Composites Science and Technology 71 (2011) 1811–1818

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Composites Science and Technology

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Interaction diagrams for carbon nanotubes under combined shortening–twisting

Bruno Faria a, Nuno Silvestre 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:Received 4 May 2011Received in revised form 27 July 2011Accepted 9 August 2011Available online 27 August 2011

Keywords:A. Carbon nanotubesB. StrengthB. Non-linear behaviourC. BucklingC. Multiscale modelling

0266-3538/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compscitech.2011.08.006

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

a b s t r a c t

A theoretical investigation on the strength and stiffness of carbon nanotubes (CNTs) under combinedshortening and twisting strains is presented. CNTs with similar length-to-diameter aspect ratios, L/D,but different atomic structures (zig-zag, armchair and chiral) have been selected. Molecular dynamics(MD) simulations have been performed to study the critical buckling behaviour and the pre-criticaland post-critical stiffness of CNTs under combined shortening–twisting conditions. The main resultsare presented in the form of interaction diagrams between the critical strain and the critical angle of twistper unit of length. An interaction equation is proposed and validated by comparison with the MD results.If shortening is more dominant than twisting, the strain energy at the onset of buckling drops consider-ably with the increase of the twisting–shortening rate. If twisting is more influential than shortening, theenergy at the onset of buckling decreases very slowly with the twisting–shortening rate. We also foundan interaction factor of 1.5 for CNTs under combined shortening–twisting, which is much lower than thevalue 2.0 commonly adopted for circular tubes at macro-scale. We conclude that CNTs are much moresensitive to buckling under shortening–twisting interaction than macro-scale tubes.

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1. Introduction

Carbon nanotubes (CNTs) can be used as basic elements fornano-drive systems, nano-actuators and nano-oscillators, forinstance as spring elements in torsional paddle oscillators ortwisting bearings in nano-electric motors. These nano-deviceshave prompted a lot of studies on the stiffness and strength ofCNTs under torsion. It is also well established that CNTs withdifferent chirality have distinct mechanical properties [1] and thattheir behaviour under twisting [2] becomes affected by theseproperties. Molecular dynamic (MD) simulations have shown thatchiral CNTs twist during stretching and stretch during twisting [3].After the key paper by Yakobson et al. [4], the attention of the sci-entific community has also been focused on the CNT bucklingbehaviour [5]. However, most of the previous MD simulations onbuckling behaviour of CNTs have been performed for pure loading(either compression or bending or torsion). The buckling behaviourof CNTs under combined loadings has only been performed in thecontext of continuum shell models [6–8] and frame models [9]. Tothe authors’ knowledge, a single MD study on the buckling andpost-buckling behaviour of CNTs under combined torsion and axialloading was reported in the literature [10]. Since CNT-based springelements in single-paddle torsional oscillator are often subjectedto combined torsion and axial loading [11,12], it is mandatory to

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study more deeply the strength and stiffness of CNTs under com-bined shortening and twisting. The main objective of this paperis to shed light on the CNT strength and stiffness under combinedshortening and twisting. The results have been rationalised in theform of interaction (ec–ac) diagrams, where ec is the critical com-pressive strain and ac is the critical angle of twist per unit of length.Additionally, an interaction formula was proposed and its resultscompared with those obtained from MD simulations.

2. Scope and procedure for MD simulations

The MD simulations have been carried out using the DL-POLY_2simulation package [13]. In these MD simulations, the newtonianequations of motion have been derived from inter-atomic forcesbased on the Tersoff–Brenner covalent potential [14,15]. Displace-ment control is adopted in the MD simulations since kinematicalquantities are imposed at the atoms located in the edges of theCNT. The origin of the cylindrical axis system (x, h, z) is locatedat the centre of the CNT mid-section, where x is the coordinatealong the longitudinal axis, h is the angular coordinate and z isthe radial coordinate. For a given CNT with radius R and length L(i.e., the minimum distance between restrained atoms), both endsections are located at x = ±L/2. The displacements in the system(x, h, z) are denoted by u, / and w, respectively. For pure shorteningdeformations, we have imposed incremental axial displacementsin opposite directions, +0.025 Å for atoms located at x = �L/2 and�0.025 Å for atoms located at x = +L/2. For pure twisting deforma-

(a) (b)

Fig. 2. CNT (6,3) under direct and inverse twisting.

