In this paper, the bandgap properties of three-dimensional holey phononic crystals with resonators are investigated by using the finite element method. The resonators are periodically arranged cubic lumps in the cubic holes connected to the matrix by narrow connectors. The influence of the geometry parameters of the resonators on the bandgap is discussed. In contrast to a system with cubic or spherical holes, which has no bandgaps, systems with resonators can exhibit complete bandgaps. The bandgaps are significantly dependent upon the geometry of the resonators. By the careful design of the shape and size of the resonator, a bandgap that is lower by an order of magnitude than the Bragg bandgap can be obtained. The vibration modes at the band edges of the lowest bandgaps are analyzed in order to understand the mechanism of the bandgap generation. It is found that the emergence of the bandgap is due to the local resonance of the resonators. Spring-mass models or spring-pendulum models are developed in order to evaluate the frequencies of the bandgap edges. The study in this paper is relevant to the optimal design of the bandgaps in light porous materials.

References

References
1.
Kushwaha
,
M. S.
,
Halevi
,
P.
,
Dobrzynski
,
L.
, and
Djafari-Rouhani
,
B.
,
1993
, “
Acoustic Band Structure of Periodic Elastic Composites
,”
Phys. Rev. Lett.
,
71
, pp.
2022
2025
.10.1103/PhysRevLett.71.2022
2.
Chen
,
C. Q.
,
Cui
,
J. Z.
,
Duan
,
H. L.
,
Feng
,
X. Q.
,
He
,
L. H.
,
Hu
,
G. K.
,
Huang
,
M. J.
,
Huo
,
Y. Z.
,
Ji
,
B. H.
,
Liu
,
B.
,
Peng
,
X. H.
,
Shi
,
H. J.
,
Sun
,
Q. P.
,
Wang
,
J. X.
,
Wang
,
Y. S.
,
Zhao
,
H. P.
,
Zhao
,
Y. P.
,
Zheng
,
Q. S.
, and
Zou
,
W. N.
,
2011
, “
Perspectives in Mechanics of Heterogeneous Solids
,”
Acta Mech. Solida Sin.
,
24
(1)
, pp.
1
26
.10.1016/S0894-9166(11)60007-4
3.
Min
,
R.
,
Wu
,
F. G.
,
Zhong
,
L. H.
,
Zhong
,
H. L.
,
Zhong
,
S.
, and
Liu
,
Y. Y.
,
2006
, “
Extreme Acoustic Band Gaps Obtained Under High Symmetry in 2D Phononic Crystals
,”
J. Phys. D: Appl. Phys.
,
39
, pp.
2272
2276
.10.1088/0022-3727/39/10/041
4.
Zhou
,
X. Z.
,
Wang
,
Y. S.
, and
Zhang
,
C. H.
,
2009
, “
Effects of Material Parameters on Elastic Band Gaps of Two-Dimensional Solid Phononic Crystals
,”
J. Appl. Phys.
,
106
, p.
014903
.10.1063/1.3159644
5.
Zhan
,
Z. Q.
, and
Wei
,
P. J.
,
2010
, “
Influence of Anisotropy on Band Gaps of 2D Phononic Crystals
,”
Acta Mech. Solida Sin.
,
23
(2)
, pp.
181
188
.10.1016/S0894-9166(10)60020-1
6.
Su
,
X. X.
,
Wang
,
Y. F.
, and
Wang
,
Y. S.
,
2012
, “
Effects of Poisson's Ratio on the Band Gaps and Defect States in Two-Dimensional Vacuum/Solid Porous Phononic Crystals
,”
Ultrasonics
,
52
, pp.
255
265
.10.1016/j.ultras.2011.08.010
7.
Kushwaha
,
M. S.
, and
Djafari-Rouhani
,
B.
,
1996
, “
Complete Acoustic Stop Bands for Cubic Arrays of Spherical Liquid Balloons
,”
J. Appl. Phys.
,
80
, pp.
3191
3195
.10.1063/1.363259
8.
Kushwaha
,
M. S.
, and
Halevi
,
P.
,
1997
, “
Stop Bands for Cubic Arrays of Spherical Balloons
,”
J. Acoust. Soc. Am.
,
101
, pp.
619
622
.10.1121/1.417964
9.
