Composite Structures 94 (2012) 1352–1358

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Composite Structures

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A molecular dynamics study on the thickness and post-critical strengthof carbon nanotubes

Nuno Silvestre a,⇑, Bruno Faria a, José N. Canongia Lopes b

a Department of Civil Engineering and Architecture, ICIST, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugalb Department of Chemical and Biological Engineering, CQE, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

a r t i c l e i n f o

Article history:Available online 4 December 2011

Keywords:Carbon nanotube (CNT)Molecular dynamics (MDs)Pre-buckling, buckling and post-bucklingShell modelsThickness

0263-8223/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruct.2011.10.029

⇑ Corresponding author. Tel.: +351 218418410; faxE-mail address: nuno.silvestre@civil.ist.utl.pt (N. S

a b s t r a c t

This paper presents a molecular dynamics (MDs) study on the linear, buckling and post-buckling behav-iour of carbon nanotubes (CNTs) under pure shortening and pure twisting. Its objectives are (i) to clarifythe issue about the most correct thickness value to adopt in the simulation of CNTs using shell modelsand (ii) to evaluate their post-critical strength. Three CNTs with similar length-to-diameter ratio but dif-ferent atomic structures (zig-zag, armchair and chiral) are selected for this study. Then, MD simulationsare performed to investigate the pre-critical, critical buckling and post-critical behaviour of CNTs underpure shortening and pure twisting. Using available analytical formulae derived from shell models, theinfluence of CNT thickness on their critical strain and critical angle of twist is investigated. Some conclu-sions are drawn regarding (i) the most appropriate choice of the thickness value to use in shell modelsand (ii) the effectiveness of post-critical stiffness and strength of CNTs.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

After the discovery of carbon nanotubes (CNTs), much researchhas been done to characterise their behaviour. Today, CNTs can beused as basic elements for nano-drive systems, nano-actuatorsand nano-oscillators, for instance as spring elements in torsionalpaddle oscillators or twisting bearings in nano-electric motors.These nano-devices have prompted a lot of studies on the stiffnessand strength of CNTs under compression and torsion. It is also wellestablished that CNTs with different chirality have distinct mechan-ical properties [1] and that their behaviour under twisting [2,3] be-comes affected by these properties. MD simulations have shownthat chiral CNTs twist during stretching and stretch during twisting[4]. These phenomena cannot be fully explained by using conven-tional isotropic shell theories. Furthermore, it is well known thatCNTs are sensitive to compression, bending and torsion, due to theirhollow configuration. MD simulations have shown that the buck-ling behaviour of a given CNT is length-dependent and may bedivided into two categories: (i) local instability for short to interme-diate lengths, and (ii) global instability for moderate to long lengths[5,6]. Nevertheless, it is known that MD analyses are computation-ally expensive and time consuming and are often limited to a max-imum number of atoms. To avoid this problem, a large amount ofinvestigations has been carried out, most of them confirming theaccuracy of continuum shell analyses [7]. It is usual to apply shell

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: +351 218497650.ilvestre).

models to study the CNT instabilities both for compression andtorsion. Silvestre and collaborators [5,8–10] have shown that shal-low shell theories (e.g., Donnell theory) are not accurate for CNTanalysis due to CNT non-shallow structure. Only more complexshell theories (e.g., Sanders theory) are capable of reproducing theresults of MD simulations. One issue that remains less clear con-cerns the adoption of CNT thickness to use in shell models. It isknown that graphene (CNT wall) possesses a membrane stiffnessC = Eh = 360 J/m2, thus it is possible to adopt either (i) E = 1.06 TPaand h = 3.40 Å (thick-walled CNT) or (ii) E = 5.5 TPa and h = 0.66 Å(thin-walled CNT) or any other E and h values that satisfyEh = 3.60 TPa Å. The first option has a clear physical meaning, sinceh = 3.40 Å is the equilibrium interlayer spacing of adjacent CNTs.The second option (h = 0.66 Å) has less physical meaning, but it iswidely accepted that it gives better results in CNT analysis usingshell models. The first main objective of this paper is to investigateif CNTs should be modelled as thin (h = 0.66 Å) or thick (h = 3.40 Å)shells. To answer this question, a MD study on the linear, bucklingand post-buckling behaviour of CNTs under either shortening ortwisting is presented. The MD buckling results are compared withestimates obtained from analytical formulae derived from shell the-ories, and the influence of CNT thickness on the critical strain andcritical angle of twist is investigated. Some conclusions are drawnregarding the most appropriate choice of the thickness value touse in shell models.

