# Wave Polarization Control in Anisotropic Locally Resonant Materials
## 2 Materials and Methods  
### Problem Formulation  
Let us consider a linear elastic solid \(\mathcal{S}\), endowed with two-dimensional periodicity, in the plane \(x_{1}-x_{2}\), composed by a matrix \((m)\) with coated cylindrical inclusions (or fibers, \(f\)). The internal fiber has an eccentricity \(e\) with respect the external coating (\(c\)) as shown in Figure 1. The in-plane periodicity of the cylindrical body allows us to study the problem under plane strain conditions. For this purpose, let \(\Omega^{\varepsilon}\) be the section of \(\mathcal{S}\) with the plane \(x_{3}=0\); it can be obtained by the periodic repetition of the unit cell \(Y^{\varepsilon}\) depicted in Figure 1b, and let \(\mathbf{x}=x_{1}\mathbf{e}_{1}+x_{2}\mathbf{e}_{2}\) be the macroscopic in-plane position vector.  
We assume the scale separation hypothesis, i.e., that \(\epsilon=\ell/L\ll 1\), \(L\) and \(\ell\) being the characteristic sizes of \(\Omega^{\varepsilon}\) and \(Y^{\varepsilon}\), respectively.  
Under the small strain and displacement hypothesis, the in-plane wave propagation at a fixed angular frequency \(\omega\) within \(\Omega^{\varepsilon}\) is governed by the Helmholtz equation:  
\[\
abla\cdot(\mathbb{D}^{\varepsilon}:\varepsilon(\mathbf{u}^{\varepsilon}))+ \rho^{\varepsilon}\omega^{2}\mathbf{u}^{\varepsilon}=\mathbf{0}\qquad\text{ in }\Omega^{\varepsilon}, \tag{1}\]  
where \(\mathbf{u}(\mathbf{x})=u_{1}(\mathbf{x})\mathbf{e}_{1}+u_{2}(\mathbf{x}) \mathbf{e}_{2}\) is the displacement field, and \(\varepsilon(\diamond)\) is the symmetric part of the gradient of \((\diamond)\), while \(\mathbb{D}^{\varepsilon}(\mathbf{x})\) and \(\rho^{\varepsilon}(\mathbf{x})\) are, respectively, the periodically varying fourth  
Figure 1: (a) Geometry of locally resonant metamaterial. (b) Unit cell with fiber and eccentric coating.  
order elastic stiffness tensor and mass density of the constituent materials. Considering constitutive isotropic materials, the elastic stiffness tensor can be expressed as  
\[\mathbb{D}^{\epsilon}=2\mu^{\epsilon}\mathbb{I}+\lambda^{\epsilon}\mathbf{I} \otimes\mathbf{I}, \tag{2}\]  
where \(\lambda^{\epsilon}(\mathbf{x})\) and \(\mu^{\epsilon}(\mathbf{x})\) are the periodically varying [PERSON]'s constants.  
To obtain a local resonant mechanism, we consider the case of soft inclusions in a stiff matrix. In particular, we assume a high stiffness contrast between the matrix and the coating, whilst we prescribe the fibers, which play the role of resonant masses only, to be stiffer than the coating. The mass densities of the three materials are supposed to be of the same order of magnitude. These hypothesis are introduced in the model as follows:  
\[\lambda^{\epsilon}=\begin{cases}\lambda_{m}&\text{in }Y_{m}^{\epsilon}\\ \epsilon^{2}\lambda_{c}&\text{in }Y_{c}^{\epsilon}\\ \epsilon^{2-N}\lambda_{f}&\text{in }Y_{f}^{\epsilon}\end{cases},\quad\mu^{ \epsilon}=\begin{cases}\mu_{m}&\text{in }Y_{m}^{\epsilon}\\ \epsilon^{2}\mu_{c}&\text{in }Y_{c}^{\epsilon}\\ \epsilon^{2-N}\mu_{f}&\text{in }Y_{f}^{\epsilon}\end{cases}\quad\text{and} \quad\rho^{\epsilon}=\begin{cases}\rho_{m}&\text{in }Y_{m}^{\epsilon}\\ \rho_{c}&\text{in }Y_{c}^{\epsilon}\\ \rho_{f}&\text{in }Y_{f}^{\epsilon}\end{cases} \tag{3}\]  
where \(\lambda_{m},\lambda_{c}\) and \(\lambda_{f}\) are of the same order of magnitude, as well as the triplets \(\mu_{m},\mu_{c},\mu_{f}\) and \(\rho_{m},\rho_{c},\rho_{f}\). The integer \(N\geq 1\) appearing as an exponent of the scaling parameter \(\epsilon\) in the fiber stiffness (3) fixes the stiffness ratio between the fibers and matrix, but will not play any role in the further developments. If \(N\geq 2\), the fibers are stiffer than the matrix, while if \(N=1\), they are stiffer than the coating but softer than the matrix.