1812 B. Faria et al. / Composites Science and Technology 71 (2011) 1811–1818

tions, we have imposed incremental rotations in opposite direc-tions, +0.5� = +p/360 rad for the atoms located at x = �L/2 and�0.5� = �p/360 rad for the atoms located at x = +L/2. This type ofkinematical imposition ensures full structural symmetry of theCNT atomic system and, for each increment, corresponds toDu0 = 0.05 Å and D/0 = 1.0� = p/180 rad between the end sectionsof the CNT. The ‘‘0’’ subscripts mean pure shortening and puretwisting, respectively. In order to impose simultaneous shorteningand twisting deformations, the following relationship is adoptedbetween the combined Du and D/,

DuDu0

� �2

þ D/D/0

� �2

¼ 1 Du ¼ Du0 cos b; D/ ¼ D/0 sin b ð1Þ

where a parameter b (in arc degrees) is used. In order to have a fulldescription of the plane (Du, D/), we have varied b from 0� to 90�with increments of 5�. Thus, a total of 19 b values are considered,and depicted in Fig. 1. For b = 0�, it is obtained Du = Du0 and D/= 0, i.e., pure shortening behaviour (without twisting). For b = 90�,it is obtained D/ = D/0 and Du = 0, i.e., pure twisting behaviour(without shortening).

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. 90 increments were performedin the MD simulations. The mean value of the strain energy V isdetermined for each increment. The Table A1 (in Appendix A)shows the mean and standard deviation values of V (before buck-ling) for the CNTs under pure shortening (b = 0�), combined short-ening–twisting (b = 45�) and pure twisting (b = 90�). The meanestimated error in the MD simulations is equal to 0.5% (b = 0�),1.4% (b = 45�) and 1.9% (b = 90�) of V.

Previous studies have shown that the buckling behaviour ofCNTs under pure shortening or pure twisting is greatly dependenton L and R [16,17]. In order to avoid such dependence as much aspossible, we have selected CNTs with similar length-to-diameteraspect ratios, having L/D around 7.0. To study the buckling behav-iour of CNTs with different chirality under combined shortening–twisting, we have considered the following zig-zag, armchair andchiral CNT structures with a total of around 350 atoms per CNT:zig-zag (8,0) with L = 46.3 Å and R = 3.1 Å, having an aspect ratioL/D = 7.4; armchair (5,5) with L = 47.3 Å and R = 3.4 Å, having anaspect ratio L/D = 7.0; chiral (6,3) with L = 41.0 Å and R = 3.1 Å,

Fig. 1. Shortening–twisting combinations adopted in the study.

having an aspect ratio L/D = 6.6. 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. 2. Inthe context of combined shortening–twisting behaviour, theresults of this analysis are deemed innovative since they havenever been issued by other researchers.

For CNTs under pure shortening, combined shortening–twistingand pure twisting, the Table A2 (in Appendix A) shows the morerelevant results (at the onset of buckling) obtained from MDsimulations, i.e., the critical strain energy Vc, the critical strainec = uc/L and the critical angle of twist per unit of length ac = /c/L.The MD simulation results presented in the next section enablethe characterisation of the non-linear behaviour of CNTs undercombined shortening and twisting.

3. Results and discussion

In accordance with the procedure explained in Section 2, theparameter b (in degrees) has been used to stipulate an ad hoccombination between shortening u and twisting /. By varying bfrom 0� to 90� with increments of 5�, a total of 19 b values havebeen considered (see Fig. 1). Recall that b = 0� corresponds to pureshortening whereas b = 90� is associated with pure twisting. Now,let us turn our attention to the behaviour of CNTs under combinedshortening and twisting, i.e., 0� < b < 90�.

In order to compare the energy V of the four CNTs, let usconsider first the shortening–twisting combination with b = 45�.Using Eq. (1), such b value means that simultaneous shorteningdisplacements Du ¼ 0:05� cos 45

�¼ 0:0354Å and twisting rota-

tions D/ ¼ p180� sin 45

�¼ 0:0123 rad are imposed to the CNT in

each increment, and the twisting–shortening rate is D/Du ¼ 0:35 rad

Å.

