

Eh

DR2k

1

1C LnmpRk

2264

375

29>=>;AkC

XNjZ1

ckjD

AjZ 0 kZ 1; 2;. ; NK1;3:3

and

mp

L

2C

n

Rk

2 !2KXNjZ1

cNjDCNx

mp

L2

2K

pkRkD

n

RN

28>:C

Eh

DR2N

1

1C LnmpRN

2264

375

2

CKWL4C

GbL2

mp

L

2C

n

RN

2 !9>=>;ANC

XNjZ1

cNjD

AjZ0;

3:4

where pkZKNq/Rk is the net pressure that is exerted on kth tube, which isassumed to be inward, the dimensionless buckling load factor N*ZNxL2/D, the

dimensionless Winkler modulus factor KWZKWL4=D, and the shear modulus




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factor GbZGbL2=D. Equations (3.3) and (3.4) can be rewritten in matrix form as

KNxmp

L2

21 0 0 . 0

0 1 0 . 0

0 0 1 . 0

0 0 0 . 1

266666664

377777775

A1

A2

A3

An

8>>>>>>>>>>>>>:

9>>>>>>>=>>>>>>>;Z

b11c12D

c13D.

c1nD

c21

Db

22

c23

D.

c2n

Dc31D

c32D

b33 .c3nD

cn1D

cn2D

cn3D. bnn

2

66666666666666664

3

77777777777777775

A1

A2

A3

An

8>>>>>>>>>>>>>:

9>>>>>>>=>>>>>>>;

;

3:5

or equivalently

KNxmp

L2

2IN!NKCN!N

! A1A2

AN

8>>>>>>>:

9>>>>=>>>>;Z0; 3:6

where

bkkZ mpL

2C n

Rk

2 !2KXNjZ1jsk

ckjDKpkRk

Dn

Rk

2C Eh

DR2k1

1C LnmpRk

2264 3752

kZ1;2; .; NK1; 3:7

and

bNNZmp

L

2C

n

RN

2 !2K

XN

jZ1

jsk

cNjDK

pNRND

n

RN

2C

Eh

DR2N

1

1C LnmpRN 2

2

64

3

75

2

CKWL4C

GbL2

mp

L

2C

n

RN

2 !: 3:8

To determine the non-zero solutions for Ak, it is necessary to equate itsdeterminate to zero. Thus, we have the characteristic equation of [C]:

det KCKNxmp

L 2I Z0: 3:9

The solution of equation (3.9) yields the buckling load of the multi-walled CNTrelative to the wavenumbers m and n.




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4. An explicit solution to the buckling of double-walled CNTsembedded in a matrix

We consider the buckling analysis of a double-walled CNT that is embedded in amatrix, which is a particular case of the multi-walled CNT that is embedded in a

matrix that is detailed in the previous section. The inner radius is RI and theouter radius is RO. We can directly obtain the characteristic equations for thebuckling load of a double-walled CNT from equations (3.3) and (3.4)

mp

L

2C

n

RI

2 !2K

c12DCNx

mp

L2

28>:

Kp1RI

D

n

RI 2

CEh

DR2I

1

1C Ln

mpRI 2

264

375

29>=>;A1C

c12D

A2Z0; 4:1

and

c21D

A1Cmp

L

2C

n

RO

2 !2K

c21DCNx

mp

L2

2K

p2ROD

n

RO

28>:C

Eh

DR2O

1

1C LnmpRO 2

264

375

2

CKWL4C

GbL2

mp

L

2C

n

RO

2 !9>=

>;A2Z0:

4:2The condition for the non-zero solution of A1 and A2 leads to a relation for thebuckling load of the double-walled CNT

Nxmp

L2

2 !2CB1CB2 N

x

mp

L2

2 !CB1B2K

c12c21D2Z0; 4:3

where

B1Zmp

L

2C

n

RI

2

!2

Kc12

D

Kp1RI

D

n

RI

2

CEh

DR

2

I

mpL

2

mp

L 2

C nRI 2

264

375

2

; 4:4

and

B2Zmp

L

2C

n

RO

2 !2K

c21DK

p2ROD

n

RO

2C

Eh

DR2O

mpL

2mp

L

2C nRO

2264

375

2

CKWL4C

GbL2

mp

L

2C

n

RO

2 !: 4:5

Note that we assume the inward pressure to be positive, and that any increase(wiKwjO0, i, jZ1, 2) or decrease (wiKwj!0) in the space between the innerand outer tubes would cause an attractive or repulsive vdW force, respectively.




