Composite Structures 131 (2015) 545–555

Contents lists available at ScienceDirect

Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Interfacial and mechanical properties of epoxy nanocomposites usingdifferent multiscale modeling schemes

http://dx.doi.org/10.1016/j.compstruct.2015.06.0140263-8223/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 (416) 978 5741; fax: +1 (416) 978 7753.E-mail address: meguid@mie.utoronto.ca (S.A. Meguid).

A.R. Alian, S.I. Kundalwal, S.A. Meguid ⇑Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada

a r t i c l e i n f o

Article history:Available online 10 June 2015

Keywords:Multiscale modelingMolecular dynamicsAtomistic-based continuumCarbon nanotubeInterfacial propertiesMechanical properties

a b s t r a c t

In this study, we investigate the interfacial and mechanical properties of carbon nanotube (CNT) rein-forced epoxy composite. The work carried out in two stages. In the first, we conducted molecular dynam-ics (MD) simulations to determine the atomic-level interfacial and mechanical properties of thetransversely isotropic representative volume element (RVE) comprised of CNT-epoxy composite. In thesecond, the Mori–Tanaka micromechanics scheme was used to scale up the mechanical properties ofthe atomic structure to the microscale level. The work was further extended and used atomistic-basedcontinuum (ABC) multiscale modeling technique, which makes use of constitutive relations derivedsolely from interatomic potentials to model the same system. Interestingly, the results of our compara-tive investigation reveals that (i) the ABC technique and MD simulation provide almost identical predic-tions for the atomic-level interfacial and mechanical properties of the nanocomposite, (ii) both modelspredict comparable bulk mechanical properties of the nanocomposite containing randomly dispersedCNTs, and (iii) they also reveal that a higher degree of orthotropy of the nanoscale representative fibersignificantly influences the bulk mechanical properties of the nanocomposite.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The remarkable physical and thermomechanical properties ofCNTs, such as low density [1], high aspect ratio [2], high Young’smodulus [3–6], high thermal conductivity [7–9], low coefficientof thermal expansion [10,11], high strength [12–14] and large frac-ture strain [13,14] have made them excellent reinforcements formultifunctional composites. It is found that few weight percent-ages of CNTs can significantly improve the interfacial and mechan-ical properties of CNT-based composites.

Several experimental studies have been carried out to investi-gate the interfacial characteristics of CNT-reinforced composites.For instance, Wagner et al. [15] estimated the interfacial shearstress between the multi-walled CNTs and the polymer based onthe fragmentation test to be as high as 500 MPa, which is morethan one order of magnitude compared with conventional compos-ites. Micro-Raman spectroscopy was used by Ajayan et al. [16] tomeasure the local mechanical behavior of single-walled CNT bun-dles in an epoxy nanocomposite. They noticed that the efficiencyof stress transfer and hence the enhancement of the mechanicalproperties is lower than expected due to sliding of the CNTs in

the agglomerated bundles. Qian et al. [2] investigated the loadtransfer in multi-walled CNT–polystyrene composites andreported that addition of 1 wt% of CNTs increases the tensile mod-ulus and strength by �39% and 25%, respectively. Schadler et al.[17] studied the interfacial characteristics of multi-walledCNT-reinforced composites with both the tension and compressionloadings. They reported that the compression modulus is higherthan that of the tensile modulus, indicating that the load transferto CNTs from the matrix is higher in compression. Cooper et al.[18] used scanning probe microscope tip to pull-out individualsingle- and multi-walled CNTs ropes from epoxy matrix. The ISSof both cases was found to be in the range of 35 to 376 MPa. Thisrelatively high value of ISS was attributed to the formation of astrong ultrathin epoxy layer at the interface. This layer exits as aresult of the formation of covalent bonds between CNTs and thesurrounding polymer molecules, which originate from the pres-ence of defects in the CNTs.

A significant number of analytical and numerical studies havealso been conducted to investigate the interfacial properties ofCNT-based composites. For example, Lordi and Yao [19] usedforce-field based molecular mechanics calculations to determinethe binding energies and sliding frictional stresses between CNTsand a range of polymer substrates, in an effort to understand thefactors governing interfacial strength. They reported that binding

546 A.R. Alian et al. / Composite Structures 131 (2015) 545–555

energies and frictional forces play only a minor role in determiningthe strength of the interface, but that helical polymer conforma-tions around the CNT are essential in developing high ISS. Liaoand Li [20] studied the interfacial characteristics of aCNT-reinforced polystyrene composite system through molecularmechanics simulations and elasticity calculations. They reportedthat the CNT-matrix bonding arises from non-bonded electrostaticand van der Waals (vdW) interactions, deformation induced bythese interactions, and mismatch in the coefficients of thermalexpansion. Their CNT pull-out simulation results suggest that theinterfacial shear stress of the CNT-polystyrene system is about160 MPa; significantly higher than conventional carbon fiber com-posite systems. Frankland et al. [21] generated stress–strain curvesof polyethylene nanocomposite reinforced with long and shortCNTs using MD simulations. Both nanocomposites were mechani-cally loaded in the axial and the transverse directions of the CNTaxis. In their study, nanocomposite reinforced with long CNTsshowed an increase in the stiffness relative to the polymer. Onthe other hand, nanocomposites reinforced with short CNTsshowed no enhancement relative to the polymer. Xiao and Zhang[22] studied the effects of CNT length and diameter on the distribu-tions of the tensile stress and interfacial shear stress of a CNT in anepoxy matrix. Their work revealed that a smaller CNT diameter hasa more effective reinforcement effect and that there exists an opti-mal tube length at which reinforcement is maximized. They alsoreported that a CNT has a greater stress transfer efficiency than asolid fiber, and provides toughness and tensile strength to theresulting nanocomposite. An analytical model has been developedby Haque and Ramasetty [23] to study the axial stress and shearstress at the interface of CNT-reinforced polymer composite mate-rials. An expression for the effective length of the CNT has alsobeen established by them for studying the load transfer efficiencyin CNT-reinforced composites. A micromechanics model has alsobeen developed by Li and Saigal [24] for assessing the interfacialshear stress transfer in CNT-reinforced polymer composites. Theirresults indicate that the stress transfer characteristics of nanocom-posites can be improved by using sufficiently long CNTs. The loadtransfer efficiency in the CNT-reinforced nanocomposites wasinvestigated by Tsai and Lu [25] using the conventional shear lagmodel and the finite element analysis. They revealed that the loadtransfer efficiency increases with the increment of the aspect ratioof CNTs. Using finite element model, Shokrieh and Rafiee [26] stud-ied the tensile behavior of embedded short CNTs in the polymermatrix in presence of vdW interactions in interphase region.They modeled the interphase using non-linear spring elementscapturing the force-distance curve of vdW interactions. Theyobserved that improvement in the Young’s modulus ofCNT-reinforced composite is negligible for lengths smaller than100 nm and saturation takes place in larger lengths on the orderof 10 lm. Recently, Kundalwal et al. [27] developed athree-phase shear lag model to analyze the stress transfer

Fig. 1. Modelling steps involved in t

characteristics of a hybrid hierarchical CNT-reinforced composite.In the first stage of their model, the transversely isotropic proper-ties of CNT-reinforced polymer composite were obtained and sub-sequently, they were used as inputs to shear lag modeldevelopment. A significant number of studies have also been car-ried out to determine the bulk mechanical properties ofCNT-based composites. These include experimental measurements(see [28] and references therein), continuum and micromechanicalanalysis [29–37], multiscale approaches [38–41], and MD simula-tions [42–45].

