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Thermal conductivity of PTFE composites filled with graphite particles and carbon fibers
Computational Materials Science 102 (2015) 45–50
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Thermal conductivity of PTFE composites filled with graphite particles and carbon fibers Jin Zunlong, Chen Xiaotang, Wang Yongqing ⇑, Wang Dingbiao School of Chemical Engineering and Energy, Zhengzhou University, Zhengzhou 450001, PR China
a r t i c l e
i n f o
Article history: Received 19 August 2014 Received in revised form 6 February 2015 Accepted 11 February 2015
Keywords: PTFE Composites Thermal properties Addition polymerization Molding
a b s t r a c t A finite element numerical model is proposed in this paper to predict the effective thermal conductivity of polytetrafluoroethylene (PTFE) composites based on the Fourier’s law of heat conduction. The reliability of the numerical model is verified using the comprehensive experimental results and theoretical models presented by predecessors. A systematic study is conducted through numerical simulation to examine the impact of graphite particle size, volume fraction, and distribution style on the thermal conductivity of PTFE composites. A prediction model of the thermal conductivity of carbon fibers reinforced PTFE composites is introduced. The results of the prediction model agree well with experimental data. A trade off between the volume fractions of graphite particles and carbon fibers is conducted. And the optimum volume fraction matching is 17.76 and 10% for graphite particles and carbon fibers, respectively. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction In petrochemical and pharmaceutical production, some pieces of heat transfer equipment frequently work under corrosive environments. Many kinds of corrosion-resistant heat exchangers have been introduced into those industrial fields, among which heat exchangers made of PTFE are the most common. PTFE has excellent chemical inertness [1]. However, pure PTFE has a relatively low thermal conductivity, which limits its practical use [2,3]. In order to enhance the performance of PTFE heat exchangers, the thermal conductivity of PTFE should be improved. One of the most significant methods is filling fillers with high thermal conductivity. As graphite particle has high thermal conductivity and great corrosion resistance, it is the best candidate filler embedded in PTFE matrix. Unfortunately, graphite particles’ embedment will weaken the mechanical performance of PTFE composites [4]. This difficulty can be overcome by adding carbon fibers into PTFE matrix at the same time, because carbon fiber has high strength, stable chemical property, and relatively high thermal conductivity [5]. Prediction the effective thermal conductivity of PTFE composites filled with graphite particles and carbon fibers is a complex problem. Many researchers have proposed theoretical and empirical methods to solve this problem.
⇑ Corresponding author. Tel./fax: +86 371 6773 9363. E-mail address: [email protected] (Y. Wang). http://dx.doi.org/10.1016/j.commatsci.2015.02.019 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
Maxwell [6,7] studied the effective thermal conductivity of PTFE composites earliest in 1873. From then on, the research on thermal conductivity of composites has been attractive. Choy and Young [8] discussed in terms of the Maxwell model generalized to the case where the inclusions are thermally anisotropic. The predicted results agreed well with the large anisotropy observed in oriented polymers. Agari and Uno [9] proposed a thermal conduction model for filled polymer with particles and predicted values were compared with experimental data. Their model was based on a generalization of parallel and series conduction models of composites. Zhang et al. [10] proposed a model to predict the thermal conductivity of filled polymer composites based on percolation theory. This model can be shown as Eq. (1) as follows,
k ¼ k2 ðkc =k2 Þð1V=1V c Þ
n
ð1Þ
where k is the thermal conductivity of composites W/m K, k2 is the thermal conductivity of filler W/m K, kc is the thermal conductivity of composites when V = Vc W/m K, V is the volume fraction of fillers, Vc is a percolation threshold, and n is a percolation exponent determined by the filler size, filler shape, and filler distribution in the matrix. Shelestova et al. [11] pointed out that an increase in the fiber length leaded to decrease in the density, resistivity, and compression strength of the composites. Kumlutas et al. [12] indicated that particle content greater than 10% exponentially improved thermal conductivity. Zhang and Liang [13] presented that the effective thermal conductivity of mixed solid media was influenced by the
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Z. Jin et al. / Computational Materials Science 102 (2015) 45–50
weight (or volume) fraction of the doped material rather than by the size or dimension of each discrete phase. With the rapid development of computer software and hardware, the numerical simulation method has been a powerful tool in almost all research fields for saving time and costs [14]. The numerical simulation can offer us details of the system behavior and help us in designing new materials with the least expenses. Several kinds of numerical simulation methods have been proposed to predict thermal and mechanical properties of polymer composites, such as finite element method (FEM) [4,7,15,16], artificial neural network (ANN) [14,17,18], adaptive neuro-fuzzy inference system (ANFIS) [14,19], and lattice Boltzmann method (LBM) [20]. The methods mentioned above have been widely adopted in many research fields, and have achieved relative accuracies. However, the models mentioned above seldom estimated the thermal conductivity of PTFE composites filled with graphite particles and carbon fibers at the same time. The present work aims to deal with this problem using finite element method and theoretical analysis. First of all, the authors conduct a detailed study on the thermal conductivity of PTFE composites filled with graphite particles using finite element method with the aid of software named ANSYSÒ. The numerical calculation results are compared with experimental data and some existing models for confirming the validity of the proposed numerical model. The effects of filler size, volume fraction, and distribution upon the thermal conductivity are explored. In the next step, the authors develop an expression for predicting the thermal conductivity of PTFE composites filled with carbon fibers and the results are compared with experimental data in previous literature. Finally, in order to obtain the maximum thermal conductivity based on satisfaction with mechanical strength, the optimum volume fractions of graphite particles and carbon fibers are given in this paper. 2. Numerical model 2.1. Establishing the numerical model Based on the Fourier’s law of heat conduction and the heat transfer process of flat plat with steady heat conduction (heat flux unchanged), the equation of the thermal conductivity of planar wall can be obtained,
k¼
qd Dt
Fig. 1. Finite element model.
used to generate meshes. Meshes are quadrilateral. In the process of mesh generating, mesh density is increased automatically in the areas of graphite particles. The mesh schematic is shown in Fig. 2. The thermal conductivity of PTFE (k2) is 0.21 W/m K, and that of the filling phase graphite (k1) is 197 W/m K. The model width (d) is 0.004 m. This model shows the finite element model of PTFE composites filled with the graphite particle size of 150 mesh. And the volume fraction of graphite particles is 10%. The boundary conditions include the heat flux on the left at q = 200 W/m2, convective air on the right at T = 298 K, and the air convection coefficient at 0.44 W/(m2 K). The numerical result of the finite element simulation is shown in Fig. 3. In accordance with the numerical simulation result diagram and Eq. (2), the thermal conductivity of this model is 0.25357 W/ m K. In order to make the simulation more accurate, a nonuniform meshing method was used. And the zones near interfaces of PTFE and graphite need finer mesh. The refined mesh is got by comparing the simulation results of models with different grid density. When the number of girds were refined to 2.53 times of that used in Fig. 2, the effective thermal conductivity was changed to 0.25635 W/m K from 0.25357 W/m K, and the relative change values is less than 1.2%. It can be concluded that the grid independent solution is achieved.