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Thus, from equation (2.9) we have c12!0 and c21!0, and therefore

B1CB22K4 B1B2K

c12c21D2

ZB1KB2

2C4

c12c21D2O0: 4:6

Thus, the solution to equation (4.3) is given by

KNxmp

L2

2Z

1

2B1CB2H

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB1KB2

2C4

c12c21D2

r : 4:7

(a) Without vdW interaction

When the vdW interaction is ignored, we have c12Zc21Zp1Zp2Z0. As thedifference between the radii of the two tubes is usually very small and KWO0and GbO0, the comparison of equations (4.4) and (4.5) gives us B2OB1. Toobtain the solution for the lowest buckling load, we take the negative sign before

the square root to allow equation (4.7) to reduce to the classical equation for thebuckling load of cylindrical shells (Timoshenko & Gere 1961):

KNxZL2

mp

2mp

L

2C

n

RI

2 !2C

Eh

DR2I

L2

mp

2 mpL

2mp

L

2C nRI

2264

375

2

: 4:8

It can be seen that the axial buckling load factor that is determined by equation(4.8) occurs on the inner tube, which is modelled as an individual cylindricalshell. However, for a double-walled CNT with a small radius, such as (5,5) or

(10,10) nanotubes, the difference between radii is not small and the effect ofROORI cannot be ignored. In this case, we have B1OB2, and the buckling loadfactor is determined by

KNxZL2

mp

2mp

L

2C

n

RO

2 !2C

Eh

DR2O

mpL

2

mpL

2C nRO

2264

375

28>:C

KWL4C

GbL2

mp

L 2C

n

RO 2

!9>=>;:

4:9

The buckling load factor that is determined by equation (4.9) will occur on theouter tube of the double-walled CNT.

(b ) With vdW interaction

We now consider a double-walled CNT that is embedded in a matrix takinginto consideration the vdW interaction. Because the vdW forces between the twotubes are equal and opposite, as shown in figure 1, the pressures that are exertedon the two tubes should satisfy the equilibrium condition p1RIZKp2RO. Because

c12!0 and c21!0, we always have B1CB2O0. Note that the equilibriumdistance between a carbon atom and a flat monolayer is around 0.34 nm(Girifalco & Lad 1956), and thus the initial pressures p1 and p2 are very small




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when the interlayer separation is taken as 0.34 nm. Hence, we have B1B2Oc12c21/D

2 and can easily arrive at

B1CB2O

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB1KB2

2C4

c12c21D2

r: 4:10

To ensure that the right-hand side of equation (4.7) is positive and lower, we takethe negative sign before the square root in equation (4.7) to obtain the solution tothe axial load that is given by

KNxmp

L2

2Z

1

2B1CB2K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB1KB2

2C4

c12c21D2

r : 4:11

Normally, the ratio (ROKRI)/RI is very small and the terms that are related tothe ratio can be neglected. Thus, we have the relationships

mp

L 2

C

n

RI 2 !2

Z

mp

L 2

C

n

RO 2

C

n

RO 2 ROCRI

RI ROKRI

RI !2

zmp

L

2C

n

RO

2 !2; 4:12

andn

RIZ

n

RO

RORIZ

n

ROC

n

RO

ROKRIRI

z

n

RO: 4:13

The substitution of equations (4.12) and (4.13) into equation (4.11) gives us anapproximate formula for the buckling load factor of a double-walled CNT that is

embedded in a matrix

KNZL2

mp

2mp

L

2C

n

RO

2 !2C

Eh

DR2O

mpL

2mp

L

2C nRO

2264

375

28>:

CKW2L4C

Gb2L2

mp

L

2C

n

RO

2 !K

c12Cc212D

Kc12Cc21

2D !2

Cc12Kc21

D

KW2L4C

Gb2L2

mp

L 2C

n

RO 2

Kp2RO

D

n

RO 2

!C

KW2L4C

Gb2L2

mp

L

2C

n

RO

2 K

p2ROD

n

RO

2 !2129>=>;:

4:14Hence, the critical buckling load factor for a double-walled CNT with a matrixthat is modelled as a Pasternak foundation can be determined by minimizing thebuckling loads that are obtained from equation (4.14). It can be observed fromequation (4.14) that the vdW force and the foundation are coupled with each

other. We now discuss the effect of the vdW interaction and the effect of both thevdW interaction and the matrix on the buckling load factor of a double-walledCNT that is embedded in a matrix.