2. Multiscale modeling

Evidently, a number of experimental, analytical and numericalworks has been reported in the literature to investigate the inter-facial and mechanical properties of CNT-based composites.Among all these, we consider the multiscale modeling techniqueis more appropriate to property evaluation of CNT-based compos-ites. It is, therefore, our objective to develop a multiscale modelcapable of characterizing both the interfacial and mechanical prop-erties of CNT-reinforced epoxy composite. This task was performedin two steps. In the first, MD simulations were used to study theinterfacial and the transversely isotropic mechanical properties ofthe nanoscale RVE containing CNT embedded in epoxy. In the sec-ond, the determined mechanical properties of the nanoscale RVEwere used to determine the bulk mechanical properties of thenanocomposite using Mori–Tanaka model. Fig. 1 demonstratesthe steps involved in the hierarchical multiscale model. To ourknowledge, among the reported multiscale models, a newly devel-oped atomistic based continuum model known here after as ABC[38,41,46] is the first which has successfully described theatomic-property relations of CNT-reinforced polymer compositein a continuum framework. In comparison to other multiscalemodels, ABC technique distinguished its novelty by incorporatingthe nanoscale transversely isotropic RVEs in determining interfa-cial and mechanical. In fact, Meguid and coworkers [38,41,46]reported both the interfacial and mechanical properties ofCNT-reinforced polymer composites using ABC technique and suc-ceeded in investigating the influence of different parameters, suchas embedded CNT length, thickness of the CNT-polymer interface,CNT diameter and LJ cut-off distance. In view of this, ourMD-Micromechanics based multiscale modeling approach wascompared with this novel ABC approach. For the sake of complete-ness, we provide below a summary of the two approaches used inmodeling the current CNT-Composite system.

2.1. Molecular dynamics modeling and simulations

This Section is devoted fully to molecular dynamics modelingand simulations. All MD runs were conducted with the aid oflarge-scale atomic/molecular massively parallel simulator

he developed multiscale model.

Table 1Parameters used in the RVEs.

Parameter Pure epoxy RVE to determineRDF

NanocompositeRVE

CNT type – (9, 9) (9, 9)CNT diameter (Å) – 12.2 12.2CNT length (Å) – 43 43RVE dimensions

(Å3)60 � 60 � 60 50 � 50 � 43 31 � 31 � 43

CNT volumefraction

– 5.5% 12.16%

Total number ofatoms

25354 12690 4340

A.R. Alian et al. / Composite Structures 131 (2015) 545–555 547

(LAMMPS) [47] by using the consistent valence force field (CVFF)[48]. Conjugate gradient algorithm was used to minimize the totalpotential energy of the initial configurations, while velocityVerlet algorithm was used to integrate the equations of motionin all MD simulations. Periodic boundary conditions were imposedon all directions of the MD unit cells. In the simulations, thenon-bonded interactions between the atoms are represented byvdW interactions and Coulombic forces. Determination of theatomic-level mechanical properties of the pure epoxy and thenanocomposite RVE was accomplished by straining the MD unitcells followed by constant-strain energy minimization. The averagevalue of the stress tensor of the MD unit cell is defined in the formof virial stress [43]; as follows

r ¼ 1V

XN

i¼1

mi

2v2

i þ Firi

� �ð1Þ

where V is the volume of the RVE; v i;mi; ri and Fi are the velocity,mass, position and force acting on the ith atom, respectively.

2.1.1. Modeling of pure epoxyTo model the matrix, we used a specific two-component epoxy

material based on a diglycidyl ether of bisphenol A (DGEBA) epoxyresin and triethylene tetramine (TETA) curing agent, which is typ-ically used in aerospace industry. During the curing process, thehydrogen atoms in the amine groups of the hardener (TETA) reactwith the epoxide groups of the resin (DGEBA) forming covalentbonds, which result in a highly cross-linked epoxy structure [49].The resin/curing agent weight ratio in the epoxy polymer was setto 100:16.7. The epoxy system was generated by packing the MDunit cell with epoxy oligomers [50]. Every epoxy oligomer consistsof six DGEBA molecules connected by one TETA molecule, asshown in Fig. 2. The main steps involved in determining the elasticmoduli of pure epoxy are as follows:

Step 1 (Minimization): The initial epoxy configuration wasoptimized to its minimum energy configuration using the con-jugate gradient algorithm. The system is considered to be opti-mized once the change in the total potential energy of thesystem between subsequent steps is less than 1.0 � 10�10

kcal/mol.Step 2 (Equilibration): The amorphous structure obtained fromthe minimization step was then equilibrated at room tempera-ture in the constant temperature and volume canonical (NVT)

(a)

(c)

Fig. 2. Chemical structures of (a) epoxy resin (DGEBA), (b) curing agent (TETA), and (cmolecule.

ensemble. The equilibration process is carried out over 25 psby using a time step of 0.5 fs. The system was then equilibratedfor another 25 ps in the isothermal–isobaric (NPT) ensemble at300 K and 1 atm to generate a compressed structure with thecorrect density and residual stresses. This equilibration stepresulted in an equilibrated amorphous structure with an aver-age density of 1.09 g/cm3. At the end, the structure is againequilibrated for 25 ps in the NVT ensemble at 300 K to obtainan equilibrated structure with minimum energy.Step 3 (Uniaxial tension): Uniaxial tensile test is carried out todetermine the elastic moduli of the epoxy material. Constantuniaxial tensile strain increments of 0.25% were applied alongparticular direction by uniformly expanding the unit cell alongthis direction and updating the atoms coordinates to fit withinthe new dimensions. After each strain increment, the MD unitcell is equilibrated using the NVT ensemble at 300 K for 5 ps.Then, the stress tensor is averaged over an interval of 5 ps toreduce the effect of fluctuations. These steps were repeatedagain in the subsequent deformation increments. The proce-dure was stopped when the total strain reached 2.5%. The samesimulation steps were adopted to determine the stress–straincurves for the other two directions and then the recordedstress–strain data were averaged over all directions.

The initial molecular configurations of an amorphous epoxy areconstructed by randomly placing 45 epoxy oligomers inside an MDcell of size 60 Å � 60 Å � 60 Å using Packmol software [50]. Thedetails of pure epoxy RVE are summarized in Table 1. The totalnumber of atoms in this cubic cell is 25354, which represent an

(b)

) cured epoxy oligomer consists of 6 DGEBA molecules connected by single TETA

-5 0 5 10 150

50

100

150

200

250

300

350

400

Radial distance from CNT axis (Å)

RD

F o

f po

lym

er a

tom

s (A

tom

s/n

m3 )

Fig. 3. RDF of the epoxy atoms around the embedded CNT (transverse cross-sectional view of the equilibrated RVE is shown in the inset).

Fig. 4. MD unit cell containing a single CNT embedded in the epoxy matrix.

