Materials Science & Engineering A 582 (2013) 225–234

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Materials Science & Engineering A

0921-50http://d

n CorrE-m

s.h.ghad

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Nonlinear analysis of coiled carbon nanotubes using the moleculardynamics finite element method

Seyed Hadi Ghaderi n, Ehsan HajiesmailiDepartment of Mechanical Engineering, Shahrood University of Technology, Shahrood 3619995161, I.R. Iran

a r t i c l e i n f o

Article history:Received 23 March 2013Received in revised form18 May 2013Accepted 21 May 2013Available online 4 June 2013

Keywords:Coiled carbon nanotubeMolecular dynamics finite element method(MDFEM)FractureNonlinear analysis

93/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.msea.2013.05.060

esponding author. Tel./fax: +98 2733 300258.ail addresses: s.h.ghaderi@shahroodut.ac.ir,eri@gmail.com (S.H. Ghaderi).

a b s t r a c t

The mechanical response of single-walled helically coiled carbon nanotubes (CCNTs) under large axialdeformations is examined using a molecular dynamics finite element method. The 3D referenceconfiguration of CCNTs is determined based on a 2D graphene layer using conformal mapping. Threesets of analyses are performed to fully describe the mechanical response of ðn;nÞ CCNTs under elongationup to the bond breaking point and compression down to the solid length or the onset of bucklinginstability. First, the strain dependency of the mechanical properties of individual CCNTs duringdeformation is investigated by calculating the stress-strain curve and the spring constant of the CCNTsfor the entire load range. Significant responses including brittle fracture under tension and bucklinginstability under compression are observed. Second, to examine the size dependence of the mechanicalproperties, several CCNTs with different geometric parameters are constructed, and their spring constant,fracture strain, fracture load, and energy storage density are determined. All CCNTs exhibit a super-elasticity of 50–66%. A comparison between the mechanical properties of CCNTs and those of carbonnanotubes (CNTs) reveals that the fracture load and energy storage per atom of CCNTs is lower than thatof the corresponding armchair CNTs.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

The ability of carbon to be tailored into virtually any basicshape on a nanometre scale provides the flexibility necessary toachieve the required properties for a specific application. Amongthe wide range of carbon morphologies, coiled carbon nanotubes(CCNTs) are of particular interest due to their unique properties.The 3D model of the CCNTs was first predicted after the discoveryof straight carbon nanotubes (CNTs) by molecular dynamics (MD)simulations [1,2]. The 3D spiral structure of CCNTs was achievedby introducing pentagonal and heptagonal defects at regularintervals to a perfect hexagonal lattice. A few years after theprediction of CCNTs, their experimental synthesis and character-isation were reported [3,4].

CCNTs promise a myriad of unique applications. They are idealcandidates for reinforcing and modifying the properties of light-weight, high-strain composites. Because of their coiled shape, theycan provide a better anchor to the matrix and a higher loadtransfer efficiency than CNTs [5]. Similarly, their superelasticproperties can enable CCNTs to function as nano-springs [6,7],

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nano-sensors, components of micro/nano-electromechanicalsystems [8,9] and nano-robots [10].

To realise the application of CCNTs, it is essential to understandtheir mechanical properties. Several experimental and theoreticalstudies have been performed to investigate the mechanical proper-ties of CCNTs. Chen et al. [6] studied the spring constant andmaximum strain of an individual doubly twinned CCNT with atubular diameter of approximately 126 nm. The CCNT was clampedbetween two atomic force microscopy cantilevers and were loadedin tension to a maximum elongation of 42%. The experimentrevealed that the behaviour of CCNT was similar to that of anelastic spring with a spring constant of 0.12 N m−1. No plasticdeformation was observed in this experiment. In another study[11], a manipulator-equipped scanning electron microscope wasused to study the mechanical and electrical properties of severalCCNTs. The results demonstrated that CCNTs could be expanded toan exceptionally high strain of 200%. Liu et al. [7] performedatomistic quantum simulations on a few ultrathin carbon nanocoilswith a tubular diameter and a pitch length as low as 1 nm under anaxial load. The CCNTs exhibited superelastic behaviour and wereable to withstand strains as high as approximately 60% in tensionand 20–35% in compression. Similarly, molecular structuralmechanics (MSM) were employed to investigate the mechanicalproperties of CCNTs in the low strain region [12]. The simulationsrevealed that as the tube diameter increased, the spring constantand shear modulus of the CCNTs also increased. In another recent

Bond AngleBending

Bond Stretching

Torsion

Non -Bonded Interaction

Fig. 1. Schematic of atomic interactions described by the molecular mechanicsmethod.