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(i) Effect of the vdW interaction

Note that again we have p2O0, c12!0, c21!0 and jc12jOjc21j. It can be seenthat the second term in the square root of equation (4.14) is positive when thecondition

KW2L4C

Gb2L2

mpL

2C

nRO

2

!%

p2ROD

nRO

2; 4:15

is satisfied.Thus, we have

Kc12Cc21

2D!

c12Cc212D

!2C

c12Kc21D

KW2L4C

Gb2L2

mp

L

2C

n

RO

2

K

p2R

OD

n

RO 2!

C

K

W2L4C

G

b2L2

mp

L 2

C

n

RO 2

K

p2R

OD

n

RO 2 !212

;

4:16

and the results from equation (4.14) show that the presence of the vdWinteraction lowers the critical buckling load factor. Again, if we have

KW2L4C

Gb2L2

mp

L

2C

n

RO

2 !O

p2ROD

n

RO

2; 4:17

then the condition that is needed for equation (4.16) to hold isKW2L4C

Gb2L2

mp

L

2C

n

RO

2 K

p2ROD

n

RO

2OK

c12Kc21D

: 4:18

In this case, equation (4.14) also results in lower critical buckling load factors dueto the presence of the vdW interaction. In contrast, if equation (4.17) holds and

KW2L4C

Gb2L2

mp

L

2C

n

RO

2 K

p2ROD

n

RO

2%K

c12Kc21D

; 4:19

then we have

Kc12Cc21

2DR

c12Cc212D

!2C

KW2L4C

Gb2L2

mp

L

2C

n

RO

2 K

p2ROD

n

RO

2 !2

Cc12Kc21

D

KW2L4C

Gb2L2

mp

L

2C

n

RO

2 K

p2ROD

n

RO

2 !12:

4:20

Hence, under the conditions of equations (4.17) and (4.19), equation (4.14)results in a higher critical buckling load factor than is achieved when the vdWinteraction is not considered.




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(ii) Effect of both the matrix and the vdW interaction

To examine the influence of both the matrix and the vdW interaction on thebuckling load, we rewrite equation (4.14) as

KNZ L2

mp

2mp

L

2C n

RO

2

!2C Eh

DR2O

mpL

2mp

L

2C nRO

2264 37528>:C

KW2L4C

Gb2L2

mp

L

2C

n

RO

2 !K

c12Cc212D

K

KW2L4C

Gb2L2

mp

L

2C

n

RO

2 !K

c12Cc212D

& '2

C 2KW2L4C

Gb2L2

mp

L 2

C

n

RO 2 !& 'c12

DCp2RO

D

n

RO 2 !2

K 2KW2L4C

Gb2L2

mp

L

2C

n

RO

2 !C

c12Kc212D

& 'p2RO

D

n

RO

2129>=>;:4:21

As the radius of multi-walled CNTs increases, the vdW coefficients cij and cjiapproach the same constant (He et al. 2005), and thus the difference between c12

and c21 is very small compared to c12. Note that c12!0 and jc12j[jp2j (jc12j isaround 12 orders of magnitude higher than jp2j; He et al. 2005), it is easy to seethat if and only if

KWL4C

GbL2

mp

L

2C

n

RO

2 !O

p2ROD

nRO

2 !2K

c12Kc21D

p2ROD

n

RO

2

p2ROD

nRO

2K

c12D

! ; 4:22

then the presence of both the elastic matrix and the vdW interaction will raisethe buckling load that is determined by equation (4.21) or equation (4.14).However, if we take GbZ0 and c12Zc21, then equation (4.14) reduces to theresult that Ru (2001b) obtained. In addition, when the vdW force is neglected, itcan be seen that equation (4.14) reduces to the classical equation for the bucklingload of the outer tube without a matrix

KNZL2

mp 2

mp

L 2C

n

RO 2

!2

CEh

DR2O

mpL

2mp

L 2C n

RO 2

264

375

28>:

9>=>;: 4:23

In this case, no influence on the buckling load comes from the matrix.




