Review on the dynamics and wave control in nonlinear periodic structures

GAO Penglin; GONG Lingyun; WANG Guoxu; LUO Yu; ZHU Junzhe; GAO Hao; MA Haibin; QU Yegao

doi:10.6052/1000-0992-24-047

Article Contents

Article Navigation > Advances in Mechanics > 2014 > Accepted Manuscript

Gao P L, Gong L Y, Wang G X, Luo Y, Zhu J Z, Gao H, Ma H B, Qu Y G. Review on the dynamics and wave control in nonlinear periodic structures. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-047

Citation:

Gao P L, Gong L Y, Wang G X, Luo Y, Zhu J Z, Gao H, Ma H B, Qu Y G. Review on the dynamics and wave control in nonlinear periodic structures. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-047

Citation:

PDF( 26632 KB)

Review on the dynamics and wave control in nonlinear periodic structures

doi: 10.6052/1000-0992-24-047 cstr: 32046.14.1000-0992-24-047

GAO Penglin^{1, 2},
GONG Lingyun^{1, 2},
WANG Guoxu^{1, 2},
LUO Yu^{1, 2},
ZHU Junzhe^{1, 2},
GAO Hao^{1, 2},
MA Haibin^{1, 2},
QU Yegao^{1, 2
,
,}

1.
State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 20024, China
2.
Institute of Vibration, Shock and Noise, Shanghai Jiao Tong University, Shanghai 200240, China

More Information

Corresponding author: quyegao@sjtu.edu.cn
Received Date: 2014-01-02
Accepted Date: 2014-03-04

Available Online: 2014-05-06

Abstract

Abstract

Periodic structures are an important form of constructing structures in natural and human engineering. The Bloch modulation caused by translational symmetry gives them unique band structures and rich time/frequency domain dynamic characteristics, providing new avenues for elastic/acoustic wave propagation regulation, novel wave-based functional device design, vibration and noise control, etc. Nonlinear effects can break through the constraints of linear theoretical frameworks and can enhance or even expand the functions of artificial periodic structures. However, nonlinear periodic systems have many difficulties in unit cell design and modeling analysis. They also face key scientific problems such as the broken of space-time invariance, the complexity of nonlinear response characteristics and mechanisms, which brings challenge to the dynamic design and practical application of nonlinear periodic structures. In response to the above problems, scholars have carried out some fruitful research by integrating multidisciplinary research methods in fileds such as mechanics, acoustics, materials science and band physics. This article aims to timely summarize the important research progress concerning about wave dynamics and control in nonlinear periodic structures, sort out the shortcomings and key problems, gather strength and promote the in-depth development of this field. First, the sources of nonlinear effects of periodic structures, unit cell structure design methods, and nonlinear dynamics modeling and analysis methods are summarized. Then, the main characteristics of nonlinear periodic structures in terms of passband, bandgap, and local energy confinement are reviewed, and the rich dynamic phenomena such as amplitude-induced band shift, wave mode coupling, low-frequency broadband bandgap, and spatial confinement of wave modes in the bandgap caused by nonlinearity are introduced. Some application explorations of nonlinear periodic structures in wave control devices, vibration and noise reduction are sorted out. Finally, in view of some shortcomings and key problems in existing research, several development directions that need special attention in future theoretical research and application exploration are prospected.
- Nonlinear periodic structure,
- dynamic design,
- nonlinear vibration,
- vibration control,
- wave control

FullText(HTML)

References(328)

