Simulation of elastic properties of single-walled carbon nanotubes

Y. Jin, F.G. Yuan*

Department of Mechanical and Aerospace Engineering, North Carolina State University, Box 7910,Raleigh, NC 27695-7921, USA

Received 4 September 2002; received in revised form 20 December 2002; accepted 15 January 2003

Abstract

In this paper, selected effective elastic moduli of single-walled carbon nanotubes are simulated numerically. This effectivemacroscopic behavior is studied using molecular dynamics (MD) simulations in which the dynamic response and mutual forceinteraction among atoms of the nanostructures are obtained when subjected to small-strain deformation. Both force and energy

approaches that link the behavior at the atomic and macroscopic scales of the nanotubes are used to predict the elastic moduliunder different deformation modes. A comparison of the elastic constants obtained from MD simulation with available experi-mental data is made.

# 2003 Elsevier Ltd. All rights reserved.

Keywords: A. Nanostructures; C. Elastic properties; Single-walled carbon nanotubes

1. Introduction

Carbon nanotubes (CNTs) were first reported byIijima [1] in 1991. Since then, research on CNTs hasbeen attracting much attention to explore their unusualelectronic and material properties [2–10]. Both experi-mental and theoretical studies indicated that CNTS arematerial with extraordinary high stiffness and axialstrength [11–19]. Due to their large aspect ratios andsmall diameters, CNTs have emerged as potentiallyattractive materials as reinforcing elements in light-weight and high strength structural composites.Studies of the mechanical behavior of CNTs have

focused mainly on the adoption of different empiricalpotentials and continuum models using elasticity the-ory. In experimented approaches, the force–displace-ment response of a nanotube is measured and the axialYoung’s modulus is obtained by comparison to anequivalent elastic beam. Treacy et al. [11] showed anaverage value 1.8 TPa (with large scatter) for the axialYoung’s modulus from the direct measurements with atransmission electron microscope of a variety of multi-walled nanotubes (MWNTs) of different inner andouter diameters using the thermal vibration analysis of

anchored tubes. The nanotubes with the smallest innerdiameter were considerably stiffer, with a Young’smodulus of 3.70 TPa. Lourie and Wagner [12] obtainedthe axial Young’s modulus for a series of temperaturesby micro-Raman spectroscopy from measurements ofcooling-induced compressive deformation of nanotubesembedded in an epoxy matrix. At 81 K, the experi-mental results gave 3 TPa for single-walled nanotubes(SWNTs) with an average radius of 0.7 nm, and 2.4 TPafor MWNTs with an average radius of 5–10 nm. Wonget al. [13] used an atomic force microscope (AFM) tomeasure force–displacement relations for anchoredMWNTs on a substrate. From a comparison to elasticbeam theory, the Young’s modulus can be extracted. Anaverage of 1.28�0.5 TPa with little dependence ofnanotube diameter was reported.In most of theoretical studies using MD simulations

for predicting elastic properties of nanotubes, Sinnott etal. [14] estimated the axial Young’s modulus for carbonfibers composed of SWNTs aligned in the direction oftubular axis. The result showed that fibers, which werecomposed of smaller radius carbon nanotubes with apacking ratio of 0.9, had a modulus of approximately1.25–1.40 TPa. Yakobson et al. [15] used a many-bodyinteratomic potential (Tersoff–Brenner potential) with acontinuum shell model and obtained the axial SWNTmodulus that ranged from 1.4 to 5.5 TPa. Note that thepaper used the value 0.066 nm as the tube wall thickness

0266-3538/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0266-3538(03)00074-5

Composites Science and Technology 63 (2003) 1507–1515

www.elsevier.com/locate/compscitech

* Corresponding author. Tel.: +1-919-515-5947; fax: +1-919-515-

5934.