ð2Þ
where k is the thermal conductivity W/m K, q is the heat flux W/m2, d is the thickness of the planar wall m, and Dt is the temperature difference at both sides of the planar wall K. The numerical model is viewed as a part of the planar wall to establish the 2D finite element model for predicting the thermal conductivity, which is shown as Fig. 1. In this model, the middle part is the PTFE composites filled with graphite particles and the upper and bottom boundaries are set as adiabatic boundary. The left and right boundaries are set as constant temperature boundary. To ensure heat balance in the region bounded by constant temperature, copper is used with the constant temperature boundary, which has a higher thermal conductivity than graphite and PTFE. Heat flow is applied on the left boundary as the hot terminal (T1), and cross-ventilation is applied on the right boundary as the cold terminal (T2). According to the Delesse law in stereoscopy [21], the filling phase is randomly distributed on the interface when the percentage of the two-phase area is equivalent to the corresponding volume fraction in the three-dimensional space. Thus, the two-phase area fraction can be expressed by the volume fraction. Adaptive techniques in the finite element method [22] are
Fig. 2. Mesh schematic (quadrilateral mesh).
47
Thermal Conductivity/ W/(m K)-1
Z. Jin et al. / Computational Materials Science 102 (2015) 45–50
0.65 0.60
results of M-E model results of N-Lmodel results of experiment results of numerical simulation
0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0
5
10
15
20
25
30
35
Volumue Fraction (%) Fig. 4. Results of numerical simulation compared with those of experiment and other prediction models. Fig. 3. Numerical simulation result diagram.
2.2. Model self-validating The numerical model is verified through self-validating. Thermal conductivity is an intrinsic property of the PTFE composites. Changing the boundary conditions during simulation does not significantly affect the simulation results of the thermal conductivity. The numerical simulation results with changed heat flux and convection boundary conditions are shown in Table 1. The relative error is less than 0.1%, which indicates that the numerical model itself is correct and reasonable. 2.3. Validity analysis of the proposed model The manufacturer produced the samples according to the experimental requirements of the authors. The test samples’ dimension is 180 90 10 mm, and the volume fractions of the graphite particles are 5%, 10%, 15%, 20%, and 30%, respectively. The plane heat source quasi-steady state method is applied to test the thermal conductivity of the five groups of test samples, which is conducted by the Heat Flow Meter (HFM436) of NETZSCHÒ Company. The microstructure of the PTFE composites is investigated by scanning electron microscope, which is conducted by the Cold Field Emission Scanning Microscope (JSM-7500F) of JEOLÒ Company. At the same time, the numerical results of the proposed model are compared with those of some previous prediction models. These prediction models are shown as below,Maxwell–Euckenm model (M–E model [9])
k ¼ k1
2k1 þ k2 þ 2Vðk2 k1 Þ 2k1 þ k2 2Vðk2 k1 Þ
ð3Þ
where k is the thermal conductivity of the composites W/m K, k1 is the thermal conductivity of the matrix W/m K, k2 is the thermal conductivity of the filled phase W/m K, and V is the filler volume fraction. Nielsen–Lewis model (N–L model [23])
k¼
1 þ ABV 1 BWV
ð4Þ
where k is the thermal conductivity of the composites W/m K, and V is the filler volume fraction. A and Vm are constants related to the size and shape of the particles (A = 1.5 and Vm = 0.637 for spherical, irregular stacked filled particles; A = 3 and Vm = 0.64 for nonspherical,
irregular
stacked-filled
particles);
W¼1þV
2
ð1V m Þ ; V 2m
k2 k 1
B ¼ k21 ; k1 is the thermal conductivity of the matrix W/m K, and k1
þA
k2 is the thermal conductivity of the filler W/m K. Results of the finite element numerical simulation are compared with those of experimental data and other prediction models. The comparison results are shown in Fig. 4. Fig. 4 shows that the error between the numerical simulation results, the experimental data, the M–E model results, and the N–L model results is acceptable, especially in low volume ratio. The comparison results indicate that the proposed finite element model is valid for predicting the thermal conductivity of PTFE composites. The numeration process of the proposed finite element model is conducted by a series of command stream coded by ANSYSÒ Parametric Design Language (APDL). Figs. 5 and 6 show that graphite particles are evenly distributed in the PTFE matrix, indicating that the test samples are reasonable. As the volume fraction of graphite particles is relative low (10%), the graphite particles show themselves a series of sea-island structures in the matrix, which verifies the hypothesis of the graphite particles state. With the increase of the volume fraction of graphite particles, they can partially connect and form continuous thermal conducting chain to improve the thermal conductivity of the PTFE composites. 3. Results and discussion 3.1. Effect of particle size and volume fraction Fig. 7 shows the thermal conductivity simulation of the graphite-filled PTFE composites with a different particle size at a given volume fraction, wherein its thermal conductivity does not simply increase or decrease monotonously, but presents extreme value points on the result curve. This result shows that a simple
Table 1 Precision analysis with changed boundary conditions. Heat-flow density (W/m2)
Convection boundary temperature (K)
Convection coefficients conv
Boundary temperature T1 (K)
Boundary temperature T2 (K)
Thermal conductivity (W/m K)
Relative deviation (%)
150 200 200
298 298 408
0.44 0.44 0.44
3.366 4.487 4.126
0.999 1.332 0.971
0.25349 0.25357 0.25357
0.02 0.012 0.012
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Z. Jin et al. / Computational Materials Science 102 (2015) 45–50
Graphite Particle
Thermal conductivity/W (m k)-1
0.58
40% Volume Fraction
0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42
50
100
150
200
250
300
Mesh Fig. 7. Relationship between graphite size and thermal conductivity. Fig. 5. Cross section of PTFE composites with SEM diagram.
