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 Lc → 0 (which corresponds to considering very large specimens of a microstructured metamaterial) the meso-and microcoefficients 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.

For the recently introduced isotropic-relaxed micromorphic generalized continuum model, we show that, under the assumption of positive-definite energy, planar harmonic waves have real velocity. We also obtain a necessary and sufficient condition for real wave velocity which is weaker than the positive definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established. Notably, we show that strong ellipticity does not imply real wave velocity in micropolar elasticity, whereas it does in isotropic linear elasticity.

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Summary To constrain seismic anisotropy under and around the Alps in Europe, we study SKS shear-wave splitting from the region densely covered by the AlpArray seismic network. We apply a technique based on measuring the splitting intensity, constraining well both the fast orientation and the splitting delay. 4 years of teleseismic earthquake data were processed, from 723 temporary and permanent broadband stations of the AlpArray deployment including ocean-bottom seismometers, providing a spatial coverage that is unprecedented. The technique is applied automatically (without human intervention), and it thus provides a reproducible image of anisotropic structure in and around the Alpine region. As in earlier studies, we observe a coherent rotation of fast axes in the western part of the Alpine chain, and a region of homogeneous fast orientation in the Central Alps. The spatial variation of splitting delay times is particularly interesting though. On one hand, there is a clear positive correlation with Alpine topography, suggesting that part of the seismic anisotropy (deformation) is caused by the Alpine orogeny. On the other hand, anisotropic strength around the mountain chain shows a distinct contrast between the Western and Eastern Alps. This difference is best explained by the more active mantle flow around the Western Alps. The new observational constraints, especially the splitting delay, provide new information on Alpine geodynamics.

SUMMARY We infer seismic azimuthal anisotropy from ambient-noise-derived Rayleigh waves in the wider Vienna Basin region. Cross-correlations of the ambient seismic field are computed for 1953 station pairs and periods from 5 to 25 s to measure the directional dependence of interstation Rayleigh-wave group velocities. We perform the analysis for each period on the whole data set, as well as in overlapping 2°-cells to regionalize the measurements, to study expected effects from isotropic structure, and isotropic–anisotropic trade-offs. To extract azimuthal anisotropy that relates to the anisotropic structure of the Earth, we analyse the group velocity residuals after isotropic inversion. The periods discussed in this study (5–20 s) are sensitive to crustal structure, and they allow us to gain insight into two distinct mechanisms that result in fast orientations. At shallow crustal depths, fast orientations in the Eastern Alps are S/N to SSW/NNE, roughly normal to the Alps. This effect is most likely due to the formation of cracks aligned with the present-day stress-field. At greater depths, fast orientations rotate towards NE, almost parallel to the major fault systems that accommodated the lateral extrusion of blocks in the Miocene. This is coherent with the alignment of crystal grains during crustal deformation occurring along the fault systems and the lateral extrusion of the central part of the Eastern Alps.