Royal Society 2005 - 3785
This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA
Overview
Download & View Royal Society 2005 - 3785 as PDF for free.
More details
- Words: 9,194
- Pages: 22
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
Buckling characteristics of embedded multi-walled carbon nanotubes K.M Liew, X.Q He and S Kitipornchai Proc. R. Soc. A 2005 461, 3785-3805 doi: 10.1098/rspa.2005.1526
References
This article cites 25 articles
Email alerting service
Receive free email alerts when new articles cite this article - sign up in the box at the top right-hand corner of the article or click here
http://rspa.royalsocietypublishing.org/content/461/2064/3785.full. html#ref-list-1
To subscribe to Proc. R. Soc. A go to: http://rspa.royalsocietypublishing.org/subscriptions
This journal is © 2005 The Royal Society
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
Proc. R. Soc. A (2005) 461, 3785–3805 doi:10.1098/rspa.2005.1526 Published online 23 September 2005
Buckling characteristics of embedded multi-walled carbon nanotubes B Y K. M. L IEW , X. Q. H E
AND
S. K ITIPORNCHAI
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-walled carbon 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, but the tube deflections are coupled with each other due to the vdW interaction. The interaction between the matrix and the outermost tube is modelled as a Pasternak foundation. Based on the proposed model, an accurate expression and a simple approximate expression are derived for the buckling load of double-walled CNTs that are embedded in a matrix. The approximate expression clearly indicates that the vdW force is coupled with the matrix parameters. A numerical simulation is carried out, and the results reveal that the increase in the number of layers leads to a decrease in the critical buckling load for multi-walled CNTs with a fixed innermost radius. In contrast, when the outermost 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-walled CNTs that are embedded in a matrix, the critical buckling load decreases first and then increases with the increase in the number of layers. This implies that there is a given number 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 their discovery in 1991 by Iijima of the NEC Laboratory in Tsukuba, Japan (Iijima 1991). 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 main theoretical methods that are used to study the mechanical behaviour of CNTs. However, the atomistic-based methods are currently limited by computing capability. 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. The computational time increases sharply with the increasing number of atoms, and Received 21 July 2004 Accepted 9 June 2005
3785
q 2005 The Royal Society
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3786
K. M. Liew and others
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/DZ9.1 contains 15 097 atoms, and the calculation of its elasto-plastic deformation up to failure 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 the continuum-beam models to analyse the constitutive behaviour of CNTs, and gave 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 the mechanical properties of single-walled CNTs, and examined their Young’s modulus and shear modulus. A truss model and an equivalent-plate model were proposed by Odegard et al. (2002) by linking computational chemistry and solid mechanics. By introducing additional rods, the energy that is associated with bond-angle variation was found to be equal to the strain energy of the rods, which allowed the authors to determine the Young’s modulus. Yakobson et al. (1996) applied a traditional continuum shell model to predict the buckling of a single-walled CNT and compared the model with an MD simulation. Their results show that the continuum shell model can obtain the buckling pattern. Based on the traditional shell model, Ru (2000, 2001a,b) proposed a continuum shell model with consideration of the van der Waals (vdW) interaction to study the buckling of double-walled CNTs. He proposed a linear proportional relationship between the variation of the vdW force and the normal deflection jump to model the vdW forces, but his model can only be applied to a doublewalled CNT. Wang et al. (2003) extended the shell model to the buckling analysis of multi-walled CNTs, but Ru’s model is not suitable for multi-walled CNTs as the effects of all of the other layers except for the adjacent layers on the vdW interaction are neglected. It is well known that CNTs have extremely good mechanical properties. Hence, many researchers have explored the possibility of increasing the strength of 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-reinforced composites is focused on the atomistic-based method (Frankland & Brenner 2000), and limited literature can be found on the experimental and continuum theory 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 of CNTs are strikingly similar to the theoretical predictions of Yakobson et al. (1996). In addition, Srivastava et al. (1999) studied the nanoplasticity of singlewalled CNTs under uniaxial compression by using tight-binding MD. Their computed critical stress is also in good agreement with the experimentally estimated range of values that was reported by Lourie et al. (1998). To address the lack of a continuum theory for the analysis of CNT-reinforced composites, a continuum model is proposed in this paper for the buckling analysis of multi-walled CNTs that are embedded in a matrix with vdW interaction taken into consideration. The interaction between the outermost tube and the matrix is modelled as a Pasternak foundation. The validity of the proposed model is demonstrated by comparing it to the existing MD simulation results. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3787
Buckling characteristics Nx
h
L
Nx van der Waals forces
RI RO
Figure 1. A continuum cylindrical shell model of a multi-walled CNT embedded in an elastic matrix under axial compression and van der Waals interaction.
2. Model development Consider an axially compressed multi-walled CNT that is embedded in an elastic matrix, as shown in figure 1, in which the individual tube is treated as a cylindrical shell of radius Ri, thickness h, and Young’s modulus E. Each tube refers to a coordinate system (x, q), where x is the axial coordinate and q the circumferential angular coordinate. The multi-walled CNT is empty inside, and the outermost tube is bonded to the matrix. The ends of all of the tubes are assumed to be simply supported. (a ) Matrix model Previous studies by Wagner et al. (1998) on the stress transfer between a CNT and a polymer matrix show that CNT–polymer adhesion is quite strong, and that not only the normal stress but also the shear stress transfers from the CNT to the polymer matrix. Thus, in the analysis of infinitesimal buckling, we assume that the relation between the pressure and the deflection of the outermost tube surface can be described by the Pasternak foundation model (Pasternak 1954) i.e. pN ðx; qÞ ZKKW wðx; qÞ C Gb V2 wðx; qÞ; ð2:1Þ where the first parameter KW is the Winkler foundation modulus (Winkler 1867), the second parameter Gb is the stiffness of the shearing layer, N is the number of layers of the outermost tube and V2 is the Laplace operator, which is defined as V2 Z Proc. R. Soc. A (2005)
v2 1 v2 C : vx 2 RN vq2
ð2:2Þ
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3788
K. M. Liew and others
This model assumes that the elastic foundation consists of a closed spaced and independent springs, where the top ends of the springs are connected to an incompressible layer that resists only transverse shear deformation. This model can describe the interaction between the pressure and deflection of the outermost tube and a shear interaction between the springs. When setting the second parameter GbZ0, the Pasternak model is reduced to the Winkler model, i.e. pN ðx; qÞ ZKKW wðx; qÞ:
ð2:3Þ
(b ) Basic formulas Based on the classical thin shell theory (Timoshenko & Gere 1961), the basic equations for the elastic buckling of a multi-walled CNT that is embedded in a matrix can be derived as the N coupled equations, i.e. 9 L1 w1 Z V41 p1 ; > > > > > > « > > = 4 ð2:4Þ Li w i Z Vi p i ; > > > > « > > > > ; 4 LN wN Z VN pN ; where wi (iZ1, 2, ., N ) is the deflection of the i th tube, pi is the pressure that is exerted on the tube i due to the vdW interaction between layers, and Li is the differential operator that is given by Li Z Di V8i KNx
v2 4 Nq v2 4 Eh v4 Vi K 2 2 Vi C 2 ; Ri vx 4 vx 2 Ri vq
ð2:5Þ
in which x is the axial and q the circumferential coordinate, NxZsx h and NqZsqh are the uniform forces per unit length in the axial and circumferential directions of the i th tube prior to buckling with sx being the axial and sq being the circumferential stress, Di is the bending stiffness of the i th tube, and V2i Z
v2 1 v2 C : vx 2 R2i vq2
ð2:6Þ
Due to the infinitesimal deflection between any two layers, the pressure at any point of the tubes can be expressed as pi ðx; yÞ Z
N X
p ij ðx; qÞ C Dpi ðx; qÞ;
ð2:7Þ
jZ1
where p ij ðx; qÞ is the initial vdW pressure contribution to the i th layer from the j th layer prior to buckling, N is the total number of layers of the multi-walled CNT, and Dpi(x, q) is the pressure increment (after buckling) that is due to the Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3789
Buckling characteristics
vdW interaction and the tube-matrix interaction, i.e. N X Dp 1j ; Dp1 Z jZ1
« Dpi Z
N X Dp ij ;
9 > > > > > > > > > > > > > > > =
> > > > > « > > > > > > N X > > 2 DpN Z Dp Nj KKW wN C Gb V wN ; > > ;
ð2:8Þ
jZ1
jZ1
where Dp ij ðx; qÞ is the pressure increment contribution to the pressure increment Dpi that is exerted on the i th layer from the j th layer. As only the infinitesimal buckling is considered, the pressure increment Dp ij ðx; qÞ that is due to the vdW interaction is assumed to be linearly proportional to the buckling deflection between two walls, i.e. Dp ij Z cij ðwi Kwj Þ; ð2:9Þ where cij is a coefficient and is determined by the derivation of the vdW forces in the sequel. Finally, we obtain the following governing buckling equations of a multi-walled CNT 9 N N X X > 4 > 4 > L1 w1 Z V1 w1 c1j K c1j V1 wj ; > > > > jZ1 jZ1 > > > > > « > > > > > N N = X X 4 4 cij K cij Vi wj ; Li wi Z Vi wi ð2:10Þ > > jZ1 jZ1 > > > > > « > > > > > N N > X X > > 4 4 6 4 cNj K cNj VN wj KKW VN wN C Gb VN wN : > LN wN Z VN wN > ; jZ1
jZ1
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 are coupled with each other due to the vdW interaction, which is characterized by the initial pressure p ij (before buckling) and the coefficient cij (after buckling). Thus, the key issue for the buckling analysis is to develop an efficient approach for 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 two tubes of a multi-walled CNT. The vdW interaction can be characterized by " ! ! # 5 2 5 2 20483s12 X ðK1Þk 10243s6 X ðK1Þk 12 p ij Z Eij K Eij6 Rj ; ð2:11Þ 4 2k C 1 2k C 1 9a4 9a k k kZ0 kZ0 Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3790
K. M. Liew and others
and
1001p3s12 13 1120p3s6 7 Eij K Eij Rj ; cij ZK 9a 4 9a 4
ð2:12Þ
where aZ1.42 ˚ A is the C–C bond length, Rj is the radius of the j th layer, the subscripts i and j denote the i th and j th layers, respectively, and Eij6 , Eij7 , Eij12 and Eij13 are the elliptical integrals.
