Evaluating the Effect of Mechanical Loading on the

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Journal of Computational and Theoretical Nanoscience Vol. 11, 1–7, 2014

Evaluating the Effect of Mechanical Loading on the Effective Thermal Conductivity of Carbon Nanotube Reinforced Polymers (a Monte-Carlo Approach) Mohsen Mazrouei1 , Hossein Jokar2 , Majid Baniassadi3 4 ∗ , Karen Abrinia3 , and Mojtaba Haghighi-Yazdi3 1

School of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran School of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran 3 School of Mechanical Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran 4 University of Strasbourg, ICube/CNRS, 2 Rue Boussingault, 67000 Strasbourg, France 2

Keywords: Thermal Conductivity, Monte-Carlo, Nanocomposite, Mechanical Loading, Homogenization.

1. INTRODUCTION The application of nanomaterials in different area of engineering has been attracting wide attention of researchers and is becoming an active frontier area in recent years. Quantum nanoscience is a new research area to the design of novel nanoscale materials which are described through quantum phenomena and principles.1–3 Nanoparticles have drawn much attention in science due to their extraordinary physical and mechanical properties.4–8 Polymer nanocomposites are a class of nanomaterials in which one or more phases with nanoscale dimensions are embedded in a polymer matrix to create a synergy between the various properties.9 10 The properties and applications of carbon nanotubes (CNTs) and related materials such as nanocomposites have been the subject of a wide range of studies in the last ∗

Author to whom correspondence should be addressed.

J. Comput. Theor. Nanosci. 2014, Vol. 11, No. 8

decade.11 12 These properties which include low mass density, high flexibility, large aspect ratio, and combination of mechanical and electrical properties of individual nanotubes make them appropriate fillers. Theoretical and experimental researches report extremely high tensile moduli and strengths, and also thermal conductivities of more than 3000 W/mK for multi-wall carbon nanotubes (MWNT) and single-wall carbon nanotubes (SWNT).13–16 Polymers with enhanced thermal and electrical conductivity play an essential role in many applications. The first use of CNTs in polymer nanocomposites was reported by Ajayan et al.17 in 1994. The enhancement of the electrical conductivity by several orders of magnitude is obtained by the addition of CNTs. The replacement of carbon black with CNTs for the preparation of electrically conducting polymer composites is expected to have a great impact on a wide range of industrial applications.18–20 Researchers are trying to use the conductivity and high aspect ratio

1546-1955/2014/11/001/007

doi:10.1166/jctn.2014.3560

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The excellent properties of carbon nanotubes (CNTs) make them appropriate fillers for polymer matrix which yield to fabricating desirable nanocomposites. In this paper, the effect of mechanical loading on thermal conductivity of CNT polymer nanocomposites is investigated. In the simulation, CNT sticks are distributed in the polymer matrix randomly according to the Monte-Carlo method. In addition the resulting volume concentrations were computed using Monte-Carlo integration method. A computer code is developed in such a way that mechanical loading is applied incrementally. This mechanical stress produces new configuration (shape) of the RVE. After any configuration, a two-step homogenization technique of Mori–Tanak was applied to determine the effective thermal conductivity and Young moduli. It was observed that the angles between CNTs and mechanical loading were decreased leading to anisotropic RVE. The results showed increase in the effective thermal conductivity and Young moduli along the direction of loading and a reduction in other directions. These results are compatible with experimental data that reveal improvement in effective thermal conductivity and Young moduli when most of the fillers of composites tend to the same alignment.

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Evaluating the Effect of Mechanical Loading on the Effective Thermal Conductivity of CNT Reinforced Polymers