3.2. Effect of particle distribution style Fig. 9 shows that when the graphite-filled volume fraction is below 40% in the PTFE matrix, a random or even distribution of graphite imposes a similar impact on the thermal conductivity of PTFE composites (the relative error is 0.3%). This impact is almost
0.60
Thermal Conductivity/ W/ (m k)-1
refinement of graphite particle size cannot effectively improve the thermal conductivity of the modification composites; and 250-mesh graphite has better modification effect on the thermal conductivity of PTFE than other particles. Therefore, 250-mesh graphite is selected as the additive in the refinement interval simulation study. Fig. 8 shows the increased volume fraction in graphite-filling, which is consistent with the increase of the effective thermal conductivity. At a relatively low graphite-filling volume fraction, the effect of different graphite particle size on thermal conductivity of PTFE composites is insignificant. With the increase of volume fraction, the effect of different graphite particle size varies and the rising speed of the thermal conductivity also increases. This finding may be attributed to the similar sea-island structure [24] formed by the graphite distribution in the matrix at a low volume fraction of graphite. In that case the graphite distribution in the matrix lacks connections and contacts. So, different graphite particle size presents few differences in low volume fraction. Moreover, the effective thermal conductivity of the composites is low. Graphite particles in some areas may be linked to form a continuous thermal conducting network when the volume fraction of graphite increases, which increases the thermal conductivity of the composites. The difficulty of the link formation varies due to the different particle size.
150mesh 200mesh 250mesh 300mesh
0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0
5% 10% 15% 20% 25% 30% 35% 40%
Volume Fraction Fig. 8. Relationship between graphite volume fraction and thermal conductivity.
negligible, indicating that the random distribution of graphite, which is close to the actual situation, can be used to replace the even distribution. Considering that the graphite volume fraction of 30% may lead to its vulnerable separation from PTFE composites and loading incapability, the optimal graphite volume fraction should be between 15% and 25% to meet the requirements of mechanical strength [25,26]. To explore the impact of graphite volume fraction on thermal conductivity of the PTFE composites, a detailed study is conducted with 250-mesh particle size and with filling volume fractions from 14% to 26% as well as a 2% interval. The simulation
Fig. 6. The EDAX SEM chemical elements analysis (JSM-7500F of JEOLÒ Company).
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Z. Jin et al. / Computational Materials Science 102 (2015) 45–50
7KHUPDO&RQGXFWLYLW\: P.
0.60
Hpt ¼ S H
even distribution random distribution
0.55
ð7Þ
pt
where H is the additive temperature gradient. The fiber aspect ratio in matrix S is defined as n = l/d; Z is the fiber axis and X and Y stand for the other two directions of the fiber. Thus,
0.50 0.45 0.40
g¼
0.35
h
n 3=2
ðn2 1Þ
1=2
nðn2 1Þ
i 1 cosh n
ð8Þ
SZ ¼ l g
0.30 0.25
SX ¼ SY ¼
0.20 0
ð9Þ g 2
ð10Þ
where S is the vector matrix, n is the fiber aspect ratio, l is the length of fiber, d is the diameter, and g is a intermediate variable. So, the average temperature gradient in the composites can be expressed as,
5% 10% 15% 20% 25% 30% 35% 40%
Volume Fraction Fig. 9. Effect of even and random distribution on thermal conductivity.
HT ¼ H þ V H Given the intermediate variable matrix,
0.38
Thermal Conductivity/ W/ (m K)-1
ð11Þ
PHVK
Imn ¼
0.36
1 ðm ¼ nÞ 0
ð12Þ
ðm–nÞ
k1 k2
0.34
A¼I
0.32
1 k1 T ¼ Iþ SS k2
ð14Þ
0.30
B¼AT
ð15Þ
where V is the volume fraction of filling fiber; m, n is the row and column number in the matrix; k1 ; k2 is the thermal conductivity of filling fiber and the PTFE matrix phase, respectively.The equivalent temperature gradient H⁄ can be expressed as,
0.28 14%
16%
18%
20%
22%
24%
ð13Þ
26%
Volume Fraction
H ¼ A TðH þ HÞ ¼ B ðH þ HÞ
ð16Þ
Hpt H ¼ ðS IÞ A T ðH þ HÞ
ð17Þ
Fig. 10. Simulation results of thermal conductivity in refinement intervals.
results are shown in Fig. 10. The obtained changing tendency of the thermal conductivity with increase of the volume fraction is generally consistent with the tendency shown in Fig. 9.
Let us to introduce the intervariable matrix,
E ¼ A T ðS IÞ
ð18Þ
So, Eq. (17) can be expressed as, 4. Carbon fibers in enhancing thermal conductivity
pt
H H ¼ E ðH þ HÞ
4.1. Prediction model of thermal conductivity
Then, Eqs. (11) and (19) can be combined as,
The carbon fibers in the composite are assumed to be randomly distributed and the composite is assumed to be isotropic. According to the Fourier’s law of heat conduction, the heat transfer is expressed as,
q ¼ kH
ð19Þ
ð5Þ
h i HT ¼ I þ VBðI þ VEÞ1 H
ð20Þ
Finally, combining Eqs. (5) and (20) will have to the expression of thermal conductivity of PTFE composites, as following,
h i1 k ¼ k2 I þ VBðI þ VEÞ1
ð21Þ
where q is heat flux, H is temperature gradient, and k is thermal conductivity. When the carbon fibers are filled into the composite, additive heat transfer will occur. The total heat transfer can be expressed as,
Considering the filling fibers’ random distribution, the PTFE composites is isotropic. The prediction expression of the thermal conductivity of PTFE composites is thus derived in this paper,
q þ qD ¼ km ðH þ HÞ
km ¼
ð6Þ
where qD is additive heat flux, km is the thermal conductivity of composites, and H is the average additive temperature gradient. Considering the temperature gradient and the total heat flux, we recur to the equivalent temperature gradient H⁄ and the vector matrix S related with the fiber aspect ratio, which were presented by Eshelby [27,28].
3 þ VðSZ 1Þ ðC þ 2DÞ 3 þ VðC þ 2DÞ
ð22Þ
where km is the thermal conductivity of the PTFE composites k þk
k þk
fZ
fX
W/m K, V is the fiber volume fraction, C ¼ SZ ðk fZk22Þk2 , D ¼ SX ðk fXk22Þk2 , k2 is the thermal conductivity of PTFE matrix phase, kfZ is the carbon fiber axial thermal conductivity W/m K, and kfX is the carbon fiber radial thermal conductivity W/m K.
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Z. Jin et al. / Computational Materials Science 102 (2015) 45–50
Thermal Conductivity/ W/ (m k)
2.0
relationship between mechanical strength and volume fractions of PTFE composites is based on the results of Ref. [30]. The optimization results indicate that the optimum volume fraction matching is 17.76 and 10% for volume fraction of graphite particles and carbon fibers, respectively. The thermal and mechanical performance of the PTFE composites under optimum volume fraction matching is shown in Table 2.
