多類型複合地震超結構之寬頻帶設計與分析

SEISMIC METAMATERIALS MADE OF AN ARRAY OF MULTIPLE CELL UNITS WITH BROAD BAND GAP

10.6652/JoCICHE.201903_31(1).0009

吳逸軒(Yi-Hsuan Wu)；汪向榮(Shiang-Jung Wang)；張國鎮(Kuo-Chun Chang)；陳東陽(Tung-Yang Chen)

地震超材料 ； 局部共振 ； 帶隙 ； seismic metamaterials ； local resonance ； band gap

中國土木水利工程學刊

31卷1期（2019 / 03 / 01）

103 - 118

繁體中文

地震超材料（seismic metamaterial）或地震超結構（seismic metastructure）為一種使用在既有建物外部的隔減振方法，其概念最早起源於電磁波領域，目的在於利用超材料的特殊性質控制或影響電磁波，而隨著研究發展與超材料尺度的增大，使得超材料的應用範疇延伸到聲學以及彈性波等領域。雖然此概念仍舊處於理論階段，但不同於傳統的隔減振方法，地震超材料具有可以使用在既有的建物外部，還能保護一個較大的區域而非單一結構物等優點，所以在近幾年，這種用以衰減或導引地震波的方法成為諸多力學領域學者所研究的新興議題。在地震波低頻的特性下，許多研究利用同一種單元結構（unit cell）以週期性排列組成地震超材料，在地震波通過時，引發單元結構內部產生局部共振（local resonance）進而導致波傳的衰減，然而多數的研究成果只能阻擋部分頻段的地震波傳。本文以Krӧdel et al.[51]的理念為基礎，提出鋼材與橡膠組成的複合圓柱能夠藉由控制子材料的體積比改變地震超材料的帶隙範圍，經過多種帶有不同特定帶隙頻率的圓柱以六角結構排列後，其多類型的複合地震超結構能夠產生一般地震超材料無法達到的寬頻帶隙，根據彈簧質量體的離散（discrete）模型與連體模型的分析，數值模擬的結果確實驗證了位於帶隙範圍2～9.6 Hz的波傳能量能夠被有效的衰減。最後提出數個值得被探討與改進的方向，提高未來在實驗與實作上的可行性。

Metamaterial is a new type of composites made of artificially structured constituents, in which it exploits local resonances to attenuate or to block propagating waves well above the characteristic size of the material structure. The concept of metamaterials was first originated in the context of optics, and applied later to thermal conduction, to acoustic waves, and more recently to seismic wave. The underlying concept relies on the design of material structure, with suitable material constituents and geometric configuration, so that the coupling interference at frequencies near local resonances will result in attenuation of the propagating waves in ways not behaved normally. Seismic metamaterial, corresponding to long-wavelength and low frequency range, is challenging in that the band width needs to be sufficiently broad. In this work, following the concept of Krodel et al. [51], we propose to use a hexagonal array of composite cylinders, made of a stiff steel core surrounded with soft rubber. To obtain broadband attenuation characteristics, the volume fraction of each composite cylinder is suitably devised so that each individual layer of composite cylinders will correspond to a different bandgap. The array, constituted by multiple layers of metamaterial, will effectively behave as a medium with wave mitigation in the infrasound regime (2-9.6 Hz). Numerical simulations based on discrete model as well as full-scale continuum model will be studied. Our finite element simulations show that within the bandgap the wave energy can be attenuated substantially. Lastly we propose a few issues that could be further explored in the future. This will add to the feasibility of practical field test in the future.