of CNTs to produce conductive plastics with exceedingly low percolation thresholds.21 Also the improvement in stiffness, strength and especially thermal conductivity due to the addition of carbon nanotubes in low polymeric matrix materials has been reported in some studies.22–24 Some particular application of nanocomposites loaded with CNT fillers could be found in the aerospace, electronics, automotive, and similar industries. For example, efficient performance of CNT-reinforced nanocomposites in conducting static electricity away from the flammable gasoline makes these materials suitable candidates to replace with the metallic fuel system components.25 The electrostatic discharge as well as the heat produced from electronic components in printed circuit boards can be conducted by even low volume fraction of added CNT. Evaluation of thermal treatment of nanocomposite materials is essential in knowing the amount of heat dissipation from them. Few studies have used continuum mechanics approach to predict the thermal properties of nanocomposites.26–28 Nishimura and Liu26 used the boundary integral equation method to analyze thermal properties of CNT based nanocomposites. They utilized fast multiple boundary element method to solve a heat conduction problem in 2-D infinite domain embedded with many rigid inclusions. Song and Youn28 found the effective thermal conductivity of carbon nanotube/polymer composites by control volume finite element method. Singh et al.29 studied equivalent thermal conductivity of CNT composites using element free Galerkin method, and concluded that the thermal conductivity of the composite is in relation with nanotube dimensions. Helmot and Sergio30 applied the well-established Mori–Tanaka scheme for modeling the overall thermal conduction behavior of composites containing reinforcements with interfacial resistances and recommended size distributions. Zhang et al.31 employed the double inclusion model to predict the thermal conductivities and elastic constants of CNTs. The work of Jan et al.32 is based on the Mori–Tanaka method to predict the effective thermal conductivity of macroscopically isotropic materials of matrix-inclusion type. Vorel et al.33 proposed the Mori–Tanaka approach to evaluate the effective thermal conductivity of carbon–carbon (C/C) plain weave textile composites. Mortazavi et al.34 used three schemes including finite element method, the Mori–Tanaka approach, and the strong contrast method for the prediction of the effective thermal conductivity and elastic modulus of isotropic random two-phase composite materials with low fillers content. In this study, it is assumed that perfect interface exists between matrix and CNTs for incorporating the effect of mechanical loading into the thermal conductivity behaviour. It should be noted that mechanical load is applied incrementally while spatial configuration is updated simultaneously with Monte-Carlo simulation. The polymer matrix is considered an elastomer with linear 2

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elastic behavior up to large strains. A two-step homogenization approach based on the Mori–Tanaka scheme is used to evaluate effective thermal conductivity after each incremental applied loading. The application of the change in thermal conductivity with respect to strain leads to designing and fabricating special sensors which can be employed for industrial purposes. The paper is organized as follows. Section 2 presents methodology including the Monte-Carlo approach, homogenization of the RVE as well as the details of mechanical loading and thermal conductivity. Finally, Section 3 includes the results and the corresponding discussion.

2. METHODOLOGY First, the details of the suggested technique to investigate the relation between the mechanical loading and thermal conductivity as well as Young’s modulus behavior are explained. Then, the topological subject of the simulation with a summary of the implementation procedure is established. Since the Young’s modulus of CNT sticks is larger than that of the polymer matrix, the elongation of sticks is neglected and it is therefore assumed that only rigid rotations and translations happen in CNT sticks. In this study, CNT fillers can be simulated as straight sticks which is a reasonable assumption that can be found in similar studies.35 36 CNT sticks are modeled as solid cylinders which are distributed inside a cubic RVE randomly. The sake of simplification in the simulation, the CNT particle can penetrate other CNTs.35–38 The Monte-Carlo integration approach is applied to obtain the volume fraction of sticks. After each new configuration of the CNTs in the RVE (explained in Section 2.3), a two-step homogenization technique of Mori–Tanaka is used to calculate thermal conductivity and Young’s modulus. 2.1. Monte Carlo Simulation Computer generated RVE has been developed to model displacement of CNT sticks within the matrix. For this purpose the following steps are taken: At first step, one point inside the block is selected randomly and this point has been considered as one of the ends of an arbitrary CNT stick. For uniform distribution of the other end point of CNT particles on the surface of a sphere, a specific algorithm is used.39 40 According to the algorithm, if the first end is located in the center of a sphere with a radius equal to the length of the CNT stick, the other end of the CNT stick is chosen randomly according to the following equation: = 2

(1)

= Arc cos2 − 1

(2)

Where and are spherical angles, and , are random numbers with uniform distribution between 0 and 1. J. Comput. Theor. Nanosci. 11, 1–7, 2014