Fig. 3a shows the variation of energy V with either axial strain e(bottom axis) or angle of twist per unit of length a (top axis).Fig. 3b displays the variation of the V-derivatives, oV/ou (rightaxis) and oV/o/ (left axis), with the imposed shortening, u (bottomaxis), or twisting angle, / (top axis). Note that the two derivativesare related by @V

@u ¼ 0:35� @V@/ since / = 0.35 u (and a = 0.35 e).

Fig. 3a shows that all V(a) (or V(e)) energy curves are quite sim-ilar. However, a more careful analysis shows that only the armchair(5,5) CNT shows a continuous energy curve for all the studiedrange of imposed strains and angles of twist. In opposition, the(8,0) and (6,3) CNTs show small sudden drops in their energycurves. The main issue at this point is to answer the followingtwo questions: (i) Why the CNT (5,5) does not exhibit suddendrops in its energy curve?; (ii) Why the energy curves of zig-zag(8,0) and chiral (6,3) CNTs show small sudden drops?

In order to answer these questions, let us look at the trends of theenergy curves: they change quadratically with either e or a, for esmaller than 0.036–0.046 and a smaller than 0.012–0.016 rad/Å.After that threshold value it seems that all energy curves startvarying linearly with e or a. These facts are better explained if we

Fig. 3. (a) Energy per atom, V, as a function of axial strain e or angle of twist per unit of length, a. (b) Energy per atom derivatives, oV/ou or oV/o/, as a function of shortening, u,or twisting angle,/.

B. Faria et al. / Composites Science and Technology 71 (2011) 1811–1818 1813

consider Fig. 3b. It shows that for each CNT the V-derivatives, oV/ouand oV/o/, increase linearly with the shortening displacement u andtwisting angle /, until they reach certain ‘‘transition points’’ denotedby the white circles in Fig. 3. After these transition points, and withthe exception of the CNT (5,5), all the V-derivative curves continueto increase with a much lower slope until they reach the maximumvalue (‘‘limit points’’, denoted by the solid circles in Fig. 3) and thenstart decreasing (causing the ‘‘peaks’’ in the oV/ou and oV/o/ curves).As mentioned before, the exception is the CNT (5,5), for which thetransition and limit points match. Those ‘‘transition points’’ corre-spond precisely to the transition between the ‘‘quadratic’’ and ‘‘lin-ear’’ trends of the energy curves, which cannot be determineddirectly from the inspection of the V curves in Fig. 3a. Indeed, allCNTs buckle when this transition point is achieved but the bucklingphenomena takes place softly, without a dynamic jump and abruptloss of strain energy. Similar evidence was identified by Zhang andShen [18] for armchair and zig-zag CNTs under pure twisting.

From the observation of Fig. 3b, we conclude that all CNTs havesimilar pre-critical stiffness (slope), but they buckle for differentcritical strains and angles of twist per unit of length: ec = 0.04200and ac = 0.01466 rad/Å for CNT (8,0), ec = 0.03659 and ac =

(8,0) (5,5)

(a) (b)Fig. 4. Critical modes (primary buckling) and post-critical shapes (secondary buckling)(b = 45�).

0.01277 rad/Å for CNT (5,5), ec = 0.04484 and ac = 0.01565 rad/Åfor CNT (6,3) under direct twisting, ec = 0.04657 and ac =0.01625 rad/Å for CNT (6,3) under inverse twisting. Moreover,buckling takes place at similar critical energies Vc: 0.11304,0.09113, 0.10785 and 0.12223 eV/atom for CNTs (8,0), (5,5),(6,3)-direct and (6,3)-inverse, respectively. We conclude thatbuckling (white circles) is triggered by the change between qua-dratic variation and non-quadratic variation of energy curves.The critical buckling mode shapes of CNTs (8,0), (5,5) and (6,3) un-der combined shortening–twisting with b = 45� are shown in Fig. 4(left side figures for each pair of configurations). It is seen that allCNTs buckle in a mixed mode: the flattened cross-sections spiralaround the tube axis with non-uniform rate (helix shape), showingan abrupt variation of twisting rotation near the kink. The forma-tion of this helix-kinked mechanism is due to the combined short-ening–twisting rate. Immediately after the primary buckling (i.e.,the path between the white circles and solid circles in Fig. 3b),all the CNTs show very small post-critical stiffness due to the for-mation of the kink. It is interesting to note that CNT (5,5), whichdoes not show any sudden drop, is the one that exhibits the steep-est descent in the post-critical branch.

