Issue 
Eur. Phys. J. Appl. Phys.
Volume 80, Number 3, December 2017



Article Number  30401  
Number of page(s)  12  
Section  Nanomaterials and Nanotechnologies  
DOI  https://doi.org/10.1051/epjap/2017170169  
Published online  29 November 2017 
https://doi.org/10.1051/epjap/2017170169
Regular Article
Finite element investigation of temperature dependence of elastic properties of carbon nanotube reinforced polypropylene
^{1}
Department of Mechanical Engineering, University of Guilan,
P.O. Box 3756,
Rasht, Iran
^{2}
Young Researchers and Elite Club, Langarud Branch, Islamic Azad University,
Langarud, Guilan, Iran
^{*} email: rouhi@iaul.ac.ir
Received:
3
May
2017
Received in final form:
18
August
2017
Accepted:
25
September
2017
Published online: 29 November 2017
This paper aims to investigate the elastic modulus of the polypropylene matrix reinforced by carbon nanotubes at different temperatures. To this end, the finite element approach is employed. The nanotubes with different volume fractions and aspect ratios (the ratio of length to diameter) are embedded in the polymer matrix. Besides, random and regular algorithms are utilized to disperse carbon nanotubes in the matrix. It is seen that as the pure polypropylene, the elastic modulus of carbon nanotube reinforced polypropylene decreases by increasing the temperature. It is also observed that when the carbon nanotubes are dispersed parallelly and the load is applied along the nanotube directions, the largest improvement in the elastic modulus of the nanotube/polypropylene nanocomposites is obtained.
© EDP Sciences, 2017
1 Introduction
Carbon nanotube (CNT) reinforced nanocomposites are a new class of composite materials which have attracted the attention from research community due to their great physical properties over ordinary composites. Some researchers have employed theoretical approaches such as molecular dynamics (MD) simulations [1–15] and the finite element (FE) method [16–41] method to investigate the mechanical behavior of nanocomposites. Although MD simulations give accurate results, the FE method has been chosen by some researchers due to the high computational cost of MD simulations. Wang et al. [20] investigated the indentation and scratch responses of epoxy matrix reinforced by silica nanoparticles. It was observed that embedding silica nanoparticles into the epoxy matrix leads to significant increase of hardness, modulus and the scratchresistance. Dai and Mishnaevsky [21] used the FE method to study the damage and fracture mechanisms in graphene reinforced nanocomposites. They showed that increasing aspect ratio, volume content and elastic properties of graphene/polymer interface layer results in increasing Young's modulus of graphene/polymer nanocomposites. They also investigated the effect of nanoclay reinforcement (localized in the fiber/matrix interface and distributed throughout the matrix) on the fatigue damage of multiscale fiber reinforced polymer composites [22]. It was observed that the multiscale composites with exfoliated nanoreinforcement and aligned nanoplatelets have higher fatigue resistance than those with intercalated/clustered and randomly dispersed ones. Chandra et al. [23] used a multiscale hybrid atomisticFE approach to investigate the tensile behavior of graphene reinforced nanocomposites. They validated the stiffness and strength obtained by their proposed model by experimental results.
The mechanical properties of polymerclay nanocomposites were investigated by Pahlavanpour et al. [24] using analytical micromechanical and 3D periodic FE simulations. They also used one and twostep homogenization models to predict the stiffness of polymerclay nanocomposites with aligned particles [25]. Silani et al. [26–28] used FE simulations to investigate the mechanical properties of exfoliated clay/epoxy nanocomposites. Bayar et al. [29] studied the influence of temperature on the mechanical properties of nanoclayreinforced polymeric nanocomposites. They showed that employing the MoriTanaka approach to predict Young's modulus of nanocomposites give more accurate results than the FE method. Almasi et al. [30] used a hierarchical multiscale approach to analyze the effects of thickness and stiffness of the interphase region on the elastic properties of the clay/epoxy nanocomposites. It was represented that the stiffness of clay/epoxy nanocomposites decreases by considering interphase layer.
The nonlinear properties of polymer/single wall carbon nanotube (SWCNTs) nanocomposite under tensile, bending and torsional loadings were investigated by Ayatollahi et al. [31]. They employed an equivalent cylindrical beam element to reduce the computational effort of the simulations. Joshi et al. [32] studied the effect of waviness of CNTs on the mechanical properties of CNT reinforced nanocomposites. It was shown that waviness of CNT significantly decreases the effective elastic modulus of nanocomposites. They also investigated the effect of pinhole defects in the structure of CNTs [33], CNT orientation [34] and CNT chirality [35] on the mechanical properties of CNT reinforced polymers.
