## 3 Asymptotic Homogenization Method: Homogeneous Problem, Local Problems and Effective Coefficients
### Random NTC Reinforced Composites  
In this section, the effect of the distribution of inclusions on the effective longitudinal Young's modulus \((E_{11}^{*})\) property is studied. For this purpose, the SAFEM-3D is implemented over different two-phase FRCs, in which the periodic unit cell has 72 similar inclusions embedded in the matrix. The effect of non-aligned inclusions is determined by their rotations with respect to the axial axis \(y_{1}\). Different fiber orientations \((\theta=0^{\circ}\) (aligned inclusions), \(10^{\circ}\), \(15^{\circ}\), \(20^{\circ}\), \(25^{\circ}\), \(30^{\circ}\), and \(35^{\circ}\) (random inclusions)) are considered. Non-alignment is implemented by dividing one-eighth of the cell into nine prisms containing a single inclusion, which are assumed to have different orientations. No contact between inclusions is allowed. Then, the inclusions are randomly oriented around the axial axis, but all of them have a fixed angle with respect to that axis. In this way, the effect of non-aligned inclusions can be described. For transversely isotropic inclusions, the elastic tensor should be rotated according to the inclusion orientation. In Figure 7, the discretization for one-eighth of the periodic cells using ANSYS (solid187 element) is shown for different inclusion shapes (discs, spheres and ellipsoids) with \(1\%\) of the inclusion volume fraction. In Figure 8, the effective longitudinal Young's modulus \((E_{11}^{*})\) as a function of the inclusion aspect ratio \(\alpha\) is shown for a two-phase FRC with aligned (\(\theta=0^{\circ}\)) and random (\(\theta=35^{\circ}\)) transversely isotropic SWNT [13] inclusions in an isotropic LaRC-SI matrix. Here, the numerical calculation is determined considering a \(1\%\) SWNT volume fraction for different aspect ratio values \(0\leq\alpha\leq 14\). The inclusion shapes for \(\alpha=0.6\), \(1\) and \(10\) can be seen in Figure 7. Additionally, a comparison between the SAFEM-3D and the deformation energy method based on the Mori-Tanaka model [12; 13] is reported. A good agreement is observed between both models. From Figure 8, it is observed that the aligned nanotubes array shows a maximum value of the longitudinal Young's modulus in the whole range of aspect ratio values. Figure 6: Effective longitudinal Young’s modulus for a two-phase FRC with an aligned isotropic SWNT nanotube [12] with a \(1\%\) volume fraction. Figure 8: Longitudinal Young’s modulus for fibers in aligned and random configurations for a 1% SWNT/LaRC-SI composite. Figure 7: Discretization for one-eighth of the periodic cells using ANSYS (Solid187 element) for different inclusion shapes with a 1% inclusion volume fraction and a 10\({}^{\circ}\) fiber orientation: (**a**) discs (\(\alpha=0.6\)), (**b**) spheres (\(\alpha=1\)) and (**c**) ellipsoids (\(\alpha=10\)). In Figure 9, the effective longitudinal [PERSON]'s modulus (\(E_{11}^{*}\)) as a function of the inclusion's aspect ratio \(\alpha\) for different fiber orientations (\(\theta=0^{\circ}(\text{aligned}),10^{\circ},\,15^{\circ},\,20^{\circ},\,25^{ \circ},\,30^{\circ}\) and \(35^{\circ}(\text{random})\)) is shown. Here, the periodic FRC is defined by transversely isotropic SWNT [13] inclusions (see Figure 7) in an isotropic LaRC-SI matrix, where a 1% SWNT volume fraction is considered. Besides, a comparison between the SAFEM-3D and the deformation energy method based on the Mori-Tanaka model [12; 13] is given for aligned and random fiber distributions. The comparisons show good agreement. From Figure 9, it is observed that the effective longitudinal Young's modulus (\(E_{11}^{*}\)) has a strong dependence on the fiber orientation. As \(\theta\) increases, the value of the longitudinal Young's modulus decreases. Maximum values of \(E_{11}^{*}\) are obtained for the aligned distribution of the inclusions. The effect of carbon nanowire (CNW) inclusions on the effective longitudinal Young's modulus (\(E_{11}^{*}\)) is now presented.