Interphase effect on the macroscopic elastic properties of non-bonded single-walled carbon nanotube composites

Composites Part B 77 (2015) 52e58

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Composites Part B journal homepage: www.elsevier.com/locate/compositesb

Interphase effect on the macroscopic elastic properties of non-bonded single-walled carbon nanotube composites Saeed Herasati a, b, Liangchi Zhang b, * a

Mechanical Engineering Department, Yasouj University, P. O. Box: 75914-353, Yasouj, Iran Laboratory for Precision and Nano Processing Technologies, School of Mechanical & Manufacturing Engineering, The University of New South Wales, NSW 2052, Australia b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 October 2014 Received in revised form 23 December 2014 Accepted 3 March 2015 Available online 11 March 2015

Compared to the small diameter of a carbon nanotube (CNT), the thickness of the CNTematrix interphase in a CNTecomposite is considerable. Hence, the interphase property can significantly influence the macroscopic properties of the composite. This paper applies an effective multi-scale method to explore such an interphase effect on the properties of nano-composites reinforced by single-walled CNTs. The method integrates the van der Waals (vdW) gap interphase, the dense interphase, and the randomly distributed wavy CNTs in a matrix to realize an accurate prediction of macroscopic properties with a nanoscopic resolution, by using a conventional finite element code commercially available. The study concluded that with the same volume fraction, increasing CNT waviness and diameter reduces the composite Young's modulus, and that ignoring either the vdW gap interphase or the dense interphase can lead to an erroneous characterization, and that both interphases can be ignored in some circumstances. Crown Copyright © 2015 Published by Elsevier Ltd. All rights reserved.

Keywords: B. Elasticity B. Interface/interphase C. Finite element analysis (FEA) Carbon nanotubeecomposite

1. Introduction Many experimental and theoretical studies on carbon nanotubes (CNTs) have shown that they have exceptional mechanical properties [1]. Hence, CNTs, both single-walled (SWCNT) and multiwalled (MWCNT), have been considered to be promising reinforcing materials to make engineering nano-composites of tailored mechanical properties. CNTematrix interphase is a transient region between the external surfaces of CNTs and the bulk matrix of a CNTecomposite. Compared to the small diameter of a CNT, the thickness of the interphase is considerable, which bonds the CNTs to the matrix and plays an important role in determining macroscopic properties of the composite [2,3]. In the case of a nonbonded CNTecomposite, there are two distinct interphase regions. One is the vdW interphase, which is the gap distance between the CNT atoms and those in the internal surface of the matrix surrounding the CNT [4]. The other is the adsorption layer interphase, which often has a greater density than the bulk matrix [5,6], hence is called a dense interphase. In some studies the vdW gap

* Corresponding author. Tel.: þ61 2 9385 6078; fax: þ61 2 9385 7316. E-mail addresses: [email protected] (S. Herasati), [email protected] (L. Zhang). http://dx.doi.org/10.1016/j.compositesb.2015.03.015 1359-8368/Crown Copyright © 2015 Published by Elsevier Ltd. All rights reserved.

interphase was assumed to be the only interphase region [7e9] with certain thickness and mechanical properties. The effect of the vdW interphase on the nano-composite properties have been investigated to a certain extent. For example, Tan et al. [10] studied analytically and reported that the vdW interphase debonds and weakens the composite when the material is subjected to a large strain. The spring element in a finite element has been used to treat CNTecomposites [11e14]. For example, Shokrieh and Rafiee [12] used spring elements in a finite element analysis to treat the vdW interactions between the CNT atoms and the surrounding matrix. They claimed that the CNTecomposite would have a nonlinear behaviour under a large strain. Using the atomistic simulation, Tsai et al. [7] characterized the vdW interphase considering the non-bonded energy between an SWCNT and matrix. They found that the interphase has a major effect on the transverse Young's modulus of aligned SWCNTecomposites. The role of the dense interphase has also been studied. Odegard et al. [3] developed a continuum-based elastic micromechanical model and concluded that the interphase influence would diminish if the diameter of a nano-filler is greater than 200 nm. Wang et al. [15] developed an FE code to include the interphase effect on the overall properties of nano-composites with different inclusion shapes. They concluded that the interphase is effective when the nano-filler diameter is less than 20 nm. Although the shape and

