## 4 Probabilistic Homogenization Analysis at the Macro-Level for Plain Weave Composites
### Effective Properties at the Macro-Level  
Figure 8 shows the homogenization analysis results based on MCS for the volume fractions of 40, 50, 60, and 70%. All the effective properties including \(E_{11,eq,macro}\), \(E_{33,eq,macro}\), and \(G_{12,eq,macro}\) followed the normal distribution. This trend was maintained, even when the volume fraction changed. However, the average and standard deviations of the stiffness increased with the volume fraction, whereas the opposite was true for the Poisson's ratio. The reason is that as the volume fraction of the tow increases, the stiffness also increases; thus resulting in differences in the standard deviation. However, Poisson's ratio is influenced more strongly by the matrix, thus exhibiting a different trend than observed for the stiffness. Particularly in the case of \(v_{23,eq,macro}\), the increase in stiffness in the direction of \(E_{33,eq,macro}\)  
\begin{table}
\begin{tabular}{c c} \hline \hline  & **Size (mm)** \\ \hline H (Overall height) & 0.225 \\ W (Overall width) & 0.9 \\ L (Overall length) & 0.9 \\ TL (Tow length) & 0.99 \\ TW (Tow width) & 0.376 \\ TT (Tow thickness) & 0.093 (VF 60\%) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Representative unit cell model size.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**No.**} & \(E_{11,low}\) & \(E_{22,low}\) & \(E_{33,low}\) & \(G_{12,low}\) & \(G_{12,low}\) & \(G_{23,low}\) & \(G_{23,low}\) & \(\
u_{12,low}\) & \(\
u_{13,low}\) & \(\
u_{23,low}\) & \(E_{m}\) & \(G_{m}\) & \(v_{m}\) \\ \hline
1 & 130.90 & 14.30 & 14.30 & 3.94 & 3.94 & 2.64 & 0.317 & 0.317 & 0.275 & 2.56 & 0.95 & 0.366 \\
2 & 149.55 & 12.91 & 12.91 & 3.61 & 3.61 & 2.41 & 0.301 & 0.301 & 0.234 & 2.51 & 866.32 & 0.334 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
999 & 150.47 & 14.13 & 14.13 & 3.80 & 3.80 & 2.56 & 0.310 & 0.310 & 0.260 & 2.56 & 0.92 & 0.359 \\
1000 & 147.37 & 13.24 & 13.24 & 3.44 & 3.44 & 2.32 & 0.324 & 0.324 & 0.258 & 2.44 & 0.83 & 0.353 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Input data extraction for the MCS of macro-level (VF 60% output data of micro-level).  
Figure 7: Repeat unit cells model for the fabric composites: (**a**) Repeat unit cell model size for fabric composites; (**b**) Finite element model of macro-level repeat unit cells for fabric composites.  
is insignificant; thus, it shows a significant decreasing tendency when the volume fraction increases. To quantify the inherent uncertainties in the properties of composites, the COV was derived from each effective property and compared. Table 9 outlines the results. The COV of the stiffness changed minimally against the volume fraction. However, the COV of \(v_{12,eq,macro}\), showing a ratio of in-plane strain stiffness increased with the volume fraction. In contrast, \(v_{23,low}\), which indicates the ratio of out-of-plane strain shows the opposite trend. Therefore, the effects of the design variables on the stiffness and Poisson's ratio of the fabric composite appear differently between the two, and they also have the same out-of-plane strain.  
Figure 8: Result of MCS for the macro-level: (**a**) Equivalent longitudinal modulus (\(E_{11,eq,macro}\)); (**b**) Equivalent transverse modulus (\(E_{33,eq,macro}\)); (**c**) Equivalent shear modulus (\(G_{12,eq,macro}\)); (**d**) Equivalent shear modulus (\(G_{23,eq,macro}\)); (**e**) Equivalent Poisson’s ratio (\(v_{12,eq,macro}\)); (**f**) Equivalent Poisson’s ratio (\(v_{23,eq,macro}\)).