Figure D-1. (a) the parameter map of the unit cell. (b) Dispersion curves for the specific cases in the map. (c) Mode shape of each case at the cut-off frequency where k=0.
Here, the dispersion curves are plotted to explain why the mass part should not be deformed. As mentioned in the previous section, the mass part in the mass-spring system is assumed as a rigid body, and the spring part is assumed to have negligible mass. If this condition is not satisfied, the unit cell no longer behaves as the discrete mass and spring; it behaves as a continuum structure. Therefore, the theoretical analysis doesn’t hold anymore. In general, it depends on the relative mass and stiffness of the spring part and mass part of the metamaterial. In other words, the mass part should be heavy and stiff enough compared to the spring part. Thus, it is important to check whether the metamaterial follows discrete model or continuum model.
Thus, the mode shape of the unit cell having various combinations of the thicknesses of the cylinder and spring is observed as shown in Figure D-1. The parameter map shows the range of the spring thickness and cylinder thickness of the unit cell in Figure 3-7(a). The spring thickness is defined as
(
3)
spring 2
t = h R- from the geometry of the unit cell illustrated in Figure 3-7(a). In Figure 3-7(d), the R3 varies from 18 mm to 19 mm, so that the thickness of the spring varies from 2 mm to 4 mm as shown in Figure D-1(a). In similar, the thickness of cylinder is defined as tcylinder =R1-R2 where R1 has a fixed value 30 mm, and R2 varies from 10 mm to 20 mm. Thus, the thickness of the cylinder varies from 10 mm to 20 mm as shown in Figure D-1(a).
The highlighted region with a red box shows the possible area of the unit cell dimension. In the possible region, the four edge points are selected as the extreme case to show the effect of breaking the rigid body assumption. The four-dispersion curve of the unit cell having the extreme parameters are plotted as shown in Figure D-1(b). It is notable that the case of (iv), whose unit cell having the thickest spring of thickness 4 mm and the thinnest cylinder of thickness 10 mm, shows significant error between analytically calculated dispersion and numerically calculated one, especially at the high frequency. The reason of this error can be seen in the mode shape of the unit cell at the cutoff frequency where k=0 as shown in Figure D-1(c). Unlike other cases, the unit cell in the case (iv) shows significant deformation on the mass part. As mentioned above, if the significant deformation occurs in the mass part, the theoretical analysis based on the discrete mass-spring system does not hold.
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Figure D-2. (a-i) Dispersion curves of the unit cells which has the different thicknesses in the cylinder and spring corresponding to the parameter map in left side.
We can find this tendency in the results from more cases as shown in Figure D-2. The dispersion curves in Figure D-2(a-i) show the analytic dispersion curve and the numerical one. Unlike the other cases, the case (h) and (i) show the noticeable error between two dispersion curves. Therefore, we can conclude that there is certain line where the mass part and spring part cannot be considered as the idle mass and spring for the discrete system. If we cross that line, the analytic prediction shows significant error compared to the numerical result at the cut-off frequency where k=0. In addition, it should be warned that the criteria can be only valid in very specific shape used in this thesis.
Although this error at the cut-off frequency, however, the analytically predicted bandgap frequency shows good agreement with the simulated one; the analytically calculated bandgap is 1190 Hz to 5219 Hz and the simulated bandgap is 1063 Hz to 5086 Hz in the case (i). Therefore, the extended mass- spring system and its analysis are still valid on the interested range, which is highlighted with the red box, of the geometry shown in Figure D-1(a).
References
[1]Zhu, R., Liu, X. N., Hu, G. K., Sun, C. T., & Huang, G. L. (2014). A chiral elastic metamaterial beam for broadband vibration suppression.Journal of Sound and Vibration,333(10), 2759-2773.
[2] 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.
[3] D’Alessandro, L., Ardito, R., Braghin, F., & Corigliano, A. (2019). Low frequency 3D ultra-wide vibration attenuation via elastic metamaterial.Scientific Reports,9(1), 8039.
[4] Carrara, M., Cacan, M. R., Toussaint, J., Leamy, M. J., Ruzzene, M., & Erturk, A. (2013).
Metamaterial-inspired structures and concepts for elastoacoustic wave energy harvesting.Smart Materials and Structures,22(6), 065004.
[5] Sugino, C., Leadenham, S., Ruzzene, M., & Erturk, A. (2016). On the mechanism of bandgap formation in locally resonant finite elastic metamaterials.Journal of Applied Physics,120(13), 134501.
[6] Garcia-Pablos, D., Sigalas, M., Montero de Espinosa, F. R., Torres, M., Kafesaki, M., & Garcia, N.
