High-frequency Homogenization of Periodic Elastic Structures

A new method of homogenization for periodic elastic structures is proposed. In the long-wavelength limit the effective homogeneous medium is introduced assuming that its dynamics, i.e. acceleration and displacement caused by propagating plane wave coincide with average acceleration and displacement of the unit cell of a given elastic structure. For 1D binary periodic structure with high acoustic contrast between the constituents, our method of homogenization leads to simple analytical formulas for the effective elastic modulus and effective mass density. These formulas are valid within a wide interval of frequencies for the long-wavelength parts of the spectrum lying well above the first acoustic band. They reproduce the exact dispersion relation with high accuracy. Negative values of one of the effective parameters are obtained within the band gaps. For the passing bands with anomalous dispersion double negative effective parameters are calculated, thus, predicting hyperbolic dispersion. The frequencies of topological transition from elliptic to hyperbolic dispersion are calculated exactly. This method of homogenization allows simple evaluation of the effective parameters and fast optimization.