FEM micromechanical modeling of nanocomposites with carbon nanotubes

Mechanical properties of carbon nanotube (CNT)based nanocomposites are broadly discussed in the literature. The influence of CNT arrangements on the elastic properties of nanocomposites based on the finiteelement method (FEM) and representative volume element (RVE) approach is presented here. This study is an application of RVE modeling in the characterization of elastic behavior of CNT polymer nanocomposites. Our main contribution is the analysis of the impact of a nanotube arrangement on the elastic properties of nanocomposite to comprehensively determine thematerial constants. While most of the articles are focused on one distribution, not all material constants are determined. Our FEM analysis is compared with micromechanical models and other results from the literature. The current work shows that nanotube arrangements lead to different results of elastic properties. The analytical micromechanical models are consistent with the numerical results only for axial Young’s modulus and Poisson’s ratio, whereas other elastic constants are lower than the numerical predictions. The results of these studies indicate that FEM can predict nanocomposite mechanical properties with good accuracy. This article is helpful and useful to comprehensively understand the influence of CNT arrangements on the elastic properties of nanocomposites.


Introduction
Since the originating work [1], specific attention, both academic and industrial, has been focused on the application of carbon nanotubes (CNTs). Due to unique properties, it is believed that few weight percentages of CNTs can significantly improve the mechanical properties of nanocomposites. Several studies containing experiments [2] and theoretical methods [3][4][5][6][7][8][9][10][11][12] have been utilized to find effective mechanical properties of CNTs. CNTs are widely used as a reinforcing phase in polymeric [13,14], ceramic [15], and metallic matrix composites [16]. However, most of them are polymeric-based composites. Recently, researchers are extensively focused on CNT/epoxy nanocomposites because of their broad applications in the electronics, aeronautics, and automotive industries. However, the increase in high-performance materials by the integration of CNTs into polymeric matrices is challenging because thermodynamic and kinetic barriers make difficult the effective dispersion of nanotubes [17].
The properties of CNT-based nanocomposite are discussed both based on experimental tests and theoretical descriptions [18][19][20][21]. To name a few, the experimental analysis of CNT polymer nanocomposites was presented by Garcia et al. [22], Yeh et al. [23], or Mikhalchan et al. [24]. However, experiments at the nano-level are of high costs. On the other hand, analytical computations can be challenging to formulate or sometimes excessively complex to solve. So to accurately characterize nanocomposites, numerical modeling is indisputably necessary to support analytical computations and experimental attempts. Researchers suggest various numerical modeling approaches to estimate the mechanical properties of CNT nanocomposites, for instance, molecular dynamics (MD), equivalent continuum models, and finite-element method (FEM). Griebel and Hamaekers [25] studied a short singlewalled CNT (SWCNT) and an infinite SWCNT embedded in a polyethylene matrix based on MD. Their results showed good agreement with the rule of mixture (ROM) in the case of the infinite CNT and with an extended ROMs in the case of the short CNT. Arash et al. [26] used MD to determine the mechanical properties of RVEs made of the short or the infinite CNTs embedded in the polymer matrix taking into account the interphase. Al-Ostaz et al. [27] applied MD to calculate the elastic properties of SWCNT-enhanced composites considering matrix, CNT, and interphase. They analyzed cases of aligned and randomly distributed SWCNTs. Anjana et al. [28] studied the effects of the volume fraction and aspect ratio on the mechanical properties of SWCNT nanocomposites using also MD. In the literature, different homogenization techniques are proposed to predict the mechanical properties of the CNT-reinforced polymer composite. Selmi et al. [29] presented a comparative study of several micromechanical models to evaluate the elastic properties of SWCNT-reinforced polymers. Among others, the stiffness enhancement of the epoxy matrix containing SWNTs was investigated for randomly oriented and fully aligned reinforcements. Fisher et al. [30] analyzed the effects of nanotube waviness on the modulus of CNT-reinforced polymers using a combined finite element and micromechanical approach. Tsai et al. [31] combined the micromechanical modeling and physical experiments to predict the modulus of CNT-reinforced nanocomposite. Hassanzadeh-Aghdam et al. [32] described the effective elastic properties of regular and random arrangements of CNTs in the polymer nanocomposites based on representative volume element (RVE). The applied RVE consisted of three phases containing CNT, polymer matrix, and interphase formed due to van der Waals forces between the CNT and the polymer. FEM is the alternative method of predicting the elastic properties of CNT-based nanocomposites which combine easy realization and cost efficiency. Liu and Chen [33,34] used RVE and FEM to estimate the elastic properties of the CNT-reinforced polymer nanocomposites. Chwał [35] and Chwał and Muc [36] applied a similar approach with various boundary conditions to calculate the mechanical properties of SWCNT-polymer nanocomposite. Zuberi and Esat [37] investigated the mechanical properties of the SWCNT-reinforced epoxy nanocomposite through FEM. Meguid et al. [38] calculated the effective properties of CNT-reinforced epoxies with the help of an atomistic-continuum (AC) model. Shokrih et al. [39] and Spanos et al. [40] joined FEM with micromechanical modeling to simulate the mechanical behavior of CNT-based nanocomposites. Malague et al. [41] proposed a procedure to assess size effects in SWCNT-polymer nanocomposites using the atomistic simulations and equivalent continuum model with a large number of CNTs. Alian et al. [42] also used multiscale modeling of CNT-epoxy nanocomposites. Alasvand Zarasvand and Golestanian [43] conducted experimental, numerical, and micromechanical studies to determine the nonlinear behavior of CNT-reinforced polymer. [44] applied the combined numerical approach and experimental verification for the prediction of the elastic behavior of CNT-reinforced polymer nanocomposites.

