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A non-linear analysis of the bending modulus of carbon nanotubes with rippling deformations
Composite Structures 69 (2005) 315–321 www.elsevier.com/locate/compstruct
A non-linear analysis of the bending modulus of carbon nanotubes with rippling deformations X. Wang
a,*
, X.Y. Wang a, Jun Xiao
b
a
b
Department of Engineering Mechanics, The School of Civil Engineering and Mechanics, Shanghai Jiaotong University, Dong Chuan Road, Shanghai 200240, PR China College of Materials Science and Technology, Nanjing University of Aeronautics and Aastronautics, Nanjing 210016, PR China Available online 10 August 2004
Abstract Recently, some experimental techniques involving bending of cantilevered nanotubes have been used to deduce their elastic bending modulus. Using these methods it was found that the value of elastic bending modulus decreases dramatically from about 1 TPa for D less than 10 nm to 100 GPa for D larger than 20 nm. This discovery defines usual belief that the bending modulus is a materialÕs intrinsic property, independent of the size of the nanotubes. This paper concerned with the non-linear moment–curvature relationship during bending of nanotubes, and the effect of ripple formation on the bending modulus. By using an advanced finite element analysis package, ABAQUS, a non-linear bending moment–curvature relationship of carbon nanotubes, which is the appearance of a rippling mode on the inner arc of the bent nanotubes, is simulated from the three-dimensional orthotropic theory of finite elasticity deformation. Utilizing the non-linear bending moment–curvature relationship, and a non-linear vibration analysis method, we capture that the rippling deformation can indeed result in effective bending modulus of carbon nanotubes decrease substantially with increasing diameter. The result carried out may be used to explain some experiment phenomena, and also guides further experiment investigating the bending behaviors of carbon nanotubes. 2004 Elsevier Ltd. All rights reserved. Keywords: Carbon nanotubes: Effective bending modulus: Non-linear bending deformation: Ripple deformation: Non-linear vibration
1. Introduction Sumio Iijima [1] first published his discovery of carbon nanotubes (CNTs) in 1991. Nanotubes have extraordinary electrical and mechanical properties in Ref. [2]. Characterization and determination of their mechanical properties are important and have been a subject of intensive research in Refs. [3–9]. The high elastic modulus (higher than 1 TPa) of the graphite sheets, of which the nanotubes are made, means that the nanotubes may be stiffer in tension and compression, and stronger than any other known material, with ben-
*
Corresponding author. Fax: +86 21 6293 2045. E-mail address: [email protected] (X. Wang).
0263-8223/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2004.07.009
eficial consequences for their application in composite bulk materials and as individual elements of nanometer scale devices. They are also remarkably flexible in bending and can undergo large elastic deformation (up to 20%) without breaking in Ref. [10]. With these distinguishing properties, nanotubes exhibit new phenomena, one of which is the formation of ripples when nanotubes are under bending in Refs. [10–14]. Wong et al. [11] pinned one end of MWNTs to molybdenum disulphide surfaces and used an atomic force microscopy (AFM) to measure the bending force versus deflection along the tube length. From the theoretical relationship between the initial bending stiffness of cantilevered nanotubes and their elastic modulus and beam dimensions, which is based on a linear (small deflection) elastic beam theory, the value of the elastic
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X. Wang et al. / Composite Structures 69 (2005) 315–321
modulus for a 32.9 nm diameter MWNT was determined as 1.26 TPa. The second method used by Poncharal in Ref. [12] is to measure the fundamental resonances included by an electric field in TEM and then calculate effective bending modulus Eeff by using the relation between resonance and modulus resulting from the linear vibration analysis of cantilevered beams. However, the result of measurement reported that the calculated effective bending modulus Eeff was found to decrease sharply, from about 1–0.1 TPa with the diameter D of carbon nanotubes increasing from 8 to 40 nm. Treacy et al. [13] obtained an effective bending modulus Eeff of 1.8 TPa (average value) for multi-walled nanotubes, and Krishnan et al. [14] obtained an effective bending modulus Eb of 1.25 (0.35/+0.45) TPa for single-walled nanotubes. The theory on which all the above measuring methods were based is true only when: (1) the material is linear elastic and isotropic; (2) the deflection is small in comparison to the tubeÕs diameter of a cantilever and (3) the bending deformation is uniform along the length without ripples mode due to bucking. Clearly, these conditions were not completely satisfied. For example, the molecular structures and wrapping direction suggest that the material would be anisotropic. On the other hand, because several techniques of determining the elastic bending modulus are based on a measurement of the bending response of nanotubes, where ripples are often accompanied, a correct understanding of the mechanics of ripple formation is vital in obtaining an accurate value of the elastic modulus. However, itÕs a pity that all of the above measurements are indirect because the small dimension of nanotubes has made it extremely difficult to exactly measure their mechanical properties directly in Ref. [15]. Thus it is also valuable to seek a numerical simulation method of obtaining the mechanical properties of carbon nanotubes. A numerical simulation of materials of nano-scale size generally use molecular dynamics (MD) method to consider the nano-effects and to give an accurate solution. However, MD simulation is limited to systems with a maximum atom number of about 109 by the scale and cost of computation. So only single-walled nanotubes with small deflection can be simulated using MD method. Existing work shows continuum mechanics results agree with MD simulation results for single and double wall tubes. For example, Yakobson et al. [16] compared the results of atomistic modeling for axially compressed buckling of single-walled nanotubes with a simple continuum shell model, and found that all the buckling patterns displayed by MD simulation can also be predicted by the continuum shell model. So a continuum model may be adopted. The authors [17] constructed a two-dimensional plain strain FEM model of a beam with rectangular across section under pure bending and obtained a bending