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5. Results and discussion

We now consider a multi-walled CNT that is embedded in an elastic matrix, asshown in figure 1. The innermost radius is RI and the outermost radius is RO.Suppose that each tube has the same length L and thickness h, and is modelled as

an individual cylindrical shell. The multi-walled CNT is subjected to thecombined action of axial compression and vdW interaction. The initial interlayerseparation between the two adjacent layers is assumed to be 0.34 nm, as this isthe value that is adopted by most published papers on the subject. For all of thenumerical examples, the bending stiffness DZ0.85 eV, EtZ360 J mK2

(Yakobson et al. 1996) and the length to the outermost radius ratio L/ROZ10.A comparison is made between the proposed continuum model and the

existing MD simulations. Considering a multi-walled CNT with an innermostradius RIZ0.34 nm without a surrounding matrix, the critical axial strains ofmulti-walled CNTs with numbers of layers that vary between two to ten are

calculated and shown in figure 2. For the double-, triple- and four-walled CNTs,the critical axial strains that are obtained with the present model are comparedwith the results that were obtained from the MD simulation by Liew et al.(2004a). It can be observed from figure 2 that the results that are obtained by thepresent model are in good agreement with those of Liew et al. (2004a), and thatthe relative errors for the critical axial strain of double-, triple- and four-walledCNTs are 0.16, 13.4 and 15%, respectively. It is worth mentioning that Yakobsonet al. (1996) obtained a critical axial strain of31Z0.05 for a single-walled CNT ofradius RZ0.477 nm by using the MD simulation, a value that is close to ourcalculated critical strain 3cZ0.0599 for a double-walled CNT with an outermostradius of ROZ0.68 nm, as shown in figure 2.

(a) Buckling of double-walled CNTs embedded in an elastic matrix

Having validated the present buckling model, we examine the buckling loadsof a double-walled CNT with an elastic matrix that is modelled as a Winklerfoundation. For all of the numerical examples in this section, the innermostradius is RIZ8.5 nm. The calculated results that are obtained by using the exactequation (4.11) and the approximate equation (4.14) are presented in figure 3for the circumferential wavenumber nZ8, and then compared with the results ofthe classical shell model equation (4.9) and Rus model (2000). As can be seen,

the various sets of results are in good agreement with each other. However, thebuckling loads that are obtained from the exact equation (4.11) (with vdWinteraction) are not much larger than those that are obtained from equation (4.9)(without vdW interaction). It should be noted from equation (2.9) that when theouter (inner) tube deflects outward (inward), the vdW force that is caused isattractive, whereas when the outer (inner) tube deflects inward (outward), thevdW force is repulsive, and thus the vdW interaction always has an effect againstbuckling. This verifies that the vdW interaction increases the critical bucklingload. In contrast, the buckling loads from the approximate equation (4.14) andRus model are smaller than those that are derived from equation (4.9). This is

because ignoring the terms that are related to the ratio of (ROKRI)/RI inequation (4.14) and Rus model leads to smaller results than those that areobtained from equation (4.9). The conclusion in Rus paper (2001a) is that the




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vdW forces do not increase the critical axial strain for infinitesimal buckling ofdouble-walled CNTs, but it is obvious that he does not consider the effect of theignorance of the terms that are related to the ratio of (ROKRI)/RI on thebuckling load.

As the vdW force is coupled with the elastic foundation, we examine the effectof both the vdW interaction and the elastic foundation by comparing equations(4.11), (4.14) and (4.23). Figure 4 shows the buckling loads with respect to the

1 7 10 110.01

0.02

0.03

0.04

0.05

0.06

0.07

criticalaxialstra

in

number of layers of the multi-walled CNT

present model

Liew et al. (2004a)

98653 42

Figure 2. Comparison of the critical axial strains obtained by the present model and the results ofthe MD simulation by Liew et al. (2004a) for multi-walled CNTs with an innermost radiusRIZ0.34 nm.

70 75 80 85 90 95 100 105 110 1151.72

1.76

1.80

1.84

1.88

1.92

1.96

bucklingload,

N*(10

5)

axial half wavenumber, m

equation (4.11)

equation (4.14)

equation (4.9)

Ru (2001a)

n=8

Figure 3. Comparison between the different solutions to the buckling loads of a double-walled CNTwith a Pasternak foundation (KWZ1!10

10 and GbZ0).




