In this work, the analysis of oblique anti-plane shear waves propagation and scattering in low frequency resonant micro-structured layered media with viscoelastic constituent layers is presented. The band structure of the infinitely periodic systems and scattering off a finite thickness slab of such media are determined using the transfer matrix method. A consistent dynamic field homogenization approach is applied, in which the micro-scale field equations are integrated and the overall macro-scale quantities are defined to be compatible with these integral forms. A reduced set of constitutive tensors is presented for general asymmetric repeating unit cells (RUC), utilizing the proposed homogenized macro-scale quantities combined with Onsager’s principle and presumed material form of elastodynamic reciprocity. This set can be further restricted by studying the form of the dispersion equation leading to a unique constitutive tensor. It is shown that for an asymmetric RUC, the full constitutive tensor, including off-diagonal elastic moduli and Willis coupling terms are required in order to match the scattering and band structure of the micro-structured media, but all the off-diagonal parameters vanish for a symmetric RUC. Therefore, it is possible to create an equivalent homogenized representation with a uniquely determined diagonal constitutive tensor for a symmetric RUC, though, as is the case also with asymmetric RUCs, all non-zero components will be wave-vector dependent. Numerical examples are presented to demonstrate the application and consistency of the proposed method. All the diagonal terms are converging to their appropriate Voigt or Reuss averages at the long-wavelength limit for all wave directions. The wave-vector dependent nature of the off-diagonal coupling constants can still be observed even in this limit. The conditions for lossy (passive) or lossless (fully elastic) systems are presented and are shown to impose weak requirements on overall constitutive tensors.

在这项工作中，提出了具有粘弹性成分层的低频共振微结构层状介质中斜反平面横波传播和散射的分析。无限周期系统的能带结构和这种介质的有限厚度板的散射是使用传递矩阵方法确定的。应用一致的动态场均匀化方法，其中对微观尺度场方程进行积分，并定义整体宏观尺度量以与这些积分形式兼容。利用建议的均质宏观量与 Onsager 原理和假定的弹性动力学互易性材料形式相结合，为一般不对称重复晶胞 (RUC) 提供了一组简化的本构张量。通过研究导致唯一本构张量的色散方程的形式，可以进一步限制该集合。结果表明，对于非对称 RUC，需要全本构张量，包括非对角弹性模量和 Willis 耦合项，以匹配微结构介质的散射和能带结构，但所有非对角参数均为零对于对称 RUC。因此，可以为对称 RUC 创建具有唯一确定的对角线本构张量的等效均质化表示，尽管与非对称 RUC 的情况一样，所有非零分量都将依赖于波矢。给出了数值例子来证明所提出方法的应用和一致性。所有对角线项都在所有波方向的长波长极限处收敛到其适当的 Voigt 或 Reuss 平均值。即使在这个限制中，仍然可以观察到非对角耦合常数的波矢量相关性质。介绍了有损（被动）或无损（完全弹性）系统的条件，并表明对整体本构张量的要求较弱。