548 A.R. Alian et al. / Composite Structures 131 (2015) 545–555

epoxy system with an average density of 1.09 g/cm3. The averageelastic constants of the epoxy were calculated from the slopeof the stress–strain curves generated from the uniaxial tensileloadings. Young’s modulus and Poisson’s ratio of pure epoxyare determined to be 1.2 GPa and 0.35, respectively.

2.1.2. CNT-epoxy interface layer thicknessThe structure of the epoxy matrix at the vicinity of the CNT

surface differs from the bulk epoxy due to the formationof an ultra-thin epoxy layer at the CNT-epoxy interface. Thisultra-thin layer at the CNT-epoxy interface, which is ignored bymany researchers, consists of a highly packed crystalline polymer,which has higher elastic properties than the amorphous bulkpolymer [18,28]. In order to obtain the actual CNT-epoxy proper-ties, the size of the RVE must be large enough to incorporate thechange of the polymer structure. The cylindrical molecularstructure of the CNT is treated as an equivalent solid cylindricalfiber [29–31,33,34,36] for determining its volume fraction in thenanocomposite RVE,

Fig. 5. Schematics of CNT-reinforced epoxy composite at

CNT volume fraction ¼LCNT � p

4 ½ðdCNT þ hvdwÞ2 � ðdCNT � tCNTÞ2�V

ð2Þ

where dCNT , LCNT and tCNT denote the respective diameter, length andthickness of a CNT, and hvdW is vdW equilibrium distance betweenthe CNT and the surrounding polymer matrix.

To determine the thickness of the interface layer, we performeda MD simulation for a system consisting of a CNT of length 43.0 Åsurrounded by 51 epoxy oligomers. The size of the periodic RVEwas 50 Å � 50 Å � 43 Å and its details are summarized inTable 1. Subsequently, the RVE is equilibrated using the same mod-eling steps as described above for pure epoxy. Fig. 3 shows theradial distribution function (RDF) of the epoxy atoms that surroundthe CNT after the equilibration. The variation of the RDF along theradial direction represents the change of the epoxy structure in thevicinity of the embedded CNT. It may be observed from Fig. 3 thatthe RDF of epoxy atoms is zero at the radial distance of 8 Å andreaches its maximum value of 360 atoms/nm3 at the radial dis-tance of 9.5 Å. Then, it starts to fluctuate around an average valueof 225 atoms/nm3. Fig. 4 shows a close view of the equilibrated

various displacements during CNT pull-out process.

A.R. Alian et al. / Composite Structures 131 (2015) 545–555 549

RVE. The obtained values of the interface thickness and the equilib-rium separation distance were used in Section 4.1 to select theappropriate RVE size.

2.1.3. CNT pull-out simulationsAmongst the most common approaches for measuring the ISS of

the CNT-polymer nanocomposite is the pull-out test. During thepull-out simulations, one-end of the fully embedded CNT isextracted from the matrix at constant pull-out rate or velocity of1� 10�5Å=fs in the NVT ensemble at 300 K [51]. The periodicboundary conditions were removed along the axial direction ofthe CNT and the matrix was constrained during the pull-out simu-lation [52]. The pull-out force and the average ISS were then deter-mined based on the work done during the pull-out test. Typicalsnapshots during CNT pull-out from epoxy matrix are depicted inFig. 5.

2.2. Effective mechanical properties of nanocomposite

MD simulations were then used as the basic unit to allow thedetermination of the effective mechanical properties of the result-ing nanocomposite using micromechanics. It is well known factthat CNT-reinforced composite behaves anisotropically when sub-jected to the different loading conditions [21,34,41,44]. Therefore,the RVE was assumed to be transversely isotropic with the 1–axisbeing the axis of symmetry. To determine its five elastic constants,the RVE is subjected to five different tests: longitudinal tension,transverse tension, in-plane tension, in-plane shear and outof-plane shear. The steps involved in the MD simulations forobtaining the elastic coefficients of the RVE are the same asadopted in the case of pure epoxy. Finally, a set of loading condi-tions were applied to the RVE to determine the corresponding fiveindependent elastic coefficients of the RVE, as listed in Table 2.

The elastic moduli of the pure epoxy and the nanocompositeRVE obtained from the MD simulations are used as inputs to themicromechanical model in order to determine the bulk mechanicalproperties of the nanocomposite. With this process firmly estab-lished in several studies [38–41], the Mori–Tanaka model [53,54]can be developed by utilizing the transversely isotropic elasticproperties of the nanocomposite RVE and the isotropic elasticproperties of the pure epoxy matrix. In case of two-phase compos-ite, where the inhomogeneity is randomly orientated in thethree-dimensional space, the following relation can be used todetermine the effective stiffness tensor [C] of the nanocomposite:

½C� ¼ ½Cm� þ vRVEð½CRVE� � ½Cm�Þð½A�½vm½I� þ vRVEh½A�i��1Þ ð3Þ

Table 2Effective elastic coefficients of the RVEs and corresponding displacement fields.

Elastic coefficients Applied strains Applied displacement

C11 e11 ¼ e u1 ¼ ex1

C33 e33 ¼ e u3 ¼ ex3

C44 e23 ¼ e=2 u2 ¼ e2 x3, u3 ¼ e

2 x2

C66 e12 ¼ e=2 u1 ¼ e2 x2, u2 ¼ e

2 x1

K23 ¼ C22þC232

e22 ¼ e33 u2 ¼ ex2, u3 ¼ ex3

Table 3LJ potential parameters used in the ABC study.

LJ interaction l (J) w (Å)

Carbon–carbon (C–C) 3:89� 10�22 3.4

Carbon–hydrogen (C–H) 4:44� 10�22 3.2

Carbon–oxygen (C–O) 4:90� 10�22 3.2

Carbon–nitrogen (C–N) 4:48� 10�22 3.3

in which the mechanical strain concentration tensor [A] is given by

½A� ¼ ½½I� þ ½SRVE�ð½Cm�Þ�1ð½CRVE� � ½Cm�Þ��1

ð4Þ

where [Cm] and [CRVE] are the stiffness tensors of the epoxy matrixand the RVE, respectively; ½I� is an identity matrix; vm and vRVE rep-resent the volume fractions of the epoxy matrix and the RVE,respectively; and ½SRVE� indicate the Eshelby tensor. The specificform of the Eshelby tensor for the RVE inclusion given by Qiu andWeng [55] is utilized herein.

It may be noted that the elastic coefficient matrix [C] directlyprovides the values of the effective elastic properties of thenanocomposite, where the RVE is aligned with the 1–axis. In caseof random orientations of CNTs, the terms enclosed with anglebrackets in Eq. (3) represent the average value of the term overall orientations defined by transformation from the local coordi-nate system of the RVE to the global coordinate system. The trans-formed mechanical strain concentration tensor for the RVEs withrespect to the global coordinates is given by

½eAijkl� ¼ tiptjqtkrtls½Apqrs� ð5Þ

where tij are the direction cosines for the transformation and aregiven by

t11 ¼ cos / cos w� sin / cos c sin w;

t12 ¼ sin / cos wþ cos / cos c sin w; t13 ¼ sin w sin c;t21 ¼ � cos / sin w� sin / cos c cos w;

t22 ¼ � sin / sin wþ cos / cos c cos w; t23 ¼ sin c cos w;

t31 ¼ sin / sin c; t32 ¼ � cos / sin c and t33 ¼ cos c

Consequently, the random orientation average of the mechanicalstrain concentration tensor h½A�i can be determined by using the fol-lowing equation [56]:

h½A�i ¼R p�p

R p0

R p=20 ½eA�ð/; c;wÞ sin cd/dcdwR p

�p

R p0

R p=20 sin cd/dcdw

ð6Þ

where /, c, and w are the Euler angles in degrees defined withrespect to three successive rotations about the principal X, Y andZ coordinates. It may be noted that the averaged mechanical strainconcentration tensors given by Eqs. (4) and (6) are used for thecases of aligned and random orientations of CNTs, respectively, inEq. (3).