The effect of CNT orientation on the mechanical properties of CNT/polymer nanocomposites was also studied by Montazeri et al. [36]. The mechanical properties of Halloysite nanotube polypropylene composite were studied by Sheidaei et al. [37] by experimental and FE approaches. Hu et al. [38] divided the representative volume elements (RVEs) to three phases, including CNT, matrix and interphase, and studied the mechanical properties of long and short CNT reinforced polymeric nanocomposites. Mohammadpour et al. [39] employed the FE method to evaluate the mechanical behavior of CNT/polymer nanocomposites with different interfacial shear strengths. Zuberi and Esat [40] investigated the mechanical properties of SWCNT/polymer nanocomposites by considering the SWCNT/polymer interphase using bonded interactions and perfect bonding model. Chanteli and Tserpes [41] analyzed the effect of CNT agglomerate on the mechanical properties of CNT/polypropylene nanocomposites. They showed that Young's modulus of CNT reinforced polypropylene is significantly affected by waviness, the multitude and the topology of the agglomerated CNTs.
The FE method is utilized herein to evaluate the temperature dependence of the elastic modulus of CNT/polypropylene nanocomposites. The elastic moduli of nanocomposites with different CNT dispersion patterns, volume fractions and aspect ratios are computed. Moreover, the effect of CNT/polypropylene interphase thickness on the elastic moduli of nanocomposites with several CNT volume fractions at different temperatures is studied.
2 Details of FE model
In this study, a 2D FE approach has been employed to investigate the mechanical properties of CNT reinforced nanocomposites. Three different dispersion patterns have been used, including random dispersion, CNTs directed along the tensile direction and CNTs directed perpendicular to the tensile direction (see Fig. 1). To randomly disperse CNTs into the polymer matrix, the Python programming language has been employed. In all of the dispersion patterns, the locations of CNTs are randomly selected. Moreover, for the randomly dispersed CNTs, the orientation is randomly chosen. The positions of CNTs are determined in such a way that no overlapping occurs between them.
Besides, different thicknesses are selected for the CNT/matrix interphase including0, half of the CNT diameter and equal to the CNT diameter (see Fig. 2). The elastic modulus of interphase is considered as twice the elastic modulus of the matrix [42]. Also, its Poisson's ratio is considered to be equal to Poisson's ratio of the matrix [42]. As it can be seen in Figure 3, CPS8RT (8node plane stress thermally coupled quadrilateral, biquadratic displacement, bilinear temperature) elements are employed to mesh the matrix in RVEs. Furthermore, CNTs are meshed by CPS8RT (8node plane stress thermally coupled quadrilateral, biquadratic displacement, bilinear temperature) elements. All of the simulations are performed by the ABAQUS software. The temperature dependent elastic moduli of polymer and CNT, which are used as the input of simulations, are extracted from [43] and [44], respectively. The considered elastic moduli of the polymer matrix and CNTs at different temperatures are given in Table 1. Moreover, Poison's ratio of the polymer and CNTs is considered as 0.42 and 0.4, respectively.
Fig. 1
Schematics of different CNT dispersion in matrix phase: (a) random dispersion, (b) CNTs directed along the tensile direction and (c) CNTs directed perpendicular to the tensile direction. 
Fig. 2
Different interphase thickness: (a) without interphase, (b) half of CNT diameter and (c) equal to CNT diameter. 
Fig. 3
Meshed model for CNT dispersion in matrix RVE. 
Considered elastic moduli of the polymer matrix and CNTs at different temperatures.
3 Results and discussion
As it was previously mentioned, the effect of embedding CNTs on the elastic modulus of CNT/polymer nanocomposites at different temperatures is investigated here. The elastic moduli of CNT/polypropylene nanocomposites with different CNT distribution patterns, volume percentages and aspect ratios are computed by using the FE method. Besides, the influence of the CNTmatrix interphase thickness on the elastic modulus of nanocomposites is explored. To obtain the elastic modulus of the nanocomposites, one sides of the constructed RVEs is restrained, while the opposite side is exposed to a displacement, δ, in tensile direction. The elastic modulus of the nanocomposite can be obtained by the following relation: (1) in which is the total reaction force which is measured on the constrained edge. In addition, L and A and are the length and crosssectional area of the RVE, respectively. The elastic moduli are calculated in the temperature range of 0–200 °C with the temperature step of 25 °C.
3.1 Effect of nanotube dispersion pattern
To reinforce the polypropylene matrix, three different patterns are used which can be observed in Figure 1. These patterns include longitudinal CNT dispersion in the loading direction, vertical CNT dispersion perpendicular to the loading direction, and random CNT dispersion. Figure 4 shows the elastic modulus of CNT reinforced polypropylene versus temperature for different CNT dispersion patterns. The CNT volume percentage is selected as 3%. Also, the aspect of CNTs is chosen as 20. The interphase thickness is considered as 0, half of the CNT diameter and equal to the CNT diameter. It is seen that as the pure polymer, the elastic modulus of CNT/polypropylene nanocomposites decreases by increasing the temperature with a nonlinear pattern. As it is predicted, due to 1D nature of nanotubes, reinforcing the nanocomposites by CNTs along the loading direction leads to the largest increase in the elastic modulus of the RVEs compared to the pure matrix. However, the curves associated with the random CNT dispersion and regular CNT dispersion perpendicular to the loading direction are approximately the same. Therefore, one can conclude that to achieve the maximum potential of the CNTs in the reinforcement of the nanocomposites, they should be dispersed regularly against the loading.