E-mail address: yuan@eos.ncsu.edu (F.G. Yuan).

while most other references use 0.34 nm instead. Lu [16]used an empirical force–constant model to determineseveral elastic moduli of single- and multi-walled nano-tubes and obtained the Young’s modulus of about 1TPa and the rotational shear modulus of about 0.5 TPa.The analysis showed that the elastic properties wereinsensitive to the radius, helicity, and the number ofwalls. However, Yao and Lordi [17] used MD simula-tions and found that changes in structure such as radiusand helicity of the SMNTs could affect the Young’smodulus because their results showed that the torsionalpotential energy, which is the dominant component oftotal potential energy, increased as the quadratic func-tion of the decreasing tube radius. Hernandez et al. [18]adopted a tight-binding method to obtain averagevalues of Young’s modulus and Poisson’s ratio ofSWNTs with different chiral vectors, which were foundto be 1.24 TPa and 0.262, respectively. Cornwell andWille [19] reported a relatively low modulus of 1.0 TPafor open-ended, free-standing SWNTs by using quen-ched molecular dynamics simulations based on theTersoff–Brenner potential. Recently Zhou et al. [20]claimed that both Young’s modulus and the wall thicknesswere independent of the radius and the helicity of SWNTs.They applied the strain energy of SWNTs directly fromelectronic band structure without introducing empiricalpotentials and continuum elasticity theory to describethe mechanical properties of SWNTs. The estimatedvalue for the axial modulus was reported as 5.0 TPa,which is 5 times larger than the value of MWNTs.In spite of the variety of theoretical studies on the

macroscopic elastic behavior of CNTs, there still remaincontroversial issues regarding the effect of geometricstructure of CNTs on elastic moduli, as evidenced bythe wide scatter among the elastic moduli reported inthe literature. The objective of this paper is to reexaminethe elastic behavior of CNTs in detail using two differ-ent approaches. The organization of this paper is asfollows. In the next section, the morphological structureof carbon nanotubes will be briefly discussed. Section 3provides the force fields and total potential energy thatare related to the interatomic potentials for MD simu-lations. In addition, the bonding and nonbonding termsin the total potential energy are described. In Section 4,several elastic moduli are determined by applying dif-ferent small-strain deformation modes. The elasticmoduli are predicted using energy and force approa-ches. Numerical results and a summary are given inSections 5 and 6.

2. Structure of carbon nanotube

Single-walled nanotubes are formed by folding a gra-phene sheet to form a hollow cylinder which is com-posed of hexagonal carbon ring units, which are also

referred to as graphene units. The fundamental carbonnanotube structure can be classified into three cate-gories: armchair, zigzag, and chiral, in terms of theirhelicity [21]. Fig. 1 shows a segment of a single graphiteplane that can be transformed into a carbon nanotubeby rolling it up into a cylinder. To describe this struc-ture, a chiral vector is defined as OA=na1+ma2, wherea1 and a2 are unit vectors for the honeycomb lattice ofthe graphene sheet, n and m are two integers, along witha chiral angle � which is the angle of the chiral vectorwith respect to the x direction shown in Fig. 1. Thischiral vector, OA, will be denoted as (n, m) which willalso specify the structure of the carbon nanotube. Vec-tor OB is perpendicular to the vector OA. To constructa CNT, we cut off the quadrangles OAB0B and roll itinto a cylinder with OB and AB0 overlapping each other.The relationship between the integers (n, m) and thenanotube radius, r, and chiral angle, � is given by

r ¼ffiffiffi3

pac�c m2 þmmþ n2

� �1=2=2� ð1Þ

� ¼ tan�1ffiffiffi3

pm= mþ 2nð Þ

h ið2Þ

where ac�c is the length of the C–C bond.

3. Force fields and total potential energy

Non-stochastic MD simulation methods are determi-nistic [22] and based on Newton’s second law, F=ma,where F is the force exerted on an atom, m is the mass, ais its acceleration. From knowledge of the force on eachatom, the acceleration of each atom in the system can bedetermined. Numerical integration of the equations ofmotion yields a trajectory that describes the positions,velocities, and accelerations of the atoms as they varywith time. From this trajectory, the average values of

Fig. 1. The graphite plane of nanotube surface coordinates.