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study [13], the mechanical properties of an individual CCNT underan axial load were examined and compared with those of a CNTwith similar dimensions. The results indicate that the CCNT wassignificantly less stiff than the CNT and exhibited a much largerpullout force when embedded in a polyethylene matrix. Thestructure of the CCNT, however, was not described in great detail.

Despite these promising reports, our knowledge of themechanical properties of CCNTs remains limited. In particular,the size-dependent mechanical properties of CCNTs and theunderlying mechanisms describing these exceptional propertiesare yet to be illuminated. In this paper, after explaining the basicsof the molecular dynamics finite element method (MDFEM), thespecial-purpose finite elements utilised in this method are pre-sented. A systematic approach is introduced to model the complex3D structure of CCNTs by performing relaxation analyses onintentionally defected CNTs. Using MDFEM, the mechanicalresponse of the CCNTs under an axial load in the high and lowstrain regions is explored. In addition, the mechanism that governsthe behaviour of CCNTs under tension and compression isexplained. The effect of the dimensions of CCNTs on their mechan-ical properties is also discussed.

2. Finite element model construction

Computational methods are powerful tools for enhancing thescientific understanding of carbon nanostructures. Among thebroad array of these methods, rigorous approaches, such asdensity functional theory (DFT) and tight binding (TB), can beused to solve any problem involving atomic or molecular motion.However, due to the significant computational requirements, theapplication of these techniques is limited to systems containing asmall number of atoms. An alternative approach is to usecontinuum mechanics, which can handle nano-mechanics pro-blems at a low computing cost, but at the expense of accuracy.Molecular mechanics (MM) calculations, also known as force fieldcalculations, can be viewed as a link between quantum andcontinuum mechanics. In MM, the atomic structure is treated asa multi-particle system, in which each atom (particle) interactsonly with the atoms that neighbour it directly. The atomicinteractions are described by the force fields or interatomicpotentials, which depend solely on the relative positions of theatoms involved.

The general expression for the total interatomic potentialenergyEt, in which the electrostatic interactions are neglected, isthe sum of energy contributions attributed to the valence orbonded interactions Eval and the non-bonded interactions Enb.This relationship is described by the following equation:

Et ¼ Eval þ Enb ¼ Er þ Eθ þ Eφ þ Eψ þ EvdW ð1Þwhere Er is bond stretching for a 2-atom interaction, Eθ isassociated with the bending bond angle for a 3-atom interaction,Eφ and Eψ account for the dihedral angle torsion and inversionbetween 4 atoms, respectively. The last term, EvdW, describes thenon-bonded van der Waals interactions for a 2-atom system and isneglected in our simulations. The kinematic parameters includingthe bond length r, the valence angle θ, the torsion angle φ and theinversion angle ψ can be stated explicitly based on the relativespatial positions of the atoms. Fig. 1 provides a schematic illustra-tion of the MM interpretation of these interactions. The atoms areconsidered to be point masses, whereas the bonds are differenttypes of springs that connect these points together.

When MM is applied to linear elastic problems, standard truss[14] and beam [12,15] finite elements, the parameters that areadjusted based on the harmonic form of the potential terms inEq. (1), can successfully predict the mechanical properties of

carbon nanostructures. However, in a straightforward applicationof truss and beam elements, some problems are inevitable. Thematerial parameters for bond bending and torsion cannot beidentified uniquely for each interaction, and there is no distinctionbetween natural and equilibrium bond angles [16]. Furthermore,standard elements with constant parameters cannot be used tomodel the nonlinear force field potentials. Likewise, if a probleminvolves geometric nonlinearities, standard elements cannot beused to model the bending and torsion interactions. To overcomethese challenges, special-purpose finite elements formulatedbased on the force field are employed [17,18], such that the bondstretching, bond angle bending and torsion are considered sepa-rately. Furthermore, the absence of rotational degrees of freedom(DOF) at the element nodes leads to a more computationallyeffective approach compared with the use of beam elements. Thehigh accuracy and effectiveness of the MDFEM, as well as numer-ous successful applications of this method in a wide range ofstudies [16–20], inspired us to adopt the MDFEM for the analysis ofCCNTs. In the following section, the force field potentials describedby DREIDING [21] are investigated, and the basics of the MDFEMare briefly reviewed.

