Tunable three-dimensional nonreciprocal transmission in a layered nonlinear elastic wave metamaterial by initial stresses

doi:10.1007/s10483-021-2808-9

Abstract

Abstract: In this work, the three-dimensional (3D) propagation behaviors in the nonlinear phononic crystal and elastic wave metamaterial with initial stresses are investigated. The analytical solutions of the fundamental wave and second harmonic with the quasi-longitudinal (qP) and quasi-shear (qS₁ and qS₂) modes are derived. Based on the transfer and stiffness matrices, band gaps with initial stresses are obtained by the Bloch theorem. The transmission coefficients are calculated to support the band gap property, and the tunability of the nonreciprocal transmission by the initial stress is discussed. This work is expected to provide a way to tune the nonreciprocal transmission with vector characteristics.

Key words: nonlinear elastic wave metamaterial, nonreciprocal transmission, threedimensional (3D) elastic wave, initial stress

2010 MSC Number:

74J30

Zhenni LI, Yize WANG, Yuesheng WANG. Tunable three-dimensional nonreciprocal transmission in a layered nonlinear elastic wave metamaterial by initial stresses. Applied Mathematics and Mechanics (English Edition), 2022, 43(2): 167-184.

TrendMD

References

[1] KHELIF, A., AOUBIZA, B., MOHAMMADI, S., ADIBI, A., and LAUDE, V. Complete band gaps in two-dimensional phononic crystal slabs. Physical Review E, 74, 046610 (2006)
[2] KUTSENKO, A. A., SHUVALOV, A. L., and NORRIS, A. N. Evaluation of the effective speed of sound in phononic crystals by the monodromy matrix method. Journal of the Acoustical Society of America, 130, 3553-3557 (2011)
[3] KUTSENKO, A. A., SHUVALOV, A. L., and NORRIS, A. N. Converging bounds for the effective shear speed in 2D phononic crystals. Journal of Elasticity, 113, 179-191 (2013)
[4] JANDRON, M. and HENANN, D. L. A numerical simulation capability for electroelastic wave propagation in dielectric elastomer composites: application to tunable soft phononic crystals. International Journal of Solids and Structures, 150, 1-21 (2018)
[5] MADEO, A., COLLET, M., MINIACI, M., BILLON, K., OUISSE, M., and NEFF, P. Modeling phononic crystals via the weighted relaxed micromorphic model with free and gradient micro-inertia. Journal of Elasticity, 130, 59-83 (2018)
[6] LAZAROV, B. S. and JENSEN, J. S. Low-frequency band gaps in chains with attached non-linear oscillators. International Journal of Non-Linear Mechanics, 42, 1186-1193 (2007)
[7] VONDREJC, J., ROHAN, E., and HECZKO, J. Shape optimization of phononic band gap structures using the homogenization approach. International Journal of Solids and Structures, 113, 147-168 (2017)
[8] DOMINO, L., TARPIN, M., PATINET, S., and EDDI, A. Faraday wave lattice as an elastic metamaterial. Physical Review E, 93, 050202 (2016)
[9] WEI, C. Q., YAN, Z. Z., ZHENG, H., and ZHANG, C. Z. RBF collocation method and stability analysis for phononic crystals. Applied Mathematics and Mechanics (English Edition), 37(5), 627-638 (2016) https://doi.org/10.1007/s10483-016-2076-8
[10] MO, C. Y., SINGH, J., RANEY, J. R., and PUROHIT, P. K. Cnoidal wave propagation in an elastic metamaterial. Physical Review E, 100, 013001 (2019)
[11] MIRANDA, E. J. P. and DOS SANTOS, J. M. C. Flexural wave band gaps in multi-resonator elastic metamaterial Timoshenko beams. Wave Motion, 91, 102391 (2019)
