# Wave Polarization Control in Anisotropic Locally Resonant Materials
## 2 Materials and Methods
### Asymptotic Homogenization
From Equation (6), it is clear that \(\mathbb{D}^{0}\) possesses minor and major symmetries, and it is possible to prove that it also satisfies the strong ellipticity condition:  
\[\exists\alpha\in\mathbb{R}^{+}\text{ such that }\forall\mathbf{a},\mathbf{b} \text{ one has }\mathbf{a}\otimes\mathbf{b}:\mathbb{D}^{0}:\mathbf{a}\otimes\mathbf{b} \geq\alpha\parallel\mathbf{a}\parallel^{2}\parallel\mathbf{b}\parallel^{2}. \tag{8}\]  
_Motion in the inclusions_--Without loss of generality, we assume that the origin of the reference system \(y_{1}-y_{2}\) coincides with the center of the fiber. Independently of the specific value of \(N\geq 1\) in (3), it is possible to show that the fiber undergoes an in-plane rigid body motion within \(Y\), i.e., one has  
\[\mathbf{u}^{0}(\mathbf{x},\mathbf{y})=\sum_{i=1}^{2}U_{i}^{0}(\mathbf{x}) \left(\mathbf{\tau}^{i}+\psi^{i}\mathbf{y}\wedge\mathbf{e}_{3}\right)\qquad\text{ in }\Omega\times Y_{f} \tag{9}\]  
where \(\mathbf{\tau}^{i}\) and \(\psi^{i}\) for \(i\in\{1,2\}\) describe, respectively, the displacement of the centroid and the in-plane rotation of the fiber. In the coating, the linearity of the problem allows to express the displacement field in the following form:  
\[\mathbf{u}^{0}(\mathbf{x},\mathbf{y})=\sum_{i=1}^{2}U_{i}^{0}(\mathbf{x})\bm {\eta}^{i}(\mathbf{y})\qquad\text{in }\Omega\times Y_{c}, \tag{10}\]  
where the functions \(\mathbf{\eta}^{i}\), for \(i\in\{1,2\}\), solve the differential problems:  
\[ \begin{cases}\
abla_{\mathbf{y}}\cdot\left[\mathbb{D}_{c}:\mathbf{\varepsilon}_{ \mathbf{y}}(\mathbf{\eta}^{i})\right]+\rho_{c}\omega^{2}\mathbf{\eta}^{i}=\mathbf{0}& \text{in }Y_{c}\\ \mathbf{\eta}^{i}=\mathbf{e}_{i}&\text{on }\partial Y_{c}\setminus\partial Y_{f}. \\ \mathbf{\eta}^{i}=\mathbf{\tau}^{i}+\psi^{i}\mathbf{y}\wedge\mathbf{e}_{3}&\text{on } \partial Y_{f}\end{cases} . \tag{11}\]  
The terms \(\mathbf{\tau}^{i}\) and \(\psi^{i}\), which appear in the boundary conditions of (11), have to be determined by enforcing the balance of linear and angular momentum of the fiber, which read  
\[ \begin{cases}\int_{\partial Y_{f}}\left[\mathbb{D}_{c}:\mathbf{\varepsilon}_{ \mathbf{y}}(\mathbf{\eta}^{i})\right]\cdot\mathbf{n}\;d\mathbf{y}+\rho_{f}|Y_{f}| \omega^{2}\mathbf{\tau}^{i}=\mathbf{0}\\ \mathbf{e}_{3}\cdot\int_{\partial Y_{f}}\mathbf{y}\wedge\left[\mathbb{D}_{c}: \mathbf{\varepsilon}_{\mathbf{y}}(\mathbf{\eta}^{i})\right]\cdot\mathbf{n}\;d\mathbf{y }+\rho_{f}\omega^{2}I_{f}\psi^{i}=\mathbf{0}\end{cases} \qquad\text{for }i,j\in\{1,2\}, \tag{12}\]  
\(I_{f}\) being the polar moment of the fib and \(\mathbf{n}\) the exterior normal to \(\partial Y_{f}\). Equations (11) and (12) define the inclusion cell problem at a given frequency \(\omega\), whose solution is given by the fields \(\mathbf{\eta}^{i}(\mathbf{y})\), \(\mathbf{\tau}^{i}\) and \(\psi^{i}\). To underline the frequency dependence of the latter, we will write in the following \(\mathbf{\eta}^{i}(\mathbf{y},\omega)\), \(\mathbf{\tau}^{i}(\omega)\) and \(\psi^{i}(\omega)\). _Homogenized equation of motion_--Following the asymptotic homogenization procedure, one can retrieve the effective Helmholtz equation of the periodic media:  
\[\
abla_{\mathbf{x}}\cdot\left[\mathbb{D}^{0}:\mathbf{\varepsilon}_{\mathbf{x}}( \mathbf{U}^{0})\right]+\omega^{2}\mathbf{\rho}^{0}(\omega)\cdot\mathbf{U}^{0}= \mathbf{0}\qquad\text{in }\Omega, \tag{13}\]where we introduced the homogenized mass density tensor \(\mathbf{\rho}^{0}(\omega)\), which turns out to be frequency dependent. The components of the mass density tensor read  
\[\rho_{ij}^{0}(\omega)=\rho_{st}\delta_{ij}+\rho_{c}\frac{|Y_{c}|}{|Y|}\left( \frac{1}{|Y_{c}|}\int_{Y_{c}}\eta_{j}^{i}(\mathbf{y},\omega)\ d\mathbf{y}-1 \right)+\rho_{f}\frac{|Y_{f}|}{|Y|}(\tau_{j}^{i}(\omega)-1), \tag{14}\]  
where \(\delta_{ij}\) is Kronecker's delta, and  
\[\rho_{st}=\rho_{m}\frac{|Y_{m}|}{|Y|}+\rho_{c}\frac{|Y_{c}|}{|Y|}+\rho_{f}\frac {|Y_{f}|}{|Y|}, \tag{15}\]  
is the static mass density of the periodic media.