# Wave Polarization Control in Anisotropic Locally Resonant Materials
## 2 Materials and Methods
### Asymptotic Homogenization  
According to the two-scale asymptotic homogenization technique, we introduce the homogenized domain \(\Omega\) and the fast variable \(\mathbf{y}=\mathbf{x}/\epsilon=y_{1}\mathbf{e}_{1}+y_{2}\mathbf{e}_{2}\), which lives in the re-scaled unit cell \(Y=Y^{\epsilon}/\epsilon\) of the periodic media. Then, the solution of Equation (1) is searched in the following form:  
\[\mathbf{u}^{\epsilon}(\mathbf{x})=\mathbf{u}^{0}\Big{(}\mathbf{x},\frac{ \mathbf{x}}{\epsilon}\Big{)}+\epsilon\ \mathbf{u}^{1}\Big{(}\mathbf{x},\frac{\mathbf{x}}{\epsilon}\Big{)}+ \epsilon^{2}\mathbf{u}^{2}\Big{(}\mathbf{x},\frac{\mathbf{x}}{\epsilon}\Big{)} +o(\epsilon^{2})\qquad\text{in }\Omega^{\epsilon}, \tag{4}\]  
where the vector fields \(\mathbf{u}^{i}(\mathbf{x},\mathbf{y})\) are defined on \(\Omega\times Y\) and are \(Y-\)periodic with respect to the second variable. Replacing the asymptotic expansion (4) in the governing Equation (1), one obtains a sequence of differential problems associated to each order of the paramater \(\epsilon\), which can be solved subsequently. Here, we briefly summarize the solution of the homogenization at the first order, i.e., the determination of the homogenized displacement field \(\mathbf{u}^{0}(\mathbf{x},\mathbf{y})\). _Motion in the matrix_--Restricting the focus on the matrix only, it is possible to prove (see [15] for further details) that the first term in the development (4) must be independent of the fast variable, namely,  
\[\mathbf{u}^{0}(\mathbf{x},\mathbf{y})=\mathbf{U}^{0}(\mathbf{x})\qquad\text{ in }\Omega\times Y_{m}. \tag{5}\]  
Then, the homogenization procedure allows to define the homogenized stiffness tensor \(\mathbb{D}^{0}\) of the periodic media, whose components can be evaluated through  
\[D^{0}_{ijhk}=\frac{1}{|Y|}\int_{Y_{m}}\Big{(}\epsilon_{\mathbf{y}}(\mathbf{\chi}^ {ij})+\mathbf{e}_{i}\odot\mathbf{e}_{j}\Big{)}:\mathbb{D}_{m}:\Big{(}\epsilon _{\mathbf{y}}(\mathbf{\chi}^{hk})+\mathbf{e}_{h}\odot\mathbf{e}_{k}\Big{)}\ d \mathbf{y}, \tag{6}\]  
for \(i,j,h,k\in\{1,2\}\). In Equation (6), functions \(\mathbf{\chi}^{ij}(\mathbf{y})\) represent the periodic displacement field in the matrix of the re-scaled unit cell \(Y\) subject to a uniform eigenstrain \(\mathbf{e}_{i}\odot\mathbf{e}_{j}\), to a periodic boundary condition on \(\partial Y\) and to the free-traction boundary condition on the internal boundary \(\partial Y_{m}\setminus\partial Y\) with the coating, i.e., \(\mathbf{\chi}^{ij}(\mathbf{y})\) solves the matrix cell problems here below:\[ \begin{cases}\
abla_{\mathbf{y}}\cdot\left[\mathbb{D}_{m}:\left(\mathbf{\varepsilon}_{ \mathbf{y}}(\mathbf{x}^{ij})+\mathbf{e}_{i}\odot\mathbf{e}_{j}\right)\right]= \mathbf{0}&\text{in }Y_{m}\\ \mathbf{\chi}^{ij}\text{ periodic}&\text{on }\partial Y\\ \left[\mathbb{D}_{m}:\left(\mathbf{\varepsilon}_{\mathbf{y}}(\mathbf{x}^{ij})+ \mathbf{e}_{i}\odot\mathbf{e}_{j}\right)\right]\cdot\mathbf{n}\text{ anti-periodic}&\text{on } \partial Y\\ \left[\mathbb{D}_{m}:\left(\mathbf{\varepsilon}_{\mathbf{y}}(\mathbf{x}^{ij})+ \mathbf{e}_{i}\odot\mathbf{e}_{j}\right)\right]\cdot\mathbf{n}=\mathbf{0}& \text{on }\partial Y_{m}\setminus\partial Y\end{cases} , \tag{7}\]  
where \(\mathbf{n}\) is the outward normal at the boundary. Therefore, the effective stiffness tensor \(\mathbb{D}^{0}\), defined by (6), can be interpreted as the homogenized stiffness of the holed periodic media. This result is a consequence of the assumed high stiffness contrast between the matrix and the inclusions.