## 4 Numerical Results  
Once the local problems have been solved, it is possible to compute the effective thermoelastic properties of the composite material. The effective elastic coefficients \(\overline{C}_{ijpq}\), the effective thermal stress \(\overline{\beta}_{ij}\) and the effective thermal conductivity \(\overline{\kappa}_{iq}\) are obtained by substituting the calculated values of \(pq\)U\({}_{k}\), \(pp\)\(R_{k}\) and \({}_{q}M\) into Equations (15), (16) and (17). The effective thermal expansion coefficient \(\overline{\alpha}_{pq}\) can be calculated by substituting \(\overline{\beta}_{ij}\) and \(\overline{C}_{ijpq}\) into Equation (4). In this section, \(\overline{C}_{ijpq}\), \(\overline{\alpha}_{pq}\) and \(\overline{\kappa}_{iq}\) are computed for two isotropic and one transversely isotropic composite materials. The first isotropic material is formed by Nicalon fibers embedded into a matrix of Barium Magnesium Aluminosilicate (Nicalon/BMAS). The second isotropic composite material is formed by Aluminum, Boron and Silicon fibers embedded within an Epoxy matrix (Al-B-Si/Epoxy). Table 2 shows the values of the elastic (Young's modulus \(E\), Poisson's ratio \(\
u\), shear modulus \(G\)) and thermal (thermal expansion \(\alpha\), thermal conductivity \(\kappa\)) constants for the fiber and matrix of the two isotropic composite materials. The transversely isotropic material is constituted by circular graphite fibers embedded into an epoxy matrix (Graphite/Epoxy). Table 3 shows the elastic and thermal constants for the fiber and the matrix of the Graphite/Epoxy composite material. The three experiments were solved for square and hexagonal cell distributions of circular fibers composite materials. Different fiber volume fraction values \(\gamma\) were used to perform the calculations. Linear (4-nodes) and quadratic (8-nodes) quadrilateral approximations were used for the FEM implementation. The results obtained were compared with the values provided by the semi-analytical expressions proposed in ref. [8]. The calculated thermoelastic effective properties are illustrated in this section. In each figure, the effective coefficients obtained by using the linear and quadratic implementations of the FEM for the square and hexagonal fiber distributions are compared. Results calculated for the square and hexagonal cells are shown in blue and red with different intensities, respectively. The values of the effective coefficients determined through the semi-analytical formulas, those calculated by using the linear FEM implementation and those obtained through the quadratic FEM formulation are shown with a square, a triangle and a point, respectively. The superscripts \(A\), 4 and 8 in the legend correspond to the values of the semi-analytical expressions, and those obtained through the linear and quadratic FEM formulations, respectively. In general, Figures 4-6 show that the effective coefficients obtained using both FEM implementations are close to the semi-analytical values. Furthermore, it can be seen that the values calculated by means of the quadratic FEM formulation are closer to the semi-analytical values than those obtained by using the linear FEM formulation. Additionally, some figures show an inset with a smaller scale to show relative differences between the semi-analytic and numerical solutions. Finally, the results obtained when performing the calculations using the square and hexagonal distributions are very similar. Due to the symmetry of the tensors and the periodic cell geometry, there are effective coefficients that yield the same results. Only effective coefficients that have different values are illustrated. Additionally, supplementary data have been included in ref. [32]. The first numerical experiment corresponds to a Nicalon/BMAS composite material. Calculated effective properties for different fiber volume fraction values \(\gamma\) are shown. As illustrated in Figure 4, semi-analytical expressions and both FEM implementations provide similar results.