本研究聚焦於地震超材料於低頻範圍(0~20 Hz)中之波動操控行為，旨在建立一套可用於三相圓球超材料之設計與分析流程，並特別探討五模微結構與粽子微結構作為中介層時的力學特性與帶隙形成機制。研究首先整合均質化方法與有限元素分析，以建立兩類微結構在不同幾何條件下的等效力學模型，並配合堆疊收斂性分析確認其可代表性。進一步導入頻散分析架構，系統性比較實體微結構模型與等效材料模型於不同晶格尺度下的波動響應差異，以評估等效材料模型於低頻局部共振情境中的適用性。
在三相超材料的參數研究部分，本研究以內核尺寸、中介層厚度、基材尺度以及粽子結構幾何比值d/a等為主要控制因子，探討其對局部共振模態、等效質量密度特性與帶隙行為的影響。同時，為更貼近實際地盤環境，本研究建立半全域土層模型，並施加P波、SH波與SV波作為入射條件，以分析不同微結構中介層於土層中可能展現的能量傳遞變化趨勢。透過上述完整的多尺度與多模型分析，本研究期望建立一套適用於五模與粽子微結構超材料的參數化設計方法，並釐清微結構型態在地震超材料低頻帶隙形成與局部共振行為中的作用，使其可作為未來地震超材料設計與相關工程應用的重要基礎。

This thesis investigates the low-frequency (0~20 Hz) wave manipulation behavior of seismic metamaterials and proposes a systematic design and analysis framework for three-phase unit cell. Special emphasis is placed on pentamode and Zongzi microstructures employed as interphases to explore their mechanical characteristics, local resonance behavior, and bandgap formation mechanisms. Homogenization theory combined with finite element analysis is used to establish equivalent mechanical models, and stacking convergence analyses are performed to verify their representativeness. Dispersion analyses are conducted to compare wave propagation responses between explicit microstructured models and equivalent material models under different geometric configurations. Parametric studies are further carried out by varying key interphase parameters to examine their effects on effective material properties, local resonance modes, and bandgap characteristics. To approximate realistic ground conditions, semi-global soil models subjected to P-, SH-, and SV-wave excitations are developed to evaluate vibration attenuation performance. The results clarify the role of interphase stiffness in bandgap formation and provide a parametric design strategy for low-frequency seismic metamaterials.

Abramowitz, M., & Stegun, I. A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. Tenth Printing. (1972).
Achaoui, Y., Antonakakis, T., Brûlé, S., Craster, R. V., Enoch, S., & Guenneau, S. Clamped seismic metamaterials: ultra-low frequency stop bands. New Journal of Physics, 19(6), 063022. (2017).
Barnwell, E. One and two-dimensional propagation of waves in periodic heterogeneous media: Transient effects and band-gap tuning (Doctoral dissertation). University of Manchester. (2015).
Bloch, F. Über die quantenmechanik der elektronen in kristallgittern. Zeitschrift für physik, 52(7-8), 555-600. (1929).
Brillouin, L. Wave propagation in periodic structures: electric filters and crystal lattices (Vol. 2): Dover publications. (1953).
Brûlé, S., Enoch, S., & Guenneau, S. Emergence of seismic metamaterials: Current state and future perspectives. Physics Letters A, 384(1), 126034. (2020).
Brûlé, S., Javelaud, E., Enoch, S., & Guenneau, S. Experiments on seismic metamaterials: molding surface waves. Physical Review Letters, 112(13), 133901. (2014).
Cottrell, A. H. The Mechanical Properties of Matter. Wiley: New York. (1964).
Dal Poggetto, V. F., & Serpa, A. L. Elastic wave band gaps in a three-dimensional periodic metamaterial using the plane wave expansion method. International Journal of Mechanical Sciences, 184, 105841. (2020).
Daradkeh, A. M., Hojat Jalali, H., & Seylabi, E. Mitigation of seismic waves using graded broadband metamaterial. Journal of Applied Physics, 132(5). (2022).
Du, Q., Zeng, Y., Huang, G. and Yang, H., Elastic metamaterial-based seismic shield for both Lamb and surface waves, AIP Advances 7, 075015. (2017).
Fang, N., Xi, D., Xu, J., Ambati, M., Srituravanich, W., Sun, C., & Zhang, X. Ultrasonic metamaterials with negative modulus. Nature Materials, 5(6), 452-456. (2006).
Favier, E., Nemati, N., & Perrot, C. Two-component versus three-component metasolids. The Journal of the Acoustical Society of America, 148(5), 3065-3074. (2020).
Favier, E., Nemati, N., Perrot, C., & He, Q.-C. Generalized analytic model for rotational and anisotropic metasolids. Journal of Physics Communications, 2(3), 035035. (2018).
Graff, K. F. Wave motion in elastic solids. Courier Corporation. (2012).
Hill, R. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids, 11(5), 357–372. (1963).
Hill, R. The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society. Section A, 65(5), 349. (1952).
Huang, H. H., & Sun, C.-T. Anomalous wave propagation in a one- dimensional acoustic metamaterial having simultaneously negative mass density and Young’s modulus. The Journal of the Acoustical Society of America, 132(4), 2887-2895. (2012).