3. Atomistic-based continuum (ABC) modeling

Recently, Meguid and coworkers [41,46] presented novel ABCmodeling schemes to investigate the interfacial and mechanicalproperties of CNT-reinforced polymer composites. In their works,the problem is formulated by using a RVE which consists of thereinforcing CNT, the surrounding polymer matrix, and the

Fig. 6. Schematic of the RVE [41].

550 A.R. Alian et al. / Composite Structures 131 (2015) 545–555

CNT/polymer interface as demonstrated in Fig. 6. The idea behindthe ABC technique is to incorporate atomistic interatomic poten-tials into a continuum framework. In this way, the interatomicpotentials introduced in the model to capture the underlying ato-mistic behavior of the different phases considered. Thus, the influ-ence of the nanophase is taken into account via appropriateatomistic constitutive formulations. For the sake of completeness,we provide a brief outline of this technique detailed in their work[41,46].

Fundamental to their proposed concept is the notion that a CNTis a geometrical space-frame-like structure and the primary bondsbetween two nearest-neighboring atoms act like load-bearingbeam members. As in traditional FE models, nodes are used to con-nect the beam elements to form the CNT structure. In this case, thenodes represent the carbon atoms and are defined by the sameatomic coordinates. They adopted the Modified Morse interatomicpair potential with an added angle bending term to describe theatomic interactions in the CNT. The parameters used for the poten-tial in their study are the same as those adopted by Belytschkoet al. [57]. The Modified Morse potential is given by:

E ¼ Es þ Eb ð7Þ

Es ¼ De 1� exp�bðr�roÞ� �2 � 1� �

ð8Þ

Eb ¼12

khðh� hoÞ2 1þ ksexticðh� hoÞ4h i

ð9Þ

where ro is the initial bond length, ho is the initial angle betweenadjacent bonds, De is the dissociation energy, b is a constant whichcontrols the ‘width’ of the potential, and kh and ksextic are the anglebending force constants.

Nonlinear rotational spring elements were used to account forthe angle-bending component, while beam elements were usedto represent the stretching component of the potential. To describethe behavior of the beam and rotational spring elements, they firstderived material models for each which accurately represents thecharacteristics of the modified Morse potential. First, by derivingthe stretching potential (Es) with respect to the change in bondlength and by utilizing the relationship, e ¼ ðr � roÞ=r, the follow-ing equation is obtained:

F ¼ 2bDeð1� exp�berÞexp�ber ð10Þ

which represents the force required to stretch a C–C bond. Thisexpression is used to describe the material behavior of the beamelements. Likewise, differentiating the angle-bending component

Fig. 7. Polymer representation and pu

of the potential (Eb) with respect to the change in rotation, the fol-lowing expression is derived:

M ¼ khDh½1þ 3ksexticðDhÞ4� ð11Þ

which represents the moment required to bend neighboring bonds.Again, this expression is used to define the stiffness of the rotationalspring elements throughout the simulation.

The LJ interatomic potential is used to describe the vdW inter-actions at the CNT/polymer interface. The LJ potential is defined as

ELJ ¼ 4l wr

� �12

� wr

� �6" #

ð12Þ

where l is the potential well depth, w is the hard sphere radius ofthe atom or the distance at which ELJ is zero, and r is the distancebetween the two atoms. In their study, the non-bonded interactionsbetween the carbon atoms in the CNT and the atoms in the polymerwere considered.

They adopted the ABC technique to explicitly model each inter-action across the interface using the concept of continuum ele-ments. In this case, each atomic interaction is represented by acontinuum truss rod which extends out from a carbon atom inthe CNT structure to an atom in the epoxy matrix. The LJ parame-ters for the interactions considered in their work are summarizedin Table 3. Again, by differentiating the potential with respect tothe separation distance, the following expression for the vdW forcebetween two interacting atoms is obtained:

FLJ ¼ 24lw

� �2

wr

� �13

� wr

� �7" #

ð13Þ

This expression is used to determine the magnitude of the force ineach interaction, which depends solely on the separation distancebetween the atoms and the type of atoms considered.

To model the surrounding epoxy matrix, a specifictwo-component epoxy is used based on a diglycidyl ether ofbisphenol A (DGEBA) and triethylene tetramine (TETA) formula-tion. The Young’s modulus and Poisson’s ratio of the epoxy matrixwas taken to be 1.07 GPa and 0.28, respectively. The epoxy wasmodeled by using higher ordered 3-D, 10-node solid tetrahedralelements with quadratic displacement behavior.

3.1. CNT pull-out

Fig. 7 shows a schematic of the displacement boundary condi-tions of the pull-out process. The nodes in the CNT are constrained

ll-out boundary conditions [46].

20

30

40

50

60

70

ISS

(MP

a)

MD ResultsABC Results

A.R. Alian et al. / Composite Structures 131 (2015) 545–555 551

from any radial displacements and an incremental axial displace-ment boundary condition is applied to the top CNT nodes to initi-ate the pull-out process. The force required to withdraw the CNTfrom the matrix is evaluated over the course of the pull-out processby summing the reaction forces at the upper CNT nodes. The corre-sponding ISS was determined by dividing the maximum pull-outforce by the initial interfacial area, A = pdl, where d and l are thediameter and length of the embedded CNT, respectively.

3.2. Effective mechanical properties of nanocomposite

Appropriate ABC formulations were developed by Wernik andMeguid [41] to determine the elastic properties of the transverselyisotropic nanoscale RVE comprised: the reinforcing CNT, the sur-rounding matrix and their interface. In order to scale-up to themacro level, the RVE was homogenized into a representative fiberexhibiting the same geometrical and mechanical characteristics insuch way that it behaves as an equivalent continuous medium. Thegeometry of the linear-elastic, homogenous representative fiber isassumed to be cylindrical, similar to that of the RVE. Transverselyisotropic elastic properties of the representative fiber were deter-mined by equating the total strain energies of the RVE and repre-sentative fiber under identical sets of loading conditions as listedin Table 2. Subsequently, the developed representative fiber is thenused with analytical and computational micromechanical model-ing techniques to determine the bulk mechanical properties ofthe nanocomposite for varied CNT concentrations and aspectratios.

0 50 100 150 2000

10

LCNT (Å)

Fig. 8. Effect of embedded CNT length on the ISS.

2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

60

Interfacial Thickness (Å)

ISS

(MP

a)

MD ResultsABC Results

Fig. 9. Effect of CNT-polymer interface thickness on the ISS.

Table 4LJ potential parameters used in the present study.