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material of SWCNTs is different from those studied by Odegard and Wang, the diameter of an SWCNT (~1 nm) is much less than the maximum effective diameter claimed by Odegard and Wang. Thus one can imagine that the interphase of SWCNTs is very effective. In their micro-mechanical model, Yang et al. [5] treated the combined vdW gap interphase and SWCNT as an inclusion, but the combined dense interphase and matrix as a new matrix. They reported that a weaker interphase would make the elastic modulus smaller, except that in the longitudinal direction of the SWCNTs. Recently, the authors [6] proposed a comprehensive nano-scale representative volume element (NRVE) for characterizing SWCNTePVC composites. This NRVE is capable of accommodating the coupled contribution of all phases associated with a CNTecomposite unit: the matrix, the dense interphase, the vdW gap interphase and the SWCNT. The vdW gap interphase and the SWCNT were regarded as an equivalent solid fibre (ESF). They then characterized the elastic properties of each phase by atomistic simulation and implemented within a three-phase continuumbased FE model. Their investigation showed that the NRVEs have transversely isotropic properties and that the vdW gap interphase is softer but the dense interphase is stiffer than the bulk matrix. They concluded that ignoring an interphase region would bring about erroneous predictions of the elastic parameters of the NRVE, particularly its transverse Young's modulus and the out-of-plane shear modulus. However, their results were limited to a special case with straight SWCNTs. In reality, nevertheless, CNTs in a composite are highly wavy because of their flexibility and extremely high aspect ratio of length to diameter [16,17]. To the best of our knowledge, the effect of interphase on the overall elastic properties of composites with wavy SWCNTs has not been studied yet. Some models have been developed to study the effect of wavy CNTs. Some of them treated the CNTs as uniform fibres of a sinusoidal wavy shape [18,19], and simplified them as a bow wavy shape [20,21]. These assumptions are inconsistent with the actual geometries of wavy CNTs in a CNTecomposite [16,17]. Recently, the authors proposed a technique for characterizing the waviness of CNTs, together with the new defined concept of waviness angle an algorithm was developed to generate wavy CNTs consistent with experimental observations [22]. Their method was verified by experimental and can be considered reliable for studying the effect of SWCNT waviness. This paper will develop a multi-scale method to explore the impact and contribution of each of the interphases outlined above on the overall elastic properties of non-bonded, randomly wavy SWCNTecomposites. 1D (aligned) and 3D distributed SWCNTs with different waviness angles will be constructed and implemented in an FE platform for stress analysis. 2. Multi-scale modelling The primary strategy of our multi-scale modelling is illustrated in Fig. 1. The basic element includes a theoretically infinitely long straight SWCNT embedded in an amorphous PVC matrix [6] and thus the shear lag effect is not studied. Since the dimension of the element is much less than 100 nm, it is called an NRVE. In a CNTecomposite, the volume fraction of CNTs is always low, rarely reaching 20% (e.g. Ref. [23]). It is therefore rational to assume, when the CNTs are dispersed well in the composite in manufacturing, that each CNT is surrounded fully by the matrix, and that the distance between individual CNTs is much larger than the thickness of an individual CNTematrix interphase. Hence, the contact between CNTs and that between CNTematrix interphases does not need to be considered in the modelling. As a result, the atomistic model of NRVE (A-NRVE) under a periodic boundary condition can be big