In addition, there are plenty of investigations devoted the linear(first-order) and buckling behaviour of CNTs, as pointed out in theexcellent review by Wang et al. [7]. However, they also mention

Fig. 1. Coordinate system and displacement nomenclature used in this work.

1 The chiral vector (n,m) represents the way a graphene sheet can be wrapped toyield a given nanotube, with the pair of indices n and m denoting the number of univectors along two directions in the honeycomb crystal lattice of graphene. Thewrapping of graphene sheets yields zig-zag CNTs if m = 0, arm-chair CNTs if n = m, andchiral CNTs if m – 0 and m – n.

N. Silvestre et al. / Composite Structures 94 (2012) 1352–1358 1353

that there is a lack of studies on the post-critical stiffness andstrength of CNTs. Bearing in mind that CNTs are molecules withvery high resilience, the existence in the literature of a limitednumber of investigations on CNT post-buckling performance issomewhat puzzling. Thus, a second main objective of this paperis to shed some light on the post-critical strength of CNTs. Thisobjective will be achieved by resorting to MD simulations of CNTsunder either shortening or twisting, and some useful remarks willbe drawn on this topic.

2. Molecular dynamics simulations

MD simulations have been carried out using the DL-POLY_2simulation package [11]. In these MD simulations, the newtonianequations of motion have been derived from inter-atomic forcesbased on the Tersoff–Brenner covalent potential [12,13]. This po-tential is a type of density-dependent force field, which has beendesigned to reproduce the properties of covalent bonding in hydro-carbons. A special feature of this potential is that it allows for bondbreaking and the associated bond hybridisation changes. The po-tential energy U is given, as a function of atomic coordinates, by

U ¼ 12

Xi–j

Uij ð1Þ

Uij ¼ URðrijÞ � BijUAðrijÞ ð2Þ

where rij is the length of the bond between atoms i and j, and UR andUA are the repulsive and attractive energy terms, respectively, givenby

URðrijÞ ¼ fcðrijÞDðeÞij

Sij � 1e�2

ffiffiffiffiffiffi2Sij

pbijðrij�RðeÞ

ijÞ ð3Þ

UAðrijÞ ¼ fcðrijÞDðeÞij Sij

Sij � 1e�2

ffiffiffiffiffiffi2Sij

pbijðrij�RðeÞ

ijÞ ð4Þ

where fC is a cut-off function given by

fcðrijÞ ¼

1 if rij < Rð1Þij

12þ 1

2 cos pðrij � Rð1Þij Þ=ðRð2Þij � Rð1Þij Þ

h iif Rð1Þij < rij < Rð2Þij

0 if rij > Rð2Þij

8>>><>>>:

ð5Þ

and RðeÞij ðe ¼ 1;2Þ are given cut-off distances. The Bij parameter rep-resents a many-body coupling of the bond between atoms i and jand their local environment.

Displacement control is adopted in the MD simulations sincekinematical quantities are imposed at the atoms located in theedges of the CNT. The origin of the cylindrical axis system (x,h,z)is located at the centre of the CNT mid-section, where x is the coor-dinate along the longitudinal axis, h is the angular coordinate and zis the radial coordinate (see Fig. 1). For a given CNT with diameter

D and length L, both end sections are located at x = ±L/2. The dis-placements in the system (x,h,z) are denoted by u, / and w,respectively.

For shortening deformations, we have imposed incremental ax-ial displacements in opposite directions, +0.025 Å for atoms lo-cated at x = �L/2 and �0.025 Å for atoms at located x = +L/2. Fortwisting deformations, we have imposed incremental rotations inopposite directions, +0.5� = +p/360 rad for the atoms located atx = �L/2 and �0.5� = �p/360 rad for the atoms located at x = +L/2.This type of kinematical imposition ensures full structural symme-try of the CNT atomic system and, for each increment, correspondsto Du = 0.05 Å and D/ = 1.0� = p/180 rad between the end sectionsof the CNT.