[12] WU, Z. J., LIU, W. Y., LI, F. M., and ZHANG, C. Band-gap property of a novel elastic metamaterial beam with X-shaped local resonators. Mechanical Systems and Signal Processing, 134, 106357 (2019)
[13] BEHRAVAN-RAD, A. and JAFARI, M. Hygroelasticity analysis of an elastically restrained functionally graded porous metamaterial circular plate resting on an auxetic material circular plate. Applied Mathematics and Mechanics (English Edition), 41(9), 1359-1380 (2020) https://doi.org/10.1007/s10483-020-2651-7
[14] ZHAO, P. C., ZHANG, K., ZHAO, C., and DENG, Z. C. Multi-resonator coupled metamaterials for broadband vibration suppression. Applied Mathematics and Mechanics (English Edition), 42(1), 53-64 (2021) https://doi.org/10.1007/s10483-021-2684-8
[15] EL-BORGI, S., FERNANDES, R., RAJENDRAN, P., YAZBECK, R., BOYD, J. G., and LAGOUDAS, D. C. Multiple bandgap formation in a locally resonant linear metamaterial beam: theory and experiments. Journal of Sound and Vibration, 488, 115647 (2020)
[16] DENG, M. X. and XIANG, Y. X. Analysis of second-harmonic generation by primary horizontal shear modes in layered planar structures with imperfect interfaces. Journal of Applied Physics, 113, 043513 (2013)
[17] IWAI, A., NAKAMURA, Y., and SAKAI, O. Enhanced generation of a second-harmonic wave in a composite of metamaterial and microwave plasma with various permittivities. Physical Review E, 92, 033105 (2015)
[18] LI, Y. F., LAN, J., LI, B. S., LIU, X. Z., and ZHANG, J. S. Nonlinear effects in an acoustic metamaterial with simultaneous negative modulus and density. Journal of Applied Physics, 120, 145105 (2016)
[19] CHAUNSALI, R., TOLES, M., YANG, J. Y., and KIM, E. Extreme control of impulse transmission by cylinder-based nonlinear phononic crystals. Journal of the Mechanics and Physics of Solids, 107, 21-32 (2017)
[20] BANERJEE, A., CALIUS, E. P., and DAS, R. Impact based wideband nonlinear resonating metamaterial chain. International Journal of Non-Linear Mechanics, 103, 138-144 (2018)
[21] LIANG, B., ZOU, X. Y., YUAN, B., and CHENG, J. C. Frequency-dependence of the acoustic rectifying efficiency of an acoustic diode model. Applied Physics Letters, 96, 233511 (2010)
[22] LIANG, B., GUO, X. S., TU, J., ZHANG, D., and CHENG, J. C. An acoustic rectifier. Nature Materials, 9, 989-992 (2010)
[23] BOECHLER, N., THEOCHARIS, G., and DARAIO, C. Bifurcation-based acoustic switching and rectification. Nature Materials, 10, 665-668 (2011)
[24] LI, X. F., NI, X., FENG, L., LU, M. H., HE, C., and CHEN, Y. F. Tunable unidirectional sound propagation through a sonic-crystal-based acoustic diode. Physical Review Letters, 106, 084301 (2011)
[25] LUO, B. B., GAO, S., LIU, J. H., MAO, Y. W., LI, Y. F., and LIU, X. Z. Nonreciprocal wave propagation in one-dimensional nonlinear periodic structures. AIP Advances, 8, 015113 (2018)
[26] GRINBERG, I., VAKAKIS, A. F., and GENDELMAN, O. V. Acoustic diode: wave non-reciprocity in nonlinearly coupled waveguides. Wave Motion, 83, 49-66 (2018)
[27] KONARSKI, S. G., HABERMAN, M. R., and HAMILTON, M. F. Frequency-dependent behavior of media containing pre-strained nonlinear inclusions: application to nonlinear acoustic metamaterials. Journal of the Acoustical Society of America, 144, 3022-3035 (2018)
[28] DARABI, A., FANG, L. Z., MOJAHED, A., FRONK, M. D., VAKAKIS, A. F., and LEAMY, M. J. Broadband passive nonlinear acoustic diode. Physical Review B, 99, 214305 (2019)