(2000). Theory and experiments on elastic band gaps.Physical Review Letters,84(19), 4349.
[7] Beli, D., Arruda, J. R. F., & Ruzzene, M. (2018). Wave propagation in elastic metamaterial beams and plates with interconnected resonators.International Journal of Solids and Structures,139-140, 105-120.
[8] Chen, Y., Hu, G., & Huang, G. (2017). A hybrid elastic metamaterial with negative mass density and tunable bending stiffness.Journal of the Mechanics and Physics of Solids,105, 179-198.
[9] Oh, J. H., Seung, H. M., & Kim, Y. Y. (2014). A truly hyperbolic elastic metamaterial lens.Applied Physics Letters,104(7), 073503.
[10] Liu, X. N., Hu, G. K., Huang, G. L., & Sun, C. T. (2011). An elastic metamaterial with simultaneously negative mass density and bulk modulus.Applied Physics Letters,98(25), 251907.
[11] Gusev, V. E., & Wright, O. B. (2014). Double-negative flexural acoustic metamaterial.New Journal of Physics,16(12), 123053.
[12] Oh, J. H., Seung, H. M., & Kim, Y. Y. (2017). Doubly negative isotropic elastic metamaterial for sub-wavelength focusing: Design and realization.Journal of Sound and Vibration,410, 169-186.
[13] Oh, J. H., Kwon, Y. E., Lee, H. J., & Kim, Y. Y. (2016). Elastic metamaterials for independent realization of negativity in density and stiffness.Scientific Reports,6(1), 23630.
[14] Wang W., Bonello, B., Djafari-Rouhani, B., Pennec, Y., & Zhao, J. (2018). Double-negative pillared elastic metamaterial. Physical Review Applied, 10(6), 064011.
[15] Wu, Y., Lai, Y., & Zhang, Z. Q. (2011). Elastic metamaterials with simultaneously negative effective shear modulus and mass density.Physical Review Letters,107(10), 105506.
67
[16] Lee, H., Oh, J. H., Seung, H. M., Cho, S. H., & Kim, Y. Y. (2016). Extreme stiffness hyperbolic elastic metamaterial for total transmission subwavelength imaging.Scientific Reports,6(1), 24026.
[17] Pendry, J. B. (2000). Negative refraction makes a perfect lens.Physical Review Letters,85(18), 3966.
[18] Zhu, R., Liu, X. N., Hu, G. K., Sun, C. T., & Huang, G. L. (2014). Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial.Nature Communications, 5(1), 5510.
[19] Aydin, K., Bulu, I., & Ozbay, E. (2007). Subwavelength resolution with a negative-index metamaterial superlens.Applied Physics Letters,90(25), 254102.
[20] Jeon, G. J., & Oh, J. H. (2021). Elastic Coiling-up-Space Metamaterial.Physical Review Applied,16(6), 064016.
[21] Liang, Z., & Li, J. (2012). Extreme acoustic metamaterial by coiling up space.Physical Review Letters,108(11), 114301.
[22] Liang, Z., Feng, T., Lok, S., Liu, F., Ng, K. B., Chan, C. H., Wang, J., Han, S., Lee, S., & Li, J.
(2013). Space-coiling metamaterials with double negativity and conical dispersion.Scientific Reports,3(1), 1614.
[23] Attarzadeh, M. A., Callanan, J., & Nouh, M. (2020). Experimental observation of nonreciprocal waves in a resonant metamaterial beam.Physical Review Applied,13(2), 021001.
[24] Brandenbourger, M., Locsin, X., Lerner, E., & Coulais, C. (2019). Non-reciprocal robotic metamaterials.Nature Communications,10(1), 4608.
[25] Chen, Y., Li, X., Nassar, H., Norris, A. N., Daraio, C., & Huang, G. (2019). Nonreciprocal wave propagation in a continuum-based metamaterial with space-time modulated resonators.Physical Review Applied,11(6), 064052.
[26] Nassar, H., Chen, H., Norris, A. N., Haberman, M. R., & Huang, G. L. (2017). Non-reciprocal wave propagation in modulated elastic metamaterials.Proceedings of the Royal Society A:
Mathematical, Physical and Engineering Sciences,473, 20170188.
[27] Nassar, H., Yousefzadeh, B., Fleury, R., Ruzzene, M., Alù, A., Daraio, C., Norris, A. N., Huang G., & Haberman, M. R. (2020). Nonreciprocity in acoustic and elastic materials.Nature Reviews Materials,5(9), 667-685.