1508 Y. Jin, F.G. Yuan /Composites Science and Technology 63 (2003) 1507–1515

properties can be estimated. Mutual atomic interactionsare described by force potentials associated with bond-ing and nonbonding phenomena. The interatomicpotential energy is the sum of bonding energy and non-bonding energy, U=Ubonding+Unonbonding. For carbonnanotubes, the nonbonding term is mainly the energy ofvan-der-Waals force, which normally has a weak influ-ence on the overall mechanical behavior among theatomic interactions of the carbon microstructure. Thevan-der-Waals force is most often modeled using theLennard–Jones 6-12 potential function [23].

Uvan-der-Waals ¼X

nonbonded pairs

Aij

r12ij�Cij

r6ij

!ð3Þ

where A and C are the atom-type dependent constants.rij=|rij|=|ri�rj| is the distance between two atoms, i andj, and ri and rj are the position vectors of ith and jthatom.The dominant part of the total potential energy, the

bonding energy, is a sum of three different interactionsamong atoms: bond stretching, bond bending, and bondtorsion.

Ubonded ¼ Ubond-stretch þUangle-bend þUtorsion ð4Þ

where the commonly used potential energy functions aregiven by

Ubond-stretch ¼X

1;2 pairs

Ks rij � r0� �2

Uangle-bend ¼Xangles

K� � � �0ð Þ2

Utorsion ¼X

1;4 pairs

K� 1� cos n�þ �0ð Þð Þ ð5Þ

In the above equations, r0 is the equilibrium distanceof the bond in this bond type, Ks is the spring constantof bond stretching, K� is the force constant of bending,yo is the rest angle for this bond, Ko is the force constantof torsion, �0 is the ideal phase for this bond type, and nis the periodicity (integer) of the bond. These coeffi-cients are determined as approximations to the detailedquantum interaction of electron wave functions under-lying the interaction of different carbon atoms. Anumerical program is developed to calculate the mutualinteraction forces determined by the gradient of thetotal potential energy of a single-walled nanotube,F=�rU. Applying Newton’s Second Law, the Verletalgorithm [24] is used to carry out the integration atspecified time steps to obtain atom trajectories.In this paper, both a force and energy approach is

adopted to predict the macroscopic elastic moduli. Inthe force approach, the elastic moduli are calculated bydirectly computing the average mechanical forcesdeveloped between carbon atoms in the nanotube. The

effective elastic moduli determination using the forceapproach can be calculated directly from the virialtheorem method [25–29] in which gives the expression ofthe stress tensor in a macroscopic system as the functionof atom coordinates and interatomic forces. Themethod provides a continuum measure of the internalmechanical interactions between atoms. The equiva-lence between the force approach and the virial theoremmethod has been proved by Theodorou and Suter [25].Meanwhile, the energy approach, which uses the secondderivative of potential energy with respect to strainunder each deformation mode, is also performed toobtain the elastic moduli. Eqs. (6) and (7) define thecalculation of elastic moduli based on force approachand energy approach, respectively:

��� ¼ �1

2V0F�ijr

�ij ð6Þ

Cijkl ¼1

V0

@2U

@"ij@"klð7Þ

In Eqs. (6) and (7), the repeated subscripts indicatethe sum of all atoms, ��� is the average atomic-levelstress tensor in the Cartesian coordinates, Fij is theinteratomic forces including bonding and nonbondingforces between two atoms i and j, "ij is the strain tensor,Cijkl is the fourth-order tensor of elastic constants. V0 isthe initial volume of the carbon nanotube, which for ahollow cylinder is given by V0=2�rhL, where L is thelength, r is the radius, and h is the assumed tube wallthickness.

4. Elastic moduli

The potential energy expression used in the energyapproach under small strain deformation is given by:

U ¼ U0 þ1

2V0Cijkl"ij"kl ð8Þ

where U is the total potential energy at equilibrium, andU0 is the initial potential energy at equilibrium beforedeformation is applied.The general elastic characteristics of the carbon

nanotubes will now be discussed. Because of the hex-agonal symmetry properties in the cylindrical surface ofnanotube, the nanotube exhibits transverse isotropy inthe �–z plane, which is defined by five independentelastic constants in the stress–strain relationships:

�rr����zz

8<:

9=; ¼

C11 C13 C13

C13 C33 C23

C13 C23 C33

24

35 "rr

"��"zz

8<:

