Spectral Element and Finite Element based Topological Edge and Localization of Protected Interface Modes in 1D Phononic Crystal Structures

In this paper, a theoretical and numerical study on the band transition and topological interface modes with topological phases for a 1D periodically piecewise continuous cross-section of phononic crystals (PCs) is established. By applying spectral element method with Timoshenko beam model, band structure and dispersion relation for flexural waves are obtained. The constitutive relation with dispersion spectra for longitudinal waves are also reported. The findings are compared and further validated by finite element based numerical simulations. Excellent agreement is found between both theoretical and numerical models. By varying geometric parameters band transition and accidental degeneracy is observed for both types of waves. Applying finite element method, polarization of wave modes is identified through Zak phase. Dirac cone plot with band inversion and mode transition frequency for both type of waves is reported. The nonlinear dispersive behavior of bending waves results in distinct Dirac cone. To confirm the existence and localization of the interface mode, finite length PC beams with varying geometric combinations are developed. Frequency response analysis through both theoretical and numerical model reveals the existence of interface mode. Interface mode is generated when mode transition frequency is common between the bandgaps of topologically distinct PCs. In addition, this interface mode is immune against the sharp edges, defects and backscattering of the waves and it provides single-way scattering free wave transport without any loss of energy. Effect of material and structural damping is also studied on the interface mode. Both theoretical and numerical results are reported side-by-side. The study provides a new insight in the design and application of topologically distinct PCs for vibration isolation, signal processing, energy storage, wave localization, energy harvesting, elastic/acoustic wave confinement and for solving other vibration related engineering problems.