Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model

In this paper the relaxed micromorphic material model for anisotropic elasticity is used to describe the dynamical behavior of a band-gap metamaterial with tetragonal symmetry. Unlike other continuum models (Cauchy, Cosserat, second gradient, classical Mindlin-Eringen micromorphic etc.), the relaxed micromorphic model is endowed to capture the main microscopic and macroscopic characteristics of the targeted metamaterial, namely, stiffness, anisotropy, dispersion and band-gaps. The simple structure of our material model, which simultaneously lives on a micro-, a meso- and a macroscopic scale, requires only the identification of a limited number of frequency-independent and thus truly constitutive parameters, valid for both static and wave-propagation analyses in the plane. The static macro- and micro- parameters are identified by numerical homogenization in static tests on the unit-cell level in [30]. The remaining inertia parameters for dynamical analyses are calibrated on the dispersion curves of the same metamaterial as obtained by a classical Bloch-Floquet analysis for two wave directions. We demonstrate via polar plots that the obtained material parameters describe very well the response of the structural material for all wave directions in the plane, thus covering the complete panorama of anisotropy of the targeted metamaterial.