References

[1]	Akbari-Farahani F, Ebrahimi-Nejad S 2024. From defect mode to topological metamaterials: A state-of-the-art review of phononic crystals & acoustic metamaterials for energy harvesting. Sensors and Actuators A: Physical, 365: 114871.
[2]	Alfahmi O, Erturk A 2024. Programmable hardening and softening cubic inductive shunts for piezoelectric structures: Harmonic balance analysis and experiments. Journal of Sound and Vibration, 571: 118029.
[3]	Alfahmi O, Sugino C, Erturk A 2022. Duffing-type digitally programmable nonlinear synthetic inductance for piezoelectric structures. Smart Materials and Structures, 31: 095044.
[4]	Allein F, Tournat V, Gusev V, et al. 2020. Linear and nonlinear elastic waves in magnetogranular chains. Physical Review Applied, 13: 024023. doi: 10.1103/PhysRevApplied.13.024023
[5]	Allein F, Tournat V, Gusev V E, et al. 2016. Tunable magneto-granular phononic crystals. Applied Physics Letters, 108: 161903. doi: 10.1063/1.4947192
[6]	Alleyne D, Cawley P 1991. A two-dimensional fourier transform method for the measurement of propagating multimode signals. The Journal of the Acoustical Society of America, 89: 1159-1168.
[7]	Alshaqaq M, Sugino C, Erturk A 2023. Digital programming of reciprocity breaking in resonant piezoelectric metamaterials. Physical Review Research, 5: 043003.
[8]	Bae M H, Oh J H 2020. Amplitude-induced bandgap: New type of bandgap for nonlinear elastic metamaterials. Journal of the Mechanics and Physics of Solids, 139: 103930.
[9]	Bae M H, Oh J H 2022. Nonlinear elastic metamaterial for tunable bandgap at quasi-static frequency. Mechanical Systems and Signal Processing, 170: 108832.
[10]	Banerjee A, Adhikari S, Hussein M I 2021. Inertial amplification band-gap generation by coupling a levered mass with a locally resonant mass. International Journal of Mechanical Sciences, 207: 106630.
[11]	Banerjee A, Calius E P, Das R 2018a. An impact based mass-in-mass unit as a building block of wideband nonlinear resonating metamaterial. International Journal of Non-Linear Mechanics, 101: 8-15.
[12]	Banerjee A, Calius E P, Das R 2018b. Impact based wideband nonlinear resonating metamaterial chain. International Journal of Non-Linear Mechanics, 103: 138-144.
[13]	Bao B, Guyomar D, Lallart M 2017. Vibration reduction for smart periodic structures via periodic piezoelectric arrays with nonlinear interleaved-switched electronic networks. Mechanical Systems and Signal Processing, 82: 230-259.
[14]	Bertoldi K, Vitelli V, Christensen J, et al. 2017. Flexible mechanical metamaterials. Nature Reviews Materials, 2: 17066. doi: 10.1038/natrevmats.2017.66
[15]	Bickham S R, Kiselev S A, Sievers A J 1993. Stationary and moving intrinsic localized modes in one-dimensional monatomic lattices with cubic and quartic anharmonicity. Physical Review B, 47: 14206-14211.
[16]	Bilal O R, Foehr A, Daraio C 2017. Bistable metamaterial for switching and cascading elastic vibrations. Proceedings of the National Academy of Sciences, 114: 4603-4606.
[17]	Boechler N, Theocharis G, Daraio C 2011. Bifurcation-based acoustic switching and rectification. Nature materials, 10: 665-668.
[18]	Boechler N, Theocharis G, Job S, et al. 2010. Discrete breathers in one-dimensional diatomic granular crystals. Physical Review Letters, 104: 244302. doi: 10.1103/PhysRevLett.104.244302
[19]	Bordiga G, Medina E, Jafarzadeh S, et al. 2024. Automated discovery of reprogrammable nonlinear dynamic metamaterials. Nature Materials, 23: 1486-1494. doi: 10.1038/s41563-024-02008-6
[20]	Bosia F, Dal Poggetto V F, Gliozzi A S, et al. 2022. Optimized structures for vibration attenuation and sound control in nature: A review. Matter, 5: 3311-3340. doi: 10.1016/j.matt.2022.07.023
[21]	Brooke D C, Umnova O, Leclaire P, et al. 2020. Acoustic metamaterial for low frequency sound absorption in linear and nonlinear regimes. Journal of Sound and Vibration, 485: 115585. doi: 10.1016/j.jsv.2020.115585
[22]	Bukhari M, Barry O 2020. Spectro-spatial analyses of a nonlinear metamaterial with multiple nonlinear local resonators. Nonlinear Dynamics, 99: 1539-1560.
[23]	Buryak A V, Trapani P D, Skryabin D V, et al. 2002. Optical solitons due to quadratic nonlinearities: From basic physics to futuristic applications. Physics Reports, 370: 63-235. doi: 10.1016/S0370-1573(02)00196-5
[24]	Cabaret J, Tournat V, Béquin P 2012. Amplitude-dependent phononic processes in a diatomic granular chain in the weakly nonlinear regime. Physical Review E, 86: 041305.
[25]	Cai C, Guo X, Yan B, et al. 2024. Modelling and analysis of the quasi-zero-stiffness metamaterial cylindrical shell for low-frequency band gap. Applied Mathematical Modelling, 135: 90-108. doi: 10.1016/j.apm.2024.06.031
[26]	Cai C, Zhou J, Wang K, et al. 2022. Metamaterial plate with compliant quasi-zero-stiffness resonators for ultra-low-frequency band gap. Journal of Sound and Vibration, 540: 117297. doi: 10.1016/j.jsv.2022.117297
[27]	Casalotti A, El-Borgi S, Lacarbonara W 2018. Metamaterial beam with embedded nonlinear vibration absorbers. International Journal of Non-Linear Mechanics, 98: 32-42.
[28]	Cha J, Daraio C 2018. Electrical tuning of elastic wave propagation in nanomechanical lattices at mhz frequencies. Nature Nanotechnology, 13: 1016-1020.
[29]	Chakraborty G, Mallik A K 2001. Dynamics of a weakly non-linear periodic chain. International Journal of Non-Linear Mechanics, 36: 375-389.
[30]	Chaunsali R, Theocharis G 2019. Self-induced topological transition in phononic crystals by nonlinearity management. Physical Review B, 100: 014302.
[31]	Chaunsali R, Xu H, Yang J, et al. 2021. Stability of topological edge states under strong nonlinear effects. Physical Review B, 103: 024106. doi: 10.1103/PhysRevB.103.024106
[32]	Chen B, Zheng Y, Dai S, et al. 2024a. Bandgap enhancement of a piezoelectric metamaterial beam shunted with circuits incorporating fractional and cubic nonlinearities. Mechanical Systems and Signal Processing, 212: 111262. doi: 10.1016/j.ymssp.2024.111262
[33]	Chen C, Li X D 2020. Microscopic fluid dynamics of a wire screen bound to a slit resonator excited by acoustic waves. Physics of Fluids, 32: 116107.
[34]	Chen W, Hao Y X, Zhang W, et al. 2024b. Vibration isolation performance of a novel metamaterials sandwich cylindrical panel by locally resonant band gap. Journal of Vibration Engineering & Technologies, 12: 6121-6136.
[35]	Chen Y, Li X, Nassar H, et al. 2019. Nonreciprocal wave propagation in a continuum-based metamaterial with space-time modulated resonators. Physical Review Applied, 11: 064052. doi: 10.1103/PhysRevApplied.11.064052
[36]	Chen Y, Wu B, Su Y, et al. 2020. Effects of strain stiffening and electrostriction on tunable elastic waves in compressible dielectric elastomer laminates. International Journal of Mechanical Sciences, 176: 105572. doi: 10.1016/j.ijmecsci.2020.105572
[37]	Chong C, Kim B, Wallace E, et al. 2024. Modulation instability and wavenumber bandgap breathers in a time layered phononic lattice. Physical Review Research, 6: 023045. doi: 10.1103/PhysRevResearch.6.023045
[38]	Cui J-G, Yang T, Chen L-Q 2018. Frequency-preserved non-reciprocal acoustic propagation in a granular chain. Applied Physics Letters, 112: 181904.
[39]	Cummings A 1984. Acoustic nonlinearities and power losses at orifices. AIAA Journal, 22: 786-792.
[40]	Darabi A, Leamy M J 2019. Tunable nonlinear topological insulator for acoustic waves. Physical Review Applied, 12: 044030.
[41]	Darabi A, Ni X, Leamy M, et al. 2020. Reconfigurable floquet elastodynamic topological insulator based on synthetic angular momentum bias. Science Advances, 6: eaba8656. doi: 10.1126/sciadv.aba8656
[42]	De Marqui Junior C, Erturk A, Inman D J 2009. An electromechanical finite element model for piezoelectric energy harvester plates. Journal of Sound and Vibration, 327: 9-25.
[43]	Deng B, Chen L, Wei D, et al. 2020a. Pulse-driven robot: Motion via solitary waves. Science Advances, 6: eaaz1166. doi: 10.1126/sciadv.aaz1166
[44]	Deng B, Mo C, Tournat V, et al. 2019a. Focusing and mode separation of elastic vector solitons in a 2d soft mechanical metamaterial. Physical Review Letters, 123: 024101. doi: 10.1103/PhysRevLett.123.024101
[45]	Deng B, Raney J R, Bertoldi K, et al. 2021. Nonlinear waves in flexible mechanical metamaterials. Journal of Applied Physics, 130: 040901. doi: 10.1063/5.0050271
[46]	Deng B, Tournat V, Wang P, et al. 2019b. Anomalous collisions of elastic vector solitons in mechanical metamaterials. Physical Review Letters, 122: 044101. doi: 10.1103/PhysRevLett.122.044101
[47]	Deng B, Wang P, He Q, et al. 2018. Metamaterials with amplitude gaps for elastic solitons. Nature Communications, 9: 3410. doi: 10.1038/s41467-018-05908-9
[48]	Deng B, Yu S, Forte A E, et al. 2020b. Characterization, stability, and application of domain walls in flexible mechanical metamaterials. Proceedings of the National Academy of Sciences, 117: 31002-31009. doi: 10.1073/pnas.2015847117
