Abstract

A modulated metamaterial that exhibits both time-periodic stiffness and mass simultaneously is presented. The metamaterial element includes a primary body that undergoes infinitesimal motion, and is connected to a dynamic-mechanism structure, involving a rotational body, and spring with a large-scale motion, which is designed to produce a time-modulated linear momentum and elastic constraint for the primary body. The non-reciprocal wave propagation is then investigated in a space–time lattice metamaterial that is constructed by coupling doubly time-modulated elements with linear springs of constant stiffness. The dispersion property shows the frequency degeneracy occurring at the center or edge of the Brillouin zone, and the unidirectional bandgap at certain frequencies. This phenomenon represents a unique property of the doubly modulated metamaterials compared to the singly modulated ones, thus may provide more promising applications to the design of non-reciprocal devices.

References

References
1.
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
. 10.1126/science.289.5485.1734
2.
Christensen
,
J.
,
Kadic
,
M.
,
Kraft
,
O.
, and
Wegener
,
M.
,
2015
, “
Vibrant Times for Mechanical Metamaterials
,”
MRS Commun.
,
5
(
3
), pp.
453
462
. 10.1557/mrc.2015.51
3.
Ma
,
G.
, and
Sheng
,
P.
,
2016
, “
Acoustic Metamaterials: From Local Resonances to Broad Horizons
,”
Sci. Adv.
,
2
(
2
), p.
e1501595
. 10.1126/sciadv.1501595
4.
Cummer
,
S. A.
,
Christensen
,
J.
, and
Alù
,
A.
,
2016
, “
Controlling Sound with Acoustic Metamaterials
,”
Nat. Rev. Mater.
,
1
(
3
), p.
16001
. 10.1038/natrevmats.2016.1
5.
Fleury
,
R.
,
Sounas
,
D. L.
,
Haberman
,
M. R.
, and
Alù
,
A.
,
2015
, “
Nonreciprocal Acoustics
,”
Acoust. Today
,
11
(
3
), pp.
14
21
.
6.
Fink
,
M.
,
Cassereau
,
D.
,
Derode
,
A.
,
Prada
,
C.
,
Roux
,
P.
,
Tanter
,
M.
,
Thomas
,
J.-L.
, and
Wu
,
F.
,
2000
, “
Time-reversed Acoustics
,”
Rep. Prog. Phys.
,
63
(
12
), pp.
1933
1995
. 10.1088/0034-4885/63/12/202
7.
Zhu
,
X.
,
Ramezani
,
H.
,
Shi
,
C.
,
Zhu
,
J.
, and
Zhang
,
X.
,
2014
, “
PT-Symmetric Acoustics
,”
Phys. Rev. X
,
4
(
3
), p.
031042
. 10.1103/PhysRevX.4.031042
8.
Popa
,
B.-I.
, and
Cummer
,
S. A.
,
2014
, “
Non-reciprocal and Highly Nonlinear Active Acoustic Metamaterials
,”
Nat. Commun.
,
5
(
1
), p.
3398
. 10.1038/ncomms4398
9.
Liang
,
B.
,
Guo
,
X. S.
,
Tu
,
J.
,
Zhang
,
D.
, and
Cheng
,
J. C.
,
2010
, “
An Acoustic Rectifier
,”
Nat. Mater.
,
9
(
12
), pp.
989
992
. 10.1038/nmat2881
10.
Gu
,
Z.-m.
,
Hu
,
J.
,
Liang
,
B.
,
Zou
,
X.-y.
, and
Cheng
,
J.-c.
,
2016
, “
Broadband Non-Reciprocal Transmission of Sound with Invariant Frequency
,”
Sci. Rep.
,
6
(
1
), p.
19824
. 10.1038/srep19824
11.
Fleury
,
R.
,
Sounas
,
D. L.
,
Sieck
,
C. F.
,
Haberman
,
M. R.
, and
Alù
,
A.