3. Solution to the buckling analysis The boundary conditions for the simply supported tubes are as follows. v2 wk Z 0; at x Z 0 and x Z L: ð3:1Þ vx 2 The deflection function that satisfies the boundary conditions equation (3.1) can be approximated by mpx wk Z Ak sin sin nq; ð3:2Þ L where Ak (kZ1, 2, ., N ) are N unknown coefficients, L is the length of the multiwalled CNT, and m and n are the axial half wavenumber and circumferential wavenumber, respectively. The substitution of equation (3.2) into equation (2.10) gives us 8 > 2 N < mp 2 n 2 2 X ckj p k Rk n 2 mp C K K C Nx > D D L Rk Rk L2 : jZ1 wk Z
2 C
Eh 6 1 4 DR2k 1 C Ln
mpRk
and 8 > < mp 2 > :
L
n C Rk
2 2
2 C
32 9 > N = X ckj 7 C A Z 0 ðk Z 1; 2; . ; N K1Þ; A 2 5 k > D j ; jZ1
N 2 p R n 2 X cNj mp CNx K K k k 2 D D RN L jZ1
32
Eh 6 1 7 5 2 4 DRN 1C Ln 2 mpRN
ð3:3Þ
9
2 > N = X cNj KW Gb mp 2 n C 4C 2 C A Z0; AN C > D j L RN L L ; jZ1 ð3:4Þ
where pkZKNq/Rk is the net pressure that is exerted on k th tube, which is assumed to be inward, the dimensionless buckling load factor N *ZNxL2/D, the dimensionless Winkler modulus factor KW ZKW L4 =D, and the shear modulus Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3791
Buckling characteristics
factor Gb ZGb L2 =D. Equations (3.3) and (3.4) can be rewritten in matrix form as 2 c c c 3 b11 12 13 . 1n D D D 7 6 78 9 2 38 9 6 6 7 A c c c 21 23 2n A 1 0 0 . 0 > 6 7> 1> 1> b22 . > > > > 6 > > > > 6 7> > > > D D D 7 > > > > 6 7 > > > 6 0 1 0 . 0 7> A A 2 2 > 7> < > < = 6 = 7> mp 2 6 6 7 c c c 31 32 3n 6 7 6 7 . b KNx Z A3 ; 6 0 0 1 . 0 7 A3 33 6 D D D 7 > > > 7> L2 6 > > > > 6 7 > > > 6 « « « « « 7> > > > > 6 7 « « > > > > 4 5> > 6 « > > 7 « « « « > > > : ; 6 ; 7: > 0 0 0 . 1 6 7 An An 6c 7 4 n1 cn2 cn3 5 . bnn D D D ð3:5Þ or equivalently
8 9 A1 > > > > > > > = mp 2 2 I KC Z 0; KNx N!N N!N > > L2 « > > > > > > : ; AN
ð3:6Þ
where 2 32 2 2 X 2 N ckj pk Rk n mp 2 n Eh 6 1 7 K C K C bkk Z 4 5 2 D D L Rk Rk DRk 1 C Ln 2 jZ1 jsk
mpRk
ðk Z 1; 2; .; N K1Þ;
ð3:7Þ
and bNN
2 32 2 2 X 2 N cNj pN RN n mp 2 n Eh 6 1 7 K Z C K C 4 5 D D L RN RN DR2N 1 C Ln 2 jZ1 mpRN K C W 4 L
G C 2b L
jsk
mp 2 L
n C RN
2 :
ð3:8Þ
To determine the non-zero solutions for Ak, it is necessary to equate its determinate to zero. Thus, we have the characteristic equation of [C ]: 2 mp det K½CKNx ½I Z 0: ð3:9Þ L The solution of equation (3.9) yields the buckling load of the multi-walled CNT relative to the wavenumbers m and n. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3792
K. M. Liew and others
4. An explicit solution to the buckling of double-walled CNTs embedded in a matrix We consider the buckling analysis of a double-walled CNT that is embedded in a matrix, 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 the outer radius is RO. We can directly obtain the characteristic equations for the buckling load of a double-walled CNT from equations (3.3) and (3.4) 8 > < mp 2 n 2 2 c 2 12 mp C K CNx > D L RI L2 : 2 32 9 > 2 = p 1 RI n Eh 6 1 c12 7 C ð4:1Þ K 4 5 A1 C A2 Z0; D RI D DR2I 1C Ln 2 > ; mpRI and 8 > < mp 2 n 2 2 c 2 p R n 2 c21 21 mp A1 C C K CNx K 2 O 2 > D D D L RO RO L : 2 C
32
Eh 6 1 KW G 7 C C 2b 4 5 4 DR2O 1C Ln 2 L L
mp 2 n C L RO
9 2 > =
A2 Z0:
> ;
mpRO
ð4:2Þ The condition for the non-zero solution of A1 and A2 leads to a relation for the buckling load of the double-walled CNT 2 2 2 c c mp mp Nx CB1 B2 K 12 221 Z0; CðB1 CB2 Þ Nx ð4:3Þ 2 2 L L D where 2 32 mp 2 2 2 2 L mp 2 n c pR n Eh 6 7 C K 12 K 1 I C B1 Z 4 2 5 ; ð4:4Þ 2 D D RI L RI DRI mp 2 C n L
RI
and 2 32 mp 2 2 2 2 L mp 2 n c pR n Eh 6 7 C K 21 K 2 O C B2 Z 4 5 D D L RO RO DR2O mp 2 C n 2 2 KW Gb mp 2 n : C 4C 2 C L RO L L
L
RO
ð4:5Þ
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 inner and outer tubes would cause an attractive or repulsive vdW force, respectively. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
Buckling characteristics
Thus, from equation (2.9) we have c12!0 and c21!0, and therefore c c c c ðB1 CB2 Þ2 K4 B1 B2 K 12 221 ZðB1 KB2 Þ2 C4 12 221 O0: D D Thus, the solution to equation (4.3) is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 c c mp KNx Z B1 CB2 H ðB1 KB2 Þ2 C4 12 221 : 2 2 L D
3793
ð4:6Þ
ð4:7Þ
(a ) Without vdW interaction When the vdW interaction is ignored, we have c12Zc21Zp1Zp2Z0. As the difference between the radii of the two tubes is usually very small and KW O0 and Gb O 0, the comparison of equations (4.4) and (4.5) gives us B2OB1. To obtain 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 the buckling load of cylindrical shells (Timoshenko & Gere 1961): 32 2 mp 2 2 2 2 2 2 2 L L mp 2 n Eh L 7 6 KNx Z C C 4 2 5 : ð4:8Þ 2 2 mp L RI DRI mp mp C RnI L 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 cylindrical shell. 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 of ROORI cannot be ignored. In this case, we have B1OB2, and the buckling load factor is determined by 8 2 32 mp 2 2 2 > 2 < mp 2 L L n 2 Eh 6 7 KNx Z C C 4 2 5 2 > 2 mp : L RO DRO mp C n L RO ð4:9Þ 9 2 > = K G mp 2 n C 2b C C W : 4 > L RO L L ; The buckling load factor that is determined by equation (4.9) will occur on the outer tube of the double-walled CNT. (b ) With vdW interaction We now consider a double-walled CNT that is embedded in a matrix taking into consideration the vdW interaction. Because the vdW forces between the two tubes are equal and opposite, as shown in figure 1, the pressures that are exerted on the two tubes should satisfy the equilibrium condition p1RIZKp2RO. Because c12!0 and c21!0, we always have B1CB2O0. Note that the equilibrium distance 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 Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3794
K. M. Liew and others
when the interlayer separation is taken as 0.34 nm. Hence, we have B1B2O c12c21/D2 and can easily arrive at rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c c B1 C B2 O ðB1 KB2 Þ2 C 4 12 221 : ð4:10Þ D To ensure that the right-hand side of equation (4.7) is positive and lower, we take the negative sign before the square root in equation (4.7) to obtain the solution to the axial load that is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 c c mp KNx Z B1 C B2 K ðB1 KB2 Þ2 C 4 12 221 : ð4:11Þ 2 2 L D Normally, the ratio (ROKRI)/RI is very small and the terms that are related to the ratio can be neglected. Thus, we have the relationships 2 2 2 2 2 mp 2 n mp 2 n n RO C R I RO KRI C Z C C RI RI L RI L RO RO 2 mp 2 n 2 z C ; ð4:12Þ L RO and n n RO n n Z Z C RI RO R I RO RO
RO KRI n z : RI RO
ð4:13Þ
The substitution of equations (4.12) and (4.13) into equation (4.11) gives us an approximate formula for the buckling load factor of a double-walled CNT that is embedded in a8matrix 2 32 mp 2 > 2 2 2 2 < L L mp 2 n Eh 6 7 C C KN Z 2 5 2 4 2 mp > L R DR mp O : O C RnO L 2 KW Gb mp 2 n c Cc21 K 12 C C 4C 2 2D L RO 2L 2L 2 2 c12 Cc21 c Kc21 KW Gb mp 2 n p 2 RO n 2 K K C 12 C C 2D D D L RO RO 2L4 2L2 9 2 2 ! 12 > = KW Gb mp 2 n p2 RO n 2 C K C C : > D L RO RO 2L4 2L2 ; ð4:14Þ Hence, the critical buckling load factor for a double-walled CNT with a matrix that is modelled as a Pasternak foundation can be determined by minimizing the buckling loads that are obtained from equation (4.14). It can be observed from equation (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 the vdW interaction and the matrix on the buckling load factor of a double-walled CNT that is embedded in a matrix. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3795
Buckling characteristics
(i) Effect of the vdW interaction Note that again we have p2O0, c12!0, c21!0 and jc12jOjc21j. It can be seen that the second term in the square root of equation (4.14) is positive when the condition 2 KW Gb mp 2 n p 2 RO n 2 % C C ; ð4:15Þ D L RO RO 2L4 2L2 is satisfied. Thus, we have 2 c12 Cc21 c12 Cc21 2 c12 Kc21 KW Gb mp 2 n K C C 2 C ! 4 2D 2D D L RO 2L 2L 2 2 ! 12 p2 RO n 2 KW Gb mp 2 n p2 RO n 2 C K C C ; K D D RO L RO RO 2L4 2L2 ð4:16Þ and the results from equation (4.14) show that the presence of the vdW interaction lowers the critical buckling load factor. Again, if we have 2 KW Gb mp 2 n p 2 RO n 2 O C C ; ð4:17Þ D L RO RO 2L4 2L2 then the condition that is needed for equation (4.16) to hold is 2 KW Gb mp 2 n p2 RO n 2 c Kc21 K C 2 C OK 12 : 4 D D L R R 2L 2L O O
ð4:18Þ
In this case, equation (4.14) also results in lower critical buckling load factors due to the presence of the vdW interaction. In contrast, if equation (4.17) holds and 2 KW Gb mp 2 n p2 RO n 2 c Kc21 K C 2 C %K 12 ; ð4:19Þ 4 D D L RO RO 2L 2L then we have c C c21 K 12 R 2D
C
c12 C c21 2D
c12 Kc21 D
2
2 2 KW Gb mp 2 n p 2 RO n 2 K C C C D L RO RO 2L4 2L2
KW 2L4
C
Gb 2
2L
mp 2 L
C
n RO
2 K
p2 RO D
2 ! 2 1
n RO
: ð4:20Þ
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 vdW interaction is not considered. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3796
K. M. Liew and others
(ii) Effect of both the matrix and the vdW interaction To examine the influence of both the matrix and the vdW interaction on the buckling load, we rewrite equation (4.14) as 8 2 32 mp 2 2 2 2 2 > < L mp 2 n Eh 6 L 7 KN Z C C 4 2 5 2 mp > L R DR 2 O mp : O C n L
2 KW Gb mp 2 n K C C 4C 2 L RO 2L 2L
RO
c12 Cc21 2D 2
2 KW Gb mp 2 n c12 Cc21 K C C K 2D L RO 2L4 2L2
2 2 KW Gb mp 2 n c12 p 2 RO n 2 C2 C C C D D L RO RO 2L4 2L2
K2
KW G C b2 4 2L 2L
mp 2 n C L RO
2
C
c12 Kc21 p2 RO 2D D
9 2 ! 12 > = n : > RO ; ð4:21Þ
