## 3 Numerical Results and Discussion
Based on the results provided by the matrix of effective elastic constants (in 10\({}^{2}\) GPa), the homogenized 3D phononic structure presents nine different elastic constants, which indicates that the artificial material has an orthorhombic crystalline anisotropy. Indeed, the anisotropy in the components of the effective elastic stiffness tensor of this structure obeys the form and the relationship of the length of the inclusion in the period of the cubic lattice, along the principal axes, \(\hat{x}\), \(\hat{y}\) and \(\hat{z}\) (see also the parallelepiped inclusion). It is evident in this non-centrosymmetric inclusion that the anisotropic response will be more noticeable if the filling fraction is increased. Finally, we present the computational results at the quasi-static limit for the effective elastic compliance constants (\(\overline{S}_{eff}\)) of several well-known phononic structures that are composed of isotropic solid materials. The case studies will consist of the individual calculation of a spherical and cubic inclusion in a cubic lattice. The solid phononic material will be composed of steel (bulk modulus (\(B\)) = 160 GPa and shear modulus (\(\mu\)) = 79.3 GPa) [41] in a glass host (\(B\) = 40 GPa and \(\mu\) = 26.2 GPa) [41] for different values of the filling fraction (see Figure 4). Here, the effective parameters were determined by using the following expressions: \(S_{11}=\left(3B+\mu\right)/\left(9B\mu\right)\), \(S_{12}=\left(25_{11}-5_{44}\right)/2\) and \(S_{44}=1/\mu\). In the case of the spherical inclusion, we use Equation (34); \(F(\mathbf{G})_{sphere}=3f\frac{sgn(\mathbf{G}\cdot\mathbf{\tau}_{0})}{( \mathbf{G}\cdot\mathbf{\tau}_{0})^{3}}-\frac{\cos(\mathbf{G}\cdot\mathbf{\tau} _{0})}{(\mathbf{G}\cdot\mathbf{\tau}_{0})^{2}}\), with \(f=\frac{4\pi}{3}\frac{r_{0}^{3}}{V_{C}}\), and \(r_{0}\) is the radius of the sphere. Figure 3: A 3D phononic crystal of a unit-cell cubic form, with an inclusion in a T-form shown in square cross-section. As \(f\) is increased in these high-symmetry structures, note that in the case when the spherical inclusions of the neighboring cells touch each other, the sphere reaches its maximum volume within the unit cell and does not completely fill it; therefore, the fill fraction \((V_{inc}/V_{c})\) covers the interval \(0\leq f\leq\frac{\pi}{6}\). This is unlike the scenario of cubic inclusions, where, by touching the inclusions of neighboring cells, the cubes consequently completely fill the unit cell, and \(f\) reaches the value of 1. As verification, we have also compared our results with those of other homogenization theories reported in the literature, considering the quasi-static limit. We have obtained the effective parameters \(\overline{\rho}_{eff}\) and \(\overline{C}_{eff}\) of a 3D phononic material comprising a prismatic inclusion in a cubic unit cell (\(a=0.01\) m), which represents a typical one-dimensional periodic structure (Figure 5). The inclusion, as in the previous case, is discretized into small cubic elements wherein the form factor is defined by Equation (34). The materials corresponding to the matrix and inclusion are aluminum (\(\rho_{AI}=2700\) kg/m\({}^{3}\); the stiffness constants (GPa) are \(C_{AI,11}=106.80\), \(C_{AI,12}=60.70\) and \(C_{AI,44}=28.20\)) [42] and silicon (\(\rho_{Si}=2330\) kg/m\({}^{3}\); the stiffness constants (GPa) are \(C_{Si,11}=166.00\), \(C_{Si,12}=63.90\) and \(C_{Si,44}=79.60\)) [42], respectively. The effective parameters obtained by this approach using the discretization of the inclusion are presented by points in Figure 6 (see panels (a)-(d)). Solid, dotted, and dashed lines indicate the effective values obtained in the long-wavelength regime, using the theories for laminated composite materials developed in previous studies [38; 43; 44] (e.g., the conventional homogenization theory and asymptotic homogenization). As can be seen in the graphs, the results obtained by the approximation presented here have good agreement with the other calculations at the filling fractions indicated.