(6,3)-direct (6,3)-inverse

(c) (d) of CNTs (8,0), (5,5), (6,3)-dir and (6,3)-inv under combined shortening–twisting

1814 B. Faria et al. / Composites Science and Technology 71 (2011) 1811–1818

Now, let us unveil the emergence of the small sudden drops inthe energy curves of zig-zag (8,0) and chiral (6,3) CNTs associatedwith the ‘‘limit points’’, mentioned earlier and denoted by solidcircles in Fig. 3. First, it should be noted that the CNTs are alreadybuckled before these sudden drops. Due to the significant twist-ing rotation and related axial shortening, the CNT cross-sectionbecomes too much ovalised and the graphene sheet becomestoo much bent in the middle of the helix shape. Then, a secondarybuckling mode takes place: the flattened CNT buckles itself in thislocalised zone. It happens like if the kink of the CNT suddenlymoves sideway due to the action of axial shortening, as it isdepicted in Fig. 4 (right side figures for each pair of configura-tions). These secondary buckling modes are associated with sud-den small drops of strain energy shown in Fig. 3a and happensimultaneously for the CNTs (8,0), (6,3)-direct and (6,3) inverse,at ec = 0.05345 and ac = 0.01866 rad/Å. In opposition to these

(a)

(c)Fig. 5. Variation of energy V with parameters c and b for a given C

CNTs, the primary buckling mode of CNT (5,5) is characterisedby lateral displacement of the kinked-helix at mid-height. Thisimperfection lowers so much the CNT post-buckling stiffness afterthe ‘‘transition point’’ that it becomes negative (decreasing slopein Fig. 3b) and no secondary buckling (‘‘limit point’’) occurs. Thisis why the energy curve of CNT (5,5) does not exhibit a uniquesudden drop.

Now, let us turn our attention to the energy curves V(c) of CNTsunder combined shortening–twisting conditions with the wholerange of b parameter values, which are depicted in Fig. 5. The aux-iliary c parameter, c = e2 + a2, is used to allow the simultaneousrepresentation of all energy curves, including those correspondingto b = 0� (pure shortening) and b = 90� (pure twisting). Each V(c)curve corresponds to a given b value and the corresponding twist-ing–shortening rate, D//Du, and the onset of buckling in eachcurve is denoted by a white circle.

(b)

(d)NT: (a) (8,0); (b) (5,5); (c) (6,3) direct; and (d) (6,3) inverse.

Fig. 6. Variation of critical strain energy Vc (at the onset of buckling) withparameter b.

B. Faria et al. / Composites Science and Technology 71 (2011) 1811–1818 1815

Independently of the CNT, it is clear that dynamic jumps hap-pen more frequently and with higher amplitudes for lower b val-ues. It means that decreasing the amount of twisting andincreasing amount of shortening leads to an increasing tendency

β =0º β =15º β =30º β =45º

Fig. 7. Critical modes of CNT (8,0) under various twisti

Fig. 8. Interaction diagrams between the critical axial strain ec and the critical twistingbetween MD results and analytical estimates.

of CNTs to undergo dynamic jumps accompanied by sudden lossesof strain energy. The evidence of high drop of energy in CNTs underpure compression has been shown by several authors [5] and willnot be detailed herein. Moreover, the existence of small suddendrops in the V(c) curves is due to either dynamic jumps of CNTsto adjacent equilibrium configurations or, for very large twistingrotation (outside the scope of this investigation) to C–C bondbreaks.