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wavenumber nZ1 and various m for a double-walled CNT with a Pasternakfoundation (KWZ1!10

10 and GbZ1!103). The buckling loads are obtained by

equations (4.11), (4.14) and (4.23), respectively, for comparison. It can beobserved that the coupled action of the vdW interaction and the elasticfoundation raises the buckling load of a double-walled CNT that is embedded inan elastic matrix. To ascertain the effect of the radius of a double-walled CNT,

figure 5 shows the critical buckling loads for double-walled CNTs with variousinner radii. The critical buckling loads are obtained by minimizing the resultsfrom equations (4.11), (4.14) and (4.23), respectively, with respect to thewavenumbers n and m. It can be seen that the difference between the criticalbuckling load that is obtained by equation (4.11) (or equations (4.14) and (4.23))is quite large for small radii, say less than 40 nm, which indicates that the sizeeffect plays an important role in the critical buckling load when the radius issmall. As the radius increases, the difference between these results becomes verysmall until it eventually vanishes. This implies that the effect of both the vdWinteraction and the elastic foundation is very small, and can be neglected for

double-walled CNTs with large radii.

(b ) Buckling of multi-walled CNTs embedded in an elastic matrix

We now consider a six-walled CNT that is embedded in an elastic foundation.For all of the numerical examples except those in table 2, the innermost radiusRIZ8.5 nm. Based on the proposed continuous model, or equation (3.5), thebuckling loads for a six-walled CNT are calculated and plotted in figures 68for various mixtures of the Winkler modulus KW and shear modulus G

b.

The buckling load is dependent on the combination of wavenumbers (m, n), and

figure 6 shows the buckling loads as a function of these wavenumbers for asix-walled CNT with KWZ0 and GbZ0. As can be seen in figure 6, the lowest

buckling load with various m for nZ0 or 1 is NxZ114 515:9. With the increase

35 40 45 50 55 60 65 70 75 809.0

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8

11.0

bucklingload,

N*(104)


equation (4.11)equation (4.21)equation (4.23)

n=8

Figure 4. Buckling loads of a double-walled CNT with a Pasternak foundation (KWZ1!103 and

GbZ1!104).




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of n from 2 to 6, the lowest buckling load with respect to n gradually rises,and thus the critical buckling load is determined to be NcrZ114 515:9. Toexamine the effect of the Winkler modulus on the buckling load, the relationshipbetween the buckling loads and the wavenumbers (m, n) is presented in figure 7for a six-walled CNT with KWZ1!10

10 and GbZ0. Again, it can be seen that

the critical buckling load is the lowest buckling load with nZ0 and is thusNcrZ139 875:2, which is higher than the critical buckling load of a six-walledCNT without an elastic matrix. Further, to examine the effect of the

10 20 30 40 50 60 70 800

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

criticalbucklingload,

N(Nm1)

inner radius,R (nm)

50 52 54 56 58 600.24

0.26

0.28

0.30

0.32equation (4.11)equation (4.14)equation (4.23)

Figure 5. Critical buckling loads versus the inner radius for a double-walled CNT with a Pasternakfoundation (KWZ1!10

10 and GbZ1!105).

50 55 60 65 70 75 80 85

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26

1.28

bucklingload,

N*(105

)


n=0n=1n

=2n=3n=4n=5n=6

Figure 6. Dependence of the buckling loads on the wavenumbers (m, n) for a six-walled CNT(KWZ0 and G

bZ0).




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shear modulus on the critical buckling load, figure 8 shows the dependence of thebuckling loads on the wavenumbers (m, n) for a six-walled CNT with KWZ1!1010 and GbZ1!10

5. The minimization of the buckling loads with respect to thewavenumbers m and n gives a critical buckling load NcrZ151 711:4, which ishigher than the critical buckling load of the six-walled CNT with a Winklerfoundation. As expected in the discussion on equation (4.14), figures 68 showthat the critical buckling load increases with the increase of the Winkler modulusand the shear modulus.

To illustrate the influence of the elastic matrix on the critical buckling load,the critical buckling loads are obtained by minimizing the buckling loads withrespect to the wavenumbers m and n for various KW and G

b and are plotted in

60 70 80 90 100 1101.35

1.40

1.45

1.50

1.55

1.60

1.65

bucklingload,

N*(105)


n=0n=1n=2n=3

n=4n=5n=6

Figure 7. Dependence of the buckling loads on the wavenumbers (m, n) for a six-walled CNT withan elastic matrix modelled as a Winkler foundation (KWZ1!10

10 and GbZ0).