LJ interaction l (J) w (Å)

C–C 10:286� 10�22 3.617

C–H 5:213� 10�22 2.45

C–O 12:767� 10�22 2.859

C–N 11:607� 10�22 3.501

4. Results and discussion

In this Section, we determine the interfacial and mechanicalproperties of the nanocomposite, and compare the same withthose predicted by the ABC multiscale model.

4.1. Interfacial properties of nanocomposite

A series of MD simulations were carried out to determine theISS of the nanoscale RVE reinforced with an individual CNT. Theeffects of such parameters as embedded CNT length, the thicknessof the CNT-polymer interface, LJ cut-off distance and CNT diameteron the ISS are investigated and discussed. To investigate the effectof the pertinent parameters on the ISS, the same CNT type and itsvolume fraction in the RVE are considered as considered by Werniket al. [46].

4.1.1. Effect of embedded CNT lengthFive different CNT lengths were considered to compare the MD

results with those of the ABC results. It is well known that a shortlength of CNT significantly influences the interfacial characteristicsof the resulting nanocomposite. Therefore, to reduce the computa-tional effort, the maximum embedded CNT length in the polymermatrix is kept limited to 200 Å. Fig. 8 shows the effect of embeddedCNT lengths on the ISS of a nanocomposite with an interfacialthickness of 3.4 Å. The ISS of the CNT-polymer composite systemexhibits a decaying length trend similar to traditional fiber com-posites. It may also be observed that the ABC model slightly underpredicts the ISS values. The subsequent MD simulations were car-ried out considering the embedded CNT length as being 200 Å.

4.1.2. Effect of CNT-polymer interfacial thicknessIn existing studies, the CNT-polymer interfacial thickness has

not yet been unambiguously defined. Several different values havebeen used in both atomistic and continuum simulations. Forinstance, Hu et al. [58] simulated the helical wrapping of one

polystyrene chain around a CNT considering only vdW interactionsusing MD simulations. The equilibrium distance between thehydrogen atoms in the polymer and carbon atoms in the CNT ran-ged from 2.851 Å to 5.445 Å. However, only one polymer chain wasconsidered, while in practical cases there may be other chainswhich also wrap around the CNT. Montazeri and Naghdabadi[59] used an interfacial thickness of 3.816 Å in their molecularstructural mechanics model of CNT-reinforced polymer compos-ites. This value corresponds to the equilibrium distance of the LJpotential. Hence, it is worthwhile to investigate the effect of differ-ent interfacial thicknesses on the ISS of CNT-reinforced polymercomposite. Fig. 9 shows the predicted ISS for the nanocompositesystem for an interfacial thickness range of 2.2–4.25 Å. This figuredemonstrates that the ISS of the CNT-polymer composite systemdecreases with the increase in the interfacial thickness. This

552 A.R. Alian et al. / Composite Structures 131 (2015) 545–555

phenomenon is attributed to the fact that vdW interactionsbetween the CNT and the epoxy become weaker with the increasein their atomic separation distance. The trend of this finding is alsowell confirmed by the ABC technique. In the ABC study, as theinterfacial thickness is decreased, a larger number of polymeratoms were included in the computational cell which increasesthe number of vdW interactions occurring over the interface andthe subsequent ISS. Comparative trends also suggest that MDapproach slightly overestimates the values of ISS over the ABCtechnique for the higher interfacial thicknesses. The subsequentMD simulations were carried out considering the CNT-polymerinterfacial thickness as being 2.2 Å.

4.1.3. Effect of LJ cut-off distanceIn most existing MD studies, the LJ interatomic potential used to

simulate vdW interactions is often truncated to reduce the compu-tational cost in such way that atom pairs whose distances aregreater than the cut-off distance have zero vdW interaction energy.The cut-off distance is often taken to be 8.5 Å (see Ref. [46] and

Table 5Material properties of the RVE and the representative fiber.

RVE CNT volume fraction C11 (GPa) C12 (Gpa

Present results 12.16% 116.4 6

5 10 15 20 250

5

10

15

20

25

30

35

40

CNT Diameter (Å)

ISS

(MP

a)

MD ResultsABC Results

Fig. 11. Effect of CNT diameter on the ISS.

4 6 8 10 12 145

10

15

20

25

30

35

40

45

LJ Cut-Off Distance (Å)

ISS

(MP

a)

MD ResultsABC Results

Fig. 10. Effect of LJ cut-off distance on the ISS.

references therein). Truncating the potential introduces a sharpdiscontinuity between atoms inside and atoms outside thecut-off radius, particularly when smaller cut-off distances are used.Therefore, here an attempt is made to investigate the effect of LJcut-off distance on the ISS of a nanocomposite. The LJ interatomicpotential is defined in Eq. (12). The corresponding LJ parametersare listed in Table 4. Fig. 10 shows the estimated MD results fora cut-off range of 6 Å to 13 Å. As expected, the values of ISS of ananocomposite system increase with the LJ cut-off distance andbecome stabilized at higher cut-off distance (> 12 Å). It may alsobe observed from Fig. 10 that both techniques predict the goodagreement between the results for a wide range of cut-off distance.

4.1.4. Effect of CNT diameterCNT diameter has a significant influence on the interfacial prop-

erties of the resulting nanocomposite. In this study, armchair CNTconfigurations are considered, with the smallest being a (5,5) CNT,and the largest an (18, 18) CNT. Fig. 11 demonstrates that the pre-dicted ISS decreases approximately linearly with the increase inthe value of the CNT diameter. This finding demonstrates theadvantage of using smaller diameter of armchair CNTs. Again, bothmultiscale modeling techniques (MD and ABC) show good agree-ment between the results for a wide range of CNT diameters. Themarginal differences occur because an identical polymer configura-tion considered in the ABC study for all simulations in order tominimize the effect of polymer distribution on the pull-out curves.

4.2. Mechanical properties of nanocomposites

A series of MD simulations were carried out to determine theelastic properties of the transversely isotropic RVE reinforced witharmchair (9,9) CNT. The details of the RVE are summarized inTable 1. In our MD simulations, we made sure that the density ofthe epoxy matrix was set to 1.09 g/cm3, which is the same as thatof the pure epoxy. Subsequently, a set of loading conditions wereapplied to the RVE to determine the corresponding five indepen-dent elastic coefficients of the RVE, as listed in Table 2. Table 5summarizes the outcome of the MD simulations. It may be

) C23 (Gpa) C33 (Gpa) C44 (Gpa) C66 (GPa)

7.5 15.1 3.8 2

0 0.01 0.02 0.03 0.04 0.050

2

4

6

8

10

12

14

CNT Volume Fraction

You

ng'

s M

odu

lus

(GP

a)

MD Multiscale ModelABC Multiscale Model

Fig. 12. Comparison of the bulk Young’s modulus of the nanocomposite containingrandomly oriented CNTs.

0 0.01 0.02 0.03 0.04 0.050

1

2

3

4

5

6

CNT Volume Fraction

Shea

r M

odul

us

(GP

a)MD Multiscale ModelABC Multiscale Model

Fig. 13. Comparison of the bulk shear modulus of the nanocomposite containingrandomly oriented CNTs.