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It is trimed partial of the links that are located outside the boundary of the simulation block/RVE by updating their second end coordinates. This algorithm is implemented in other works such as microstructure reconstruction that has been reported.41–43 2.2. Homogenization The solution is based on a two-step homogenization. The RVE is decomposed into grains, each one considered as a two-phase composite.44 At the first step, it is assumed that the RVE is decomposed into grains as many as the number of fillers with each grain encapsulating only one filler. Homogenization of each grain was performed using the Mori–Tanaka method. The following equation presents the volume fraction of the RVE: N i i vf frve = (3) Vtot Here vfi denotes the volume of the ith CNT, N is the total number of grains which corresponds to the number of fillers and Vtot stands for the total volume of the RVE. Volume of each grain can be proportional to vfi and in this study, all of the CNTs have the same length and diameter and hence all of the grains have the same volume. Another assumption is that volume fraction of the RVE is equal to the volume fraction of the grains. Therefore, it can be concluded that:40 vfi

=

frve = fg =

(4) 1

(5)

Where vgi is the volume of grain and is the Coefficient between vgi and vfi and fg stands for volume fraction of grain. In the second step, homogenization of the whole RVE is performed. After calculating thermal conductivity along every CNT’s axis inside each grain the thermal conductivity of the grains is transferred from local to global coordinates. In order to evaluate the effective thermal conductivity of the RVE, the sum of the products of each thermal conductivity weighted by its associated filler volume is divided by the total volume of the fillers. More details will be explained in Section 2.4. The two-step homogenization approach is illustrated in Figure 1.40 2.3. Mechanical Loading Mechanical loading can influence the mechanical behavior of heterogeneous polymer materials containing CNT sticks. When a mechanical force is applied on the carbon nanotube reinforced polymers, a new configuration of ingredients inside the medium occurs. This new configuration of CNT sticks changes the thermal conductivity and Young’s modulus in different directions. J. Comput. Theor. Nanosci. 11, 1–7, 2014

The polymer matrix is considered to be an “elastomer” which has linear elastic behaviour up to very large strains. The displacement field is used to find the new configuration of ingredients inside the medium after mechanical loading. The displacement field is related to the strain field through the strain–displacement equations and then the strain components are related to stresses. On nano-scale, mechanical properties of the nanocomposites change suddenly, but on a larger scale average properties can be seen. In larger scale, the non-homogenous medium represents an effective behavior that could belong to homogenous equivalent medium (HEM). After each new configuration the approach of two-step tailor-made homogenization technique, which is appropriate for computer simulations, is suggested to determine the effective stiffness and thermal conductivity of the RVEs in our modeling. The mechanical loading process and obtaining the new configuration of sticks are as follows: The load is applied incrementally. At each increment the effective stiffness of the typical RVE is updated. The displacement field is also calculated based on the stiffness tensor obtained at the previous increment. In order to obtain the new configuration of sticks; the stiffness tensor is updated according to the following equation:40 N i i lf ce two-step ceff = N i (6) i lf two-step

is the stiffness tensor, lf is the length of Where ceff sticks, ce is global stiffness tensor of each grain, and N is number of grains. In order to understand how new configuration is achieved, 3 concepts are presented here: • relationship between the linear elasticity equation and displacement field of a homogenized heterogeneous medium • determination of spherical angles of the CNTs with unidirectional stress ( x • the dependence of anisotropic components on unidirectional stress. 3

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vgi

Fig. 1. The RVE is decomposed into grains. Each of grain is homogenized (first Step). Next, a second homogenization is accomplished for all the grains (second step).44


For infinitesimal range of strains, continuum mechanics expresses in the form below ij = 0 5Uij + Uji

i j = 1 2 3

(7)

U is the increment of the displacement vector and the following indices are the spatial derivative. Incremental components of the displacement field are derived by considering constant strains and fixing the point which coincides with the center of global reference frame to prevent rotation and displacement u = x xx + y xy + z xz

(8)

v = y yy + x xy + z yz

(9)

w = z zz + x xz + y yz

(10)

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Where u v and w are incremental components of the displacement in the three directions, respectively. The coordinates of the two ends of CNTs are transformed into the new coordinates based on the following equations Xn = xn 1 + xx + yn xy + zn xz

(11)

Yn = yn 1 + yy + xn xy + zn yz

(12)

Zn = zn 1 + zz + yn yz + xn xz

(13)

