Anti-plane Shear Waves in Layered Composites: Band Structure and   Anomalous Wave-refraction
  For oblique anti-plane shear waves in periodic layered elastic composites, it is shown that negative energy refraction is accompanied by positive phase-velocity refraction and positive energy refraction is accompanied by negative phase-velocity refraction, and that, this happens over a broad range of frequencies. The composite's unit cell may consist of any number of layers of any variable mass-density and elastic shear modulus (with large discontinuities).   Explicit series expressions for displacement, velocity, strain and stress components, and energy-flux fields are given, and group-velocity vector is calculated. The approach is based on a mixed variational principle where the displacement and stress components are viewed as independent fields subject to arbitrary variation. These fields are hence approximated independently, thereby ensuring the necessary continuity conditions. The resulting computational method yields the composite's frequency band structure and the associated mode shapes, in terms of the wave-vector components for any desired number of frequency bands.   The general results are illustrated using a two-phase and a three-phase unit cell with piecewise constant properties. It is shown that on their second frequency pass-bands, only the components of the phase and group velocities normal to the layers are antiparallel, while the components along the layers are parallel. Therefore, both the two-phase and the three-phase composites display negative energy refraction with positive phase-velocity refraction and positive phase-velocity refraction with negative energy refraction, depending on how the composite is interfaced with a homogeneous solid.   The presented method is applicable and effective also when some or all of the layers in a unit cell have spatially varying properties. 
Sia Nemat-Nasser
cond-mat.mtrl-sci