For comparison purposes, Fig. 6 shows onset-of-buckling curvesas a function of b for (8,0), (5,5), (6,3)-direct and (6,3)-inverseCNTs. In the 0� 6 b 6 60� range (when shortening is dominant rel-ative to twisting), the strain energy at the onset of buckling (Vc)drops considerably with the increase of b. When twisting is moreinfluential than shortening, in the 60� 6 b 6 90� range, Vc decreasesat a slower rate. Zhang and Shen [10] also found a similar behav-iour for the (8,8) CNTs under combined twisting–shortening: intheir MD study, the strain energy at the onset of buckling increasedwhen shortening got more dominant than twisting. It is also possi-ble to conclude that, regardless of b, the (6,3) CNT under inversetwisting has always the highest Vc, the (5,5) CNT exhibits alwaysthe lowest Vc, and the (8,0) and (6,3)-direct CNTs exhibit interme-diate values. These facts mean that the amount of twisting–shortening rate D//Du affects the critical strain energy (Vc) but

β =60º β =75º β =90º

ng–shortening rates (b in degree; D//Du in rad/Å).

angle per unit of length ac, obtained from (a) MD simulations and (b) comparison

Fig. 9. Post-critical modes of CNT (8,0) under various twisting–shortening rates (b in degree; D//Du in rad/Å).

1816 B. Faria et al. / Composites Science and Technology 71 (2011) 1811–1818

does not influence the relative behaviour of the CNTs under thecombined strains: the chiral and armchair CNTs being the mostand less energetic CNTs, respectively.

The critical buckling mode shapes of CNT (8,0) under varioustwisting–shortening rates (b; D//Du) are shown in Fig. 7. Asthe twisting–shortening rate D//Du increases, it is seen thatthe CNT deformation is less localised. A pure kink (localiseddeformation) exists if D/ = 0 and a helix flattened shape (distrib-uted deformation) exists if Du = 0. Between these limit situations,the critical mode is mixed: the kink mode component is prevalentfor D//Du > 0.20 rad/Å, the helix mode is dominant for D//Du > 0.20 rad/Å, and a true coupled mode exists for 0.20 < D//Du < 0.60 rad/Å.

Fig. 8a shows the interaction diagram between the critical axialstrain ec and the critical angle of twist per unit of length ac. Itexhibits non-linear behaviour. Albeit the similar aspect ratio, L/D,of all studied CNTs, the interaction curves display different valuesfor distinct CNT chirality. As the twisting–shortening rate D//Dudecreases, the interaction curves ec–ac of CNTs (8,0) and (5,5) be-come closer and coincide for b = 0� (D//Du ? 0). An analogousconclusion can be reached for the interaction curves ec–ac of CNTs(6,3)-direct and (6,3)-inverse. It should be mentioned that Zhangand Shen [10] also found qualitatively similar curves for the Fc–Tc

interaction of CNTs (8,8) and (14,0). If a non-linear interactivecurve is used to fit the MD data (minimising the error usingLSM–Least Square Method), the following expression is obtained,

ec

e0c

� �1:5

þ ac

a0c

� �1:5

¼ 1 ð2Þ

where e0c is the critical strain for pure shortening and a0

c is the crit-ical twisting angle per unit of length for pure twisting (circles lyingin the vertical and horizontal axes of Fig. 8a). The curves corre-sponding to Eq. (2) are drawn in Fig. 8b and follow the MD datafairly well. For a given value of the twisting–shortening rate, D//Du, the following expression should be used with Eq. (2) to deter-mine the values of ec and ac,

ac

ec¼ D/

Duð3Þ

In order to solve the system of Eqs. (2), (3), we only need thevalues of e0

c (pure shortening), a0c (pure twisting) and the applied

twisting–shortening rate D//Du. These values can be determinedapproximately from available continuum mechanics formulae [16].

In Equation (2), and regardless of the CNT chirality, the interac-tion coefficient for buckling of CNTs under combined shortening–twisting is equal to 1.5. To the authors’ knowledge, this evidence(interaction factor of 1.5) has never been achieved by other inves-tigations in the field of CNTs. This fact means that CNTs possess ashortening–twisting interaction curve between the linear interac-tion (coefficient 1.0) and quadratic interaction (coefficient 2.0). Itshould be mentioned that circular tubes at macro scale display ashortening–twisting interaction coefficient close to 2.0 (quadraticinteraction). Due to the fact that the interaction coefficient of CNTsis much lower (25%) than that of macro-scale tubes, we concludethat CNTs are much more sensitive to buckling under shorten-ing–twisting interaction than macro-scale tubes.

