Describing metamaterials with generalized continuum models : the relaxed micromorphic approach
Metamaterials are attracting today growing attention in the scientific community due to their numerous possible astonishing applications. By their intrinsic nature, metamaterials show strong heterogeneities at the level of the microstructure and, except for few particular cases, their mechanical behavior is definitely anisotropic. This is often the case when dealing with band-gap metamaterials, since it is very likely that the band-gap properties strongly depend on the direction of propagation of the traveling wave. In our work, we propose a transparent framework for the study of anisotropy in metamaterials through the introduction of the anisotropic relaxed micromorphic model. In this model, only fourth order elastic tensors are featured, so remaining in the framework of the class of symmetries of classical continuum mechanics. Notwithstanding this simple formulation, the anisotropic relaxed micromorphic model allows to realistically describe the anisotropic mechanical behavior of a large class of metamaterials, including those exhibiting band-gap behaviors. We support such claims by providing precise examples of the efficacy of such model on actual metamaterials, by showing that the anisotropic relaxed micromorphic model is able to globally reproduce i) the dispersion curves for all direction of propagation ii) the band gap properties as function of the direction of propagation iii) the polar diagrams of the velocity for all modes of interest (also those at higher frequencies).