# Wave Polarization Control in Anisotropic Locally Resonant Materials
## 1 Introduction  
Locally resonant metamaterials have attracted a deep interest in recent years for their peculiar properties concerning wave propagation. In particular, periodic materials with heavy, stiff inclusions with a soft coating embedded in a stiff matrix have been demonstrated to have broad spectral gaps at low frequency, see, for example [1, 2]. The intervals of frequency inside which waves are attenuated can have different applications in vibration isolation and impact absorption [3, 4, 5, 6, 7, 8].  
The physical mechanism of local resonance with the corresponding spectral gaps can be associated with the concept of an \"effective dynamic mass density\" that becomes negative in certain frequency intervals [9, 10]. Asymptotic homogenization is a mathematical technique widely employed to study the effective homogenized properties of periodic metamaterials [11, 12, 13, 14]. Under proper hypotheses on the constituent materials, the method was employed in [15] to study the propagation of elastic waves in binary locally resonant metamaterials with cylindrical soft inclusions periodically distributed in a stiff matrix. In the range of validity of the homogenization, the authors also proved that the intervals of negative effective mass density correspond to the band gaps of the metamaterial. The same approach was followed also to characterize the dynamics of ternary locally resonant metamaterials and plates [2, 16].  
The anisotropic properties of metamaterials can be exploited to achieve peculiar dispersive properties besides those related to the formation of band gaps. In [17], the authors studied wave propagation through a locally resonant metamaterial characterized by anisotropic dynamic mass density in the presence of an interface and derived the conditions leading to negative refraction. Another important aspect is the polarization of propagating waves, which depends on the anisotropic properties of materials. Non-isotropic locally resonant materials can be designed to achieve mode conversion, for instance, from longitudinal to shear waves or from longitudinal or flexural to torsional waves [18, 19, 20], for several interesting applications [21].  
In this work, we make use of the asymptotic homogenization technique to study the in-plane propagation of elastic waves in a ternary locally resonant metamaterial made of a stiff matrix and cylindrical eccentric coated inclusions, which is characterized by an effective anisotropic mass. After presenting the main hypotheses concerning the geometry of the metamaterial and the stiffness contrast of the constituent materials, Section 2 is devoted to developing the homogenization procedure and obtaining the explicit expressions of the effective stiffness and mass tensors, which are both anisotropic. This allows to highlight the effect of the different material and geometric parameters and opens the way to metamaterial optimization.  
The elastic wave propagation within the periodic media is then investigated in Section 3, where we show that the dispersion properties of the metamaterial depend on the signature of the effective mass. Particular emphasis is devoted to the case of an indefinite mass tensor, which leads to the formation of polarization bands that can be exploited for manipulating the polarization of elastic waves.  
Finally, in Section 4, we perform several transmission analyses to validate the results predicted through asymptotic homogenization.