Previous studies have shown that the buckling behaviour ofCNTs under pure shortening or pure twisting is greatly dependenton L and D [5,6]. In order to avoid such dependence as much as pos-sible, we have selected CNTs with similar length-to-diameter as-pect ratios, L/D = 7.0. To study the buckling behaviour of CNTswith different chirality under pure shortening and pure twisting,we have considered the following zig-zag, armchair and chiralCNT structures1:

(i) Zig-zag (8,0) with L = 46.3 Å and D = 6.2 Å, having an aspectratio L/D = 7.4 and 352 atoms.

(ii) Armchair (5,5) with L = 47.3 Å and D = 6.8 Å, having anaspect ratio L/D = 7.0 and 380 atoms.

(iii) Chiral (6,3) with L = 41.0 Å and D = 6.2 Å, having an aspectratio L/D = 6.6 and 336 atoms.

These are depicted in Fig. 2a. Since the chiral CNT (6,3) displaysan asymmetric lattice atomic structure, it has been analysed underdirect and inverse twisting rotation conditions, as seen in Fig. 2b.The results of this specific analysis are deemed innovative sincethey have never been issued by other researchers.

All simulations were performed at a temperature of 298.15 Kusing a Berendsen thermostat. The newtonian equations of motionwere integrated using the Verlet leapfrog predictor–corrector algo-rithm. A time step of 1 fs was used and, after each increment of Duand D/, the CNT was fully relaxed for a period of 30.000 time steps.The configurational (or strain) energy, V, of the CNT calculated ineach increment corresponds to the average value of the energywithin the last 10.000 time steps. Ninety increments were per-formed in the simulations of the CNT under pure shortening and180 increments were used for the analysis of CNT under pure twist-ing. The mean value of the strain energy V is determined for eachincrement. The Table 1 shows the mean and standard deviationvalues of V (before buckling) for the CNTs under pure shortening

t

(a) (b) Fig. 2. (a) CNTs (8,0), (5,5), (6,3) and (b) CNT (6,3) under direct and inverse twisting.

Table 1Average and standard deviation values of strain energy V of some CNTs under pureshortening and pure twisting.

CNT Pure shortening, V (eV/atom) Pure twisting, V (eV/atom)

Average Sd. dev Average Sd. dev.

(8,0) 0.22421 0.00116 0.07985 0.00113(5,5) 0.22674 0.00109 0.05167 0.00112(6,3) direct 0.27089 0.00121 0.05478 0.00119(6,3) inverse 0.27089 0.00121 0.07301 0.00117

1354 N. Silvestre et al. / Composite Structures 94 (2012) 1352–1358

and pure twisting. The mean estimated error in the dynamics sim-ulations is equal to 0.5% (pure shortening) and 1.9% (pure twisting)of V. The MD simulation results are presented and discussed in thefollowing sections.

3. CNTS under pure shortening

The three CNTs have been subjected to axial shortening and theresults obtained from MD simulations are shown in Fig. 3. Fig. 3ashows the variation of the strain energy per atom, V, as a functionof the axial strain e = u/L. Fig. 4b displays the variation of the forceper atom, F, with the axial shortening u. In each increment, theforce is calculated as the 1st derivative of the energy w.r.t. the axialshortening u, i.e., F = oV/ou (note that 1.0 eV/Å = 1.602 nN).

From Fig. 3a, it is clear that the energy V grows quadraticallywith the strain e and, at buckling, it suddenly drops. These dropsare related to the buckling of CNTs (8,0), (5,5) and (6,3) for thecritical strains ec = 0.081, ec = 0.080, ec = 0.094, respectively. The

