This paper provides a detailed comparison of the two-scale homogenization method and of the Bloch-Floquet theory for the determination of band-gaps in locally resonant metamaterials. A medium composed by a stiff matrix with soft inclusions with 2D periodicity is considered and the equivalent mass density of the homogenized medium is explicitly obtained both for in-plane and out-of-plane wave propagation through two-scale asymptotic expansion. The intervals of frequency where the effective mass is negative identify the band-gaps of the material. The Bloch-Floquet problem is then considered and, through an asymptotic analysis, its is shown that it leads to the same prediction of the band-gaps. The results are confirmed by some examples and the limits of the asymptotic approach are explicitly given and numerically verified.

中文翻译：

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二维局部共振超材料带隙预测的均质化方法和 Bloch-Floquet 理论

本文详细比较了用于确定局部共振超材料中带隙的两尺度均质化方法和 Bloch-Floquet 理论。考虑了由具有二维周期性的软夹杂物的刚性基体组成的介质，并且通过双尺度渐近扩展明确获得了平面内和平面外波传播的均质介质的等效质量密度。有效质量为负的频率间隔确定了材料的带隙。然后考虑 Bloch-Floquet 问题，并通过渐近分析表明它导致了对带隙的相同预测。结果通过一些例子得到证实，渐近方法的极限被明确给出并进行了数值验证。