Abstract

Locally resonant materials allow for wave propagation control in the subwavelength regime. Even though these materials do not need periodicity, they are usually designed as periodic systems since this allows for the application of the Bloch theorem and analysis of the entire system based on a single unit cell. However, geometries that are invariant to translation result in equations of motion with periodic coefficients only if we assume plane wave propagation. When wave fronts are cylindrical or spherical, a system realized through tessellation of a unit cell does not result in periodic coefficients and the Bloch theorem cannot be applied. Therefore, most studies of periodic locally resonant systems are limited to plane wave propagation. In this article, we address this limitation by introducing a locally resonant effective phononic crystal composed of a radially varying matrix with attached torsional resonators. This material is not geometrically periodic but exhibits effective periodicity, i.e., its equations of motion are invariant to radial translations, allowing the Bloch theorem to be applied to radially propagating torsional waves. We show that this material can be analyzed under the already developed framework for metamaterials. To show the importance of using an effectively periodic system, we compare its behavior to a system that is not effectively periodic but has geometric periodicity. We show considerable differences in transmission as well as in the negative effective properties of these two systems. Locally resonant effective phononic crystals open possibilities for subwavelength elastic wave control in the near field of sources.

References

1.
Hussein
,
M. I.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2014
, “
Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook
,”
ASME Appl. Mech. Rev.
,
66
(
4
), p.
040802
.
2.
Khelif
,
A.
, and
Adibi
,
A.
,
2015
,
Phononic Crystals: Fundamentals and Applications
,
Springer
,
New York
.
3.
Sigalas
,
M.
, and
Economou
,
E. N.
,
1993
, “
Band Structure of Elastic Waves in Two Dimensional Systems
,”
Solid State Commun.
,
86
(
3
), pp.
141
143
.
4.
Matlack
,
K. H.
,
Bauhofer
,
A.
,
Krödel
,
S.
,
Palermo
,
A.
, and
Daraio
,
C.
,
2016
, “
Composite 3D-Printed Metastructures for Low-Frequency and Broadband Vibration Absorption
,”
Proc. Natl. Acad. Sci.
,
113
(
30
), pp.
8386
8390
.
5.
Pierce
,
C. D.
,
Willey
,
C. L.
,
Chen
,
V. W.
,
Hardin
,
J. O.
,
Berrigan
,
J. D.
,
Juhl
,
A. T.
, and
Matlack
,
K. H.
,
2020
, “
Adaptive Elastic Metastructures From Magneto-Active Elastomers
,”
Smart Mater. Struct.
,
29
(
6
), p.
065004
.
6.
Nimmagadda
,
C.
, and
Matlack
,
K. H.
,
2019
, “
Thermally Tunable Band Gaps in Architected Metamaterial Structures
,”
J. Sound Vib.
,
439
, pp.
29
42
.
7.
Bertoldi
,
K.
,
2017
, “
Harnessing Instabilities to Design Tunable Architected Cellular Materials
,”
Annu. Rev. Mater. Res.
,
47
(
1
), pp.
51
61
.
8.
Yang
,
S.
,
Page
,
J. H.
,
Liu
,
Z.
,
Cowan
,
M. L.
,
Chan
,
C. T.
, and
Sheng
,
P.
,
2004
, “
Focusing of Sound in a 3D Phononic Crystal
,”
Phys. Rev. Lett.
,
93
(
2
), p.
024301
.
9.
Sukhovich
,
A.
,
Merheb
,
B.
,
Muralidharan
,
K.
,
Vasseur
,
J. O.
,
Pennec
,
Y.
,
Deymier
,
P. A.
, and
Page
,
J. H.
,
2009
, “
Experimental and Theoretical Evidence for Subwavelength Imaging in Phononic Crystals
,”
Phys. Rev. Lett.
,
102
(
15
), p.
154301
.
10.
Süsstrunk
,
R.
, and
Huber
,
S. D.
,
2015
, “
Observation of Phononic Helical Edge States in a Mechanical Topological Insulator
,”
Science
,
349
(
6243
), pp.
47
50
.
11.
Mousavi
,
S. H.
,
Khanikaev
,
A. B.
, and
Wang
,
Z.
,
2015
, “
Topologically Protected Elastic Waves in Phononic Metamaterials
,”
Nat. Commun.
,
6
(
1
), pp.
1
7
.
12.
Wang
,
P.
,
Lu
,
L.
,
Bertoldi
,
K.
,
John
,
H.
, and
Paulson
,
A.