moment–curvature relationship. But this model did not take into account the dimensional Poisson effect. Moreover, the material orientation of the model varies from carbon nanotubes, which are axially symmetric. A round cross sectional beam model of three dimensions have been constructed, and a bending moment–curvature relationship, which is more likely to be the truly bending constitution of MWNTs have been obtained by using an advanced finite element analysis package, ABAQUS. Thus the effective bending modulus of carbon nanotubes is obtained by using non-linear vibration theory. The result carried out in the paper gives some explanations for the measured result in Ref. [12], indicates that a rippling deformation can indeed result in the effective bending modulus of carbon nanotubes decreasing substantially with increasing diameter, and that the result from classical linear vibration theory may be invalid in such measurements. 2. Non-linear vibration analysis Fig. 1 shows an experiment phenomenon measuring the fundamental resonance included by an electric field in TEM and effective bending modulus Eeff calculated from the linear vibration analysis of cantilevered beams by Poncharal in Ref. [12]. Based on the beam theory, the bending vibration equation with a harmonic excitation force F(x, t) can be expressed as _ tÞ þ qA€ M 00 ðx; tÞ þ 2lwðx; wðx; tÞ ¼ F ðx; tÞ
ð1aÞ
~ F ðx; tÞ ¼ GðxÞ cosðxtÞ
ð1bÞ
where M(x, t) expresses the bending moment, w(x, t) the beam deflection function, l the damping coefficient, F(x, t) the excitation load measured per unit length, _ tÞ the partial derivatives ow(x, t)/ and w 0 (x, t) and wðx; ox and ow(x, t)/ot, respectively. The corresponding boundary conditions for a cantilevered beam are expressed as wð0; tÞ ¼ w0 ð0; tÞ ¼ 0
ð2aÞ
w00 ðL; tÞ ¼ w000 ðL; tÞ ¼ 0
ð2bÞ
To examine the effect of the rippling formation on bending modulus of carbon nanotubes, we give a nonlinear vibration analysis that takes into account the relatively large deformation corresponding to rippling. Taking into account the relatively and locally large deformation appearing in the rippling formation of the nanotubes, it should be noticed the effect of the higher order terms in the constitutive relation on the resonance frequency relation to the elastic bending modulus. Because the bending curvature j(x, t) is a function of the beam deflection w(x, t), it is necessary to get a non-linear relation between M(x, t) and j(x, t) from the full three-
X. Wang et al. / Composite Structures 69 (2005) 315–321
317
Fig. 1. Carbon nanotubeÕs resonance shown in (a) and (b) under compatible frequency shown in (c), and the relationship of bending modulus and diameter shown in (d) [10].
dimensional theory of finite deformation. Thus, substituting the non-linear relation between M(x, t) and j(x, t) into Eq. (1) yields a non-linear differential equation governing the deflection function w(x, t). The structure of carbon nanotubes can be considered as arising from the folding of one or more layers of layers of graphite to form a cylinder composed of carbon hexagons shown in Fig. 2 from Ref. [18], so the stressstrain relationship of graphite can be used to express the constitution of carbon nanotubes with an appropriate orientation. The experiments up to now do indicate that the axial elasticity modulus of carbon nanotubes is close to the elasticity modulus along the basal plane of graphite in Ref. [14]. To better express the materialÕs main direction of carbon nanotubes, the constitution in a cylindrical coordinate system is represented as
8 9 2 36:5 rr > > > > > > 6 > > > rh > > > 6 15 > > > 6 = 6 15 z ¼ 109 6 6 0 > > > shz > 6 > > > > 6 > > > srz > 4 0 > > > > : ; 0 srh
15
15
0
0
1060
180
0
0
180 0
1060 0 0 220
0 0
0 0
0 0
0 0 2:25 0
Fig. 2. Nanotube geometries: armchair (top), zigzag (middle), chiral (bottom) [16].
38 9 er > > > > > > 7 > > 0 7> eh > > > > > > 7> < 7 0 ez = 7 ðPaÞ 0 7 chz > > > 7> > > 7> > > > 0 5> crz > > > > ; : > 2:25 crh 0
Using the same ratio of length to diameter of nanotubes (L/D = 500) as that of experimental samples in Ref. [12] to performing the finite element analysis is extremely challenging because the elements in finite element mesh must have dimensions significantly smaller than the spatial period of rippling. However, theoretically, under the pure bending condition the bending mo-
ð3Þ
ment on each cross section is the same and the beamÕs neutral axis has a constant curvature. Thus, a beam with smaller aspect ratio (L/D500) can be used to simulate the rippling deformation of the beam under the pure bending condition. It should be pointed out that in the FEM numerical simulation the exact application of the boundary condition of the beam subjected to pure
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X. Wang et al. / Composite Structures 69 (2005) 315–321
bending is very difficult, due to the rotation and the changing unknown stress distribution of the end sections. An approach is to construct a four-point bending beam. A commercial finite element code ABAQUS was used to simulate the rippling formation of bent nanotubes. It can automatically choose proper loading increments and converge criteria in a non-linear analysis. Fig. 3(a) shows the undeformed mesh of pure bending portion of a solid cylindrical model for carbon nanotubes, Fig. 3(b) shows the configuration of a simulated this carbon nanotubes bent with the rippling deformation, and Fig. 3(c) shows overall view of a bending nanotubes with ripples by means of a measuring. It is plotted in Fig. 4 the bending moment M versus the bending curvature j at each loading step for the four-point bending solid cylinder with the length-todiameter ratios L/D = 10, and 20 from FEM simulating result. According to the beam bending theory, we know that M(x, t) should be an odd function j(x, t). Using a polynomial up to the ninth order, the discrete points in Fig. 4 are fitted as
Fig. 4. Two results derived from different ratios to L/D, where represents L/D = 10, d represents L/D = 20.