As the radius of multi-walled CNTs increases, the vdW coefficients cij and cji approach 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 is around 12 orders of magnitude higher than jp2j; He et al. 2005), it is easy to see that if and only if 2 KW Gb mp 2 n O C 2 C 4 L RO L L
p2 RO D
n RO
2 2
c12 Kc21 p2 RO n 2 D D RO ; 2 c p2 RO n K 12 RO D D K
ð4:22Þ
then the presence of both the elastic matrix and the vdW interaction will raise the 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 the result that Ru (2001b) obtained. In addition, when the vdW force is neglected, it can be seen that equation (4.14) reduces to the classical equation for the buckling load of the outer tube without a matrix 8 2 32 9 mp 2 > 2 2 > 2 < = L mp 2 n 2 Eh 6 L 7 KN Z C C ð4:23Þ 4 2 5 >: mp > L RO DR2O mp 2 n : ; C L RO In this case, no influence on the buckling load comes from the matrix. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
Buckling characteristics
3797
5. Results and discussion We now consider a multi-walled CNT that is embedded in an elastic matrix, as shown 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 the combined action of axial compression and vdW interaction. The initial interlayer separation between the two adjacent layers is assumed to be 0.34 nm, as this is the value that is adopted by most published papers on the subject. For all of the numerical examples, the bending stiffness DZ0.85 eV, EtZ360 J m K2 (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 innermost radius RIZ0.34 nm without a surrounding matrix, the critical axial strains of multi-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 compared with 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 the present model are in good agreement with those of Liew et al. (2004a), and that the relative errors for the critical axial strain of double-, triple- and four-walled CNTs are 0.16, 13.4 and 15%, respectively. It is worth mentioning that Yakobson et al. (1996) obtained a critical axial strain of 31Z0.05 for a single-walled CNT of radius RZ0.477 nm by using the MD simulation, a value that is close to our calculated critical strain 3cZ0.0599 for a double-walled CNT with an outermost radius 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 loads of a double-walled CNT with an elastic matrix that is modelled as a Winkler foundation. For all of the numerical examples in this section, the innermost radius is RIZ8.5 nm. The calculated results that are obtained by using the exact equation (4.11) and the approximate equation (4.14) are presented in figure 3 for the circumferential wavenumber nZ8, and then compared with the results of the classical shell model equation (4.9) and Ru’s model (2000). As can be seen, the various sets of results are in good agreement with each other. However, the buckling loads that are obtained from the exact equation (4.11) (with vdW interaction) 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 the outer (inner) tube deflects outward (inward), the vdW force that is caused is attractive, whereas when the outer (inner) tube deflects inward (outward), the vdW force is repulsive, and thus the vdW interaction always has an effect against buckling. This verifies that the vdW interaction increases the critical buckling load. In contrast, the buckling loads from the approximate equation (4.14) and Ru’s 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 in equation (4.14) and Ru’s model leads to smaller results than those that are obtained from equation (4.9). The conclusion in Ru’s paper (2001a) is that the Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3798
K. M. Liew and others 0.07 present model Liew et al. (2004a)
critical axial strain
0.06 0.05 0.04 0.03 0.02 0.01
1
2
7 3 4 5 6 8 9 number of layers of the multi-walled CNT
10
11
Figure 2. Comparison of the critical axial strains obtained by the present model and the results of the MD simulation by Liew et al. (2004a) for multi-walled CNTs with an innermost radius RIZ0.34 nm.
1.96 equation (4.11) equation (4.14) equation (4.9) Ru (2001a)
buckling load, N * (×10 5)
1.92 1.88
n=8 1.84 1.80 1.76 1.72 70
75
80
85 90 95 100 axial half wavenumber, m
105
110
115
Figure 3. Comparison between the different solutions to the buckling loads of a double-walled CNT Z 1 !1010 and GbZ0). with a Pasternak foundation (KW
vdW forces do not increase the critical axial strain for infinitesimal buckling of double-walled CNTs, but it is obvious that he does not consider the effect of the ignorance of the terms that are related to the ratio of (ROKRI)/RI on the buckling load. As the vdW force is coupled with the elastic foundation, we examine the effect of 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 Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3799
Buckling characteristics 11.0 equation (4.11) equation (4.21) equation (4.23)
buckling load, N * (×104)
10.8 10.6 10.4
n=8
10.2 10 9.8 9.6 9.4 9.2 9.0 35
40
45
50 55 60 65 70 axial half wavenumber, m
75
80
Figure 4. Buckling loads of a double-walled CNT with a Pasternak foundation (KW Z 1 !103 and 4 GbZ1!10 ).
wavenumber nZ1 and various m for a double-walled CNT with a Pasternak foundation (KW Z 1 !1010 and GbZ1!103). The buckling loads are obtained by equations (4.11), (4.14) and (4.23), respectively, for comparison. It can be observed that the coupled action of the vdW interaction and the elastic foundation raises the buckling load of a double-walled CNT that is embedded in an 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 various inner radii. The critical buckling loads are obtained by minimizing the results from equations (4.11), (4.14) and (4.23), respectively, with respect to the wavenumbers n and m. It can be seen that the difference between the critical buckling 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 size effect plays an important role in the critical buckling load when the radius is small. As the radius increases, the difference between these results becomes very small until it eventually vanishes. This implies that the effect of both the vdW interaction 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 radius RIZ8.5 nm. Based on the proposed continuous model, or equation (3.5), the buckling loads for a six-walled CNT are calculated and plotted in figures 6–8 for various mixtures of the Winkler modulus KW and shear modulus Gb . 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 a six-walled CNT with KW Z 0 and Gb Z 0. As can be seen in figure 6, the lowest buckling load with various m for nZ0 or 1 is Nx Z 114 515:9. With the increase Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3800
K. M. Liew and others 4.5 0.32 equation (4.11) equation (4.14) equation (4.23)
critical buckling load, N (N m –1)
4.0 0.30
3.5 3.0
0.28
2.5 0.26 2.0 0.24 50
1.5
52
54
56
58
60
1.0 0.5 0
10
20
30 40 50 inner radius, R (nm)
60
70
80
Figure 5. Critical buckling loads versus the inner radius for a double-walled CNT with a Pasternak Z 1 !1010 and GbZ1!105). foundation (KW
1.28 n=0 n=1 n=2 n=3 n=4 n=5 n=6
buckling load, N * (×105)
1.26 1.24 1.22 1.20 1.18 1.16 1.14 1.12 50
55
60 65 70 75 axial half wavenumber, m
80
85
Figure 6. Dependence of the buckling loads on the wavenumbers (m, n) for a six-walled CNT (KW Z 0 and Gb Z 0).
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 Ncr Z 114 515:9. To examine the effect of the Winkler modulus on the buckling load, the relationship between the buckling loads and the wavenumbers (m, n) is presented in figure 7 Z 1 !1010 and Gb Z 0. Again, it can be seen that for a six-walled CNT with KW the critical buckling load is the lowest buckling load with nZ0 and is thus Ncr Z 139 875:2, which is higher than the critical buckling load of a six-walled CNT without an elastic matrix. Further, to examine the effect of the Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3801
Buckling characteristics 1.65 n =0 n =1 n =2 n =3 n =4 n =5 n =6
buckling load, N * (×105 )
1.60 1.55 1.50 1.45 1.40 1.35 60
70
80 90 axial half wavenumber, m
100
110
Figure 7. Dependence of the buckling loads on the wavenumbers (m, n) for a six-walled CNT with Z 1 !1010 and Gb Z 0). an elastic matrix modelled as a Winkler foundation (KW
1.85 buckling load, N * (×105 )
1.80 1.75 1.70
n=0 n=1 n=2 n=3 n=4 n=5 n=6
1.65 1.60 1.55 1.50 60 65 70 75 80 85 90 95 100 105 110 115 axial half wavenumber, m
Figure 8. Dependence of the buckling loads on the wavenumbers (m, n) for a six-walled CNT with Z 1 !1010 and Gb Z 1 !105 ). an elastic matrix modelled as a Pasternak foundation (KW
shear modulus on the critical buckling load, figure 8 shows the dependence of the Z1! buckling loads on the wavenumbers (m, n) for a six-walled CNT with KW 10 5 10 and Gb Z 1 !10 . The minimization of the buckling loads with respect to the wavenumbers m and n gives a critical buckling load Ncr Z 151 711:4, which is higher than the critical buckling load of the six-walled CNT with a Winkler foundation. As expected in the discussion on equation (4.14), figures 6–8 show that the critical buckling load increases with the increase of the Winkler modulus and 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 with and Gb and are plotted in respect to the wavenumbers m and n for various KW Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3802
K. M. Liew and others
critical buckling loads, N * (×10 5)
2.0 1.9 1.8 1.7 1.6 1.5
G*b =0 G*b =1× 105 G*b =2× 105 G*b =3× 105 G*b =4× 105 G*b =5× 105
1.4 1.3 1.2 1.1 1.0 –1
0
1
2 3 4 5 6 7 8 * (×1010) Winkler modulus, KW
9
10 11
Figure 9. Critical buckling loads of a six-walled CNT versus the elastic foundation modulus.