[49]	Deng B, Zanaty M, Forte A E, et al. 2022a. Topological solitons make metamaterials crawl. Physical Review Applied, 17: 014004. doi: 10.1103/PhysRevApplied.17.014004
[50]	Deng B, Zareei A, Ding X, et al. 2022b. Inverse design of mechanical metamaterials with target nonlinear response via a neural accelerated evolution strategy. Advanced Materials, 34: 2206238. doi: 10.1002/adma.202206238
[51]	Devaux T, Tournat V, Richoux O, et al. 2015. Asymmetric acoustic propagation of wave packets via the self-demodulation effect. Physical Review Letters, 115: 234301. doi: 10.1103/PhysRevLett.115.234301
[52]	Deymier P A. Acoustic metamaterials and phononic crystals//: Springer Science & Business Media, 2013
[53]	Donahue C M, Anzel P W J, Bonanomi L, et al. 2014. Experimental realization of a nonlinear acoustic lens with a tunable focus. Applied Physics Letters, 104: 014103. doi: 10.1063/1.4857635
[54]	Dong E, Cao P, Zhang J, et al. 2022. Underwater acoustic metamaterials. National Science Review, 10: nwac246.
[55]	Dong H-W, Shen C, Liu Z, et al. 2024. Inverse design of phononic meta-structured materials. Materials Today, 80: 824-855. doi: 10.1016/j.mattod.2024.09.012
[56]	Fan L, Ge H, Zhang S-y, et al. 2013. Nonlinear acoustic fields in acoustic metamaterial based on a cylindrical pipe with periodically arranged side holes. The Journal of the Acoustical Society of America, 133: 3846-3852. doi: 10.1121/1.4803904
[57]	Fang X, Lacarbonara W, Cheng L 2024. Advances in nonlinear acoustic/elastic metamaterials and metastructures. Nonlinear Dynamics.
[58]	Fang X, Sheng P, Wen J, et al. 2022. A nonlinear metamaterial plate for suppressing vibration and sound radiation. International Journal of Mechanical Sciences, 228: 107473. doi: 10.1016/j.ijmecsci.2022.107473
[59]	Fang X, Wen J, Benisty H, et al. 2020. Ultrabroad acoustical limiting in nonlinear metamaterials due to adaptive-broadening band-gap effect. Physical Review B, 101: 104304. doi: 10.1103/PhysRevB.101.104304
[60]	Fang X, Wen J, Bonello B, et al. 2017a. Ultra-low and ultra-broad-band nonlinear acoustic metamaterials. Nature Communications, 8: 1288. doi: 10.1038/s41467-017-00671-9
[61]	Fang X, Wen J, Bonello B, et al. 2017b. Wave propagation in one-dimensional nonlinear acoustic metamaterials. New Journal of Physics, 19: 053007. doi: 10.1088/1367-2630/aa6d49
[62]	Fang X, Wen J, Yin J, et al. 2016. Broadband and tunable one-dimensional strongly nonlinear acoustic metamaterials: Theoretical study. Physical Review E, 94: 052206. doi: 10.1103/PhysRevE.94.052206
[63]	Fang X, Wen J, Yu D, et al. 2018. Bridging-coupling band gaps in nonlinear acoustic metamaterials. Physical Review Applied, 10: 054049. doi: 10.1103/PhysRevApplied.10.054049
[64]	Fermi E, Pasta P, Ulam S, et al. 1955. Studies of the nonlinear problems. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States).
[65]	Flach S, Gorbach A V 2008. Discrete breathers — advances in theory and applications. Physics Reports, 467: 1-116.
[66]	Fleury R, Khanikaev A B, Alù A 2016. Floquet topological insulators for sound. Nature Communications, 7: 11744.
[67]	Florijn B, Coulais C, van Hecke M 2014. Programmable mechanical metamaterials. Physical Review Letters, 113: 175503.
[68]	Frandsen N M M, Bilal O R, Jensen J S, et al. 2016. Inertial amplification of continuous structures: Large band gaps from small masses. Journal of Applied Physics, 119.
[69]	Frandsen N M M, Jensen J S 2017. Modal interaction and higher harmonic generation in a weakly nonlinear, periodic mass–spring chain. Wave Motion, 68: 149-161.
[70]	Fraternali F, Carpentieri G, Amendola A, et al. 2014. Multiscale tunability of solitary wave dynamics in tensegrity metamaterials. Applied Physics Letters, 105: 201903. doi: 10.1063/1.4902071
[71]	Fraternali F, Senatore L, Daraio C 2012. Solitary waves on tensegrity lattices. Journal of the Mechanics and Physics of Solids, 60: 1137-1144.
[72]	Frazier M J, Kochmann D M 2017. Band gap transmission in periodic bistable mechanical systems. Journal of Sound and Vibration, 388: 315-326.
[73]	Fronk M D, Fang L, Packo P, et al. 2023. Elastic wave propagation in weakly nonlinear media and metamaterials: A review of recent developments. Nonlinear Dynamics, 111: 10709-10741. doi: 10.1007/s11071-023-08399-6
[74]	Fronk M D, Leamy M J 2017. Higher-order dispersion, stability, and waveform invariance in nonlinear monoatomic and diatomic systems. Journal of Vibration and Acoustics, 139: 051003.
[75]	Ganesh R, Gonella S 2015. From modal mixing to tunable functional switches in nonlinear phononic crystals. Physical Review Letters, 114: 054302.
[76]	Ganeshan S, Sun K, Das Sarma S 2013. Topological zero-energy modes in gapless commensurate Aubry-André-Harper models. Physical Review Letters, 110: 180403.
[77]	Gao L, Mak C M, Ma K W, et al. 2024. Mechanisms of multi-bandgap inertial amplification applied in metamaterial sandwich plates. International Journal of Mechanical Sciences, 277: 109424. doi: 10.1016/j.ijmecsci.2024.109424
[78]	Gao N, Zhang Z, Deng J, et al. 2022a. Acoustic metamaterials for noise reduction: A review. Advanced Materials Technologies, 7: 2100698. doi: 10.1002/admt.202100698
[79]	Gao P, Climente A, Sánchez-Dehesa J, et al. 2017. Theoretical study of platonic crystals with periodically structured N-beam resonators. Journal of Applied Physics, 123: 091707.
[80]	Gao P, Climente A, Sánchez-Dehesa J, et al. 2019a. Single-phase metamaterial plates for broadband vibration suppression at low frequencies. Journal of Sound and Vibration, 444: 108-126. doi: 10.1016/j.jsv.2018.12.022
[81]	Gao P, Qu Y, Christensen J 2022b. Non-Hermitian elastodynamics in gyro-odd continuum media. Communications Materials, 3: 74.
[82]	Gao P, Torrent D, Cervera F, et al. 2019b. Majorana-like zero modes in Kekulé distorted sonic lattices. Physical Review Letters, 123: 196601. doi: 10.1103/PhysRevLett.123.196601
[83]	Gao P, Willatzen M, Christensen J 2020. Anomalous topological edge states in non-Hermitian piezophononic media. Physical Review Letters, 125: 206402.
[84]	Gao Y, Wang L 2022. Nonlocal active metamaterial with feedback control for tunable bandgap and broadband nonreciprocity. International Journal of Mechanical Sciences, 219: 107131.
[85]	Gao Y, Wang L 2023. Broad bandgap active metamaterials with optimal time-delayed control. International Journal of Mechanical Sciences, 254: 108449.
[86]	Gao Y, Wang L 2024. An active tunable piezoelectric metamaterial beam for broadband vibration suppression by optimization. Acta Mechanica Sinica, 40: 523235.
[87]	Geniet F, Leon J 2002. Energy transmission in the forbidden band gap of a nonlinear chain. Physical Review Letters, 89: 134102.
[88]	Gillman A, Fuchi K, Buskohl P R 2018. Truss-based nonlinear mechanical analysis for origami structures exhibiting bifurcation and limit point instabilities. International Journal of Solids and Structures, 147: 80-93.
[89]	Gilpin W, Bull M S, Prakash M 2020. The multiscale physics of cilia and flagella. Nature Reviews Physics, 2: 74-88.
[90]	Gliozzi A S, Miniaci M, Bosia F, et al. 2015. Metamaterials-based sensor to detect and locate nonlinear elastic sources. Applied Physics Letters, 107: 161902. doi: 10.1063/1.4934493
[91]	Gong C, Fang X, Cheng L 2023. Band degeneration and evolution in nonlinear triatomic metamaterials. Nonlinear Dynamics, 111: 97-112.
[92]	Gong L, Zhang G, Gao P, et al. 2025. Tunable nonlinear piezoelectric metabeams for multimode vibration suppression. International Journal of Mechanical Sciences, doi: https://doi.org/10.1016/j.ijmecsci.2025.110238.
[93]	Guddala S, Ramakrishna S A 2016. Optical limiting by nonlinear tuning of resonance in metamaterial absorbers. Optics Letters, 41: 5150-5153.
[94]	Guo X, Gusev V E, Tournat V, et al. 2019. Frequency-doubling effect in acoustic reflection by a nonlinear, architected rotating-square metasurface. Physical Review E, 99: 052209. doi: 10.1103/PhysRevE.99.052209
[95]	Guo X, Lissek H, Fleury R 2020. Improving sound absorption through nonlinear active electroacoustic resonators. Physical Review Applied, 13: 014018.
[96]	Hatanaka D, Yamaguchi H 2020. Real-space characterization of cavity-coupled waveguide systems in hypersonic phononic crystals. Physical Review Applied, 13: 024005.
[97]	Heckl M A 1964. Investigations on the vibrations of grillages and other simple beam structures. The Journal of the Acoustical Society of America, 36: 1335-1343.
[98]	Herbold E B, Nesterenko V F 2013. Propagation of rarefaction pulses in discrete materials with strain-softening behavior. Physical Review Letters, 110: 144101.
[99]	Hersh A S, Walker B E, Celano J W 2003. Helmholtz resonator impedance model, part 1: Nonlinear behavior. AIAA Journal, 41: 795-808.
[100]	Hsu C W, Zhen B, Stone A D, et al. 2016. Bound states in the continuum. Nature Reviews Materials, 1: 16048. doi: 10.1038/natrevmats.2016.48
[101]	Hu B, Fang X, Cheng L, et al. 2023. Attenuation of impact waves in a nonlinear acoustic metamaterial beam. Nonlinear Dynamics, 111: 15801-15816. doi: 10.1007/s11071-023-08689-z