9=; ð9Þ

� �z ¼ G�z��z; � rz ¼ Grz�rz; � r� ¼ Grz�r� ð10Þ

Y. Jin, F.G. Yuan /Composites Science and Technology 63 (2003) 1507–1515 1509

where G�z ¼Ez

2 1þv�zð Þ

Eq. (9) can be expressed in terms of engineering con-stants as

"rr"��"zz

8<:

9=; ¼

1

Er�vzrEz

�vzrEz

vzrEz

1

Ez�v�zEz

vzrEz

v�zEz

1

Ez

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

�rr����zz

8<:

9=; ð11Þ

In the following, several elastic moduli are determinedunder small strain deformation by applying deforma-tion modes [27,28]:

(1) Calculation of E2 and vz�

To calculate the axial Young’s modulus, Ez, andthe Poisson’s ratio, vz� the atoms are displaced byuz="0zzz. The average strain and stresses are:

"zz ¼ "0zz;

�zz 6¼ 0; any other �ij ¼ 0ð12Þ

Through MD simulation, the longitudinal elastic

modulus, Ez, can be calculated by a force andenergy approach using the following relationships:

Ez ¼�zz"0zz

��Force approach

Ez ¼1

V0

@2U

@"02zz��Energy approach

ð13Þ

The Poisson’s ratio is obtained as:

vz� ¼ �"��"0zz

ð14Þ

(2) Calculation of E� and v�z

If a radial displacement ur ¼ "0��r is applied toevery atom of the nanotube, then we obtain onlya hoop stress and can calculate a pure hoop stressin which:

"�� ¼ "0��

��� 6¼ 0; any other �ij ¼ 0ð15Þ

Hence, E� and v�z are given by:

E� ¼���

"0����Force approach

E� ¼1

V0

@2U

@"02

��

��Energy approachð16Þ

v�z ¼ �"zz

"0��

ð17Þ

(3) Calculation of the Rotational Shear Modulus G�z

The required displacement mode in this case isu� ¼ �0

�zz which is applied to all atoms. Theresulting strain and stresses are then given by:

��z ¼ �0�z; any other "ij ¼ 0

��z 6¼ 0; any other �ij ¼ 0ð18Þ

Then the rotational shear modulus G�z can then

be calculated by:

G�z ¼��z

�0�z

��Force approach

G�z ¼1

V0

@2U

@�02�z

��Energy approach

ð19Þ

(4) Calculation of C23 and C33

Here the carbon atoms are under deformationuz ¼ "0zzz and lateral displacements are con-strained. The average strain state imposed is:

"zz ¼ "0zz; any other "ij ¼ 0 ð20Þ

Based on the relationship of stress and strain, C23

and C33 are given by:

C23 ¼���"0zz

C33 ¼�zz"0zz

9>>=>>;��Force approach

C33 ¼1

V0

@2U

@"02zz��Energy approach

ð21Þ

Although C13 can be calculated using a force

approach, this modulus is small, on the order of 1GPa, which is within the numerical error boundsof the present analysis. Therefore, the determi-nation of C13 is not attempted.

Note that, in the energy approach, the initial potentialenergy, U0, is obtained from the MD simulation byminimizing the energy of the nanotube without anyexternally applied strain state. For each deformationmode, three constant strain values, 0.0005, 0.001, and0.0015 are applied to calculate the total potential ener-gies. A least-squares fit for the four potential energies isused to obtain the elastic moduli.In the force approach, only an applied strain value of

0.001 for each deformation mode is used to calculate theelastic moduli. Note that the stresses calculated fromEq. (6) need to be transformed into polar coordinatesfor calculating some of the elastic moduli.