A.R. Alian et al. / Composite Structures 131 (2015) 545–555 553

observed from the results that the elastic properties of the RVE aresignificantly higher than those of the neat epoxy. In the next step ofmultiscale model, nanocomposite RVE will be used as an effectivefiber in the Mori–Tanaka model to determine the bulk mechanicalproperties of the nanocomposite at the microscale level (see Fig. 1).

Considering the epoxy as the matrix phase and the transverselyisotropic nanoscale RVE as the reinforcement, the bulk mechanicalproperties of the nanocomposite were determined by following themicromechanical modeling approach developed in Section 2.2.Practically, the orientations of the CNT reinforcements in the poly-mer matrix can vary over the volume of the nanocomposite.Therefore, studying the properties of nanocomposites reinforcedwith randomly oriented CNTs is of a great importance. For such

0 0.01 0.02 0.03 0.04 0.050

20

40

60

80

CNT Volume Fraction (a)

EL (G

Pa)

0 0.01 0.02 0.03 0.04 0.050

0.2

0.4

0.6

0.8

1

CNT Volume Fraction (c)

GL (G

Pa)

MD Results

Fig. 14. Comparison of the bulk (a) longitudinal Young’s ðELÞ, (b) transverse Youngnanocomposite predicted by the present multiscale model with those predicted by the

investigation, nanoscale RVEs are considered to be completely ran-domly dispersed in the epoxy matrix over the volume of thenanocomposite. The present multiscale model results are com-pared with those predicted by Wernik and Meguid [41] forCNT-reinforced composites containing randomly oriented trans-versely isotropic representative fibers. It may be noted that bothmodels consider the maximum CNT volume fraction in thenanocomposite up to 5%. As expected, the randomly dispersed caseprovides the isotropic elastic properties for the resulting nanocom-posite. Figs. 12 and 13 illustrate the variations of the bulk Young’sand shear moduli of the nanocomposite with the CNT loading,respectively. The results predicted by both the multiscale modelsare in good agreement for the lower CNT loadings. The marginaldifference between the results at the higher CNT loadings areattributed to the fact that the equilibrium distance between thecarbon atoms of a CNT and the atoms of the polymer matrix isassumed as3.8 Å in Ref. [41], while in the present study it is consid-ered as 2.2 Å. In addition, a uniform distribution of the polymernodal density (99 nodes/nm3) around the embedded CNT isassumed in Ref. [41], while the polymer nodal density obtainedfrom the MD simulations is found to vary along the radial direc-tion, as shown in Fig. 3.

New fabrication techniques utilizing CNT fibers and arrays haveemerged as means of producing nanocomposites with aligned CNTreinforcements [60]. Hence, the estimation of the bulk mechanicalproperties of the nanocomposite containing aligned RVEs along the1-axis would be an important study. Therefore, the work was fur-ther extended to determine the bulk mechanical properties of thenanocomposite. The present multiscale model results are also com-pared with those predicted by Wernik and Meguid [41] forCNT-reinforced composites containing aligned representativefibers. For the comparison purpose, we considered their case of

0 0.01 0.02 0.03 0.04 0.050

5

10

15

20

CNT Volume Fraction (b)

ET (G

Pa)

0 0.01 0.02 0.03 0.04 0.050

0.2

0.4

0.6

0.8

1

CNT Volume Fraction (d)

GT (G

Pa)

ABC Results

’s ðET Þ, (c) longitudinal shear ðGLÞ, and (d) transverse shear ðGT Þ moduli of theABC model.

554 A.R. Alian et al. / Composite Structures 131 (2015) 545–555

long CNTs embedded in the epoxy matrix having aspect ratio of1000. Fig. 14 shows the variations of the elastic moduli of thenanocomposite against the CNT volume fraction. All moduli exhibitsimilar trends; with increasing CNT volume fraction, all moduliincrease approximate linearly. It may be observed fromFig. 14(a) and (b) that the ABC multiscale model significantly overpredicts the effective values of EL and ET over those of the presentmultiscale model. This phenomenon is mainly attributed to theeffect of the degree of orthotropy of the nanoscale effective fibers(i.e., RVE and representative fiber) on the resulting nanocomposite.On the other hand, the present multiscale model slightly over pre-dicts the effective values of GL and GT over those estimated by theABC model. At this juncture it is important to mention that theeffective transverse Young’s modulus of composite is matrix dom-inant. In contrast to this, the ABC model significantly over esti-mates the values of ET . Therefore, the present multiscale model isconsidered to be an appropriate technique on the conservative sideto determine the bulk elastic properties of the nanocomposite con-taining either randomly or aligned dispersed CNTs. In addition, it isimportant to note that the present multiscale model represents thereal nanocomposite structure with the minimum assumptionscompared to that of the ABC model which model the atomic struc-ture as an equivalent beam and truss elements. Thus, it can beinferred from these comparisons that the present multiscale modelcan be reliably applied to determine the interfacial and mechanicalproperties of nanocomposites.

5. Conclusions

In this article, we developed a multiscale model to determinethe interfacial and bulk mechanical properties of CNT-reinforcedepoxy composites. Two aspects of the work were examined. First,MD simulations were carried out to determine the interfacial andthe transversely isotropic elastic properties of the resulting nanos-cale RVEs. Second, the developed RVEs were, in turn, used withanalytical micromechanical technique of the Mori–Tanaka type todetermine the bulk elastic properties of the nanocomposite.Furthermore, we carried out a detailed comparison between theMD and ABC predictions considering pertinent parameters, suchas embedded CNT length, thickness of the CNT-polymer interface,CNT diameter, LJ cut-off distance and orientations of dispersedCNTs. The following is a summary of our findings:

1. ISS results predicted by the MD and ABC techniques agree wellwith each other for a wide range of structural and geometricalparameters.

2. Both multiscale approaches indicate that the increase in thelength of a CNT leads to a decrease in the values of ISS of theresulting nanocomposite.

3. They also indicate that the values of ISS of a nanocomposite sys-tem increase with the LJ cut-off distance and become stabilizedat higher cut-off value.

4. The bulk elastic properties of the nanocomposite containingrandomly dispersed CNTs predicted by these techniques agreewith each other.

5. The ABC model over predicts the longitudinal and transverseYoung’s moduli of the nanocomposite containing aligned CNTs.

Acknowledgements

The authors wish to acknowledge the financial support pro-vided by the Natural Sciences and Engineering Research Councilof Canada and the Discovery Accelerator Supplement.

References

[1] Gao GH, Cagin T, Goddard WA. Energetics, structure, mechanical andvibrational properties of single-walled carbon nanotubes. Nanotechnology1998;9:184–91. http://dx.doi.org/10.1088/0957-4484/9/3/007.

[2] Qian D, Dickey EC, Andrews R, Rantell T. Load transfer and deformationmechanisms in carbon nanotube-polystyrene composites. Appl Phys Lett2000;76:2868–70. http://dx.doi.org/10.1063/1.126500.

[3] Treacy MMJ, Ebbesen TW, Gibson JM. Exceptionally high Young’s modulusobserved for individual carbon nanotubes. Nature 1996;381:678–80. http://dx.doi.org/10.1038/381678a0.