Finally, Fig. 9 shows the post-critical deflected shapes of the(8,0) CNT under various twisting–shortening rates (b; D//Du).For decreasing twisting–shortening rate D//Du, the (8,0) CNT ismore prone to lateral deflection, which is a direct consequence ofa kink formation. For D/ = 0, the CNT shows a huge lateral deflec-tion at mid-height section (kink) for post-critical stages. Thismechanism exhibits null post-critical stiffness: the shortening dis-placement increases under constant force. For Du = 0, the CNT doesnot bend laterally since no kink develops at post-critical stages.This mechanism possesses positive post-critical stiffness: thetwisting rotation increases under increasing torque. For combinedtwisting–shortening (cases between these limit situations), thecritical mode is mixed. However, the mechanism exhibits neithernull nor positive post-critical stiffness, but rather negative post-critical stiffness: the CNT deformation increases under decreasingcombined force–torque.

4. Conclusion

A theoretical study on the strength and stiffness of CNTs undercombined shortening and twisting was presented. CNTs with sim-ilar aspect ratio L/D but different atomic structures (zig-zag, arm-chair and chiral) were selected for the MD simulations. Firstly,the pre-critical, critical and post-critical behaviour of CNTs undercombined shortening and twisting was thoroughly investigated.The main results were shown in the form of diagrams of interac-tion between the critical axial strain (ec) and the critical angle oftwist per unit of length (ac). An interaction equation was proposedand its estimates were compared with MD results. The followingconclusions can be drawn:

B. Faria et al. / Composites Science and Technology 71 (2011) 1811–1818 1817

(i) All CNTs have similar pre-critical stiffness (chiral CNTs exhi-bit slightly higher stiffness). As far as buckling is concerned,armchair CNTs and chiral CNTs under direct twisting aremore sensitive to buckling than zig-zag CNTs and chiralCNTs under inverse twisting;

(ii) Except for the armchair CNTs, all other CNTs have qualita-tively similar post-critical curves: a small nearly horizontalpath is followed by a sudden drop of force correspondingto a dynamic jump (sudden loss of strain energy) to anotherequilibrium configuration, followed by another almost hori-zontal path. The armchair CNT does not exhibit dynamicjump but its post-critical curve always decreases at a higherrate than the other CNTs. Armchair CNTs have the lowest(more negative) post-critical stiffness;

(iii) If shortening is more dominant than twisting, the strainenergy at the onset of buckling drops considerably withthe increase of the twisting–shortening rate. If twisting ismore influential than shortening, the energy at the onsetof buckling decreases very slowly with the twisting–short-ening rate;

(iv) Regardless of the CNT structure orientation, the interactiondiagrams ec–ac have similar qualitative non-linear shapeswith an interaction coefficient equal to 1.5 that is indepen-dent of the CNT chirality. However, there are different inter-action curves ec–ac for each CNT structure due to theinfluence of the CNT chirality on the values of e0

c (criticalstrain for pure shortening) and a0

c (critical twisting angleper unit of length for pure twisting).

(v) The interaction curves of armchair and zig-zag CNTs divergefor increasing twisting–shortening rate. The interactioncurves of chiral CNTs under direct and inverse twisting alsomove slightly away for increasing twisting–shortening rates.

Table A1Average and standard deviation values of strain energy V of some CNTs under pure shorte

Pure shortening (b = 0�) Shortenin

V (eV/atom) V (eV/ato

CNT Average Sd. Dev Average(8,0) 0.22421 0.00116 0.09664(5,5) 0.22674 0.00109 0.07302(6,3) direct 0.27089 0.00121 0.08735(6,3) inverse 0.27089 0.00121 0.09963

Table A2Values of critical strain energy Vc, critical strain ec and critical angle of twist per unit oftwisting.

b (�) CNT (8,0) CNT (5,5)

Vc (eV/atom) ec (–) ac (rad/Å) Vc (eV/atom) ec (–) ac (rad/Å)

0 0.25289 0.08099 0 0.26293 0.08025 05 0.23827 0.07853 0.00240 0.23347 0.07574 0.00231

