Equivalent Nonlocal Medium Of Two-Dimensional Phononic Crystals

The unique properties and wide potential applications of phononic crystals have made the associated study one of the active research areas in condensed physics, acoustics, mechanics and the other related disciplines. Of various aspects of the research, the one into the average properties, including the average fluctuating properties of phononic crystals, based on the principle of uniformity, is the problem people pay much attention to. At present, most equivalent medium models are established in low-frequency limits, and the equivalent material parameters got from these models change with frequency, and are regarded as constants when the frequency is extremely low. By this way can the equivalent material parameters be obtained, but it is difficult to get dynamic field equations in a real sense. Therefore, it will be of significant importance in the research of phononic crystals to develop a rational equivalent uniform continuous medium model and establish dynamic field equations in a real sense.Nonlocal elasticity theory, which considers microstructure effect, succeeded in studying wave propagation in solids when the wavelength approximates the lattice's characteristic length. Based on this fact, we suggest that, when the wavelength approximates phononic crystals' periodic characteristic length, the two-dimensional phononic crystals could be modeled as a nonlocal elastic medium, and that the nonlocal elastic modoli are determined by matching the dispersion relations. Taking the two-dimensional solid-solid phononic crystals for example, we calculate the equivalent nonlocal elastic moduli as well as the dispersion curves of bulk and surface waves, with consideration of the first and second bands. The detailed analysis leads us to the following conclusions:1) From the comparison of the dispersion curves of bulk and surface waves with those acquired by adopting plane wave expansion method, it is proved that the present model not only is reasonable in physics and has good approximation in mathematics.2) The model not only yields the nonlocal elastic moduli but also gives the dynamic field equations. This lays the theoretical foundation for research of wave propagation in phononic crystals within the frame of continuous medium mechanics. 3) The model gives the approximate relations between the dispersion curves of the bulk P and SV waves of two-dimension phononic crystals, and of the bulk and surface waves. This is likely to reflect an inherent relationship between them.