,
2015
, “
Topological Phononic Crystals With One-Way Elastic Edge Waves
,”
Phys. Rev. Lett.
,
115
(
10
), p.
104302
.
13.
Wang
,
Y.-T.
,
Luan
,
P.-G.
, and
Zhang
,
S.
,
2015
, “
Coriolis Force Induced Topological Order for Classical Mechanical Vibrations Related Content
,”
New J. Phys.
,
17
(
7
), p.
073031
.
14.
Brillouin
,
L.
,
1946
,
Wave Propagation in Periodic Structures; Electric Filters and Crystal Lattices
,
McGraw-Hill
,
New York
.
15.
Liu
,
Z.
,
Zhang
,
X.
,
Mao
,
Y.
,
Zhu
,
Y. Y.
,
Yang
,
Z.
,
Chan
,
C. T.
, and
Sheng
,
P.
,
2000
, “
Locally Resonant Sonic Materials
,”
Science
,
289
(
5485
), pp.
1734
1736
.
16.
Hussein
,
M. I.
, and
Frazier
,
M. J.
,
2013
, “
Metadamping: An Emergent Phenomenon in Dissipative Metamaterials
,”
J. Sound Vib.
,
332
(
20
), pp.
4767
4774
.
17.
Huang
,
H. H.
,
Sun
,
C. T.
, and
Huang
,
G. L.
,
2009
, “
On the Negative Effective Mass Density in Acoustic Metamaterials
,”
Int. J. Eng. Sci.
,
47
(
4
), pp.
610
617
.
18.
Den Hartog
,
J. P.
,
1947
,
Mechanical Vibrations
,
McGraw-Hill Book Compnay, Inc.
,
New York
.
19.
Christensen
,
J.
, and
García De Abajo
,
F. J.
,
2012
, “
Anisotropic Metamaterials for Full Control of Acoustic Waves
,”
Phys. Rev. Lett.
,
108
(
12
), p.
124301
.
20.
Zhu
,
R.
,
Liu
,
X. N.
,
Hu
,
G. K.
,
Sun
,
C. T.
, and
Huang
,
G. L.
,
2014
, “
Negative Refraction of Elastic Waves at the Deep-Subwavelength Scale in a Single-Phase Metamaterial
,”
Nat. Commun.
,
5
(
1
), p.
5510
.
21.
Ge Kaina
,
N.
,
Lemoult
,
F.
,
Fink
,
M.
, and
Lerosey
,
G.
,
2015
, “
Negative Refractive Index and Acoustic Superlens From Multiple Scattering in Single Negative Metamaterials
,”
Nature
,
525
(
7567
), pp.
77
81
.
22.
Arretche
,
I.
, and
Matlack
,
K. H.
,
2020
, “
Effective Phononic Crystals for Non-Cartesian Elastic Wave Propagation
,”
Phys. Rev. B
,
102
(
13
), p.
134308
.
23.
Torrent
,
D.
, and
Sánchez-Dehesa
,
J.
,
2009
, “
Radial Wave Crystals: Radially Periodic Structures From Anisotropic Metamaterials for Engineering Acoustic or Electromagnetic Waves
,”
Phys. Rev. Lett.
,
103
(
6
), p.
064301
.
24.
Hvatov
,
A.
, and
Sorokin
,
S.
,
2018
, “
On Application of the Floquet Theory for Radially Periodic Membranes and Plates
,”
J. Sound Vib.
,
414
, pp.
15
30
.
25.
Haisheng
,
S.
,
Liqiang
,
D.
,
Shidan
,
L.
,
Wei
,
L.
,
Shaogang
,
L.
,
Weiyuan
,
W.
,
Dongyan
,
S.
, and
Dan
,
Z.
,
2014
, “
Propagation of Torsional Waves in a Thin Circular Plate of Generalized Phononic Crystals
,”
J. Phys. D: Appl. Phys.
,
47
(
29
), p.
295501
.
26.
Yeh
,
P.
,
Yariv
,
A.
, and
Maron
,
E.
,
1978
, “
Theory of Bragg Fiber
,”
J. Opt. Soc. Am.
,
68
(
9
), pp.
1196
1201
.
27.
Shu
,
H.
,
Zhao
,
L.
,
Shi
,
X.
,
Liu
,
W.
,
Shi
,
D.
, and
Kong
,
F.
,
2015
, “
Torsional Wave Propagation in a Circular Plate of Piezoelectric Radial Phononic Crystals
,”
J. Appl. Phys.
,
118
(
18
), p.