figure 9. It is very clear from figure 9 that the critical buckling loads increase as and Gb increase. As can be seen, the critical buckling load rises most quickly KW for Gb Z 0. As Gb increases, the increase of the critical with the increase of KW buckling load slows gradually with the increase of KW . However, with any value of Gb , the difference between any two critical buckling loads becomes smaller until all of the critical buckling loads approach the same with the increase of KW constant, as shown in figure 9. Two cases are considered to examine the dependence of the critical buckling load on the number of layers of a multi-walled CNT. The first case is that of a multi-walled CNT with a fixed innermost radius of 8.5 nm. Table 1 presents the Z 0, 1!1010, 2!1010 and critical buckling loads with any combination of KW 5 5 Gb Z 0, 1!10 , 2!10 for two- to ten-walled CNTs that are embedded in an elastic matrix. Again, it can be seen that the critical buckling load increases with and Gb for all of the CNTs. The critical buckling load the increase of KW decreases as the layers increase from two to ten for any combination of KW Z 0, 10 10 5 5 1!10 , 2!10 and Gb Z 0, 1!10 , 2!10 . This is because the radius of the outermost tube increases as the total number of layers increases, and the critical buckling load lowers as the radius increases (Allen & Bulson 1980). Note that we assume that the entire CNT buckles when one tube buckles, and that critical buckling always occurs on the outermost tube, or the tube next to the outermost tube when the outermost tube is bonded with the matrix. In the second case, the critical buckling loads of a multi-walled CNT with a fixed 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 a fixed innermost radius of 8.5 nm, table 2. For multi-walled CNTs without a Z 0 and Gb Z 0), the critical buckling load rises (or in other words, matrix (KW the capability against buckling increases) as the number of layers increases. However, it is interesting to note that the critical buckling load rises first and then drops with the increase of the number of layers for multi-walled CNTs Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3803
Buckling characteristics
Table 1. Critical buckling loads Nx (N mK1) for multi-walled CNTs with a fixed innermost radius fixed as RIZ8.5 nm Gb Z 0
Gb Z 1 !105
Gb Z 2 !105
total number of layers KW Z0
Z KW Z KW 10 1 !10 2 !1010 KW Z0
Z KW Z KW 10 1 !10 2 !1010 KW Z0
Z KW Z KW 10 1 !10 2 !1010
2 3 4 5 6 7 8 9 10
2.9194 2.4134 2.1333 1.9552 1.8311 1.7386 1.6657 1.6055 1.5539
3.6847 2.8395 2.4115 2.1559 1.9861 1.8642 1.7717 1.6978 1.6363
4.3750 3.2005 2.6386 2.3154 2.1069 1.9610 1.8526 1.7677 1.6984
1.6159 1.5853 1.5557 1.5270 1.4991 1.4719 1.4453 1.4190 1.3930
3.7164 2.9174 2.4824 2.2131 2.0313 1.9001 1.8001 1.7205 1.6547
2.4874 2.1234 1.9331 1.8122 1.7260 1.6592 1.6044 1.5575 1.5160
4.3868 3.2563 2.6906 2.3576 2.1401 1.9872 1.8732 1.7841 1.7116
3.3156 2.6075 2.2573 2.0485 1.9084 1.8066 1.7278 1.6637 1.6095
4.9695 3.5351 2.8574 2.4715 2.2251 2.0546 1.9294 1.8327 1.7549
Table 2. Critical buckling loads Nx (N mK1) for multi-walled CNTs with a fixed outermost radius ROZ11.56 nm Gb Z 0
Gb Z 1 !105
Gb Z 2 !105
total number of layers KW Z0
Z KW Z KW 10 1 !10 2 !1010 KW Z0
Z KW Z KW 10 1 !10 2 !1010 KW Z0
Z KW Z KW 10 1 !10 2 !1010
2 3 4 5 6 7 8 9 10 11 12 13 14 15
1.8952 1.7149 1.6257 1.5779 1.5527 1.5415 1.5397 1.5445 1.5539 1.5667 1.5816 1.5973 1.6130 1.6275
2.3874 2.0289 1.8514 1.7517 1.6930 1.6591 1.6413 1.6347 1.6363 1.6440 1.6558 1.6704 1.6864 1.7024
2.8627 2.3214 2.0546 1.9032 1.8115 1.7554 1.7221 1.7046 1.6984 1.7006 1.7091 1.7218 1.7374 1.7540
1.2298 1.2488 1.2683 1.2884 1.3090 1.3299 1.3510 1.3722 1.3930 1.4131 1.4319 1.4490 1.4637 1.4760
2.3657 2.0526 1.8848 1.7848 1.7233 1.6858 1.6647 1.6554 1.6547 1.6604 1.6707 1.6843 1.6996 1.7152
1.7431 1.5925 1.5278 1.4981 1.4864 1.4854 1.4913 1.5020 1.5160 1.5320 1.5493 1.5666 1.5832 1.5981
2.8379 2.3397 2.0815 1.9297 1.8353 1.7760 1.7397 1.7197 1.7116 1.7122 1.7194 1.7313 1.7461 1.7624
2.2454 1.9194 1.7670 1.6852 1.6396 1.6154 1.6048 1.6037 1.6095 1.6202 1.6342 1.6504 1.6673 1.6838
3.2892 2.6030 2.2558 2.0541 1.9290 1.8494 1.7995 1.7698 1.7549 1.7509 1.7550 1.7650 1.7791 1.7955
that are embedded in a matrix, which means that due to the coupling effects of the vdW force and the matrix, there is a given number of layers at which the critical buckling load is lowest for a multi-walled CNT that is embedded 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 the matrix parameters KW Z 1 !1010 and Gb Z 0; KW Z 2 !1010 and Gb Z 0; KW Z0 10 10 10 10 a n d Gb Z 1 !10 ; KW Z 1 !10 a n d Gb Z 1 !10 ; KW Z 2 !10 a nd Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3804
K. M. Liew and others
Gb Z 1 !1010 ; KW Z 0 and Gb Z 2 !1010 ; KW Z 1 !1010 and Gb Z 2 !1010 ; and 10 10 KW Z 2 !10 and Gb Z 2 !10 , respectively. The corresponding lowest critical buckling loads are 1.5397, 1.6547, 1.4854, 1.6347, 1.7116, 1.6037, 1.6984 and 1.7509 N mK1, respectively, as shown in table 2.
6. Conclusions By using the continuum cylindrical shell theory, an efficient methodology has been established for the buckling analysis of multi-walled CNTs that are bonded with a matrix. To describe the interaction between the CNT and the matrix, the matrix 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 interaction between two adjacent layers is described and the coefficient simply treated 2 as cZ ð320 ergs cmK2 Þ=ð0:16d Þ, 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 derived for the buckling analysis of double-walled CNTs that are embedded in a matrix. The derived expressions clearly indicate the effects of the vdW interaction and the 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 vdW interaction and the matrix parameters on the critical buckling load of multiwalled CNTs. The numerical results reveal that the critical buckling load decreases with the increase in the number of layers for multi-walled CNTs with a fixed innermost radius; the critical buckling load increases with the increase in the number of layers for multi-walled CNTs with a fixed outermost radius without a matrix; and the critical buckling load decreases first and then increases with the increase in the number of layers for multi-walled CNTs with a fixed outermost radius that are embedded in a matrix. That implies that there is a worst number of layers for a multi-walled CNT that is embedded in a matrix at which the critical buckling load is the lowest, and that this number depends on the matrix parameters. The work described in this paper was partially supported by a grant from the City University of Hong Kong (project no. 7200037).
References Allen, H. G. & Bulson, P. S. 1980 Background to buckling, ch. 7. UK: McGraw-Hill. Bower, C., Rosen, R., Jin, L., Han, J. & Zhou, O. 1999 Appl. Phys. Lett. 74, 3317–3319. (doi:10. 1063/1.123330.) Frankland, S. J. V. & Brenner, D. W. 2000 Mal. Rec. Symp. Proc. 593, 199–204. Girifalco, L. A. & Lad, R. A. 1956 J. Chem. Phys. 25, 693–697. (doi:10.1063/1.1743030.) Govindjee, S. & Sackman, J. L. 1999 Solid State Commun. 110, 227–230. (doi:10.1016/S00381098(98)00626-7.) Harik, V. M. 2002 Comp. Mater. Sci. 24, 328–342. (doi:10.1016/S0927-0256(01)00255-5.) He, X. Q., Kitipornchai, S. & Liew, K. M. 2005 J. Mech. Phys. Solid 53, 303–326. (doi:10.1016/ j.jmps.2004.08.003.) Hernandez, E., Goze, C., Bernier, P. & Rubio, A. 1998 Phys. Rev. Lett. 80, 4502–4505. (doi:10. 1103/PhysRevLett.80.4502.) Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
Buckling characteristics
3805
Iijima, S. 1991 Nature 354, 56–58. (doi:10.1038/354056a0.) Jin, L., Bower, C. & Zhou, O. 1998 Appl. Phys. Lett. 73, 1197–1199. (doi:10.1063/1.122125.) Lau, K. T., Chipara, M., Ling, H. Y. & Hui, D. 2004 Composites: Part B 35, 95–101. (doi:10.1016/j. compositesb.2003.08.008.) Li, C. & Chou, T. W. 2003a Int. J. Solids Struct. 40, 2487–2499. (doi:10.1016/S00207683(03)00056-8.) Li, C. & Chou, T. W. 2003b Compos. Sci. Technol. 63, 1517–1524. (doi:10.1016/S02663538(03)00072-1.) Liew, K. M., Wong, C. H., He, X. Q., Tan, M. J. & Meguid, S. A. 2004a Phys. Rev. B 69, 115 429. (doi:10.1103/PhysRevB.69.115429.) Liew, K. M., He, X. Q. & Wong, C. H. 2004b Acta Mater. 52, 2521–2527. (doi:10.1016/j.actamat. 2004.01.043.) Lourie, O., Cox, D. M. & Wagner, H. D. 1998 Phys. Rev. Lett. 81, 1638. (doi:10.1103/ PhysRevLett.81.1638.) Odegard, G. M., Gates, T. S., Nicholson, L. M. & Wise, K. E. 2002 Compos. Sci. Technol. 62, 1869–1880. (doi:10.1016/S0266-3538(02)00113-6.) Pasternak, P. L. 1954. On a new method of analysis of an elastic foundation by mean of two foundation constants. Gps. Izd. Lit. Po Strait I Arkh., Moscow. Ru, C. Q. 2000 J. Appl. Phys. 87, 7227–7231. (doi:10.1063/1.372973.) Ru, C. Q. 2001a J. Mech. Phys. Solids 49, 1265–1279. (doi:10.1016/S0022-5096(00)00079-X.) Ru, C. Q. 2001b J. Appl. Phys. 89, 3426–3433. (doi:10.1063/1.1347956.) Sanchez-Portal, D., Artacho, E. & Soler, J. M. 1999 Phys. Rev. B 59, 12 678–12 688. (doi:10.1103/ PhysRevB.59.12678.) Schadler, L. S., Giannaris, S. C. & Ajayan, P. M. 1998 Appl. Phys. Lett. 73, 3842–3844. (doi:10. 1063/1.122911.) Srivastava, D., Menon, M. & Cho, K. 1999 Phys. Rev. Lett. 83, 2973. (doi:10.1103/PhysRevLett. 83.2973.) Timoshenko, S. P. & Gere, J. M. 1961 Theory of elastic stability. New York: McGraw-Hill. Wagner, H. D., Lourie, O., Feldman, Y. & Tenne, R. 1998 Appl. Phys. Lett. 72, 188–190. (doi:10. 1063/1.120680.) Wang, C. Y., Ru, C. Q. & Mioduchowski, A. 2003 Int. J. Solids Struct. 40, 3893–3911. (doi:10. 1016/S0020-7683(03)00213-0.) Winkler, E. 1867 Die Lehre von der Elasticitaet und Festigkeit, Prag, Dominicus. Yakobson, B. I., Brabec, C. J. & Bernholc, J. 1996 Phys. Rev. Lett. 76, 2511–2514. (doi:10.1103/ PhysRevLett.76.2511.)