2.1. The DREIDING force field

In this section, the DREDING potentials applicable to carbonstructures with sp2 hybridisation, such as graphene and nano-tubes, are discussed. According to DREIDING, the interactionpotentials that account for nonlinearities at large deformationsare described as follows:

Er ¼De e−β rij−reð Þ−1h i2

ð2Þ

Eθ ¼12Cijk cos θijk− cos θ0j

� �2ð3Þ

Eφ ¼12Vijkl 1− cos njk φijkl−φ

0jk

� �h in o: ð4Þ

The harmonic forms of these potentials, suitable for smalldeformations, are described as follows:

Elinr ¼ 12ke rij−re� �2 ð5Þ

Elinθ ¼ 12Kijk θijk−θ0j

� �2ð6Þ

Fig. 2. Molecular dynamics finite elements and the atomic kinematics: (a) a 2-nodebond stretching element where rij is the distance between two atoms, I and J; (b) a3-node bond angle bending element; (c) a 4-node torsion element; and (d) thedefinition of the valence angle, θijk , and the torsion angle, φijkl , based on the unitvectors provided in parts (a)-(c).

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Elinφ ¼ 14Vijkln

2jk φijkl−φ0

jk

� �2: ð7Þ

where rij is the bond length; θijk is the valance angle between twobonds, IJ and JK; φijkl is the dihedral torsion angle defined by thetwo planes, IJK and JKL, as shown in Fig. 2; and re, θ0j and φ0

jk aretheir equilibrium values, respectively. The equilibrium values andother constants in Eqs. (2)–(7) are as follows [21]:

De ¼ 0:7295 aJ; β¼ 22:365 nm−1; ke ¼ 729:79 N⋅m−1; re ¼ 0:139 nm

Cijk ¼ 0:926 aJ; Kijk ¼ 0:695 aJ; θ0j ¼ π=3; Vijkl ¼ 0:1737;

njk ¼ 2; φ0jk ¼ 0 or π

2.2. Molecular dynamics finite element method

In MDFEM, each n-atom interaction is considered as an n-nodefinite element with only three translational DOF at each node. Thestiffness and the residual force of each element are calculatedanalytically using standard interatomic force fields. There aremany force fields for describing atomic interactions, for example,the case of the DREIDING force field, n∈f2; 3; 4g. In this study, the

development of the finite elements is presented based on theDREIDING force field.

The finite elements associated with each interaction are shownin Fig. 2. Because no DOF rotations are considered for the nodes,the bond bending angle and torsion moments are described asforce couples acting on the nodes of relevant elements. Theresidual force on each element is obtained from the first derivativeof the related energy E of the element with respect to the nodalcoordinates. The MDFEM can be used under static and dynamicconditions. The stiffness matrix of the element, required for thestatic analyses employed in this paper, is obtained from thederivative of the residual force vector of the element with respectto the nodal coordinates. For the sake of brevity, only the residualforce vector for each element is discussed herein.

2.2.1. 2-node element for bond stretchingFor a 2-node element shown in Fig. 2 .a, the nodal residual

forces are described as follows:

FI ¼−FJ ¼−Fijuij ð8Þwhere uij is the unit vector from node I to J and Fij is the forcemagnitude, obtained from the derivative of the force potential.From the Morse function in Eq. (2), the following relationship isobtained:

Fij ¼ −2βDe e−2β rij−reð Þ−e−β rij−reð Þh ið9Þ

The harmonic form of Eq. (5) can be described as follows:

F linij ¼ ke rij−re� �

: ð10Þ

where ke is the stiffness of the linear element.

2.2.2. 3-node element for bond angle bendingThe nodal residual forces for a 3-node element are essentially

described by the bending moment when the bond angle θijkdeviates from its equilibrium value θ0j as follows:

FI ¼ Fijkui; FK ¼ Fkjiuk; FJ ¼ −ðFI þ FK Þ ð11Þ

where ui and uk, are unit vectors in the IJK plane and perpendi-cular to the IJ and JK bonds, respectively, as shown in Fig. 2.b. Thederivative of ∂Eθ=∂θijk yields the bending angle moment M. Usingthe lever arm rij, Fijk is found using the harmonic cosine formobtained from Eq. (3) as follows:

Fijk ¼ −Cijk

rijsin θijk cos θijk− cos θ0j

� �: ð12Þ

The linear force F linijk derived from the harmonic angle form inEq. (6) is provided as follows:

