## 3 Results and Discussion
### Analysis of the Variation Mechanism According to the Relationship between the CNT and RVE Lengths
As shown in this graph, increasing \(C_{l}/R_{L}\) resulted in a slight increase in the effective properties, which then sharply increased after this ratio reached 1. This finding suggests that \(E_{1}\) increased because the CNTs fully support both opposite faces of the RVE in the \(x\)-direction. Figure 5b depicts the variation in \(E_{1}\) according to the CNT and RVE lengths and normalized based on \(E_{1}\) at RVE = 3 \(\upmu\)m. As shown in this graph, the effective properties increased as the RVE length increased. Based on the above analysis results, the main variables affecting the effective properties of the RVE were identified as the CNT and RVE lengths, and further examination of these properties is necessary. Figure 2: CNT arrangements inside the RVE: (**a**) \(R_{L}>C_{l}\) and (**b**) \(R_{L}=C_{l}\). To determine the geometrical relationship between the main variables and effective properties, we first examined the geometrical relationship among the main variables. For this purpose, a simplified two-parameter approach was proposed that can calculate the effective properties using the CNT and RVE lengths considering the arrangement of CNTs inside the RVE, as shown in Figure 6. First, it was assumed that the longitudinal elastic modulus \(E_{1}\) of the RVE is governed by the number of CNT filaments contained in the RVE, and the size of \(E_{1}\) is proportional to the ratio of the CNT length to the RVE length. Accordingly, the effective properties of the RVE are determined by the CNT length fraction and its specific elastic modulus. This process is described in Equations (1) and (2):  
Figure 4: Models according to changes in the RVE length: RVE = (a) 2, (b) 4, and (c) 8 \(\upmu\)m.  
Figure 5: Changes in the effective properties according to changes in the CNT and RVE lengths: (a) variation in the effective properties according to \(C_{l}/R_{L}\); (b) variation in the effective properties according to the RVE length. \[m_{i}=\frac{CNT_{L}(Length\ of\ CNT)}{RVE_{L}(Size\ of\ RVE)}(0<m\leq 1);\ \text{ratio of the $i-$ th CNT filament length to the RVE length} \tag{1}\]  
\[E_{1}=\sum_{i=1}^{CNT\ number}E_{1,i}=E_{1,1}+E_{1,2}+E_{1,3}\ \cdots \tag{2}\]  
where \(E_{1,i}\) is the specific elastic modulus in cases in which only one \(i\)-th CNT filament is present inside the RVE. To calculate the effective properties of the RVE using Equation (1), the specific elastic modulus, which is an effective property that exists when a single CNT filament is present in the RVE, is required. Figure 7 shows the variations in the effective properties according to increases in the CNT length when a single filament CNT is arranged inside the RVE. As illustrated in Figure 7a-d, the longitudinal elastic modulus increased as \(C_{l}/R_{L}\) increased, and the CNT length significantly affected the longitudinal elastic modulus. Using the calculated data, an equation to calculate the longitudinal elastic modulus according to \(C_{l}/R_{L}\) was derived. Figure 6: Simplified RVE model. Figure 7: Changes in the effective properties of a CNT inside an RVE according to an RVE length of (**a**) 3, (**b**) 5, (**c**) 8, and (**d**) 10 \(\upmu\)m RVE. When several CNT filaments are arranged inside an RVE, the longitudinal elastic modulus can be calculated using the specific elastic modulus determined previously and Equation (2). Figure 8 presents the longitudinal elastic modulus (COV = 0.1) calculated using this process. It was identified that as the RVE length increases, \(E_{1}\) also gradually increases. Accordingly, the effective properties of the RVE are using the relationship between the RVE length and CNT. It is necessary to determine this relationship for specifying the RVE length of the CNT/PVDF nanocomposites.