10 0.23003 0.07657 0.00471 0.20534 0.07071 0.0043515 0.20777 0.07198 0.00673 0.16909 0.06630 0.0062020 0.19844 0.06901 0.00877 0.15235 0.06053 0.0076925 0.17702 0.06362 0.01036 0.13952 0.05551 0.0090430 0.16217 0.05892 0.01187 0.12410 0.05030 0.0101435 0.14792 0.05396 0.01319 0.10813 0.04498 0.0109940 0.12503 0.04715 0.01381 0.08960 0.03883 0.0113745 0.11304 0.04200 0.01466 0.09113 0.03659 0.0127750 0.10992 0.03818 0.01588 0.07815 0.03122 0.0129955 0.09539 0.03221 0.01606 0.07287 0.02726 0.0135960 0.08929 0.02754 0.01665 0.06465 0.02270 0.0137365 0.08379 0.02282 0.01708 0.06612 0.01964 0.0147070 0.08517 0.01884 0.01807 0.06161 0.01553 0.0148975 0.08669 0.01453 0.01893 0.06299 0.01203 0.0156780 0.08831 0.00994 0.01968 0.06737 0.00843 0.0167085 0.09019 0.00508 0.02028 0.06607 0.00423 0.0168990 0.09561 0 0.02111 0.06499 0 0.01696

The CNT (6,3) behaves differently for inverse and directtwisting: the direct twisting of CNT (6,3) implied the ‘‘curl-ing’’ of the CNT helix structure and prompted the onset ofbuckling, while the inverse twisting of CNT (6,3) impliedthe ‘‘straightening’’ the CNT helix structure and delayedthe occurrence of buckling.

(vi) To the authors’ knowledge, it is the first time that an inter-action factor of 1.5 has been obtained from MD simulationsof CNTs. Since circular tubes at macro-scale display a short-ening–twisting interaction coefficient close to 2.0 (quadraticinteraction), which is 25% higher than the one obtained forCNTs, we conclude that CNTs are much more sensitive tobuckling under shortening–twisting interaction thanmacro-scale tubes.

(vii) The observed here off-axis instability and buckling is cor-rectly detected in present simulations (and some earlierwork such as [4]), but can be fully missed if a smaller heli-cal-symmetry based periodic cell is used [19,20].

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).

Appendix A

See Tables A1 and A2.

ning, combined shortening–twisting and pure twisting.

g–twisting (b = 45�) Pure twisting (b = 90�)

m) V (eV/atom)

Sd. Dev. Average Sd. Dev.0.00115 0.07985 0.001130.00119 0.05167 0.001120.00111 0.05478 0.001190.00118 0.07301 0.00117

length ac for CNTs under pure shortening, combined shortening–twisting and pure

CNT (6,3) – dir CNT (6,3) – inv

Vc (eV/atom) ec (–) ac (rad/Å) Vc (eV/atom) ec (–) ac (rad/Å)

0.31389 0.09390 0 0.31389 0.09390 00.27801 0.08869 0.00271 0.27887 0.08869 0.002710.24325 0.08287 0.00510 0.29386 0.09007 0.005540.20332 0.07539 0.00705 0.24980 0.08246 0.007710.18601 0.07105 0.00903 0.22340 0.07678 0.009750.16883 0.06632 0.01079 0.19817 0.07074 0.011510.14132 0.05914 0.01192 0.18058 0.06548 0.013200.12638 0.05394 0.01318 0.16914 0.06094 0.014890.11696 0.04951 0.01450 0.14708 0.05418 0.015870.10785 0.04484 0.01565 0.12223 0.04657 0.016250.09546 0.03919 0.01630 0.10878 0.04076 0.016960.08778 0.03427 0.01709 0.10881 0.03707 0.018480.07741 0.02866 0.01733 0.09656 0.03110 0.018800.08422 0.02577 0.01929 0.09658 0.02680 0.020060.07489 0.02002 0.01920 0.09292 0.02169 0.020800.07873 0.01578 0.02056 0.09315 0.01673 0.021790.07357 0.01038 0.02054 0.09695 0.01165 0.023060.07799 0.00542 0.02163 0.09762 0.00595 0.023750.06812 0 0.02043 0.08854 0 0.02299

1818 B. Faria et al. / Composites Science and Technology 71 (2011) 1811–1818

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