Refraction Characteristics of Phononic Crystals
  The refraction properties of phononic crystals are revealed by examining the anti-plane shear waves in doubly periodic elastic composites with unit cells containing rectangular and/or elliptical inclusions. The band-structure, group velocity, and energy-flux vector are calculated using a powerful variational method which accurately and efficiently yields all the field quantities over multiple frequency pass-bands. Equifrequency contours and energy-flux vectors are calculated as functions of the wave-vector. By superimposing the energy-flux vectors on equifrequency contours in the plane of the wave-vector components, and supplementing this with a three-dimensional graph of the corresponding frequency surface,a wealth of information is extracted essentially at a glance. This way it is shown that a composite with even a simple square unit cell containing a central circular inclusion can display negative or positive energy and phase-velocity refractions, or simply performs a harmonic vibration (standing wave), depending on the frequency and the wave-vector. Moreover that the same composite when interfaced with a suitable homogeneous solid can display: 1. negative refraction with negative phase-velocity refraction; 2. negative refraction with positive phase-velocity refraction; 3. positive refraction with negative phase-velocity refraction; 4. positive refraction with positive phase-velocity refraction; or even 5. complete reflection with no energy transmission, depending on the frequency, and direction and the wave length of the plane-wave which is incident from the homogeneous solid to the interface. By comparing our results with those obtained using the Rayleigh quotient and, for the layered case, with the exact solutions, the remarkable accuracy and the convergence rate of the present solution method are demonstrated. MatLab codes with comments will be provided. 
Sia Nemat-Nasser
cond-mat.mtrl-sci