## 4 Numerical Results
The results obtained for the square and hexagonal cell distributions are also similar, differing slightly for large volume fraction values. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \(\mathbf{E}\) & \(\mathbf{\
u}\) & \(\mathbf{G}\) & \(\mathbf{\alpha\times 10^{-6}}\) & \(\mathbf{\kappa}\) \\  & (GPa) & & (GPa) & (1/c) & (\(W_{\textit{fmK}}\)) \\ \hline Nicalon & 200 & 0.299 & 77 & 3.2 & 0.9 \\ Al-B-Si & 73 & 0.23 & 29.675 & 5 & 0.9 \\ BMAS & 106 & 0.233 & 43 & 2.7 & 0.198 \\ Epoxy & 3.43 & 0.35 & 1.270 & 60 & 0.198 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Elastic and thermal constants of the fibers and matrices of the Nicalon/BMAS and Al-B-Si/Epoxy composite materials. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \(\mathbf{E_{t}}\) & \(\mathbf{E_{a}}\) & \(\mathbf{v_{t}}\) & \(\mathbf{v_{a}}\) & \(\mathbf{G}\) & \(\mathbf{\alpha_{t}\times 10^{-6}}\) & \(\mathbf{\alpha_{a}\times 10^{-6}}\) \\  & (GPa) & (GPa) & \(\mathbf{v_{t}}\) & (GPa) & (1/c) & (1/c) \\ \hline Graphite & 9.66 & 345 & 0.20 & 0.29 & 2.07 & 9 & \(-\)0.36 \\ Epoxy & 3.456 & 3.456 & 0.35 & 0.35 & 1.28 & 40.9 & 40.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Elastic and thermal constants of the fiber and the matrix of the Graphite/Epoxy composite material. Figure 4: Effective properties for Nicalon/BMAS composite material for square and hexagonal distributions. Figure 5: Effective properties for AI-B-Si/Epoxy composite material for square and hexagonal distributions. From the computed elastic coefficients, it is possible to calculate the effective [PERSON]'s modulus and the effective Poisson's ratio. Table 4 shows the effective [PERSON]'s modulus (\(E^{4}\)) and the effective Poisson's ratio (\(\
u^{4}\)) values obtained by using the linear FEM approximation and the results reported in ref. [23] (\(E\) and \(\
u\)). As illustrated in Table 4, the estimated values of \(E^{4}\) and \(\
u^{4}\) are very similar to those reported in ref. [23]. The effective thermal expansion for the Nicalon/BMAS composite material was also calculated in ref. [23] for a square cell distribution. Dots were used in Figure 7 to illustrate the effective thermal expansion coefficients reported in ref. [23]. The results calculated through the linear SAFEM model presented in this work are illustrated with square symbols. The relative difference between the value of \(\overline{\pi}_{22}\) and \(\overline{\pi}_{11}\) can not be appreciated within the  
Figure 6: Effective properties for Graphite/Epoxy composite material for square and hexagonal distributions. scale shown in Figure 7. Note that for a volume fraction of \(\gamma=0\), the effective thermal expansion values correspond to the thermal expansion coefficient of the matrix. The second example corresponds to the Al-B-Si/Epoxy composite material. This is also an isotropic composite material. Effective elastic properties \(\overline{C}\), effective thermal expansion \(\overline{\pi}\) and effective thermal conductivity \(\overline{\kappa}\) are illustrated in Figure 5 for square and hexagonal distributions. Calculations for the linear and quadratic FEM approximations, and the semi-analytical expressions are preformed for each coefficient. Results obtained from both FEM implementations are very similar to those estimated when applying the expressions proposed in [8]. The last example belongs to a transversely isotropic composite material formed by Graphite fibers embedded into an Epoxy matrix. Effective thermal properties for Graphite/Epoxy are shown in Figure 6. As in the previous experiments, the values obtained from the quadratic FEM formulation are closer to the semi-analytical values than those obtained by using the linear FEM formulation. In addition, the figure illustrates that there is little variation between the results obtained when performing the calculations using the square and hexagonal distributions. From the calculated effective coefficients, it is possible to find the effective Young's modulus and Poisson's ratio.