Volume Fraction of *UDSKLWH Volume Fraction of *UDSKLWH Volume Fraction of*UDSKLWH Volume Fraction of*UDSKLWH Volume Fraction of*UDSKLWH Volume Fraction of*UDSKLWH Volume Fraction of*UDSKLWH
1.8 1.6 1.4 1.2 1.0 0.8
5. Conclusions
0.6 0.4 0.2
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
Volume Fraction of Carbon Fibers Fig. 11. Thermal conductivity of the PTFE composites.
Table 2 Comparison with PTFE and PTFE composites.
PTFE PTFE composites
Thermal conductivity (W/(m K))
Tensile strength (MPa)
0.21 1.353
23.35 20.21
When the carbon fiber volume fraction is low (usually less than 10%), the predicted results by Eq. (22) are consistent with the experimental results of Chen and Wang [29], with a deviation of about 8%. Based on the calculation results in numerical model, these numerical results are then feed into Eq. (22). So the thermal conductivity of the PTFE composites filled with graphite and carbon fibers at the same time is obtained. And the results are shown in Fig. 11. Results show that the thermal conductivity of the PTFE composites improved with the increase of volume fraction of the carbon fibers, which can be attributed to its high thermal conductivity. When carbon fibers are added in the composites, the fiber structure can be linked with the graphite in the matrix to form a network of thermal conductivity. More carbon fibers mean more links of thermal conductivity networks in the matrix, which improve thermal conductivity of the PTFE composites. 4.2. Optimum volume fraction matching As it is known that the addition of graphite particles will weaken the mechanical strength of PTFE composites. This difficulty can be overcome by adding carbon fibers into PTFE matrix at the same time. There is a trade-off between the volume fractions of graphite particles and carbon fibers. Based on the basis of meeting the requirements of mechanical strength, it is expected that the PTFE composites have higher thermal performance. In order to optimize the volume ratios of graphite particles and carbon fibers, we write a program in MATLABÓ to conduct the optimization analysis. The objective function seeks to maximize the thermal conductivity of the PTFE composites. And the objective function is inclined to suffer the constraints of the relationship of volume fractions and thermal conductivity, the relationship of volume fractions and mechanical strength, and the mechanical strength greater than 20 MPa. According to the results of numerical simulation, these numerical results are then feed into Eq. (22). So the relationship of thermal conductivity of the PTFE composites filled with graphite and carbon fibers at the same time is obtained. And the
(1) The feasibility of the finite element simulation method presented in this paper is verified through experiments and the commonly used prediction models introduced by predecessors. (2) A systematic study is conducted on the effect of graphite particles size, graphite filled volume fraction, and its distribution on thermal conductivity of the PTFE composites. And 250-mesh graphite has better modification effect than other size particles. (3) A prediction model of thermal conductivity of PTFE composites filled with carbon fibers is presented. (4) The optimum volume fraction matching is introduced. The results indicate that the optimum volume fraction matching is 17.76 and 10% for volume fraction of graphite particles and carbon fibers, respectively.
Acknowledgment The authors are grateful to the China Postdoctoral Science Foundation funded project (Grant Nos. 201104400 and 2014M552011) for financial support. References [1] P. Gonon, A. Sylvestre, J. Appl. Phys. 92 (2002) 4581–4589. [2] S. Thomas, V.N. Deepu, P. Mohanan, M.T. Sehastian, J. Am. Ceram. Soc. 91 (2008) 1971–1975. [3] F. Xiang, H. Wang, X. Yao, J. Eur. Ceram. Soc. 26 (2006) 1999–2002. [4] W.Z. Cai, S.T. Tu, G.L. Tao, J. Thermoplast. Compos. 18 (2005) 241–253. [5] Y. Agari, A. Ueda, S. Nagai, J. Appl. Polym. Sci. 43 (1991) 1117–1142. [6] J. Maxwell, Electricity and Magnetism, Clarendon, Oxford, 1873. [7] Y. Yin, S.T. Tu, J. Reinf. Plast. Compos. 21 (2002) 1619–1627. [8] C.L. Choy, K. Young, Polymer 18 (1977) 769–776. [9] Y. Agari, T. Uno, J. Appl. Polym. Sci. 32 (1986) 5705–5712. [10] G.Q. Zhang, Y.P. Xia, H. Wang, Y. Tao, G.L. Tao, S.T. Tu, H.P. Wu, J. Compos. Mater. 44 (2010) 963–970. [11] V.A. Shelestova, V.V. Serafimovich, P.N. Grakovich, Mech. Compos. Mater. 38 (2002) 125–131. [12] D. Kumlutas, I.H. Tavman, M.T. Coban, Compos. Sci. Technol. 63 (2003) 113– 117. [13] Y.P. Zhang, X.G. Liang, Mater. Des. 12 (1995) 91–95. [14] H. Fazilat, M. Ghatarband, S. Mazinani, Comput. Mater. Sci. 58 (2012) 31–37. [15] Z.G. Li, H. Chen, L.F. Cai, Z.H. Zhu, Y.S. Wang, Y. Zhang, J. Reinf. Plast. Compos. 31 (2013) 1586–1598. [16] X.T. Li, X.Y. Fan, Y.D. Zhu, J. Li, J.M. Adams, S. Shen, H.Z. Li, Comput. Mater. Sci. 63 (2012) 207–213. [17] Z. Zhang, P. Klein, K. Friedrich, Compos. Sci. Technol. 62 (2002) 1001–1009. [18] Z. Zhang, K. Friedrich, Compos. Sci. Technol. 63 (2003) 2029–2044. [19] A.H. Mesbahi, D. Semnani, K.S. Nouri, Composites Part B. 43 (2012) 549–558. [20] F. Zhou, G.X. Cheng, Comput. Mater. Sci. 92 (2014) 157–165. [21] F. Dell’lsola, M. Guarascil, K. Hutter, Arch. Appl. Mech. 70 (2000) 323–337. [22] Y. Kondratyuk, R. Stevenson, Soc. Ind. Appl. Math. 46 (2008) 747–775. [23] T.B. Lewis, L.E. Nielsen, J. Appl. Polym. Sci. 14 (1970) 1449–1471. [24] H.K. He, L. Chen, Y. Zhang, Fiber Polym. 15 (2014) 1941–1949. [25] H.X. Xu, X.H. Yuan, X.N. Cheng, Lubr. Eng. 32 (2007) 88–118. [26] B.B. Difallah, M. Kharrat, M. Dammak, G. Monteil, Mater. Des. 34 (2012) 782– 787. [27] J.D. Eshelby, J. Elasticity 5 (1975) 321–335. [28] J.D. Eshelby, Noordhoff Int. Publ. (1974) 69–84. [29] C.H. Chen, Y.C. Wang, Mech. Mater. 23 (1996) 217–228. [30] P.N. Grakovich, V.A. Shelestova, Sci. China Math. 44 (2001) 292–296.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Thermal conductivity of PTFE composites filled with graphite particles and carbon fibers Jin Zunlong, Chen Xiaotang, Wang Yongqing ⇑, Wang Dingbiao School of Chemical Engineering and Energy, Zhengzhou University, Zhengzhou 450001, PR China
a r t i c l e
i n f o
Article history: Received 19 August 2014 Received in revised form 6 February 2015 Accepted 11 February 2015
Keywords: PTFE Composites Thermal properties Addition polymerization Molding
