基于一致场积分的局部共振粘弹性层状介质斜向反平面剪切波的整体动力特性
Overall dynamic properties of locally resonant viscoelastic layered media based on consistent field integration for oblique anti-plane shear waves
摘要：
本文研究了具有粘弹性分层结构的微结构介质中的斜向反平面剪切波在低频共振情况下的传播和散射问题。利用传输矩阵方法，计算了无限周期系统的能带结构和有限厚平板的散射。一个一致的动态场均匀化的方法，其中的微尺度场方程的集成和整体的宏观尺度量被定义为与这些积分形式兼容。利用提出的均匀化宏观尺度量，结合Onsager原理和假定的弹性动力学互易性的物质形式，提出了一套适用于一般非对称重复单胞（RUC）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，全本构张量需要匹配的散射和能带结构的微结构介质，但所有的非对角参数消失的对称RUC。因此，有可能创建具有用于对称RUC的唯一确定的对角本构张量的等效均化表示，但是，与非对称RUC的情况一样，所有非零分量将是波矢量相关的。数值算例验证了该方法的适用性和一致性。所有对角项在所有波方向的长波长极限处都收敛到它们相应的Voigt或Reuss平均值。即使在这种极限下，仍然可以观察到非对角耦合常数的波矢依赖性。有损或无损系统的条件，并施加弱的要求，整体本构张量。
英文摘要：
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 are required 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 or lossless systems are presented and are shown to impose weak requirements on overall constitutive tensors.
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