1510 Y. Jin, F.G. Yuan /Composites Science and Technology 63 (2003) 1507–1515

5. Numerical results

Fig. 2 shows a schematic of a single-walled nanotubewith armchair helicity (n, n). In the present analysis,focus is placed on the elastic moduli of armchair nano-tubes ranging from n=6–20. The various geometricdimensions of the nanotubes are listed in Table 1. Theaspect ratio of the nanotubes, which is the length-to-radius ratio, is set to be greater than 10 to ensure thatedge effects can be neglected. A key issue here is that thewall thickness of the nanotubes needs to be specified inthe continuum sense before predicting the values ofelastic moduli for the single-walled carbon nanotubes.For multi-walled nanotubes, the distance between anytwo walls is close to that of the interlayer separationdistance in graphite, which is 0.34 nm. This interlayerseparation distance is used as the effective wall thicknessof SWNTs [16, 18]. The initial atoms velocities, vi, arechosen randomly from a Maxwell–Boltzmann distribu-tion at a given temperature which gives the probabilitythat an atom v has a velocity v�i in the a direction givenby

p v�i� �

¼mi

2�kBT

� �1=2

exp �1

2

miv�2

i

kBT

" #ð22Þ

where kB is the Boltzmann constant.

The temperature is related to the atom velocities bythe relation

T ¼1

3N

XNi¼1

pi�� ��2mi

ð23Þ

In the present model, a temperature of 20 �C is appliedand the various atomic parameters used in the MDsimulations are listed in Table 2 [29].An MD simulation was performed to obtain a mini-

mum energy state of the nanotube without any applieddeformation. The simulation was run for approximately5 ps with a time step of 1 fs after which the energyattained a constant minimum value. Then differenttypes of deformation modes discussed in Section 4 wereapplied to the nanotube by scaling atomic positionswith applying boundary conditions along the appro-priate coordinates. The nanotube reaches an equili-brium state over approximately 5 ps. After theequilibrium a 5 ps equilibrium period is carried out. Byintegrating molecular dynamic equations, the stressesand total potential energy are calculated from the aver-aged results from this equilibrium period. Then thevalue of elastic moduli can be obtained using force andenergy approach through Eqs. (6) and (7), respectively.Fig. 3 examines the partition of bonding energies for

the nanotube with n=12 under four strain values "zz0 .

Note that the nonbonding energy due to van-der-Waalsforce is several orders of magnitude smaller than thoserelated to the bonding energies. From the figurealthough the torsional energy has a dominant contribu-tion to the total potential energy, the energy is virtuallyconstant. Thus there is little influence of the torsionalenergy in predicting the modulus Ez because the con-tribution of the constant energy value is eliminatedthrough differentiation. In Fig. 4, the values of Ez and

Table 1

Geometry properties of single-walled nanotubes from n=6–20

(n, n)

Number

of atoms

Radius

(nm)

Length

(nm)

(6, 6)

456 0.407 4.427

(7, 7)

616 0.475 5.165

(8, 8)

800 0.543 5.903

(9, 9)

1008 0.611 6.641

(10, 10)

1240 0.678 7.379

(11,11)

1496 0.746 8.116

(12, 12)

1776 0.814 8.854

(13, 13)

2080 0.882 9.592

(14, 14)

2408 0.950 10.330

(15, 15)

2760 1.018 11.068

(16, 16)

3136 1.085 11.806

(17, 17)

3536 1.153 12.544

(18, 18)

3960 1.221 13.281

(19, 19)

4408 1.289 14.019

(20, 20)

4880 1.357 14.757

Table 2

Material properties of single-walled tubes used in the MD simulations

[29]

Constants

mi=19.9�10�27 kg

ac�c=0.142 nm

Ks=700 (Kcal/mol)/A2

K�=100 Kca/mol

K�=0.05 Kcal/mol

A=1.05�1019 (A12 Kcal/mol)

C=�6.73�109 (A12 Kcal/mol)

r0 =0.142 nm

�0=120�

�0=180�

n=2

kB =1.9872�10�3 Kcal/(mol K)

Fig. 2. A schematic for single-walled carbon nanotube with an arm-

chair (n, n) configuration.