a b s t r a c t A finite element numerical model is proposed in this paper to predict the effective thermal conductivity of polytetrafluoroethylene (PTFE) composites based on the Fourier’s law of heat conduction. The reliability of the numerical model is verified using the comprehensive experimental results and theoretical models presented by predecessors. A systematic study is conducted through numerical simulation to examine the impact of graphite particle size, volume fraction, and distribution style on the thermal conductivity of PTFE composites. A prediction model of the thermal conductivity of carbon fibers reinforced PTFE composites is introduced. The results of the prediction model agree well with experimental data. A trade off between the volume fractions of graphite particles and carbon fibers is conducted. And the optimum volume fraction matching is 17.76 and 10% for graphite particles and carbon fibers, respectively. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction In petrochemical and pharmaceutical production, some pieces of heat transfer equipment frequently work under corrosive environments. Many kinds of corrosion-resistant heat exchangers have been introduced into those industrial fields, among which heat exchangers made of PTFE are the most common. PTFE has excellent chemical inertness [1]. However, pure PTFE has a relatively low thermal conductivity, which limits its practical use [2,3]. In order to enhance the performance of PTFE heat exchangers, the thermal conductivity of PTFE should be improved. One of the most significant methods is filling fillers with high thermal conductivity. As graphite particle has high thermal conductivity and great corrosion resistance, it is the best candidate filler embedded in PTFE matrix. Unfortunately, graphite particles’ embedment will weaken the mechanical performance of PTFE composites [4]. This difficulty can be overcome by adding carbon fibers into PTFE matrix at the same time, because carbon fiber has high strength, stable chemical property, and relatively high thermal conductivity [5]. Prediction the effective thermal conductivity of PTFE composites filled with graphite particles and carbon fibers is a complex problem. Many researchers have proposed theoretical and empirical methods to solve this problem.
⇑ Corresponding author. Tel./fax: +86 371 6773 9363. E-mail address: [email protected] (Y. Wang). http://dx.doi.org/10.1016/j.commatsci.2015.02.019 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
Maxwell [6,7] studied the effective thermal conductivity of PTFE composites earliest in 1873. From then on, the research on thermal conductivity of composites has been attractive. Choy and Young [8] discussed in terms of the Maxwell model generalized to the case where the inclusions are thermally anisotropic. The predicted results agreed well with the large anisotropy observed in oriented polymers. Agari and Uno [9] proposed a thermal conduction model for filled polymer with particles and predicted values were compared with experimental data. Their model was based on a generalization of parallel and series conduction models of composites. Zhang et al. [10] proposed a model to predict the thermal conductivity of filled polymer composites based on percolation theory. This model can be shown as Eq. (1) as follows,
k ¼ k2 ðkc =k2 Þð1V=1V c Þ
n
ð1Þ
where k is the thermal conductivity of composites W/m K, k2 is the thermal conductivity of filler W/m K, kc is the thermal conductivity of composites when V = Vc W/m K, V is the volume fraction of fillers, Vc is a percolation threshold, and n is a percolation exponent determined by the filler size, filler shape, and filler distribution in the matrix. Shelestova et al. [11] pointed out that an increase in the fiber length leaded to decrease in the density, resistivity, and compression strength of the composites. Kumlutas et al. [12] indicated that particle content greater than 10% exponentially improved thermal conductivity. Zhang and Liang [13] presented that the effective thermal conductivity of mixed solid media was influenced by the
46
Z. Jin et al. / Computational Materials Science 102 (2015) 45–50
weight (or volume) fraction of the doped material rather than by the size or dimension of each discrete phase. With the rapid development of computer software and hardware, the numerical simulation method has been a powerful tool in almost all research fields for saving time and costs [14]. The numerical simulation can offer us details of the system behavior and help us in designing new materials with the least expenses. Several kinds of numerical simulation methods have been proposed to predict thermal and mechanical properties of polymer composites, such as finite element method (FEM) [4,7,15,16], artificial neural network (ANN) [14,17,18], adaptive neuro-fuzzy inference system (ANFIS) [14,19], and lattice Boltzmann method (LBM) [20]. The methods mentioned above have been widely adopted in many research fields, and have achieved relative accuracies. However, the models mentioned above seldom estimated the thermal conductivity of PTFE composites filled with graphite particles and carbon fibers at the same time. The present work aims to deal with this problem using finite element method and theoretical analysis. First of all, the authors conduct a detailed study on the thermal conductivity of PTFE composites filled with graphite particles using finite element method with the aid of software named ANSYSÒ. The numerical calculation results are compared with experimental data and some existing models for confirming the validity of the proposed numerical model. The effects of filler size, volume fraction, and distribution upon the thermal conductivity are explored. In the next step, the authors develop an expression for predicting the thermal conductivity of PTFE composites filled with carbon fibers and the results are compared with experimental data in previous literature. Finally, in order to obtain the maximum thermal conductivity based on satisfaction with mechanical strength, the optimum volume fractions of graphite particles and carbon fibers are given in this paper. 2. Numerical model 2.1. Establishing the numerical model Based on the Fourier’s law of heat conduction and the heat transfer process of flat plat with steady heat conduction (heat flux unchanged), the equation of the thermal conductivity of planar wall can be obtained,
k¼
qd Dt
Fig. 1. Finite element model.
used to generate meshes. Meshes are quadrilateral. In the process of mesh generating, mesh density is increased automatically in the areas of graphite particles. The mesh schematic is shown in Fig. 2. The thermal conductivity of PTFE (k2) is 0.21 W/m K, and that of the filling phase graphite (k1) is 197 W/m K. The model width (d) is 0.004 m. This model shows the finite element model of PTFE composites filled with the graphite particle size of 150 mesh. And the volume fraction of graphite particles is 10%. The boundary conditions include the heat flux on the left at q = 200 W/m2, convective air on the right at T = 298 K, and the air convection coefficient at 0.44 W/(m2 K). The numerical result of the finite element simulation is shown in Fig. 3. In accordance with the numerical simulation result diagram and Eq. (2), the thermal conductivity of this model is 0.25357 W/ m K. In order to make the simulation more accurate, a nonuniform meshing method was used. And the zones near interfaces of PTFE and graphite need finer mesh. The refined mesh is got by comparing the simulation results of models with different grid density. When the number of girds were refined to 2.53 times of that used in Fig. 2, the effective thermal conductivity was changed to 0.25635 W/m K from 0.25357 W/m K, and the relative change values is less than 1.2%. It can be concluded that the grid independent solution is achieved.
