Multi-displacement continuum modelling of the metamaterial plate with periodical arranged resonators

Abstract

• The wave propagation in a metamaterial plate with periodically arranged resonators is studied. • The additional displacement fields are introduced to characterize the response of resonators. • The function variation principle is used to derive the governing equations of Bloch waves. • The propagation along symmetric direction and any oblique direction are both considered. • The influences of interface conditions on the dispersion feature under the oblique propagation are first discussed. Multi-displacement continuum modelling of a two-dimensional (2D) elastic metamaterials plate with periodically arranged local resonator over the surface of the plate is studied in this paper. The additional displacement fields are introduced to model the response of the local resonators. The continuous conditions between the adjacent unit-cells are used to reflect the periodicity of the microstructured continuum and resultantly turned into the constraint conditions between the additional displacement field and the other continuous field. The dispersion features of the multiple-displacement coupled wave propagating along the high symmetrical direction and any oblique direction are both studied numerically. It is found that the multi-displacement coupled waves can be divided into the coupled longitudinal wave and the coupled transversal wave when propagating along the highly symmetric direction but cannot be divided into the coupled longitudinal wave and the coupled transversal wave when propagating along any oblique direction. The effects of boundary conditions on the dispersion of acoustic and optical branches of coupled waves are discussed in detail. At last, the influences of the parameters of resonator on the dispersion feature of the multi-displacement coupled waves are investigated numerically.