53

enough to avoid any artificial effect of interphase or CNT contacts. Although, A-NRVE has been characterized with the aid of molecular dynamics (MD) and molecular mechanics (MM) [6], the authors have proposed an efficient and flexible continuum based threephase model [6] enables to characterize a minimum NRVE dimension confined by the external diameter of the dense interphase layer, called 3P-NRVE model, as shown graphically in Fig. 1. With 3P-NRVE model, we can evaluate the transversely isotropic elastic properties of the 3P-NRVE, which takes into account all the atomistic interactions. When the 3P-NRVE is applied to a greater scale analysis, it can be treated as a homogeneous solid NRVE (SNRVE). In this way, the calculation will be efficient but the lower scale (atomistic and interphase) effects are already included. A series of S-NRVEs can then be assembled to build up a randomly wavy SWCNT-fibre, as illustrated in Fig. 1. Compared to a square cross section, a circular cross section fibre can be a better representative of CNTs. In an FE modelling, however, it needs a fine mesh generation on the cross section. In contrast, square cross section fibres can be assembled with a series of single cubic homogenized elements which need much less computational cost while including all nanoscopic effects. A micro-scale representative volume element (MRVE) can therefore be obtained by assembling such SWCNT-fibres with the surrounding PVC matrix. Such an MRVE is ready to be used for an efficient finite element analysis but with the atomistic and interphase effects counted. More details about the SNRVE of the above strategy will be explained in the following sections. 2.1. Elastic properties of S-NRVE The elastic properties of S-NRVE can be obtained by transferring those of 3P-NRVE using the method developed by the authors previously [6]. A 3P-NRVE includes three phases: the ESF, the dense interphase and the bulk matrix, as illustrated in Fig. 1. To explore the diameter effect on the properties of the 3P-NRVE, a variety of SWCNTs, with typical diameters ranging from 6.76 Å of SWCNT (5, 5) to 26.18 Å of SWCNT (20, 20), were used to construct the corresponding 3P-NRVEs. The elastic properties of ESF were from our previous work [6]. It has been shown that the dense interphase of PVC has a thickness of 3 Å, Young's modulus of 17 GPa, and Poisson's ratio of 0.40 [6]. The Young's modulus and Poisson's ratio of the bulk PVC are 3.6 GPa and 0.38, respectively [24]. The overall elastic properties of 3P-NRVE can therefore be obtained by a 3D finite element analysis as explained in detail in Ref. [6]. This gives rise to the five elastic moduli of the corresponding S-NRVE, as listed in Table 1. Rows AeD represent the schemes of: (A) all phases included, (B) vdW gap interphase excluded, (C) dense interphase excluded, and (D) both dense interphase and vdW gap excluded. The elastic properties of the bulk PVC from our previous atomistic simulation [6] and the experimental results available [24] are Em ¼ 3.6 GPa and nm ¼ 0.38 [25]. 2.2. Elastic properties of MRVE The wavy SWCNT-fibres could be generated based on the algorithm developed by Ref. [22] in which a wavy SWCNT-fibre is assembled as a cord of a sequential random walking segments. Each cord starts from a boundary plane of an MRVE and continues in a random direction (it in most cases goes across another boundary plane). Each new point on the cord is generated on the base of a right circular cone, see Fig. 2. The apex of the cone is located at the end point of the last segment and the axis of the cone is coincident with the orientation of that segment. With a pre-determined walking distance of ds, the only variable is the waviness angle, qmax, which is the maximum angle of the cones that can be

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Fig. 1. A schematic diagram of the multi-scale modelling strategy for characterising SWCNTecomposites from the atomistic scale NRVE to the micro-scale MRVE. The MRVE is readily to be used for an efficient finite element analysis but with the atomistic and interphase effects counted.

determined from the experimental images [16], as demonstrated by Ref. [22]. The cone angle of the ith segment, qi, is the deflection angle and has a randomly normal distribution. An appropriate walking distance is to take it as the CNT diameter [22]. In this study ds ¼ 3 nm is used. The wavy SWCNT-fibres are then assembled with the PVC matrix in a cubic MRVE. In this study, a uniform displacement boundary condition (Dirichlet condition) has been applied on the MRVEs. It is noted that compared to the uniform displacement condition, a periodic

Table 1 Elastic properties of transversely isotropic 3P-NRVEs containing SWCNTs of different diameters. SWCNT

Ro (Å) Cross section (Å2) 18  18

(5, 5)

6

(10, 10)

9.3

24.6  24.6

(15, 15) 12.6

31.2  31.2

(20, 20) 15.6

37.2  37.2

Elastic modules (GPa)

(A) (B) (C) (D) (A) (B) (C) (D) (A) (B) (C) (D) (A) (B) (C) (D)

Poisson's ratio

Exx, Eyy Ezz

Gyz, Gxz nzy, nzx

nxy

11.27 15.54 5.91 6.96 10.11 12.61 6.10 7.04 6.96 7.43 4.83 5.09 5.10 4.34 3.57 2.80

1.97 4.24 0.75 2.40 1.74 5.05 0.79 3.05 1.53 5.74 0.65 3.56 1.40 5.88 0.62 3.68

0.58 0.65 0.49 0.52 0.66 0.73 0.57 0.59 0.78 0.85 0.69 0.74 0.79 0.86 0.78 0.85

221.33 233.02 214.87 226.90 231.63 237.82 226.97 233.24 214.66 220.22 210.91 216.51 200.66 219.94 197.47 216.78

0.29 0.33 0.25 0.30 0.29 0.32 0.27 0.31 0.30 0.32 0.28 0.30 0.30 0.32 0.28 0.30

(A): all phases included; (B): vdW gap excluded; (C): dense interphase excluded; (D): both dense interphase and vdW gap excluded.