[102]	Hu G, C. M. Austin A, Sorokin V, et al. 2021. Metamaterial beam with graded local resonators for broadband vibration suppression. Mechanical Systems and Signal Processing, 146: 106982. doi: 10.1016/j.ymssp.2020.106982
[103]	Huang S, Li Y, Zhu J, et al. 2023. Sound-absorbing materials. Physical Review Applied, 20: 010501. doi: 10.1103/PhysRevApplied.20.010501
[104]	Huang S, Zhou Z, Li D, et al. 2020. Compact broadband acoustic sink with coherently coupled weak resonances. Science Bulletin, 65: 373-379. doi: 10.1016/j.scib.2019.11.008
[105]	Hussein M I, Leamy M J, Ruzzene M 2014. Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook. Applied Mechanics Reviews, 66: 040802.
[106]	Ingard U, Ising H 1967. Acoustic nonlinearity of an orifice. The Journal of the Acoustical Society of America, 42: 6-17.
[107]	Ingård U, Labate S 1950. Acoustic circulation effects and the nonlinear impedance of orifices. The Journal of the Acoustical Society of America, 22: 211-218.
[108]	Jeon G J, Oh J H 2021. Nonlinear acoustic metamaterial for efficient frequency down-conversion. Physical Review E, 103: 012212.
[109]	Jhang K-Y, Kim K-C 1999. Evaluation of material degradation using nonlinear acoustic effect. Ultrasonics, 37: 39-44.
[110]	Jian Y, Tang L, Hu G, et al. 2022. Design of graded piezoelectric metamaterial beam with spatial variation of electrodes. International Journal of Mechanical Sciences, 218: 107068. doi: 10.1016/j.ijmecsci.2022.107068
[111]	Jiao W, Gonella S 2018a. Intermodal and subwavelength energy trapping in nonlinear metamaterial waveguides. Physical Review Applied, 10: 024006.
[112]	Jiao W, Gonella S 2018b. Mechanics of inter-modal tunneling in nonlinear waveguides. Journal of the Mechanics and Physics of Solids, 111: 1-17.
[113]	Jiao W, Gonella S 2021. Wavenumber-space band clipping in nonlinear periodic structures. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477: 20210052.
[114]	Jiao W, Shu H, Tournat V, et al. 2024. Phase transitions in 2d multistable mechanical metamaterials via collisions of soliton-like pulses. Nature Communications, 15: 333. doi: 10.1038/s41467-023-44293-w
[115]	Jin L, Khajehtourian R, Mueller J, et al. 2020. Guided transition waves in multistable mechanical metamaterials. Proceedings of the National Academy of Sciences, 117: 2319-2325. doi: 10.1073/pnas.1913228117
[116]	Jin Y, Torrent D, Rouhani B D, et al. 2025. The 2024 phononic crystals roadmap. Journal of Physics D: Applied Physics, 58: 113001. doi: 10.1088/1361-6463/ad9ab2
[117]	Jing X, Sun X 2002. Sound-excited flow and acoustic nonlinearity at an orifice. Physics of Fluids, 14: 268-276.
[118]	Job S, Santibanez F, Tapia F, et al. 2009. Wave localization in strongly nonlinear hertzian chains with mass defect. Physical Review E, 80: 025602. doi: 10.1103/PhysRevE.80.025602
[119]	Johnson K L 1982. One hundred years of hertz contact. Proceedings of the Institution of Mechanical Engineers, 196: 363-378.
[120]	Kadic M, Bückmann T, Schittny R, et al. 2013. Metamaterials beyond electromagnetism. Reports on Progress in Physics, 76: 126501. doi: 10.1088/0034-4885/76/12/126501
[121]	Khajehtourian R, Kochmann D M 2021. A continuum description of substrate-free dissipative reconfigurable metamaterials. Journal of the Mechanics and Physics of Solids, 147: 104217.
[122]	Khomeriki R, Lepri S, Ruffo S 2004. Nonlinear supratransmission and bistability in the fermi-pasta-ulam model. Physical Review E, 70: 066626.
[123]	Khoo I C, Wang Y K 1976. Multiple time scale analysis of an anharmonic crystal. Journal of Mathematical Physics, 17: 222-227.
[124]	Kim B L, Chong C, Hajarolasvadi S, et al. 2023. Dynamics of time-modulated, nonlinear phononic lattices. Physical Review E, 107: 034211. doi: 10.1103/PhysRevE.107.034211
[125]	Kim E, Li F, Chong C, et al. 2015. Highly nonlinear wave propagation in elastic woodpile periodic structures. Physical Review Letters, 114: 118002. doi: 10.1103/PhysRevLett.114.118002
[126]	Kim E, Yang J 2019. Review: Wave propagation in granular metamaterials. Functional Composites and Structures, 1: 012002.
[127]	Komkin A, Bykov A, Mironov M 2020. Experimental study of nonlinear acoustic impedance of circular orifices. The Journal of the Acoustical Society of America, 148: 1391-1403.
[128]	Korpas L M, Yin R, Yasuda H, et al. 2021. Temperature-responsive multistable metamaterials. ACS Applied Materials & Interfaces, 13: 31163-31170.
[129]	Krushynska A O, Miniaci M, Bosia F, et al. 2017. Coupling local resonance with bragg band gaps in single-phase mechanical metamaterials. Extreme Mechanics Letters, 12: 30-36. doi: 10.1016/j.eml.2016.10.004
[130]	Kumar R, Ezhilarasi D 2023. A state-of-the-art survey of model order reduction techniques for large-scale coupled dynamical systems. International Journal of Dynamics and Control, 11: 900-916.
[131]	Kushwaha M S, Halevi P, Dobrzynski L, et al. 1993. Acoustic band structure of periodic elastic composites. Physical Review Letters, 71: 2022-2025. doi: 10.1103/PhysRevLett.71.2022
[132]	Laly Z, Atalla N, Meslioui S-A 2018. Acoustical modeling of micro-perforated panel at high sound pressure levels using equivalent fluid approach. Journal of Sound and Vibration, 427: 134-158.
[133]	Laly Z, Atalla N, Meslioui S-A, et al. 2019. Sensitivity analysis of micro-perforated panel absorber models at high sound pressure levels. Applied Acoustics, 156: 7-20. doi: 10.1016/j.apacoust.2019.06.025
[134]	Lan J, Li Y, Yu H, et al. 2017. Nonlinear effects in acoustic metamaterial based on a cylindrical pipe with ordered helmholtz resonators. Physics Letters A, 381: 1111-1117. doi: 10.1016/j.physleta.2017.01.036
[135]	Larbi W, Deü J-F 2019. Reduced order finite element formulations for vibration reduction using piezoelectric shunt damping. Applied Acoustics, 147: 111-120.
[136]	Lazarov B S, Jensen J S 2007. Low-frequency band gaps in chains with attached non-linear oscillators. International Journal of Non-Linear Mechanics, 42: 1186-1193.
[137]	Leon J 2003. Nonlinear supratransmission as a fundamental instability. Physics Letters A, 319: 130-136.
[138]	Lepri S, Casati G 2011. Asymmetric wave propagation in nonlinear systems. Physical Review Letters, 106: 164101.
[139]	Li F, Anzel P, Yang J, et al. 2014. Granular acoustic switches and logic elements. Nature Communications, 5: 5311. doi: 10.1038/ncomms6311
[140]	Li J, Chan C T 2004. Double-negative acoustic metamaterial. Physical Review E, 70: 055602.
[141]	Li Z-N, Wang Y-Z, Wang Y-S 2020. Tunable nonreciprocal transmission in nonlinear elastic wave metamaterial by initial stresses. International Journal of Solids and Structures, 182-183: 218-235.
[142]	Li Z-N, Wang Y-Z, Wang Y-S 2021. Tunable mechanical diode of nonlinear elastic metamaterials induced by imperfect interface. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477: 20200357.
[143]	Li Z-N, Yuan B, Wang Y-Z, et al. 2019. Diode behavior and nonreciprocal transmission in nonlinear elastic wave metamaterial. Mechanics of Materials, 133: 85-101. doi: 10.1016/j.mechmat.2019.03.010
[144]	Liang B, Guo X S, Tu J, et al. 2010. An acoustic rectifier. Nature Materials, 9: 989-992. doi: 10.1038/nmat2881
[145]	Librandi G, Tubaldi E, Bertoldi K 2021. Programming nonreciprocity and reversibility in multistable mechanical metamaterials. Nature Communications, 12: 3454.
[146]	Liu E, Fang X, Wen J 2022a. Harmonic and shock wave propagation in bistable periodic structure: Regularity, randomness, and tunability. Journal of Vibration and Control, 28: 3332-3343.
[147]	Liu J, Wang Y, Yang S, et al. 2024. Customized quasi-zero-stiffness metamaterials for ultra-low frequency broadband vibration isolation. International Journal of Mechanical Sciences, 269: 108958. doi: 10.1016/j.ijmecsci.2024.108958
[148]	Liu Z, Fang H, Xu J, et al. 2023. Discriminative transition sequences of origami metamaterials for mechanologic. Advanced Intelligent Systems, 5: 2200146. doi: 10.1002/aisy.202200146
[149]	Liu Z, Shan S, Cheng L 2022b. Nonlinear-lamb-wave-based plastic damage detection assisted by topologically designed metamaterial filters. Structural Health Monitoring, 22: 1828-1843.
[150]	Liu Z, Zhang X, Mao Y, et al. 2000. Locally resonant sonic materials. Science, 289: 1734-1736. doi: 10.1126/science.289.5485.1734
[151]	Macías-Díaz J E 2008. Numerical study of the transmission of energy in discrete arrays of sine-gordon equations in two space dimensions. Physical Review E, 77: 016602.
[152]	Man Y, Boechler N, Theocharis G, et al. 2012. Defect modes in one-dimensional granular crystals. Physical Review E, 85: 037601. doi: 10.1103/PhysRevE.85.037601
[153]	Manktelow K, Leamy M J, Ruzzene M 2011. Multiple scales analysis of wave–wave interactions in a cubically nonlinear monoatomic chain. Nonlinear Dynamics, 63: 193-203.
