Frequency- and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model

In this paper, we explore the use of micromorphic-type interface conditions for the modeling of a finite-sized metamaterial. We show how finite-domain boundary value problems can be approached in the framework of enriched continuum mechanics (relaxed micromorphic model) by imposing continuity of macroscopic displacement and of generalized tractions, as well as additional conditions on the micro-distortion tensor and on the double-traction. The case of a metamaterial slab of finite width is presented, its scattering properties are studied via a semi-analytical solution of the relaxed micromorphic model and compared to a direct finite-element simulation encoding all details of the selected microstructure. The reflection and transmission coefficients obtained via the two methods are presented as a function of the frequency and of the direction of propagation of the incident wave. We find excellent agreement for a large range of frequencies going from the long-wave limit to frequencies beyond the first band-gap and for angles of incidence ranging from normal to near-parallel incidence. The present paper sets the basis for a new viewpoint on finite-size metamaterial modeling enabling the exploration of meta-structures at large scales.