Mðx; tÞ ¼ EIjð1 a3 D2 j2 þ a5 D4 j4 a7 D6 j6 þ a9 D8 j8 Þ ð4Þ 3
where j = j(x, t), a3 = 1.755 · 10 , a5 = 2.0122 · 106, a7 = 1.115 · 109 and a9 = 2.266 · 1011. The two order of partial derivatives to x variable in Eq. (4), gives M 00 ðx; tÞ ¼ EI½j00 ð1 3a3 D2 j2 þ 5a5 D4 j4 7a7 D6 j6 2
þ 9a9 D8 j8 Þ þ ðj0 Þ ð6a3 D2 j þ 20a5 D4 j3 42a7 D6 j5 þ 72a9 D8 j7 Þ
ð5Þ Based on the large deflection deformation, the relation between the bending curvature j(x, t) and the beam deflection w(x, t) can be expressed as jðx; tÞ ¼
w00 ðx; tÞ
½1 þ w0 ðx; tÞ2 3=2 3 15 2 4 ¼ w00 1 ðw0 Þ þ ðw0 Þ 2 8 2
ffi w00 ðx; tÞ½1 rðw0 Þ
ð6Þ
where r = 1.5. Substituting Eq. (6) into Eq. (5), yields M 00 ðx; tÞ ¼ EIfw0000 3a3 D2 ½2w00 ðw000 Þ2 þ ðw00 Þ2 w0000
r½2ðw00 Þ3 þ 6w0 w00 w000 þ ðw0 Þ2 w0000 g
ð7Þ
Substituting Eq. (7) into Eq. ((1), we obtain a non-linear vibration equation which is expressed as EIw0000 þ 2lw_ þ qA€ w ¼ EIN ðwÞ þ F ðx; tÞ
ð8aÞ
where N ðwÞ ¼ 3a3 D2 ½2w00 ðw000 Þ2 þ ðw00 Þ2 w0000 þ r½2ðw00 Þ3 Fig. 3. FEM model of carbon nanotubes and the rippling configuration. (a) shows the undeformed meshes of the purely bending part, (b) shows the rippling deformed shape of the bent carbon nanotubes from FEM simulation, and (c) shows TEM image of a bent nanotube with ripples experimentally obtained [10].
þ 6w0 w00 w000 þ ðw0 Þ2 w0000
ð8bÞ
Making all the variables in Eq. (8) dimensionless pffiffiffiffiffiffiffiffiffiffiffiffiffi by using the characteristic length L, time L2 qA=EI and force EI/L3, gives
X. Wang et al. / Composite Structures 69 (2005) 315–321
€ _ þ w ¼ N þ F ðx ; t Þ 0000 þ 2 lw w
ð9aÞ
~ t Þ F ðx ; t Þ ¼ Gðx Þ cosðx
ð9bÞ
b2 ¼
Z
1 3
pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ w=L; x ¼ x=L; t ¼ t EI=qA=L2 ; w pffiffiffiffiffiffiffiffiffiffiffi 3 ¼ lL EIqA; N ¼ N =L ; G ¼ GL3 =EI l
0
¼ 20:1939
ð10Þ
2
2
2
~ tÞ u þ 2e vu_ þ € u ¼ e N u þ e g cosðx
000 2
00
ð11Þ
ð12aÞ
00 2 0000
N u ¼ 2a3 ðD=LÞ ½2u ðu Þ þ ðu Þ u
3
2
þ r½2ðu00 Þ þ 6u0 u0000 u000 þ ðu0 Þ u0000
ð12bÞ
In the above formulae, u(x*, t*) and g(x*) are, respectively, expanded as 1 1 X X qn ðt Þ/n ðx Þ; gðx Þ ¼ gn /n ðx Þ ð13Þ uðx ; t Þ ¼ n¼1
n¼1
where /n for n = 1, 2, 3, . . . represent the normalized mode functions of the cantilevered beam from the linear vibration analysis, which satisfies the following formula. Z 1 /i ðx Þ/j ðx Þdx ¼ 0 ði 6¼ jÞ 0 ð14Þ Z 1 /i ðx Þ/j ðx Þdx ¼ 1
ði ¼ jÞ
0
Substituting Eq. (13) into Eq. (12) and utilizing Eq. (14), we have 2 ~ t Þ € q1 þ 2e2 vq_ 1 þ ðx1 Þ q1 ¼ e2 aq31 þ e2 g1 cosðx
where Z b1 ¼ 0
1 2
2
00
00 00 /1 ½2/001 ð/000 1 Þ þ ð/1 Þ /1 dx ¼ 119:6
ð17Þ
ð793:533D=LÞ þ 30:3555
where 2
3a3 ðD=LÞ2 119:6 þ r 20:1939 2
Poncharal [12], based on linear analysis of cantilevered beam, given the first order of inherence frequency corresponding to the fundamental mode of vibration x1 ffi 3:516. It is expected that the non-linearity may ~ to deviate slightly from cause the resonance frequency x x1 in Ref. [12]. By means of the perturbation method of ~ for multi-scales in Ref. [19], the resonance frequency x the non-linear oscillations can be determined. Introducing a small perturbation parameter e expands w into w = eu. From Eqs. (8) and (10), it is seen that non-linear term N is of the third order in w. To make the exciting _ be both of the same force F and the damping force 2lw ¼ e2 v and order as the non-linear term N , we give l 3 Gðx Þ ¼ e gðxA Þ, so that the non-linear oscillation equation (9) becomes 0000
ð16bÞ 2
The boundary conditions (2) is rewritten as 00 ð1; t Þ ¼ w 000 ð1; t Þ ¼ 0 w
00
a ¼ 3a3 ðD=LÞ b1 þ rb2
ð0; t Þ ¼ w 0 ð0; t Þ ¼ 0; w
2
0 00 000 00 00 /1 ½2/001 ð/000 1 Þ þ 6/1 /1 /1 þ ð/1 Þ /1 dx
where
319
ð15Þ
ð16aÞ
It is seen that Eq. (15) is a standard third-order nonlinear oscillation equation [17] which is yielded by the parameter a. Assume that the fundamental resonance frequency in the non-linearity analysis is ~ ¼ x1 re1 x
ð18Þ
the solution for the third-order non-linear oscillation equation (15), and dimensionless excitation force can be, respectively, expressed as q1 ðt ; eÞ ¼ q10 ðT 0 ; T 1 Þ þ e1 q11 ðT 0 ; T 1 Þ þ . . . :
ð19aÞ
~ t Þ ¼ e21 g1 cosðx1 T 0 rT 1 Þ e2 g1 cosðx
ð19bÞ
Substituting Eq. (19) into Eq. (15), and comparing coefficients of the identical power of e1, we have 2
D20 q11 þ ðx1 Þ q11 ¼ 2D0 D1 q10 2vD0 q10 þ aq310 þ g1 cosðx1 T 0 rT 1 Þ 2
D20 q10 þ ðx1 Þ q10 ¼ 0
ð20aÞ ð20bÞ
where Dn ¼ oTo n ðn ¼ 0; 1Þ. The solution of Eq. (20) can be easily obtained. Substituting the solution of Eq. (20) into Eq. (19a), the solution for the third-order non-linear oscillation equation (15) is given by q1 ¼ ap cosðx1 t þ bÞ þ OðeÞ
ð21Þ
where ap represents the maximum vibration amplitude, and is expressed as g ap ¼ 1 ð22Þ 2x1 v and b ¼ r rT 1
ð23Þ
In Eq. (23), r represents a parameter variable, and is written as 2 3a 2 3a g1 3ag21 a ¼ ¼ ð24Þ r¼ 2 8x1 p 8x1 2x1 v 32x3 1 v Thus, the first order of resonance frequency with the physical dimension in the non-linearity analysis is given by sffiffiffiffiffiffi ~ 1 EI x ~¼ 2 x ð25Þ qA L
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X. Wang et al. / Composite Structures 69 (2005) 315–321
Substituting a measured resonance frequency x1 based ~ in Eq. (25), on linear cantilevered beam theory for x yields sffiffiffiffiffiffiffiffiffi x1 Eeff I ~¼ 2 ð26Þ x qA L In the above formula, Eeff represents the effective bending modulus of carbon nanotubes with rippling deformation. From Eqs. (25) and (26), we have 2 2 2 ~ ~ Eeff x1 re2 x x ¼ ¼ ¼ E x1 x1 x1 !2 !2 3ag1 e2 3G2 1 ¼ 1 ¼ 1 a 32ðx1 Þ4 l2 32ðx1 Þ4 l2 " 2 #2 G1 4 ¼ 1 6:137 10 a l ( 2 )2 G 2 ¼ 1 ½3876:44 ðD=LÞ þ 0:0186 1 l
Fig. 6. The corresponding relation between Eeff/E and D/L when K 1 =l ¼ 1.