ð2Þ
where k is the thermal conductivity W/m K, q is the heat flux W/m2, d is the thickness of the planar wall m, and Dt is the temperature difference at both sides of the planar wall K. The numerical model is viewed as a part of the planar wall to establish the 2D finite element model for predicting the thermal conductivity, which is shown as Fig. 1. In this model, the middle part is the PTFE composites filled with graphite particles and the upper and bottom boundaries are set as adiabatic boundary. The left and right boundaries are set as constant temperature boundary. To ensure heat balance in the region bounded by constant temperature, copper is used with the constant temperature boundary, which has a higher thermal conductivity than graphite and PTFE. Heat flow is applied on the left boundary as the hot terminal (T1), and cross-ventilation is applied on the right boundary as the cold terminal (T2). According to the Delesse law in stereoscopy [21], the filling phase is randomly distributed on the interface when the percentage of the two-phase area is equivalent to the corresponding volume fraction in the three-dimensional space. Thus, the two-phase area fraction can be expressed by the volume fraction. Adaptive techniques in the finite element method [22] are
Fig. 2. Mesh schematic (quadrilateral mesh).
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Thermal Conductivity/ W/(m K)-1
Z. Jin et al. / Computational Materials Science 102 (2015) 45–50
0.65 0.60
results of M-E model results of N-Lmodel results of experiment results of numerical simulation
0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0
5
10
15
20
25
30
35
Volumue Fraction (%) Fig. 4. Results of numerical simulation compared with those of experiment and other prediction models. Fig. 3. Numerical simulation result diagram.
2.2. Model self-validating The numerical model is verified through self-validating. Thermal conductivity is an intrinsic property of the PTFE composites. Changing the boundary conditions during simulation does not significantly affect the simulation results of the thermal conductivity. The numerical simulation results with changed heat flux and convection boundary conditions are shown in Table 1. The relative error is less than 0.1%, which indicates that the numerical model itself is correct and reasonable. 2.3. Validity analysis of the proposed model The manufacturer produced the samples according to the experimental requirements of the authors. The test samples’ dimension is 180 90 10 mm, and the volume fractions of the graphite particles are 5%, 10%, 15%, 20%, and 30%, respectively. The plane heat source quasi-steady state method is applied to test the thermal conductivity of the five groups of test samples, which is conducted by the Heat Flow Meter (HFM436) of NETZSCHÒ Company. The microstructure of the PTFE composites is investigated by scanning electron microscope, which is conducted by the Cold Field Emission Scanning Microscope (JSM-7500F) of JEOLÒ Company. At the same time, the numerical results of the proposed model are compared with those of some previous prediction models. These prediction models are shown as below,Maxwell–Euckenm model (M–E model [9])
k ¼ k1
2k1 þ k2 þ 2Vðk2 k1 Þ 2k1 þ k2 2Vðk2 k1 Þ
ð3Þ
where k is the thermal conductivity of the composites W/m K, k1 is the thermal conductivity of the matrix W/m K, k2 is the thermal conductivity of the filled phase W/m K, and V is the filler volume fraction. Nielsen–Lewis model (N–L model [23])
k¼
1 þ ABV 1 BWV
ð4Þ
where k is the thermal conductivity of the composites W/m K, and V is the filler volume fraction. A and Vm are constants related to the size and shape of the particles (A = 1.5 and Vm = 0.637 for spherical, irregular stacked filled particles; A = 3 and Vm = 0.64 for nonspherical,
irregular
stacked-filled
particles);
W¼1þV
2
ð1V m Þ ; V 2m
k2 k 1
B ¼ k21 ; k1 is the thermal conductivity of the matrix W/m K, and k1
þA
k2 is the thermal conductivity of the filler W/m K. Results of the finite element numerical simulation are compared with those of experimental data and other prediction models. The comparison results are shown in Fig. 4. Fig. 4 shows that the error between the numerical simulation results, the experimental data, the M–E model results, and the N–L model results is acceptable, especially in low volume ratio. The comparison results indicate that the proposed finite element model is valid for predicting the thermal conductivity of PTFE composites. The numeration process of the proposed finite element model is conducted by a series of command stream coded by ANSYSÒ Parametric Design Language (APDL). Figs. 5 and 6 show that graphite particles are evenly distributed in the PTFE matrix, indicating that the test samples are reasonable. As the volume fraction of graphite particles is relative low (10%), the graphite particles show themselves a series of sea-island structures in the matrix, which verifies the hypothesis of the graphite particles state. With the increase of the volume fraction of graphite particles, they can partially connect and form continuous thermal conducting chain to improve the thermal conductivity of the PTFE composites. 3. Results and discussion 3.1. Effect of particle size and volume fraction Fig. 7 shows the thermal conductivity simulation of the graphite-filled PTFE composites with a different particle size at a given volume fraction, wherein its thermal conductivity does not simply increase or decrease monotonously, but presents extreme value points on the result curve. This result shows that a simple
Table 1 Precision analysis with changed boundary conditions. Heat-flow density (W/m2)
Convection boundary temperature (K)
Convection coefficients conv
Boundary temperature T1 (K)
Boundary temperature T2 (K)
Thermal conductivity (W/m K)
Relative deviation (%)
150 200 200
298 298 408
0.44 0.44 0.44
3.366 4.487 4.126
0.999 1.332 0.971
0.25349 0.25357 0.25357
0.02 0.012 0.012
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Z. Jin et al. / Computational Materials Science 102 (2015) 45–50
Graphite Particle
Thermal conductivity/W (m k)-1
0.58
40% Volume Fraction
0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42
50
100
150
200
250
300
Mesh Fig. 7. Relationship between graphite size and thermal conductivity. Fig. 5. Cross section of PTFE composites with SEM diagram.