1. 簡廷宇,黃瑜琛,吳逸軒,李冠慧,翁崇寧,陳東陽(2019)。新型態外部隔減震技術-地震超材料之設計與分析。中國土木水利工程學刊
   連結：
2. http://www.cnl.com.tw/uploads/files/rubber_char_6.pdf.
3. http://www.geotechdata.info/parameter/soil-young's-modulus.html.
4. http://nptel.ac.in/courses/105106055/Mod2/Lecture4.pdf.
5. http://www.matweb.com/search/DataSheet.aspx?MatGUID=cbe7a469897a47eda563816c86a73520&ckck=1.
6. Achaoui, Y.,Antonakakis, T.,Brule, S.,Craster, R.,Enoch, S.,Guenneau, S.(2017).Clamped seismic metamaterials: ultra-low frequency stop bands.New Journal of Physics,19(6),063022.
7. Achaoui, Y.,Ungureanu, B.,Enoch, S.,Brûlé, S.,Guenneau, S.(2016).Seismic waves damping with arrays of inertial resonators.Extreme Mechanics Letters,8,30-37.
8. Auld, B. A.(1973).Acoustic Fields and Waves in Solids.РиполКлассик.
9. Avilés, J.,Sánchez-Sesma, J.(1983).Piles as barriers for elastic waves.Journal of the Geotechnical Engineering Division,109(9),1133-1146.
10. Bloch, F., Über die quantenmechanik der elektronen in kristallgittern, Zeitschrift Für Physik, Vol. 52, No. 7, pp. 555-600 (1929).
11. Bradly, C. J.,Cracknell, A. P.(1972).The Mathematical Theory of Symmetry in Solids.
12. Bragg, W. L., “The diffraction of short electromagnetic waves by a crystal,” Proceedings of the Cambridge Philosophical Society, Vol. 17, No. 1, pp. 43-57 (1913).
13. Brûlé, S.,Javelaud, E.,Enoch, S.,Guenneau, S.(2014).Experiments on seismic metamaterials: Molding surface waves.Physical Review Letters,112(13),133901.
14. Chen, T.,Tsai, Y.-L.(2013).A derivation for the acoustic material parameters in transformation domains.Journal of Sound and Vibration,332(4),766-779.
15. Chen, T.,Weng, C.-N.(2009).Invisibility cloak with a twin cavity.Optics Express,17(10),8614-8620.
16. Chen, T.,Weng, C.-N.,Chen, J.-S.(2008).Cloak for curvilinearly anisotropic media in conduction.Applied Physics Letters,93(11),114103.
17. Colombi, A.,Colquitt, D.,Roux, P.,Guenneau, S.,Craster, R. V.(2016).A seismic metamaterial: The resonant metawedge.Scientific Reports,6(1),27717.
18. Colombi, A.,Guenneau, S.,Roux, P.,Craster, R. V.(2015).Transformation seismology: composite soil lenses for steering surface elastic Rayleigh waves.Scientific Reports,6,25320.
19. Colquitt, D. J.,Colombi, A.,Craster, R. V.,Roux, P.,Guenneau, S. R. L.(2017).Seismic metasurfaces: Sub-wavelength resonators and Rayleigh wave interaction.Journal of the Mechanics and Physics of Solids,99,379-393.
20. Conca, C.,Planchard, J.,Vanninathan, M.(1995).Fluids and Periodic Structures.John Wiley & Sons.
21. Diatta, A.,Achaoui, Y.,Brûlé, S.,Enoch, S.,Guenneau, S.(2016).Control of Rayleigh-like waves in thick plate Willis metamaterials.AIP Advances,6(12),121707.
22. Du, Q.,Zeng, Y.,Huang, G.,Yang, H.(2017).Elastic metamaterial-based seismic shield for both Lamb and surface waves.AIP Advances,7(7),075015.
23. Finocchio, G.,Casablanca, O.,Ricciardi, G.,Alibrandi, U.,Garescì, F.,Chiappini, M.,Azzerboni, B.(2014).Seismic metamaterials based on isochronous mechanical oscillators.Applied Physics Letters,104(19),191903.
24. Gao, G. Y.,Li, Z. Y.,Qiu, Ch,Yue, Z. Q.(2006).Three-dimensional analysis of rows of piles as passive barriers for ground vibration isolation.Soil Dynamics and Earthquake Engineering,26(11),1015-1027.
