Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics

TL;DR

This paper analyzes the anisotropy classes of elastic tensors in the relaxed micromorphic model, establishing macroscopic consistency conditions and asymptotic relations that facilitate the modeling of anisotropic metamaterials.

Contribution

It introduces a simplified anisotropic relaxed micromorphic model with fewer independent components and derives homogenization formulas linking micro- and macro-scale properties.

Findings

Homogenization formula relates micro- and macro-stiffness in the relaxed model.

Standard micromorphic models lack a similar homogenization relation.

Relaxed micromorphic model is promising for characterizing anisotropic metamaterials.

Abstract

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.