Zhou
,
X. Z.
,
Wang
,
Y. S.
, and
Zhang
,
C. Z.
,
2010
, “
Three-Dimensional Sonic Band Gaps Tuned by Material Parameters
,”
Appl. Mech. Mater.
,
29–32
, pp.
1797
1802
.10.4028/www.scientific.net/AMM.29-32.1797
10.
Goffaux
,
C.
, and
Vigneron
,
J. P.
,
2001
, “
Spatial Trapping of Acoustic Waves in Bubbly Liquids
,”
Phys. B
,
296
, pp.
195
200
.10.1016/S0921-4526(00)00800-0
11.
Zhang
,
X.
,
Liu
,
Z. Y.
,
Liu
,
Y. Y.
, and
Wu
,
F. G.
,
2003
, “
Elastic Wave Band Gaps for Three-Dimensional Phononic Crystals With Two Structural Units
,”
Phys. Lett. A
,
313
, pp.
455
460
.10.1016/S0375-9601(03)00807-7
12.
Kuang
,
W. M.
,
Hou
,
Z. L.
,
Liu
,
Y. Y.
, and
Li
,
H.
,
2006
, “
The Band Gaps of Cubic Phononic Crystals With Different Shapes of Scatterers
,”
J. Phys. D: Appl. Phys.
,
39
, pp.
2067
2071
.10.1088/0022-3727/39/10/014
13.
Khelif
,
A.
,
Hsiao
,
F. L.
,
Choujaa
,
A.
,
Benchabane
,
S.
, and
Laude
,
V.
,
2010
, “
Octave Omnidirectional Band Gap in a Three-Dimensional Phononic Crystals
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
,
57
, pp.
1621
1625
.10.1109/TUFFC.2010.1592
14.
Sainidou
,
R.
,
Stefanou
,
N.
, and
Modinos
,
A.
,
2002
, “
Formation of Absolute Frequency Gaps in Three-Dimensional Solid Phononic Crystals
,”
Phys. Rev. B
,
66
, p.
212301
.10.1103/PhysRevB.66.212301
15.
Tanaka
,
Y.
,
Tomoyasu
,
Y.
, and
Tamura
,
S.
,
2000
, “
Band Structure of Acoustic Waves in Phononic Lattice: Two-Dimensional Composites With Large Acoustic Mismatch
,”
Phys. Rev. B
,
62
, pp.
7387
7392
.10.1103/PhysRevB.62.7387
16.
Wu
,
T. T.
,
Huang
,
Z. G.
, and
Liu
,
S. Y.
,
2005
, “
Surface Acoustic Wave Band Gaps in Micro-Machined Air/Silicon Phononic Structures—Theoretical Calculation and Experiment
,”
Z. Kristallogr.
,
220
, pp.
841
847
.10.1524/zkri.2005.220.9-10.841
17.
Wang
,
Y. F.
,
Wang
,
Y. S.
, and
Su
,
X. X.
,
2011
, “
Large Bandgaps of Two-Dimensional Phononic Crystals With Cross-Like Holes
,”
J. Appl. Phys.
,
110
, p.
113520
.10.1063/1.3665205
18.
Liu
,
Z. Y.
,
Zhang
,
X. X.
,
Mao
,
Y. W.
,
Zhu
,
Y. Y.
,
Yang
,
Z. Y.
,
Chan
,
C. T.
, and
Sheng
,
P.
,
2000
, “
Locally Resonant Sonic Materials
,”
Science
,
289
, pp.
1734
1736
.10.1126/science.289.5485.1734
19.
Liu
,
Z. Y.
,
Chan
,
C. T.
, and
Sheng
,
P.
,
2002
, “
Three-Component Elastic Wave Band-Gap Material
,”
Phys. Rev. B
,
65
, p.
165116
.10.1103/PhysRevB.65.165116
20.
Yan
,
Z. Z.
, and
Wang
,
Y. S.
,
2006
, “
Wavelet-Based Method for Calculating Elastic Band Gaps of Two-Dimensional Phononic Crystals
,”
Phys. Rev. B
,
74
, p.
224303
.10.1103/PhysRevB.74.224303
21.
Yan
,
Z. Z.
, and
Wang
,
Y. S.
,
2008
, “
Wavelet Method for Calculating the Defect State of Two-Dimensional Phononic Crystals
,”
Acta Mech. Solida Sinica
,
21
(2)
, pp.