Where n = 1 or 2 and Xn , Yn , Zn are new positions. The relationships between the spherical coordinates and Cartesian coordinates of a CNT stick are computed with x2 − x1 2 + y2 − y1 2 sin = 0≤≤ (14) z2 − z1 y − y1 tan 1 = 2 0 ≤ 1 ≤ 2 (15) x2 − x1 Applying the Hooke’s law, the relation between effective stiffness tensor of a homogeneous medium, Ceff , the average incremental stress and strain tensors and , can be written as follows: = Seff

(16)

−1 Where Seff = Ceff and is referred to as the effective compliance tensor that is a 6 × 6 matrix for any anisotropic material. When the compliance tensor of the RVE is updated at each increment, it is seen that the terms of anisotropy are very small compared to other terms and therefore those terms can be ignored and the RVE can be taken analogous to an isotropic medium. After applying the mechanical loading the new spherical coordinates can be obtained in terms of the old coordinates and the applied strains components 1 1+ xx x 2 sin ≈ (17) +y 2 z 1+ zz

tan 1 ≈ 4

1 + yy tan1 1 + xx

(18)

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The simplifications are that shearing strains are zero and zz = yy < 0 which are the realistic assumptions. xx > 0 and it is concluded that

sin > sin

(19)

tan 1

(20)

< tan 1

The last two inequalities express that a unidirectional tensile stress reduces the sticks’ angles with respect to the X-axis and aligns the CNT sticks with the X-axis. 2.4. Thermal Conductivity The change in thermal conductivity caused by the application of mechanical loading in the X-direction is now investigated. Mean field micromechanics approaches are simple ways for evaluation of the effective thermal and mechanical properties. The Mori–Tanaka method is one of these approaches. This method predicts the behavior of composites containing reinforcements at non-dilute volume fractions via dilute inhomogeneities that are subjected to effective matrix fields.45 Hatta and Taya46 introduced the Eshelby-like tensor for the solution of thermal conductivity problem. The carbon nanotubes are assumed similar to Prolate spheroid (a1 = a2 < a3 ). This assumption yields the following Eshelby tensor 2 1/2 a 3 a3 a21 a3 −1 a3 s11 = s22 (21) 1 − cosh 3/2 a1 a21 a1 2a23 − a21 s33 = 1 − s22

(22)

Where a1 , a2 and a3 are principal half axes of Prolate spheroid. The dilute inclusion concentration tensor, Adil , can be achieved in analogy to Hill’s47 equations as: −1

Adil = I + skm−1 ki − km

(23)

Where I is the second order identity tensor, ki and km are the thermal conductivity tensors of the inclusion and matrix, respectively, and S is the Eshelby tensor.48 Eventually, the Mori–Tanaka macroscopic thermal conductivity tensor (ke of the nanocomposite with randomly oriented inclusions can be expressed by:34 −1

ke = 1 − vf km + vf ki Adil 1 − vf I + vAdil

(24)

Where vf stands for volume fraction. The above equations give the thermal conductivity of a grain along carbon nanotube (local coordinates). In order to calculate thermal conductivity of a grain in global coordinates the following transformation tensor is used and its angles were showed in Figure 2.49 ⎤ ⎡ cos 1 cos − sin 1 cos 1 sin sin 1 sin ⎦ T = ⎣ sin 1 cos cos 1 (25) − sin 0 cos ke = T KT t J. Comput. Theor. Nanosci. 11, 1–7, 2014

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RVEs which had different volume fractions are analyzed. The plots of changing thermal conductivity in three directions and Young modulus in x-direction versus the applied strain are shown in Figures 3 and 4, respectively. (a)

Fig. 2.

Relationship between the local and global coordinate systems.

Where ke is thermal conductivity tensor of the grain in global coordinates and T t is transpose of matrix T . After obtaining all thermal conductivity tensors of the grains, they are averaged based on the following equation (the second step of homogenization) N i i lf ke Keff = N i (26) i lf

(b)

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Where Keff is the effective thermal conductivity tensor of the RVE and lfi is length of the ith filler.40

