# Random Fiber Array Generation Considering Actual Noncircular Fibers with a Particle-Shape Library
## 3 Results and Discussion
### Comparison of the Results between Actual Noncircular Fibers and Circular Fibers
#### 3.2.1 Comparison of the Effective Elastic Properties  
To calculate the effective elastics properties of RVEs, we employed a popular computational homogenization scheme [29] with a set of periodic boundary conditions (PBCs) showing high convergence rate [30] on RVEs. Its details are summarized in Appendix A. As a result, longitudinal modulus \(E_{1}\), transverse modulus \(E_{2}\) and \(E_{3}\), out-of-plane shear modulus \(G_{12}\), in-plane shear modulus \(G_{23}\), and Poisson's ratio \(\
u_{12}\) and \(\
u_{23}\) were calculated by using Abaqus and MATLAB. Furthermore, for each RFR; RSE algorithm; and noncircular, circular fiber shape, 100 RVEs were used for each \(V_{f}\). As constituent materials of the composite, two composite materials: M55J/M18 [22] and T300/PR319 [31] were used. Fibers are transversely isotropic materials and matrices are isotropic materials as shown in Table 3. We chose these two materials due to the large difference of the relative ratios of elastic modulus **RR** between fiber and matrix defined as  
\[\textbf{RR}=\{\textbf{RR}_{L},\ \textbf{RR}_{T}\}^{T}=\{RR_{1},\ RR_{2},\ RR_{3},\ RR_{4},\ RR_{5}\}^{T}= \left\{ \begin{array}{llll}E_{1}^{f},\ \ v_{12}^{f},\ \ \frac{E_{2}^{f}}{E_{2}^{m}},\ \frac{G_{23}^{f}}{G_{23}^{m}},\ \frac{G_{12}^{f}}{G_{12}^{m}} \end{array} \right\}^{T} \tag{4}\]  
where \(\textbf{RR}_{L}=\{RR_{1},\ RR_{2}\}\) and \(\textbf{RR}_{T}=\{RR_{3},\ RR_{4},\ RR_{5}\}\). Here, \(\textbf{RR}_{L}\) indicates the relative ratio of the longitudinal elastic moduli, and \(\textbf{RR}_{T}\) indicates the relative ratio of the transverse elastic moduli. In the case of M55J/M18, the relative ratio of the all transverse moduli, including \(RR_{3}\), \(RR_{4}\), and \(RR_{5}\), are 1.8229, 2.1923, and 14.1314, respectively, which are much lower than those of T300/PR319, such as 15.7895, 19.9184, and 42.6258. Figures 7 and 8 show the effective elastic properties (\(E_{1}\), \(E_{2}\), \(G_{23}\), \(G_{12}\)) according to \(V_{f}\) from 5% to 60% of M55J/M18 and T300/PR319 by the computational homogenization, respectively. Figures 9 and 10 show the histogram and normal distribution in terms of \(\mu\) and \(\sigma\) of the effective elastic properties for \(V_{f}=\) 60% of M55J/M18 and T300/PR319, respectively. Tables 4 and 5 show the \(\mu\) and relative error of the effective elastic properties for \(V_{f}=\) 60%, 55%, and 45% of M55J/M18 and T300/PR319, respectively. From the results, it can be seen that both materials show little deviation in \(E_{1}\), but the deviation of RVE with noncircular fiber is greater than that of RVE with circular fibers as seen in Figures 9a and \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{\(V_{f}^{req}\)} & \multicolumn{2}{c}{**RSE**} \\ \cline{2-3}  & **Circular Fiber** & **Noncircular Fiber** \\ \cline{2-3}  & (\(\mu\) (\%), \(\sigma\) (\(\times 10^{-3}\)),\(\sigma\)/\(\mu\) (\(\times 10^{-5}\))) & (\(\mu\) (\%), \(\sigma\) (\(\times 10^{-3}\)),\(\sigma\)/\(\mu\) (\(\times 10^{-5}\))) \\ \hline
60\% & (60.2397, 0.0114, 0.0188) & (59.9654, 2.0530, 3.4236) \\
55\% & (55.0578, 0.0100, 0.0181) & (54.9282, 1.7879, 3.2550) \\
45\% & (44.6939, 0.0113, 0.0253) & (44.9997, 1.7071, 3.7936) \\
35\% & (34.9779, 0.0120, 0.0342) & (35.0199, 1.9037, 5.4360) \\
25\% & (25.2617, 0.0120, 0.0476) & (25.0375, 1.8032, 7.2020) \\
15\% & (14.8981, 0.0087, 0.0585) & (15.0151, 1.9072, 12.7021) \\
5\% & (5.1820, 0.0061, 0.1174) & (5.0660, 1.9280, 38.0575) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._