Fig. 2. The schematic diagram of the random walking segments of a CNT cord.

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boundary gives more accurate results for a small RVE. However, when the dimensions of RVE exceed a certain size, the size effect vanishes and both boundary conditions give similar results [26]. Although the MRVE of this study has a micro-scale dimension, it contains many CNTs (see Fig. 3) and can be assumed as a big RVE. Thus it is not necessary to use a periodic boundary with a lot of extra constraint equations. To determine the minimum acceptable size of MRVE, in a sensitivity analysis, the dimension of MRVE includes CNT (10, 10) with a fixed waviness angle of 30 were increased and the equivalent overall elastic constant was evaluated. The sensitivity analysis showed that, due to the size effect, the overall Young's modulus of MRVE change with the variation of its dimensions, but it converges to a certain quantity when the dimensions reach about 100 times those of the S-NRVE side length that we used as a criterion for all MRVEs. More details about the RVE size can be found in Ref. [27]. Although the CNTs that are generated by algorithm cross the boundary planes, the final dimensions of MRVE were increased by 1% with pure matrix to exhibit a reduced size of CNTs due to the non-periodic boundary condition requirements. The target volume fraction has been counted in the new extended MRVE dimensions. The finite element models for stress analysis are then generated with the aid of the commercial ANSYS Workbench and transferred to ANSYS-APDL. Since the S-NRVE elements have transversely isotropic properties, the local coordinate system of each element was directed in parallel with the neutral axis (axis of symmetry) of the CNT cord with a script file developed. To study the interphase effect under different waviness angles, SWCNT (10, 10) is selected because this size is typical in the synthesized SWCNTs. However, when studying the interphase effect with varying SWCNT diameter, the waviness angle is fixed. To explore the role of CNT alignment, both aligned SWCNTs, denoted as the 1D-Scheme for convenience of discussion, and randomly distributed in three-dimensional directions, denoted as the 3DScheme, are investigated with a fixed volume fraction of 0.35% but a variety of waviness angles (e.g. Fig. 3). To generate the 1D-Schemes, the algorithm constrains each SWCNT to a narrow cube (bounded by four planes of x0 ± Dx/2 and y0 ± Dy/2) with the same length of RVE in z-direction. The values of Dx and Dy depend on the average

55

maximum amplitude of the aligned SWCNTs from experimental images [28]. In this study, the local RVEs for the 1D-Scheme fibres are limited to Dx ¼ Dy ¼ 20 Å. More details about the generation of 1D- and 3D-Schemes have been explained in Ref. [22]. 3. Stressestrain analysis The stiffness matrix components of an MRVE are determined based on the generalized Hook's law for the stressestrain relation, i.e.

slm ¼ Clmnk 3nk ;

(1)

where Clmnk is the modulus tensor and slm, 3nk are the stress and strain tensors, respectively. Since both the stress and strain tensors are symmetric, the Voigt notation (i.e., 1 ¼ xx, 2 ¼ yy, 3 ¼ zz, 4 ¼ xz, 5 ¼ yz, 6 ¼ xy) is convenient to use, which gives rise to

si ¼ Cij 3j

i; j ¼ 1 : 6;

(2)

where si, Cij and 3j are stress, stiffness matrix and strain components respectively. We assume that the MRVE models with 1D- and 3D-Schemes have transversely isotropic and isotropic properties respectively. Under this condition, the shear terms are decoupled from the normal stress/strain terms [29]. In this study, the shear moduli of composites are not considered. Thus the reduced form of Eq. (2) (i.e. shear effects not included) is more convenient. Thus

si ¼ Cij 3j ;

i; j ¼ 1 : 3;

(3)

The stiffness matrix, Cij, in Eq. (3) includes 9 components. With the aid of the developed script file mentioned above, a small normal strain of 0.2% is applied to one face of the cubic MRVE model (e.g. mdirection) while keeping the other faces fixed in their normal directions (3m ¼ 0.002, 3n ¼ 0, n s m). All nodes on the faces are free in their tangential directions (si ¼ 0, i ¼ 4:6). The total normal forces of each face are then calculated from the nodes involved. The average normal stresses are subsequently calculated by dividing the total normal reaction force by the associated area. By applying each normal strain, three components of a column of the stiffness matrix

Fig. 3. 1D and 3D-Schemes of SWCNTecomposites with a volume fraction of 0.35% and waviness angles of (a) qmax ¼ 0 , (b) qmax ¼ 30 , and (c) qmax ¼ 50 .

