# 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
#### 3.2.2 Comparison of the Maximum Von Mises Stress in Matrix  
In this section, the distribution of the von Mises stress (\(\sigma_{v}\)) in the matrix is compared when 1% of the tensile strain is given with PBCs in the transverse direction (2-direction) by using Abaqus. At this time, the stress was calculated at the centroid of the element. Furthermore, as in Section 3.2.1, for each RFR; RSE algorithm; and noncircular, circular fiber shape, 100 RVEs were used for each \(V_{f}\). Figure 11 shows an example of a contour sample of von Mises stress in the matrix of T300/PR319 with \(V_{f}=60\%\). Figure 11 and Table 7 show the maximum von Mises stress value in terms of the \(\mu\) and error bar in the matrix for diverse \(V_{f}\) of M55J/M18 and T300/PR319, respectively. The results show that the RVE generated by RFR has a higher maximum von Mises stress value in the matrix than those generated by RSE. As mentioned in Section 3.2.1, it is because the minimum distance between the fibers of RFR is smaller than that of the RSE, so that it has larger maximum von Mises stress in the matrix. Furthermore, as the \(V_{f}\) increases, \(\mu\) of the maximum von Mises stress tends to increase. However, the variation of \(\mu\) is sensitive to the minimum distance between the fibers as reported in other work [32]. In this work, we set the minimum distance of RFR as \(l_{\min}=0.07\tau_{f}\) and \(l_{\max}=0.08\tau_{f}\) and that of RSE as in Table 1. Thus, the trend of the maximum stress could be altered by adjusting the minimum distance.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(V_{f}\)} & \multirow{2}{*}{**Methodology**} & \multicolumn{2}{c}{**M55J/M18**} & \multicolumn{2}{c}{**T300/PR319**} \\ \cline{3-6}  & & \(\frac{E_{2}\
u_{32}}{E_{3}\
u_{23}}\) & \(\frac{E_{2}}{2(1+\
u_{23})G_{23}}\) & \(\frac{E_{2}\
u_{32}}{E_{3}\
u_{23}}\) & \(\frac{E_{2}}{2(1+\
u_{23})G_{23}}\) \\ \hline
65\% (master RVE) & RSE & 1.000000 & 1.000307 & 0.999998 & 0.999942 \\
60\% & RFR & 1.000001 & 1.000471 & 0.999911 & 1.019233 \\  & RSE & 1.000001 & 0.999690 & 0.999974 & 0.999263 \\
55\% & RFR & 1.000000 & 1.000729 & 0.999875 & 1.020811 \\  & RSE & 0.999999 & 0.999596 & 1.000068 & 0.998706 \\
45\% & RFR & 1.000000 & 1.000837 & 0.999891 & 1.018256 \\  & RSE & 1.000000 & 1.000124 & 1.000003 & 1.001005 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Symmetry of the compliance matrix and anisotropic ratio of M55J/M18 and T300/PR319 with noncircular fibers.  
Figure 11: Von Mises stress with 1% tensile strain for \(V_{f}\) = 60% of T300/PR319: (**a**) RVEs with noncircular fibers and (**b**) RVEs with circular fibers.  
Figure 12: Comparison of maximum von Mises stress in the matrix for \(V_{f}\) ranging from 5% to 60%: (**a**) M55J/M18 and (**b**) T300/PR319.  
In addition, as \(\mathbf{RR}_{T}\) increases (i.e., T300/PR319), the RVE composed of noncircular fibers tends to a larger maximum von Mises stress in the matrix and wider deviation than the RVEs composed of circular fibers. In the case of T300/PR319 with \(V_{f}=60\%\), the deviations of RFR and RSE according to fiber shape are 8.83%, 9.68%, respectively. Furthermore, it can be seen that the maximum deviation is 12.69% when \(V_{f}=45\%\) with the RSE algorithm in T300/PR319. However, in the case of M55J/M18 with a lower \(\mathbf{RR}_{T}\), the maximum deviation of von Mises stress is quite minimal, less than 2%. Thus, it is concluded that the effect of shape on the maximum stress is critical as the \(\mathbf{RR}_{T}\) is larger.