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Complex dispersion relations and evanescent waves in periodic beams via the extended differential quadrature method

机译：扩展差分求积法求周期光束中的复色散关系和e逝波

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Because of computation difficulties, investigations about the complex band structures and the evanescent wave modes in phononic crystals are very limited. In this paper, a novel k(omega) method, referred to as the Extended Differential Quadrature Element Method (EDQEM), is successfully developed to investigate the complex dispersion relations and the evanescent wave modes in periodic beams. At first, based on the Bloch-Floquet theorem and the two widely used beam theories, i.e., the Euler-Bernoulli beam theory and the Timoshenko beam theory, the EDQEM is developed to solve the dispersion equations of flexural waves in periodic beams. Comparisons with other related investigations are conducted to validate the correctness of the proposed method. Furthermore, considering three important factors, the shape of the unit cell, the pattern of the sampling point as well as the number of the sampling point, the convergence of the proposed method is investigated. Second, with the help of the EDQEM, complex dispersion relations of periodic beams are investigated and wave mode analysis is conducted, from which all possible waves, including propagative waves, purely evanescent waves and complex waves, in the complex dispersion curves are discussed. It is found that complex wave modes in periodic beams arise from two situations: (1) at the boundary of the first Brillouin zone and (2) within the first Brillouin zone. These complex wave modes are the transition modes between two propagative waves, or between the propagative wave and the complex wave. When the damping effect is included, all waves in periodic beams transfer into the complex waves. And, band gaps are not truly apparent anymore.
机译：由于计算困难，关于声子晶体中的复带结构和van逝波模的研究非常有限。本文成功地开发了一种新的k（Ω）方法，称为扩展微分正交元素法（EDQEM），以研究周期光束中的复色散关系和van逝波模。首先，基于Bloch-Floquet定理和两种广泛使用的光束理论，即Euler-Bernoulli光束理论和Timoshenko光束理论，开发了EDQEM来求解周期光束中弯曲波的色散方程。与其他相关研究进行了比较，以验证所提出方法的正确性。此外，考虑到三个重要因素，即晶胞的形状，采样点的模式以及采样点的数量，研究了该方法的收敛性。其次，借助EDQEM，研究了周期光束的复数色散关系，并进行了波模分析，从中讨论了复数色散曲线中所有可能的波，包括传播波，纯渐逝波和复数波。发现周期性波束中的复杂波模是由两种情况引起的：（1）在第一个布里渊区的边界处，以及（2）在第一个布里渊区内。这些复波模式是两个传播波之间或传播波与复波之间的过渡模式。当包括阻尼作用时，周期光束中的所有波都转变为复波。而且，带隙不再是真正明显的。

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