Kassa and Arumugam
According to the literature review, it is observed that the influence of different parameters including CNT wall number, length, aspect ratio, waviness, alignment, distribution, van der Waals forces, interphases, and matrix modulus on nanocomposite mechanical properties has been studied. The alignment of CNTs is a significant structural parameter that affects the elastic properties of CNT-based nanocomposites. Therefore, the main goal of this investigation is to determine the elastic properties of CNT-epoxy nanocomposites considering the arrangements of nanoreinforcement. Our main input is the analysis of the effect of a nanotube arrangement on the elastic properties of nanocomposite to comprehensively determine the material constants. According to our knowledge, most of the studies are focused on one distribution or not all material constants are determined. Our predictions are also compared to other modeling techniques or experimental data. Here we focus on nanocomposites having the two most common distributions of reinforcement, namely, square and hexagonal. Since the global mechanical response of the CNT nanocomposite is expected, the continuum mechanics approach is adequate. This work involves the numerical homogenization procedure of RVE and the FEM. Theoretical models describing effective mechanical properties of CNT-based nanocomposites are still a current issue.

Materials and methods
The current analysis considers the influence of reinforcement distributions in the representative subregion on the effective elastic properties of the nanocomposite. According to the experimental observations, in CNT-polymer nanocomposite the reinforcement distribution is usually unknown. These nanocomposites cannot offer high mechanical properties. On the other hand, theoretical computations indicate an excellent increase in the mechanical properties even for a small amount of CNT in nanocomposites. Sometimes the theoretical predictions and experimental data can be in error by orders of magnitude [31]. However, some techniques can be applied to align the nanotube distributions in a polymeric matrix. Here to estimate the influence of CNT arrangement on the elastic properties of nanocomposite, some idealization of the CNT distribution in the matrix is assumed. There are many ways to idealize the cross section of a composite. We focus on nanocomposites having the two most common distributions of reinforcement considered in microcomposites, namely, square and hexagonal [45][46][47] (Figure 1). Based on the work by Hill [48], for perfectly distributed fibers due to symmetry and periodicity, one representative array can be selected to analyze the composite at the microscale. Thus, from the whole specimen, a representative subregion called RVE is selected ( Figure 1). Moreover, this RVE as a volume of material statistically characterizes a homogeneous material. For these arrangements, the transversely isotropic material with aligned and uniformly distributed long CNTs is adopted in this work. Here we assume the isotropic planes (2 and 3) are perpendicular to the axial direction (1) of the CNTs (Figure 1).
Since the purpose of the analysis is to evaluate the material constants for nanocomposite, the prepared model should be able to simulate the stress-strain behavior. For finite element-based simulation, the RVE is selected from nanocomposite with long SWCNTs, aligned and uniformly distributed in the epoxy matrix. We have decided to consider CNT as a long fiber instead of a short one according to the experimental results presented in the work of Garcia et al. [22]. They have concluded that nanocomposites having highly aligned CNTs should be considered as a typical long-fiber composite.