[4] Krishnan A, Dujardin E, Ebbesen EW, Yianilos PN, Treacy MMJ. Young’smodulus of single-walled nanotubes. Phys Rev B 1998;58:14013–9. http://dx.doi.org/10.1103/PhysRevB.58.14013.

[5] Shen L, Li J. Transversely isotropic elastic properties of single-walled carbonnanotubes. Phys Rev B 2004;69:045414. http://dx.doi.org/10.1103/PhysRevB.69.045414.

[6] Batra RC, Sears A. Uniform radial expansion/contraction of carbon nanotubesand their transverse elastic moduli. Model Simul Mater Sci Eng2007;12:835–44. http://dx.doi.org/10.1088/0965-0393/15/8/001.

[7] Hone J, Whitney C, Piskoti C, Zettl A. Thermal conductivity of single-walledcarbon nanotubes. Phys Rev B 1999;59:2514–6. http://dx.doi.org/10.1103/PhysRevB.59.R2514.

[8] Berber S, Kwon YK, Tomanek D. Unusually high thermal conductivity of carbonnanotubes. Phys Rev Lett 2000;84:4613–6. http://dx.doi.org/10.1103/PhysRevLett. 84.4613.

[9] Yang X-S. Modelling heat transfer of carbon nanotubes. Model Simul Mater SciEng 2005;13:893–902. http://dx.doi.org/10.1088/0965-0393/13/6/008.

[10] Jiang H, Liu B, Huang Y, Hwang KC. Thermal expansion of single wall carbonnanotubes. ASME J Eng Mater Technol 2004;126:265–70. http://dx.doi.org/10.1115/1.1752925.

[11] Kwon Y, Berber S, Tomanek D. Thermal contraction of carbon fullerenes and2nanotubes. Phys Rev Lett 2004;92:015901. http://dx.doi.org/10.1103/PhysRevLett. 92.015901.

[12] Yakobson BI, Brabec CJ, Bernholc J. Nanomechanics of carbon tubes:instabilities beyond linear response. Phys Rev Lett 1996;76:2511–4. http://dx.doi.org/10.1103/PhysRevLett. 76.2511.

[13] Wong EW, Sheehan PE, Lieber CM. Nanobeam mechanics: elasticity, strength,and toughness of nanorods and nanotubes. Science 1997;277:1971–5. http://dx.doi.org/10.1126/science.277.5334.197.

[14] Yu MF, Lourie O, Dyer MJ, Moloni K, Kelly T, Ruoff RS. Strength and breakingmechanism of multiwalled carbon nanotubes under tensile load. Science2000;287:637–40. http://dx.doi.org/10.1126/science.287.5453.637.

[15] Wagner HD, Lourie O, Feldman Y, Tenne R. Stress-induced fragmentation ofmultiwall carbon nanotubes in a polymer matrix. Appl Phys Lett1998;72:188–90. http://dx.doi.org/10.1063/1.120680.

[16] Ajayan PM, Schadler LS, Giannaris C, Rubio A. Single-walled carbon nanotube-polymer composites: Strength and weakness. Adv Mater 2000;12:750–3.http://dx.doi.org/10.1002/(SICI)1521-4095(200005)12:10<750::AID-ADMA750>3.0.CO;2-6.

[17] Schadler LS, Giannaris SC, Ajayan PM. Load transfer in carbon nanotube epoxycomposites. Appl Phys Lett 1998;73:3842–4. http://dx.doi.org/10.1063/1.122911.

[18] Cooper CA, Cohen SR, Barber AH, Wagner HD. Detachment of nanotubes from apolymer matrix. Appl Phys Lett 2002;81:3873–5. http://dx.doi.org/10.1063/1.1521585.

[19] Lordi V, Yao N. Molecular mechnaics of binding in carbon-nanotube-polymercomposites. J. Mater. Res. 2000;15:2770–9. http://dx.doi.org/10.1557/JMR.2000.0396.

[20] Liao K, Li S. Interfacial characteristics of a carbon nanotube-polystyrenecomposite system. Appl Phys Lett 2001;79:4225–7. http://dx.doi.org/10.1063/1.1428116.

[21] Frankland SJV, Harik VM, Odegard GM, Brenner DW, Gates TS. The stress–strain behavior of polymer–nanotube composites from molecular dynamicssimulation. Compos Sci Technol 2003;63:1655–61. http://dx.doi.org/10.1016/S0266-3538(03)00059-9.

[22] Xiao KQ, Zhang LC. The stress transfer efficiency of a single-walled carbonnanotube in epoxy matrix. J Mater Sci 2004;39:4481–6. http://dx.doi.org/10.1023/B:JMSC.0000034141.48785.d2.

[23] Haque A, Ramasetty A. Theoretical study of stress transfer in carbon nanotubereinforced polymer matrix composites. Compos Struct 2005;71:68–77. http://dx.doi.org/10.1016/j.compstruct.2004.09.029.

[24] Li K, Saigal S. Micromechanical modeling of stress transfer in carbon nanotubereinforced polymer composites. Mater Sci Eng A 2007;457:44–57. http://dx.doi.org/10.1016/j.msea.2006.12.018.

[25] Tsai J-L, Lu T-C. Investigating the load transfer efficiency in carbon nanotubesreinforced nanocomposites. Compos Struct 2009;90:172–9. http://dx.doi.org/10.1016/j.compstruct.2009.03.004.

[26] Shokrieh MM, Rafiee R. Investigation of nanotube length effect on thereinforcement efficiency in carbon nanotube based composites. ComposStruct 2010;92:2415–20. http://dx.doi.org/10.1016/j.compstruct.2010.02.018.

[27] Kundalwal SI, Ray MC, Meguid SA. Shear lag model for regularly staggeredshort fuzzy fiber reinforced composite. ASME J Appl Mech 2014;81:091001/1.http://dx.doi.org/10.1115/1.402780.

A.R. Alian et al. / Composite Structures 131 (2015) 545–555 555

[28] Coleman JN, Khan U, Blau WJ, Gun’ko YK. Small but strong: a review of themechanical properties of carbon nanotube–polymer composites. Carbon2006;44:1624–52. http://dx.doi.org/10.1016/j.carbon.2006.02.03.

[29] Thostenson ET, Chou TW. On the elastic properties of carbon nanotube- basedcomposites: modeling and characterization. J Phys D: Appl Phys2003;36:573–82.

[30] Odegard GM, Gates TS, Wise KE, Park C, Siochi EJ. Constitutive modeling ofnanotube-reinforced polymer composites. Compos Sci Technol2003;63:1671–87. http://dx.doi.org/10.1016/S0266-3538(03)00063-0.

[31] Lusti HR, Gusev AA. Finite element predictions for the thermoelastic propertiesof nanotube reinforced polymers. Model Simul Mater Sci Eng2004;12:S107–19. http://dx.doi.org/10.1088/0965-0393/12/3/S05.

[32] Tserpes KI, Papanikos P. Continuum modeling of carbon nanotube-basedsuper-structures. Compos Struct 2009;91:131–7. http://dx.doi.org/10.1016/j.compstruct.2009.04.039.

[33] Kundalwal SI, Ray MC. Micromechanical analysis of fuzzy fiber reinforcedcomposites. Int J Mech Mater Des 2011;7:149–66. http://dx.doi.org/10.1007/s10999-011-9156-4.