F linijk ¼Kijk

rijθijk−θ0j

� �ð13Þ

The bond angle θijk is defined according to the orientation ofthe unit vectors uij and ujk as follows:

θijk ¼ acosð−uij⋅ujkÞ ð14Þ

Considering the moment balance of forces FI and FK about J, theother force magnitude Fkji is obtained as follows:

Fkji ¼rjkrij

Fijk ð15Þ

2.2.3. 4-node element for torsionThe nodal residual forces for a 4-node element (see Fig. 2.c)

reflect the torsion moment opposing any deviation from theequilibrium value φ0

jk of the dihedral angle φijkl from the

Fig. 3. Stress-strain curve of a ð5;5Þ CNT obtained using the proposed methodcompared with the density functional theory (DFT) results reported in [22].

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equilibrium. These four residual forces include:

FI ¼ Fiijkuijk

FL ¼ Fljklujkl

FJ ¼ Fjijkuijk þ Fjjklujkl

FK ¼ Fkijkuijk þ Fkjklujkl: ð16ÞThe first index of the force magnitudes refers to the node

involved and the remaining indices describe the IJK and JKLplanes. The force magnitude, Fiijk, is determined fromFiijk ¼ T=ðrij sin θijkÞ; where T is the torsion moment obtained fromthe derivative of the torsion energy with respect to φijkl,T ¼ ∂Eφ=∂φijkl, and rij sin θijk is the lever arm. In this manner, Eq.(4) is transformed into the following equation:

Fiijk ¼ −Vijklnjk

2rij sin θijksin njk φijkl−φ

0jk

� �h ið17Þ

and Eq. (17) into the following equation:

F liniijk ¼ −Vijkln2

jk

2rij sin θijkφijkl−φ

0jk

� �: ð18Þ

There are two values for the equilibrium dihedral angle φ0jk:

0 and π, according to the relative orientation of IJK and JKL planes.Based on this consideration

φijkl− φ0jk ¼

asin ujk uijk � ujkl� �� �

; uijkujkl≥0−asin ujk uijk � ujkl

� �� �; uijkujklo0

(ð19Þ

Considering the equilibrium of moments and forces, the fol-lowing equations are obtained:

Fljkl ¼ −rij sin θijkrkl sin θjkl

Fiijk

Fjjkl ¼ −rkl cos θjkl

rjkFljkl

Fkijk ¼−rij cos θijk

rjkFiijk

Fjijk ¼ −ðFiijk þ FkijkÞFkjkl ¼−ðFljkl þ FjjklÞ ð20Þ

It should be noted that to avoid coupling the energy terms, theinstantaneous magnitudes of the lever arms and angles are used inEqs. (12), (13), (15), (17), and (20). In contrast, if the equilibriumvalues of angles and bond length were used in the calculations oflarge deformations, as is performed in [19], the actual angle bendand torsion moments produced by the nodal forces would be verydiffer from the results obtained using ∂Eθ=∂θijk and ∂Eφ=∂φijkl,respectively.

2.3. Development and verification of finite elements

According to the formulation described in the previous section,three ABAQUS user elements (UELs) corresponding to the 2-,3- and 4-node elements were developed. The nonlinear systemof equations was then solved using the ABAQUS/standard solver.The individual elements are examined by performing verificationtests at large deformations and are validated by comparing thenumerical and analytical results from the force field. Furthermore,to assess the reliability of the present MM model and its para-meters for the analysis of CCNTs, straight CNTs were consideredfirst because of the wealth of literature available for thesematerials. For example, the stress-strain curve of a ð5;5Þ armchairCNT is obtained under tension and compared to the results of theDFT simulations reported in [22]. First, the near-to-equilibriumconfiguration is constructed using a MATLAB code. The relaxedconfiguration, representing the equilibrium state for the atomicstructure with no external load, is then obtained using a staticanalysis. Finally, one end is fixed while the other end is

incrementally displaced to stretch the CNT until fracture. Theresulting stress-strain curve is compared with the DFT results inFig. 3. In this case, the CNT is viewed as a thin cylinder with a wallthickness of 0.34 nm. These results suggests that the use of theDREIDING force field in formulating the finite elements in theMDFEM provides good agreement with the DFT predictions [22]over a wide range of strains. When the modified Morse potential[23] was employed in the MDFEM, a Young’s modulus of 1047 GPawas obtained, which is similar to the value of 1032 GPa deter-mined using the DFT approach [22]. When the DREIDING forcefield was employed, the MDFEM generated a Young’s modulus of866 GPa.