Proc. R. Soc. A (2005)
Buckling characteristics of embedded multi-walled carbon nanotubes K.M Liew, X.Q He and S Kitipornchai Proc. R. Soc. A 2005 461, 3785-3805 doi: 10.1098/rspa.2005.1526
References
This article cites 25 articles
Email alerting service
Receive free email alerts when new articles cite this article - sign up in the box at the top right-hand corner of the article or click here
http://rspa.royalsocietypublishing.org/content/461/2064/3785.full. html#ref-list-1
To subscribe to Proc. R. Soc. A go to: http://rspa.royalsocietypublishing.org/subscriptions
This journal is © 2005 The Royal Society
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
Proc. R. Soc. A (2005) 461, 3785–3805 doi:10.1098/rspa.2005.1526 Published online 23 September 2005
Buckling characteristics of embedded multi-walled carbon nanotubes B Y K. M. L IEW , X. Q. H E
AND
S. K ITIPORNCHAI
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-walled carbon 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, but the tube deflections are coupled with each other due to the vdW interaction. The interaction between the matrix and the outermost tube is modelled as a Pasternak foundation. Based on the proposed model, an accurate expression and a simple approximate expression are derived for the buckling load of double-walled CNTs that are embedded in a matrix. The approximate expression clearly indicates that the vdW force is coupled with the matrix parameters. A numerical simulation is carried out, and the results reveal that the increase in the number of layers leads to a decrease in the critical buckling load for multi-walled CNTs with a fixed innermost radius. In contrast, when the outermost 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-walled CNTs that are embedded in a matrix, the critical buckling load decreases first and then increases with the increase in the number of layers. This implies that there is a given number 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 their discovery in 1991 by Iijima of the NEC Laboratory in Tsukuba, Japan (Iijima 1991). 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 main theoretical methods that are used to study the mechanical behaviour of CNTs. However, the atomistic-based methods are currently limited by computing capability. 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. The computational time increases sharply with the increasing number of atoms, and Received 21 July 2004 Accepted 9 June 2005
3785
q 2005 The Royal Society
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3786
K. M. Liew and others
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/DZ9.1 contains 15 097 atoms, and the calculation of its elasto-plastic deformation up to failure 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 the continuum-beam models to analyse the constitutive behaviour of CNTs, and gave 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 the mechanical properties of single-walled CNTs, and examined their Young’s modulus and shear modulus. A truss model and an equivalent-plate model were proposed by Odegard et al. (2002) by linking computational chemistry and solid mechanics. By introducing additional rods, the energy that is associated with bond-angle variation was found to be equal to the strain energy of the rods, which allowed the authors to determine the Young’s modulus. Yakobson et al. (1996) applied a traditional continuum shell model to predict the buckling of a single-walled CNT and compared the model with an MD simulation. Their results show that the continuum shell model can obtain the buckling pattern. Based on the traditional shell model, Ru (2000, 2001a,b) proposed a continuum shell model with consideration of the van der Waals (vdW) interaction to study the buckling of double-walled CNTs. He proposed a linear proportional relationship between the variation of the vdW force and the normal deflection jump to model the vdW forces, but his model can only be applied to a doublewalled CNT. Wang et al. (2003) extended the shell model to the buckling analysis of multi-walled CNTs, but Ru’s model is not suitable for multi-walled CNTs as the effects of all of the other layers except for the adjacent layers on the vdW interaction are neglected. It is well known that CNTs have extremely good mechanical properties. Hence, many researchers have explored the possibility of increasing the strength of 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-reinforced composites is focused on the atomistic-based method (Frankland & Brenner 2000), and limited literature can be found on the experimental and continuum theory 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 of CNTs are strikingly similar to the theoretical predictions of Yakobson et al. (1996). In addition, Srivastava et al. (1999) studied the nanoplasticity of singlewalled CNTs under uniaxial compression by using tight-binding MD. Their computed critical stress is also in good agreement with the experimentally estimated range of values that was reported by Lourie et al. (1998). To address the lack of a continuum theory for the analysis of CNT-reinforced composites, a continuum model is proposed in this paper for the buckling analysis of multi-walled CNTs that are embedded in a matrix with vdW interaction taken into consideration. The interaction between the outermost tube and the matrix is modelled as a Pasternak foundation. The validity of the proposed model is demonstrated by comparing it to the existing MD simulation results. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3787
Buckling characteristics Nx
h
L
Nx van der Waals forces
RI RO
Figure 1. A continuum cylindrical shell model of a multi-walled CNT embedded in an elastic matrix under axial compression and van der Waals interaction.
2. Model development Consider an axially compressed multi-walled CNT that is embedded in an elastic matrix, as shown in figure 1, in which the individual tube is treated as a cylindrical shell of radius Ri, thickness h, and Young’s modulus E. Each tube refers to a coordinate system (x, q), where x is the axial coordinate and q the circumferential angular coordinate. The multi-walled CNT is empty inside, and the outermost tube is bonded to the matrix. The ends of all of the tubes are assumed to be simply supported. (a ) Matrix model Previous studies by Wagner et al. (1998) on the stress transfer between a CNT and a polymer matrix show that CNT–polymer adhesion is quite strong, and that not only the normal stress but also the shear stress transfers from the CNT to the polymer matrix. Thus, in the analysis of infinitesimal buckling, we assume that the relation between the pressure and the deflection of the outermost tube surface can be described by the Pasternak foundation model (Pasternak 1954) i.e. pN ðx; qÞ ZKKW wðx; qÞ C Gb V2 wðx; qÞ; ð2:1Þ where the first parameter KW is the Winkler foundation modulus (Winkler 1867), the second parameter Gb is the stiffness of the shearing layer, N is the number of layers of the outermost tube and V2 is the Laplace operator, which is defined as V2 Z Proc. R. Soc. A (2005)
v2 1 v2 C : vx 2 RN vq2
ð2:2Þ
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3788
K. M. Liew and others
This model assumes that the elastic foundation consists of a closed spaced and independent springs, where the top ends of the springs are connected to an incompressible layer that resists only transverse shear deformation. This model can describe the interaction between the pressure and deflection of the outermost tube and a shear interaction between the springs. When setting the second parameter GbZ0, the Pasternak model is reduced to the Winkler model, i.e. pN ðx; qÞ ZKKW wðx; qÞ:
ð2:3Þ
(b ) Basic formulas Based on the classical thin shell theory (Timoshenko & Gere 1961), the basic equations for the elastic buckling of a multi-walled CNT that is embedded in a matrix can be derived as the N coupled equations, i.e. 9 L1 w1 Z V41 p1 ; > > > > > > « > > = 4 ð2:4Þ Li w i Z Vi p i ; > > > > « > > > > ; 4 LN wN Z VN pN ; where wi (iZ1, 2, ., N ) is the deflection of the i th tube, pi is the pressure that is exerted on the tube i due to the vdW interaction between layers, and Li is the differential operator that is given by Li Z Di V8i KNx
v2 4 Nq v2 4 Eh v4 Vi K 2 2 Vi C 2 ; Ri vx 4 vx 2 Ri vq
ð2:5Þ
in which x is the axial and q the circumferential coordinate, NxZsx h and NqZsqh are the uniform forces per unit length in the axial and circumferential directions of the i th tube prior to buckling with sx being the axial and sq being the circumferential stress, Di is the bending stiffness of the i th tube, and V2i Z
v2 1 v2 C : vx 2 R2i vq2
ð2:6Þ
Due to the infinitesimal deflection between any two layers, the pressure at any point of the tubes can be expressed as pi ðx; yÞ Z
N X
p ij ðx; qÞ C Dpi ðx; qÞ;
ð2:7Þ
jZ1
where p ij ðx; qÞ is the initial vdW pressure contribution to the i th layer from the j th layer prior to buckling, N is the total number of layers of the multi-walled CNT, and Dpi(x, q) is the pressure increment (after buckling) that is due to the Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3789
Buckling characteristics
vdW interaction and the tube-matrix interaction, i.e. N X Dp 1j ; Dp1 Z jZ1
« Dpi Z
N X Dp ij ;
9 > > > > > > > > > > > > > > > =
> > > > > « > > > > > > N X > > 2 DpN Z Dp Nj KKW wN C Gb V wN ; > > ;
ð2:8Þ
jZ1
jZ1
where Dp ij ðx; qÞ is the pressure increment contribution to the pressure increment Dpi that is exerted on the i th layer from the j th layer. As only the infinitesimal buckling is considered, the pressure increment Dp ij ðx; qÞ that is due to the vdW interaction is assumed to be linearly proportional to the buckling deflection between two walls, i.e. Dp ij Z cij ðwi Kwj Þ; ð2:9Þ where cij is a coefficient and is determined by the derivation of the vdW forces in the sequel. Finally, we obtain the following governing buckling equations of a multi-walled CNT 9 N N X X > 4 > 4 > L1 w1 Z V1 w1 c1j K c1j V1 wj ; > > > > jZ1 jZ1 > > > > > « > > > > > N N = X X 4 4 cij K cij Vi wj ; Li wi Z Vi wi ð2:10Þ > > jZ1 jZ1 > > > > > « > > > > > N N > X X > > 4 4 6 4 cNj K cNj VN wj KKW VN wN C Gb VN wN : > LN wN Z VN wN > ; jZ1
jZ1
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 are coupled with each other due to the vdW interaction, which is characterized by the initial pressure p ij (before buckling) and the coefficient cij (after buckling). Thus, the key issue for the buckling analysis is to develop an efficient approach for 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 two tubes of a multi-walled CNT. The vdW interaction can be characterized by " ! ! # 5 2 5 2 20483s12 X ðK1Þk 10243s6 X ðK1Þk 12 p ij Z Eij K Eij6 Rj ; ð2:11Þ 4 2k C 1 2k C 1 9a4 9a k k kZ0 kZ0 Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3790
K. M. Liew and others
and
1001p3s12 13 1120p3s6 7 Eij K Eij Rj ; cij ZK 9a 4 9a 4
ð2:12Þ
where aZ1.42 ˚ A is the C–C bond length, Rj is the radius of the j th layer, the subscripts i and j denote the i th and j th layers, respectively, and Eij6 , Eij7 , Eij12 and Eij13 are the elliptical integrals.
