  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.
