## 3 Results and Discussion
### Analysis of the Variation in Effective Properties According to the RVE Length  
To identify the changes in the effective properties of the RVE according to the CNT length fraction, the RVE effective properties according to length COVs of 0.1, 0.2, and 0.3 were calculated and are shown in Figure 12. It is evident that even among RVEs of identical length, the effective properties increased as the COV increased. For a detailed analysis, the length fraction of CNTs according to changes in the COV was analyzed, as shown in Figure 13. Figure 13a-c show the CNT length fraction when the COV is 0.1, 0.2, and 0.3, respectively, based on a 3 \(\upmu\)m RVE. It was identified that as the COV increased, the maximum length fraction of the CNTs gradually decreased and the proportion of CNTs with a length greater than 2 \(\upmu\)m increased. Figure 13d-f show the CNT length fraction according to changes in the COV based on a 5 \(\upmu\)m RVE. A trend similar to that found for the 3 \(\upmu\)m RVE was observed, and the maximum CNT length increased as the COV increased. This trend was also observed for the 8 \(\upmu\)m RVE shown in Figure 13g-i for COVs of 0.1, 0.2, and 0.3, and 10 \(\upmu\)m RVE shown in Figure 13j-l for COVs of 0.1, 0.2, and 0.3, respectively. Figure 11: Changes in the effective properties according to increase in the RVE length: (**a**) increasing rate of the effective properties based on a 3 \(\upmu\)m RVE according to increasing RVE length; (**b**) difference in the rate of change in the effective properties with increasing RVE length. Figure 12: Changes in the effective properties according to changes in the RVE length and COV. Figure 13: CNT length fraction according to changing COV of **(a)** 3(COV = 0.1), **(b)** 3(COV = 0.2), **(c)** 3, **(d)** 5(COV = 0.1), **(e)** 5(COV = 0.2), **(f)** 5(COV = 0.3), **(g)** 8(COV = 0.1), **(h)** 8(COV = 0.2), **(i)** 8(COV = 0.3), **(j)** 10(COV = 0.1), **(k)** 10(COV = 0.2), and **(l)** 10(COV = 0.3) \(\mu\)m.  
Considering the above results, the length fraction of CNTs greater than 2 \(\upmu\)m according to changes in the COV was analyzed, as shown in Figure 14. As the COV increased, the length fraction of CNTs greater than 2 \(\upmu\)m increased. Moreover, in RVEs of the same length, this parameter increased as the COV increased. Further, as the RVE length increased, the difference in the length fractions of CNTs greater than 2 \(\upmu\)m between COVs became smaller. This observation can be explained based on the relationships between the CNT length fraction and COV, and between the arrangement conditions of the CNTs inside the RVE and the RVE length. As shown in Figure 1, the length fraction of CNTs greater than 2 \(\upmu\)m increases as the COV increases, and the rate of this increase also increases. Furthermore, as the RVE length increases, the probability that the CNTs are not cut and are present inside the RVE increases, and the difference in the length fraction of CNTs greater than 2 \(\upmu\)m decreases even if the COV changes. To analyze the changes in the effective properties according to increases in the RVE length, the rate of change was examined based on the effective properties of the 3 \(\upmu\)m RVE shown in Figure 8. First, the rate of change in the effective properties according to the RVE length is shown in Figure 15a. When the rate of change in the effective properties in a 3 \(\upmu\)m RVE was 100%, this rate gradually increased as the RVE length increased and reached 118-120% for a 10 \(\upmu\)m RVE. For a more detailed analysis, Figure 15b shows the rate of increase for an RVE that is 1 \(\upmu\)m less than the target RVE. From 3 to 4 \(\upmu\)m RVEs, this rate increased by approximately 5-6%, but as the RVE gradually increased beyond this point, this rate gradually decreased. From 9 to 10 \(\upmu\)m RVEs, the change was approximately 1%.