(a)ε

∂ ∂ε

∂ ∂ε

∂ ∂ε

Fig. 3. (a) Variation of energy V with axial strain e,

zig-zag and armchair CNTs have similar aspect ratios (L/D around7) and exhibit similar critical strains. The very small difference be-tween the ec values means that the critical strain does not appearto be closely dependent on these CNT structures. However, the chi-ral CNT has also a similar aspect ratio (L/D around 7) but it pos-sesses a much higher critical strain (17.5% larger). It means thatthe critical strain of chiral CNTs depends on the atomic structure(CNT chirality). The critical buckling mode shapes of CNTs (8,0),(5,5) and (6,3) under pure shortening are shown in Fig. 4 (on theleft side of each pair of structures). It is seen that all CNTs bucklelocally, similarly to a cylindrical thin shell. Nevertheless, bucklingoccurs for different number of half-waves. The post-critical modeshapes of the CNTs are also shown in Fig. 4 (on the right side ofeach pair of structures). It is seen that they are slightly different:the CNTs (8,0) and (6,3) deflect laterally while the CNT (5,5) shiftsto a kink at mid span mixed with lateral deflection.

Based on Sanders shell model, Silvestre et al. [8] proposed thefollowing formula to calculate the critical strain ec for local and glo-bal buckling of CNTs,

ec ¼h

1:7R�Max 0:6; 1� L10R

ffiffihR

qh iif L

D � 3:8ffiffiRh

q

2 pRL

� �2 if LD � 3:8

ffiffiRh

q

8><>:

ð6Þ

where h and R are the CNT thickness and radius, respectively.Comparing the aspect ratio L/D with its cross-section property

(3.8 � (R/h)0.5), a given CNT may be classified as short or long.The first expression corresponds to local buckling for short CNTswhile the second one corresponds to global buckling of long CNTs.There has been much discussion concerning the graphene

(b)

Å

u Å

and (b) variation of force F with shortening u.

Fig. 4. Critical modes and post-critical shapes of CNTs (8,0), (5,5), (6,3) under shortening.

(b)(a)α φ

Fig. 5. (a) Variation of energy V with the angle of twist per unit of length a, and (b) variation of torque T with the twisting angle /.

N. Silvestre et al. / Composite Structures 94 (2012) 1352–1358 1355

thickness value. Yakobson et al. [14] proposed h = 0.66 Å. However,recent studies [15,16] have shown that this value is slightly low forbuckling analyses of CNTs. Huang et al. [15] proposed h = 0.734 Åwhile Wang et al. [16] suggested h = 0.76 Å. Following these inves-tigations, we have adopted h = 0.75 Å. Eq. (6) stipulates that theseCNTs are short (i.e. they fail by local buckling), despite L/D � 7.0 isclose to the transitional value (3.8 � (R/h)0.5 � 8.0). From Eq. (6),we obtain ec = 0.085, ec = 0.078 and ec = 0.085 for the CNTs (8,0),(5,5) and (6,3), respectively. These values show good agreementwith the MD results obtained for the zig-zag and armchair CNTs.For the chiral CNT, the analytical value of ec is 10% lower thanthe MD result, meaning that continuum shell models are moreaccurate for zigzag and armchair CNTs than for chiral ones. Thisconclusion is in line with previous ones showing the anisotropyof chiral CNTs.

Fig. 3b shows that F increases linearly with u and the curves ex-hibit a huge drop when the CNTs buckle. It is interesting to notethat both zig-zag (8,0) and armchair (5,5) CNTs possess the samepre-critical axial stiffness oF/ou = 0.039 eV/Å2 atom, while the chi-ral (6,3) CNT has a higher pre-critical axial stiffness oF/ou = 0.046 eV/Å2 atom. In this case, it is also possible to use contin-uum mechanics formulas to assess the CNT pre-critical axial stiff-ness, which is given by

@F@u¼ Eð2pRhÞ

Lð7Þ

Using E = 1.06 TPa and h = 3.40 Å (thick-walled CNT), one ob-tains oF/ou = 0.043 eV/Å2 atom for zig-zag (8,0) and armchair(5,5) CNTs, and oF/ou = 0.051 eV/Å2 atom for the chiral (6,3) CNT.

