## 3 Numerical Results and Discussion
In this case, we will consider an Indium (\(In\)) inclusion in an Iron (\(Fe\)) matrix corresponding to the tetragonal and cubic symmetries, respectively. The material parameters of \(Fe\) are [39]: \(\rho_{Fe}=7870\) kg/m\({}^{3}\); the stiffness constants (GPa) are \(C_{Fe,11}=231.4\), \(C_{Fe,12}=134.7\) and \(C_{Fe,44}=116.4\). Meanwhile, those of \(In\) are [40]: \(\rho_{In}=7290\) kg/m\({}^{3}\); the stiffness constants (GPa) are \(C_{In,11}=45.2\), \(C_{In,33}=44.9\), \(C_{In,12}=40.0\), \(C_{In,13}=41.2\), \(C_{In,44}=6.52\) and \(C_{In,66}=12.0\). For this structure, the fill fraction corresponds to a fixed value of 0.25. The computational results  
Figure 1: A 3D phononic crystal of a unit-cell cubic form, with a cube inclusion. of Equation (33) for the nonzero elements of the homogenized matrices \(\overline{\rho}_{eff}\) (kg/m\({}^{3}\)) and \(\overline{C}_{eff}=\overline{S}_{eff}^{-1}\) (10\({}^{2}\) GPa) are:  
\[\overline{\rho}_{eff}=\left( \begin{array}{cccc}7720.5&0&0\\ 0&7720.5&0\\ 0&0&7720.5\end{array} \right),\]  
and  
\[\overline{C}_{eff}=\left( \begin{array}{cccccc}1.58&0.906&0.911&0&0&0\\ 0.906&1.58&0.911&0&0&0\\ 0.911&0.911&1.570&0&0&0\\ 0&0&0&0.635&0&0\\ 0&0&0&0&0.635&0\\ 0&0&0&0&0.685\end{array} \right).\]  
According to these results, the 3D solid phononic crystal has a tetragonal symmetry (6 independent values of elastic stiffness constants in \(\overline{C}_{eff}\)). The matrix \(\overline{\rho}_{eff}\) is isotropic, being characterized by the scalar [3]:  
\[\rho_{eff}=\rho_{a}f+\rho_{b}(1-f), \tag{35}\]  
which, as has been demonstrated, is the standard average for periodic structures constituted of solid-solid elastic materials. Another scenario is depicted in Figure 2, showing the geometry of a 3D phononic crystal of a unit-cell cubic form and one centered inclusion in the form of a parallelepiped of \(In\) in a \(Fe\) host. Here, the filling fraction has a value of 0.078. Thus, from (33), we find that the array of values for the effective density (kg/m\({}^{3}\)) and stiffness constants matrices (10\({}^{2}\) GPa) are:  
\[\overline{\rho}_{eff}=\left( \begin{array}{cccc}7824.2&0&0\\ 0&7824.2&0\\ 0&0&7824.2\end{array} \right),\]  
and  
\[\overline{C}_{eff}=\left( \begin{array}{cccccc}2.07&1.18&1.20&0&0&0\\ 1.18&2.01&1.19&0&0&0\\ 1.20&1.19&2.09&0&0&0\\ 0&0&0&0.0962&0&0\\ 0&0&0&0&1.017&0\\ 0&0&0&0&0&0.0967\end{array} \right).\]  
The effective mass density turns out to be diagonal, and it adjusts to Equation (35) again. On the other hand, the effective stiffness constants exhibit nine independent values that are shown in matrix-form for \(\overline{C}_{eff}\); therefore, the three-dimensional solid-solid  
Figure 2: A 3D phononic crystal of a unit-cell cubic form with an inclusion in the form of a parallelepiped. phononic media behaves like an orthorhombic crystal, which is due to the parallelepiped form of the inclusion. Now we are going to focus our attention on presenting the computational results for the effective mass density and elastic stiffness tensors of a 3D phononic crystal with a cubic lattice, with a centered inclusion of \(Int\) in a T-form (not centrosymmetric) of a square cross-section (chosen for computational convenience), where the host material is \(Fe\). The scenario is depicted in Figure 3.  
For this specific design of artificial material, the corresponding filling fraction is 0.081. The corresponding effective parameters values calculated from Equation (33) are:  
\[\overline{p}_{eff}=\left( \begin{array}{cccc}7822.9&0&0\\ 0&7822.9&0\\ 0&0&7822.9\end{array} \right),\]  
and  
\[\overline{C}_{eff}=\left( \begin{array}{cccccc}2.057&1.198&1.185&0&0&0\\ 1.198&2.088&1.187&0&0&0\\ 1.185&1.187&2.014&0&0&0\\ 0&0&0&0.959&0&0\\ 0&0&0&0.921&0\\ 0&0&0&0&0&1.024\end{array} \right).\]  
As can be seen, the resulting effective mass density matrix (in kg/m\({}^{3}\)) indicates a linear behavior for its three spatial components that shows good agreement with Equation (35), as it is again a periodic structure with solid components.