ð27Þ Eq. (27) expresses that the effective bending modulus Eeff decreases with increasing ratio (D/L), and with increasing excitation load-to-damping ratio (G1 =l ), which is similar to the measured results in Ref. [10] shown in Fig. 1(d).
Fig. 7. The corresponding relation between Eeff/E and D/L when K 1 =l ¼ 2.
3. Results and conclusion Figs. 5–9 show the changes rules of the effective bending modulus-to-YoungÕs modulus ratio Eeff/E with increasing the diameter-to-height ratio (D/L), and different excitation load-to-damping ratio G1 =l . From Figs. 5–9, it is seen that when the excitation load-to-damping ratio G1 =l is a given value the effective bending modulus ratio Eeff/E can decreases as the diameter-to-height ratio (D/L) increases. Comparing Figs. 5–9, it is seen Fig. 8. The corresponding relation between Eeff/E and D/L when K 1 =l ¼ 3.
Fig. 5. The corresponding relation between Eeff/E and D/L when K 1 =l ¼ 0:5.
that the larger the excitation load-to-damping ratio G1 =l is, the more sharply the effective bending modulus-to-YoungÕs modulus ratio Eeff/E drops. But, in Refs. [12], author does not present the discussions of the effect of the excitation load-to-damping ratio G1 =l on the effective bending modulus of carbon nanotubes. It is seen that the non-linear term in the third-order non-linear linear equation (15) is yielded by the parameter a that is based on the effect of the higher order term––rw00 (w 0 )2 in the expression (6) of curvature j. When the parameter a equals zero, the third-order
X. Wang et al. / Composite Structures 69 (2005) 315–321
321
mechanical properties directly in Refs. [10–14]. The present work presents a numerical simulation model, based on non-linear analysis for obtaining the bending modulus of carbon nanotubes and provides a valuable guide for further experimental studies in this area.
References
Fig. 9. The corresponding relation between Eeff/E and D/L when K 1 =l ¼ 4.
non-linear oscillation equation (15) will become a standard linear vibration equation in Ref. [12]. Assuming that a equals zero, the condition using linear vibration equation to obtain a measured resonance frequency x1 can be expressed as beams of diameter-to-length (D/ L) < 144. Carbon nanotubes of diameter-to-length (D/ L) measured in Ref. [10] approximately equals 500. Evidently, this does not satisfy the condition using a linear vibration equation. This condition, (D/L) < 144, suggests that one should be particularly cautious in using the results of the linear vibration theory to obtain a measured resonance frequency x1 . This paper obtains the coefficients (a3, a5, a7, a9) describing the non-linear relation between bending moment M(x) and curvature j in Eq. (4) by using a polynomial up to the ninth order to fit the discrete points in Fig. 4. This is different from those in Ref. [15]. The above different phenomenon is mainly because the discrete points describing the non-linear relationship between bending moment M(x) and curvature j shown in Fig. 4 are obtained by means of a three-dimensional finite element for solid cylinders. The bending moment–curvature relationship in Ref. [17] is obtained by using a two dimensional plain strain FEM model that did not take into account the three-dimensional Poisson effect. On the other hand, because the material orientation of the model varies from carbon nanotubes, which are axially symmetric, a round cross sectional cylinder model of three-dimensions constructed, and the corresponding bending moment–curvature relationship are more likely to simulate the truly bending constitution of carbon nanotubes. However, the small dimension of carbon nanotubes makes it extremely difficult to exactly measure their
[1] Iijima S. Helical microtubes of graphitic carbon. Nature 1991;354: 56–8. [2] Dresselhaus MS, Dresselhaus G, Avouris PH. Carbon nanotubes. Topics Appl Phys 2001;80:287–332. [3] Lau KT, Hui D. The revolutionary creation of new advanced materials-carbon nanotube composites. Composites: Part B 2002;33:263–77. [4] Lau KT. Interfacial bonding characteristics of nanotube/polymer composites. Chem Phys Lett 2003;370:399–405. [5] Ru CQ. Effective bending stiffness of carbon nanotubes. Phys Rev B 2000;62:9973–6. [6] Zhan CT, Zhong C, Yang O. Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective YoungÕs moduli dependent on layer number. Phys Rev B 2002; 65:233407-1-4. [7] Falvo M, Taylor R, Helser A, Chi V, Brooks F, Washburn S, et al. nanometer-scale rolling and sliding of carbon nanotubes. Nature 1999;397:236–8. [8] Kuzumaki T, Hayashi T, Ichinose H, Miyazawa K, Ito K, Ishida Y. In-situ observed deformation of carbon nanotubes. Philos Mag A 1998;77:1461–9. [9] Yu MF, Lourie O, Dyer M, Moloni K, Kelly TF, Ruoff RS. Strength and breaking mechanism of multi-walled carbon nanotubes under tensile load. Science 2000;287:637–9. [10] Falvo MR, Clary GJ, Taylor RM, Chi V, Brooks Jr FP, Washburn S, Superfine R. Bending and buckling of carbon nanotubes under larger strain. Nature 1997;389:582–4. [11] Wong EW, Sheehan PE, Liebert CM. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes and carbon nanotubes. Science 1997;277:1971–3. [12] Poncharal P, Wang ZL, Ugarte D, de Heer WA. Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 1999;283:1513–6. [13] Treacy MMJ, Ebbesen TW, Gibson JM. Exceptionally high YoungÕs modulus observed for individual carbon nanotubes. Nature 1996;38:678. [14] Krishnan A, Dujardin E, Ebbesen TW, et al. YoungÕs modulus of single-walled nanotubes.. Phys Rev B 1998;58:14013–9. [15] Ebbesen TW. Phys Today 1996;49(6):26. [16] Yakobson BI, Brabec CJ, Berholc J. Nanomechanics of carbon tubes: instabilities beyond the linear response. Phys Rev Lett 1996;76:2511–4. [17] Liu JZ, Zheng QS, Jiang Q. Effect of a rippling mode on resonances of carbon nanotubes. Phys Rev Lett 2001;86:4843–6. [18] Rao CNR, Satishkumar BC, Govindaraj A, Nath M. Chemphyschem 2001;2:78–105. [19] Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: Wiley; 1997.
