地震超材料的研究近年來有許多學者進行著相關的研究，包含了一些週期排列諧振器共振板的研究，相對地，週期排列諧振器共振殼上的研究仍在起步階段，本論文基於Timoshenko梁、Mindlin板考量了一階剪力影響的理論來與古典梁、古典板做比較，進而探討了共振帶隙、布拉格帶隙的成因與剪力對帶隙所產生的變化，發現高頻區域下的頻散行為與古典梁、古典板在同一頻率下有較低的波數，即是考量剪力變形所造成的差異，同時由於考量一階剪力影響所以可以對厚度比以及理論中的剪力修正係數做變化，其中在改變厚度比的分析中發現改變厚度比中的長、寬、高的任一數值，會影響共振帶隙上邊界的位置，例如將高度增加會使得共振帶隙上邊界往低頻處移動，將高度減少會使得帶隙上邊界往高頻處移動，並且發現共振帶隙下邊界基本上固定不會移動，而變化剪力修正係數中發現剪力修正係數的變化對帶隙幾乎沒有變化產生。
將上述得到的結果延伸到殼理論當中，即本論文所關注的曲率對帶隙的影響，為此推導出受曲率影響的撓曲波殼頻散方程組，首先對於週期排列於球殼上的晶格堆積問題做介紹了解如何處理週期性結構的頻散問題，藉此與撓曲板頻散現象從單曲率的圓柱球殼到雙曲率的圓球殼來進行一系列的曲率變化對帶隙影響分析，在與撓曲板結果比較後，發現在有曲率的影響下帶隙會往高處移動，並且隨著曲率半徑愈小，帶隙位置愈高，其中共振帶隙會會變大而布拉格帶隙會變小，而從單曲率延伸到雙曲率後，發現帶隙位置又移動到更加高頻的位置，同時共振帶隙變得更大，布拉格帶隙變得更小，最後透過有限元程式驗證頻散結果，證明上述的分析是正確的，同時透過全域模擬地震情形下，發現彈簧質量諧振器確實有發揮出地震消能的作用。

Seismic metamaterials have been widely studied in recent years. Among the numerous physical models, flexural plates with periodically attached resonators could be a simplified and realistic model to simulate the complicated physical behavior of seismic motion of the ground. In the same time, the research that curved plates or shells with periodic resonators are an emerging subject in which the curvature of the plates are taken into account. Here in this MS thesis we will compare local resonance bandgap and Bragg bandgap in Timoshenko beams and Mindlin plates, which consider the effects of shearing deformations in contrast to the classical beam and plate theories. Our results show that wave number will be decreased in the high-frequency range when shear effect is taken into account. In addition, there is a wide-range in thickness ratio with Timoshenko beams and Mindlin plates and we find out that the bandgap in local resonance will increase as the thickness ratio increases. According to the trends in numerical simulations, we can derive the governing equations and periodic lattices in curve plates. With numerical analysis, the results show that there is an additional wave propagation gap in low-frequency range caused by curvature. As the radius of curvature smaller, local resonance bandgap and Bragg bandgap will be occurred in high frequency with bigger bandgap range and smaller band gap range, respectively. Lastly, we utilize a seismic acceleration in COMSOL model to verify spring-mass resonator can attenuate the seismic energy effectively.

Beli, D., et al. (2018). Wave propagation in elastic metamaterial beams and plates with interconnected resonators. International Journal of Solids and Structures 139-140: 105-120.

Bloch, F. (1929). Über die Quantenmechanik der Elektronen in Kristallgittern. Zeitschrift für Physik 52(7): 555-600.

Brûlé, S., et al. (2014). Experiments on Seismic Metamaterials: Molding Surface Waves. Physical Review Letters 112(13): 133901.

Braile, L. (2010). Seismic Wave Demonstrations and Animation, Purdue University 1-15.

Brillouin, L. O. (2003). Wave propagation in periodic structures : electric filters and crystal lattices. Mineola, N.Y., Dover Publications.

Chopra, A. K. (2007). Dynamics of Structures, Pearson Education.

Colombi, A., et al. (2016a). A seismic metamaterial: The resonant metawedge. Scientific Reports 6(1): 27717.

Colombi, A., et al. (2016b). Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances. Scientific Reports 6: 19238-19238.

Fang, N., et al. (2006). Ultrasonic metamaterials with negative modulus. Nature Materials 5(6): 452-456.

Ferreira, A. H. R., et al. (2017). Semi-analytical Formulation for Sound Transmission Loss Analysis through a Thick Plate with Periodically attached Spring-mass Resonators. International Conference on Structural Engineering Dynamics

Ferreira, A., et al. (2019). Flexural wave band gaps in locally resonant elastic
metamaterial plate using Reissner-Mindlin theory. International Symposium on
Dynamic Problems of Mechanics 2019-0153

Graff, K. F. (1975). Wave Motion in Elastic Solids. Columbus, Ohio State University Press.

Hoefakker, J. (2010). Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks, Eburon.

Kaneko, T. (1975). On Timoshenko's correction for shear in vibrating beams. Journal of Physics D: Applied Physics 8(16): 1927-1936.

Lee, S. H., et al. (2010). Composite Acoustic Medium with Simultaneously Negative Density and Modulus. Physical Review Letters 104(5): 054301.

Liu, Z., et al. (2000). Locally Resonant Sonic Materials. Science 289(5485): 1734-1736.

Mindlin, R. D. (1951). Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates. Journal of Applied Mechanics 18(1): 31-38.

Nateghi, A. et al. (2017). Wave propagation in locally resonant cylindrically curved metamaterial panels. International Journal of Mechanical Sciences 127: 73-90.

Pendry, J. B. (2000). Negative Refraction Makes a Perfect Lens. Physical Review Letters 85(18): 3966-3969.

Pendry, J. B., et al. (1996). Extremely Low Frequency Plasmons in Metallic Mesostructures. Physical Review Letters 76(25): 4773-4776.

Pendry, J. B., et al. (2006). Controlling Electromagnetic Fields. Science 312(5781): 1780-1782.

Soedel, W., et al. (2005). Vibrations of Shells and Plates, Third Edition. Journal of The Acoustical Society of America 117: 1683-1683.

Timoshenko, S., et al. (1961). Theory of Elastic Stability. New York, McGraw-Hill.

Towhata, I., et al. (2008). Geotechnical Earthquake Engineering, Springer Science & Business Media.

Veselago, V. G.(1968). The electrodynamics of substances with simultaneously
negative values of and μ, Soviet Physics Uspekhi 10(4): 509.

Xiao, Y., et al. (2012a). Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators. Journal of Physics D: Applied Physics 45(19): 195401.

Xiao, Y., et al. (2012b). Sound transmission loss of metamaterial-based thin plates with multiple subwavelength arrays of attached resonators. Journal of Sound and Vibration 331(25): 5408-5423.

Yao, D., et al. (2022). Flexural wave mitigation in metamaterial cylindrical curved shells with periodic graded arrays of multi-resonator. Mechanical Systems and Signal Processing 168: 108721.

寧彥傑(2016)，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。

李冠慧(2019)，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。

簡廷宇(2019)，具帶隙效應之層狀基礎於隔減震之應用，成功大學土木工程學系碩士論文。

趙乙謙(2021)，預應力與阻尼系統對局部共振薄板之撓曲波帶隙的影響，成功大學土木工程學系碩士論文。