Structures made of functionally graded materials (FGM) are successful attempts to enhance the mechanical properties of homogeneous materials. On the other hand, periodically architected structures provide phenomenal opportunities to design structures much lighter than their bulk counterparts showing exceptional mechanical characteristics. In the present study, to utilize the advantages of both technologies, wave propagation properties of functionally graded metamaterials (FGMM), i.e., periodically architected structures made of FGM, are investigated for three different planar topologies, and their vibration filtering performances are analyzed. The mathematical formulations to obtain the equation of motion for the FGMM are developed using the finite element method, and Floquet–Bloch’s theorem is employed to find their dispersion curves. Periodically architected structures with hexagonal, rectangular, and triangular unit cells are considered, and the effects of the FGM on their stop-band percentages are investigated. A comparison between band structures of pure steel (St), pure alumina ( $$\hbox {Al}_{2}\hbox {O}_{3}$$ Al 2 O 3 ) and $$\hbox {St}{-}\hbox {Al}_{2}\hbox {O}_{3}$$ St - Al 2 O 3 reveals that using FGM in the periodically architected structures can greatly enhance wave propagation properties by opening new stop-band regions leading to structures with much more versatility and tunability. The material distribution is assumed to vary according to both power-law and exponential-law rules along the beam axis and thickness, and the effects of Young’s modulus ratio, density ratio, relative density, and non-negative power-law exponent are scrutinized on the bandgap properties. The results indicate that periodically architected structures made of FGM exhibit much higher percentages of stop-bands, and playing with corresponding FGM parameters can tune this value for desired engineering needs. In addition, a mathematical approach is presented to investigate the polarization of the studied FGMM in the longitudinal, transverse, and rotational directions, and to measure the effects of material distribution on the polarization of the first three branches of the dispersion curves. It is revealed that the polarization factors of the three first dispersion branches are mainly geometry-dependent and change slightly with the material distribution.

中文翻译：

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平面功能梯度超材料中的可调谐弹性波传播

由功能梯度材料 (FGM) 制成的结构是提高均质材料机械性能的成功尝试。另一方面，定期构建的结构为设计比大体积结构轻得多的结构提供了绝佳的机会，这些结构显示出卓越的机械特性。在本研究中，为了利用这两种技术的优势，针对三种不同的平面拓扑研究了功能梯度超材料 (FGMM)，即由 FGM 制成的周期性结构的波传播特性，并分析了它们的振动过滤性能。使用有限元方法开发了获得 FGMM 运动方程的数学公式，并使用 Floquet-Bloch 定理来找到它们的色散曲线。考虑了具有六边形、矩形和三角形晶胞的周期性架构结构，并研究了 FGM 对其阻带百分比的影响。纯钢 (St)、纯氧化铝 ( $$\hbox {Al}_{2}\hbox {O}_{3}$$ Al 2 O 3 ) 和 $$\hbox {St} 的能带结构比较{-}\hbox {Al}_{2}\hbox {O}_{3}$$ St - Al 2 O 3 表明在周期性构造的结构中使用 FGM 可以通过打开新的阻带极大地增强波的传播特性区域导致结构具有更多的多功能性和可调性。假定材料分布根据沿着梁轴和厚度的幂律和指数律规则以及杨氏模量比、密度比、相对密度、和非负幂律指数的带隙特性被仔细检查。结果表明，由 FGM 制成的周期性建筑结构表现出更高的阻带百分比，并且使用相应的 FGM 参数可以调整该值以满足所需的工程需求。此外，还提出了一种数学方法来研究所研究的 FGMM 在纵向、横向和旋转方向上的极化，并测量材料分布对色散曲线前三个分支极化的影响。结果表明，三个第一色散分支的偏振因子主要取决于几何形状，并随材料分布而略有变化。结果表明，由 FGM 制成的周期性建筑结构表现出更高的阻带百分比，并且使用相应的 FGM 参数可以调整该值以满足所需的工程需求。此外，还提出了一种数学方法来研究所研究的 FGMM 在纵向、横向和旋转方向上的极化，并测量材料分布对色散曲线前三个分支极化的影响。结果表明，三个第一色散分支的偏振因子主要取决于几何形状，并随材料分布而略有变化。结果表明，由 FGM 制成的周期性建筑结构表现出更高的阻带百分比，并且使用相应的 FGM 参数可以调整该值以满足所需的工程需求。此外，还提出了一种数学方法来研究所研究的 FGMM 在纵向、横向和旋转方向上的极化，并测量材料分布对色散曲线前三个分支极化的影响。结果表明，三个第一色散分支的偏振因子主要取决于几何形状，并随材料分布而略有变化。提出了一种数学方法来研究所研究的 FGMM 在纵向、横向和旋转方向上的极化，并测量材料分布对色散曲线前三个分支极化的影响。结果表明，三个第一色散分支的偏振因子主要取决于几何形状，并随材料分布而略有变化。提出了一种数学方法来研究所研究的 FGMM 在纵向、横向和旋转方向上的极化，并测量材料分布对色散曲线前三个分支极化的影响。结果表明，三个第一色散分支的偏振因子主要取决于几何形状，并随材料分布而略有变化。