Bandgap characteristics analysis and graded design of a novel metamaterial for flexural wave suppression

doi:10.1007/s10483-025-3204-7

摘要/Abstract

Abstract:

A novel elastic metamaterial is proposed with the aim of achieving low-frequency broad bandgaps and bandgap regulation. The band structure of the proposed metamaterial is calculated based on the Floquet-Bloch theorem, and the boundary modes of each bandgap are analyzed to understand the effects of each component of the unit cell on the bandgap formation. It is found that the metamaterials with a low elastic modulus of ligaments can generate flexural wave bandgaps below 300 Hz. Multi-frequency vibrations can be suppressed through the selective manipulation of bandgaps. The dual-graded design of metamaterials that can significantly improve the bandgap width is proposed based on parametric studies. A new way that can regulate the bandgap is revealed by studying the graded elastic modulus in the substrate. The results demonstrate that the nonlinear gradient of the elastic modulus in the substrate offers better bandgap performance. Based on these analyses, the proposed elastic metamaterials can pave the way for multi-frequency vibration control, low-frequency bandgap broadening, and bandgap tuning.

Key words: metamaterial, flexural wave bandgap, local resonance, graded design

中图分类号:

O421⁺.5

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 1-24.

Fan YANG, Zhaoyang MA, Xingming GUO. Bandgap characteristics analysis and graded design of a novel metamaterial for flexural wave suppression[J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 1-24.

图/表 19

Parameter	$l b / m$	$h b / m$	$l e / m$	$h e / m$	$l L / m$	$h L / m$	$R t / m$	$R L / m$
Value	0.055	0.01	0.005	0.01	0.008	0.001	0.005	0.003

Material	$ρ / (kg ⋅ m − 3)$	$E / Pa$	$ν$
Substrate	2 700	$7 × 1010$	0.33
Elastic block	100	$2.1 × 106$	0.46
Ligament	100	Varying	0.33