Y. Jin, F.G. Yuan /Composites Science and Technology 63 (2003) 1507–1515 1511

E� obtained for different armchair single-wailed carbonnanotubes with radii ranging from 0.407 to 1.357 nmare plotted using the energy and force approaches. It isclearly shown that Ez and E� have little dependence onthe nanotube radius in both the energy and force

approaches. The prediction of Ez, however, does show aslight increase in magnitude as the value of n increases.The prediction for Ez and E� differ only slightly usingthe energy and force approach for all the nanotubesstudied. This suggests that the carbon nanotube isessentially isotropic in the plane of cylindrical surface.In Fig. 4(a), difference in the average values of Ez andE� based on the energy approach is about 0.9%, wherethe average value of Ez is predicted as 1.347 TPa and theaverage value of E� is 1.353 TPa. In Fig. 4(b), differencebetween the average values of Ez and E� based on theforce approach is around 0.36%, where the averagevalue of Ez is 1.236 TPa and E� is 1.242 TPa. These dataconfirm that Ez and E� exhibit a little dependence on theradii of the armchair nanotubes studied using both theenergy and force approaches. The values for the pre-dicted Ez and E� moduli may be summarized as:

Ez¼E�¼1:350� 0:012 TPa ��Force approach

1:238� 0:006 TPa ��Energy approach

�ð24Þ

Fig. 5. Poisson’s ratios ��z and �z� versus radius of armchair single-

walled carbon nanotube.

Fig. 4. Ez and E� distribution versus radius of armchair single-walled

carbon nanotube. (n, n) is the chiral vector: (a) Energy approach;

(b) Force approach.

Fig. 3. Partition of bonding potential energies of a single-walled

nanotube (n=12) under different strains "0zz.

Fig. 6. Shear modulus G�z versus radius of armchair single-walled

carbon nanotube.

1512 Y. Jin, F.G. Yuan /Composites Science and Technology 63 (2003) 1507–1515

The above values also show that the two approachesyield similar prediction of moduli, with the energyapproach giving an average value that is 9% larger incomparison with that obtained using the forceapproach. Figs. 5 and 6 show the Poisson’s ratios v�z, vz�and shear modulus G�z as a function of carbon nano-tube radii, respectively. From Fig. 5, the average valuesof Poisson’s ratios are obtained as �z�=0.261 andv�z=0.259. Utilizing the relationship for the shearmodulus in a transversely isotropic material,G�z ¼

Ez

2 1þv�zð Þ, a comparison between the direct numerical

results for G�z and the values derived from the elastic

Table 3

Comparison between the numerical result and the theoretical result

derived from G�z=Ez/[2(l+��z)]

Energy approach

Force approach

Numerical result of G�z (TPa)

0.547 0.492

Theoretical result of G�z (TPa)

0.536 0.491

Relative error

2.0% 0.2%

Table 4

Elastic moduli of single-walled carbon nanotubes

(n,n)

Ez E� C33 C23 G�z vz� v�z

Energy approach

(6,6)

1.324 1.376 1.428 0.551

(7,7)

1.336 1.370 1.429 0.545

(8,8)

1.339 1.365 1.429 0.544

(9,9)

1.335 1.359 1.428 0.544

(10,10)

1.338 1.355 1.428 0.550

(11,11)

1.338 1.354 1.428 0.548

(12,12)

1.336 1.352 1.427 0.547

(13,13)

1.354 1.350 1.427 0.547

(14,14)

1.348 1.349 1.426 0.544

(15,15)

1.350 1.346 1.426 0.543

(16,16)

1.366 1.345 1.425 0.547

(17,17)

1.350 1.344 1.424 0.548

(18,18)

1.357 1.343 1.424 0.553

(19,19)

1.360 1.342 1.424 0.548

(20,20)

1.370 1.342 1.423 0.545

Average

1.347

�0.013

1.353

�0.010

1.426

�0.002

0.547

�0.003

Force approach

(6,6)

1.217 1.245 1.432 0.254 0.483

(7,7)

1.227 1.240 1.432 0.251 0.487

(8,8)

1.232 1.239 1.432 0.249 0.489

(9,9)

1.235 1.241 1.432 0.247 0.492

(10,10)

1.235 1.241 1.431 0.246 0.493

(11,11)

1.236 1.244 1.431 0.245 0.494

(12,12)

1.238 1.241 1.430 0.244 0.493

(13,13)

1.240 1.245 1.430 0.243 0.494

(14,14)

