首先將研究超常材料的等校材料性質，本文以彈簧與質量塊組成的週期系統為基礎，提出單共振模型，藉由結合Liu等人[8]與Oh等人[10]的等效材料性質方法，計算此模型的等效質量與等效彈簧常數，並可得到等效材料性質為與頻率有關的方程式。首先以Oh等人[10]方法將模型等效為單質量彈簧系統，並可得出此模型的色散關係，接著以Liu等人[8]方法計算等效系統的等效材料性質。最後與模型色散曲線對應，等效負材料性質的頻率區間是否皆為色散能隙區間，驗證本文等效材料性質方法的正確性。
接著提出隨時空變化之超常材料模型，由此模型可以使彈性波具有不可逆波傳行為。不可逆波傳行為代表在空間中兩點以相反方向傳遞的能量將帶有不對稱的波傳資訊。隨時空變化之超常材料色散曲線可由晶格動力學以理論推導，為了改善以理論推導的流程之複雜性，本文提出結合時域有限差分與微擾法的數值方法，快速的計算出此模型的色散曲線，並利用有限差分法模擬此模型的波傳行為，進一步證明彈性波的不可逆波傳行為。

First, this research discusses dynamically effective properties of metamaterials. A single resonance model, i.e., periodic spring mass system with sub-structure , was proposed.
Based on the dynamically effective properties method proposed from Liu et al. [8] and Oh et al. [10], the dynamically effective mass and elastic property of the model are derived and found to be frequency dependent. With Oh et al.’s [10] method, the model is equivalent to a single mass spring system, and the dispersion relation of the model could be obtained. Then the dynamically effective material properties of the equivalent system could be calculated by Liu et al.’s [8] method. Corresponding to the model dispersion curve to see whether the band gap always coincides with the frequency range of negative effective properties, the correctness of dynamically effective properties method proposed from this research was verified.
Second, a spatio-temporal metamaterial model was proposed to demonstrate nonreciprocal wave propagation for elastic wave. Nonreciprocal wave propagation means the transmission of energy in opposite directions between any two points in space will have asymmetric wave propagation information. The dispersion relation of the model could be theoretically calculated using lattice dynamics. In order to improve the complexity of the theoretical derivation processes, a numerical simulation combining finite difference time-domain method and perturbation method was proposed. With numerical perturbation method, the model dispersion curve could be quickly and numerically calculated. The wave propagation behavior inside the model was simulated using finite difference method and the nonreciprocal wave propagation of the elastic waves were further illustrated.

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