Extracting method of dynamic effective parameters for plate-type elastic metamaterials with decentralized resonant mass

Abstract

In this paper, a method is presented to obtain both in-plane and out-plane dynamic effective parameters of plate-type elastic metamaterials with decentralized resonant mass. The method is derived by plate-type elastic metamaterials with concentrated resonant mass, and the effective material properties are obtained by averaging local physical field in each phase when prescribed displacements are applied on the boundaries of the unit cells. The accuracy of the method is verified, since the frequency ranges of negative mass density are the same with those of band gaps. In addition, the starting and ending of negative mass density are analyzed by eigenstates on the bandgap edges. Multi-negative parameters are produced by abundant resonances exhibited in the unit cells when synergetic motions are generated by the decentralized resonant mass. The proposed method is extended to determine the effective properties of the plate-type elastic metamaterials with decentralized resonant mass. Compared with the calculated effective properties of the model with concentrated resonant mass, more negative mass density regions are formed both in-plane and out-plane for the model with decentralized resonant mass. Moreover, negative moduli are also generated by monopolar and quadrupolar resonances, and the negative hybrid dispersion bands are produced by the combination of negative mass density and negative moduli. The calculated results show that the proposed method is convenient to determine both in-plane and out-plane dynamic effective parameters of plate-type elastic metamaterials with complex internal structures, and easier than the method based on the Mie scattering solution. Therefore, the proposed method will be helpful in designing plate-type elastic metamaterials with resonant micro-structures.