A non-linear analysis of the bending modulus of carbon nanotubes with rippling deformations X. Wang
a,*
, X.Y. Wang a, Jun Xiao
b
a
b
Department of Engineering Mechanics, The School of Civil Engineering and Mechanics, Shanghai Jiaotong University, Dong Chuan Road, Shanghai 200240, PR China College of Materials Science and Technology, Nanjing University of Aeronautics and Aastronautics, Nanjing 210016, PR China Available online 10 August 2004
Abstract Recently, some experimental techniques involving bending of cantilevered nanotubes have been used to deduce their elastic bending modulus. Using these methods it was found that the value of elastic bending modulus decreases dramatically from about 1 TPa for D less than 10 nm to 100 GPa for D larger than 20 nm. This discovery defines usual belief that the bending modulus is a materialÕs intrinsic property, independent of the size of the nanotubes. This paper concerned with the non-linear moment–curvature relationship during bending of nanotubes, and the effect of ripple formation on the bending modulus. By using an advanced finite element analysis package, ABAQUS, a non-linear bending moment–curvature relationship of carbon nanotubes, which is the appearance of a rippling mode on the inner arc of the bent nanotubes, is simulated from the three-dimensional orthotropic theory of finite elasticity deformation. Utilizing the non-linear bending moment–curvature relationship, and a non-linear vibration analysis method, we capture that the rippling deformation can indeed result in effective bending modulus of carbon nanotubes decrease substantially with increasing diameter. The result carried out may be used to explain some experiment phenomena, and also guides further experiment investigating the bending behaviors of carbon nanotubes. 2004 Elsevier Ltd. All rights reserved. Keywords: Carbon nanotubes: Effective bending modulus: Non-linear bending deformation: Ripple deformation: Non-linear vibration
1. Introduction Sumio Iijima [1] first published his discovery of carbon nanotubes (CNTs) in 1991. Nanotubes have extraordinary electrical and mechanical properties in Ref. [2]. Characterization and determination of their mechanical properties are important and have been a subject of intensive research in Refs. [3–9]. The high elastic modulus (higher than 1 TPa) of the graphite sheets, of which the nanotubes are made, means that the nanotubes may be stiffer in tension and compression, and stronger than any other known material, with ben-
*
Corresponding author. Fax: +86 21 6293 2045. E-mail address: [email protected] (X. Wang).
0263-8223/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2004.07.009
eficial consequences for their application in composite bulk materials and as individual elements of nanometer scale devices. They are also remarkably flexible in bending and can undergo large elastic deformation (up to 20%) without breaking in Ref. [10]. With these distinguishing properties, nanotubes exhibit new phenomena, one of which is the formation of ripples when nanotubes are under bending in Refs. [10–14]. Wong et al. [11] pinned one end of MWNTs to molybdenum disulphide surfaces and used an atomic force microscopy (AFM) to measure the bending force versus deflection along the tube length. From the theoretical relationship between the initial bending stiffness of cantilevered nanotubes and their elastic modulus and beam dimensions, which is based on a linear (small deflection) elastic beam theory, the value of the elastic
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X. Wang et al. / Composite Structures 69 (2005) 315–321
modulus for a 32.9 nm diameter MWNT was determined as 1.26 TPa. The second method used by Poncharal in Ref. [12] is to measure the fundamental resonances included by an electric field in TEM and then calculate effective bending modulus Eeff by using the relation between resonance and modulus resulting from the linear vibration analysis of cantilevered beams. However, the result of measurement reported that the calculated effective bending modulus Eeff was found to decrease sharply, from about 1–0.1 TPa with the diameter D of carbon nanotubes increasing from 8 to 40 nm. Treacy et al. [13] obtained an effective bending modulus Eeff of 1.8 TPa (average value) for multi-walled nanotubes, and Krishnan et al. [14] obtained an effective bending modulus Eb of 1.25 (0.35/+0.45) TPa for single-walled nanotubes. The theory on which all the above measuring methods were based is true only when: (1) the material is linear elastic and isotropic; (2) the deflection is small in comparison to the tubeÕs diameter of a cantilever and (3) the bending deformation is uniform along the length without ripples mode due to bucking. Clearly, these conditions were not completely satisfied. For example, the molecular structures and wrapping direction suggest that the material would be anisotropic. On the other hand, because several techniques of determining the elastic bending modulus are based on a measurement of the bending response of nanotubes, where ripples are often accompanied, a correct understanding of the mechanics of ripple formation is vital in obtaining an accurate value of the elastic modulus. However, itÕs a pity that all of the above measurements are indirect because the small dimension of nanotubes has made it extremely difficult to exactly measure their mechanical properties directly in Ref. [15]. Thus it is also valuable to seek a numerical simulation method of obtaining the mechanical properties of carbon nanotubes. A numerical simulation of materials of nano-scale size generally use molecular dynamics (MD) method to consider the nano-effects and to give an accurate solution. However, MD simulation is limited to systems with a maximum atom number of about 109 by the scale and cost of computation. So only single-walled nanotubes with small deflection can be simulated using MD method. Existing work shows continuum mechanics results agree with MD simulation results for single and double wall tubes. For example, Yakobson et al. [16] compared the results of atomistic modeling for axially compressed buckling of single-walled nanotubes with a simple continuum shell model, and found that all the buckling patterns displayed by MD simulation can also be predicted by the continuum shell model. So a continuum model may be adopted. The authors [17] constructed a two-dimensional plain strain FEM model of a beam with rectangular across section under pure bending and obtained a bending
