In this study, the wave propagation properties of lattice metamaterials with Koch fractal structures are investigated in terms of band structures and directional wave propagation. The analytical models of lattice metamaterials are established using the finite element method, and the dispersion relation is solved using the Bloch’s theorem. The band structures of the lattice metamaterials with different numbers of iterations are studied, and the group velocities at a selected frequency are calculated to analyze the directional wave propagation characteristics. Furthermore, dynamic responses of the finite structures are calculated using commercial finite element software to verify the band gaps and directional wave propagation behaviors in the lattice metamaterials. The results show that multiple and low band gaps are present in the lattice materials with various geometric parameters of the Koch fractal, and the position of the lowest band gap decreases as the number of iterations increases. The results indicate the potential applications of lattice metamaterials with Koch fractals for vibration isolation and multi-functional design.

中文翻译：

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Koch分形在格形超材料中的弹性波传播

在这项研究中，从带结构和定向波传播角度研究了具有科赫分形结构的晶格超材料的波传播特性。利用有限元方法建立了晶格超材料的解析模型，并利用布洛赫定理求解了色散关系。研究了具有不同迭代次数的晶格超材料的能带结构，并计算了选定频率下的群速度，以分析方向波的传播特性。此外，使用商业有限元软件计算有限结构的动力响应，以验证晶格超材料中的带隙和定向波传播行为。结果表明，具有不同Koch分形几何参数的晶格材料中存在多个带​​隙和低带隙，并且随着迭代次数的增加，最低带隙的位置减小。结果表明具有Koch分形的晶格超材料在隔振和多功能设计中的潜在应用。