,
2014
, “
Sound Isolation and Giant Linear Nonreciprocity in a Compact Acoustic Circulator
,”
Science
,
343
(
6170
), pp.
516
519
. 10.1126/science.1246957
12.
Baz
,
A.
,
2018
, “
Active Nonreciprocal Acoustic Metamaterials Using a Switching Controller
,”
J. Acoust. Soc. Am.
,
143
(
3
), pp.
1376
1384
. 10.1121/1.5026510
13.
Mojahed
,
A.
,
Bunyan
,
J.
,
Tawfick
,
S.
, and
Vakakis
,
A. F.
,
2019
, “
Tunable Acoustic Nonreciprocity in Strongly Nonlinear Waveguides with Asymmetry
,”
Phys. Rev. Appl.
,
12
(
3
), p.
034033
. 10.1103/PhysRevApplied.12.034033
14.
Leamy
,
M. J.
,
Darabi
,
A.
,
Fang
,
L.
,
Fronk
,
M.
, and
Vakakis
,
A.
,
2019
, “
Passive Non-reciprocity in Asymmetrical, Hierarchical, Nonlinear Metamaterials
,”
J. Acoust. Soc. Am.
,
145
(
3
), pp.
1725
1725
. 10.1121/1.5101338
15.
Trainiti
,
G.
, and
Ruzzene
,
M.
,
2016
, “
Non-reciprocal Elastic Wave Propagation in Spatiotemporal Periodic Structures
,”
New J. Phys.
,
18
(
8
), p.
083047
. 10.1088/1367-2630/18/8/083047
16.
Nassar
,
H.
,
Xu
,
X. C.
,
Norris
,
A. N.
, and
Huang
,
G. L.
,
2017
, “
Modulated Phononic Crystals: Non-reciprocal Wave Propagation and Willis Materials
,”
J. Mech. Phys. Solids
,
101
, pp.
10
29
. 10.1016/j.jmps.2017.01.010
17.
Wallen
,
S. P.
, and
Haberman
,
M. R.
,
2019
, “
Nonreciprocal Wave Phenomena in Spring-Mass Chains with Effective Stiffness Modulation Induced by Geometric Nonlinearity
,”
Phys. Rev. E
,
99
(
1
), p.
013001
. 10.1103/PhysRevE.99.013001
18.
Yi
,
K.
,
Collet
,
M.
, and
Karkar
,
S.
,
2017
, “
Frequency Conversion Induced by Time-Space Modulated Media
,”
Phys. Rev. B
,
96
(
10
), p.
104110
. 10.1103/PhysRevB.96.104110
19.
Ansari
,
M. H.
,
Attarzadeh
,
M. A.
,
Nouh
,
M.
, and
Karami
,
M. A.
,
2018
, “
Application of Magnetoelastic Materials in Spatiotemporally Modulated Phononic Crystals for Nonreciprocal Wave Propagation
,”
Smart Mater. Struct.
,
27
(
1
), p.
015030
. 10.1088/1361-665X/aa9d3d
20.
Nassar
,
H.
,
Chen
,
H.
,
Norris
,
A. N.
, and
Huang
,
G. L.
,
2017
, “
Non-reciprocal Flexural Wave Propagation in a Modulated Metabeam
,”
Extreme Mech. Lett.
,
15
, pp.
97
102
. 10.1016/j.eml.2017.07.001
21.
Yi
,
K.
,
Ouisse
,
M.
,
Sadoulet-Reboul
,
E.
, and
Matten
,
G.
,
2019
, “
Active Metamaterials with Broadband Controllable Stiffness for Tunable Band Gaps and non-Reciprocal Wave Propagation
,”
Smart Mater. Struct.
,
28
(
6
), p.
065025
. 10.1088/1361-665X/ab19dc
22.
Shui
,
L.-Q.
,
Yue
,
Z.-F.
,
Liu
,
Y.-S.
,
Liu
,
Q.-C.
,
Guo
,
J.-J.
, and
He
,
X.-D.