Huang, H. H., & Sun, C. T. Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density. New Journal of Physics, 11(1), 013003. (2009).
Kadic, M., Bückmann, T., Schittny, R., Gumbsch, P., and Wegener, M. Pentamode metamaterials with independently tailored bulk modulus and mass density. Physical Review Applied, 2, 054007. (2014)
Kadic, M., Bückmann, T., Stenger, N., Thiel, M. and Wegener, M. On the practicability of pentamode mechanical metamaterials. Applied Physics Letters, 100, 191901. (2012).
Kittel, C., & McEuen, P. Introduction to Solid State Physics. John Wiley & Sons. (2018).
Kuo, C.H., Lin, Z.Y., Hsu, S.T., Chen, J.H., Chen, J.S. and T. Chen, Pyramid-shaped metamaterials with tunable mechanical properties and broad frequency bandgaps, forthcoming, ASCE Journal of Engineering Mechanics, accepted, (2025).
Kushwaha, M. S., Halevi, P., Martinez, G., Dobrzynski, L., & Djafari-Rouhani, B. Theory of acoustic band structure of periodic elastic composites. Physical Review B, 49(4), 2313. (1994).
Liu, Z., Chan, C. T., & Sheng, P. Analytic model of phononic crystals with local resonances. Physical Review B, 71(1), 014103. (2005).
Liu, X. N., Hu, G. K., Huang, G. L., & Sun, C. T. An elastic metamaterial with simultaneously negative mass density and bulk modulus. Applied physics letters, 98(25). (2011).
Liu, Z., Zhang, X., Mao, Y., Zhu, Y. Y., Yang, Z., Chan, C. T., & Sheng, P. Locally resonant sonic materials. science, 289(5485), 1734-1736. (2000).
Milton, G. W., & Cherkaev, A. Which elasticity tensors are realizable? Journal of Engineering Materials and Technology, 117, 483–493. (1995).
Miniaci, M., Krushynska, A., Bosia, F., & Pugno, N. M. Large scale mechanical metamaterials as seismic shields. New Journal of Physics, 18(8), 083041. (2016).
Norris, A. N. Mechanics of elastic networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470, 20140522. (2014).
Norris, A. N. Poisson's ratio in cubic materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 462, 3385–3405. (2006).
Pendry, J. B. Negative refraction makes a perfect lens. Physical Review Letters, 85(18), 3966. (2000).
Pendry, J. B., Holden, A., Stewart, W., & Youngs, I. Extremely low frequency plasmons in metallic mesostructures. Physical Review Letters, 76(25), 4773. (1996).
Shukla, M. The shear bounds of the cubic polycrystal and its experimental shear modulus. Journal of Physics D: Applied Physics, 15, L177–L180. (1982).
Smith, D. R., Padilla, W. J., Vier, D., Nemat-Nasser, S. C., & Schultz, S. Composite medium with simultaneously negative permeability and permittivity. Physical Review Letters, 84(18), 4184. (2000).
Thomas, T. Y. On the stress–strain relations for cubic crystals. Proceedings of the National Academy of Sciences, 55, 235–239. (1966).
Thomson, W. Elements of a mathematical theory of elasticity. Philosophical Transactions of the Royal Society of London, 146, 481–498. (1856).
Timoshenko, S. History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures. Courier Corporation. (1983).
Veselago, V. G. Electrodynamics of substances with simultaneously negative and. Usp. Fiz. Nauk, 92(7), 517. (1967).
Walpole, L. J. Orthotropically textured elastic aggregates of cubic crystals. J. Mech. Phys. Solids, 35, 497–517. (1987).
Walpole, L. J. The elastic shear moduli of cubic crystal. Journal of Physics D: Applied Physics, 19, 457–462. (1986).
Walpole, L. J. Fourth rank tensors of the thirty-two crystal classes: multiplication tables. Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, 391, 149–179. (1984a).
Walpole, L. J. The stress-strain law of a textured aggregate of cubic crystals. J. Mech. Phys. Solids, 33, 363–370. (1984b).
Wu, Y., Lai, Y., & Zhang, Z.-Q. Elastic metamaterials with simultaneously negative effective shear modulus and mass density. Physical Review Letters, 107(10), 105506. (2011).
Zener, C. Elasticity and anelasticity of metals. University of Chicago Press. (1948).
Zheng, Q. S., & Chen, T. New perspective on Poisson's ratios of elastic solids. Acta Mechanica, 150(3), 191-195. (2001).
李冠慧，地震超材料設計之減震模擬及效益評估，成功大學土木工程學系碩士論文 (2019)。
林宗穎，週期排列之橢圓及粽子結構超材料之共振代隙與消能機制，成功大學土木工程學系碩士論文 (2023)。
洪頤謙，具柔緩或剛性圓柱內核之二相超材料的局部共振行為，成功大學土木工程學系碩士論文 (2025)。
郭騏豪，類粽子形狀週期排列微結構之超材料機械性質探討，成功大學土木工程學系碩士論文 (2022)。
寧彥傑，具負等效質量慣性矩之微極彈性模型設計，成功大學土木工程學系碩士論文 (2016)。
簡廷宇、黃瑜琛、吳逸軒、李冠慧、翁崇寧、陳東陽，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， 31(4), 395-410. (2019).