It is interesting to note that if we use E = 4.80 TPa and h = 0.75 Å(thin-walled CNT), we obtain exactly the same pre-critical axialstiffness values of the thick-walled CNT. This is due to the fact thatCNT wall membrane stiffness C = Eh = 360 J/m2, and Eq. (7) could bewritten as

@F@u¼ 2pRc

Lð8Þ

which depends only on the CNT radius R and length L.After buckling (u > ucr), the force applied to the CNTs shows a

high scatter, but it always varies around a constant (average) value,Fpc. It should be mentioned that Liew et al. [17] also found a similarscatter around a mean force value after CNT buckling. Moreover,Zhang and Shen [18] also found that the post-critical force of theCNT under shortening remains uniform after buckling. In order tofind the value of the post-critical force, Fpc, it is preferable to resortto the V(e) curves shown in Fig. 3a. After buckling, all V(e) curvescease their quadratic variation with e and start showing a clear lin-ear dependence with e. At first glance, the three linear paths seemto have similar slopes, @V/oe. Using a linear trend line to fit the en-ergy data, the values 1.73, 2.08 and 1.96 eV/atom have been ob-tained for oV/@e of the post-critical CNTs (8,0), (5,5) and (6,3),respectively. Since the post-critical force is given by Fpc = oV/(L@e), values of Fpc = 0.037, 0.044 and 0.048 eV/Å atom have beenobtained for the CNTs (8, 0), (5,5) and (6,3), respectively.

Concerning the pre-critical, critical and post-critical behaviourof CNTs under shortening conditions, a few remarks can be made.The zig-zag and armchair CNTs have similar pre-critical stiffness,lower than that of the chiral CNT. As far as buckling is concerned,

1356 N. Silvestre et al. / Composite Structures 94 (2012) 1352–1358

both zig-zag and armchair CNTs have analogous critical strains,much lower than that of the chiral CNT. Regarding the post-criticalforce, the differences between the CNTs are obvious: the armchairis the ‘‘weakest’’ CNT while the chiral is the ‘‘strongest’’ CNT.Regarding the post-critical stiffness, all CNTs are similar as theypossess ‘‘null’’ post-critical stiffness. In fact, the force F varies con-sistently around a uniform Fpc value.

4. CNTS under pure twisting

The three CNTs have been subjected to pure twisting and the re-sults obtained from MD simulations are shown in Fig. 6. Recall thatthe chiral CNT (6,3) has been analysed twice, for direct and inversetwisting (see Fig. 2b). Fig. 5a shows the variation of energy peratom, V, as a function of the angle of twist per unit of length,a = //L. Fig. 5b displays the variation of torque per atom, T, as afunction of the twisting angle, /. At each increment, the torque iscalculated as the first derivative of the energy w.r.t. the twistingangle /, i.e., T = @V/o/ (note that 1.0 eV = 1.602 nNÅ).

Fig. 5a shows that all V(a) energy curves are quite similar. How-ever, a more careful analysis shows that only the zig-zag CNT (8,0)shows two small sudden drops in its V(a) curve at ac = 0.056 rad/Åand ac = 0.065 rad/Å, whereas the CNTs (5,5) and (6,3) show a con-tinuous behaviour in the studied range of angles of twist. Thebehaviour of the CNTs (5,5) and (6,3) V(a) curves contrasts withthe previously discussed shortening V(e) curves where large sud-den drops were always found. The main issue at this point is to an-swer the following two questions:

� Why the CNTs (5,5) and (6,3) do not exhibit sudden drops intheir V(a) curves?� Why the V(a) curve of zig-zag CNT (8,0) shows two small sud-

den drops?

In order to answer the first question, let us look at the trends ofthe V(a) curves: they change quadratically with a, for a smallerthan 0.020–0.025 rad/Å. After that threshold value it seems thatall V(a) curves start varying linearly with a. These facts are betterexplained if we consider Fig. 5b. The figure shows that for each CNTthe torque T increases linearly with the twisting angle /, until itreaches a certain limit and then starts decreasing (causing the‘‘peaks’’ in the T(/) curves). These limit values correspond preciselyto the transition between the ‘‘quadratic’’ and ‘‘linear’’ trends of theV(a) curves � (a � 0.022 rad/Å) = (/ � 1 rad)/(L � 45 Å) � which

Fig. 6. Critical mode shapes of CNTs (8,0), (5,5), (6,3)-dir and (6,3)-inv undertwisting.

cannot be determined directly from the inspection of the V(a)curves in Fig. 5a. Indeed, all CNTs (including the zig-zag CNT(8,0)) buckle when this limit is achieved but the buckling phenom-ena takes place softly, without a dynamic jump and abrupt loss ofstrain energy. Similar evidence was identified by Zhang and Shen[18] for armchair and zig-zag CNTs under twisting.

