Homogenization of Two-Dimensional Phononic Crystals at Low Frequencies

Effective velocities of elastic waves propagating in two-dimensional phononic crystal at low frequencies are analysed theoretically, and exact analytical formulas for effective velocities of elastic waves are derived according to the method presented by Krokhin et al. [Phys. Rev. Lett. 91 （2003） 264302]. Numerical calculations for phononic crystals consisted of array of Pb cylinders emSedded in epoxy show that the composites have distinct anisotropy at low filling fraction. The anisotropy increases as the filling fraction increases, while as the filling fraction closes to the limitation, the anisotropy decreases.     

Homogenization of Three-Dimensional Periodic Solid-Solid Elastic Composites

We develop an approach to homogenize three-dimensional periodic solid-solid elastic composites with cubic lattice at low frequencies, by using plane wave expansion and perturbation theory with respect to the long wavelength limit. Based on the fact that the two shear waves propagating along lattice axis are degenerated, we derive formulae for effective velocities parallel and normal to the lattice axis, from which three independent effective elastic moduli are calculated, respectively. Theoretical results, which take into account the multiple scattering and the structure of the periodic medium, are in good agreement with the previous isotropic theory at high-symmetry directions.