3. Solution to the buckling analysis The boundary conditions for the simply supported tubes are as follows. v2 wk Z 0; at x Z 0 and x Z L: ð3:1Þ vx 2 The deflection function that satisfies the boundary conditions equation (3.1) can be approximated by mpx wk Z Ak sin sin nq; ð3:2Þ L where Ak (kZ1, 2, ., N ) are N unknown coefficients, L is the length of the multiwalled CNT, and m and n are the axial half wavenumber and circumferential wavenumber, respectively. The substitution of equation (3.2) into equation (2.10) gives us 8 > 2 N < mp 2 n 2 2 X ckj p k Rk n 2 mp C K K C Nx > D D L Rk Rk L2 : jZ1 wk Z
2 C
Eh 6 1 4 DR2k 1 C Ln
mpRk
and 8 > < mp 2 > :
L
n C Rk
2 2
2 C
32 9 > N = X ckj 7 C A Z 0 ðk Z 1; 2; . ; N K1Þ; A 2 5 k > D j ; jZ1
N 2 p R n 2 X cNj mp CNx K K k k 2 D D RN L jZ1
32
Eh 6 1 7 5 2 4 DRN 1C Ln 2 mpRN
ð3:3Þ
9
2 > N = X cNj KW Gb mp 2 n C 4C 2 C A Z0; AN C > D j L RN L L ; jZ1 ð3:4Þ
where pkZKNq/Rk is the net pressure that is exerted on k th tube, which is assumed to be inward, the dimensionless buckling load factor N *ZNxL2/D, the dimensionless Winkler modulus factor KW ZKW L4 =D, and the shear modulus Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3791
Buckling characteristics
factor Gb ZGb L2 =D. Equations (3.3) and (3.4) can be rewritten in matrix form as 2 c c c 3 b11 12 13 . 1n D D D 7 6 78 9 2 38 9 6 6 7 A c c c 21 23 2n A 1 0 0 . 0 > 6 7> 1> 1> b22 . > > > > 6 > > > > 6 7> > > > D D D 7 > > > > 6 7 > > > 6 0 1 0 . 0 7> A A 2 2 > 7> < > < = 6 = 7> mp 2 6 6 7 c c c 31 32 3n 6 7 6 7 . b KNx Z A3 ; 6 0 0 1 . 0 7 A3 33 6 D D D 7 > > > 7> L2 6 > > > > 6 7 > > > 6 « « « « « 7> > > > > 6 7 « « > > > > 4 5> > 6 « > > 7 « « « « > > > : ; 6 ; 7: > 0 0 0 . 1 6 7 An An 6c 7 4 n1 cn2 cn3 5 . bnn D D D ð3:5Þ or equivalently
8 9 A1 > > > > > > > = mp 2 2 I KC Z 0; KNx N!N N!N > > L2 « > > > > > > : ; AN
ð3:6Þ
where 2 32 2 2 X 2 N ckj pk Rk n mp 2 n Eh 6 1 7 K C K C bkk Z 4 5 2 D D L Rk Rk DRk 1 C Ln 2 jZ1 jsk
mpRk
ðk Z 1; 2; .; N K1Þ;
ð3:7Þ
and bNN
2 32 2 2 X 2 N cNj pN RN n mp 2 n Eh 6 1 7 K Z C K C 4 5 D D L RN RN DR2N 1 C Ln 2 jZ1 mpRN K C W 4 L
G C 2b L
jsk
mp 2 L
n C RN
2 :
ð3:8Þ
To determine the non-zero solutions for Ak, it is necessary to equate its determinate to zero. Thus, we have the characteristic equation of [C ]: 2 mp det K½CKNx ½I Z 0: ð3:9Þ L The solution of equation (3.9) yields the buckling load of the multi-walled CNT relative to the wavenumbers m and n. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3792
K. M. Liew and others
4. An explicit solution to the buckling of double-walled CNTs embedded in a matrix We consider the buckling analysis of a double-walled CNT that is embedded in a matrix, 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 the outer radius is RO. We can directly obtain the characteristic equations for the buckling load of a double-walled CNT from equations (3.3) and (3.4) 8 > < mp 2 n 2 2 c 2 12 mp C K CNx > D L RI L2 : 2 32 9 > 2 = p 1 RI n Eh 6 1 c12 7 C ð4:1Þ K 4 5 A1 C A2 Z0; D RI D DR2I 1C Ln 2 > ; mpRI and 8 > < mp 2 n 2 2 c 2 p R n 2 c21 21 mp A1 C C K CNx K 2 O 2 > D D D L RO RO L : 2 C
32
Eh 6 1 KW G 7 C C 2b 4 5 4 DR2O 1C Ln 2 L L
mp 2 n C L RO
9 2 > =
A2 Z0:
> ;
mpRO
ð4:2Þ The condition for the non-zero solution of A1 and A2 leads to a relation for the buckling load of the double-walled CNT 2 2 2 c c mp mp Nx CB1 B2 K 12 221 Z0; CðB1 CB2 Þ Nx ð4:3Þ 2 2 L L D where 2 32 mp 2 2 2 2 L mp 2 n c pR n Eh 6 7 C K 12 K 1 I C B1 Z 4 2 5 ; ð4:4Þ 2 D D RI L RI DRI mp 2 C n L
RI
and 2 32 mp 2 2 2 2 L mp 2 n c pR n Eh 6 7 C K 21 K 2 O C B2 Z 4 5 D D L RO RO DR2O mp 2 C n 2 2 KW Gb mp 2 n : C 4C 2 C L RO L L
L
RO
ð4:5Þ
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 inner and outer tubes would cause an attractive or repulsive vdW force, respectively. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
Buckling characteristics
Thus, from equation (2.9) we have c12!0 and c21!0, and therefore c c c c ðB1 CB2 Þ2 K4 B1 B2 K 12 221 ZðB1 KB2 Þ2 C4 12 221 O0: D D Thus, the solution to equation (4.3) is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 c c mp KNx Z B1 CB2 H ðB1 KB2 Þ2 C4 12 221 : 2 2 L D
3793
ð4:6Þ
ð4:7Þ
(a ) Without vdW interaction When the vdW interaction is ignored, we have c12Zc21Zp1Zp2Z0. As the difference between the radii of the two tubes is usually very small and KW O0 and Gb O 0, the comparison of equations (4.4) and (4.5) gives us B2OB1. To obtain 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 the buckling load of cylindrical shells (Timoshenko & Gere 1961): 32 2 mp 2 2 2 2 2 2 2 L L mp 2 n Eh L 7 6 KNx Z C C 4 2 5 : ð4:8Þ 2 2 mp L RI DRI mp mp C RnI L 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 cylindrical shell. 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 of ROORI cannot be ignored. In this case, we have B1OB2, and the buckling load factor is determined by 8 2 32 mp 2 2 2 > 2 < mp 2 L L n 2 Eh 6 7 KNx Z C C 4 2 5 2 > 2 mp : L RO DRO mp C n L RO ð4:9Þ 9 2 > = K G mp 2 n C 2b C C W : 4 > L RO L L ; The buckling load factor that is determined by equation (4.9) will occur on the outer tube of the double-walled CNT. (b ) With vdW interaction We now consider a double-walled CNT that is embedded in a matrix taking into consideration the vdW interaction. Because the vdW forces between the two tubes are equal and opposite, as shown in figure 1, the pressures that are exerted on the two tubes should satisfy the equilibrium condition p1RIZKp2RO. Because c12!0 and c21!0, we always have B1CB2O0. Note that the equilibrium distance 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 Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3794
K. M. Liew and others
when the interlayer separation is taken as 0.34 nm. Hence, we have B1B2O c12c21/D2 and can easily arrive at rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c c B1 C B2 O ðB1 KB2 Þ2 C 4 12 221 : ð4:10Þ D To ensure that the right-hand side of equation (4.7) is positive and lower, we take the negative sign before the square root in equation (4.7) to obtain the solution to the axial load that is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 c c mp KNx Z B1 C B2 K ðB1 KB2 Þ2 C 4 12 221 : ð4:11Þ 2 2 L D Normally, the ratio (ROKRI)/RI is very small and the terms that are related to the ratio can be neglected. Thus, we have the relationships 2 2 2 2 2 mp 2 n mp 2 n n RO C R I RO KRI C Z C C RI RI L RI L RO RO 2 mp 2 n 2 z C ; ð4:12Þ L RO and n n RO n n Z Z C RI RO R I RO RO
RO KRI n z : RI RO
ð4:13Þ
The substitution of equations (4.12) and (4.13) into equation (4.11) gives us an approximate formula for the buckling load factor of a double-walled CNT that is embedded in a8matrix 2 32 mp 2 > 2 2 2 2 < L L mp 2 n Eh 6 7 C C KN Z 2 5 2 4 2 mp > L R DR mp O : O C RnO L 2 KW Gb mp 2 n c Cc21 K 12 C C 4C 2 2D L RO 2L 2L 2 2 c12 Cc21 c Kc21 KW Gb mp 2 n p 2 RO n 2 K K C 12 C C 2D D D L RO RO 2L4 2L2 9 2 2 ! 12 > = KW Gb mp 2 n p2 RO n 2 C K C C : > D L RO RO 2L4 2L2 ; ð4:14Þ Hence, the critical buckling load factor for a double-walled CNT with a matrix that is modelled as a Pasternak foundation can be determined by minimizing the buckling loads that are obtained from equation (4.14). It can be observed from equation (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 the vdW interaction and the matrix on the buckling load factor of a double-walled CNT that is embedded in a matrix. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3795
Buckling characteristics
(i) Effect of the vdW interaction Note that again we have p2O0, c12!0, c21!0 and jc12jOjc21j. It can be seen that the second term in the square root of equation (4.14) is positive when the condition 2 KW Gb mp 2 n p 2 RO n 2 % C C ; ð4:15Þ D L RO RO 2L4 2L2 is satisfied. Thus, we have 2 c12 Cc21 c12 Cc21 2 c12 Kc21 KW Gb mp 2 n K C C 2 C ! 4 2D 2D D L RO 2L 2L 2 2 ! 12 p2 RO n 2 KW Gb mp 2 n p2 RO n 2 C K C C ; K D D RO L RO RO 2L4 2L2 ð4:16Þ and the results from equation (4.14) show that the presence of the vdW interaction lowers the critical buckling load factor. Again, if we have 2 KW Gb mp 2 n p 2 RO n 2 O C C ; ð4:17Þ D L RO RO 2L4 2L2 then the condition that is needed for equation (4.16) to hold is 2 KW Gb mp 2 n p2 RO n 2 c Kc21 K C 2 C OK 12 : 4 D D L R R 2L 2L O O
ð4:18Þ
In this case, equation (4.14) also results in lower critical buckling load factors due to the presence of the vdW interaction. In contrast, if equation (4.17) holds and 2 KW Gb mp 2 n p2 RO n 2 c Kc21 K C 2 C %K 12 ; ð4:19Þ 4 D D L RO RO 2L 2L then we have c C c21 K 12 R 2D
C
c12 C c21 2D
c12 Kc21 D
2
2 2 KW Gb mp 2 n p 2 RO n 2 K C C C D L RO RO 2L4 2L2
KW 2L4
C
Gb 2
2L
mp 2 L
C
n RO
2 K
p2 RO D
2 ! 2 1
n RO
: ð4:20Þ
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 vdW interaction is not considered. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3796
K. M. Liew and others
(ii) Effect of both the matrix and the vdW interaction To examine the influence of both the matrix and the vdW interaction on the buckling load, we rewrite equation (4.14) as 8 2 32 mp 2 2 2 2 2 > < L mp 2 n Eh 6 L 7 KN Z C C 4 2 5 2 mp > L R DR 2 O mp : O C n L
2 KW Gb mp 2 n K C C 4C 2 L RO 2L 2L
RO
c12 Cc21 2D 2
2 KW Gb mp 2 n c12 Cc21 K C C K 2D L RO 2L4 2L2
2 2 KW Gb mp 2 n c12 p 2 RO n 2 C2 C C C D D L RO RO 2L4 2L2
K2
KW G C b2 4 2L 2L
mp 2 n C L RO
2
C
c12 Kc21 p2 RO 2D D
9 2 ! 12 > = n : > RO ; ð4:21Þ