,
2015
, “
Novel Composites with Asymmetrical Elastic Wave Properties
,”
Compos. Sci. Technol.
,
113
(
Supplement C
), pp.
19
30
. 10.1016/j.compscitech.2015.03.007
23.
Nanda
,
A.
, and
Karami
,
M. A.
,
2018
, “
One-way Sound Propagation via Spatio-Temporal Modulation of Magnetorheological Fluid
,”
J. Acoust. Soc. Am.
,
144
(
1
), pp.
412
420
. 10.1121/1.5048181
24.
Yi
,
K.
,
Collet
,
M.
, and
Karkar
,
S.
,
2018
, “
Reflection and Transmission of Waves Incident on Time-Space Modulated Media
,”
Phys. Rev. B
,
98
(
5
), p.
054109
. 10.1103/PhysRevB.98.054109
25.
Yi
,
K.
,
Karkar
,
S.
, and
Collet
,
M.
,
2018
, “
One-Way Energy Insulation Using Time-Space Modulated Structures
,”
J. Sound Vib.
,
429
, pp.
162
175
. 10.1016/j.jsv.2018.05.017
26.
Lurie
,
K. A.
,
2007
,
An Introduction to the Mathematical Theory of Dynamic Materials
,
Springer
,
New York, NY
.
27.
Cassedy
,
E. S.
, and
Oliner
,
A. A.
,
1963
, “
Dispersion Relations in Time-Space Periodic Media: Part I-Stable Interactions
,”
Proc. IEEE
,
51
(
10
), pp.
1342
1359
. 10.1109/PROC.1963.2566
28.
Cassedy
,
E. S.
,
1967
, “
Dispersion Relations in Time-Space Periodic Media: Part II-Unstable Interactions
,”
Proc. IEEE
,
55
(
7
), pp.
1154
1168
. 10.1109/PROC.1967.5775
29.
Attarzadeh
,
M. A.
, and
Nouh
,
M.
,
2018
, “
Non-reciprocal Elastic Wave Propagation in 2D Phononic Membranes with Spatiotemporally Varying Material Properties
,”
J. Sound Vib.
,
422
, pp.
264
277
. 10.1016/j.jsv.2018.02.028
30.
Swinteck
,
N.
,
Matsuo
,
S.
,
Runge
,
K.
,
Vasseur
,
J. O.
,
Lucas
,
P.
, and
Deymier
,
P. A.
,
2015
, “
Bulk Elastic Waves with Unidirectional Backscattering-Immune Topological States in a Time-Dependent Superlattice
,”
J. Appl. Phys.
,
118
(
6
), p.
063103
. 10.1063/1.4928619
31.
Vila
,
J.
,
Pal
,
R. K.
,
Ruzzene
,
M.
, and
Trainiti
,
G.
,
2017
, “
A Bloch-Based Procedure for Dispersion Analysis of Lattices with Periodic Time-Varying Properties
,”
J. Sound Vib.
,
406
(
Supplement C
), pp.
363
377
. 10.1016/j.jsv.2017.06.011
32.
Nassar
,
H.
,
Chen
,
H.
,
Norris
,
A. N.
,
Haberman
,
M. R.
, and
Huang
,
G. L.
,
2017
, “
Non-reciprocal Wave Propagation in Modulated Elastic Metamaterials
,”
Proc. R. Soc. A: Math. Phys. Eng. Sci.
,
473
(
2202
), p.
20170188
. 10.1098/rspa.2017.0188
33.
Milton
,
G. W.
, and
Mattei
,
O.
,
2017
, “
Field Patterns: a new Mathematical Object
,”
Proc. R. Soc. A: Math. Phys. Eng. Sci.
,
473
(
2198
), p.
20160819
. 10.1098/rspa.2016.0819
34.
Riva
,
E.
,
Marconi
,
J.
,
Cazzulani
,
G.
, and
Braghin
,
F.
,
2019
, “
Generalized Plane Wave Expansion Method for non-Reciprocal Discretely Modulated Waveguides
,”
J. Sound Vib.
,
449
, pp.