2.4. CCNT model construction scheme

The first step in understanding the behaviour of CCNTs underan external load is to determine the reference structure. Unlikestraight CNTs that have a simple translational symmetry, CCNTshave a screw type symmetry and their construction procedure isnot straightforward. The main difference between CCNT and CNT isthat the former must contain non-hexagonal rings [2]. Thesubstitution of some hexagons by pentagons and heptagons inthe carbon structures leads to positive and negative curvatures,respectively [24]. These non-hexagonal ring patterns are respon-sible for the bending of a straight CNT and its transformation into aCCNT [25].

To construct a CCNT, Ihara et al. [2] cut a nanotorus into piecesand stretched it in the direction of the fibre axis. Alternatively,Akagi et al. [26] drew a development map including pentagonalholes of a pristine graphene sheet and formed it into a CCNT bycoupling the sides of the holes together. László and Rassat [27]approached the construction of CCNTs using a topological coordi-nate method for fullerenes and the null space embedding ofgraphs. Chung et al. [28] summarised the aforementioned con-struction and classification methods and proposed a more generalscheme for obtaining CCNTs with various conformations [29]. Liuet al. [7] introduced a pair of heptagons and a pair of pentagons ontwo sides of a CNT to form a basic building block and thenconstructed the entire CCNT by assembling several building blockstogether.

Fig. 4. The modelling steps used to describe a CCNT: (a) the preparation of a development map on a pristine graphene sheet and the introduction of holes in the pattern,(b) the rolling of the graphene sheet to construct a defective CNT and (c) the coupling of the sides with defects to form a CCNT.

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A facile and systematic approach for the construction of CCNTsincorporating only hexagonal, heptagonal and pentagonal ringswas proposed in [12]. This method is also implemented in thisstudy, and the CCNT is constructed in three steps:

1.

An array of hexagonal holes is introduced into a pristinegraphene sheet (Fig. 4.a).

2.

The graphene sheet is rolled to form a defective CNT (Fig. 4.b)by mapping it onto a cylinder.

3.

The initial stress of bonds is released using a relaxationanalysis, and then the two sides of each hole are joinedtogether. The resulting CNT transforms into a CCNT (Fig. 4.c).

The difference between the initial length, r, of the elementsconnecting the two sides of each hole in a defective CNT and itsequilibrium length, re ¼ 0:139 nm, leads to an initially high stress.When this stress is released using a relaxation analysis, each atommoves to its equilibrium position and the two sides of each holeare joined together. Because of the large deformations involved,

the third step is broken up into two relaxation stages. In the firststage, a linear bond stretching element described by Eq. (10) witha stiffness of ke ¼ 729:79 N⋅m−1 is employed, and a nonlinear staticanalysis is performed to release the initial high stress of theelements connecting the two sides of each hole to obtain anapproximate near-to-relaxed configuration for the CCNT. The finalfree-free reference structure of the CCNT is obtained in the secondrelaxation stage using another nonlinear static analysis, in whichnonlinear bond stretching elements described by Eq. (9) are usedto direct each atom to its equilibrium position.

Fig. 4 illustrates the three steps employed in the modellingprocedure of a CCNT. The structure shown in this figure has beenpreviously studied and referred to as 36I [6,10,4,1] coiled CNT in[2] and helix C360 in [26].

In the proposed modelling procedure, the sharp and openedges at each hole define the two pentagonal and heptagonalcarbon rings at the outer and inner sides of the CCNT, respectively.To specify the dimensions of each hole, only one vector is needed,which is referred to as the hole vector H. In addition, two other

PitchMax. radius

Median radiusMean radius

Min.radius

Tube radius

Fig. 5. Illustration of the reference configuration of a (6, 6) CCNT and the associated geometric parameters: (a) top view and (b) side view.

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vectors, referred to as the arrangement vector A and the rollingvector R, are required to determine the array of holes. The rollingvector also represents the rolling direction and thus determinesthe tube diameter and chirality of the CCNT. These three vectors,which are necessary to fully define the atomic structure of theCCNT, are illustrated in Fig. 4. They can be expressed in terms ofbasic vectors a1 and a2 as ðn;mÞ in the following manner:

V¼ n a1 þm a2 : ð21ÞThe morphological characteristics of the CCNTs, which are

constructed based on the procedure presented herein, arecontrolled by these three vectors. The rolling vector R determinesthe tube diameter. The length and direction of the arrangement forhole vectors A and H, are compared with those of the rollingvector R, to determine the pitch and diameter of the coil.