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in Eq. (3) are determined. Assuming that XY is the plane of symmetry, the transversely isotropic elastic moduli of 1D-Schemes associated with the normal stress/strain can be determined by the compliance matrix below, which is the inverse of the stiffness matrix.

2

S ¼ C 1

1 6 E 6 11 6 6 n21 ¼6 6 E 11 6 6 4 n31  E33



n21 E11

1 E11 

n31 E33

3 n31 E33 7 7 7 n31 7 7:  E33 7 7 7 1 5 

(4)

E33

where Eij and nij represent the Young's modulus and Poisson's ratios respectively. It is noted that in the case of transversely isotropic materials, n31 =E33 ¼ n13 =E11 and E11 ¼ E22, therefore only one of them are shown in Eq. (4). In the case of isotropic 3D-Schemes, there are only two independent elastic moduli which can be determined by

S ¼ C 1

2 1 14 n ¼ E n

n 1 n

3 n n 5 1

diameter. They should converge to case A, after a big enough diameter of CNTs. This is consistent with the findings of Wang et al. [15] and Odegard et al. [3] for other materials and shapes of inclusions. Fig. 4 also shows that when the volume fraction of SWCNTs in a CNTecomposite is a constant, the macroscopic Young's modulus of the composite decreases with increasing the waviness and CNT diameter. This is supported by the results of Tsai et al. [7]. In other word, with a given CNT concentration and waviness angle, smaller SWCNTs are more superior reinforcing fillers. Fig. 5a presents the variation of the longitudinal EC/Em of the composite with aligned SWCNT (10, 10), that is a 1D-Scheme. It shows that both the vdW gap interphase and the dense interphases can be ignored without a remarkable effect on the overall longitudinal Young's modulus of the composite. Fig. 5b illustrates that the result above is consistent in the whole range of SWCNT diameters examined. When the waviness angle is zero, EC/Em in Figs. 4a and 5a represents straight SWCNTs. Here, we consider that SWCNTs are long (e.g. longer than 1 mm); thus the upper bond of EC/Em can be obtained simply with the rule of mixtures [30],

(5)

Using the method above, three normal strains are applied separately and all 9 components of Eq. (5) are determined. Although in this case a normal strain in one direction is sufficient to evaluate the Young's modulus of the material, the results of three normal strains were averaged for more accuracy. 4. Results and discussion Fig. 4a shows the variation of the relative Young's modulus (Ec/ Em) of the CNTecomposite using the 3D-Scheme of SWCNT (10, 10). Each point in the figure is an average of three samples. The results show that the Young's modulus will be overestimated if the effect of vdW gap interphase is ignored but will be underestimated if the dense interphase effect is neglected. When both the vdW gap interphase and dense interphases are ignored, the results approach those including all phases. This indicates that with SWCNT (10, 10), the overall influence of the interphases as a whole is insignificant. Fig. 4b shows that SWCNT diameter has a big influence on the composite properties. When the diameter is greater than SWCNT (10, 10), the interphase effect becomes greater. A closer look at Fig. 4b shows that the upper and lower bonds, which are cases B and C respectively, approach each other with increasing the SWCNT

EC ¼ Em

 h

 ECNT  1 $yCNT  1; Em

(6)

where EC, Em, ECNT and yCNT represent the Young's modulus of composite, matrix, CNT and the volume fraction of CNTs, respectively. Coefficient h takes a value of 1 for unidirectional CNTecomposites and 0.2 for random 3D distributed CNTecomposites. We assume that the CNT and half of the surrounding vdW gap is a solid fibre. In the case of SWCNT (10, 10), the diameter of the fibre is 15.81 Å with Young's modulus of ECNT¼ 680 GPa. Thus Eq. (6) gives rise to EC =Em j3D ¼ 1:136 and EC =Em 1D ¼ 1:694 which is very close to our results of 1.13 and 1.60 for 1D and 3D distribution of CNTs. Fig. 6a shows the effect of waviness angle on the transverse EC/ Em of composites with aligned SWCNT (10, 10). It can be seen that when the waviness angle is small, Scheme (A) consideration (i.e., all interphases included) brings about a greater Young's modulus than the Scheme (D) consideration (without an interphase). The discrepancy vanishes as the waviness angle increases. It is a fact that an SWCNT has small deflection stiffness and a high aspect ratio of length to diameter. Hence, an SWCNT in a composite normally has a high waviness angle [16,17]. This seems to indicate that the interphase effect on the transverse modulus is negligible in the case of SWCNT (10, 10). Fig. 6b shows that for SWCNTs of a diameter