3.2. Effect of particle distribution style Fig. 9 shows that when the graphite-filled volume fraction is below 40% in the PTFE matrix, a random or even distribution of graphite imposes a similar impact on the thermal conductivity of PTFE composites (the relative error is 0.3%). This impact is almost
0.60
Thermal Conductivity/ W/ (m k)-1
refinement of graphite particle size cannot effectively improve the thermal conductivity of the modification composites; and 250-mesh graphite has better modification effect on the thermal conductivity of PTFE than other particles. Therefore, 250-mesh graphite is selected as the additive in the refinement interval simulation study. Fig. 8 shows the increased volume fraction in graphite-filling, which is consistent with the increase of the effective thermal conductivity. At a relatively low graphite-filling volume fraction, the effect of different graphite particle size on thermal conductivity of PTFE composites is insignificant. With the increase of volume fraction, the effect of different graphite particle size varies and the rising speed of the thermal conductivity also increases. This finding may be attributed to the similar sea-island structure [24] formed by the graphite distribution in the matrix at a low volume fraction of graphite. In that case the graphite distribution in the matrix lacks connections and contacts. So, different graphite particle size presents few differences in low volume fraction. Moreover, the effective thermal conductivity of the composites is low. Graphite particles in some areas may be linked to form a continuous thermal conducting network when the volume fraction of graphite increases, which increases the thermal conductivity of the composites. The difficulty of the link formation varies due to the different particle size.
150mesh 200mesh 250mesh 300mesh
0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0
5% 10% 15% 20% 25% 30% 35% 40%
Volume Fraction Fig. 8. Relationship between graphite volume fraction and thermal conductivity.
negligible, indicating that the random distribution of graphite, which is close to the actual situation, can be used to replace the even distribution. Considering that the graphite volume fraction of 30% may lead to its vulnerable separation from PTFE composites and loading incapability, the optimal graphite volume fraction should be between 15% and 25% to meet the requirements of mechanical strength [25,26]. To explore the impact of graphite volume fraction on thermal conductivity of the PTFE composites, a detailed study is conducted with 250-mesh particle size and with filling volume fractions from 14% to 26% as well as a 2% interval. The simulation
Fig. 6. The EDAX SEM chemical elements analysis (JSM-7500F of JEOLÒ Company).
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Z. Jin et al. / Computational Materials Science 102 (2015) 45–50
7KHUPDO&RQGXFWLYLW\: P.
0.60
Hpt ¼ S H
even distribution random distribution
0.55
ð7Þ
pt
where H is the additive temperature gradient. The fiber aspect ratio in matrix S is defined as n = l/d; Z is the fiber axis and X and Y stand for the other two directions of the fiber. Thus,
0.50 0.45 0.40
g¼
0.35
h
n 3=2
ðn2 1Þ
1=2
nðn2 1Þ
i 1 cosh n
ð8Þ
SZ ¼ l g
0.30 0.25
SX ¼ SY ¼
0.20 0
ð9Þ g 2
ð10Þ
where S is the vector matrix, n is the fiber aspect ratio, l is the length of fiber, d is the diameter, and g is a intermediate variable. So, the average temperature gradient in the composites can be expressed as,
5% 10% 15% 20% 25% 30% 35% 40%
Volume Fraction Fig. 9. Effect of even and random distribution on thermal conductivity.
HT ¼ H þ V H Given the intermediate variable matrix,
0.38
Thermal Conductivity/ W/ (m K)-1
ð11Þ
PHVK
Imn ¼
0.36
1 ðm ¼ nÞ 0
ð12Þ
ðm–nÞ
k1 k2
0.34
A¼I
0.32
1 k1 T ¼ Iþ SS k2
ð14Þ
0.30
B¼AT
ð15Þ
where V is the volume fraction of filling fiber; m, n is the row and column number in the matrix; k1 ; k2 is the thermal conductivity of filling fiber and the PTFE matrix phase, respectively.The equivalent temperature gradient H⁄ can be expressed as,
0.28 14%
16%
18%
20%
22%
24%
ð13Þ
26%
Volume Fraction
H ¼ A TðH þ HÞ ¼ B ðH þ HÞ
ð16Þ
Hpt H ¼ ðS IÞ A T ðH þ HÞ
ð17Þ
Fig. 10. Simulation results of thermal conductivity in refinement intervals.
results are shown in Fig. 10. The obtained changing tendency of the thermal conductivity with increase of the volume fraction is generally consistent with the tendency shown in Fig. 9.
Let us to introduce the intervariable matrix,
E ¼ A T ðS IÞ
ð18Þ
So, Eq. (17) can be expressed as, 4. Carbon fibers in enhancing thermal conductivity
pt
H H ¼ E ðH þ HÞ
4.1. Prediction model of thermal conductivity
Then, Eqs. (11) and (19) can be combined as,
The carbon fibers in the composite are assumed to be randomly distributed and the composite is assumed to be isotropic. According to the Fourier’s law of heat conduction, the heat transfer is expressed as,
q ¼ kH
ð19Þ
ð5Þ
h i HT ¼ I þ VBðI þ VEÞ1 H
ð20Þ
Finally, combining Eqs. (5) and (20) will have to the expression of thermal conductivity of PTFE composites, as following,
h i1 k ¼ k2 I þ VBðI þ VEÞ1
ð21Þ
where q is heat flux, H is temperature gradient, and k is thermal conductivity. When the carbon fibers are filled into the composite, additive heat transfer will occur. The total heat transfer can be expressed as,
Considering the filling fibers’ random distribution, the PTFE composites is isotropic. The prediction expression of the thermal conductivity of PTFE composites is thus derived in this paper,
q þ qD ¼ km ðH þ HÞ
km ¼
ð6Þ
where qD is additive heat flux, km is the thermal conductivity of composites, and H is the average additive temperature gradient. Considering the temperature gradient and the total heat flux, we recur to the equivalent temperature gradient H⁄ and the vector matrix S related with the fiber aspect ratio, which were presented by Eshelby [27,28].
3 þ VðSZ 1Þ ðC þ 2DÞ 3 þ VðC þ 2DÞ
ð22Þ
where km is the thermal conductivity of the PTFE composites k þk
k þk
fZ
fX
W/m K, V is the fiber volume fraction, C ¼ SZ ðk fZk22Þk2 , D ¼ SX ðk fXk22Þk2 , k2 is the thermal conductivity of PTFE matrix phase, kfZ is the carbon fiber axial thermal conductivity W/m K, and kfX is the carbon fiber radial thermal conductivity W/m K.
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Z. Jin et al. / Computational Materials Science 102 (2015) 45–50
Thermal Conductivity/ W/ (m k)
2.0
relationship between mechanical strength and volume fractions of PTFE composites is based on the results of Ref. [30]. The optimization results indicate that the optimum volume fraction matching is 17.76 and 10% for volume fraction of graphite particles and carbon fibers, respectively. The thermal and mechanical performance of the PTFE composites under optimum volume fraction matching is shown in Table 2.