moment–curvature relationship. But this model did not take into account the dimensional Poisson effect. Moreover, the material orientation of the model varies from carbon nanotubes, which are axially symmetric. A round cross sectional beam model of three dimensions have been constructed, and a bending moment–curvature relationship, which is more likely to be the truly bending constitution of MWNTs have been obtained by using an advanced finite element analysis package, ABAQUS. Thus the effective bending modulus of carbon nanotubes is obtained by using non-linear vibration theory. The result carried out in the paper gives some explanations for the measured result in Ref. [12], indicates that a rippling deformation can indeed result in the effective bending modulus of carbon nanotubes decreasing substantially with increasing diameter, and that the result from classical linear vibration theory may be invalid in such measurements. 2. Non-linear vibration analysis Fig. 1 shows an experiment phenomenon measuring the fundamental resonance included by an electric field in TEM and effective bending modulus Eeff calculated from the linear vibration analysis of cantilevered beams by Poncharal in Ref. [12]. Based on the beam theory, the bending vibration equation with a harmonic excitation force F(x, t) can be expressed as _ tÞ þ qA€ M 00 ðx; tÞ þ 2lwðx; wðx; tÞ ¼ F ðx; tÞ
ð1aÞ
~ F ðx; tÞ ¼ GðxÞ cosðxtÞ
ð1bÞ
where M(x, t) expresses the bending moment, w(x, t) the beam deflection function, l the damping coefficient, F(x, t) the excitation load measured per unit length, _ tÞ the partial derivatives ow(x, t)/ and w 0 (x, t) and wðx; ox and ow(x, t)/ot, respectively. The corresponding boundary conditions for a cantilevered beam are expressed as wð0; tÞ ¼ w0 ð0; tÞ ¼ 0
ð2aÞ
w00 ðL; tÞ ¼ w000 ðL; tÞ ¼ 0
ð2bÞ
To examine the effect of the rippling formation on bending modulus of carbon nanotubes, we give a nonlinear vibration analysis that takes into account the relatively large deformation corresponding to rippling. Taking into account the relatively and locally large deformation appearing in the rippling formation of the nanotubes, it should be noticed the effect of the higher order terms in the constitutive relation on the resonance frequency relation to the elastic bending modulus. Because the bending curvature j(x, t) is a function of the beam deflection w(x, t), it is necessary to get a non-linear relation between M(x, t) and j(x, t) from the full three-
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317
Fig. 1. Carbon nanotubeÕs resonance shown in (a) and (b) under compatible frequency shown in (c), and the relationship of bending modulus and diameter shown in (d) [10].
dimensional theory of finite deformation. Thus, substituting the non-linear relation between M(x, t) and j(x, t) into Eq. (1) yields a non-linear differential equation governing the deflection function w(x, t). The structure of carbon nanotubes can be considered as arising from the folding of one or more layers of layers of graphite to form a cylinder composed of carbon hexagons shown in Fig. 2 from Ref. [18], so the stressstrain relationship of graphite can be used to express the constitution of carbon nanotubes with an appropriate orientation. The experiments up to now do indicate that the axial elasticity modulus of carbon nanotubes is close to the elasticity modulus along the basal plane of graphite in Ref. [14]. To better express the materialÕs main direction of carbon nanotubes, the constitution in a cylindrical coordinate system is represented as
8 9 2 36:5 rr > > > > > > 6 > > > rh > > > 6 15 > > > 6 =
15
15
0
0
1060
180
0
0
180 0
1060 0 0 220
0 0
0 0
0 0
0 0 2:25 0
Fig. 2. Nanotube geometries: armchair (top), zigzag (middle), chiral (bottom) [16].
38 9 er > > > > > > 7 > > 0 7> eh > > > > > > 7> < 7 0 ez = 7 ðPaÞ 0 7 chz > > > 7> > > 7> > > > 0 5> crz > > > > ; : > 2:25 crh 0
Using the same ratio of length to diameter of nanotubes (L/D = 500) as that of experimental samples in Ref. [12] to performing the finite element analysis is extremely challenging because the elements in finite element mesh must have dimensions significantly smaller than the spatial period of rippling. However, theoretically, under the pure bending condition the bending mo-
ð3Þ
ment on each cross section is the same and the beamÕs neutral axis has a constant curvature. Thus, a beam with smaller aspect ratio (L/D500) can be used to simulate the rippling deformation of the beam under the pure bending condition. It should be pointed out that in the FEM numerical simulation the exact application of the boundary condition of the beam subjected to pure
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X. Wang et al. / Composite Structures 69 (2005) 315–321
bending is very difficult, due to the rotation and the changing unknown stress distribution of the end sections. An approach is to construct a four-point bending beam. A commercial finite element code ABAQUS was used to simulate the rippling formation of bent nanotubes. It can automatically choose proper loading increments and converge criteria in a non-linear analysis. Fig. 3(a) shows the undeformed mesh of pure bending portion of a solid cylindrical model for carbon nanotubes, Fig. 3(b) shows the configuration of a simulated this carbon nanotubes bent with the rippling deformation, and Fig. 3(c) shows overall view of a bending nanotubes with ripples by means of a measuring. It is plotted in Fig. 4 the bending moment M versus the bending curvature j at each loading step for the four-point bending solid cylinder with the length-todiameter ratios L/D = 10, and 20 from FEM simulating result. According to the beam bending theory, we know that M(x, t) should be an odd function j(x, t). Using a polynomial up to the ninth order, the discrete points in Fig. 4 are fitted as
Fig. 4. Two results derived from different ratios to L/D, where represents L/D = 10, d represents L/D = 20.