[28] Wu, L. Y., Chen, L. W., & Liu, C. M. (2009). Acoustic energy harvesting using resonant cavity of a sonic crystal.Applied Physics Letters,95(1), 013506.
[29] Jo, S. H., Yoon, H., Shin, Y. C., Choi, W., Park, C. S., Kim, M., & Youn, B. D. (2020). Designing a phononic crystal with a defect for energy localization and harvesting: Supercell size and defect location.International Journal of Mechanical Sciences,179, 105670.
[30] Park, C. S., Shin, Y. C., Jo, S. H., Yoon, H., Choi, W., Youn, B. D., & Kim, M. (2019). Two- dimensional octagonal phononic crystals for highly dense piezoelectric energy harvesting.Nano Energy,57, 327-337.
[31] Lv, H., Tian, X., Wang, M. Y., & Li, D. (2013). Vibration energy harvesting using a phononic crystal with point defect states.Applied Physics Letters,102(3), 034103.
[32] Xiao, Y., Wen, J., & Wen, X. (2012). Broadband locally resonant beams containing multiple periodic arrays of attached resonators.Physics Letters A,376(16), 1384-1390.
[33] Tian, Y., Wu, J. H., Li, H., Gu, C., Yang, Z., Zhao, Z., & Lu, K. (2019). Elastic wave propagation in the elastic metamaterials containing parallel multi-resonators.Journal of Physics D: Applied Physics,52(39), 395301.
[34] Oh, J. H., Qi, S., Kim, Y. Y., & Assouar, B. (2017). Elastic metamaterial insulator for broadband low-frequency flexural vibration shielding.Physical Review Applied,8(5), 054034.
[35] Oh, J. H., & Assouar, B. (2016). Quasi-static stop band with flexural metamaterial having zero rotational stiffness.Scientific Reports,6(1), 33410.
[36] Chen, Y. Y., Hu, G. K., & Huang, G. L. (2016). An adaptive metamaterial beam with hybrid shunting circuits for extremely broadband control of flexural waves.Smart Materials and Structures,25(10), 105036.
[37] Chen, Y. Y., Huang, G. L., & Sun, C. T. (2014). Band gap control in an active elastic metamaterial with negative capacitance piezoelectric shunting.Journal of Vibration and Acoustics,136(6), 061008.
[38] Wang, T., Sheng, M. P., Guo, Z. W., & Qin, Q. H. (2016). Flexural wave suppression by an acoustic metamaterial plate.Applied Acoustics,114, 118-124.
[39] Bae, M. H., Choi, W., Ha, J. M., Kim, M., & Seung, H. M. (2022). Extremely low frequency wave localization via elastic foundation induced metamaterial with a spiral cavity.Scientific Reports,12(1), 3993.
[40] Mu, D., Wang, K., Shu, H., & Lu, J. (2022). Low frequency broadband bandgaps in elastic metamaterials with two-stage inertial amplification and elastic foundations.Journal of Physics D:
Applied Physics.55, 345302.
[41] Lee, H., Oh, J. H., Seung, H. M., Cho, S. H., & Kim, Y. Y. (2016). Extreme stiffness hyperbolic elastic metamaterial for total transmission subwavelength imaging.Scientific Reports,6(1), 1-12.
69
[42] Lee, S. W., Shin, Y. J., Park, H. W., Seung, H. M., & Oh, J. H. (2021). Full-wave tailoring between different elastic media: a double-unit elastic metasurface.Physical Review Applied,16(6), 064013.
[43] Lee, S. W., & Oh, J. H. (2020). Single-layer elastic metasurface with double negativity for anomalous refraction.Journal of Physics D: Applied Physics,53(26), 265301.
[44] Xu, X., Barnhart, M. V., Fang, X., Wen, J., Chen, Y., & Huang, G. (2019). A nonlinear dissipative elastic metamaterial for broadband wave mitigation.International Journal of Mechanical Sciences,164, 105159.
[45] Graff, K. F. (1975) Wave Motion in Elastic Solids, Dover Publications Inc., New York.
[46] Park, H. W., & Oh, J. H. (2019). Study of abnormal group velocities in flexural metamaterials.Scientific Reports,9(1), 1-13.
[47] Qi, S., & Assouar, B. (2017). Acoustic energy harvesting based on multilateral metasurfaces.Applied Physics Letters,111(24), 243506.
[48] Jin, Y., Wang, W., Khelif, A., & Djafari-Rouhani, B. (2021). Elastic metasurfaces for deep and robust subwavelength focusing and imaging.Physical Review Applied,15(2), 024005.