172
181
. 10.1016/j.jsv.2019.03.001
35.
Wang
,
Y.
,
Yousefzadeh
,
B.
,
Chen
,
H.
,
Nassar
,
H.
,
Huang
,
G.
, and
Daraio
,
C.
,
2018
, “
Observation of Nonreciprocal Wave Propagation in a Dynamic Phononic Lattice
,”
Phys. Rev. Lett.
,
121
(
19
), p.
194301
. 10.1103/PhysRevLett.121.194301
36.
Chen
,
Y.
,
Li
,
X.
,
Nassar
,
H.
,
Norris
,
A. N.
,
Daraio
,
C.
, and
Huang
,
G.
,
2019
, “
Nonreciprocal Wave Propagation in a Continuum-Based Metamaterial with Space-Time Modulated Resonators
,”
Phys. Rev. Appl.
,
11
(
6
), p.
064052
. 10.1103/PhysRevApplied.11.064052
37.
Trainiti
,
G.
,
Xia
,
Y.
,
Marconi
,
J.
,
Cazzulani
,
G.
,
Erturk
,
A.
, and
Ruzzene
,
M.
,
2019
, “
Time-Periodic Stiffness Modulation in Elastic Metamaterials for Selective Wave Filtering: Theory and Experiment
,”
Phys. Rev. Lett.
,
122
(
12
), p.
124301
. 10.1103/PhysRevLett.122.124301
38.
Attarzadeh
,
M. A.
,
Callanan
,
J.
, and
Nouh
,
M.
,
2020
, “
Experimental Observation of Nonreciprocal Waves in a Resonant Metamaterial Beam
,”
Phys. Rev. Appl.
,
13
(
2
), p.
021001
. 10.1103/PhysRevApplied.13.021001
39.
Reed
,
E. J.
,
Soljacic
,
M.
, and
Joannopoulos
,
J. D.
,
2003
, “
Reversed Doppler Effect in Photonic Crystals
,”
Phys. Rev. Lett.
,
91
(
13
), p.
133901
. 10.1103/PhysRevLett.91.133901
40.
Danas
,
K.
,
Kankanala
,
S. V.
, and
Triantafyllidis
,
N.
,
2012
, “
Experiments and Modeling of Iron-Particle-Filled Magnetorheological Elastomers
,”
J. Mech. Phys. Solids
,
60
(
1
), pp.
120
138
. 10.1016/j.jmps.2011.09.006
41.
Gump
,
J.
,
Finkler
,
I.
,
Xia
,
H.
,
Sooryakumar
,
R.
,
Bresser
,
W. J.
, and
Boolchand
,
P.
,
2004
, “
Light-Induced Giant Softening of Network Glasses Observed Near the Mean-Field Rigidity Transition
,”
Phys. Rev. Lett.
,
92
(
24
), p.
245501
. 10.1103/PhysRevLett.92.245501
42.
Huang
,
J.
, and
Zhou
,
X.
,
2019
, “
A Time-Varying Mass Metamaterial for non-Reciprocal Wave Propagation
,”
Int. J. Solids Struct.
,
164
, pp.
25
36
. 10.1016/j.ijsolstr.2018.12.029
43.
Attarzadeh
,
M. A.
,
Maleki
,
S.
,
Crassidis
,
J. L.
, and
Nouh
,
M.
,
2019
, “
Non-reciprocal Wave Phenomena in Energy Self-Reliant Gyric Metamaterials
,”
J. Acoust. Soc. Am.
,
146
(
1
), pp.
789
801
. 10.1121/1.5114916
44.
Yao
,
S.
,
Zhou
,
X.
, and
Hu
,
G.
,
2010
, “
Investigation of the Negative-Mass Behaviors Occurring Below a cut-off Frequency
,”
New J. Phys.
,
12
(
10
), p.
103025
. 10.1088/1367-2630/12/10/103025
You do not currently have access to this content.