Mðx; tÞ ¼ EIjð1 a3 D2 j2 þ a5 D4 j4 a7 D6 j6 þ a9 D8 j8 Þ ð4Þ 3
where j = j(x, t), a3 = 1.755 · 10 , a5 = 2.0122 · 106, a7 = 1.115 · 109 and a9 = 2.266 · 1011. The two order of partial derivatives to x variable in Eq. (4), gives M 00 ðx; tÞ ¼ EI½j00 ð1 3a3 D2 j2 þ 5a5 D4 j4 7a7 D6 j6 2
þ 9a9 D8 j8 Þ þ ðj0 Þ ð6a3 D2 j þ 20a5 D4 j3 42a7 D6 j5 þ 72a9 D8 j7 Þ
ð5Þ Based on the large deflection deformation, the relation between the bending curvature j(x, t) and the beam deflection w(x, t) can be expressed as jðx; tÞ ¼
w00 ðx; tÞ
½1 þ w0 ðx; tÞ2 3=2 3 15 2 4 ¼ w00 1 ðw0 Þ þ ðw0 Þ 2 8 2
ffi w00 ðx; tÞ½1 rðw0 Þ
ð6Þ
where r = 1.5. Substituting Eq. (6) into Eq. (5), yields M 00 ðx; tÞ ¼ EIfw0000 3a3 D2 ½2w00 ðw000 Þ2 þ ðw00 Þ2 w0000
r½2ðw00 Þ3 þ 6w0 w00 w000 þ ðw0 Þ2 w0000 g
ð7Þ
Substituting Eq. (7) into Eq. ((1), we obtain a non-linear vibration equation which is expressed as EIw0000 þ 2lw_ þ qA€ w ¼ EIN ðwÞ þ F ðx; tÞ
ð8aÞ
where N ðwÞ ¼ 3a3 D2 ½2w00 ðw000 Þ2 þ ðw00 Þ2 w0000 þ r½2ðw00 Þ3 Fig. 3. FEM model of carbon nanotubes and the rippling configuration. (a) shows the undeformed meshes of the purely bending part, (b) shows the rippling deformed shape of the bent carbon nanotubes from FEM simulation, and (c) shows TEM image of a bent nanotube with ripples experimentally obtained [10].
þ 6w0 w00 w000 þ ðw0 Þ2 w0000
ð8bÞ
Making all the variables in Eq. (8) dimensionless pffiffiffiffiffiffiffiffiffiffiffiffiffi by using the characteristic length L, time L2 qA=EI and force EI/L3, gives
X. Wang et al. / Composite Structures 69 (2005) 315–321
€ _ þ w ¼ N þ F ðx ; t Þ 0000 þ 2 lw w
ð9aÞ
~ t Þ F ðx ; t Þ ¼ Gðx Þ cosðx
ð9bÞ
b2 ¼
Z
1 3
pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ w=L; x ¼ x=L; t ¼ t EI=qA=L2 ; w pffiffiffiffiffiffiffiffiffiffiffi 3 ¼ lL EIqA; N ¼ N =L ; G ¼ GL3 =EI l
0
¼ 20:1939
ð10Þ
2
2
2
~ tÞ u þ 2e vu_ þ € u ¼ e N u þ e g cosðx
000 2
00
ð11Þ
ð12aÞ
00 2 0000
N u ¼ 2a3 ðD=LÞ ½2u ðu Þ þ ðu Þ u
3
2
þ r½2ðu00 Þ þ 6u0 u0000 u000 þ ðu0 Þ u0000
ð12bÞ
In the above formulae, u(x*, t*) and g(x*) are, respectively, expanded as 1 1 X X qn ðt Þ/n ðx Þ; gðx Þ ¼ gn /n ðx Þ ð13Þ uðx ; t Þ ¼ n¼1
n¼1
where /n for n = 1, 2, 3, . . . represent the normalized mode functions of the cantilevered beam from the linear vibration analysis, which satisfies the following formula. Z 1 /i ðx Þ/j ðx Þdx ¼ 0 ði 6¼ jÞ 0 ð14Þ Z 1 /i ðx Þ/j ðx Þdx ¼ 1
ði ¼ jÞ
0
Substituting Eq. (13) into Eq. (12) and utilizing Eq. (14), we have 2 ~ t Þ € q1 þ 2e2 vq_ 1 þ ðx1 Þ q1 ¼ e2 aq31 þ e2 g1 cosðx
where Z b1 ¼ 0
1 2
2
00
00 00 /1 ½2/001 ð/000 1 Þ þ ð/1 Þ /1 dx ¼ 119:6
ð17Þ
ð793:533D=LÞ þ 30:3555
where 2
3a3 ðD=LÞ2 119:6 þ r 20:1939 2
Poncharal [12], based on linear analysis of cantilevered beam, given the first order of inherence frequency corresponding to the fundamental mode of vibration x1 ffi 3:516. It is expected that the non-linearity may ~ to deviate slightly from cause the resonance frequency x x1 in Ref. [12]. By means of the perturbation method of ~ for multi-scales in Ref. [19], the resonance frequency x the non-linear oscillations can be determined. Introducing a small perturbation parameter e expands w into w = eu. From Eqs. (8) and (10), it is seen that non-linear term N is of the third order in w. To make the exciting _ be both of the same force F and the damping force 2lw ¼ e2 v and order as the non-linear term N , we give l 3 Gðx Þ ¼ e gðxA Þ, so that the non-linear oscillation equation (9) becomes 0000
ð16bÞ 2
The boundary conditions (2) is rewritten as 00 ð1; t Þ ¼ w 000 ð1; t Þ ¼ 0 w
00
a ¼ 3a3 ðD=LÞ b1 þ rb2
ð0; t Þ ¼ w 0 ð0; t Þ ¼ 0; w
2
0 00 000 00 00 /1 ½2/001 ð/000 1 Þ þ 6/1 /1 /1 þ ð/1 Þ /1 dx
where
319
ð15Þ
ð16aÞ
It is seen that Eq. (15) is a standard third-order nonlinear oscillation equation [17] which is yielded by the parameter a. Assume that the fundamental resonance frequency in the non-linearity analysis is ~ ¼ x1 re1 x
ð18Þ
the solution for the third-order non-linear oscillation equation (15), and dimensionless excitation force can be, respectively, expressed as q1 ðt ; eÞ ¼ q10 ðT 0 ; T 1 Þ þ e1 q11 ðT 0 ; T 1 Þ þ . . . :
ð19aÞ
~ t Þ ¼ e21 g1 cosðx1 T 0 rT 1 Þ e2 g1 cosðx
ð19bÞ
Substituting Eq. (19) into Eq. (15), and comparing coefficients of the identical power of e1, we have 2
D20 q11 þ ðx1 Þ q11 ¼ 2D0 D1 q10 2vD0 q10 þ aq310 þ g1 cosðx1 T 0 rT 1 Þ 2
D20 q10 þ ðx1 Þ q10 ¼ 0
ð20aÞ ð20bÞ
where Dn ¼ oTo n ðn ¼ 0; 1Þ. The solution of Eq. (20) can be easily obtained. Substituting the solution of Eq. (20) into Eq. (19a), the solution for the third-order non-linear oscillation equation (15) is given by q1 ¼ ap cosðx1 t þ bÞ þ OðeÞ
ð21Þ
where ap represents the maximum vibration amplitude, and is expressed as g ap ¼ 1 ð22Þ 2x1 v and b ¼ r rT 1
ð23Þ
