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The conception of new metamaterials showing unorthodox behaviors with respectto elastic wavepropagation has become possible in recent years thanks topowerful dynamical homogenization techniques. Such methods effectively allow todescribe the behavior of an infinite medium generated by periodicallyarchitectured base materials. Nevertheless, when it comes to the study of thescattering properties of finite-sized structures, dealing with the correctboundary conditions at the macroscopicscale becomes challenging. In this paper,we show how finite-domain boundary value problems canbe set-up in the frameworkof enriched continuum mechanics (relaxed micromorphic model) by imposingcontinuity of macroscopic displacement and of generalized traction whennon-local effects areneglected.The case of a metamaterial slab of finite widthis presented, its scattering properties are studied viaa semi-analyticalsolution of the relaxed micromorphic model and compared to numericalsimulationsencoding all details of the selected microstructure. The reflectioncoefficient obtained via the twomethods is presented as a function of thefrequency and of the direction of propagation of the incidentwave. We findexcellent agreement for a large range of frequencies going from the long-wavelimitto frequencies beyond the first band-gap and for angles of incidenceranging from normal to nearparallel incidence. The case of a semi-infinitemetamaterial is also presented and is seen to be areliable measure of theaverage behavior of the finite metastructure. A tremendous gain in termsofcomputational time is obtained when using the relaxed micromorphic model forthe study of theconsidered metastructure.

Relaxed micromorphic broadband scattering for finite-size meta-structures -- a detailed development