Volume Fraction of *UDSKLWH Volume Fraction of *UDSKLWH Volume Fraction of*UDSKLWH Volume Fraction of*UDSKLWH Volume Fraction of*UDSKLWH Volume Fraction of*UDSKLWH Volume Fraction of*UDSKLWH
1.8 1.6 1.4 1.2 1.0 0.8
5. Conclusions
0.6 0.4 0.2
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
Volume Fraction of Carbon Fibers Fig. 11. Thermal conductivity of the PTFE composites.
Table 2 Comparison with PTFE and PTFE composites.
PTFE PTFE composites
Thermal conductivity (W/(m K))
Tensile strength (MPa)
0.21 1.353
23.35 20.21
When the carbon fiber volume fraction is low (usually less than 10%), the predicted results by Eq. (22) are consistent with the experimental results of Chen and Wang [29], with a deviation of about 8%. Based on the calculation results in numerical model, these numerical results are then feed into Eq. (22). So the thermal conductivity of the PTFE composites filled with graphite and carbon fibers at the same time is obtained. And the results are shown in Fig. 11. Results show that the thermal conductivity of the PTFE composites improved with the increase of volume fraction of the carbon fibers, which can be attributed to its high thermal conductivity. When carbon fibers are added in the composites, the fiber structure can be linked with the graphite in the matrix to form a network of thermal conductivity. More carbon fibers mean more links of thermal conductivity networks in the matrix, which improve thermal conductivity of the PTFE composites. 4.2. Optimum volume fraction matching As it is known that the addition of graphite particles will weaken the mechanical strength of PTFE composites. This difficulty can be overcome by adding carbon fibers into PTFE matrix at the same time. There is a trade-off between the volume fractions of graphite particles and carbon fibers. Based on the basis of meeting the requirements of mechanical strength, it is expected that the PTFE composites have higher thermal performance. In order to optimize the volume ratios of graphite particles and carbon fibers, we write a program in MATLABÓ to conduct the optimization analysis. The objective function seeks to maximize the thermal conductivity of the PTFE composites. And the objective function is inclined to suffer the constraints of the relationship of volume fractions and thermal conductivity, the relationship of volume fractions and mechanical strength, and the mechanical strength greater than 20 MPa. According to the results of numerical simulation, these numerical results are then feed into Eq. (22). So the relationship of thermal conductivity of the PTFE composites filled with graphite and carbon fibers at the same time is obtained. And the
(1) The feasibility of the finite element simulation method presented in this paper is verified through experiments and the commonly used prediction models introduced by predecessors. (2) A systematic study is conducted on the effect of graphite particles size, graphite filled volume fraction, and its distribution on thermal conductivity of the PTFE composites. And 250-mesh graphite has better modification effect than other size particles. (3) A prediction model of thermal conductivity of PTFE composites filled with carbon fibers is presented. (4) The optimum volume fraction matching is introduced. The results indicate that the optimum volume fraction matching is 17.76 and 10% for volume fraction of graphite particles and carbon fibers, respectively.
Acknowledgment The authors are grateful to the China Postdoctoral Science Foundation funded project (Grant Nos. 201104400 and 2014M552011) for financial support. References [1] P. Gonon, A. Sylvestre, J. Appl. Phys. 92 (2002) 4581–4589. [2] S. Thomas, V.N. Deepu, P. Mohanan, M.T. Sehastian, J. Am. Ceram. Soc. 91 (2008) 1971–1975. [3] F. Xiang, H. Wang, X. Yao, J. Eur. Ceram. Soc. 26 (2006) 1999–2002. [4] W.Z. Cai, S.T. Tu, G.L. Tao, J. Thermoplast. Compos. 18 (2005) 241–253. [5] Y. Agari, A. Ueda, S. Nagai, J. Appl. Polym. Sci. 43 (1991) 1117–1142. [6] J. Maxwell, Electricity and Magnetism, Clarendon, Oxford, 1873. [7] Y. Yin, S.T. Tu, J. Reinf. Plast. Compos. 21 (2002) 1619–1627. [8] C.L. Choy, K. Young, Polymer 18 (1977) 769–776. [9] Y. Agari, T. Uno, J. Appl. Polym. Sci. 32 (1986) 5705–5712. [10] G.Q. Zhang, Y.P. Xia, H. Wang, Y. Tao, G.L. Tao, S.T. Tu, H.P. Wu, J. Compos. Mater. 44 (2010) 963–970. [11] V.A. Shelestova, V.V. Serafimovich, P.N. Grakovich, Mech. Compos. Mater. 38 (2002) 125–131. [12] D. Kumlutas, I.H. Tavman, M.T. Coban, Compos. Sci. Technol. 63 (2003) 113– 117. [13] Y.P. Zhang, X.G. Liang, Mater. Des. 12 (1995) 91–95. [14] H. Fazilat, M. Ghatarband, S. Mazinani, Comput. Mater. Sci. 58 (2012) 31–37. [15] Z.G. Li, H. Chen, L.F. Cai, Z.H. Zhu, Y.S. Wang, Y. Zhang, J. Reinf. Plast. Compos. 31 (2013) 1586–1598. [16] X.T. Li, X.Y. Fan, Y.D. Zhu, J. Li, J.M. Adams, S. Shen, H.Z. Li, Comput. Mater. Sci. 63 (2012) 207–213. [17] Z. Zhang, P. Klein, K. Friedrich, Compos. Sci. Technol. 62 (2002) 1001–1009. [18] Z. Zhang, K. Friedrich, Compos. Sci. Technol. 63 (2003) 2029–2044. [19] A.H. Mesbahi, D. Semnani, K.S. Nouri, Composites Part B. 43 (2012) 549–558. [20] F. Zhou, G.X. Cheng, Comput. Mater. Sci. 92 (2014) 157–165. [21] F. Dell’lsola, M. Guarascil, K. Hutter, Arch. Appl. Mech. 70 (2000) 323–337. [22] Y. Kondratyuk, R. Stevenson, Soc. Ind. Appl. Math. 46 (2008) 747–775. [23] T.B. Lewis, L.E. Nielsen, J. Appl. Polym. Sci. 14 (1970) 1449–1471. [24] H.K. He, L. Chen, Y. Zhang, Fiber Polym. 15 (2014) 1941–1949. [25] H.X. Xu, X.H. Yuan, X.N. Cheng, Lubr. Eng. 32 (2007) 88–118. [26] B.B. Difallah, M. Kharrat, M. Dammak, G. Monteil, Mater. Des. 34 (2012) 782– 787. [27] J.D. Eshelby, J. Elasticity 5 (1975) 321–335. [28] J.D. Eshelby, Noordhoff Int. Publ. (1974) 69–84. [29] C.H. Chen, Y.C. Wang, Mech. Mater. 23 (1996) 217–228. [30] P.N. Grakovich, V.A. Shelestova, Sci. China Math. 44 (2001) 292–296.