104
109
.10.1007/s10338-008-0813-6
22.
Sigalas
,
M. M.
, and
García
,
N.
,
2000
, “
Theoretical Study of Three Dimensional Elastic Band Gaps With the Finite-Difference Time-Domain Method
,”
J. Appl. Phys.
,
87
, pp.
3122
3125
.10.1063/1.372308
23.
Chandra
,
H.
,
Deymier
,
P. A.
, and
Vasseur
,
J. O.
,
2004
, “
Elastic Wave Propagation Along Waveguides in Three-Dimensional Phononic Crystals
,”
Phys. Rev. B
,
70
, p.
054302
.10.1103/PhysRevB.70.054302
24.
Yang
,
S. X.
,
Page
,
J. H.
,
Liu
,
Z. Y.
,
Cowan
,
M. L.
,
Chan
,
C. T.
, and
Sheng
,
P.
,
2004
, “
Focusing of Sound in a 3D Phononic Crystals
,”
Phys. Rev. Lett.
,
93
, p.
024301
.10.1103/PhysRevLett.93.024301
25.
Papanikolaou
,
N.
,
Psarobas
,
I. E.
, and
Stefanou
,
N.
,
2010
, “
Absolute Spectral Gaps for Infrared Light and Hypersound in Three-Dimensional Metallodielectric Phoxonic Crystals
,”
Appl. Phys. Lett.
,
96
, p.
231917
.10.1063/1.3453448
26.
Wang
,
G.
,
Wen
,
X. S.
,
Wen
,
J. H.
,
Shao
,
L. H.
, and
Liu
,
Y. Z.
,
2004
, “
Two-Dimensional Locally Resonant Phononic Crystals With Binary Structures
,”
Phys. Rev. Lett.
,
93
, p.
154302
.10.1103/PhysRevLett.93.154302
27.
Wang
,
G.
,
Shao
,
L. H.
,
Liu
,
Y. Z.
, and
Wen
,
J. H.
,
2006
, “
Accurate Evaluation of Lowest Band Gaps in Ternary Locally Resonant Phononic Crystals
,”
Chin. Phys.
,
15
, pp.
1843
1848
.10.1088/1009-1963/15/8/036
28.
Li
,
J. B.
,
Wang
,
Y. S.
, and
Zhang
C. Z.
,
2012
, “
Dispersion Relations of a Periodic Array of Fluid-Filled Holes Embedded in an Elastic Solid
,”
J. Comput. Acoust.
,
20
(
4
) p.
1250014
.10.1142/S0218396X12500142
29.
Wang
X. C.
,
2003
,
Finite Element Method
,
Tsinghua University Press
,
Beijing
(in Chinese).
30.
Mohammadi
,
S.
,
Eftekhar
,
A. A.
,
Khelif
,
A.
,
Moubchir
,
H.
,
Westafer
,
R.
,
Hunt
,
W. D.
, and
Adibi
A.
,
2007
, “
Complete Phononic Bandgaps and Bandgap Maps in Two-Dimensional Silicon Phononic Crystal Plates
,”
Electron. Lett.
,
43
, pp.
898
899
.10.1049/el:20071159
31.
Timoshenko
,
S.
, and
Woinowsky-Krieger
,
S.
,
1959
,
Theory of Plates and Shells
,
2nd ed.
,
McGraw-Hill Inc
.,
Singapore
.
32.
Dreizler
,
R. M.
, and
Lüdde
,
C. S.
,
2011
,
Theoretical Mechanics: Theoretical Physics 1
,
1st ed.
,
Springer
,
Berlin
.
33.
Thomson
,
W. T.
,
1972
,
Theory of Vibration With Applications
,
Prentice-Hall, Inc.
,
Upper Saddle River, NJ
.
34.
Bao
S. H.
, and
Gong
Y. Q.
,
2006
,
Structural Mechanics
,
Wuhan University of Technology Press, Wuhan, PRC
.
35.
Timoshenko
,
S. P.
, and
Goodier
J. N.
,
1970
,
Theory of Elasticity
,
3rd ed.
,
McGraw-Hill Inc
.,
New York
.
You do not currently have access to this content.