60 65 70 75 80 85 90 95 100 105 110 1151.50

1.55

1.60

1.65

1.70

1.75

1.80

1.85

n=0n=1n=2n=3n=4n=5

n=6

bucklingload,

N*(105)


Figure 8. Dependence of the buckling loads on the wavenumbers (m, n) for a six-walled CNT with

an elastic matrix modelled as a Pasternak foundation (KWZ1!10

10

and GbZ1!10

5

).




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figure 9. It is very clear from figure 9 that the critical buckling loads increase asKW and G

b increase. As can be seen, the critical buckling load rises most quickly

with the increase of KW for GbZ0. As G

b increases, the increase of the critical

buckling load slows gradually with the increase of KW. However, with any valueof Gb, the difference between any two critical buckling loads becomes smallerwith the increase ofKW until all of the critical buckling loads approach the same

constant, as shown in figure 9.Two cases are considered to examine the dependence of the critical bucklingload on the number of layers of a multi-walled CNT. The first case is that of amulti-walled CNT with a fixed innermost radius of 8.5 nm. Table 1 presents thecritical buckling loads with any combination of KWZ0, 1!10

10, 2!1010 andGbZ0, 1!10

5, 2!105 for two- to ten-walled CNTs that are embedded in anelastic matrix. Again, it can be seen that the critical buckling load increases withthe increase of KW and G

b for all of the CNTs. The critical buckling load

decreases as the layers increase from two to ten for any combination of KWZ0,1!1010, 2!1010 and GbZ0, 1!10

5, 2!105. This is because the radius of the

outermost tube increases as the total number of layers increases, and the criticalbuckling load lowers as the radius increases (Allen & Bulson 1980). Note that weassume that the entire CNT buckles when one tube buckles, and that criticalbuckling always occurs on the outermost tube, or the tube next to the outermosttube when the outermost tube is bonded with the matrix.

In the second case, the critical buckling loads of a multi-walled CNT with afixed outermost radius of 11.56 nm are derived and presented in table 2.The results can be compared with those in table 1 for multi-walled CNTs with afixed innermost radius of 8.5 nm, table 2. For multi-walled CNTs without amatrix (KWZ0 and G

bZ0), the critical buckling load rises (or in other words,

the capability against buckling increases) as the number of layers increases.However, it is interesting to note that the critical buckling load rises firstand then drops with the increase of the number of layers for multi-walled CNTs

1 0 1 2 3 4 5 6 7 8 9 10 111.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

criticalbuckling

loads,N

*

(105)

Winkler modulus, K*W (1010)

G*b=0

G*b=1105

G*b=2105

G*b=3105

G*b=4105

G*b=5105

Figure 9. Critical buckling loads of a six-walled CNT versus the elastic foundation modulus.




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that are embedded in a matrix, which means that due to the couplingeffects of the vdW force and the matrix, there is a given number of layers atwhich the critical buckling load is lowest for a multi-walled CNT that isembedded in a matrix. For example, the lowest critical buckling loads occur on

the eight-, ten-, seven-, nine-, ten-, nine-, ten-, and eleven-walled CNTs for thematrix parameters KWZ1!1010 and GbZ0; K

WZ2!10

10 and GbZ0; KWZ0

and GbZ1!1010; KWZ1!10

10 a nd GbZ1!1010; KWZ2!10

10 a nd

Table 1. Critical buckling loads Nx (N mK1) for multi-walled CNTs with a fixed innermost radius

fixed as RIZ8.5 nm

totalnumberof layers

GbZ0 GbZ1!10

5 GbZ2!105

KWZ0

KWZ

1!1010KWZ

2!1010 KWZ0

KWZ

1!1010KWZ

2!1010 KWZ0

KWZ

1!1010KWZ

2!1010

2 1.6159 2.9194 3.7164 2.4874 3.6847 4.3868 3.3156 4.3750 4.96953 1.5853 2.4134 2.9174 2.1234 2.8395 3.2563 2.6075 3.2005 3.53514 1.5557 2.1333 2.4824 1.9331 2.4115 2.6906 2.2573 2.6386 2.85745 1.5270 1.9552 2.2131 1.8122 2.1559 2.3576 2.0485 2.3154 2.47156 1.4991 1.8311 2.0313 1.7260 1.9861 2.1401 1.9084 2.1069 2.22517 1.4719 1.7386 1.9001 1.6592 1.8642 1.9872 1.8066 1.9610 2.05468 1.4453 1.6657 1.8001 1.6044 1.7717 1.8732 1.7278 1.8526 1.92949 1.4190 1.6055 1.7205 1.5575 1.6978 1.7841 1.6637 1.7677 1.8327

