  In this paper, we study the anisotropy classes of the fourth order elastic
tensors of the relaxed micromorphic model, also introducing their second order
counterpart by using a Voigt-type vector notation. In strong contrast with the
usual micromorphic theories, in our relaxed micromorphic model only classical
elasticity-tensors with at most 21 independent components are studied together
with rotational coupling tensors with at most 6 independent components. We show
that in the limit case $L_c\rightarrow 0$ (which corresponds to considering
very large specimens of a microstructured metamaterial the meso- and
micro-coefficients of the relaxed model can be put in direct relation with the
macroscopic stiffness of the medium via a fundamental homogenization formula.
We also show that a similar homogenization formula is not possible in the case
of the standard Mindlin-Eringen-format of the anisotropic micromorphic model.
Our results allow us to forecast the successful short term application of the
relaxed micromorphic model to the characterization of anisotropic mechanical
metamaterials.