In Eq. (23), r represents a parameter variable, and is written as 2 3a 2 3a g1 3ag21 a ¼ ¼ ð24Þ r¼ 2 8x1 p 8x1 2x1 v 32x3 1 v Thus, the first order of resonance frequency with the physical dimension in the non-linearity analysis is given by sffiffiffiffiffiffi ~ 1 EI x ~¼ 2 x ð25Þ qA L
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X. Wang et al. / Composite Structures 69 (2005) 315–321
Substituting a measured resonance frequency x1 based ~ in Eq. (25), on linear cantilevered beam theory for x yields sffiffiffiffiffiffiffiffiffi x1 Eeff I ~¼ 2 ð26Þ x qA L In the above formula, Eeff represents the effective bending modulus of carbon nanotubes with rippling deformation. From Eqs. (25) and (26), we have 2 2 2 ~ ~ Eeff x1 re2 x x ¼ ¼ ¼ E x1 x1 x1 !2 !2 3ag1 e2 3G2 1 ¼ 1 ¼ 1 a 32ðx1 Þ4 l2 32ðx1 Þ4 l2 " 2 #2 G1 4 ¼ 1 6:137 10 a l ( 2 )2 G 2 ¼ 1 ½3876:44 ðD=LÞ þ 0:0186 1 l
Fig. 6. The corresponding relation between Eeff/E and D/L when K 1 =l ¼ 1.
ð27Þ Eq. (27) expresses that the effective bending modulus Eeff decreases with increasing ratio (D/L), and with increasing excitation load-to-damping ratio (G1 =l ), which is similar to the measured results in Ref. [10] shown in Fig. 1(d).
Fig. 7. The corresponding relation between Eeff/E and D/L when K 1 =l ¼ 2.
3. Results and conclusion Figs. 5–9 show the changes rules of the effective bending modulus-to-YoungÕs modulus ratio Eeff/E with increasing the diameter-to-height ratio (D/L), and different excitation load-to-damping ratio G1 =l . From Figs. 5–9, it is seen that when the excitation load-to-damping ratio G1 =l is a given value the effective bending modulus ratio Eeff/E can decreases as the diameter-to-height ratio (D/L) increases. Comparing Figs. 5–9, it is seen Fig. 8. The corresponding relation between Eeff/E and D/L when K 1 =l ¼ 3.
Fig. 5. The corresponding relation between Eeff/E and D/L when K 1 =l ¼ 0:5.
that the larger the excitation load-to-damping ratio G1 =l is, the more sharply the effective bending modulus-to-YoungÕs modulus ratio Eeff/E drops. But, in Refs. [12], author does not present the discussions of the effect of the excitation load-to-damping ratio G1 =l on the effective bending modulus of carbon nanotubes. It is seen that the non-linear term in the third-order non-linear linear equation (15) is yielded by the parameter a that is based on the effect of the higher order term––rw00 (w 0 )2 in the expression (6) of curvature j. When the parameter a equals zero, the third-order
X. Wang et al. / Composite Structures 69 (2005) 315–321
321
mechanical properties directly in Refs. [10–14]. The present work presents a numerical simulation model, based on non-linear analysis for obtaining the bending modulus of carbon nanotubes and provides a valuable guide for further experimental studies in this area.
References
Fig. 9. The corresponding relation between Eeff/E and D/L when K 1 =l ¼ 4.
non-linear oscillation equation (15) will become a standard linear vibration equation in Ref. [12]. Assuming that a equals zero, the condition using linear vibration equation to obtain a measured resonance frequency x1 can be expressed as beams of diameter-to-length (D/ L) < 144. Carbon nanotubes of diameter-to-length (D/ L) measured in Ref. [10] approximately equals 500. Evidently, this does not satisfy the condition using a linear vibration equation. This condition, (D/L) < 144, suggests that one should be particularly cautious in using the results of the linear vibration theory to obtain a measured resonance frequency x1 . This paper obtains the coefficients (a3, a5, a7, a9) describing the non-linear relation between bending moment M(x) and curvature j in Eq. (4) by using a polynomial up to the ninth order to fit the discrete points in Fig. 4. This is different from those in Ref. [15]. The above different phenomenon is mainly because the discrete points describing the non-linear relationship between bending moment M(x) and curvature j shown in Fig. 4 are obtained by means of a three-dimensional finite element for solid cylinders. The bending moment–curvature relationship in Ref. [17] is obtained by using a two dimensional plain strain FEM model that did not take into account the three-dimensional Poisson effect. On the other hand, because the material orientation of the model varies from carbon nanotubes, which are axially symmetric, a round cross sectional cylinder model of three-dimensions constructed, and the corresponding bending moment–curvature relationship are more likely to simulate the truly bending constitution of carbon nanotubes. However, the small dimension of carbon nanotubes makes it extremely difficult to exactly measure their
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