[154]	Manktelow K, Leamy M J, Ruzzene M 2013a. Comparison of asymptotic and transfer matrix approaches for evaluating intensity-dependent dispersion in nonlinear photonic and phononic crystals. Wave Motion, 50: 494-508.
[155]	Manktelow K, Narisetti R K, Leamy M J, et al. 2013b. Finite-element based perturbation analysis of wave propagation in nonlinear periodic structures. Mechanical Systems and Signal Processing, 39: 32-46. doi: 10.1016/j.ymssp.2012.04.015
[156]	Manktelow K L, Leamy M J, Ruzzene M 2013c. Topology design and optimization of nonlinear periodic materials. Journal of the Mechanics and Physics of Solids, 61: 2433-2453.
[157]	Maradudin A A, Mazur P, Montroll E W, et al. 1958. Remarks on the vibrations of diatomic lattices. Reviews of Modern Physics, 30: 175-196. doi: 10.1103/RevModPhys.30.175
[158]	Marathe A, Chatterjee A 2006. Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales. Journal of Sound and Vibration, 289: 871-888.
[159]	Mead D J 1971. Vibration response and wave propagation in periodic structures. Journal of Engineering for Industry, 93: 783-792.
[160]	Meaud J 2020. Nonlinear wave propagation and dynamic reconfiguration in two-dimensional lattices with bistable elements. Journal of Sound and Vibration, 473: 115239.
[161]	Melling T H 1973. The acoustic impendance of perforates at medium and high sound pressure levels. Journal of Sound and Vibration, 29: 1-65.
[162]	Merkel A, Tournat V, Gusev V 2011. Experimental evidence of rotational elastic waves in granular phononic crystals. Physical Review Letters, 107: 225502.
[163]	Miniaci M, Gliozzi A S, Morvan B, et al. 2017. Proof of concept for an ultrasensitive technique to detect and localize sources of elastic nonlinearity using phononic crystals. Physical Review Letters, 118: 214301. doi: 10.1103/PhysRevLett.118.214301
[164]	Mojahed A, Bunyan J, Tawfick S, et al. 2019. Tunable acoustic nonreciprocity in strongly nonlinear waveguides with asymmetry. Physical Review Applied, 12: 034033. doi: 10.1103/PhysRevApplied.12.034033
[165]	Mosquera-Sánchez J A, Alfahmi O, Erturk A, et al. 2024. Broadening the frequency response of a duffing-type piezoelectric shunt by means of negative capacitance. Journal of Sound and Vibration, 578: 118344. doi: 10.1016/j.jsv.2024.118344
[166]	Mosquera-Sánchez J A, De Marqui C 2024. Broadband and multimode attenuation in duffing- and nes-type piezoelectric metastructures. International Journal of Mechanical Sciences, 270: 109084.
[167]	Mosquera-Sánchez J A, De Marqui Jr C 2021. Dynamics and wave propagation in nonlinear piezoelectric metastructures. Nonlinear Dynamics, 105: 2995-3023.
[168]	Nadkarni N, Arrieta A F, Chong C, et al. 2016. Unidirectional transition waves in bistable lattices. Physical Review Letters, 116: 244501. doi: 10.1103/PhysRevLett.116.244501
[169]	Nadkarni N, Daraio C, Kochmann D M 2014. Dynamics of periodic mechanical structures containing bistable elastic elements: From elastic to solitary wave propagation. Physical Review E, 90: 023204.
[170]	Narisetti R K, Leamy M J, Ruzzene M 2010. A perturbation approach for predicting wave propagation in one-dimensional nonlinear periodic structures. Journal of Vibration and Acoustics, 132: 031001.
[171]	Narisetti R K, Ruzzene M, Leamy M J 2011. A perturbation approach for analyzing dispersion and group velocities in two-dimensional nonlinear periodic lattices. Journal of Vibration and Acoustics, 133: 061020.
[172]	Narisetti R K, Ruzzene M, Leamy M J 2012. Study of wave propagation in strongly nonlinear periodic lattices using a harmonic balance approach. Wave Motion, 49: 394-410.
[173]	Nash L M, Kleckner D, Read A, et al. 2015. Topological mechanics of gyroscopic metamaterials. Proceedings of the National Academy of Sciences, 112: 14495-14500. doi: 10.1073/pnas.1507413112
[174]	Nassar H, Chen H, Norris A N, et al. 2017. Non-reciprocal wave propagation in modulated elastic metamaterials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473: 20170188. doi: 10.1098/rspa.2017.0188
[175]	Nassar H, Yousefzadeh B, Fleury R, et al. 2020. Nonreciprocity in acoustic and elastic materials. Nature Reviews Materials, 5: 667-685. doi: 10.1038/s41578-020-0206-0
[176]	Nesterenko V F 1983. Propagation of nonlinear compression pulses in granular media. Journal of Applied Mechanics and Technical Physics, 24: 733-743.
[177]	Nesterenko V F 2018. Waves in strongly nonlinear discrete systems. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376: 20170130.
[178]	Nesterenko V F, Daraio C, Herbold E B, et al. 2005. Anomalous wave reflection at the interface of two strongly nonlinear granular media. Physical Review Letters, 95: 158702. doi: 10.1103/PhysRevLett.95.158702
[179]	Ni X, Rizzo P, Yang J, et al. 2012. Monitoring the hydration of cement using highly nonlinear solitary waves. NDT & E International, 52: 76-85.
[180]	Norris A N 2008. Acoustic cloaking theory. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 464: 2411-2434.
[181]	Ou H, Hu L, Wang Y, et al. 2024. High-efficient and reusable impact mitigation metamaterial based on compression-torsion coupling mechanism. Journal of the Mechanics Physics of Solids, 186: 105594. doi: 10.1016/j.jmps.2024.105594
[182]	Pal A, Sitti M 2023. Programmable mechanical devices through magnetically tunable bistable elements. Proceedings of the National Academy of Sciences, 120: e2212489120.
[183]	Pal R K, Vila J, Leamy M, et al. 2018. Amplitude-dependent topological edge states in nonlinear phononic lattices. Physical Review E, 97: 032209. doi: 10.1103/PhysRevE.97.032209
[184]	Palermo A, Celli P, Yousefzadeh B, et al. 2020. Surface wave non-reciprocity via time-modulated metamaterials. Journal of the Mechanics and Physics of Solids, 145: 104181. doi: 10.1016/j.jmps.2020.104181
[185]	Panigrahi S R, Feeny B F, Diaz A R 2017. Wave–wave interactions in a periodic chain with quadratic nonlinearity. Wave Motion, 69: 65-80.
[186]	Park S-H 2013a. Acoustic properties of micro-perforated panel absorbers backed by helmholtz resonators for the improvement of low-frequency sound absorption. Journal of Sound and Vibration, 332: 4895-4911.
[187]	Park S-H 2013b. A design method of micro-perforated panel absorber at high sound pressure environment in launcher fairings. Journal of Sound and Vibration, 332: 521-535.
[188]	Patil G U, Matlack K H 2021. Wave self-interactions in continuum phononic materials with periodic contact nonlinearity. Wave Motion, 105: 102763.
[189]	Patil G U, Matlack K H 2022. Review of exploiting nonlinearity in phononic materials to enable nonlinear wave responses. Acta Mechanica, 233: 1-46.
[190]	Peng F 2018. Sound absorption of a porous material with a perforated facing at high sound pressure levels. Journal of Sound and Vibration, 425: 1-20.
[191]	Pernas-Salomón R, Shmuel G 2020. Fundamental principles for generalized willis metamaterials. Physical Review Applied, 14: 064005.
[192]	Plattner L 2004. Optical properties of the scales of morpho rhetenor butterflies: Theoretical and experimental investigation of the back-scattering of light in the visible spectrum. Journal of the Royal Society Interface, 1: 49-59.
[193]	Popa B-I, Cummer S A 2014. Non-reciprocal and highly nonlinear active acoustic metamaterials. Nature Communications, 5: 3398.
[194]	Porter M A, Kevrekidis P G, Daraio C 2015. Granular crystals: Nonlinear dynamics meets materials engineering. Physics Today, 68: 44-50.
[195]	Qu R, Guo J, Fang Y, et al. 2023. Broadband quasi-perfect sound absorption by a metasurface with coupled resonators at both low- and high-amplitude excitations. Mechanical Systems and Signal Processing, 204: 110782. doi: 10.1016/j.ymssp.2023.110782
[196]	Qu S, Gao N, Tinel A, et al. 2022. Underwater metamaterial absorber with impedance-matched composite. Science Advances, 8: eabm4206. doi: 10.1126/sciadv.abm4206
[197]	Raney J R, Nadkarni N, Daraio C, et al. 2016. Stable propagation of mechanical signals in soft media using stored elastic energy. Proceedings of the National Academy of Sciences, 113: 9722-9727. doi: 10.1073/pnas.1604838113
[198]	Richoux O, Tournat V, Le Van Suu T 2007. Acoustic wave dispersion in a one-dimensional lattice of nonlinear resonant scatterers. Physical Review E, 75: 026615.
[199]	Rienstra S W, Singh D K 2018. Nonlinear asymptotic impedance model for a helmholtz resonator of finite depth. AIAA Journal, 56: 1792-1802.
[200]	Rosa M I N, Leamy M J, Ruzzene M 2023. Amplitude-dependent edge states and discrete breathers in nonlinear modulated phononic lattices. New Journal of Physics, 25: 103053.
[201]	Rosa M I N, Pal R K, Arruda J R F, et al. 2019. Edge states and topological pumping in spatially modulated elastic lattices. Physical Review Letters, 123: 034301. doi: 10.1103/PhysRevLett.123.034301
[202]	Sadeghi S, Allison S R, Bestill B, et al. 2021. Tmp origami jumping mechanism with nonlinear stiffness. Smart Materials and Structures, 30: 065002. doi: 10.1088/1361-665X/abf5b2
[203]	Scheibner C, Irvine W T M, Vitelli V 2020. Non-hermitian band topology and skin modes in active elastic media. Physical Review Letters, 125: 118001.