25. Gazalet, S.,Dupont, S.,Kastelik, J. C.,Rolland, Q.,Djafari-Rouhani, B.(2013).A tutorial survey on waves propagating in periodic media: Electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains.Wave Motion,50(3),619-654.
26. Haupt, W. A.(1981).Model test on screening of surface waves.International Society for Soil Mechanics and Geotechnical Engineering,10,215-222.
27. Hill, R.(1963).Elastic properties of reinforced solids: Some theoretical principles.Journal of the Mechanics and Physics of Solids,11(5),357-372.
28. Huang, G. L.,Sun, C. T.(2010).Band gaps in a multiresonator acoustic metamaterial.Journal of Vibration and Acoustic,132,031003.
29. Huang, H. H.,Sun, C. T.(2011).Locally resonant acoustic metamaterials with 2D anisotopic effective mass density.Philosophical Magazine,91(6),981-996.
30. Huang, H. H.,Sun, C. T.(2009).Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density.New Journal of Physics,11,013003.
31. Huang, H.,Sun, C.,Huang, G.(2009).On the negative effective mass density in acoustic metamaterials.International Journal of Engineering Science,47(4),610-617.
32. Huang, J.,Shi, Z.(2013).Application of periodic theory to rows of piles for horizontal vibration attenuation.International Journal of Geomechanics,13(2),132-142.
33. Huang, J.,Shi, Z.(2013).Attenuation zones of periodic pile barriers and its application in vibration reduction for plane waves.Journal of Sound and Vibration,332(19),4423-4439.
34. 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(4),040802.
35. John, S.(1987).Strong localization of photons in certain disordered dielectric superlattices.Physical Review Letters,58(23),2486-2489.
36. Kattis, S. E.,Polyzos, D.,Beskos, D. E.(1999).Modelling of pile wave barriers by effective trenches and their screening effectiveness.Soil Dynamics and Earthquake Engineerung,18(1),1-10.
37. Kim, S. H.,Das, M.(2012).Seismic waveguide of metamaterials.Modern Physics Letters B,26(17),1250105.
38. Krӧdel, S.,Thomé, N.,Daraio, C.(2015).Wide band-gap seismic metastructure.Extreme Mechanics Letters,4,111-117.
39. Kushwaha, M. S.,Halevi, P.,Dobrzynski, L.,Djafari-Rouhani, B.(1993).Acoustic band structure of periodic elastic composites.Physical Review Letters,71(13),2022-2025.
40. Leonhardt, U.(2006).Optical conformal mapping.Science,312(5781),1777-1780.
41. Liao, S.,Sangrey, DA(1978).Use of piles as isolation barriers.Journal of the Geotechnical Engineering Division,104(9),1139-1152.
42. Liu, Y.,Sun, X. Z.,Chen, S. T.(2013).Band gap structures in two-dimensional super porous phononic crystals.Ultrasonics,53(2),518-524.
43. Liu, Z.,Zhang, X.,Mao, Y.,Zhu, Y. Y.,Yang, Z.,Chan, C. T.,Sheng, P.(2000).Locally resonant sonic materials.Science,289(5485),1734-1736.
44. Manger, G. E.(1963).Porosity and Bulk Density of Sedimentary Rocks.United States Geological Survey.
45. Milton, G. W.,Briane, M.,Willis, J. R.(2006).On cloaking for elasticity and physical equations with a transformation invariant form.New Journal of Physics,8(10),248.
46. Miniaci, M.,Krushynska, A.,Bosia, F.,Pugno, N. M.(2016).Large scale mechanical metamaterials as seismic shields.New Journal of Physics,18(8),083041.
47. Norris, A. N.(2015).Acoustic cloaking.Acoustics Today,11(1),38-46.
48. Palermo, A.,Krödel, S.,Marzani, A.,Daraio, C.(2016).Engineered metabarrier as shield from seismic surface waves.Scientific Reports,6,39356.
