Topological edge modeling and localization of protected interface modes in 1D phononic crystals for longitudinal and bending elastic waves

Abstract

In this paper, a theoretical and numerical study on the band transition and topological interface modes with topological phases is established for a 1D periodic cross-sections of phononic crystals (PCs) consisting of circular aluminum beams. The longitudinal waves band structure is studied by the transfer matrix method while the modified transfer matrix method with the Euler-Bernoulli beam model is established for studying the dispersion relation of bending waves. The initial five Bloch bands are presented and finite element validation with good agreement between the models has been reported. The variation in geometric parameters result into band transitions and accidental degeneracies has been observed at the lower and higher wave modes. The geometric phases of the Bloch bands are determined through numerical calculations of the Zak phase where symmetric and unsymmetric edge-mode states are distinguished. The Dirac cone plots indicate that band transitions and exchange in wave mode polarizations are successive in nature and their existence increases with increasing bandgaps. Thus, the number of mode transition frequency increases. Geometric phases provide information about the interface modes. These interface modes are generated when mode transition frequency is common between the bandgaps of a topological PC and they are validated through finite unit cell based frequency response study. Furthermore, through a time transient analysis the passband, bandgap and interface mode frequencies are differentiated and localization with robustness of interface mode is reported through the Q-factor calculation and spatial distribution of displacement fields. The effects of damping and material loss factor on the interface modes are also considered. In conclusion, this study mainly focuses on the existence of multiple interface modes for a topological PC at the lower and higher frequency regions that it can be further extended for solving multiple vibration related engineering problems.