Besides, it is seen that the difference between curves is not affected by temperature. In other words, the effect of CNT dispersion pattern on the elastic modulus of CNT/polypropylene nanocomposites does not change at larger temperatures. For example, for polypropylene matrix reinforced by 3% volume fraction of CNT and for the interphase thickness as half of the CNT diameter at the temperature of 25 °C, the regular dispersion of CNTs along the loading direction leads to 37.5% improvement of the polypropylene elastic modulus. While, two other dispersion patterns result in about 20% increase of the polypropylene elastic modulus. Moreover, at the temperature of 175 °C, the associated improvements are 38.7% for regular dispersion along the loading direction and 20% for two other dispersion patterns. Comparing the calculated results, it can be observed that the differences between the improvement percentages due to different CNT dispersion patterns do not significantly change at different temperatures.
The effect of CNT dispersion pattern on the stress distribution in the nanocomposite can be investigated from Figure 5. Here, the CNTs with the aspect ratio of 20 and volume percentage of 3% are dispersed into the polymer matrix by different dispersion algorithm. Besides, the interphase thickness is equal to the CNT diameter. It can be seen that CNTs possess larger stresses than the matrix. This can be related to their larger strengths. However, all of the CNTs do not tolerate the maximum value of stress, which have been given at the side of figures. In other words, most of the CNTs bear only the average stresses. Comparing the figures, it can be concluded that the maximum stresses of RVEs with longitudinal dispersion of CNTs along the tension are larger than those of the other two patterns. The most critical part of matrix is at the tips of CNTs.
Fig. 4
Elastic modulus of polypropylene matrix reinforced by CNTs with different distribution patterns versus temperature (3% volume percent, thickness of interphase = (a) 0, (b) half of SWCNT diameter and (c) SWCNT diameter). 
Fig. 5
Equivalent vonMises stress contours for polypropylene matrix reinforced by CNTs ((a) random distribution, (b) regular distribution along the loading and (c) regular distribution perpendicular to the loading, CNT aspect ratio = 20, CNT volume fraction = 3%, interphase thickness = CNT diameter, temperature = 100 °C). 
3.2 Effect of nanotube volume percentage
The effect of nanotube volume fraction on the elastic modulus of CNT/polypropylene nanocomposites can be investigated in Figure 6 for random CNT dispersion, regular CNT dispersion along the loading direction and regular CNT dispersion perpendicular to the loading direction. The aspect ratio of CNTs is considered as 20, and the CNT/matrix interphase thickness is equal to half of the CNT diameter. Additionally, CNTs are dispersed in the polypropylene matrix with the volume fractions of 1%, 3% and 5%. It is observed that for all of the CNT dispersion patterns, the elastic modulus of CNT/polypropylene nanocomposites increases by increasing the CNT volume fraction. This can be related to the larger elastic modulus of CNTs compared to the elastic modulus of pure polymer.
As the previous subsection, the effect of CNT volume fraction on the elastic properties of CNT reinforced polypropylene is almost the same at different temperatures. For example, for the regular dispersion of CNTs along the loading direction at the temperature of 50 °C, the improvement percentages of 1%, 3% and 5% CNT volume fractions are obtained as 12.27%, 37.84% and 69.55%, respectively. At the temperature of 150 °C, the corresponding values are computed as 12.31%, 38.46% and 70.77%.
Comparing Figure 7, in which the stress distributions of nanocomposites are shown for the CNT volume fraction of 5% and CNT aspect ratio of 20, with Figure 5a, it can be concluded that increasing CNT volume fraction leads to decreasing the maximum stress in the RVEs. This is due to larger density of CNTs in the RVEs with the larger CNT volume fraction which leads to dividing a same load between a larger number of CNTs. So, a smaller load is carried by each of the nanotubes.
Fig. 6
Elastic modulus of polypropylene matrix reinforced by CNTs with different volume fractions versus temperature: (a) random CNT dispersion, (b) regular CNT dispersion along the loading direction and (c) regular CNT dispersion perpendicular to the loading direction, interphase thickness = half of the CNT diameter. 
Fig. 7
Equivalent vonMises stress contours for polypropylene matrix reinforced by CNTs (random distribution, CNT aspect ratio = 20, CNT volume fraction = 5%, interphase thickness = CNT diameter, temperature = 100 °C). 
3.3 Effect of interphase thickness