[204]	Sen Gupta G 1970. Natural flexural waves and the normal modes of periodically-supported beams and plates. Journal of Sound and Vibration, 13: 89-101.
[205]	Sepehri S, Mashhadi M M, Fakhrabadi M M S 2022. Wave propagation in fractionally damped nonlinear phononic crystals. Nonlinear Dynamics, 110: 1683-1708.
[206]	Serra-Garcia M, Lydon J, Daraio C 2016. Extreme stiffness tunability through the excitation of nonlinear defect modes. Physical Review E, 93: 010901.
[207]	Settimi V, Lepidi M, Bacigalupo A 2021. Nonlinear dispersion properties of one-dimensional mechanical metamaterials with inertia amplification. International Journal of Mechanical Sciences, 201: 106461.
[208]	Shan S, Wen F, Cheng L 2021. Purified nonlinear guided waves through a metamaterial filter for inspection of material microstructural changes. Smart Materials and Structures, 30: 095017.
[209]	Shen H, Wen J, Yu D, et al. 2013. Control of sound and vibration of fluid-filled cylindrical shells via periodic design and active control. Journal of Sound and Vibration, 332: 4193-4209. doi: 10.1016/j.jsv.2013.03.007
[210]	Sheng P, Fang X, Yu D, et al. 2024. Mitigating aeroelastic vibration of strongly nonlinear metamaterial supersonic wings under high temperature. Nonlinear Dynamics.
[211]	Sigalas M M, Economou E N 1992. Elastic and acoustic wave band structure. Journal of Sound and Vibration, 158: 377-382.
[212]	Silva P B, Leamy M J, Geers M G D, et al. 2019a. Emergent subharmonic band gaps in nonlinear locally resonant metamaterials induced by autoparametric resonance. Physical Review E, 99: 063003. doi: 10.1103/PhysRevE.99.063003
[213]	Silva T, Tan D, De Marqui C, et al. 2019b. Vibration attenuation in a nonlinear flexible structure via nonlinear switching circuits and energy harvesting implications. Journal of Intelligent Material Systems and Structures, 30: 965-976. doi: 10.1177/1045389X19828835
[214]	Silva T M, Clementino M A, De Marqui Jr C, et al. 2018. An experimentally validated piezoelectric nonlinear energy sink for wideband vibration attenuation. Journal of Sound and Vibration, 437: 68-78. doi: 10.1016/j.jsv.2018.08.038
[215]	Singh D K, Rienstra S W 2014. Nonlinear asymptotic impedance model for a helmholtz resonator liner. Journal of Sound and Vibration, 333: 3536-3549.
[216]	Singhal T, Kim E, Kim T-Y, et al. 2017. Weak bond detection in composites using highly nonlinear solitary waves. Smart Materials and Structures, 26: 055011. doi: 10.1088/1361-665X/aa6823
[217]	Sivian L J 1935. Acoustic impedance of small orifices. The Journal of the Acoustical Society of America, 7: 94-101.
[218]	Snee D D J M, Ma Y-P 2019. Edge solitons in a nonlinear mechanical topological insulator. Extreme Mechanics Letters, 30: 100487.
[219]	Spadoni A, Daraio C 2010. Generation and control of sound bullets with a nonlinear acoustic lens. Proceedings of the National Academy of Sciences, 107: 7230-7234.
[220]	Su J, Rupp J, Garmory A, et al. 2015. Measurements and computational fluid dynamics predictions of the acoustic impedance of orifices. Journal of Sound and Vibration, 352: 174-191. doi: 10.1016/j.jsv.2015.05.009
[221]	Su W P, Schrieffer J R, Heeger A J 1979. Solitons in polyacetylene. Physical Review Letters, 42: 1698-1701.
[222]	Sugimoto N 1992. Propagation of nonlinear acoustic waves in a tunnel with an array of helmholtz resonators. Journal of Fluid Mechanics, 244: 55-78.
[223]	Sugimoto N 1996. Acoustic solitary waves in a tunnel with an array of helmholtz resonators. The Journal of the Acoustical Society of America, 99: 1971-1976.
[224]	Sugino C, Ruzzene M, Erturk A 2020. Digitally programmable resonant elastic metamaterials. Physical Review Applied, 13: 061001.
[225]	Sun Y, Zheng H, Han Q, et al. 2024. Non-contact electromagnetic controlled metamaterial beams for low-frequency vibration suppression. International Journal of Solids and Structures, 290: 112667. doi: 10.1016/j.ijsolstr.2024.112667
[226]	Swinteck N Z, Muralidharan K, Deymier P A 2013. Phonon scattering in one-dimensional anharmonic crystals and superlattices: Analytical and numerical study. Journal of Vibration and Acoustics, 135: 041016.
[227]	Talebi Bidhendi M R 2022. Band gap transmission in a periodic network of coupled buckled beams. International Journal of Solids and Structures, 252: 111766.
[228]	Tam C K W, Auriault L 1999. Jet mixing noise from fine-scale turbulence. AIAA Journal, 37: 145-153.
[229]	Tam C K W, Ju H, Jones M G, et al. 2010. A computational and experimental study of resonators in three dimensions. Journal of Sound and Vibration, 329: 5164-5193. doi: 10.1016/j.jsv.2010.06.005
[230]	Tam C K W, Ju H, Walker B E 2008. Numerical simulation of a slit resonator in a grazing flow under acoustic excitation. Journal of Sound and Vibration, 313: 449-471.
[231]	Tam C K W, Kurbatskii K A, Ahuja K K, et al. 2001. A numerical and experimental investigation of the dissipation mechanisms of resonant acoustic liners. Journal of Sound and Vibration, 245: 545-557. doi: 10.1006/jsvi.2001.3571
[232]	Tam C K W, Pastouchenko N N, Jones M G, et al. 2014. Experimental validation of numerical simulations for an acoustic liner in grazing flow: Self-noise and added drag. Journal of Sound and Vibration, 333: 2831-2854. doi: 10.1016/j.jsv.2014.02.019
[233]	Tamura S, Hurley D C, Wolfe J P 1988. Acoustic-phonon propagation in superlattices. Physical Review B, 38: 1427-1449.
[234]	Tang Y, He W, Xin F, et al. 2020. Nonlinear sound absorption of ultralight hybrid-cored sandwich panels. Mechanical Systems and Signal Processing, 135: 106428. doi: 10.1016/j.ymssp.2019.106428
[235]	Tempelman J R, Matlack K H, Vakakis A F 2021. Topological protection in a strongly nonlinear interface lattice. Physical Review B, 104: 174306.
[236]	Theocharis G, Boechler N, Daraio C. Nonlinear periodic phononic structures and granular crystals//P. A. DEYMIER. Acoustic metamaterials and phononic crystals. Berlin, Heidelberg: Springer, 2013: 217-251.
[237]	Theocharis G, Kavousanakis M, Kevrekidis P G, et al. 2009. Localized breathing modes in granular crystals with defects. Physical Review E, 80: 066601. doi: 10.1103/PhysRevE.80.066601
[238]	Tian W, Yang Z, Li M, et al. 2025. Theoretical modeling and mechanism analysis of nonlinear metastructure for supersonic aeroelastic suppression. Mechanical Systems and Signal Processing, 224: 111931. doi: 10.1016/j.ymssp.2024.111931
[239]	Tian W, Zhao T, Gu Y, et al. 2022a. Nonlinear flutter suppression and performance evaluation of periodically embedded nonlinear vibration absorbers in a supersonic fgm plate. Aerospace Science and Technology, 121: 107198. doi: 10.1016/j.ast.2021.107198
[240]	Tian W, Zhao T, Yang Z 2022b. Supersonic meta-plate with tunable-stiffness nonlinear oscillators for nonlinear flutter suppression. International Journal of Mechanical Sciences, 229: 107533.
[241]	Tian Y, Shen Y, Rao D, et al. 2019. Metamaterial improved nonlinear ultrasonics for fatigue damage detection. Smart Materials and Structures, 28: 075038. doi: 10.1088/1361-665X/ab2566
[242]	Tournat V, Gusev V E, Castagnède B 2004. Self-demodulation of elastic waves in a one-dimensional granular chain. Physical Review E, 70: 056603.
[243]	Trainiti G, Xia Y, Marconi J, et al. 2019. Time-periodic stiffness modulation in elastic metamaterials for selective wave filtering: Theory and experiment. Physical Review Letters, 122: 124301. doi: 10.1103/PhysRevLett.122.124301
[244]	Vakakis A F, King M E 1998. Resonant oscillations of a weakly coupled, nonlinear layered system. Acta Mechanica, 128: 59-80.
[245]	Vasiliev V V, Barynin V A, Razin A F 2012. Anisogrid composite lattice structures – development and aerospace applications. Composite Structures, 94: 1117-1127.
[246]	Vasios N, Deng B, Gorissen B, et al. 2021. Universally bistable shells with nonzero gaussian curvature for two-way transition waves. Nature Communications, 12: 695. doi: 10.1038/s41467-020-20698-9
[247]	Veenstra J, Gamayun O, Guo X, et al. 2024. Non-reciprocal topological solitons in active metamaterials. Nature, 627: 528-533. doi: 10.1038/s41586-024-07097-6
[248]	Vila J, Paulino G H, Ruzzene M 2019. Role of nonlinearities in topological protection: Testing magnetically coupled fidget spinners. Physical Review B, 99: 125116.
[249]	Wallen S P, Boechler N 2017. Shear to longitudinal mode conversion via second harmonic generation in a two-dimensional microscale granular crystal. Wave Motion, 68: 22-30.
[250]	Wang K, Zhou J, Cai C, et al. 2019. Mathematical modeling and analysis of a meta-plate for very low-frequency band gap. Applied Mathematical Modelling, 73: 581-597. doi: 10.1016/j.apm.2019.04.033
[251]	Wang T, Touzé C, Li H, et al. 2024a. Nonlinear dispersion relationships and dissipative properties of damped metamaterials embedding bistable attachments. Nonlinear Dynamics.
[252]	Wang T, ul Islam T, Steur E, et al. 2024b. Programmable metachronal motion of closely packed magnetic artificial cilia. Lab on a Chip, 24: 1573-1585. doi: 10.1039/D3LC00956D