184904
.
28.
Xu
,
Z.
,
Wu
,
F.
, and
Guo
,
Z.
,
2012
, “
Low Frequency Phononic Band Structures in Two-Dimensional Arc-Shaped Phononic Crystals
,”
Phys. Lett. A
,
376
(
33
), pp.
2256
2263
.
29.
Ma
,
T.
,
Chen
,
T.
,
Wang
,
X.
,
Li
,
Y.
, and
Wang
,
P.
,
2014
, “
Band Structures of Bilayer Radial Phononic Crystal Plate With Crystal Gliding
,”
J. Appl. Phys.
,
116
(
10
), p.
104505
.
30.
Xiao
,
Y.
,
Mace
,
B. R.
,
Wen
,
J.
, and
Wen
,
X.
,
2011
, “
Formation and Coupling of Band Gaps in a Locally Resonant Elastic System Comprising a String With Attached Resonators
,”
Phys. Lett. A
,
375
(
12
), pp.
1485
1491
.
31.
Wang
,
G.
,
Wen
,
X.
,
Wen
,
J.
, and
Liu
,
Y.
,
2006
, “
Quasi-One-Dimensional Periodic Structure With Locally Resonant Band Gap
,”
ASME J. Appl. Mech.
,
73
(
1
), pp.
167
170
.
32.
Wachel
,
J. C.
, and
Szenasi
,
F. R.
,
1993
, “
Analysis of Torsional Vibrations in Rotating Machinery
,”
Proceedings of the Twenty-Second Turbomachinery Symposium
,
Dallas, TX
,
Sept. 14–16
, pp.
127
151
.
33.
Ma
,
G.
,
Fu
,
C.
,
Wang
,
G.
,
Del Hougne
,
P.
,
Christensen
,
J.
,
Lai
,
Y.
, and
Sheng
,
P.
,
2016
, “
Polarization Bandgaps and Fluid-Like Elasticity in Fully Solid Elastic Metamaterials
,”
Nat. Commun.
,
7
(
1
), pp.
1
8
.
34.
Yu
,
D.
,
Liu
,
Y.
,
Wang
,
G.
,
Cai
,
L.
, and
Qiu
,
J.
,
2006
, “
Low Frequency Torsional Vibration Gaps in the Shaft With Locally Resonant Structures
,”
Phys. Lett. A
,
348
(
3–6
), pp.
410
415
.
35.
Ma
,
G.
, and
Sheng
,
P.
,
2016
, “
Acoustic Metamaterials: From Local Resonances to Broad Horizons
,”
Sci. Adv.
,
2
(
2
), p.
e1501595
.
36.
Yu
,
D.
,
Liu
,
Y.
,
Wang
,
G.
,
Zhao
,
H.
, and
Qiu
,
J.
,
2006
, “
Flexural Vibration Band Gaps in Timoshenko Beams With Locally Resonant Structures
,”
J. Appl. Phys.
,
100
(
12
), p.
124901
.
37.
Nouh
,
M. A.
,
Aldraihem
,
O. J.
, and
Baz
,
A.
,
2016
, “
Periodic Metamaterial Plates With Smart Tunable Local Resonators
,”
J. Intell. Mater. Syst. Struct.
,
27
(
13
), pp.
1829
1845
.
38.
Krödel
,
S.
,
Thomé
,
N.
, and
Daraio
,
C.
,
2015
, “
Wide Band-Gap Seismic Metastructures
,”
Extreme Mech. Lett.
,
4
, pp.
111
117
.
39.
Wu
,
Y.
,
Lai
,
Y.
, and
Zhang
,
Z.-Q.
,
2011
, “
Elastic Metamaterials With Simultaneously Negative Effective Shear Modulus and Mass Density
,”
Phys. Rev. Lett.
,
107
(
10
), p. 124901.
40.
Liu
,
X. N.
,
Hu
,
G. K.
,
Sun
,
C. T.
, and
Huang
,
G. L.
,
2011
, “
Wave Propagation Characterization and Design of Two-Dimensional Elastic Chiral Metacomposite
,”
J. Sound Vib.
,
330
(
11
), pp.
2536
2553
.
41.
Al Ba’ba’a
,
H.
,
Callanan
,
J.
, and
Nouh
,
M.
,
2019
, “
Emergence of Pseudo-Phononic Gaps in Periodically Architected Pendulums
,”
Front. Mater.
,
6
, p.
119
.
You do not currently have access to this content.