具共振器機械超穎材料之應用性探討

Abstract

超穎材料為人造的材料，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，透過震動或機構等方式，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的性質。 本文延伸兼具負質量密度與負楊氏模數的雙負理論模型建構出實際機械模型，並將實際機械模型以不同尺度進行討論，再利用數值模擬獲得所有尺寸機械模型的質量與剛度，代入頻散方程式繪製頻散曲線討論實際機械模型的波傳行為，接著透過有限元分析軟體ABAQUS來分析特定頻率下的波傳現象。另一方面，本文也將一維雙負的理論模型推廣成二維雙負的理論模型，並推導其運動方程式與繪製頻散曲線討論二維理論模型的波傳行為。 根據數值模擬的結果，本研究設計之實際機械模型，分別在不同的尺度下，特定的頻率範圍負載下，都可以展現出特殊的向後波波傳現象，進而得知所有尺度的機械模型都能同時兼具負質量密度與負楊氏模數。在二維理論模型方面，可以從其頻散曲線亦發現負斜率的曲線，此負斜率曲線可能具有特殊的波傳現象。最後也發現機械模型，可以有效地阻隔特定頻率範圍內的波傳遞，此概念可應用於工程上減緩衝擊波的破壞。

This article presents methods for modeling, analysis, and design of practical metamaterial with simultaneously negative mass density and Young’s modulus. Metamaterials are man-made materials that make objects exhibit behavior defferent from the general laws of physics by changing the geometry and dimensions. Metamaterial research extends from the electromagnetic into acoustic and solid mechanics. By defferent 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. The thesis id divided into three parts. First, through theoretical metamaterial model having double negativities, spring-mass system and trusses construct a practical model. We separate the practical model into four different sizes and expect to find the negative slope line in the dispersion curve. The effect of the practical model is explicitly confirmed by analysis of wave propagation using numerical simulation in ABAQUS. By numerical simulation, we find the practical mode having anomalous wave propagation called backward wave. Sencod, we create two-dimensional acoustic metamaterial having simultaneously negative mass density and Young’s modulus. The dipersion curve of two-dimensional model also has the negative slope line which represents special wave propagation phenomenon. Finally, we try to apply the practical model to the present engineering field.