本研究探討具備二維週期性排列之三相超材料的彈性波傳行為，聚焦包覆層對局域共振與帶隙之影響。為突破傳統等向性假設，本文於包覆層引入異向性(anisotropic)材料，並建立平面波展開法解析框架。推導中，透過勁度張量座標轉換，成功將空間週期材料之傅立葉展開由無窮級數精確簡化為有限項，這顯示本研究成功克服了圓柱異向性在平面波展開法中複雜的數學形式。
數值結果顯示，異向性特徵的引入能誘發顯著的能帶跳躍現象，並放大完全帶隙峰值。相較於傳統設計，此方法在不改變材料配比與總重的前提下，大幅提升了波動傳播的調控自由度與方向性設計能力。
在物理實現上，本研究利用有限元素法建構兩種等向性材料交替堆疊的層狀微結構來實現正交異向性；進一步賦予層狀微結構任意傾角，即可重現完全異向性特徵。有限元素法與平面波展開法計算結果高度一致，驗證了理論的準確性。本研究證實，僅需微調微結構幾何即可有效操控波的傳遞與衰減，為低頻減震與地震防護等工程應用提供極具潛力之新途徑。

This study investigates the elastic wave propagation behavior of two-dimensional periodic three-phase metamaterials, focusing on the influence of the coating layer on local resonance and bandgaps. To transcend traditional isotropic assumptions, this work introduces anisotropic materials into coatings and establishes an analytical framework based on the plane wave unfolding (PWE) method. During the derivation, by utilizing the coordinate transformation of stiffness tensors, the Fourier expansion of the spatially periodic material is successfully simplified from an infinite series to precise finite terms. This achievement demonstrates that the proposed method effectively overcomes the complex mathematical challenges associated with cylindrical anisotropy within the PWE framework.
Numerical results indicate that the introduction of anisotropy can induce significant band jumping phenomena and amplify the peaks of complete bandgaps. Compared to traditional designs, this approach substantially enhances the degrees of freedom for wave control and directional design capability without altering material ratios or total weight.
For physical implementation, this study employs the FEM to construct a layered microstructure composed of two alternating isotropic materials to achieve orthogonal anisotropy; furthermore, by assigning arbitrary inclination angles to the layered microstructure, fully anisotropic characteristics can be reproduced. The high consistency between the FEM and PWE results validates the accuracy of the theory. This research confirms that wave transmission and attenuation can be effectively manipulated simply by fine-tuning the microstructure geometry, offering a promising new avenue for engineering applications such as low-frequency vibration reduction and seismic protection.

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