Modeling a labyrinthine acoustic metamaterial through an inertia-augmented relaxed micromorphic approach

Abstract

We present an inertia-augmented relaxed micromorphic model that enriches the relaxed micromorphic model previously introduced by the authors via a term Curl [Formula: see text] in the kinetic energy density. This enriched model allows us to obtain a good overall fitting of the dispersion curves while introducing the new possibility of describing modes with negative group velocity that are known to trigger negative refraction effects. The inertia-augmented model also allows for more freedom on the values of the asymptotes corresponding to the cut-offs. In the previous version of the relaxed micromorphic model, the asymptote of one curve (pressure or shear) is always bounded by the cut-off of the following curve of the same type. This constraint does not hold anymore in the enhanced version of the model. While the obtained curves’ fitting is of good quality overall, a perfect quantitative agreement must still be reached for very small wavelengths that are close to the size of the unit cell.

Multi-element Metamaterial’s Design Through the Relaxed Micromorphic Model

“…The kinetic energy K and strain energy W of the reduced relaxed micromorphic model are shown in equations ( 1)-( 2) [43,109,112,135]:…”

“…It has been largely shown that the bulk mechanical response of mechanical metamaterials can be effectively explored using an elastic-and inertia-augmented micromorphic model (reduced relaxed micromorphic model) which is able to describe the main metamaterials' fingerprint characteristics (anisotropy, dispersion, band-gaps, size-effects, etc. ), while keeping a reduced structure (free of unnecessary parameters) [3,43,44,83,100,101,102,109,112,113,114,115,135]. This model can be linked a posteriori to real metamaterials' microstructures via an inverse fitting procedure.…”

Green’s functions for the isotropic planar relaxed micromorphic model — Concentrated force and concentrated couple

“…The relaxed micromorphic is a new generalized continuum model that allows to describe size-effects and band-gaps appearing in metamaterials (Rizzi et al, 2021a;Voss et al, 2023;Rizzi et al, 2022a,c;Ramirez et al, 2023;Demore et al, 2022;Rizzi et al, 2022d;Madeo et al, 2015) (in its dynamic setting). The relaxed micromorphic model belongs to the family of micromorphic models (Mindlin, 1964;Eringen, 1999) where the kinematics are given by the classical displacement 𝑢 ∶ 𝛺 → R 3 and the non-symmetric micro-distortion 𝑃 ∶ 𝛺 → R 3×3 .…”

“…The above mentioned advantages have led to a multitude of investigations in short-time from the numerical side (Sky et al, 2024(Sky et al, , 2021(Sky et al, , 2022bSchröder et al, 2022;Sarhil et al, 2023), from the modelling side (Rizzi et al, 2021a;Voss et al, 2023;Rizzi et al, 2022a,c;Ramirez et al, 2023;Demore et al, 2022;Rizzi et al, 2022d;Madeo et al, 2015), analytical solutions (Rizzi et al, 2022b(Rizzi et al, , 2021d, regularity of solutions (Knees et al, 2023b,a), and many others.…”

Design and experimental validation of a finite-size labyrinthine metamaterial for vibro-acoustics: enabling upscaling towards large-scale structures

“…The band gap region is broad since the unit cell's design mostly exploits the Bragg-scattering mechanism. This unit cell is an optimized version of the cell presented in Voss et al [24], such that it was stable enough to be manufactured and mechanically loaded. The thickness of the beams was set equal to the thickness of the air gap throughout the unit cell and the number of air gaps in the radial direction was reduced.…”