参考文献 55

[1]	LEE, J. and KIM, Y. Y. Elastic metamaterials for guided waves: from fundamentals to applications. Smart Materials and Structures, 32(12), 123001 (2023)
[2]	MUHAMMAD and LIM, C. W. From photonic crystals to seismic metamaterials: a review via phononic crystals and acoustic metamaterials. Archives of Computational Methods in Engineering, 29(2), 1137–1198 (2022)
[3]	MA, F. Y., WANG, C., LIU, C. R., and WU, J. H. Structural designs, principles, and applications of thin-walled membrane and plate-type acoustic/elastic metamaterials. Journal of Applied Physics, 129(23), 231103 (2021)
[4]	KUSHWAHA, M. S., HALEVI, P., DOBRZYNSKI, L., and DJAFARI-ROUHANI, B. Acoustic band structure of periodic elastic composites. Physical Review Letters, 71(13), 2022–2025 (1993)
[5]	LIU, Z. Y., ZHANG, X. X., MAO, Y. W., ZHU, Y. Y., YANG, Z. Y., CHAN, C. T., and SHENG, P. Locally resonant sonic materials. Science, 289(5485), 1734–1736 (2000)
[6]	RUPIN, M., LEMOULT, F., LEROSEY, G., and ROUX, P. Experimental demonstration of ordered and disordered multiresonant metamaterials for Lamb waves. Physical Review Letters, 112(23), 234301 (2014)
[7]	LI, H. Z., LIU, X. C., LIU, Q., LI, S., YANG, J. S., TONG, L. L., SHI, S. B., SCHMIDT, R., and SCHROEDER, K. U. Sound insulation performance of double membrane-type acoustic metamaterials combined with a Helmholtz resonator. Applied Acoustics, 205, 109297 (2023)
[8]	LIN, Q. H., LIN, Q. L., WANG, Y. H., and DI, G. Q. Sound insulation performance of sandwich structure compounded with a resonant acoustic metamaterial. Composite Structures, 273, 114312 (2021)
[9]	XIAO, Z. Q., GAO, P. L., WANG, D. W., HE, X., and WU, L. Z. Ventilated metamaterials for broadband sound insulation and tunable transmission at low frequency. Extreme Mechanics Letters, 46, 101348 (2021)
[10]	SANGIULIANO, L., REFF, B., PALANDRI, J., WOLF-MONHEIM, F., PLUYMERS, B., DECKERS, E., DESMET, W., and CLAEYS, C. Low frequency tyre noise mitigation in a vehicle using metal 3D printed resonant metamaterials. Mechanical Systems and Signal Processing, 179, 109335 (2022)
[11]	LI, Z. W., HU, H., and WANG, X. D. A new two-dimensional elastic metamaterial system with multiple local resonances. International Journal of Mechanical Sciences, 149, 273–284 (2018)
[12]	ZHOU, Y. C., YE, L., and CHEN, Y. Investigation of novel 3D-printed diatomic and local resonant metamaterials with impact mitigation capacity. International Journal of Mechanical Sciences, 206, 106632 (2021)
[13]	LIU, Z. Y., CHAN, C. T., and SHENG, P. Analytic model of phononic crystals with local resonances. Physical Review B, 71(1), 014103 (2005)
[14]	LIU, X. N., HU, G. K., HUANG, G. L., and SUN, C. T. An elastic metamaterial with simultaneously negative mass density and bulk modulus. Applied Physics Letters, 98(25), 251907 (2011)
[15]	LIU, Y. Q., SU, X. Y., and SUN, C. T. Broadband elastic metamaterial with single negativity by mimicking lattice systems. Journal of the Mechanics and Physics of Solids, 74, 158–174 (2015)
[16]	LU, L., RU, C. Q., and GUO, X. M. Vibration isolation of few-layer graphene sheets. International Journal of Solids and Structures, 185, 78–88 (2020)
[17]	LU, L., RU, C. Q., and GUO, X. M. Negative effective mass of a filled carbon nanotube. International Journal of Mechanical Sciences, 134, 174–181 (2017)
[18]	CHEN, J. S., SHARMA, B., and SUN, C. T. Dynamic behaviour of sandwich structure containing spring-mass resonators. Composite Structures, 93(8), 2120–2125 (2011)
[19]	NOUH, M., ALDRAIHEM, O., and BAZ, A. Vibration characteristics of metamaterial beams with periodic local resonances. Journal of Vibration and Acoustics-Transactions of the ASME, 136(6), 061012 (2014)
