Anti-plane shear waves in periodic elastic composites: band structure and anomalous wave refraction

For anti-plane shear waves in periodic elastic composites, it is shown that negative energy refraction can be accompanied by positive phase-velocity refraction and positive energy refraction can be accompanied by negative phase-velocity refraction, and that this can happen over a broad range of frequencies. Hence, in general, negative refraction does not necessarily require antiparallel group and phase-velocity vectors. Details are given for layered composites and the results are extended to, and illustrated for, two-dimensional periodic composites, revealing a wealth of information about the refractive characteristics of this class of composites. The composite's unit cell may consist of any number of constituents of any variable mass density and elastic modulus, admitting large discontinuities. A powerful variational-based solution method is used that applies to one-, two-and three-dimensional composites, irrespective of their constituents being homogeneous or heterogeneous. The calculations are direct, accurate and efficient, yielding the band structure, group-velocity, energy-flux and phase-velocity vectors as functions of the frequency and wavevector components, over an entire frequency band.