[253]	Wang Y-F, Wang Y-Z, Wu B, et al. 2020. Tunable and active phononic crystals and metamaterials. Applied Mechanics Reviews, 72: 040801. doi: 10.1115/1.4046222
[254]	Wang Y, Yousefzadeh B, Chen H, et al. 2018. Observation of nonreciprocal wave propagation in a dynamic phononic lattice. Physical Review Letters, 121: 194301. doi: 10.1103/PhysRevLett.121.194301
[255]	Wei L-S, Wang Y-Z, Wang Y-S 2020. Nonreciprocal transmission of nonlinear elastic wave metamaterials by incremental harmonic balance method. International Journal of Mechanical Sciences, 173: 105433.
[256]	Wen Z, Jin Y, Gao P, et al. 2022. Topological cavities in phononic plates for robust energy harvesting. Mechanical Systems and Signal Processing, 162: 108047. doi: 10.1016/j.ymssp.2021.108047
[257]	Wu B, Destrade M, Chen W 2020. Nonlinear response and axisymmetric wave propagation in functionally graded soft electro-active tubes. International Journal of Mechanical Sciences, 187: 106006.
[258]	Wu B, Jiang W, Jiang J, et al. 2024. Wave manipulation in intelligent metamaterials: Recent progress and prospects. Advanced Functional Materials, 34: 2316745. doi: 10.1002/adfm.202316745
[259]	Wu G, Lu Z, Xu X, et al. 2019. Numerical investigation of aeroacoustics damping performance of a helmholtz resonator: Effects of geometry, grazing and bias flow. Aerospace Science and Technology, 86: 191-203. doi: 10.1016/j.ast.2019.01.007
[260]	Wu Z, Wang K W 2019. On the wave propagation analysis and supratransmission prediction of a metastable modular metastructure for non-reciprocal energy transmission. Journal of Sound and Vibration, 458: 389-406.
[261]	Xia Y, Ruzzene M, Erturk A 2019. Dramatic bandwidth enhancement in nonlinear metastructures via bistable attachments. Applied Physics Letters, 114: 093501.
[262]	Xia Y, Ruzzene M, Erturk A 2020. Bistable attachments for wideband nonlinear vibration attenuation in a metamaterial beam. Nonlinear Dynamics, 102: 1285-1296.
[263]	Xiao Y, Wen J, Wen X 2012. Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators. Journal of Physics D: Applied Physics, 45: 195401.
[264]	Xiao Z, Gao P, He X, et al. 2023. Multifunctional acoustic metamaterial for air ventilation, broadband sound insulation and switchable transmission. Journal of Physics D: Applied Physics, 56: 044006. doi: 10.1088/1361-6463/acaa44
[265]	Xiao Z, Gao P, He X, et al. 2024. Lightweight cellular multifunctional metamaterials with superior low-frequency sound absorption, broadband energy harvesting and high load-bearing capacity. Materials & Design, 241: 112912.
[266]	Xiao Z, Gao P, Wang D, et al. 2021. Ventilated metamaterials for broadband sound insulation and tunable transmission at low frequency. Extreme Mechanics Letters, 46: 101348. doi: 10.1016/j.eml.2021.101348
[267]	Xu L, Xiang Z 2024. Chaotic metastructures for frequency self-conversion. Mechanical Systems and Signal Processing, 206: 110927.
[268]	Xu Q, Wang J, Lv Y, et al. 2023. Vibration characteristics of linear and nonlinear dissipative elastic metamaterials rotor with geometrical nonlinearity. International Journal of Non-Linear Mechanics, 157: 104543. doi: 10.1016/j.ijnonlinmec.2023.104543
[269]	Xu X, Barnhart M V, Fang X, et al. 2019. A nonlinear dissipative elastic metamaterial for broadband wave mitigation. International Journal of Mechanical Sciences, 164: 105159. doi: 10.1016/j.ijmecsci.2019.105159
[270]	Xue Y, Li J, Wang Y, et al. 2023. Broadband vibration attenuation in nonlinear meta-structures with magnet coupling mechanism: Theory and experiments. Communications in Nonlinear Science and Numerical Simulation, 127: 107543. doi: 10.1016/j.cnsns.2023.107543
[271]	Xue Y, Li J, Wang Y, et al. 2024. Widely tunable magnetorheological metamaterials with nonlinear amplification mechanism. International Journal of Mechanical Sciences, 264: 108830. doi: 10.1016/j.ijmecsci.2023.108830
[272]	Yablonovitch E 1987. Inhibited spontaneous emission in solid-state physics and electronics. Physical Review Letters, 58: 2059-2062.
[273]	Yang D, Guo X, Zhang W, et al. 2024. Non-linear dynamics and bandgap control in magneto-rheological elastomers metamaterials with inertial amplification. Thin-Walled Structures, 204: 112237. doi: 10.1016/j.tws.2024.112237
[274]	Yao D, Xiong M, Luo J, et al. 2022. Flexural wave mitigation in metamaterial cylindrical curved shells with periodic graded arrays of multi-resonator. Mechanical Systems and Signal Processing, 168: 108721. doi: 10.1016/j.ymssp.2021.108721
[275]	Yao S, Zhou X, Hu G 2010. Investigation of the negative-mass behaviors occurring below a cut-off frequency. New Journal of Physics, 12: 103025.
[276]	Yasuda H, Korpas L M, Raney J R 2020. Transition waves and formation of domain walls in multistable mechanical metamaterials. Physical Review Applied, 13: 054067.
[277]	Yi J, Chen C Q 2024. Delocalization and higher-order topology in a nonlinear elastic lattice. New Journal of Physics, 26: 063004.
[278]	Yi J, Zhang Y, Chen C Q 2023. Tunable mode conversion in a mechanical metamaterial via second harmonic generation. Journal of Sound and Vibration, 565: 117911.
[279]	Yilmaz C, Hulbert G M, Kikuchi N 2007. Phononic band gaps induced by inertial amplification in periodic media. Physical Review B, 76: 054309.
[280]	Yin D, Yi K, Liu Z, et al. 2022. Design of cylindrical metashells with piezoelectric materials and digital circuits for multi-modal vibration control. Frontiers in Physics, 10: 958141. doi: 10.3389/fphy.2022.958141
[281]	Yoon S, Kim A, Cantwell W J, et al. 2023. Defect detection in composites by deep learning using solitary waves. International Journal of Mechanical Sciences, 239: 107882. doi: 10.1016/j.ijmecsci.2022.107882
[282]	Yousefzadeh B, Phani A S 2016. Supratransmission in a disordered nonlinear periodic structure. Journal of Sound and Vibration, 380: 242-266.
[283]	Yu D, Wen J, Shen H, et al. 2012. Propagation of flexural wave in periodic beam on elastic foundations. Physics Letters A, 376: 626-630. doi: 10.1016/j.physleta.2011.11.056
[284]	Yu M, Fang X, Wen J, et al. 2024. Robust nonlinear elastic metamaterial enabled by collision damping. Mechanics of Advanced Materials and Structures, 31: 3630-3637. doi: 10.1080/15376494.2023.2180557
[285]	Yu M, Fang X, Yu D 2021. Combinational design of linear and nonlinear elastic metamaterials. International Journal of Mechanical Sciences, 199: 106422.
[286]	Zareei A, Deng B, Bertoldi K 2020. Harnessing transition waves to realize deployable structures. Proceedings of the National Academy of Sciences, 117: 4015-4020.
[287]	Zhang J, Romero-García V, Theocharis G, et al. 2021a. High-amplitude sound propagation in acoustic transmission-line metamaterial. Applied Physics Letters, 118: 104102. doi: 10.1063/5.0040702
[288]	Zhang Q, Fang H, Xu J 2020. Programmable stopbands and supratransmission effects in a stacked miura-origami metastructure. Physical Review E, 101: 042206.
[289]	Zhang Q, Guo D, Hu G 2021b. Tailored mechanical metamaterials with programmable quasi-zero-stiffness features for full-band vibration isolation. Advanced Functional Materials, 31: 2101428.
[290]	Zhang Q, Rudykh S 2024. Propagation of solitary waves in origami-inspired metamaterials. Journal of the Mechanics and Physics of Solids, 187: 105626.
[291]	Zhang X, Yu H, He Z, et al. 2021c. A metamaterial beam with inverse nonlinearity for broadband micro-vibration attenuation. Mechanical Systems and Signal Processing, 159: 107826. doi: 10.1016/j.ymssp.2021.107826
[292]	Zhang Y, Deshmukh A, Wang K-W 2023. Embodying multifunctional mechano-intelligence in and through phononic metastructures harnessing physical reservoir computing. Advanced Science, 10: 2305074.
[293]	Zhang Z, Gao P, Liu W, et al. 2022. Structured sonic tube with carbon nanotube-like topological edge states. Nature Communications, 13: 5096. doi: 10.1038/s41467-022-32777-0
[294]	Zhao B, Thomsen H R, Pu X, et al. 2024. A nonlinear damped metamaterial: Wideband attenuation with nonlinear bandgap and modal dissipation. Mechanical Systems and Signal Processing, 208: 111079. doi: 10.1016/j.ymssp.2023.111079
[295]	Zhao C, Zhang K, Zhao P, et al. 2023a. Finite-amplitude nonlinear waves in inertial amplification metamaterials: Theoretical and numerical analyses. Journal of Sound and Vibration, 560: 117802. doi: 10.1016/j.jsv.2023.117802
[296]	Zhao J, Zhou G, Zhang D, et al. 2023b. Integrated design of a lightweight metastructure for broadband vibration isolation. International Journal of Mechanical Sciences, 244: 108069. doi: 10.1016/j.ijmecsci.2022.108069
[297]	Zhao J, Zhou H, Yi K, et al. 2023c. Ultra-broad bandgap induced by hybrid hardening and softening nonlinearity in metastructure. Nonlinear Dynamics, 111: 17687-17707. doi: 10.1007/s11071-023-08808-w
[298]	Zhao T, Yang Z, Tian W 2023d. Tunable nonlinear metastructure with periodic bi-linear oscillators for broadband vibration suppression. Thin-Walled Structures, 191: 110975.
