Abstract The present research is an effort to enhance and control locally resonant bandgaps in periodic metamaterials of hexagonal and triangular topologies by introducing spider-web and snowflake-inspired hierarchies to the conventional geometries. Due to their low connectivity number, hexagonal lattice structures are unable to exhibit locally resonant bandgaps. By introducing the spider-web hierarchy, the connectivity number of the structure is increased, and the constituting beams act as local resonators. Thus, the hexagonal lattice structure can attenuate propagating waves in desirable frequency ranges by localizing the wave energy in the constituent beams. As a result, better wave-filtering performance is achieved by introducing hierarchy to hexagonal lattices. Furthermore, the snowflake hierarchical triangular lattice structures exhibit new tunable attenuation zones for which the locations depend on the mechanical and geometrical parameters. The finite element method and Bloch’s theorem are applied to analyze the wave propagation in the considered architected structures. Moreover, theoretical formulations are presented to predict the locations of each locally resonant bandgaps. It is shown that by changing the mechanical and geometrical parameters of the added hierarchical structures, the locally resonant bandgaps can be perfectly tuned to the desired frequencies in the long-wavelength region. Finally, two theoretical diagrams are proposed to represent initial design concepts of tunable acoustic/elastic metamaterials of hexagonal and triangular unit cells with maximum bandgap widths in low frequencies. To present a more comprehensive analysis of the wave propagation in hierarchical structures, the effects of imaginary wave-vector components on the dispersion curves and attenuation behavior of each topology are also investigated. The results of the current study enable the engineers to design architected periodic structures with the ability to present bandgaps in desired frequencies, which has been the goal of structural engineers for years.

中文翻译：

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可调局部共振超材料的研究：蜘蛛网和雪花层次结构的影响

摘要 本研究致力于通过将蜘蛛网和雪花启发的层次结构引入到传统几何结构中来增强和控制六角形和三角形拓扑结构的周期性超材料中的局部共振带隙。由于它们的连接数低，六边形晶格结构无法表现出局部共振带隙。通过引入蜘蛛网层次结构，增加了结构的连接数，并且构成的梁充当了局部谐振器。因此，六边形晶格结构可以通过将波能量定位在组成光束中来衰减所需频率范围内的传播波。因此，通过在六边形晶格中引入层次结构，可以实现更好的滤波性能。此外，雪花分层三角形晶格结构表现出新的可调衰减区，其位置取决于机械和几何参数。应用有限元方法和布洛赫定理来分析所考虑的建筑结构中的波传播。此外，还提出了理论公式来预测每个局部谐振带隙的位置。结果表明，通过改变添加的分层结构的机械和几何参数，可以将局部谐振带隙完美地调谐到长波长区域中的所需频率。最后，提出了两个理论图来表示在低频具有最大带隙宽度的六角形和三角形晶胞的可调声学/弹性超材料的初始设计概念。为了对分层结构中的波传播进行更全面的分析，还研究了虚波矢量分量对每个拓扑的色散曲线和衰减行为的影响。当前研究的结果使工程师能够设计出能够在所需频率下呈现带隙的​​建筑周期结构，这是结构工程师多年来的目标。