As the radius of multi-walled CNTs increases, the vdW coefficients cij and cji approach 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 is around 12 orders of magnitude higher than jp2j; He et al. 2005), it is easy to see that if and only if 2 KW Gb mp 2 n O C 2 C 4 L RO L L
p2 RO D
n RO
2 2
c12 Kc21 p2 RO n 2 D D RO ; 2 c p2 RO n K 12 RO D D K
ð4:22Þ
then the presence of both the elastic matrix and the vdW interaction will raise the 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 the result that Ru (2001b) obtained. In addition, when the vdW force is neglected, it can be seen that equation (4.14) reduces to the classical equation for the buckling load of the outer tube without a matrix 8 2 32 9 mp 2 > 2 2 > 2 < = L mp 2 n 2 Eh 6 L 7 KN Z C C ð4:23Þ 4 2 5 >: mp > L RO DR2O mp 2 n : ; C L RO In this case, no influence on the buckling load comes from the matrix. Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
Buckling characteristics
3797
5. Results and discussion We now consider a multi-walled CNT that is embedded in an elastic matrix, as shown 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 the combined action of axial compression and vdW interaction. The initial interlayer separation between the two adjacent layers is assumed to be 0.34 nm, as this is the value that is adopted by most published papers on the subject. For all of the numerical examples, the bending stiffness DZ0.85 eV, EtZ360 J m K2 (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 innermost radius RIZ0.34 nm without a surrounding matrix, the critical axial strains of multi-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 compared with 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 the present model are in good agreement with those of Liew et al. (2004a), and that the relative errors for the critical axial strain of double-, triple- and four-walled CNTs are 0.16, 13.4 and 15%, respectively. It is worth mentioning that Yakobson et al. (1996) obtained a critical axial strain of 31Z0.05 for a single-walled CNT of radius RZ0.477 nm by using the MD simulation, a value that is close to our calculated critical strain 3cZ0.0599 for a double-walled CNT with an outermost radius 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 loads of a double-walled CNT with an elastic matrix that is modelled as a Winkler foundation. For all of the numerical examples in this section, the innermost radius is RIZ8.5 nm. The calculated results that are obtained by using the exact equation (4.11) and the approximate equation (4.14) are presented in figure 3 for the circumferential wavenumber nZ8, and then compared with the results of the classical shell model equation (4.9) and Ru’s model (2000). As can be seen, the various sets of results are in good agreement with each other. However, the buckling loads that are obtained from the exact equation (4.11) (with vdW interaction) 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 the outer (inner) tube deflects outward (inward), the vdW force that is caused is attractive, whereas when the outer (inner) tube deflects inward (outward), the vdW force is repulsive, and thus the vdW interaction always has an effect against buckling. This verifies that the vdW interaction increases the critical buckling load. In contrast, the buckling loads from the approximate equation (4.14) and Ru’s 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 in equation (4.14) and Ru’s model leads to smaller results than those that are obtained from equation (4.9). The conclusion in Ru’s paper (2001a) is that the Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3798
K. M. Liew and others 0.07 present model Liew et al. (2004a)
critical axial strain
0.06 0.05 0.04 0.03 0.02 0.01
1
2
7 3 4 5 6 8 9 number of layers of the multi-walled CNT
10
11
Figure 2. Comparison of the critical axial strains obtained by the present model and the results of the MD simulation by Liew et al. (2004a) for multi-walled CNTs with an innermost radius RIZ0.34 nm.
1.96 equation (4.11) equation (4.14) equation (4.9) Ru (2001a)
buckling load, N * (×10 5)
1.92 1.88
n=8 1.84 1.80 1.76 1.72 70
75
80
85 90 95 100 axial half wavenumber, m
105
110
115
Figure 3. Comparison between the different solutions to the buckling loads of a double-walled CNT Z 1 !1010 and GbZ0). with a Pasternak foundation (KW
vdW forces do not increase the critical axial strain for infinitesimal buckling of double-walled CNTs, but it is obvious that he does not consider the effect of the ignorance of the terms that are related to the ratio of (ROKRI)/RI on the buckling load. As the vdW force is coupled with the elastic foundation, we examine the effect of 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 Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3799
Buckling characteristics 11.0 equation (4.11) equation (4.21) equation (4.23)
buckling load, N * (×104)
10.8 10.6 10.4
n=8
10.2 10 9.8 9.6 9.4 9.2 9.0 35
40
45
50 55 60 65 70 axial half wavenumber, m
75
80
Figure 4. Buckling loads of a double-walled CNT with a Pasternak foundation (KW Z 1 !103 and 4 GbZ1!10 ).
wavenumber nZ1 and various m for a double-walled CNT with a Pasternak foundation (KW Z 1 !1010 and GbZ1!103). The buckling loads are obtained by equations (4.11), (4.14) and (4.23), respectively, for comparison. It can be observed that the coupled action of the vdW interaction and the elastic foundation raises the buckling load of a double-walled CNT that is embedded in an 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 various inner radii. The critical buckling loads are obtained by minimizing the results from equations (4.11), (4.14) and (4.23), respectively, with respect to the wavenumbers n and m. It can be seen that the difference between the critical buckling 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 size effect plays an important role in the critical buckling load when the radius is small. As the radius increases, the difference between these results becomes very small until it eventually vanishes. This implies that the effect of both the vdW interaction 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 radius RIZ8.5 nm. Based on the proposed continuous model, or equation (3.5), the buckling loads for a six-walled CNT are calculated and plotted in figures 6–8 for various mixtures of the Winkler modulus KW and shear modulus Gb . 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 a six-walled CNT with KW Z 0 and Gb Z 0. As can be seen in figure 6, the lowest buckling load with various m for nZ0 or 1 is Nx Z 114 515:9. With the increase Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3800
K. M. Liew and others 4.5 0.32 equation (4.11) equation (4.14) equation (4.23)
critical buckling load, N (N m –1)
4.0 0.30
3.5 3.0
0.28
2.5 0.26 2.0 0.24 50
1.5
52
54
56
58
60
1.0 0.5 0
10
20
30 40 50 inner radius, R (nm)
60
70
80
Figure 5. Critical buckling loads versus the inner radius for a double-walled CNT with a Pasternak Z 1 !1010 and GbZ1!105). foundation (KW
1.28 n=0 n=1 n=2 n=3 n=4 n=5 n=6
buckling load, N * (×105)
1.26 1.24 1.22 1.20 1.18 1.16 1.14 1.12 50
55
60 65 70 75 axial half wavenumber, m
80
85
Figure 6. Dependence of the buckling loads on the wavenumbers (m, n) for a six-walled CNT (KW Z 0 and Gb Z 0).
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 Ncr Z 114 515:9. To examine the effect of the Winkler modulus on the buckling load, the relationship between the buckling loads and the wavenumbers (m, n) is presented in figure 7 Z 1 !1010 and Gb Z 0. Again, it can be seen that for a six-walled CNT with KW the critical buckling load is the lowest buckling load with nZ0 and is thus Ncr Z 139 875:2, which is higher than the critical buckling load of a six-walled CNT without an elastic matrix. Further, to examine the effect of the Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3801
Buckling characteristics 1.65 n =0 n =1 n =2 n =3 n =4 n =5 n =6
buckling load, N * (×105 )
1.60 1.55 1.50 1.45 1.40 1.35 60
70
80 90 axial half wavenumber, m
100
110
Figure 7. Dependence of the buckling loads on the wavenumbers (m, n) for a six-walled CNT with Z 1 !1010 and Gb Z 0). an elastic matrix modelled as a Winkler foundation (KW
1.85 buckling load, N * (×105 )
1.80 1.75 1.70
n=0 n=1 n=2 n=3 n=4 n=5 n=6
1.65 1.60 1.55 1.50 60 65 70 75 80 85 90 95 100 105 110 115 axial half wavenumber, m
Figure 8. Dependence of the buckling loads on the wavenumbers (m, n) for a six-walled CNT with Z 1 !1010 and Gb Z 1 !105 ). an elastic matrix modelled as a Pasternak foundation (KW
shear modulus on the critical buckling load, figure 8 shows the dependence of the Z1! buckling loads on the wavenumbers (m, n) for a six-walled CNT with KW 10 5 10 and Gb Z 1 !10 . The minimization of the buckling loads with respect to the wavenumbers m and n gives a critical buckling load Ncr Z 151 711:4, which is higher than the critical buckling load of the six-walled CNT with a Winkler foundation. As expected in the discussion on equation (4.14), figures 6–8 show that the critical buckling load increases with the increase of the Winkler modulus and 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 with and Gb and are plotted in respect to the wavenumbers m and n for various KW Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3802
K. M. Liew and others
critical buckling loads, N * (×10 5)
2.0 1.9 1.8 1.7 1.6 1.5
G*b =0 G*b =1× 105 G*b =2× 105 G*b =3× 105 G*b =4× 105 G*b =5× 105
1.4 1.3 1.2 1.1 1.0 –1
0
1
2 3 4 5 6 7 8 * (×1010) Winkler modulus, KW
9
10 11
Figure 9. Critical buckling loads of a six-walled CNT versus the elastic foundation modulus.