Fig. 4. The relative Young's modulus of the CNTecomposite with 3D-Schemes, (a) with SWCNT (10, 10), and (b) with a verity of SWCNTs from SWCNT (5, 5) to (15, 15) under a given qmax ¼ 30 .

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Fig. 5. Variation of the relative longitudinal Young's modulus of the composite with aligned SWCNTs. (a) Using SWCNT(10,10) but varying qmax, and (b) using qmax ¼ 30 but varying SWCNT diameters.

Fig. 6. Variation of the relative transverse Young's modulus of the composites with aligned SWCNTs. (a) Using SWCNT(10,10) but varying qmax, and (b) using qmax ¼ 30 but varying SWCNT diameters.

smaller than (10, 10), the interphase plays an important role on the transverse modulus. The smaller the diameter, the greater the interphase effect becomes. It must be pointed out further that the present study focuses on non-bonded SWCNTecomposites only. When chemical bond takes place between functionalized CNTs and matrix, such as those discussed by Refs. [31,32], the mechanical properties of the interphases will change. In this case, the effects of interphases, waviness and alignment on the macroscopic behaviour of an SWCNTecomposite will also be altered. This will need to be studied further. 5. Conclusions This paper has investigated the effects of SWCNTematrix interphases, SWCNT alignment, and SWCNT waviness on the macroscopic elastic properties of SWCNTecomposites. A truly multi-scale characterization method has been applied to enable an efficient continuum mechanics analysis involving atomistic effects of the materials. The influential factors studied include the vdW gap interphase, the dense interphase, the waviness of SWCNTs and the nanotube diameter. The study concludes that ignoring the contribution of the vdW gap interphase will overestimate the macroscopic Young's modulus of an SWCNTecomposite, but that neglecting the role of the dense interphase will underestimate the modulus. Due to the opposite effect of the vdW gap interphase and the dense interphase, the overall interphase does not alter the longitudinal Young's modulus of aligned SWCNTecomposites. The same statement is valid for the case of 3D-Schemes with SWCNT (10, 10) and smaller.

References [1] Shokrieh MM, Rafiee R. A review of the mechanical properties of isolated carbon nanotubes and carbon nanotube composites. Mech Compos Mater 2010;46(2):155e72. [2] Sevostianov I, Kachanov M. Effect of interphase layers on the overall elastic and conductive properties of matrix composites. Applications to nanosize inclusion. Int J Solids Struct 2007;44(3e4):1304e15. [3] Odegard GM, Clancy TC, Gates TS. Modeling of the mechanical properties of nanoparticle/polymer composites. Polymer 2005;46(2):553e62. [4] Wernik JM, Cornwell-Mott BJ, Meguid SA. Determination of the interfacial properties of carbon nanotube reinforced polymer composites using atomistic-based continuum model. Int J Solids Struct 2012;49(13): 1852e63. [5] Yang S, Yu S, Kyoung W, Han D-S, Cho M. Multiscale modeling of sizedependent elastic properties of carbon nanotube/polymer nanocomposites with interfacial imperfections. Polymer 2012;53(2):623e33. [6] Herasati S, Zhang LC, Ruan HH. A new method for characterizing the interphase regions of carbon nanotube composites. Int J Solids Struct 2014;51(9): 1781e91. [7] Tsai JL, Tzeng SH, Chiu YT. Characterizing elastic properties of carbon nanotubes/polyimide nanocomposites using multi-scale simulation. Compos Part B Eng 2010;41(1):106e15. [8] Shokrieh MM, Rafiee R. Prediction of mechanical properties of an embedded carbon nanotube in polymer matrix based on developing an equivalent long fiber. Mech Res Commun 2010;37(2):235e40. [9] Hu N, Fukunaga H, Lu C, Kameyama M, Yan B. Prediction of elastic properties of carbon nanotube reinforced composites. Proc R Soc A Math Phys Eng Sci 2005;461(2058):1685e710. [10] Tan H, Jiang LY, Huang Y, Liu B, Hwang KC. The effect of van der Waals-based interface cohesive law on carbon nanotube-reinforced composite materials. Compos Sci Technol 2007;67(14):2941e6. [11] Spanos KN, Georgantzinos SK, Anifantis NK. Investigation of stress transfer in carbon nanotube reinforced composites using a multi-scale finite element approach. Compos Part B Eng 2014;63:85e93. [12] Shokrieh MM, Rafiee R. Investigation of nanotube length effect on the reinforcement efficiency in carbon nanotube based composites. Compos Struct 2010;92(10):2415e20.