3. RESULTS AND DISCUSSION The CNTs are considered as cylinders with 2 nm diameter and 200 nm length which are uniformly distributed. The dimension of cubic RVE is 1000 × 1000 × 1000 nm3 . Thermal conductivity of CNTs is about 3200 (w/m · K). The Poisson’s ratios of CNTs vary from 0.12 to 0.350 and Young’s moduli is between 100 GPa up to ∼ 1 Tpa.51 When isotropy is assumed, normal values of the Poisson’s ratio and Young’s modulus of 0.22 and 250 GPa are considered for CNT sticks. Also the Poisson’s ratios, Young’s modulus and thermal conductivity of matrix are 0.497, 5 MPa and 0.4 (w/m · K), respectively. The famous statistical method of Monte-Carlo integration based on repeated random sampling inside the RVE is used to calculate the volume fraction of distributed permeable sticks randomly. In this approach, the points are randomly distributed within the considered RVE. If suitably small loading increments are taken, the assumption of elastic region within the matrix will not defect the algorithm even in large strain. In the computer code developed in this paper, one can enter any arbitrary input stress and set the increment whereby the algorithm updates the configuration and saves the data results. Normal unidirectional tensile stress values of 0, 5, 10 and 15 MPa are applied in the X-direction by taking 0.01 increment stresses on sample RVE. 8 generated J. Comput. Theor. Nanosci. 11, 1–7, 2014

(c)

Fig. 3. The diagrams of the effective thermal conductivity in (a) Xdirection, (b) Y -direction, and (c) Z-direction versus the applied strain for different volume fractions.

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Fig. 4. The diagram of the Young modulus in x-direction versus the applied strain for different volume fractions.

It is clear from the graphs of Figure 3(a) that thermal conductivity is increased along the direction of mechanical load and Figures 3(b) and (c) show a decreasing trend in other directions. The curves in Figure 4 show an increase in Young modulus in the direction of mechanical loading. The reason of improvement in thermal conductivity and Young modulus along the mechanical loading direction could be interpreted by alignment of the fillers after mechanical loading. Some experimental results have also been obtained by other authors that could validate results of this work.52–56 Experimental data obtained by Choi et al.52 have indicated that epoxy/MWCNT nanocomposites under a 25 T magnetic field has led to a 10% increase in thermal conductivity which is caused by the alignment of MWCNTs. Also, the work of Hong et al.53 shows that the thermal conductivity of heat transfer nanofluids containing CNTs and Fe2 O3 was increased after alignment by an external magnetic field. These improvements are an evidence which verifies the present theoretical results. Ghose et al.54 generated EVA/MWCNT samples with significant alignment, and observed thermal conductivity parallel to alignment in aligned sample higher than in the arbitrary oriented CNTs. Enhancement of the thermal conductivity (10–20 times higher than the polymer matrix) was also reported by Borca-Tasciuc et al.55 for composites made by infiltrating poly-dimethyl siloxane (PDMS) in aligned MWCNT arrays. Moreover, Huang et al.56 obtained highly thermal conductive silicone elastomer/CNT composites using in situ injection of polymers into CNT arrays. They found that the thermal conductivity of the aligned composites with only a 0.4 vol.% loading is 116% and 105% higher than that of pure elastomer S160 and dispersed composites, respectively, which is a good validation for presented work again.

4. SUMMARY AND CONCLUSION In this paper, the effect of mechanical loading on thermal conductivity of nanocomposites was investigated. 6

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The polymer matrix was reinforced with carbon nanotubes (CNTs) which were considered as prolate spheroid (a1 = a2 < a3 ). CNT sticks were distributed in the matrix randomly by applying the Monte-Carlo simulation. A computer code was developed in such a way that mechanical loading was applied incrementally. By choosing appropriate increment in mechanical loading the assumption of elastic matrix could not defect the algorithm even in the large strain. In addition, the resulting volume concentrations were computed according to the MonteCarlo integration method. Mechanical stress caused a new configuration of CNT’s in the RVE, and consequently this change influenced the effective thermal conductivity and effective stiffness. In order to evaluate effective thermal conductivity, a two-step homogenization technique of Mori–Tanaka was utilized. It is worth mentioning that the deviation of CNTs towards the direction of load is seen because of mechanical stresses; this leads to an anisotropic RVE. The obtained graphs indicate that effective thermal conductivity and Young moduli are increased along with mechanical stresses. Also it is observed that these mechanical properties are reduced in other directions. Experimental studies also show an improvement in effective thermal conductivity and Young moduli when fillers of composites are mostly in the same direction. The application of change in thermal conductivity in three directions with respect to strain leads to designing and fabricating special sensors used for industrial purposes. The paper deals with a specific type of nanomaterials. Other type of nanomaterials and different nano-fabrication methods are provided in Refs. [57–59]. Quantum theory is a capable research area to characterize nanostructures but even the interpretation of modern quantum theory seems still to be a dilemma.60 Acknowledgments: Authors are grateful to Dr. Aminizadeh and Dr. Mosadegh for their useful guidance during this paper.

