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, the relaxed micromorphic model is endowed to capture all 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. The 3 macro-parameters are obtained by imposing periodic boundary conditions thus mimicking the structure at large. The 3 micro-parameters can be uniquely identified from unit-cells, which (i) represent the unit-cells with maximal stiffness and (ii) preserve its tetragonal symmetry. Both conditions (i) and (ii) are built on the inherent rationale of the relaxed micromorphic model. The missing mesoscopic elastic parameters directly follow from a recently developed harmonic-mean type micro-macro homogenization rule, which establishes the general relation between the elasticities in the micromorphic model on its three scales. 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.
Publisher URL: http://arxiv.org/abs/1709.07054