In this sect, the effect of CNT/matrix interphase thickness on the elastic modulus of the nanocomposites is investigated. Nanotubes with the aspect ratio of 20 are randomly dispersed in the polymer matrix. The CNT volume fractions are considered as 1%, 3% and 5%. Figure 8 shows the elastic modulus of CNT/polypropylene nanocomposites against temperature. The CNT/polypropylene interphase thicknesses are equal to 0, half of the CNT diameter and CNT diameter. It is observed that increasing the interphase thickness leads to increasing the elastic modulus of the nanocomposites. The effect of the interphase thickness on RVEs' elastic modulus is more prominent for the volume fraction of 5%. However, for the volume fraction of 1%, the curves are almost coincident. Therefore, one can conclude that the effect of CNT/matrix interphase thickness on the elastic properties of CNT/polypropylene nanocomposites is negligible for small CNT volume fractions. Besides, for two other considered volume fractions, increasing the temperature leads to converging the curves associated with different CNT/matrix interphase thicknesses. So, it can be concluded that at a sufficiently large temperature, the elastic modulus of the CNT/polypropylene nanocomposites is not affected by CNT/matrix interphase thickness.
Represented in Figure 9 is the stress distribution plot for CNT reinforced polypropylene. The CNT aspect ratio and volume percentage are 20 and 3%, respectively. Also, no interphase is considered between CNT and the matrix. Comparing with Figure 5a, it is observed that the maximum stress in the RVE without CNT/matrix interphase is almost twice the maximum stress of the RVE in which the interphase thickness is equal to the CNT diameter. So, it can be concluded that the interphase can decrease the stresses which are created during loading.
Fig. 8
Elastic modulus of polypropylene matrix reinforced by CNTs with different volume fractions versus temperature (a) random distribution, (a) 1%, (b) 3% and (c) 5% volume fraction. 
Fig. 9
Equivalent vonMises stress contours for matrix reinforced by CNTs (random distribution, CNT aspect ratio = 20, CNT volume fraction = 3%, without interphase, temperature = 100 °C). 
3.4 Effect of nanotube aspect ratio
Considering the CNT aspect ratio as 10–50, in this subsection, the effect of CNT aspect ratio on the mechanical properties of CNT/polypropylene nanocomposites is evaluated. CNTs with the volume fraction of 3% are dispersed regularly parallel to the loading direction. Figure 10 shows the elastic modulus of CNT/polypropylene nanocomposites with different CNT aspect ratios. One can find that employing CNTs with larger aspect ratios to reinforce the polypropylene matrix leads to nanocomposites with larger elastic modulus. The effect of CNT aspect ratio on the elastic modulus of CNT/polypropylene nanocomposite is more pronounced for the CNT volume fraction of 5%.
In Figure 11, CNTs with the aspect ratio of 40 and volume fraction of 3% are employed to reinforce the nanocomposites. Comparing this figure with Figure 5a, the effect of CNT aspect ratio in the stress distribution over the RVE can be investigated. It can be seen that RVEs reinforced by CNTs with larger aspect ratios possess larger maximum stress. Moreover, the stress concentration in matrix at the tips of the CNTs decreases for the CNTs with larger aspect ratios.
Fig. 10
Elastic modulus of polypropylene matrix reinforced by 1% volume percent CNTs with different aspect ratios versus temperature, regular CNT dispersion along the loading direction, (a) 1%, (b) 3% and (c) 5%. 
Fig. 11
Equivalent vonMises stress contours for matrix reinforced by CNTs (random distribution, CNT aspect ratio = 40, CNT volume fraction = 3%, interphase thickness = CNT diameter, temperature = 100 °C). 
4 Conclusion
The temperature dependence of the elastic modulus of CNT reinforced polypropylene was evaluated here in. The random and regular algorithms were used to disperse the nanotubes into the polypropylene matrix. Besides, the influences of CNT aspect ratio and volume percentages on the elastic modulus of CNT/polypropylene nanocomposites were studied. It was shown that the elastic modulus of nanocomposites increases by increasing both of the mentioned parameters. The effect of CNT aspect ratio on the elastic properties of nanocomposites is more significant for larger aspect ratios. It was shown that the improvement of the elastic modulus of pure matrix caused by embedding CNTs with different volume percentages and dispersion patters is almost the same. In addition, considering three different CNT/polypropylene interphase thicknesses, it was observed that the effect of this parameter on the nanocomposite elastic modulus is approximately negligible for the small CNT volume percentages.
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Cite this article as: Masoud Ahmadi, Reza Ansari, Saeed Rouhi, Finite element investigation of temperature dependence of elastic properties of carbon nanotube reinforced polypropylene, Eur. Phys. J. Appl. Phys. 80, 30401 (2017)
All Tables
Considered elastic moduli of the polymer matrix and CNTs at different temperatures.
All Figures
Fig. 1
Schematics of different CNT dispersion in matrix phase: (a) random dispersion, (b) CNTs directed along the tensile direction and (c) CNTs directed perpendicular to the tensile direction. 

