The elastic wave propagation through the finite and infinite periodic laminated structure of micropolar elasticity

The reflection and transmission coefficients through multilayered structure of finite thickness and the dispersive relations of Bloch waves in one-dimension infinite periodic structure (phononic crystals) consisting of two repetitive different micropolar slabs are studied. Due to the additional microrotational degree of freedom, the modes of elastic waves in a micropolar solid are different from that in the traditional elastic medium. The transfer matrix and the stiffness matrix of each slab and further one single cell are obtained for both in-plane and anti-plane situations. The dispersive equations are derived by using Bloch theorem. The relation between the minimum eigenvalue of transfer matrix of single cell and the stop bands are presented. The reflection and transmission coefficients through finite multilayered structure are also calculated based on the stiffness matrix method to avoid the numerical instability of the transfer matrix method. The energy fluxes carried by the reflection and transmission waves are estimated and the energy conservation is checked to validate the numerical results. Based on the numerical results, the influence of micropolar parameters on the dispersive feature of Bloch waves are discussed. It is found that the dispersive feature and the bandgap of Bloch waves in the micropolar phononic crystal are evidently different from that in the classic phononic crystal. The dispersive feature and bandgap of Bloch wave at high frequency are more sensitive to the micropolar constants than at low frequency range.