## 3 Asymptotic Homogenization Method: Homogeneous Problem, Local Problems and Effective Coefficients
### Aligned NTC-Reinforced Composites
Here, the constituent properties of the composite are considered as transversely isotropic materials, and cylindrical inclusions (see Figure 4b) are assumed. In addition, a comparison between the analytical AHM and the semi-analytical approaches of SAFEM-2D [46] and SAFEM-3D are reported. An analysis of the relative error is also shown. As can be seen, good agreement between the AHM and the SAFEM-2D is observed. However, a major difference is observed for the SAFEM-3D approach. This is because the SAFEM-3D calculations are developed considering that the fiber's length in the \(y_{3}\) axial direction is equal to 0.49 within one-eighth of the cubic cell \(S\) with an edge length of 0.5. Furthermore, it is observed that, as the inclusion volume fraction increases, the relative errors increase, and this is more noticeable for \(C_{3333}^{*}\). Here, the SAFEM-3D model is an approximation to the long-fiber case. On the other hand, as expected, an increase of the inclusion volume fraction leads to an increment in the effective properties, because the elastic properties of the inclusions are higher than those of the matrix. Table 4 reports the effective elastic moduli of a two-phase FRC computed by a semi-analytical method (SAM) [50], the self-consistent method (SCM) [50] and the SAFEM-3D as a function of the aspect ratios (\(\alpha\)) and the volume fractions (\(v_{f}\)). In addition, an analysis  
of the relative error between the SAFEM-3D and the other two models (SAM and SCM) is also shown. Here, a two-phase FRC with transversely isotropic ellipsoidal inclusions (Al 6061) embedded in a matrix [50] is assumed; see Table 2. A good agreement among the three models is observed, with a relative error below 1% in every case. This result is to be expected because small inclusion volume fractions are assumed. On the other hand, it is observed that all the effective elastic properties increase as the volume fraction of the ellipsoidal inclusions decreases. This fact is caused by a much stiffer matrix and more matrix material within the system. Table 5 illustrates the effective elastic coefficients of a three-phase FRC for different volume fractions of the inclusions and computed relative errors (Error 1 and Error 2), as in Table 4. Here, the composite material is reinforced by ellipsoidal (\(\alpha=0.666\)) and spherical (\(\alpha=1\)) inclusions. For the computations, the constituents materials of the SiC matrix, spherical inclusions (Al 6061) and ellipsoidal inclusions (matrix [50]) are given in Table 2. In this analysis, the configuration of these inclusions within the composite is performed to validate the SAFEM-3D model within the range aspect ratios and volume fractions used  
in this work. Additionally, it is worth mentioning that the distribution and shape of the fibers provide a broader view on the scope of the SAFEM-3D model. As can be seen in Table 5, a good agreement is observed between the models. Moreover, the effective elastic properties decrease as the volume fraction of both inclusions increase. This fact is due to a stiffer SiC matrix being considered; therefore, there is less matrix material in the system. Here, the relative errors are in general below 2.2%. Figure 6 illustrates the longitudinal Young's modulus \((E_{11}^{*})\) as a function of the nanotube length, for a two-phase composite reinforced by aligned isotropic SWNT nanotubes [12] in a LaRC-SI polymeric matrix. Here, the SWNT nanotubes are simulated by ellipsoidal inclusions within a matrix. In addition, a comparison between the SAFEM-3D and the continuous-equivalent method [12] is reported. The numerical calculations are performed considering a SWNT nanotube volume fraction equal to 0.01. As can be seen, very good agreement is observed.