彈性超穎材料(負楊氏模數模型) 波傳行為探討與實驗分析

Abstract

超穎材料(Metamaterial)為人造的材料，藉由幾何設計與尺寸的改變，使物體展現出與一般物理定律不同的行為，起初從電磁波的研究逐漸發展至聲學及固體力學等領域。超穎材料的學理關鍵在於材料內的微結構，而彈性超穎材料主要利用外力激發使材料內的微結構產生局部共振，透過振動或機構等方式，進而在等效模型中得到負質量密度、負楊氏模數、負體積模數等不存在於自然界中的性質。 本文延伸等效負楊氏模數模型針對剪力波發展超穎結構樑模型，推導其運動方程式並繪製頻散曲線討論波傳行為，接著透過有限元素分析軟體COMSOL，分析超穎結構樑模型系統參數對於頻率響應的影響。最後實際設計超穎結構並利用3D列印機製作試體，進行一維振動實驗觀察波傳現象觀察是否在目標頻率段具有波傳衰減行為。 根據數值模擬及實驗結果，本研究設計之週期性超穎結構樑，由頻域分析得出在280Hz~300Hz在正弦外力激振下具有振幅衰減行為。透過一維振動實驗發現當外力激振頻率落在局部共振器的頻率時衰減振幅可減少約80%。由上述結果得知本文提出之模型，實際中可以有效地阻隔特定頻率範圍內的剪力波傳遞，期許此概念可應用於彈性波濾波及減震功能。

This article presents methods for modeling, analysis, and design of metamaterial beams with extreme Young’s modulus. Metamaterials are man-made materials that make objects exhibit behavior different from the general laws of physics by changing the geometry and dimensions. Metamaterial research extends from the electromagnetic into acoustics and solid mechanics. By different mechanism such as translational or rotational vibration, Elastic solid metamaterials would be the equivalent models of media having negative mass density, negative Young's modulus, or negative bulk modulus in excitation force. Objects can be made vibration absorption when utilizing those phenomena. The thesis is divided into three parts. First, through a combination of Euler-Bernoulli beam, spring-mass system and trusses construct a theoretical model. From a unit cell of an infinite metamaterial beam (meta-beam), governing equations are derived using the extended Hamilton principle. By Bloch-Floquet theory for periodic structures we except to find the stop-band in the dispersion curve created by resonators. We uses the incoming elastic wave in the beam to resonant the spring-mass system. We expect that the system resonance creates additional bending moments to stop the wave propagation in meta-beam. Second, we design the possible practical meta-beam. The effect of the meta-beam is explicitly confirmed by analysis of wave propagation using numerical simulations in COMSOL. By numerical simulation, we expect to find the actual working mechanism in meta-beams and their transient responses. Finally, the practical designs and their dynamic behaviors are examined and discussed using numerical simulations.