From the observation of Fig. 5b, we conclude that all CNTs havesimilar pre-critical stiffness around @T/o/ = 0.190–0.200 eV/atom.It is also possible to use continuum mechanics formulas to assessthe CNT pre-critical torsional stiffness ( G = E/2(1 + m), m = 0.19),which is given by

@T@/¼

G p2 Rþ h

2

� �4 � R� h2

� �4h i

Lð9Þ

Using E = 1.06 TPa and h = 3.40 Å (thick-walled CNT), one ob-tains oT/@/ = 0.226 eV/atom for the zig-zag CNT (8,0), oT/@/= 0.260 eV/atom for the armchair CNT (5,5), and @T/o/= 0.268 eV/atom for the chiral CNT (6,3). Considering the CNTsthick-walled, the estimated torsional stiffness of the zig-zag CNT(8,0) is only 10% higher than the MD value, but the torsional stiff-ness predictions of the other CNTs are 30% higher than the MD val-ues. On the other hand, if we introduce the thin-walled CNTproperties (E = 4.80 TPa and h = 0.75 Å) in Eq. (9) one obtains oT/@/ = 0.176 eV/atom for the zig-zag (8,0) CNT, oT/o/ = 0.210 eV/atom for the armchair CNT (5,5), and @T/o/ = 0.209 eV/atom forthe chiral CNT (6,3). These predictions, using thin-walled CNTproperties, are much more accurate (5% maximum difference) thanthe thick-walled CNT counterparts.

From the observation of Fig. 5b, we also conclude that the CNTsbuckle for slightly different critical angles of twist: /c = 1.01 rad forCNT (8,0), /c = 0.84 rad for CNT (5,5), /c = 0.94 rad for direct twist-ing of CNT (6,3), /c = 1.05 rad for inverse twisting of CNT (6,3).From these results, we can anticipate that armchair CNTs are moresusceptible to buckling than the other configurations. Furthermore,we conclude that chiral CNTs are more sensitive to buckling whenthey are submitted to direct twisting (Fig. 2a) rather than inversetwisting (Fig. 2b). The critical buckling mode shapes of CNTs(8,0), (5,5) and (6,3) under pure twisting are shown in Fig. 6.

It is seen that all of them show a non-uniform helix deformedshape, consisting of an ovalised cross-section that spirals aroundthe longitudinal axis between its ends. Moreover, the chirality ofthe CNT (6,3) must be correlated with distinct buckling behaviourunder inverse and direct twisting (see Fig. 2b): the direct twistingof CNT (6,3) implies an increased ‘‘curling’’ of the CNT helix struc-ture, which means that one will reach /c sooner. In the case of in-verse twisting, one is ‘‘straightening’’ the CNT helix structure and‘‘delaying’’ the occurrence of buckling, which means that one willreach /c later.

Based on Sanders shell model, Silvestre [9] proposed the follow-ing formula to calculate the critical angle of twist /c (in rad) for lo-cal and global buckling of CNTs,

/c ¼1:7 h

R

� �1:25 LR

� �0:5 if LD � 4:36

ffiffiRh

q

23

hR

� �1:5 LR

� �1:0 if LD � 4:36

ffiffiRh

q

8><>:

ð10Þ

Unlike in the shortening case, twisted CNTs always buckle lo-cally, mostly with an ovalised buckle. There is also a limit aspectratio (4.36 � (R/h)0.5) in twisting, but this corresponds to the tran-sition between ranges where the boundary conditions influence ornot the critical angle of twist. If the actual aspect ratio L/D is lowerthan that limit, the CNT is classified as short, the boundary condi-tions (end caps) influence the /c values and Eq. (10.1) applies. Con-versely, if the actual aspect ratio L/D is higher than that limit, theCNT is classified as long, the boundary conditions (end caps) donot influence the /c values and Eq. (10.2) applies.

αc=0.056 rad/Å αc=0.065 rad/Å

(a) (b)

1st break of C-C bonds 2nd break of C-C bonds

Fig. 7. Post-buckling shapes of CNT (8,0) under twisting: (a) 1st break of two bonds (ac = 0.056 rad/Å) and (b) 2nd break of two bonds (ac = 0.065 rad/Å).

