Reflection and transmission of elastic waves from an immersed Willis slab at oblique incidence

Elastic metamaterials allow for the control of wave propagation by exploiting local resonances in their internal structure, which leads to unconventional effective properties. However, complex phenomena like non-reciprocity and asymmetry require advanced homogenized models like the Willis model, which introduces additional coupling terms in the elastodynamics equations. The identification of the effective properties of the Willis medium has been predominantly restricted to 1D-wave propagation problems, and therefore here we extend its application to a heterogeneous, viscoelastic meta-slab under oblique wave incidence. In such a configuration, the key challenge in the parameter identification arises from the response of the medium, which is made more complex due to polarization coupling and a larger number of effective parameters. To address this, we adapt the stiffness matrix method to this configuration, enabling the analytical determination of reflection and transmission coefficients. We show how the partial inverse identification of some effective parameters can be analytically obtained by exploiting polarization decoupling at normal incidence. A numerical case study of a meta-slab with a resonant inclusion demonstrates how Willis coupling allows for the prescription of the asymmetric behavior of the meta-slab with a single set of effective parameters. Altogether, the reported methodology and results pave the way towards the complete identification of the effective properties of the Willis medium.