On the emergence of negative effective density and modulus in 2-phase phononic crystals
On the emergence of negative effective density and modulus in 2-phase phononic crystals
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DOI：
10.1016/j.jmps.2019.02.016
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发表时间：
2018-11
影响因子：
5.3
通讯作者：
A. Mokhtari;Yan Lu;A. Srivastava
中科院分区：
文献类型：
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作者：
A. Mokhtari;Yan Lu;A. Srivastava
在这篇文章中，我们报道了传统上被认为是非局部共振的两相声子单胞的超材料性质，包括负的和奇异的有效性质。这里报道的负的有效材料性质发生在均化极限以下，因此，是对整体行为的可接受的描述。首先通过描述的基于水平集的新的拓扑优化过程揭示了实现这一目标的材料属性组合。优化过程表明，其中一个相同时比另一个相更轻和更硬的两相单胞导致了动态行为，它具有局部共振复合材料的所有伴随特征，包括远远低于均化极限的负有效性能。我们使用Craig-Bampton分解进一步研究了这一点，并阐明了这些性质是通过单胞的基本内部模形和刚体模之间的相互作用而出现的。通过对一维两相晶胞的显式数值计算，我们发现只有上述特定的材料性质组合才会出现负的有效性质。此外，我们还给出了支持这一结论的证明。这一概念也适用于二维单胞，我们证明了由两个材料相组成的适当设计的六角形单胞在适当的频率范围内表现出负的有效剪切模数和密度，在该频率范围内它也表现出负的折射率。与一维情形一样，我们证明了在二维情形下的齐次化描述也是严格的。本文的一个重要结论是，这类预期会导致负性质的单胞可以从经典的单胞(具有显式局部共振相位的三相单胞)扩展到也包括拓扑上更简单的两相单胞。
In this paper we report metamaterial properties including negative and singular effective properties for what would traditionally be considered non locally resonant 2-phase phononic unit cells. The negative effective material properties reported here occur well below the homogenization limit and are, therefore, acceptable descriptions of overall behavior. The material property combinations which make this possible were first revealed by a novel level set based topology optimization process which we describe. The optimization process revealed that a 2-phase unit cell in which one of the phases is simultaneously lighter and stiffer than the other results in dynamic behavior which has all the attendant characteristics of a locally resonant composite including negative effective properties far below the homogenization limit. We investigate this further using the Craig–Bampton decomposition and clarify that these properties emerge through an interplay between the fundamental internal modeshape of the unit cell and a rigid body mode. Through explicit numerical calculations on 1-D, 2-phase unit cells, we show that negative effective properties only appear for the specific material property combination mentioned above. Furthermore, we provide a proof which supports this conclusion. The concept is also shown to hold for 2-D unit cells where we show that an appropriately designed hexagonal unit cell made of 2 material phases exhibits negative effective shear modulus and density in an appropriate frequency regime in which it also exhibits negative refraction. As in the 1-D case, we show that the homogenized description in the 2-D case is rigorous as well. An important conclusion of this paper is that the class of unit cells expected to result in negative properties can be expanded beyond the classic unit cell (three-phase unit cells with an explicit locally resonant phase) to include topologically simpler 2-phase unit cells as well.