[20]	ZHU, R., LIU, X. N., HU, G. K., SUN, C. T., and HUANG, G. L. A chiral elastic metamaterial beam for broadband vibration suppression. Journal of Sound and Vibration, 333(10), 2759–2773 (2014)
[21]	YAN, Z. M., XIAO, H. J., LIU, Y. Y., and TAN, T. Band-gap dynamics and programming for low-frequency broadband elastic metamaterial. Composite Structures, 291, 115535 (2022)
[22]	CHEN, Y. M., FANG, X., WANG, J., FILIPPI, M., and CARRERA, E. An analysis of band gap characteristics of metamaterial plates with dual helix cells. Mechanics of Advanced Materials and Structures, 31(1), 92–102 (2024)
[23]	YANG, F., MA, Z. Y., and GUO, X. M. Bandgap characteristics of the two-dimensional missing rib lattice structure. Applied Mathematics and Mechanics (English Edition), 43(11), 1631–1640 (2022) https://doi.org/10.1007/s10483-022-2923-6
[24]	ZHANG, Z. and HAN, X. K. A new hybrid phononic crystal in low frequencies. Physics Letters A, 380(45), 3766–3772 (2016)
[25]	AN, X. Y., LAI, C. L., HE, W. P., and FAN, H. L. Three-dimensional chiral meta-plate lattice structures for broad band vibration suppression and sound absorption. Composites Part B: Engineering, 224, 109232 (2021)
[26]	KHEYBARI, M., DARAIO, C., and BILAL, O. R. Tunable auxetic metamaterials for simultaneous attenuation of airborne sound and elastic vibrations in all directions. Applied Physics Letters, 121(8), 081702 (2022)
[27]	AN, X. Y., LAI, C. L., FAN, H. L., and ZHANG, C. Z. 3D acoustic metamaterial-based mechanical metalattice structures for low-frequency and broadband vibration attenuation. International Journal of Solids and Structures, 191, 293–306 (2020)
[28]	GERARD, N. J., OUDICH, M., XU, Z. P., YAO, D. S., CUI, H. C., NAIFY, C. J., IKEI, A., ROHDE, C. A., ZHENG, X. Y., and JING, Y. Three-dimensional trampolinelike behavior in an ultralight elastic metamaterial. Physical Review Applied, 16(2), 024015 (2021)
[29]	YAO, D. H., XIONG, M. K., LUO, J. Y., and YAO, L. Y. Flexural wave mitigation in metamaterial cylindrical curved shells with periodic graded arrays of multi-resonator. Mechanical Systems and Signal Processing, 168, 108721 (2022)
[30]	ZHANG, L. W., BAI, Z. H., ZHANG, Q., JIN, Y., and CHEN, Y. F. On vibration isolation performance and crashworthiness of a three-dimensional lattice metamaterial. Engineering Structures, 292, 116510 (2023)
[31]	LU, X. C., WU, X. B., XIANG, H. R., SHEN, J., LI, Y. J., LI, Y. B., and WANG, X. S. Triple tunability of phononic bandgaps for three-dimensional printed hollow sphere lattice metamaterials. International Journal of Mechanical Sciences, 221, 107166 (2022)
[32]	ASSOUAR, M. B. and OUDICH, M. Enlargement of a locally resonant sonic band gap by using double-sides stubbed phononic plates. Applied Physics Letters, 100(12), 123506 (2012)
[33]	LI, S. B., CHEN, T. N., WANG, X. P., LI, Y. G., and CHEN, W. H. Expansion of lower-frequency locally resonant band gaps using a double-sided stubbed composite phononic crystals plate with composite stubs. Physics Letters A, 380(25-26), 2167–2172 (2016)
[34]	XIAO, Y., WEN, J. H., and WEN, X. S. Longitudinal wave band gaps in metamaterial-based elastic rods containing multi-degree-of-freedom resonators. New Journal of Physics, 14, 033042 (2012)
[35]	CHEN, H., LI, X. P., CHEN, Y. Y., and HUANG, G. L. Wave propagation and absorption of sandwich beams containing interior dissipative multi-resonators. Ultrasonics, 76, 99–108 (2017)
[36]	JIN, Y., JIA, X. Y., WU, Q. Q., HE, X., YU, G. C., WU, L. Z., and LUO, B. L. Design of vibration isolators by using the Bragg scattering and local resonance band gaps in a layered honeycomb meta-structure. Journal of Sound and Vibration, 521, 116721 (2022)
[37]	ZHANG, P. and TO, A. C. Broadband wave filtering of bioinspired hierarchical phononic crystal. Applied Physics Letters, 102(12), 121910 (2013)