[299]	Zheng S, Li T, Zhao J, et al. 2022. Deployment impact experiment and dynamic analysis of modular truss antenna. International Journal of Aerospace Engineering, 2022: 2038932.
[300]	Zheng Y, Qu Y, Dai S, et al. 2024. Mitigating vibration and sound radiation with a digital piezoelectric meta-shell in heavy fluids. Journal of Sound and Vibration, 573: 118221. doi: 10.1016/j.jsv.2023.118221
[301]	Zhou J, Dou L, Wang K, et al. 2019. A nonlinear resonator with inertial amplification for very low-frequency flexural wave attenuations in beams. Nonlinear Dynamics, 96: 647-665. doi: 10.1007/s11071-019-04812-1
[302]	Zhou J, Wang K, Xu D, et al. 2017. Local resonator with high-static-low-dynamic stiffness for lowering band gaps of flexural wave in beams. Journal of Applied Physics, 121: 044902. doi: 10.1063/1.4974299
[303]	Zhou S, Zhang R, Cheng Y, et al. 2024a. Non-reciprocity and selective wave amplification in nonlinear inertia-induced autoparametric periodic structures. Applied Mathematical Modelling, 130: 457-471. doi: 10.1016/j.apm.2024.03.005
[304]	Zhou W, Li Y, Yan G, et al. 2024b. Quasi-full bandgap generating mechanism by coupling negative stiffness and inertial amplification. European Journal of Mechanics - A/Solids, 103: 105143. doi: 10.1016/j.euromechsol.2023.105143
[305]	Zhou W, Wang Y-Z 2024. Metamaterial robot driven by nonlinear elastic waves with stop band and nonreciprocal crawling. Nonlinear Dynamics, 112: 5825-5845.
[306]	Zhou W J, Li X P, Wang Y S, et al. 2018. Spectro-spatial analysis of wave packet propagation in nonlinear acoustic metamaterials. Journal of Sound and Vibration, 413: 250-269. doi: 10.1016/j.jsv.2017.10.023
[307]	Zhou X, Liu X, Hu G 2012. Elastic metamaterials with local resonances: An overview. Theoretical and Applied Mechanics Letters, 2: 041001.
[308]	Zhou X, Zhang W, Geng X, et al. 2025. Broadband sound absorption of micro-perforated sandwich panels with hierarchical honeycomb core at high sound pressure levels. Composite Structures, 354: 118794. doi: 10.1016/j.compstruct.2024.118794
[309]	Zhu J, Gao H, Dai S, et al. 2023a. Multilayer structures for high-intensity sound energy absorption in low-frequency range. International Journal of Mechanical Sciences, 247: 108197. doi: 10.1016/j.ijmecsci.2023.108197
[310]	Zhu J, Qu Y, Gao H, et al. 2022. Nonlinear sound absorption of helmholtz resonators with serrated necks under high-amplitude sound wave excitation. Journal of Sound and Vibration, 537: 117197. doi: 10.1016/j.jsv.2022.117197
[311]	Zhu J, Qu Y, Gao H, et al. 2024. A tunable sound absorber with perfect sound absorption for suppressing acoustic waves of different intensities. Journal of Sound and Vibration, 577: 118306. doi: 10.1016/j.jsv.2024.118306
[312]	Zhu W, Deng W, Liu Y, et al. 2023b. Topological phononic metamaterials. Reports on Progress in Physics, 86: 106501. doi: 10.1088/1361-6633/aceeee
[313]	Zhuang Y, Li Q, Yang D, et al. 2024. Vibration suppression of power cabin for underwater vehicle based on mechanical metamaterial flanges. Ocean Engineering, 310: 118540. doi: 10.1016/j.oceaneng.2024.118540
[314]	陈毅, 刘晓宁, 向平, 等. 2016. 五模材料及其水声调控研究. 力学进展, 46: 201609 (Chen Y, Liu X, Xiang P, et al. 2016. Pentamode material for underwater acoustic wave control. Advances in Mechanics, 46: 201609). doi: 10.6052/1000-0992-16-010 Chen Y, Liu X, Xiang P, et al. 2016. Pentamode material for underwater acoustic wave control. Advances in Mechanics, 46: 201609. doi: 10.6052/1000-0992-16-010
[315]	方鑫 2018. 非线性声学超材料中弹性波传播理论及其减振应用研究. 国防科技大学 (Fang, X. 2018. Nonlinear acoustic metamaterials: theory of elastic wave propagation and applications on vibration reduction. National University of Defense Technology). Fang, X. 2018. Nonlinear acoustic metamaterials: theory of elastic wave propagation and applications on vibration reduction. National University of Defense Technology.
[316]	耿琳琳, 袁锦波, 程文, 等. 2024. 非厄米力学系统基本原理与研究进展. 力学进展, 54: 1-60 (Geng L L, Yuan J B, Cheng W, et al. 2024. Fundamental principles and research progress of non-Hermitian mechanical systems. Advances in Mechanics, 54: 1-60). doi: 10.6052/1000-0992-23-049 Geng L L, Yuan J B, Cheng W, et al. 2024. Fundamental principles and research progress of non-Hermitian mechanical systems. Advances in Mechanics, 54: 1-60. doi: 10.6052/1000-0992-23-049
[317]	胡更开, 刘晓宁. 弹性超材料设计与波动控制//科学出版社: 科学出版社, 2024.
[318]	胡海岩, 田强, 张伟, 等. 2013. 大型网架式可展开空间结构的非线性动力学与控制. 力学进展, 43: 390-414 (Hu H, Tian Q, Zhang W, et al. Nonlinear dynamics and control of large deployable space structures composed of trusses and meshes. 2013. Advances in Mechanics, 43: 390-414). Hu H, Tian Q, Zhang W, et al. Nonlinear dynamics and control of large deployable space structures composed of trusses and meshes. 2013. Advances in Mechanics, 43: 390-414.
[319]	马大猷. 1996. 高声强下的微穿孔板. 声学学报, 21: 10-14 (Maa D. 1996. Microperforated panel at high sound intensity. Acta Acustica, 21: 10-14). Maa D. 1996. Microperforated panel at high sound intensity. Acta Acustica, 21: 10-14.
[320]	孟光, 周徐斌, 苗军. 2016. 航天重大工程中的力学问题. 力学进展, 46: 267-322 (Meng G, Zhou X, Miao J. 2016. Mechanical problems in momentous projects of aerospace engineering. Advances in Mechanics, 46: 201606). Meng G, Zhou X, Miao J. 2016. Mechanical problems in momentous projects of aerospace engineering. Advances in Mechanics, 46: 201606.
[321]	苏常伟, 梁冉, 王雪仁, 等. 2023. 水下航行器线谱振动噪声研究进展. 舰船科学技术, 45: 1-8 (Su C, Liang R, Wang X, et al. 2023. Research progress of line spectrum vibration and noise of underwater vehicle. Ship Science and Technology, 45: 1-8). doi: 10.3404/j.issn.1672-7649.2023.08.001 Su C, Liang R, Wang X, et al. 2023. Research progress of line spectrum vibration and noise of underwater vehicle. Ship Science and Technology, 45: 1-8 doi: 10.3404/j.issn.1672-7649.2023.08.001
[322]	孙秀婷, 钱佳伟, 齐志凤, 等. 2023. 非线性隔振及时滞消振方法研究进展. 力学进展, 53: 308-356 (Sun X T, Qian J W, Qi Z F, et al. 2023. Review on research progress of nonlinear vibration isolation and time-delayed suppression method. Advances in Mechanics, 53: 308-356). doi: 10.6052/1000-0992-22-048 Sun X T, Qian J W, Qi Z F, et al. 2023. Review on research progress of nonlinear vibration isolation and time-delayed suppression method. Advances in Mechanics, 53: 308-356 doi: 10.6052/1000-0992-22-048
[323]	王凯, 周加喜, 蔡昌琦, 等. 2022. 低频弹性波超材料的若干进展. 力学学报, 54: 2678-2694 (Wang K, Zhou Jia, Cai C, et al. 2022. Review of low-frequency elastic wave metamaterials. Chinese Journal of Theoretical and Applied Mechanics, 54: 2678-2694). doi: 10.6052/0459-1879-22-108 Wang K, Zhou Jia, Cai C, et al. 2022. Review of low-frequency elastic wave metamaterials. Chinese Journal of Theoretical and Applied Mechanics, 54: 2678-2694. doi: 10.6052/0459-1879-22-108
[324]	王育人, 缪旭弘, 姜恒, 等. 2017. 水下吸声机理与吸声材料. 力学进展, 47: 92-121 (Wang Y R, Miao X H, Jiang H, et al. 2017. Review on underwater sound absorption materials and mechanisms. Advances in Mechanics, 47: 201703). Wang Y R, Miao X H, Jiang H, et al. 2017. Review on underwater sound absorption materials and mechanisms. Advances in Mechanics, 47: 201703
[325]	吴林志, 熊健, 马力, 等. 2012. 新型复合材料点阵结构的研究进展. 力学进展, 42: 41-67 (Wu L, Xiong J, Ma L, et al. 2012. Processes in the study of novel composite sandwich panels with lattice truss cores. Advances in Mechanics, 42: 41-67). doi: 10.6052/1000-0992-2012-1-lxjzJ2011-095 Wu L, Xiong J, Ma L, et al. 2012. Processes in the study of novel composite sandwich panels with lattice truss cores. Advances in Mechanics, 42: 41-67. doi: 10.6052/1000-0992-2012-1-lxjzJ2011-095
[326]	易凯军, 陈洋洋, 朱睿. 2022. 力电耦合主动超材料及其弹性波调控. 科学通报, 67: 1290-1304 (Yi K J, Chen Y Y, Zhu R, et al. 2022. Electromechanical active metamaterials and their applications in controlling elastic wave propagation. Chinese Science Bulletin, 67: 1290-1304). doi: 10.1360/TB-2021-0573 Yi K J, Chen Y Y, Zhu R, et al. 2022. Electromechanical active metamaterials and their applications in controlling elastic wave propagation. Chinese Science Bulletin, 67: 1290-1304 doi: 10.1360/TB-2021-0573
[327]	尹剑飞, 蔡力, 方鑫, 等. 2022. 力学超材料研究进展与减振降噪应用. 力学进展, 52: 508-586 (Yin J F, Cai L, Fang X, et al. 2022. Review on research progress of mechanical metamaterials and their applications in vibration and noise control. Advances in Mechanics, 52: 508-586). doi: 10.6052/1000-0992-22-005 Yin J F, Cai L, Fang X, et al. 2022. Review on research progress of mechanical metamaterials and their applications in vibration and noise control. Advances in Mechanics, 52: 508-586 doi: 10.6052/1000-0992-22-005
[328]	袁毅, 游镇宇, 陈伟球. 2021. 压电超构材料及其波动控制研究: 现状与展望. 力学学报, 53: 2101-2116 (Yuan Y, You Z, Chen W. 2021. Piezoelectric metamaterials and wave control: status quo and prospects. Chinese Journal of Theoretical and Applied Mechanics, 53: 2101-2116). doi: 10.6052/0459-1879-21-198 Yuan Y, You Z, Chen W. 2021. Piezoelectric metamaterials and wave control: status quo and prospects. Chinese Journal of Theoretical and Applied Mechanics, 53: 2101-2116. doi: 10.6052/0459-1879-21-198