49. Pendry, J. B.(2000).Negative refraction makes a perfect lens.Physical Review Letters,85(18),3966-3969.
50. Pendry, J. B.,Holden, A. J.,Robbins, D. J.,Stewart, W.(1999).Magnetism from conductors and enhanced nonlinear phenomena.IEEE Transactions on Microwave Theory and Techniques,47(11),2075-2084.
51. Pendry, J. B.,Holden, A.,Stewart, W.,Youngs, I.(1996).Extremely low frequency plasmons in metallic mesostructures.Physical Review Letters,76(25),4773-4776.
52. Pendry, J. B.,Schurig, D.,Smith, D. R.(2006).Controlling electromagnetic fields.Science,312(5781),1780-1782.
53. Pennec, Y.,Djafari-Rouhani, B.,Larabi, H.,Vasseur, J.,Hladky-Hennion, A. C.(2009).Phononic crystals and manipulation of sound.Physica Status Solidi C,6(9),2080-2085.
54. Richart, F. E.,Hall, J. R.,Woods, R. D.(1970).Vibrations of Soils and Foundations.Englewood Cliffs, New Jersey:Prentice-Hall.
55. Sheng, P.(2006).Introduction to Wave Scattering, Localization and Mesoscopic Phenomena.
56. Shi, Z. F.,Cheng, Z. B.,Xiong, C.(2010).A new seismic isolation method by using a periodic foundation.Earth and Space
57. Sigalas, M. M.,Economou, E. N.(1992).Elastic and acoustic wave band structure.Journal of Sound and Vibration,158(2),377-382.
58. Sun, M. M.,Liang, G. Q.,Xia, T. D.,Chen, X. L.,Qian, J. L.,Fang, H. J.,Li, X. M.,Wan, X. L.(2014).Ground vibration isolation of multiple scattering by using rows of tubular piles as barriers.Shock and Vibration,894213
59. Ungureanu, B.,Achaoui, Y.,Enoch, S.,Brûlé, S.,Guenneau, S.(2016).Auxetic-like metamaterials as novel earthquake protections.The European Physical Journal Applied Metamaterials,2(17)
60. Vasseur, J. O.,Djafari-Rouhani, B.,Dobrzynski, L.,Kushwaha, M. S.,Halevi, P.(1994).Complete acoustic band gaps in periodic fibre reinforced composite materials: the carbon/epoxy composite and some metallic systems.Journal of Physics: Condensed Matter,6(42),8759-8770.
61. Veselago, V. G.(1968).The electrodynamics of substances with simultaneously negative values of and μ.Soviet Physics Uspekhi,10(4),509-514.
62. Wagner, P. R.,Dertimanis, V. K.,Chatzi, E. N.,Beck, J. L.(2018).Robust-to-uncertainties optimal design of seismic metamaterials.Journal of Engineering Mechanics,144(3),04017181.
63. Walser, R. M.(2001).Electromagnetic metamaterials.International Society for Optics and Photonics
64. Wilcox, C.(1978).Theory of Bloch waves.Journal D’Analyse Mathématique,33(1),146-167.
65. Yablonovitch, E.(1987).Inhibited spontaneous emission in solid-state physics and electronics.Physical Review Letters,58(20),2059-2062.
66. Yan, Y.,Cheng, Z.,Menq, F.,Mo, Y. L.,Tang, Y.,Shi, Z.(2015).Three dimensional periodic foundations for base seismic isolation.Smart Materials and Structures,24(7),075006.
67. Yao, S.,Zhou, X.,Hu, G.(2008).Experimental study on negative effective mass in a 1D mass–spring system.New Journal of Physics,10(4),043020.
68. Zeng, Y.,Xu, Y.,Deng, K.,Zeng, Z.,Yang, H.,Muzamil, M.,Du, Q.(2018).Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space.Journal of Applied Physics,123,214901.
69. 小栗富士雄,小栗達男(2012).標準機械設計圖表便覽.眾文圖書股份有限公司.
70. 張博威(2013)。臺南，國立成功大學土木工程研究所。
71. 欒丕綱,陳啟昌(2010).光子晶體：從蝴蝶翅膀到奈米光子學.五南圖書出版股份有限公司.

1. 蘇于琪,陳東陽,張國鎮,張文忠,林正洪,林子剛,汪向榮,吳東諭(2022)。地震超材料的隔減震技術。結構工程，37(3)，66-80。
2. 蘇于琪,游忠翰,陳東陽,張國鎮,汪向榮,李冠慧(2020)。地震超材料設計之減震分析及效益評估。中國土木水利工程學刊，32(7)，597-607。
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