figure 9. It is very clear from figure 9 that the critical buckling loads increase as and Gb increase. As can be seen, the critical buckling load rises most quickly KW for Gb Z 0. As Gb increases, the increase of the critical with the increase of KW buckling load slows gradually with the increase of KW . However, with any value of Gb , the difference between any two critical buckling loads becomes smaller until all of the critical buckling loads approach the same with the increase of KW constant, as shown in figure 9. Two cases are considered to examine the dependence of the critical buckling load on the number of layers of a multi-walled CNT. The first case is that of a multi-walled CNT with a fixed innermost radius of 8.5 nm. Table 1 presents the Z 0, 1!1010, 2!1010 and critical buckling loads with any combination of KW 5 5 Gb Z 0, 1!10 , 2!10 for two- to ten-walled CNTs that are embedded in an elastic matrix. Again, it can be seen that the critical buckling load increases with and Gb for all of the CNTs. The critical buckling load the increase of KW decreases as the layers increase from two to ten for any combination of KW Z 0, 10 10 5 5 1!10 , 2!10 and Gb Z 0, 1!10 , 2!10 . This is because the radius of the outermost tube increases as the total number of layers increases, and the critical buckling load lowers as the radius increases (Allen & Bulson 1980). Note that we assume that the entire CNT buckles when one tube buckles, and that critical buckling always occurs on the outermost tube, or the tube next to the outermost tube when the outermost tube is bonded with the matrix. In the second case, the critical buckling loads of a multi-walled CNT with a fixed 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 a fixed innermost radius of 8.5 nm, table 2. For multi-walled CNTs without a Z 0 and Gb Z 0), the critical buckling load rises (or in other words, matrix (KW the capability against buckling increases) as the number of layers increases. However, it is interesting to note that the critical buckling load rises first and then drops with the increase of the number of layers for multi-walled CNTs Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3803
Buckling characteristics
Table 1. Critical buckling loads Nx (N mK1) for multi-walled CNTs with a fixed innermost radius fixed as RIZ8.5 nm Gb Z 0
Gb Z 1 !105
Gb Z 2 !105
total number of layers KW Z0
Z KW Z KW 10 1 !10 2 !1010 KW Z0
Z KW Z KW 10 1 !10 2 !1010 KW Z0
Z KW Z KW 10 1 !10 2 !1010
2 3 4 5 6 7 8 9 10
2.9194 2.4134 2.1333 1.9552 1.8311 1.7386 1.6657 1.6055 1.5539
3.6847 2.8395 2.4115 2.1559 1.9861 1.8642 1.7717 1.6978 1.6363
4.3750 3.2005 2.6386 2.3154 2.1069 1.9610 1.8526 1.7677 1.6984
1.6159 1.5853 1.5557 1.5270 1.4991 1.4719 1.4453 1.4190 1.3930
3.7164 2.9174 2.4824 2.2131 2.0313 1.9001 1.8001 1.7205 1.6547
2.4874 2.1234 1.9331 1.8122 1.7260 1.6592 1.6044 1.5575 1.5160
4.3868 3.2563 2.6906 2.3576 2.1401 1.9872 1.8732 1.7841 1.7116
3.3156 2.6075 2.2573 2.0485 1.9084 1.8066 1.7278 1.6637 1.6095
4.9695 3.5351 2.8574 2.4715 2.2251 2.0546 1.9294 1.8327 1.7549
Table 2. Critical buckling loads Nx (N mK1) for multi-walled CNTs with a fixed outermost radius ROZ11.56 nm Gb Z 0
Gb Z 1 !105
Gb Z 2 !105
total number of layers KW Z0
Z KW Z KW 10 1 !10 2 !1010 KW Z0
Z KW Z KW 10 1 !10 2 !1010 KW Z0
Z KW Z KW 10 1 !10 2 !1010
2 3 4 5 6 7 8 9 10 11 12 13 14 15
1.8952 1.7149 1.6257 1.5779 1.5527 1.5415 1.5397 1.5445 1.5539 1.5667 1.5816 1.5973 1.6130 1.6275
2.3874 2.0289 1.8514 1.7517 1.6930 1.6591 1.6413 1.6347 1.6363 1.6440 1.6558 1.6704 1.6864 1.7024
2.8627 2.3214 2.0546 1.9032 1.8115 1.7554 1.7221 1.7046 1.6984 1.7006 1.7091 1.7218 1.7374 1.7540
1.2298 1.2488 1.2683 1.2884 1.3090 1.3299 1.3510 1.3722 1.3930 1.4131 1.4319 1.4490 1.4637 1.4760
2.3657 2.0526 1.8848 1.7848 1.7233 1.6858 1.6647 1.6554 1.6547 1.6604 1.6707 1.6843 1.6996 1.7152
1.7431 1.5925 1.5278 1.4981 1.4864 1.4854 1.4913 1.5020 1.5160 1.5320 1.5493 1.5666 1.5832 1.5981
2.8379 2.3397 2.0815 1.9297 1.8353 1.7760 1.7397 1.7197 1.7116 1.7122 1.7194 1.7313 1.7461 1.7624
2.2454 1.9194 1.7670 1.6852 1.6396 1.6154 1.6048 1.6037 1.6095 1.6202 1.6342 1.6504 1.6673 1.6838
3.2892 2.6030 2.2558 2.0541 1.9290 1.8494 1.7995 1.7698 1.7549 1.7509 1.7550 1.7650 1.7791 1.7955
that are embedded in a matrix, which means that due to the coupling effects of the vdW force and the matrix, there is a given number of layers at which the critical buckling load is lowest for a multi-walled CNT that is embedded 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 the matrix parameters KW Z 1 !1010 and Gb Z 0; KW Z 2 !1010 and Gb Z 0; KW Z0 10 10 10 10 a n d Gb Z 1 !10 ; KW Z 1 !10 a n d Gb Z 1 !10 ; KW Z 2 !10 a nd Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
3804
K. M. Liew and others
Gb Z 1 !1010 ; KW Z 0 and Gb Z 2 !1010 ; KW Z 1 !1010 and Gb Z 2 !1010 ; and 10 10 KW Z 2 !10 and Gb Z 2 !10 , respectively. The corresponding lowest critical buckling loads are 1.5397, 1.6547, 1.4854, 1.6347, 1.7116, 1.6037, 1.6984 and 1.7509 N mK1, respectively, as shown in table 2.
6. Conclusions By using the continuum cylindrical shell theory, an efficient methodology has been established for the buckling analysis of multi-walled CNTs that are bonded with a matrix. To describe the interaction between the CNT and the matrix, the matrix 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 interaction between two adjacent layers is described and the coefficient simply treated 2 as cZ ð320 ergs cmK2 Þ=ð0:16d Þ, 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 derived for the buckling analysis of double-walled CNTs that are embedded in a matrix. The derived expressions clearly indicate the effects of the vdW interaction and the 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 vdW interaction and the matrix parameters on the critical buckling load of multiwalled CNTs. The numerical results reveal that the critical buckling load decreases with the increase in the number of layers for multi-walled CNTs with a fixed innermost radius; the critical buckling load increases with the increase in the number of layers for multi-walled CNTs with a fixed outermost radius without a matrix; and the critical buckling load decreases first and then increases with the increase in the number of layers for multi-walled CNTs with a fixed outermost radius that are embedded in a matrix. That implies that there is a worst number of layers for a multi-walled CNT that is embedded in a matrix at which the critical buckling load is the lowest, and that this number depends on the matrix parameters. The work described in this paper was partially supported by a grant from the City University of Hong Kong (project no. 7200037).
References Allen, H. G. & Bulson, P. S. 1980 Background to buckling, ch. 7. UK: McGraw-Hill. Bower, C., Rosen, R., Jin, L., Han, J. & Zhou, O. 1999 Appl. Phys. Lett. 74, 3317–3319. (doi:10. 1063/1.123330.) Frankland, S. J. V. & Brenner, D. W. 2000 Mal. Rec. Symp. Proc. 593, 199–204. Girifalco, L. A. & Lad, R. A. 1956 J. Chem. Phys. 25, 693–697. (doi:10.1063/1.1743030.) Govindjee, S. & Sackman, J. L. 1999 Solid State Commun. 110, 227–230. (doi:10.1016/S00381098(98)00626-7.) Harik, V. M. 2002 Comp. Mater. Sci. 24, 328–342. (doi:10.1016/S0927-0256(01)00255-5.) He, X. Q., Kitipornchai, S. & Liew, K. M. 2005 J. Mech. Phys. Solid 53, 303–326. (doi:10.1016/ j.jmps.2004.08.003.) Hernandez, E., Goze, C., Bernier, P. & Rubio, A. 1998 Phys. Rev. Lett. 80, 4502–4505. (doi:10. 1103/PhysRevLett.80.4502.) Proc. R. Soc. A (2005)
Downloaded from rspa.royalsocietypublishing.org on 4 November 2009
Buckling characteristics
3805
Iijima, S. 1991 Nature 354, 56–58. (doi:10.1038/354056a0.) Jin, L., Bower, C. & Zhou, O. 1998 Appl. Phys. Lett. 73, 1197–1199. (doi:10.1063/1.122125.) Lau, K. T., Chipara, M., Ling, H. Y. & Hui, D. 2004 Composites: Part B 35, 95–101. (doi:10.1016/j. compositesb.2003.08.008.) Li, C. & Chou, T. W. 2003a Int. J. Solids Struct. 40, 2487–2499. (doi:10.1016/S00207683(03)00056-8.) Li, C. & Chou, T. W. 2003b Compos. Sci. Technol. 63, 1517–1524. (doi:10.1016/S02663538(03)00072-1.) Liew, K. M., Wong, C. H., He, X. Q., Tan, M. J. & Meguid, S. A. 2004a Phys. Rev. B 69, 115 429. (doi:10.1103/PhysRevB.69.115429.) Liew, K. M., He, X. Q. & Wong, C. H. 2004b Acta Mater. 52, 2521–2527. (doi:10.1016/j.actamat. 2004.01.043.) Lourie, O., Cox, D. M. & Wagner, H. D. 1998 Phys. Rev. Lett. 81, 1638. (doi:10.1103/ PhysRevLett.81.1638.) Odegard, G. M., Gates, T. S., Nicholson, L. M. & Wise, K. E. 2002 Compos. Sci. Technol. 62, 1869–1880. (doi:10.1016/S0266-3538(02)00113-6.) Pasternak, P. L. 1954. On a new method of analysis of an elastic foundation by mean of two foundation constants. Gps. Izd. Lit. Po Strait I Arkh., Moscow. Ru, C. Q. 2000 J. Appl. Phys. 87, 7227–7231. (doi:10.1063/1.372973.) Ru, C. Q. 2001a J. Mech. Phys. Solids 49, 1265–1279. (doi:10.1016/S0022-5096(00)00079-X.) Ru, C. Q. 2001b J. Appl. Phys. 89, 3426–3433. (doi:10.1063/1.1347956.) Sanchez-Portal, D., Artacho, E. & Soler, J. M. 1999 Phys. Rev. B 59, 12 678–12 688. (doi:10.1103/ PhysRevB.59.12678.) Schadler, L. S., Giannaris, S. C. & Ajayan, P. M. 1998 Appl. Phys. Lett. 73, 3842–3844. (doi:10. 1063/1.122911.) Srivastava, D., Menon, M. & Cho, K. 1999 Phys. Rev. Lett. 83, 2973. (doi:10.1103/PhysRevLett. 83.2973.) Timoshenko, S. P. & Gere, J. M. 1961 Theory of elastic stability. New York: McGraw-Hill. Wagner, H. D., Lourie, O., Feldman, Y. & Tenne, R. 1998 Appl. Phys. Lett. 72, 188–190. (doi:10. 1063/1.120680.) Wang, C. Y., Ru, C. Q. & Mioduchowski, A. 2003 Int. J. Solids Struct. 40, 3893–3911. (doi:10. 1016/S0020-7683(03)00213-0.) Winkler, E. 1867 Die Lehre von der Elasticitaet und Festigkeit, Prag, Dominicus. Yakobson, B. I., Brabec, C. J. & Bernholc, J. 1996 Phys. Rev. Lett. 76, 2511–2514. (doi:10.1103/ PhysRevLett.76.2511.)
Proc. R. Soc. A (2005)
Related Documents
Royal Society 2005 - 3785
June 2020 5
Royal Society 2005 3059
June 2020 0
Royal Society 2006 3555
June 2020 3
Nasa 163712main Griffin Royal Society
October 2019 10
Umi-umd-3785
May 2020 14