The aligned long CNT-polymer nanocomposite consists of three phases, namely, polymer matrix, cylindrical CNTs, and circular cavities ( Figure 2). A similar model was proposed by Selmi et al. [29]. We do not take into account interphase, assuming perfect bonds between CNTs and epoxy. It is beyond the scope of this article and will be addressed in a future study. In the case of a square array, the nanotube is in the RVE center. Whereas for a hexagonal array, RVE has modified sides, i.e., with one nanotube at its center and a quarter of a nanotube at each corner ( Figure 3).
Here both the CNTs and the polymeric matrix are assumed as an isotropic, homogenous, and linearly elastic material. The typical material data for SWCNT and epoxy resin [36] have been used in the finite element analysis (FEA; Table 1).
To compute the effective elastic properties of CNTepoxy nanocomposite, the 3-D RVE was built. The full RVEs for square and hexagonal arrangements are presented in Figure 3. The following notations have been applied for RVE: a 1the length (along the axis of CNT), a 2the width, a 3the high of RVE, R ithe inner radius, and tthe thickness of CNT. Due to the symmetry of the reinforced RVEs, a quarter of the RVEs was modeled in the ABAQUS package ( Figure 2). Mostly the volume fraction of CNTs in nanocomposites is low [20]. At a higher CNT fraction, the mechanical properties of the nanocomposite were observed to deteriorate due to the formation of CNT agglomerates, which act as stress concentrators. In this work, we use the low-volume fraction of CNT, i.e., only 2.75% vol. Therefore, we consider the cylindrical CNT which has an outer diameter of 10 nm and thickness of 0.3 nm. The calculated CNT volume fraction considers only the area occupied by the cylindrical nanotube without considering the inner cavity.
After the geometrical calculations, the numerical FE model of RVEs was built. The details of the models are listed in Table 2. The general structural analysis was conducted.
Numerical RVEs were built using eight-node linear hexahedron finite elements. A double layer of the elements on the CNT model was generated (Figure 4).  The displacements and tractions applied at the boundaries of the nanocomposite RVE are presented as [45]: where Sthe bounding surface, x jthe surface coordinate, and n jthe component of the normal vector to S. The applications of appropriate displacement u i give uniform stresses σ ij used for calculating nanocomposite material constant.
In the micromechanical analysis, the geometry of fiber arrangement in the matrix is an essential condition to develop a material model for the analysis. In the present study, it is assumed that the fibers and the matrix are the only two phases in the composite, the fibers and matrix are perfectly bonded, the fibers are continuous and parallel with a cylindrical cross section and have a uniform diameter along its length, and the fibers and matrix comply with Hooke's law. The effective Hook's law for a composite is defined as: where the overbar denotes the average over the unit cell and σ ij and ε ij are the average stress and strain tensor, respectively, C ⁎ [ ]is the effective elastic moduli in which a total number of independent components is controlled by the prescribed symmetry, i, j = 1, 2, 3. The stress-strain The transversely isotropic material is characterized by a set of five equations having five effective independent stiffness moduli C 11 ⁎ , C 12 ⁎ , C 22 ⁎ , C 23 ⁎ , and C 44 ⁎ . The effective elastic moduli can be expressed in terms of five independent engineering constants, such as the axial and transverse Young's moduli E A ⁎ and E T ⁎ , the axial and transverse Poisson's ratios ν A ⁎ and ν T ⁎ , and the axial shear modulus G A ⁎ as follows [49]: where Effective elastic properties of the transverse isotropic CNT-polymer nanocomposite were estimated through a series of numerical simulations ( Table 3). Involving the symmetry of RVE, the displacements were applied on the characteristic faces of one-fourt RVE, namely, x 1 = 0, x 1 = a 1 , x 2 = 0, x 2 = a 2 , x 3 = 0, and x 3 = a 3 (Figure 3). The displacements in x 1 , x 2 , and x 3 directions are denoted as u 1 , u 2 , and u 3 , respectively. The assumed small displacement is symbolized as e.
The effective properties are computed according to the following relations: Axial elongation E A and ν A u 1 = 0 u 1 = e u 2 = 0 u 3 = 0 Transverse elongation E T and ν T u 1 = 0 • For load acting in transverse directionthe transverse modulus and Poisson's ratio are: • For axial shear loadingthe axial shear modulus is: The average stresses and strains were numerically calculated using the volume averaging method: where Nthe total number of integration points, σ ij kthe stress component, and ε ij kthe strain component at integration points k in volume V k .