1.239 1.241 1.429 0.243 0.494

(15,15)

1.238 1.244 1.427 0.242 0.495

(16,16)

1.241 1.243 1.428 0.242 0.495

(17,17)

1.241 1.240 1.428 0.242 0.495

(18,18)

1.241 1.243 1.428 0.241 0.492

(19.19)

1.241 1.245 1.427 0.241 0.495

(20,20)

1.243 1.243 1.427 0.241 0.495

Average

1.236

�0.007

1.242

�0.002

1.430

�0.002

0.245

�0.004

0.492

�0.004

Poisson’s ratios

(6,6)

0.257 0.267

(7,7)

0.260 0.264

(8,8)

0.265 0.267

(9,9)

0.267 0.265

(10,10)

0.255 0.262

(11,11)

0.258 0.255

(12,12)

0.260 0.262

(13,13)

0.261 0.253

(14,14)

0.258 0.263

(15,15)

0262 0.252

(16,16)

0.262 0.255

(17,17)

0.261 0.263

(18,18)

0.262 0.255

(19,19)

0.263 0.248

(20,20)

0.266 0.253

Average

0.261

�0.003

0.259

�0.006

The elastic constants are in the units of TPa. C23 is obtained from the

force approach. z-direction is the axis direction of the nanotube.

Fig. 7. C23 versus radius of armchair single-walled carbon nanotube

using force approach.

Fig. 8. C33 versus radius of armchair single-walled carbon nanotube.

Y. Jin, F.G. Yuan /Composites Science and Technology 63 (2003) 1507–1515 1513

relation using Ez and v�z is shown in Table 3. The com-parison shows a close agreement and yields furtherconfirmation of the transverse isotropy of the nanotube.In Fig. 6, the rotational shear modulus G�z is predictedfrom the two approaches. Again, the modulus is shownto be relatively insensitive to the radius of the nanotube,although the force approach does predict a slightincrease of the modulus which reaches a plateau as thevalue of n becomes larger. The prediction of averagerotational shear modulus G�z using energy approachgives a value that is 11% greater than that obtainedusing the force approach. The average value obtainedusing force approach, which is 0.492 TPa, is close to theprediction by Lu, which is 0.455 TPa in average [16].Figs. 7 and 8 depict the predictions of the C23 and C33

elastic coefficients versus nanotubes radii. The averagevalue of C33 is 1.426 TPa with standard deviation 0.14%using the energy approach. Using the force approach,C33, demonstrated an average value of 1.430 TPa with astandard deviation 0.14%. All elastic moduli predictedusing the force and energy approaches are listed inTable 4.

6. Summary

The elastic moduli of armchair SWNTs using MDsimulations have been calculated. Since single-wallednanotubes have a single carbon atom in the thicknessdirection, the continuum engineering constants that arerelated to the radial direction, Er, vzr, and Gr� (or Grz),are dependent on the assumed cross-sectional areaassociated with the nanotube and are yet to be clearlydefined. It has been shown from simulations using botha force and energy approach that the overall elasticconstants of SWNTs are insensitive to the nanotubemorphology such as nanotube radii ranging from 0.407to 1.357 nm. Thus the effect of curvature on the elasticconstants can be neglected. In general, the energyapproach predicts larger values of modulus than thoseobtained using the force approach. The only exceptionto this is in the prediction of C33. In addition, the forceapproach is expected to provide a more accurate pre-diction of moduli because the energy approach requiresan interpolating equation to be generated for severaltotal potential energy values followed by differentiationto obtain elastic moduli. Furthermore, the forceapproach can predict the nonlinear stress–strain beha-vior without assuming that the total potential energy ina quadratic form described in Eq. (8). The assumptionof transversely isotropic properties on the cylinder sur-face of the single-walled nanotube is confirmed bynumerical calculations. The values of the in-planeYoung’s modulus, rotational shear modulus, and in-plane Poisson’s ratio are comparable to existing theo-retical and experimental results.

Acknowledgements

The research is supported by the NASA LangleyResearch Center. Valuable suggestions by Mr. E. Sae-ther are gratefully acknowledged.

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