In the text 
Fig. 2
Different interphase thickness: (a) without interphase, (b) half of CNT diameter and (c) equal to CNT diameter. 

In the text 
Fig. 3
Meshed model for CNT dispersion in matrix RVE. 

In the text 
Fig. 4
Elastic modulus of polypropylene matrix reinforced by CNTs with different distribution patterns versus temperature (3% volume percent, thickness of interphase = (a) 0, (b) half of SWCNT diameter and (c) SWCNT diameter). 

In the text 
Fig. 5
Equivalent vonMises stress contours for polypropylene matrix reinforced by CNTs ((a) random distribution, (b) regular distribution along the loading and (c) regular distribution perpendicular to the loading, CNT aspect ratio = 20, CNT volume fraction = 3%, interphase thickness = CNT diameter, temperature = 100 °C). 

In the text 
Fig. 6
Elastic modulus of polypropylene matrix reinforced by CNTs with different volume fractions versus temperature: (a) random CNT dispersion, (b) regular CNT dispersion along the loading direction and (c) regular CNT dispersion perpendicular to the loading direction, interphase thickness = half of the CNT diameter. 

In the text 
Fig. 7
Equivalent vonMises stress contours for polypropylene matrix reinforced by CNTs (random distribution, CNT aspect ratio = 20, CNT volume fraction = 5%, interphase thickness = CNT diameter, temperature = 100 °C). 

In the text 
Fig. 8
Elastic modulus of polypropylene matrix reinforced by CNTs with different volume fractions versus temperature (a) random distribution, (a) 1%, (b) 3% and (c) 5% volume fraction. 

In the text 
Fig. 9
Equivalent vonMises stress contours for matrix reinforced by CNTs (random distribution, CNT aspect ratio = 20, CNT volume fraction = 3%, without interphase, temperature = 100 °C). 

In the text 
Fig. 10
Elastic modulus of polypropylene matrix reinforced by 1% volume percent CNTs with different aspect ratios versus temperature, regular CNT dispersion along the loading direction, (a) 1%, (b) 3% and (c) 5%. 

In the text 
Fig. 11
Equivalent vonMises stress contours for matrix reinforced by CNTs (random distribution, CNT aspect ratio = 40, CNT volume fraction = 3%, interphase thickness = CNT diameter, temperature = 100 °C). 

In the text 
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