This document analyzes how the mechanical properties of rubber coatings on inclusions influence wave propagation in acoustic metamaterials (AM). AM are composites that can attenuate waves at certain frequencies. The study models AM as periodic structures and finds that frequency gaps where waves are attenuated occur due to local resonance of inclusions vibrating at their resonant frequencies. Numerical results show that an incompressible rubber coating creates a single frequency gap at twice the frequency of the lowest gap for a compressible rubber coating. The conclusion is that the mechanical properties of rubber coatings significantly impact wave propagation characteristics in AM, and viscoelastic or nonlinear rubber behavior needs modeling for real AM.

1. ‘Exotic’ wave propagation in
acoustic metamaterials
A.O. Krushynska, V.G. Kouznetsova, M.G.D. Geers
/ Mechanics of Materials, Department of Mechanical Engineering
Introduction
Acoustic metamaterials (AM) – composites with identical
coated inclusions – can attenuate waves and have nega-
tige refractive index at certain frequencies. They can be
used for acoustic shielding from low-frequency noise and
cloaking of objects from sound. Unusual wave propagation
in AM is governed by mechanical properties of inclusions.
Aim: analyze the influence of the inclusion coating
(rubber) behavior on the formation of frequency gaps –
frequencies on which waves are attenuated in AM.
Method
AM is modeled as a periodic structure characterized by a
unit cell (Fig.1). Wave attenuation occurs due to the local
resonance effect: the energy of the wave in matrix is
grabbed by inclusions vibrating at resonant frequencies
(Fig.2). Frequency gaps emerge for harmonics without
radial symmetry in the displacement field (Fig.3-4). The
more inclusions, the wider frequency gaps.
Numerical results
Wave propagation spectrum (frequency vs. wavelength) is
plotted to find frequency gaps. It is well studied for AM with
a linear elastic ‘compressible’ rubber coating (Fig. 3).
For more realistic incompressible behavior for rubber
(Fig.4), there is exactly one frequency gap at twice higher
frequencies compared to the lowest gap in Fig.3.
Conclusion and further research
1. Mechanical properties of the rubber coating influence
the wave propagation characteristics significantly.
2. Accounting for viscoelastic or nonlinear rubber beha-
vior is required to model wave attenuation in real AM.
Fig.3. Wave characteristics for AM with ‘compressible’ rubber;
Rin=5mm, Rex=7.5mm, L=15.5mm; filling fraction of inclusions 73 %.
Fig.4. Wave characteristics for AM from Fig.3 with ‘incompressible’
rubber coating for inclusions.
max 0
Fig.2. Wave attenuation in AM due to the local resonance effect
t=t1 ; wave propa-
gates in matrix
t=t2 ; inclusions
start to vibrate
t=t3 ; inclusions
grab the wave energy
t=t4; wave in
matrix disappears
0 time t
wave
amplitude
t1 t2 t3 t4
Fig.1. Structure of a typical AM and a representative unit cell
incident wave