58

S. Herasati, L. Zhang / Composites Part B 77 (2015) 52e58

[13] Giannopoulos GI, Georgantzinos SK, Anifantis NK. A semi-continuum finite element approach to evaluate the Young's modulus of single-walled carbon nanotube reinforced composites. Compos Part B Eng 2010;41(8):594e601. [14] Tahan Latibari S, Mehrali M, Mottahedin L, Fereidoon A, Metselaar HSC. Investigation of interfacial damping nanotube-based composite. Compos Part B Eng 2013;50:354e61. [15] Wang HW, Zhou HW, Peng RD, Mishnaevsky Jr L. Nanoreinforced polymer composites: 3D FEM modeling with effective interface concept. Compos Sci Technol 2011;71(7):980e8. [16] Qian D, Dickey EC, Andrews R, Rantell T. Load transfer and deformation mechanisms in carbon nanotubeepolystyrene composites. Appl Phys Lett 2000;76(20):2868e70. [17] Shaffer MSP, Windle AH. Fabrication and characterization of carbon nanotube/ poly(vinyl alcohol) composites. Adv Mater 1999;11(11):937e41. [18] Karami G, Garnich M. Micromechanical study of thermoelastic behavior of composites with periodic fiber waviness. Compos Part B Eng 2005;36(3):241e8. [19] Tsai CH, Zhang C, Jack DA, Liang R, Wang B. The effect of inclusion waviness and waviness distribution on elastic properties of fiber-reinforced composites. Compos Part B Eng 2011;42(1):62e70. [20] Nafar Dastgerdi J, Marquis G, Salimi M. The effect of nanotubes waviness on mechanical properties of CNT/SMP composites. Compos Sci Technol 2013;86(0):164e9. [21] Shao LH, Luo RY, Bai SL, Wang J. Prediction of effective moduli of carbon nanotube-reinforced composites with waviness and debonding. Compos Struct 2009;87(3):274e81.

[22] Herasati S, Zhang L. A new method for characterizing and modeling the waviness and alignment of carbon nanotubes in composites. Compos Sci Technol 2014;100:136e42. [23] Cebeci H, Villoria RGd, Hart AJ, Wardle BL. Multifunctional properties of high volume fraction aligned carbon nanotube polymer composites with controlled morphology. Compos Sci Technol 2009;69(15e16):2649e56. [24] Dyment J, Ziebland H. The tensile properties of some plastics at low temperatures. J Appl Chem 1958;8(4):203e6. [25] http://www2.shimadzu.com/applications/Autograph/ MeasurementPoissonsRatioPVC.pdf. [26] Terada K, Hori M, Kyoya T, Kikuchi N. Simulation of the multi-scale convergence in computational homogenization approaches. Int J Solids Struct 2000;37(16):2285e311. [27] Gitman IM, Askes H, Sluys LJ. Representative volume: existence and size determination. Eng Fract Mech 2007;74(16):2518e34. [28] Ginga NJ, Chen W, Sitaraman SK. Waviness reduces effective modulus of carbon nanotube forests by several orders of magnitude. Carbon 2014;66:57e66. [29] Kaw AK. Mechanics of composite materials. Taylor & Francis Group; 2006. [30] Piggott M. Load bearing fibre composites. 2nd ed. US: Springer; 2002. [31] Frankland SJV, Caglar A, Brenner DW, Griebel M. Molecular simulation of the influence of chemical cross-links on the shear strength of carbon nanotubeepolymer interfaces. J Phys Chem B 2002;106(12):3046e8. [32] Mylvaganam K, Zhang LC, Cheong WCD. The effect of interface chemical bonds on the behaviour of nanotubeepolyethylene composites under nano-particle impacts. J Comput Theor Nanosci 2007;4(1):122e6.