## 3 Results and Discussion  
### Analysis of the Variation Mechanism According to the Relationship between the CNT and RVE Lengths  
To identify the geometrical relationship between the effective properties of the RVE and the RVE and CNT lengths, we first analyzed the arrangement of CNTs inside the RVE. When selecting the RVE, an arrangement model considering three conditions according to the RVE length could be derived. The first condition considers that the RVE is longer than the CNTs such that some CNTs are located inside the RVE. The second condition considers that the RVE and CNTs have identical lengths such that some CNTs fully meet both opposite faces of the RVE in the \(x\)-direction. The third condition considers CNTs that are longer than the RVE such that they are cut. In this study, the third condition was not considered because it does not satisfy the general selection criterion for RVEs in which the RVE should be larger than its internal components. Figure 2 shows the conditions under which the CNTs can be arranged in the RVE. First, Figure 2a depicts the condition in which the RVE is longer than the CNTs such that some CNTs are located inside the RVE. The cases in which the CNTs can be arranged inside and outside the RVE are denoted as 1 to 9. Among these, 3 to 2 are conditions in which the CNTs can be arranged inside the RVE. Of these, 3 and 2 show that the CNTs crossing the RVE edge are cut, and only the CNTs inside the RVE are arranged. In 4 to 6, the CNTs are arranged inside the RVE without being cut. The conditions under which the CNTs are not arranged inside the RVE are 1, 2, 8, and 9, wherein the CNTs are arranged outside the RVE and therefore do not exist inside the RVE. Second, Figure 2b shows the condition in which the RVE and CNTs have the same length. All cases in which the CNTs can be arranged inside and outside the RVE are denoted as 1 to 3 and 2 to 6, respectively. Further, 1, 2, 8, and 9 are conditions under which the CNTs are not arranged inside the RVE, whereas 3, 2, and 6 are conditions under which the CNTs can be arranged inside the RVE. Of these, 3 and 2 show that the CNTs crossing the RVE edge are cut, and only the CNTs inside the RVE are arranged. In 6, the CNTs fully meet both opposite faces of the RVE. Figure 1: CNT length fraction according to COV. These arrangements of CNTs according to the RVE length also affect the effective properties of the RVE; as the length of the CNTs inside the RVE increases, the effective properties increase. Therefore, the impact on the effective properties is smaller when the CNTs are cut than when they are not. Accordingly, a detailed analysis of the variation in the effective properties according to the RVE length and CNT length is necessary. To observe the variations in the effective properties according to the RVE and CNT lengths, cube RVEs were modeled to perform the subsequent analysis. First, as shown in Figure 3, RVEs were modeled in which monofilament CNTs of different lengths were arranged in RVEs of the same length. Elements of C3D8 with eight nodes per element and six degrees of freedom per node were used for this model. There are 27,000 elements and 29,791 nodes in the model. For the boundary condition, one surface in the \(x\)-axis direction was completely constrained to derive effective properties \(E_{1}\), and a displacement condition was applied to the opposite surface in the \(x\)-axis direction as much as the RVE length. Second, as depicted in Figure 4, RVEs were modeled in which the RVE length was increased while maintaining 3 wt% of CNTs inside the RVEs, and the effective properties were then calculated. Figure 5 shows the effective properties calculated as described above. Figure 5a shows the variations in the effective properties (\(E_{1}\)) according to the CNT length (\(C_{l}\)) and RVE length (\(R_{L}\)), whose values were normalized based on \(E_{1}\) of \(C_{l}/R_{L}\) = 0.17.