References 1. P. K. Bose, N. Paitya, S. Bhattacharya, D. De, S. Saha, K. M. Chatterjee, S. Pahari, and K. P. Ghatak, Quantum Matter 1, 89 (2012). 2. M. Narayanan and A. J. Peter, Quantum Matter 1, 53 (2012). 3. N. Paitya, S. Bhattacharya, D. De, and K. P. Ghatak, Quantum Matter 1, 63 (2012). 4. M. Vörös and A. Gali, J. Comput. Theor. Nanosci. 9, 1906 (2012). 5. M. Sargolzaei, S. Keshavarz, and M. Esmaeilzadeh, J. Comput. Theor. Nanosci. 10, 587 (2013). 6. K. Momeni and S. M. Z. Mortazavi, J. Comput. Theor. Nanosci. 9, 1670 (2012). 7. T. Ono, Y. Fujimoto, and S. Tsukamoto, Quantum Matter 1, 4 (2012). 8. B. Tüzün and A. Erkoç, Quantum Matter 1, 136 (2012). 9. F. Mollaamin and M. Monajjemi, J. Comput. Theor. Nanosci. 9, 597 (2012). 10. N. Nouri and S. Ziaei-Rad, J. Comput. Theor. Nanosci. 9, 2144 (2012).

J. Comput. Theor. Nanosci. 11, 1–7, 2014

Mazrouei et al.

Evaluating the Effect of Mechanical Loading on the Effective Thermal Conductivity of CNT Reinforced Polymers 36. F. Du, J. Fischer, and K. Winey, Phys. Rev. B 72 (2005). 37. N. Hu, Y. Karube, M. Arai, T. Watanabe, C. Yan, Y. Li, Y. Liu, and H. Fukunaga, Carbon 48, 680 (2010). 38. N. Hu, Y. Karube, C. Yan, Z. Masuda, and H. Fukunaga, Acta Mater. 56, 2929 (2008). 39. E. Weisstein, in. 40. A. Ghazavizadeh, M. Baniassadi, M. Safdari, A. A. Atai, S. Ahzi, S. A. Patlazhan, J. Gracio, and D. Ruch, J. Comput. Theor. Nanosci. 8, 2087 (2011). 41. D. S. Li, M. Baniassadi, H. Garmestani, S. Ahzi, M. M. R. Taha, and D. Ruch, J. Comput. Theor. Nanosci. 7, 1462 (2010). 42. M. Baniassadi, H. Garmestani, D. S. Li, S. Ahzi, M. Khaleel, and X. Sun, Acta Mater. 59, 30 (2011). 43. H. Garmestani, M. Baniassadi, D. S. Li, M. Fathi, and S. Ahzi, International Journal of Theoretical and Applied Multiscale Mechanics 1, 134 (2009). 44. O. Pierard, C. Friebel, and I. Doghri, Compos. Sci. Technol. 64, 1587 (2004). 45. T. Mori and K. Tanaka, Acta Metallurgica 21, 571 (1973). 46. H. Hiroshi and T. Minoru, International Journal of Engineering Science 24, 1159 (1986). 47. R. Hill, Journal of the Mechanics and Physics of Solids 13, 213 (1965). 48. J. D. Eshelby, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 241, 376 (1957). 49. F. T. Fisher, Mechanical Engineering, Northwestern University (2002). 50. Z.-C. Tu and Z.-C. Ou-Yang, Phys. Rev. B 65 (2002). 51. M.-F. Yu, O. Lourie, M. J. Dyer, K. Moloni, T. F. Kelly, and R. S. Ruoff, Science 637 (2000), www.sciencemag.org. 52. E. S. Choi, J. S. Brooks, D. L. Eaton, M. S. Al-Haik, M. Y. Hussaini, H. Garmestani, D. Li, and K. Dahmen, J. Appl. Phys. 94, 6034 (2003). 53. H. Hong, B. Wright, J. Wensel, S. Jin, X. R. Ye, and W. Roy, Synth. Met. 157, 437 (2007). 54. S. Ghose, K. A. Watson, D. C. Working, J. W. Connell, J. G. Smith, Jr, and Y. P. Sun, Compos. Sci. Technol. 68, 1843 (2008). 55. T. Borca-Tasciuc, M. Mazumder, Y. Son, S. K. Pal, L. S. Schadler, and P. M. Ajayan, J. Nanosci. Nanotech. 7, 1581 (2007). 56. H. Huang, C. H. Liu, Y. Wu, and S. Fan, Adv. Mater. 17, 1652 (2005). 57. A. Herman, Reviews in Theoretical Science 1, 3 (2013). 58. E. L. Pankratov and E. A. Bulaeva, Reviews in Theoretical Science 1, 58 (2013). 59. Q. Zhao, Reviews in Theoretical Science 1, 83 (2013). 60. A. Khrennikov, Reviews in Theoretical Science 1, 34 (2013).