Fig. 6. Variations in the geometric parameters of (n, n) CCNTs as a function of n.

Fig. 7. Boundary conditions applied to CCNTs under an axial load.

3. Results and discussion

3.1. Geometry of the CCNTs

Based on the proposed modelling approach, several CCNTssimilar to those constructed by Liu et al. [7], are modelled. Amongthe wide variety of possible structures, the CCNTs are constructedfrom the ðn;nÞ armchair CNTs with the widest possible holes andthe minimum possible pitch. These CCNTs are referred to as theðn;nÞ CCNTs. The rolling, hole and arrangement vectors for ðn;nÞCCNTs are referred to as ðn;nÞ, ð−nþ 1;−1Þ and ð−n;nþ 1Þ, respec-tively. The structure of a ð6;6Þ CCNT is illustrated in Fig. 5.

The ðn;nÞ CCNTs exhibit a special morphology. The heptagonaland pentagonal carbon rings are placed at the innermost andoutermost surfaces of the CCNT, respectively, resulting in signifi-cant curvature in the vicinity of the heptagons and pentagons.Consequently, the CCNT looks like a polygon when viewed fromthe top. The sharp points at the outer surface and the highcurvature of ðn;nÞ CCNTs imply that this family of CCNTs is theleast efficient for carrying load and storing energy. This considera-tion can be used in the interpretation of the data, i.e., the resultscan be viewed as an estimate of the minimum ultimate load, strainand energy storage capacity of the CCNTs with the same pitch andradii but different ring patterns.

Fig. 5 demonstrates the structure of a free-free ð6;6Þ CCNT andits associated geometric parameters, revealing that the end type ofthe CCNTs is plain and open. The radii of the CCNT are calculatedbased on the distance between the atoms and the axis of the CCNT.The pitch is the distance between the tube centre and one coil tothe tube centre of the adjacent coil, which is calculated by dividingthe length of the CCNT by the number of pitches. The tubediameter of the CCNT is considered to be equal to the diameterof the corresponding CNT, i.e., the defective armchair CNT basedon which the CCNT is constructed. Fig. 6 shows the change in thegeometric parameters of (n, n) CCNTs as a function of n. For thisspecial type of CCNT, the coil radii, excluding the minimum radius,

are proportional to n. To obtain CCNTs with the smallest possibleinner radius, heptagonal rings constructed on the inner surface ofthe CCNTs should be as close to one another as possible. Conse-quently, the heptagonal ring pattern is the same for all (n, n) CCNTsand the minimum coil radius remains nearly constant with n.The helix angle and pitch of the CCNTs are proportional to 1/nbecause each hole of the defective CNT is circumferentially shiftedby one cell with respect to the previous hole, while the tubecircumference is proportional to n.

3.2. Mechanical response of CCNTs under tension and compression

The deformation behaviour and changes in the mechanicalproperties of individual CCNTs during deformation are investi-gated. Accordingly, the CCNTs are stretched to the failure point andcompressed to the onset of buckling instability or in the absence of

Fig. 8. Axial force-displacement curves obtained for ðn;nÞ CCNTs.

Fig. 9. Illustration of failure for a ð6;6Þ CCNT subject to an axial load. The breakingof bonds under tension is initiated in the heptagonal carbon rings at the innersurface of the CCNT.

Fig. 10. The reference and bucked configurations of ð6;6Þ CCNT under compression.

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buckling, the compression is continued until contact took placebetween the adjacent coils. In these analyses, one end of the CCNTis fixed, while the other end is displaced along the CCNT axis. Thelast row of atoms at the two ends of the CCNT is rigidly attached totwo reference points. All DOF of a reference point at one end areretained as fixed, while the other reference points are allowed tomove parallel to the CCNT axis, and all other DOF are restrained(see Fig. 7). The imposed boundary conditions closely simulate theexperimental conditions in which the clamps can effectivelyprevent the coils from rotation [6].

Fig. 8 illustrates the force displacement curves obtained byperforming nonlinear static analyses on ð6;6Þ, ð10;10Þ, and ð14;14ÞCCNTs. Each CCNT is nearly three pitches long. The resultsdemonstrate the brittle nature of fracture of the CCNTs. Thefracture under tension is initiated in the heptagonal carbon ringsplaced at the inner surface of the CCNT and grows rapidly towardsthe pentagonal rings at the outer surface of the CCNT.

