近年不少學者投入彈性超材料中負等效質量密度、負等效體積模數、負等效剪力模數等設計及應用相關研究。為了使彈性超材料發揮更佳的過濾向量波傳遞特性，本文將先介紹一些基本理論、等效參數公式以及波傳特性間的探討，接著再根據上述理論關係式利用有限元素軟體COMSOL數值分析模擬，其中我們先分析一僅使用三種自然界材料就具備雙負頻帶彈性超材料晶格系統，其頻散結構圖會出現雙負頻帶，且內質量結構之運動模態多變，晶格內會出現負的等效參數，同時建立一連體系統來驗證波傳的預測及分析，另外我們調整晶格內部幾何來產生一具有三負頻帶彈性超材料晶格系統，同樣分析其晶格內部特性，其頻散結構圖會出現新的第三負頻且整體頻散結構圖中可使用頻寬變大，而在第三負頻區間會產生新的塗料層旋轉共振運動模態，同時此三負頻區間會出現超異向性體、類流體、類不可壓縮體的濾波功能特性，因此本文所提出的模型有效地增加濾波的頻寬範圍及濾波的功能特性。

The objective of this work is to design elastic metamaterials that can behave effectively as an medium with negative mass density and negative elastic modulus. Two geometric configurations of periodic array of unit cell, made of three constituent materials, that can achieve the bi-negative property, are proposed in this thesis. We utilize finite finite element methods to study the wave propagation characteristics, and to derive the effective parameters of the periodic array. In our first demonstration, an elastic metamaterial lattice system with double negative dispersion bands is achieved for certain frequency range. We demonstrate that this internal microstructure can give rise to particular resonance modes leading to negative effective parameters. In addition, by adjusting the geometriic dimension of unit cell, a second metamaterial model with three negative bands is proposed. Our numerical result confirms that the dispersion diagram can actually induce an additional negative band, that is associated to rotational resonance. Such resonant mode may result in negative effective shear modulus. The proposed model can have applications in the wave-filtering and other novel applications of practical interest.

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