A multiscale semi-analytical approach towards plane wave analysis of local resonance metamaterials, by Ashwin Sridhar (Eindhoven University of Technology)

DENORMS Action’s Workshop “Modelling of high performance acoustic structures Porous media, metamaterials and sonic crystals”, Rome, 24-25th January 2017

Website of DENORMS Action

Programme of the Workshop

Acoustic metamaterials (i.e. subwavelength absorbers), metasurfaces and sonic crystals for audible sound manipulation

Speaker: Ashwin Sridhar (Eindhoven University of Technology)

Abstract: 

The design of subwavelength metamaterials is important for low frequency applications especially in areas of sound and vibration isolation. Local resonance is the only dominant phenomena at this length scale. Although the phenomenon has been well understood, there is still a lack of analysis tools for the general design of local resonance metamaterials. The dispersion spectra computed from Finite Element (FE/FEM) models of the unit cell under Bloch boundary conditions provides the most general approach for analyzing the elastodynamic behavior of metamaterials, but the computational cost of such calculations is highly prohibitive and moreover the results obtained are in form of band diagrams which provides a more qualitative picture and not precise characterization of the unit cell, therefore the approach is not ideal towards engineering design and rapid prototyping.

Since the analysis is restricted to the subwavelength regime, simpler models can be developed by employing the extended first order homogenization framework [1], which introduces a relaxed scale separation principle that approximates the Bloch boundary condition with periodic boundaries (the Brillouin zone is assumed to extend to infinity) while maintaining the full dynamic description of the microstructure. This assumption is valid for Local Resonance metamaterials with a relatively stiffer matrix compared to the resonating inclusions. Based on this multiscale formulation, a homogenized enriched continuum can be derived in which the standard Cauchy continuum is enriched with additional scalar field variables, each representing the modal amplitude of the respective local resonance eigenmode and, associated with it, the corresponding micro balance law.

The enriched continuum model provides a compact description of the macroscopic dynamics of subwavelength metamaterials. A powerful analysis technique can be realized by Fourier transforming the enriched continuum equations to obtain the reduced dispersion equation. The resulting eigenvalue problem is significantly cheaper to solve compared to those of the full FE models with Bloch boundary condition. Furthermore, under certain approximations, the equation can be solved analytically whereby closed form expressions for the Local Resonance bandgaps can be obtained. This allows the characterization of the bandgaps in terms of a set of enriched material parameters. These parameters are obtained via numerical static and modal analysis of the unit cell. The proposed approach therefore combines the advantage of numerical methods such as FEM in its ability to model complex topology and the closed form solutions for bandgaps obtained from the reduced dispersion equation. Hence it can be termed “semi-analytical”. This forms an efficient analysis tool for the general design of Local Resonance metamaterials.

[1] A. Sridhar, V.G. Kouznetsova, M.G.D. Geers, Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum, Computational Mechanics, (2016) DOI 10.1007/s00466-015-1254-y