Fig. 9 illustrates the failure of a (6, 6) CCNT under tension andthe location of the broken bonds. Under a tensile deformation, themajority of the load is absorbed by the bonds placed at the innersurface of the CCNT, whereas during compression, the load is moreevenly distributed among the bonds at the inner and outersurfaces. Interestingly, unlike the CNTs, the CCNTs undergo agradual buckling process under compression, which is illustratedin Fig. 10. For CCNTs with a larger n that exhibit a small pitches anda large tube diameter adjacent to the coils in contact with oneanother, the coil reaches its solid length and no buckling isobserved.

The derivatives of the force displacement curves multiplied bythe number of pitches, i.e., the spring constant k (axial stiffness)for one pitch of the CCNTs, are plotted in Fig. 11. As shown in this

figure, stiffness k varies with the axial deformation. As the CCNT isstretched, the length of the bonds and the pitch of the CCNTincrease. These two factors have opposite effects on the stiffness ofthe CCNT during elongation. An increase in the pitch as a resultof elongation increases the stiffness of the CCNTs, as in the case ofcontinuum helical springs [30]. In contrast, a small increase in thebond length throughout elongation leads to a remarkable decreasein the stiffness of the nonlinear bond stretching element (see

Fig. 11. Variations in the spring constant k for one pitch of ðn;nÞ CCNTs under axialdisplacement.

Fig. 12. Bond stretching energy, force and stiffness, as determined using theDREIDING force field.

Fig. 13. The spring constant k for a single pitch of a (6, 6) CCNT as a function of axialdisplacement obtained using linear and nonlinear bond stretching elements anddetermined using DREIDING.

Fig. 14. Spring constant of ðn;nÞ CCNTs for small tensile strains predicted in thisstudy compared with the reported results from tight-binding (TB), density func-tional theory (DFT) and molecular structural mechanics (MSM) calculations.

S.H. Ghaderi, E. Hajiesmaili / Materials Science & Engineering A 582 (2013) 225–234232

Fig. 12). This bond softening phenomenon results in a decrease inthe stiffness, k, for the whole CCNT, which is similar to the case ofCNTs subjected to elongation. For the ðn;nÞ CCNTs with a larger nand a smaller pitch, the effect of bond softening due to stretchingbecomes stronger than the effect of increasing the pitch. Thus forðn;nÞ CCNTs with a higher n, the spring constant k mainlydecreases as the CCNT is stretched (Fig. 11).

To determine the effect of bond softening, the ð6;6Þ CCNT isfurther analysed using the harmonic form of the bond stretchingpotential corresponding to DREIDING (Eq. (5)). At this potential,the bond stretching force varies linearly with the bond length suchthat the bond stiffness remains constant (Fig. 10). The results ofthis analysis compared with those obtained using the nonlinearMorse function (Eq. (2)) are illustrated in Fig. 13. When the bondstiffness is constant, the CCNT spring constant, k, increasesgradually, while the CCNT is deformed. This result demonstratesthat the decrease in the spring constant, k, in Fig. 11 is due to bondsoftening. The two curves for the compression region, however,

are identical, indicating that the bonds experience a small varia-tion in length when the CCNT is loaded under compression. Liuet al. [7] examined the variation of average C–C bond length inCCNTs subjected to axial deformation and demonstrated thatunder compression, the average bond length is only slightlyreduced and the strain energy is stored mainly due to bond anglevariations.

3.3. Size dependent mechanical properties of CCNTs

Several CCNTs are analysed to investigate how the springconstant, fracture load, fracture strain, and energy storage densityof ðn;nÞ CCNTs change as a function of n. The spring constant k foreach CCNT is measured by stretching the CCNT by 0.1 nm. Theslope of the force displacement curve multiplied by the number of

Fig. 16. Energy storage density of (n, n) CCNTs under an axial load compared withthat of CNTs.

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pitches is the spring constant for a single pitch of the CCNT. EachCCNT has nearly three pitches in these analyses. Fig. 14 shows thespring constant, k, for small deformations obtained using theproposed model and compared with those reported in the litera-ture. Although the DREIDING force field provides a good predictionfor the mechanical properties of straight and coiled CNTs in thehigh strain region, it slightly underestimates the stiffness of thesenanostructures in the low strain region. Accordingly, the springconstant of the CCNTs calculated using the modified Morsepotential is in good agreement with the TB and DFT results. Asshown in Fig. 14, the spring constant of ðn;nÞ CCNTs graduallyincreases with n, which suggests that the increase in the tubediameter has a predominant effect rather than the decrease in theCCNT pitch.