[34] Kundalwal SI, Ray MC. Effective properties of a novel composite reinforcedwith short carbon fibers and radially aligned carbon nanotubes. Mech Mater2012;53:47–60. http://dx.doi.org/10.1016/j.mechmat.2012.05.008.

[35] Tserpes KI, Chanteli A. Parametric numerical evaluation of the effective elasticproperties of carbon nanotube-reinforced polymers. Compos Struct2013;99:366–74. http://dx.doi.org/10.1016/j.compstruct.2012.12.004.

[36] Kundalwal SI, Ray MC. Effect of carbon nanotube waviness on the effectivethermoelastic properties of a novel continuous fuzzy fiber reinforcedcomposite. Compos Part B: Eng 2014;57:199–209. http://dx.doi.org/10.1016/j.compositesb.2013.10.003.

[37] Wang JF, Liew KM. On the study of elastic properties of CNT-reinforcedcomposites based on element-free MLS method with nanoscale cylindricalrepresentative volume element. Compos Struct 2015;124:1–9. http://dx.doi.org/10.1016/j.compstruct.2015.01.006.

[38] Meguid SA, Wernik JM, Cheng ZQ. Atomistic-based continuum representationof the effective properties of nano-reinforced epoxies. Int J Solids Struct2010;47:1723–36. http://dx.doi.org/10.1016/j.ijsolstr.2010.03.009.

[39] Shokrieh MM, Rafiee R. On the tensile behavior of an embedded carbonnanotube in polymer matrix with non-bonded interphase region. ComposStruct 2010;92:647–52. http://dx.doi.org/10.1016/j.compstruct.2009.09.033.

[40] Ayatollahi MR, Shadlou S, Shokrieh MM. Multiscale modeling for mechanicalproperties of carbon nanotube reinforced nanocomposites subjected todifferent types of loading. Compos Struct 2011;93:2250–9. http://dx.doi.org/10.1016/j.compstruct.2011.03.013.

[41] Wernik JM, Meguid SA. Multiscale micromechanical modeling of theconstitutive response of carbon nanotube –reinforced structural adhesives.Int J Solids Struct 2014;51:2575–89. http://dx.doi.org/10.1016/j.ijsolstr.2014.03.009.

[42] Griebel M, Hamaekers J. Molecular dynamics simulations of the elastic moduliof polymer–carbon nanotube composites. Comput Methods Appl Mech Eng2004;193:1773–88. http://dx.doi.org/10.1016/j.cma.2003.12.025.

[43] Qi D, Hinkley J, He G. Molecular dynamics simulation of themal andmechanical properties of polyimide-carbon-nanotube composites. ModelSimul Mater Sci Eng 2005;13:493–507. http://dx.doi.org/10.1088/0965-0393/13/4/002.

[44] Al-Ostaz A, Pal G, Mantena PR, Cheng A. Molecular dynamics simulation ofSWCNT-polymer nanocomposite and its constituents. J Mater Sci2008;43:164–73. http://dx.doi.org/10.1007/s10853-007-2132-6.

[45] Talukdar K, Mitra AK. Comparative MD simulation study on the mechanicalproperties of a zigzag single-walled carbon nanotube in the presence of Stone-Thrower-Wales defects. Compos Struct 2010;92:1701–5. http://dx.doi.org/10.1016/j.compstruct.2009.12.008.

[46] Wernik JM, Cornwell-Mott BJ, Meguid SA. Determination of the interfacialproperties of carbon nanotube reinforced polymer composites usingatomistic-based continuum model. Int J Solids Struct 2012;49:1852–63.http://dx.doi.org/10.1016/j.ijsolstr.2012.03.024.

[47] Plimpton S. Fast parallel algorithms for short-range molecular dynamics. JComput Phys 1995;117:1–19. http://dx.doi.org/10.1006/jcph.1995.1039.

[48] Dauber-Osguthorpe P, Roberts VA, Osguthorpe DJ, Wolff J, Genest M, HaglerAT. Structure and energetics of ligand binding to proteins: Escherichia colidihydrofolate reductase-trimethoprim, a drug-receptor system. Proteins1998;4:31–47. http://dx.doi.org/10.1002/prot.340040106.

[49] Vignoud L, David L, Sixou B, Vigier G. Influence of electron irradiation on themobility and on the mechanical properties of DGEBA/TETA epoxy resins.Polymer 2001;42:4657–65. http://dx.doi.org/10.1016/S0032-3861(00)00791-6.

[50] Martínez L, Andrade R, Birgin EG, Martínez JM. PACKMOL: a package forbuilding initial configurations for molecular dynamics simulations. J ComputPhys 2009;30:2157–64. http://dx.doi.org/10.1002/jcc.21224.

[51] Yang L, Tong L, He X. MD simulation of carbon nanotube pullout behavior andits use in determining mode I delamination toughness. Comput Mater Sci2012;55:356–64. http://dx.doi.org/10.1016/j.commatsci.2011.12.014.

[52] Li Y, Liu Y, Peng X, Yan C, Liu S, Hu N. Pull-out simulations on interfacialproperties of carbon nanotube-reinforced polymer nanocomposites. ComputMater Sci 2011;50:1854–60. http://dx.doi.org/10.1016/j.commatsci.2011.01.029.

[53] Mori T, Tanaka K. Average stress in matrix and average elastic energy ofmaterials with misfitting inclusions. Acta Metall 1973;21:571–4. http://dx.doi.org/10.1016/0001-6160(73)90064-3.

[54] Benveniste Y. A new approach to the application of Mori–Tanaka’s theory incomposite materials. Mech Mater 1987;6:147–57. http://dx.doi.org/10.1016/0167-6636(87)90005-6.

[55] Qui YP, Weng GJ. On the application of Mori–Tanaka’s theory involvingtransversely isotropic spheroidal inclusions. Int J Eng Sci 1990;28:1121–37.http://dx.doi.org/10.1016/0020-7225(90)90112-V.

[56] Marzari N, Ferrari M. Textural and micromorphological effects on the overallelastic response of macroscopically anisotropic composites. J Appl Mech1992;59:269–75. http://dx.doi.org/10.1115/1.2899516.

[57] Belytschko T, Xiao SP, Schatz GC, Ruoff RS. Atomistic simulations of nanotubefracture. Phys Rev B 2002;65:1–8. http://dx.doi.org/10.1103/PhysRevB.65.235430.

[58] Hu N, Fukunaga N, Lu C, Kameyama M, Yan M. Prediction of elastic propertiesof carbon nanotube reinforced composites. Proc R Soc London A2005;461:1685–710. http://dx.doi.org/10.1098/rspa.2004.1422.

[59] Montazeri A, Naghdabadi R. Investigation the stability of SWCNT-polymercomposites in the presence of CNT geometrical defects using multiscalemodeling. In: Proceedings of the Fourth International Conference onMultiscale Materials Modeling, 2008, p. 163–166.

[60] Bradford PD, Wang X, Zhao H, Maria J-P, Jia Q, Zhu YT. A novel approach tofabricate high volume fraction nanocomposites with long aligned carbonnanotubes. Compos Sci Technol 2010;70:1980–5. http://dx.doi.org/10.1016/j.compscitech.2010.07.020.