## 3 Asymptotic Homogenization Method: Homogeneous Problem, Local Problems and Effective Coefficients
### Aligned NTC-Reinforced Composites  
In this section, the effect properties of two and three-phase FRCs with aligned inclusions (discs, spheres, ellipsoids and hollow and massive cylinders) are studied. The discretization for one-eighth of the periodic cells using ANSYS (Solid187 element) is shown; see Figure 2. In Figure 3, the effective longitudinal Young's modulus \((E_{11}^{*})\), transverse Young's modulus \((E_{22}^{*})\), transverse shear modulus \((\mu_{23}^{*})\), Poisson's ratio \((\
u_{12}^{*})\), plain strain bulk modulus \((\kappa_{23}^{*})\) and axial shear modulus \((\mu_{12}^{*})\) are reported for a two-phase RFC (graphite/epoxy) [11] as a function of the volume fraction. Furthermore, comparisons between the SAFEM-3D and the Mori-Tanaka method (MT) [11] are shown for three different types of inclusions--discs, spheres and ellipsoids--with aspect ratios of \(\alpha=0.5\), \(\alpha=1\) and \(\alpha=2\), respectively. As can be seen from these comparisons, the effective properties \(E_{11}^{*}\), \(E_{22}^{*}\), \(\mu_{23}^{*}\), \(\mu_{12}^{*}\) and \(\kappa_{23}^{*}\) are in good agreement for volume fractions less than 0.3; however, a notable difference is shown when the volume fractions are greater than 0.3. On the one hand, there are small differences in the whole range of volume fraction for \(\
u_{12}^{*}\). \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & **Epoxy**[11] & **Graphite**[11] & **LaRC-SI**[12; 13; 14] & **SWNT**[14] & **Matrix**[46] & **Inclusion**[46] \\ \hline \(\mathrm{C}_{1111}\) & **8.93** & **234.23** & **8.14** & **3024** & **94.23** & **483.68** \\ \(\mathrm{C}_{1122}\) & **6.47** & **12.14** & **5.42** & **1008** & **40.38** & **99.06** \\ \(\mathrm{C}_{1133}\) & **6.47** & **12.14** & **5.42** & **1008** & **40.38** & **99.06** \\ \(\mathrm{C}_{3333}\) & **8.93** & **28.25** & **8.14** & **3024** & **94.23** & **483.68** \\ \(\mathrm{C}_{2323}\) & **2.46** & **16.12** & **2.71** & **2016** & **53.84** & **384.61** \\ \(\mathrm{C}_{1212}\) & **2.46** & **20.00** & **2.71** & **2016** & **53.84** & **384.61** \\ \hline  & **SiC**[50] & **AI** 6061**[50] & **Matrix**[50] & **SWNT**[12] & **SWNT**[13] & \\ \hline \(\mathrm{C}_{1111}\) & **474.20** & **110.50** & **265.30** & **1074.8** & **586.1** & \\ \(\mathrm{C}_{1122}\) & **98.00** & **57.10** & **83.70** & **220.1** & **17.09** & \\ \(\mathrm{C}_{1133}\) & **98.00** & **57.10** & **83.70** & **220.1** & **17.09** & \\ \(\mathrm{C}_{3333}\) & **474.20** & **110.50** & **265.30** & **1074.8** & **49.96** & \\ \(\mathrm{C}_{2323}\) & **188.10** & **26.70** & **90.80** & **854.7** & **4.94** & \\ \(\mathrm{C}_{1212}\) & **188.10** & **26.70** & **90.80** & **854.7** & **34.4** & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constituents’ elastic properties \([GPa]\). SWNT: single-walled nanotube. tion, a comparison between the SAFEM-3D and three different homogenization schemes based on the Mori-Tanaka method (MT) [14] are reported. Here, the SWNT isotropic reinforcement [14] embedded in an isotropic matrix has an aspect ratio of \(\alpha=12.3\). The SWNT reinforcement is designed with a hollow cylindrical geometry (see Figure 4a) simulating the carbon nanotube. It can be observed that the SAFEM-3D results are close to those reported by the MT models. Furthermore, there are two types of concavity in the effective behaviors according to different approaches. Table 3 shows the effective elastic moduli of a two-phase FRC (Inclusion [46]/matrix [46]) for three different values of the inclusion's volume fraction: \(v_{f}=0.05\), \(v_{f}=0.2\) and \(v_{f}=0.35\)  
Figure 4: Discretization for one-eighth of the periodic cells using ANSYS (Solid187 element) with (**a**) hollow and (**b**) massive cylinders. Figure 5: Normalized longitudinal Young’s modulus \((E_{11}^{*}/E_{11}^{(2)})\) as a function of the inclusion’s volume fraction for a two-phase fiber-reinforced composite (FRC) (LaRC-SI/SWNT) with hollow and massive nanotubes in an aligned configuration and aspect ratio of \(\alpha=12.3\).