[29] CHEN, Y. J., WU, B., SU, Y. P., and CHEN, W. Q. Tunable two-way unidirectional acoustic diodes: design and simulation. Journal of Applied Mechanics, 86, 031010 (2019)
[30] WALLEN, S. P. and HABERMAN, M. R. Nonreciprocal wave phenomena in spring-mass chains with effective stiffness modulation induced by geometric nonlinearity. Physical Review E, 99, 031001 (2019)
[31] CHATTERJEE, M., DHUA, S., CHATTOPADHYAY, A., and SAHU, S. A. Reflection and refraction for three-dimensional plane waves at the interface between distinct anisotropic half-spaces under initial stresses. International Journal of Geomechanics, 16, 04015099 (2016)
[32] ZHANG, Z., HAN, X. K., and JI, G. M. Mechanism for controlling the band gap and the flat band in three component phononic crystals. Journal of Physics and Chemistry of Solids, 123, 235-241 (2018)
[33] GUO, X., JI, S. S., LIU, H., and REN, K. Dispersion relations of elastic waves in three-dimensional cubical piezoelectric phononic crystal with initial stresses and mechanically and dielectrically imperfect interfaces. Applied Mathematical Modelling, 69, 405-424 (2019)
[34] FOMENKO, S. I., GOLUB, M. V., CHEN, A., WANG, Y. S., and ZHANG, C. Band-gap and pass-band classification for oblique waves propagating in a three-dimensional layered functionally graded piezoelectric phononic crystal. Journal of Sound and Vibration, 439, 219-240 (2019)
[35] LI, Z. N., WANG, Y. Z., and WANG, Y. S. Three-dimensional nonreciprocal transmission in a layered nonlinear elastic wave metamaterial. International Journal of Non-Linear Mechanics, 125, 103531 (2020)
[36] WANG, Y. Z., LI, F. M., and KISHIMOTO, K. Effects of the initial stress on the propagation and localization properties of Rayleigh waves in randomly disordered layered piezoelectric phononic crystals. Acta Mechanica, 216, 291-300 (2011)
[37] GUO, X. and WEI, P. J. Dispersion relations of elastic waves in one-dimensional piezoelectric phononic crystal with initial stresses. International Journal of Mechanical Sciences, 106, 231-244 (2016)
[38] BARNWELL, E. G., PARNELL, W. J., and ABRAHAMS, I. D. Antiplane elastic wave propagation in pre-stressed periodic structures; tuning, band gap switching and invariance. Wave Motion, 63, 98-110 (2016)
[39] ROSE, J. L. Ultrasonic Waves in Solid Media, Cambridge University Press, Cambridge (1999)
[40] NORRIS, A. N. Symmetry conditions for third order elastic moduli and implications in nonlinear wave theory. Journal of Elasticity, 25, 247-257 (1991)
[41] DENG, M. X. Cumulative second-harmonic generation of Lamb-mode propagation in a solid plate. Journal of Applied Physics, 85, 3051-3058 (1999)
[42] ROKHLIN, S. I. and WANG, L. Stable recursive algorithm for elastic wave propagation in layered anisotropic media: stiffness matrix method. Journal of the Acoustical Society of America, 112, 822-834 (2002)
[43] TAN, E. L. Stiffness matrix method with improved efficiency for elastic wave propagation in layered anisotropic media. Journal of the Acoustical Society of America, 118, 3400-3403 (2005)
[44] CHATTOPADHYAY, A. Wave reflection and refraction in triclinic crystalline media. Archive of Applied Mechanics, 73, 568-579 (2004)
[45] QUINTANILLA, F. H., LOWE, M. J. S., and CRASTER, R. V. Full 3D dispersion curve solutions for guided waves in generally anisotropic media. Journal of Sound and Vibration, 363, 545-559 (2016)