N. Silvestre et al. / Composite Structures 94 (2012) 1352–1358 1357

Using h = 0.75 Å (thin-walled CNT), a transitional aspect ratio of9.0 is determined and the CNTs are classified as short (recall that L/D � 7.0). From the use of Eq. (10.1), the following approximatedvalues of the critical angle of twist are obtained: /c = 1.07 rad forCNT (8,0), /c = 0.95 rad for CNT (5,5), /c = 1.12 rad for CNT (6,3).These approximate values agree fairly well with the MD results,showing errors around 10%. However, Eq. (10) does not dependon the direction of twisting and the approximate result for the chi-ral CNT (6,3) is closer to the MD value for inverse twisting (Fig. 1b).Using h = 3.40 Å (thick-walled), a transitional aspect ratio of 4.3 isdetermined and the CNTs are classified as long (recall that L/D � 7.0). From the use of Eq. (10.2), the following approximatedvalues of the critical angle of twist are obtained: /c = 11.4 rad forCNT (8,0), /c = 9.2 rad for CNT (5,5), /c = 10.1 rad for CNT (6,3).These values are much higher (about 10 times!) than the MD re-sults. Additionally, the CNTs buckle in local modes, as seen inFig. 5, rather than in global modes. Once again, we conclude thatCNTs should be modelled as thin-walled shells.

Also unlike in the shortening case, the torque T does not dropsuddenly when the CNT buckles: after buckling T decreases propor-tionally with the increase of /. This means that all CNTs have anegative post-critical stiffness. At first glance, the four linear pathsseem to have a similar negative slope @T/o/, that is, similar post-critical stiffness. Using a linear regression to fit the torque data,the values �0.038, �0.036, �0.034 and �0.071 eV/atom are ob-tained for the slope oT/@/ of the post-critical paths correspondingto CNTs (8,0), (5,5), (6,3) under direct twisting and (6,3) under in-verse twisting, respectively. It is surprising to observe that thepost-critical stiffness of the chiral CNT (6,3) under inverse twistingis twice that of the other CNTs. It means that chiral CNTs under in-verse twisting are the stiffer ones for buckling (higher /c) but theyare also the slender ones in post-critical regime (lower @T/o/).Again, there is a correlation between the twisting direction of chi-ral CNTs and their post-buckling stiffness.

However, the negative post-critical @T/o/ trends stop at twist-ing angle around 1.60–1.70 rad. After reaching the values 1.72,1.61, 1.68 and 1.59 rad, corresponding to CNTs (8,0), (5,5), (6,3)under direct twisting and (6,3) under inverse twisting, respec-tively, the torque T starts to increase with /, at a slow @T/o/ rate.Similar trends were found by Zhang and Shen [18] for armchairand zig-zag CNTs under twisting. It is also very interesting to notethat Yao and Han [19] found an analogous post-buckling behaviourusing an elastic shell model. They found that the post-bifurcationbehaviour is unstable (descending post-critical path) but becomesstable (ascending post-critical path) after a local minimum of thetorque-twisting angle curve. New linear regressions in this stable

region (1.60 rad < / < 3.14 rad) yield positive post-critical stiffnessvalues, oT/@/, of 0.00, +0.024, +0.018 and +0.014 eV/atom, corre-sponding to CNTs (8,0), (5,5), (6,3) under direct twisting and(6,3) under inverse twisting, respectively. Concerning the pre-crit-ical, critical and post-critical behaviour of CNTs under twisting, afew conclusions can be forwarded. All CNTs have similar pre-criti-cal stiffness. As far as buckling is concerned, armchair CNTs aremore susceptible to buckling than the other configurations andchiral CNTs are more sensitive to buckling when they are submit-ted to direct twisting. All CNTs have qualitatively similar post-crit-ical T(/) curves: a clear descending branch after buckling followedby a slight increasing path for large twisting rotations. All the CNTshave negative post-critical stiffness, but the chiral CNT (6,3) underinverse twisting is the frailest CNT.