[38]	HAO, S. T., SHENG, H., LIU, X. S., LI, H. Q., LI, S. H., and DING, Q. Low-frequency and broadband vibration absorption of a metamaterial plate with acoustic black hole resonators. Thin-Walled Structures, 202, 112073 (2024)
[39]	LI, X. P., CHEN, Y. Y., HU, G. K., and HUANG, G. L. A self-adaptive metamaterial beam with digitally controlled resonators for subwavelength broadband flexural wave attenuation. Smart Materials and Structures, 27(4), 045015 (2018)
[40]	LIN, L. F., LU, Z. Q., ZHAO, L., ZHENG, Y. S., DING, H., and CHEN, L. Q. Vibration isolation of mechatronic metamaterial beam with resonant piezoelectric shunting. International Journal of Mechanical Sciences, 254, 108448 (2023)
[41]	BELI, D., RUZZENE, M., and DE MARQUI, C. Bridging-coupling phenomenon in linear elastic metamaterials by exploiting locally resonant metachain isomers. Physical Review Applied, 14(3), 034032 (2020)
[42]	BANERJEE, A., DAS, R., and CALIUS, E. P. Frequency graded 1D metamaterials: a study on the attenuation bands. Journal of Applied Physics, 122(7), 075101 (2017)
[43]	LIU, C. C. and REINA, C. Broadband locally resonant metamaterials with graded hierarchical architecture. Journal of Applied Physics, 123(9), 095108 (2018)
[44]	WANG, Y. H., YANG, J., CHEN, Z. X., LIN, Y., GONG, L. P., ZHANG, S. W., LI, W. H., and SUN, S. S. Investigation of a magnetorheological elastomer metamaterial sandwich beam with tunable graded stiffness for broadband vibration attenuation. Smart Materials and Structures, 32(6), 065022 (2023)
[45]	AN, X. Y., YUAN, X. F., SUN, G. Q., HE, W. P., LAI, C. L., HOU, X. X., and FAN, H. L. Sandwich plate-type metastructures with periodic graded resonators for low-frequency and broadband vibration attenuation. Ocean Engineering, 298, 117229 (2024)
[46]	LI, X. F., CHENG, S. L., YANG, H. Y., YAN, Q., WANG, B., XIN, Y. J., SUN, Y. T., DING, Q., YAN, H., and LI, Y. J. Analysis of low frequency vibration attenuation and wave propagation mechanism of graded maze structure. Physica B: Condensed Matter, 649, 414519 (2023)
[47]	JIANG, W. F., YIN, M., LIAO, Q. H., XIE, L. F., and YIN, G. F. Three-dimensional single-phase elastic metamaterial for low-frequency and broadband vibration mitigation. International Journal of Mechanical Sciences, 190, 106023 (2021)
[48]	BANERJEE, A. Flexural waves in graded metabeam lattice. Physics Letters A, 388, 127057 (2021)
[49]	JIAN, Y. P., HU, G. B., TANG, L. H., TANG, W., ABDI, M., and AW, K. C. Analytical and experimental study of a metamaterial beam with grading piezoelectric transducers for vibration attenuation band widening. Engineering Structures, 275, 115091 (2023)
[50]	JIAN, Y. P., TANG, L. H., HU, G. B., LI, Z. Y., and AW, K. C. Design of graded piezoelectric metamaterial beam with spatial variation of electrodes. International Journal of Mechanical Sciences, 218, 107068 (2022)
[51]	SCHIMIDT, C. S., DE OLIVEIRA, L. P. R., and DE MARQUI, C. Vibro-acoustic performance of graded piezoelectric metamaterial plates. Composite Structures, 327, 117656 (2024)
[52]	YANG, N., LI, N. B., WANG, L., and LI, B. W. Thermal rectification and negative differential thermal resistance in lattices with mass gradient. Physical Review B, 76(2), 020301 (2007)
[53]	MELO, F., JOB, S., SANTIBANEZ, F., and TAPIA, F. Experimental evidence of shock mitigation in a Hertzian tapered chain. Physical Review E, 73(4), 041305 (2006)
[54]	HU, G. B., AUSTIN, A. C. M., SOROKIN, V., and TANG, L. H. Metamaterial beam with graded local resonators for broadband vibration suppression. Mechanical Systems and Signal Processing, 146, 106982 (2021)
[55]	KRUSHYNSKA, A. O., KOUZNETSOVA, V. G., and GEERS, M. G. D. Visco-elastic effects on wave dispersion in three-phase acoustic metamaterials. Journal of the Mechanics and Physics of Solids, 96, 29–47 (2016)