Each CCNT is stretched until failure, so the load and displace-ment corresponding to fracture are called the fracture load andfracture displacement, respectively. Fig. 15 shows the fracturestrain of the CCNTs in which the fracture displacement is dividedby the initial length of the CCNT. The CCNTs show exceptionalsuperelasticity under tension with a fracture strain in the range of50% and 66%. The superelastic behaviour for CCNTs was alsopredicted in [7] and observed experimentally in [6].

The maximum load that a CCNT or CNT can withstand before thefracture occurs is called the fracture load. As shown in Fig. 15, thefracture load of (n, n) CCNTs is much lower than that of thecorresponding CNTs and increases at a very slow rate in comparisonto (n, n) CNTs. When a CCNT is stretched, a transverse shear and atorsional shear proportional to the coil radius are induced. Themajority of load is absorbed by the limited number of bonds placedat the inner surface near to the axis of the CCNT where the shearingforce is maximum. Thus, as n increases, a small increase in thenumber of load carrying bonds takes place. In contrast, the coilradius and the torsional shear of the load increase gradually, whichexplains why the fracture load is nearly constant as a function of n.

When a CCNT is stretched, elastic strain energy is stored, whichcan be readily calculated for each n- node element using thecorresponding analytical expression for the n_atom interactionpotential. The energy storage capacity of the CCNT is the ultimatestrain energy stored in the CCNT, i.e., the energy differencebetween the equilibrium position and the position immediately

Fig. 15. Fracture load and strain of ðn;nÞ CCNTs compared with those ofarmchair CNTs.

prior to fracture. The energy storage density is the energy storagecapacity divided by the number of atoms. Fig. 16 provides a plot ofthe energy storage density for (n, n) CCNTs as a function of n. Theresults demonstrate a decrease in the energy storage, whichsuggests that CCNTs with a smaller n index can store higher strainenergy. The majority of energy is stored in the bonds at the innersurface of the CCNT where the greatest curvature is observed. Anincrease in n leads to a very small increase in the number of loadcarrying bonds, whereas the number of atoms steadily increases.Thus, the energy storage density is smaller for the (n, n) CCNTswith a large n compared with that of CCNTs with a smaller n. Theseresults suggest that smaller CCNTs are more efficient for loadcarrying and energy storage purposes.

The mechanical properties of ðn;nÞ CCNTs compared with thoseof ðn;nÞ CNTs are illustrated in Fig. 15. Unlike those of armchairCNTs, the fracture load and fracture strain of CCNTs vary as afunction of n. The CCNTs can withstand a higher elongation, whilethe CNTs can carry a higher load. As expected, because there aremany non-load-carrying bonds in CCNTs, the energy storagedensity of the CCNTs is lower than that of CNTs by approximatelyone order of magnitude.

4. Conclusions

In this paper, the mechanical response of CCNTs under highcompression and tensile loads were investigated using an atomis-tic based finite element method. Finite elements were developedto account for the bond stretching, bending and torsion. Asystematic approach was employed to construct several ðn;nÞCCNTs, which were elongated until failure or otherwisecompressed until buckling occurred or until the adjacent coilscame in contact with one another. The most interesting observa-tions include the following:

�

The force-displacement relationship is nearly linear for a widerange of strains.

�

The fracture is brittle and initiated in the heptagonal carbonrings at the inner surface of the CCNT, growing towards thepentagonal rings at the outer surface of the CCNT.

�

CCNTs exhibit exceptional superelasticity under tension with afracture strain in the range of 50% and 66%.

�

CCNTs with a smaller index n can store higher strain energyand carry a higher load.

�

The energy storage density of the ðn;nÞ CCNTs is one order ofmagnitude lower than that of corresponding armchair CNTs.The CCNTs can withstand higher strains, whereas the CNTs cancarry a higher load.

It should be noted that the geometry of CCNTs considered inthis study is the least efficient for the purpose of carrying load and

S.H. Ghaderi, E. Hajiesmaili / Materials Science & Engineering A 582 (2013) 225–234234

storing energy compared with that of CCNTs with a similar pitchand radii but different morphologies. Therefore, the resultspresented can be viewed as an estimate of the minimum possiblespring constant, fracture load, fracture strain, and energy storagedensity for CCNTs.

Appendix A. Supplementary Information

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.psychres.2013.05.018.

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