[49] Tol, S., Degertekin, F. L., & Erturk, A. (2016). Gradient-index phononic crystal lens-based enhancement of elastic wave energy harvesting.Applied Physics Letters,109(6), 063902.
[50] Hyun, J., Choi, W., & Kim, M. (2019). Gradient-index phononic crystals for highly dense flexural energy harvesting.Applied Physics Letters,115, 173901.
[51] Allam, A., Sabra, K., & Erturk, A. (2021). Sound energy harvesting by leveraging a 3D-printed phononic crystal lens.Applied Physics Letters,118, 103504.
[52] Jo, S. H., Yoon, H., Shin, Y. C., & Youn, B. D. (2020). A graded phononic crystal with decoupled double defects for broadband energy localization.International Journal of Mechanical Sciences,183, 105833.
[53] Li, Y., Chen, T., Wang, X., Ma, T., & Jiang, P. (2014). Acoustic confinement and waveguiding in two-dimensional phononic crystals with material defect states.Journal of Applied Physics,116(2), 024904.
[54] Wu, L. Y., Chen, L. W., & Liu, C. M. (2009). Acoustic energy harvesting using resonant cavity of a sonic crystal.Applied Physics Letters,95(1), 013506.
[55] Jo, S. H., Yoon, H., Shin, Y. C., & Youn, B. D. (2021). An analytical model of a phononic crystal with a piezoelectric defect for energy harvesting using an electroelastically coupled transfer matrix.International Journal of Mechanical Sciences,193, 106160.
[56] Jiang, P., Wang, X. P., Chen, T. N., & Zhu, J. (2015). Band gap and defect state engineering in a multi-stub phononic crystal plate.Journal of Applied Physics,117(15), 154301.
[57] Jo, S. H., Yoon, H., Shin, Y. C., Kim, M., & Youn, B. D. (2020). Elastic wave localization and harvesting using double defect modes of a phononic crystal.Journal of Applied Physics,127(16), 164901.
[58] Ma, T. X., Fan, Q. S., Li, Z. Y., Zhang, C., & Wang, Y. S. (2020). Flexural wave energy harvesting by multi-mode elastic metamaterial cavities.Extreme Mechanics Letters,41, 101073.
[59] Lv, X. F., Fang, X., Zhang, Z. Q., Huang, Z. L., & Chuang, K. C. (2019). Highly localized and efficient energy harvesting in a phononic crystal beam: Defect placement and experimental validation.Crystals,9(8), 391.
[60] Colombi, A., Roux, P., & Rupin, M. (2014). Sub-wavelength energy trapping of elastic waves in a metamaterial.The Journal of the Acoustical Society of America,136(2), EL192-EL198.
[61] Wen, Z., Jin, Y., Gao, P., Zhuang, X., Rabczuk, T., & Djafari-Rouhani, B. (2022). Topological cavities in phononic plates for robust energy harvesting.Mechanical Systems and Signal Processing,162, 108047.
[62] Lv, H., Tian, X., Wang, M. Y., & Li, D. (2013). Vibration energy harvesting using a phononic crystal with point defect states.Applied Physics Letters,102(3), 034103.
[63] Aalami, B., & Atzori, B. (1974). Flexural vibrations and Timoshenko's beam theory. AIAA Journal, 12(5), 679-685.
[64] Veselago, V. G. (1967). The electrodynamics of substances with simultaneously negative values of e and m. Soviet Physics Uspekhi10(4), 509-514.
[65] Ren, X., Das, R., Tran, P., Ngo, T. D., & Xie, Y. M. (2018). Auxetic metamaterials and structures:
a review. Smart Materials and Structures, 27(2), 023001.
[66] Lv, C., Krishnaraju, D., Konjevod, G., Yu, H., & Jiang, H. (2014). Origami based mechanical metamaterials. Scientific Reports, 4, 5979.
[67] Wang, P., Casadei, F., Shan, S., Weaver, J. C., & Bertoldi, K. (2014). Harnessing buckling to design tunable locally resonant acoustic metamaterials. Physical Review Letters, 113, 014301.
[68] Li, Y., Li, W., Han, T., Zheng, X., Li, J., Li, B., Fan, S. & Qiu, C. W. (2021). Transforming heat transfer with thermal metamaterials and devices. Nature Reviews Materials, 6, 488-507.
[69] Oudich, M., & Li, Y. (2017). Tunable sub-wavelength acoustic energy harvesting with a metamaterial plate. Journal of Physics D: Applied Physics, 50, 315104.
71
Acknowledgement
Thank you for everyone!