Received: 12 June 2013. Accepted: 9 July 2013.

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11. V. Sgobba, G. M. Aminur Rahman, C. Ehli, and D. M. Guldi, Covalent and non-covalent approaches toward multifunctional carbon nanotube materials, Fullerenes: Principles and Applications, The Royal Society of Chemistry (2007), Chap. 11, pp. 329–379. 12. P. R. Bandaru, J. Nanosci. Nanotech. 7, 1239 (2007). 13. S. Berber, Y. K. Kwon, and D. Tomanek, Phys. Rev. Lett. 84, 4613 (2000). 14. C. Yu, L. Shi, Z. Yao, D. Li, and A. Majumdar, Nano Lett. 5, 1842 (2005). 15. M. M. J. Treacy, T. W. Ebbesen, and J. M. Gibson, Nature 381, 678 (1996). 16. A. Krishnan, E. Dujardin, T. W. Ebbesen, P. N. Yianilos, and M. M. J. Treacy, Phys. Rev. B 58, 14013 (1998). 17. P. M. Ajayan, O. Stephan, C. Colliex, and D. Trauth, Science (New York, N.Y.) 265, 1212 (1994). 18. N. Grossiord, J. Loos, O. Regev, and C. Koning, Chem. Mater. 18, 1089 (2006). 19. M. H. Al-Saleh and U. Sundararaj, Carbon 47, 2 (2009). 20. W. Bauhofer and J. Z. Kovacs, Compos. Sci. Technol. 69, 1486 (2009). 21. B. E. Kilbride, J. N. Coleman, J. Fraysse, P. Fournet, M. Cadek, A. Drury, S. Hutzler, S. Roth, and W. J. Blau, J. Appl. Phys. 92, 4024 (2002). 22. D. Qian, E. C. Dickey, R. Andrews, and T. Rantell, Appl. Phys. Lett. 76, 2868 (2000). 23. L. S. Schadler, S. C. Giannaris, and P. M. Ajayan, Appl. Phys. Lett. 73, 3842 (1998). 24. Z. Han and A. Fina, Prog. Polym. Sci. 36, 914 (2011). 25. P. P. Gerbino and K. C. Arabe, news. thomasnet.com (2003). 26. N. Nishimura and Y. J. Liu, Computational Mechanics 35, 1 (2004). 27. J. Zhang, M. Tanaka, T. Matsumoto, and A. Guzik, JSME 47, 181 (2004). 28. Y. S. Song and J. R. Youn, Carbon 44, 710 (2006). 29. I. V. Singh, M. Tanaka, and M. Endo, Computational Mechanics 39, 719 (2007). 30. H. J. Böhm and S. Nogales, Compos. Sci. Technol. 68, 1181 (2008). 31. Z. H. Zhang, N. Yu, and W. H. Chao, IMECE (2008). 32. J. Stránský, J. Vorel, J. Zeman, and M. Šejnoha, Micromachines 2, 129 (2011). 33. J. Vorel, J. Sýkora, and M. Šejnoha, Two Step Homogenization of Effective Thermal Conductivity for Macroscopically Orthotropic C/C Composites (2008). 34. B. Mortazavi, M. Baniassadi, J. Bardon, and S. Ahzi, Composites Part B: Engineering 45, 1117 (2013). 35. T. C. Theodosiou and D. A. Saravanos, Compos. Sci. Technol. 70, 1312 (2010).