In order to answer the second question, about the emergence oftwo small sudden drops in the V(a) curve of zig-zag CNT (8,0) (seeFig. 5a), it is mandatory to understand that they occur for verylarge twisting rotations (ac = 0.056 rad/Å and ac = 0.065 rad/Å)when the CNT is already buckled. Due to the significant twistingrotation, the CNT cross-section becomes too much ovalised andthe graphene sheet becomes too much bent in localised zones ofthe helix shape. In these zones, some CAC bonds are much ten-sioned and, for increasing twist rotation, they reach their limitstrength and break. The breaking of these bonds is associated withsudden small drops of strain energy V(a). For ac = 0.056 rad/Å, twoCAC bonds break in the zig-zag CNT (8,0) structure, leading to thefirst drop seen in the V(a) curve of Fig. 5a. The Fig. 7a shows thebroken CAC bond visible in the bottom half of the twisted CNT –the other broken CAC bond is hidden in the top half of the CNT,not shown in Fig. 7a. If the rotation continues to increase, two addi-tional CAC bonds break for ac = 0.065 rad/Å (see Fig. 7b), implyingthe second drop seen in the V(a) curve of Fig. 5a. The visible bondbreak is located in the top half of the CNT while the other is hiddenin the bottom half of the CNT.

5. Conclusion

This paper was devoted to study the strength and stiffness ofCNTs under pure shortening and pure twisting. The choice of themost correct thickness value for simulation of CNTs using shellmodels was explained. In order to evaluate these properties, CNTswith similar aspect ratio L/D but different atomic structures (zig-zag, armchair and chiral) were selected. Firstly, MD simulationswere performed to study the buckling behaviour of the CNTsunder pure shortening and pure twisting conditions, as well astheir pre-critical and post-critical stiffness. In this context,

1358 N. Silvestre et al. / Composite Structures 94 (2012) 1352–1358

available formulae based on continuum mechanics were applied topredict the critical values of axial strain for pure shortening androtation angle for pure twisting. Concerning the behaviour of CNTsand the choice of the most correct thickness, the following conclu-sions are due:

(i) Pre-critical stiffness – For pure shortening, the zig-zag andarmchair CNTs possess lower pre-critical stiffness than thechiral CNTs. For pure twisting, all the CNTs have similarpre-critical stiffness.

(ii) Critical buckling – For pure shortening, zig-zag and armchairCNTs have much lower critical strains than the chiral CNT.For pure twisting, armchair CNTs exhibit a lower criticalangle of twist than the other CNTs. In this case, chiral CNTsunder direct twisting are more sensitive to buckling.

(iii) Post-critical stiffness – For pure shortening, all CNTs havesimilar (nearly null) post-critical stiffness. However, thearmchair CNT has the lower force level while the chiralCNT displays the higher force level. For pure twisting, allCNTs have negative post-critical stiffness (unstable behav-iour) that becomes positive (stable behaviour) for increasingrotation. However, the chiral CNT under inverse twisting ismore sensitive than the other CNTs. Finally, it should bementioned that an increase of twisting rotation componentleads to increasingly and highly tensioned CAC bonds atlocalised zones of the CNT helix shape. When these bondsreached their limit strength, they broke and the correspond-ing strain energy V(a) exhibited sudden small drops. Thesedrops were always lower than the ones associated withthe buckling of CNT under shortening.

(iv) CNT thickness – The results obtained from MD simulationsand shell model analyses have shown that CNTs should beclassified as thin-walled shells. The thickness h = 0.75 Åwas shown to provide results that compare fairly well withMD results. A thickness value between 0.66 and 0.75 Åshould be used when resorting to shell models.

(v) Effect of CNT structure – The chirality of the CNT (6,3) wascorrelated with distinct buckling behaviour under inverseand direct twisting: the direct twisting of CNT (6,3) impliedthe ‘‘curling’’ of the CNT helix structure and prompted theonset of buckling, while the inverse twisting of CNT (6,3)implied the ‘‘straightening’’ the CNT helix structure anddelayed the occurrence of buckling.

Acknowledgements

The authors gratefully acknowledged the financial support gi-ven by the Portuguese Foundation for Science and Technology(FCT), in the context of the project entitled ‘‘Modelling and Analysisof Nanostructures: Carbon Nanotubes and Nanocomposites’’(PTDC/ECM/103490/2008).

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