10 1.3930 1.5539 1.6547 1.5160 1.6363 1.7116 1.6095 1.6984 1.7549

Table 2. Critical buckling loads Nx (N mK1) for multi-walled CNTs with a fixed outermost radius

ROZ11.56 nm

totalnumberof layers

GbZ0 GbZ1!10

5 GbZ2!105

KWZ0

KWZ

1!1010KWZ

2!1010 KWZ0

KWZ

1!1010KWZ

2!1010 KWZ0

KWZ

1!1010KWZ

2!1010

2 1.2298 1.8952 2.3657 1.7431 2.3874 2.8379 2.2454 2.8627 3.28923 1.2488 1.7149 2.0526 1.5925 2.0289 2.3397 1.9194 2.3214 2.60304 1.2683 1.6257 1.8848 1.5278 1.8514 2.0815 1.7670 2.0546 2.25585 1.2884 1.5779 1.7848 1.4981 1.7517 1.9297 1.6852 1.9032 2.05416 1.3090 1.5527 1.7233 1.4864 1.6930 1.8353 1.6396 1.8115 1.92907 1.3299 1.5415 1.6858 1.4854 1.6591 1.7760 1.6154 1.7554 1.84948 1.3510 1.5397 1.6647 1.4913 1.6413 1.7397 1.6048 1.7221 1.79959 1.3722 1.5445 1.6554 1.5020 1.6347 1.7197 1.6037 1.7046 1.769810 1.3930 1.5539 1.6547 1.5160 1.6363 1.7116 1.6095 1.6984 1.754911 1.4131 1.5667 1.6604 1.5320 1.6440 1.7122 1.6202 1.7006 1.7509

12 1.4319 1.5816 1.6707 1.5493 1.6558 1.7194 1.6342 1.7091 1.755013 1.4490 1.5973 1.6843 1.5666 1.6704 1.7313 1.6504 1.7218 1.765014 1.4637 1.6130 1.6996 1.5832 1.6864 1.7461 1.6673 1.7374 1.779115 1.4760 1.6275 1.7152 1.5981 1.7024 1.7624 1.6838 1.7540 1.7955




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GbZ1!1010; KWZ0 and G

bZ2!10

10; KWZ1!1010 and GbZ2!10

10; andKWZ2!10

10 and GbZ2!1010, respectively. The corresponding lowest critical

buckling loads are 1.5397, 1.6547, 1.4854, 1.6347, 1.7116, 1.6037, 1.6984 and1.7509 N mK1, respectively, as shown in table 2.

6. Conclusions

By using the continuum cylindrical shell theory, an efficient methodology hasbeen established for the buckling analysis of multi-walled CNTs that are bondedwith a matrix. To describe the interaction between the CNT and the matrix, thematrix is modelled as a Pasternak foundation. In contrast to the work of Ru(2000, 2001a,b) and Wang et al. (2003), in which only the vdW interactionbetween two adjacent layers is described and the coefficient simply treatedas cZ320 ergs cmK2=0:16 d

2, this work has adopted mathematical expressions

to predict the vdW interaction between any two layers of a multi-walled CNT.An accurate expression and a simple approximate expression have been derivedfor the buckling analysis of double-walled CNTs that are embedded in a matrix.The derived expressions clearly indicate the effects of the vdW interaction andthe matrix parameters on the critical buckling load.

Using the proposed cylindrical shell model and the explicit expressions,numerical simulations have been carried out to examine the effects of the vdWinteraction and the matrix parameters on the critical buckling load of multi-walled CNTs. The numerical results reveal that the critical buckling loaddecreases with the increase in the number of layers for multi-walled CNTs with afixed innermost radius; the critical buckling load increases with the increase in

the number of layers for multi-walled CNTs with a fixed outermost radiuswithout a matrix; and the critical buckling load decreases first and then increaseswith the increase in the number of layers for multi-walled CNTs with a fixedoutermost radius that are embedded in a matrix. That implies that there is aworst number of layers for a multi-walled CNT that is embedded in a matrix atwhich the critical buckling load is the lowest, and that this number depends onthe matrix parameters.

The work described in this paper was partially supported by a grant from the City University ofHong Kong (project no. 7200037).

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