Study of band gaps of two-dimensional phononic crystals with criss-crossed elliptical holes
Nan GAO1(),Jian LI1,Rong-hao BAO1,2,3,*(),Wei-qiu CHEN1,2,3
1. Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China 2. Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou 310027, China 3. Soft Matter Research Center, Zhejiang University, Hangzhou 310027, China
The band gaps of elastic wave in two-dimensional phononic crystals with criss-crossed elliptical holes and the tunability of band gaps induced by the artificially introduced inclusions were analyzed through finite element method and experimental method. When the in-plane excitation is uniformly applied along the thickness of a thin phononic crystal plate, the dynamic response of the plate can be approximated as a plane-stress problem. Thin phononic plates with elliptical holes arrayed in a criss-cross pattern were analyzed with a two-dimensional finite element model based on the contrarian thinking in order to calculate the wave transmission along the ΓX direction of the reciprocal lattice. The effects of the ratio of major axis to minor axis, the porosity of holes, and the artificially introduced line defects on the band gaps and the wave propagation characteristics were systematically analyzed. The testing samples with the same dimensions as the numerical models were produced to conduct the transmission experiments. The numerical results accorded well with the experimental results. Results show that the width of band gap enlarges with the increasing of the ratio of major axis to minor axis, and the band gap can be reversely controlled by the insertion of line defects.
Nan GAO,Jian LI,Rong-hao BAO,Wei-qiu CHEN. Study of band gaps of two-dimensional phononic crystals with criss-crossed elliptical holes. Journal of ZheJiang University (Engineering Science), 2019, 53(4): 811-818.
Fig.1Thin plate with criss-crossed elliptical holes
Fig.2First Brillouin zone and irreducible Brillouin zone of squared lattice
材料
ρ/（g·cm?3）
E/GPa
μ
亚克力（PMMA）
1.18
2.7
0.4
Tab.1Mechanical properties of testing sample
Fig.3Band structure of phononic plate with criss-crossed elliptical holes $(\phi = 50{\text{%}}, \kappa = 3.0)$
Fig.4Locations of incentive point and receiving point
Fig.5Comparison of band gap with transmission spectrum along ΓX direction
Fig.6PMMA sample with criss-crossed elliptical holes
Fig.7Schema of wave propagation test system
Fig.8Wave propagation test setup
Fig.9Transmission spectrum along ΓX direction by numerical method and experiment for phononic crystals with circular holes $(\phi = 50{\text{%}} ,\;\kappa = 1.0)$
Fig.10Effect of ratio of major axis to minor axis on band gap along ΓX direction
Fig.11Effect of porosity on band gap along ΓX direction
Fig.12Wave modes at lower edge of band gap with different ratios of major axis to minor axis when $\phi =50{\text{%}}$
Fig.13Mass-ligament models
Fig.14Contour plots of displacement for phononic crystal plate
Fig.15Two kinds of line defect
Fig.16Effects of line defects on transmission
Fig.17Effects of defects on wave propagation under 10.1 kHz
[1]
SIGALAS M M, ECONOMOU E N Elastic and acoustic wave band structure[J]. Journal of Sound and Vibration, 1992, 158 (2): 377- 382
doi: 10.1016/0022-460X(92)90059-7
[2]
KUSHWAHA M S, HALEVI P, MARTíNEZ G, et al Theory of acoustic band structure of periodic elastic composites[J]. Physical Review B, 1994, 49 (4): 2313- 2322
doi: 10.1103/PhysRevB.49.2313
[3]
MOHAMMADI S, EFTEKHAR A A, KHELIF A, et al Complete phononic bandgaps and bandgap maps in two-dimensional silicon phononic crystal plates[J]. Electronics Letters, 2007, 43 (16): 898- 899
doi: 10.1049/el:20071159
[4]
REINKE C M, SU M F, OLSSON R H, et al Realization of optimal bandgaps in solid-solid, solid-air, and hybrid solid-air-solid phononic crystal slabs[J]. Applied Physics Letters, 2011, 98 (6): 055405
[5]
WANG Y F, WANG Y S, SU X X Large bandgaps of two-dimensional phononic crystals with cross-like holes[J]. Journal of Applied Physics, 2011, 110 (11): 2059
[6]
DOWLING J P Sonic band structure in fluids with periodic density variations[J]. Journal of the Acoustical Society of America, 1992, 91 (5): 2539- 2543
doi: 10.1121/1.402990
[7]
SIGALAS M M, SOUKOULIS C M Elastic-wave propagation through disordered and/or absorptive layered systems[J]. Physical Review B, 1995, 51 (5): 2780
doi: 10.1103/PhysRevB.51.2780
[8]
AO X, CHAN C T Complex band structures and effective medium descriptions of periodic acoustic composite systems[J]. Physical Review B, 2009, 80 (23): 308- 310
[9]
LIU Z, CHAN C T, SHENG P, et al Elastic wave scattering by periodic structures of spherical objects: theory and experiment[J]. Physical Review B, 2000, 62 (4): 2446- 2457
doi: 10.1103/PhysRevB.62.2446
[10]
GARCIA-PABLOS D, SIGALAS M, FR M D E, et al Theory and experiments on elastic band gaps[J]. Physical Review Letters, 2000, 84 (19): 4349
doi: 10.1103/PhysRevLett.84.4349
[11]
SIGALAS M M, GARCíA N Theoretical study of three dimensional elastic band gaps with the finite-difference time-domain method[J]. Journal of Applied Physics, 2000, 87 (6): 3122- 3125
doi: 10.1063/1.372308
[12]
MEAD D J A general theory of harmonic wave propagation in linear periodic systems with multiple coupling[J]. Journal of Sound and Vibration, 1973, 27 (2): 235- 260
doi: 10.1016/0022-460X(73)90064-3
[13]
LANGLET P, HLADKY‐HENNION A, DECARPIGNY J Analysis of the propagation of plane acoustic waves in passive periodic materials using the finite element method[J]. Journal of the Acoustical Society of America, 1995, 98 (5): 2792- 2800
doi: 10.1121/1.413244
[14]
黄屹澜, 高楠, 鲍荣浩, 等 周期结构后屈曲新算法及其应用[J]. 计算力学学报, 2016, 33 (4): 509- 515 HUANG Yi-lan, GAO Nan, BAO Rong-hao, et al A novel algorithm for post-buckling analysis of periodic structures and its application[J]. Chinese Journal of Computational Mechanics, 2016, 33 (4): 509- 515
[15]
MARTíNEZSALA R, SANCHO J, SáNCHEZ J V, et al Sound attenuation by sculpture[J]. Nature, 1995, 378 (6554): 241
[16]
YANG C L, ZHAO S D, WANG Y S Experimental evidence of large complete bandgaps in zig-zag lattice structures[J]. Ultrasonics, 2017, 74: 99- 105
doi: 10.1016/j.ultras.2016.10.004
[17]
李建宝. 声子晶体带隙调控的数值与实验研究[D]. 北京: 北京交通大学, 2011. LI Jian-bao. Numerical and experimental investigation on band gap engineering of phononic crystals [D]. Beijing: Beijing Jiaotong University, 2011.
[18]
BERTOLDI K, BOYCE M C Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures[J]. Physical Review B, 2008, 77 (5): 439- 446
[19]
LANDAU L D, LIFSHITZ E M. Theory of elasticity [M]. Berlin: Springer, 1959.