Comparative analysis and discussion
The results of the effective engineering constants of CNTpolymer nanocomposites are listed in Figure 5. The numerical values are compared with micromechanical models such as the Vanin model [50], the Chamis model [51], and the Halpin-Tsai model [52] in the case of long CNTs embedded in the epoxy matrix. In micromechanical computations, we used data presented in Tables 1 and 2. Moreover, to discuss FEM results, some literature findings are placed in Figure 5 regarding other modeling techniques such as the AC and MD. Among others, only some works have been selected, which consider polymeric matrix reinforced with a small amount of SWCNTs. Being aware that theoretical predictions, in general, overestimate experimental findings, we also verified the results with the experimental ones from the literature, e.g., ref. [22].  from FEM are higher than the micromechanical predictions but lower than the results from AC model [42].
All micromechanical relations and models involving continuum assumption result in higher values of E T ⁎ than MD results from the work [28]. The current FE model also leads to the relatively high value of the axial shear modulus G A ⁎ which is not predicted by the micromechanical models. This inconsistency needs further consideration from the authors. The lower discrepancy is observed for the transverse shear modulus G T ⁎ . The transverse Poisson's ratio ν T ⁎ calculated numerically for the square and hexagonal arrays shows higher values than computed according to micromechanical models. There is a lack of data from other models ( Figure 5). This discrepancy also needs further consideration from the authors. The experimental results for CNT-epoxy nanocomposites still inform much about lower elastic properties. For example, according to the work by Garcia et al. [22], the axial Young's modulus is almost three times lower than our FEM results. In general, the modeling predictions indicate higher values of effective engineering constants mostly because of assumed idealizations.
To assess the influence of the CNT's addition on the mechanical properties of nanocomposite, the results for the square and hexagonal arrays are normalized with pure matrix moduli. The normalized values for the axial (E E  Figure 6.
The applied matrix properties are as follows: E m = 3.2 GPa and G m = 1.231 GPa. The theoretical predictions confirm the increase in mechanical properties for the polymeric nanocomposite with the addition of CNTs. Improvement in mechanical properties is visible in the axial and the transverse direction; however, the highest increase is for the axial Young's modulusmore than ninefold regardless of the CNT arrays in the composite. The current FEA also revealed more than threefold increase in the axial shear modulus. However, this result is not confirmed in micromechanical calculations, so the authors will consider further analysis. The normalized transverse moduli rise but not as impressive as in the axial direction. The transverse stiffness modulus grows about 22-30%, whereas the transverse shear modulus  increases about 10% for the hexagonal array and only 1% for the square array.
The current values are comparable with FEM results in the work by Liu and Chen [33]. They observed a 10-fold increase in E A ⁎ for the 5% vol. of CNTs in the nanocomposite. However, they reported more than threefold rise in the transverse Young's modulus, which is not confirmed by the present analysis. The discrepancy in the transverse properties of nanocomposites may suggest that the transverse moduli are more sensitive to fluctuations in volume fractions of CNTs than the axial ones. As presented in the work [34], the impact of volume fraction on the transverse properties is less prominent in nanocomposites having stiffer matrices than the polymeric ones. the current work indicates a 9-fold rise. The experimental values listed in the article by Garcia et al. [22] for highly aligned CNT-polymer nanocomposites also indicated a high increase in the axial Young's modulus (more than threefold). From these FEM analyses and literature comparison, it is not possible to judge which predictions are better.

Conclusions
The FE modeling was carried out for two different CNT arrangements, namely, square and hexagonal. By applying various boundary conditions on RVE, the effective material properties were calculated. Our findings are compared with micromechanical results. Besides, some literature results are also presented and discussed. Numerical results showed a high increase in the elastic properties; however, during modeling of the nanocomposite, a lot of assumption is made, which are mostly a great challenge to realize during the sample preparations such as, e.g., perfect alignment and strong interphase. Some findings are not confirmed both in the micromechanical modeling and in the literature review, and it needs further consideration. We have also analyzed results from different techniques applied in nanocomposite modeling. However, at this point, it is not possible to judge which predictions are better. It is still worthwhile to note the discrepancy between the FEM models and experimental observations. The comparison of results with the experimental ones revealed that CNT/polymer nanocomposites are efficient in load carrying; however, experiments do not fulfill theoretical predictions. However, the nanocomposite FEM modeling is very important and can lead to potentially valuable research. The modeling results are in the current interest of searching a method to gain the theoretical predictions of CNT's dispersion and distribution in the polymeric matrix and functionalization of CNT.
Funding information: The paper publication has received funding from Cracow University of Technology.