## 5 Elliptical Fiber Cross Section  
In this section, the thermoelastic effective coefficients are calculated for a composite material formed by fibers with an elliptical cross section embedded into a matrix with a square distribution as illustrated in Figure 9a. Figure 9b shows the periodic cell \(Y\) for the material described and the quarter of the periodic cell is indicated with dashed lines. The ellipse is defined by the length of its semi-axes, denoted by \(a\) and \(b\) in Figure 9b. It is possible to calculate the volume fraction occupied by the fiber using the following relation \(\gamma=ab\pi\). Figure 8: Effective thermal expansion coefficients for Graphite/Epoxy calculated using the linear FEM and those reported in ref. [23]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\gamma\) & \(E_{a}\) & \(E_{a}^{4}\) & \(v_{31}\) & \(v_{31}^{4}\) \\ \hline
0 & 3.46 & 3.46 & 0.350 & 0.350 \\
0.2 & 71.78 & 71.66 & 0.318 & 0.337 \\
0.4 & 140.12 & 139.97 & 0.287 & 0.325 \\
0.5 & 208.41 & 208.28 & 0.257 & 0.313 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: [PERSON]’s modulus (\(E_{a}^{4}\)) and [PERSON]’s ratio (\(\
u_{31}^{4}\)) for the Graphite/Epoxy composite material calculated by using the linear FEM approximation and those reported in [23]. Figure 9: (**a**) Material composed of fibers with an elliptical cross section, (**b**) periodic cell \(Y\). The isotropic compound formed by Nicalon fibers in a BMAS matrix was used to perform the calculations. The values of the elastic and thermal properties of the fiber and the matrix are listed in Table 2. The values summarized in the following tables show the results calculated through the model presented in this work. The first column shown in each table corresponds to the values of the fiber volume fraction (\(\gamma\)) used during the numerical simulations. The results obtained by applying the linear and quadratic FEM are shown for each effective coefficient. Table 6 shows the elastic effective coefficients calculated by using linear and quadratic FEM approximations for different values of \(\gamma\). The calculations were performed using an aspect ratio for the ellipse of \(\
icefrac{{*}}{{b}}=2\). Finally, the differences between the different approximations used in this work are shown in each table. Table 7 shows the calculated values of the thermal effective coefficients (thermal expansion and thermal conductivity) for the Nicalon/BMAS composite material formed by elliptical fibers. Additionally, the difference between FEM approximations are also illustrated. On the one hand, composites with circular cross-section fibers have the same values for the effective coefficients \(\overline{C}_{2323}\) and \(\overline{C}_{1313}\). On the other hand, when the composite material is made with elliptical cross-section fibers, \(\overline{C}_{2323}\) and \(\overline{C}_{1313}\) do not have the same value. Similarly, the thermal expansion coefficients \(\overline{\pi}_{11}\) and \(\overline{\pi}_{22}\) and thermal conductivity coefficients \(\overline{\kappa}_{11}\) and \(\overline{\kappa}_{22}\) have the same values for composites with circular cross-section fibers and different values for composites with elliptical cross-section fibers, as observed in Tables 6 and 7. The aforementioned difference between effective coefficients is expected because the fiber is not evenly distributed in the \(y_{1}\) and \(y_{2}\) directions.