Effective Velocity For Elastic Wave Propagation In Two-dimensional Phononic Crystals At Low Frequencies

In recent years, there have been many researches on elastic wave propagation in phononic crystals (PCs). On the one hand, full band gaps have caught much attention. The full band gaps are frequency domains in which propagation of sound and elastic waves are forbidden, numerous applications of which could be realized. On the other hand, like the negative refraction phenomenon existing in photonic crystals, negative refraction of PCs draws many scholars'attention. In addition, with the homogeneous property in low frequencies, PCs can be also utilized in the frequency region well below the gap, which has also received considerable attention. This paper studies the effective velocity for elastic waves propagation in two-dimensional (2D) PCs at low frequencies.In the long-wavelength, elastic waves propagating in the inhomogeneous elastic medium act like propagating in a homogeneous medium, which is the result of averaging over the wavelength, because it covers many lattice periods. Therefore, the effective velocity could be studied and analyzed at low frequencies.For periodic solid-solid medium, the three polarized wave modes are coupled each other. For 2D PCs with axial symmetry in periodic plane, modes polarized in-plane (L wave and SV wave) and the mode polarized out-plane (SH wave) are decoupled. Based on plane wave expansion, we first derived the wave equations of elastic waves propagating in 2D PCs, and then obtained the predicted sound velocities from the slope of the first acoustic band along the direction of detection. The results calculated by this method match well with those results by the finite element method or the analytical formulas derived by Q. Ni, which has been employed to calculate the slowness curves of PVC with square lattice of cylindrical pores. Numerical results show that the effective velocities of PCs are anisotropic. The influences of material parameters, filling fraction and lattice shapes on the anisotropy are analyzed. We find that PCs with rectangular lattices have much higher anisotropy than those with square lattices. For long-wavelength elastic waves propagating in the periodic plane of 2D PCs with square lattices, slowness curve of SV bulges in, L curve bulges out, and the SH curve is a circle. The slowness curves exhibit four-fold symmetry. For long wavelength elastic waves propagating in the periodic plane of PCs with rectangular lattices, for all three propagating modes, not only L wave and SV wave, but also SH wave, the effective velocities are distinctly anisotropic and the slowness curves exhibit two-fold symmetry. The anisotropy increases as the filling fraction increases or as the width to length ratio of the lattice decreases, and high anisotropy can be obtained in PCs with large contrast between material parameters, which is much higher in rectangular lattice than in square lattice with the same material parameters. Owing to these dependences, the effective velocity can be controlled in engineering.The liquid system for 2D PCs has also been studied. The effective speed of sound is anisotropic in PCs with rectangular lattices and the anisotropy mainly relates to the densities of the materials. Since the similarity of wave equations in liquid system and SH mode, the anisotropy of SH wave is mainly affected by elastic constant C44, which can be helpful in the derivation of empirical formulas for effective velocities and equivalent physical parameters in 2D PCs with rectangular lattices.