Periodically alternated elastic support induced topological phase transition in phononic crystal beam systems

Determining strong robustness in flexural wave propagation in beam systems has always been a promising achievement with various real-life applications such as energy harvesting and waveguiding. In this manuscript, the bending characteristics of a phononic crystal beam on periodically alternated linear elastic support systems with topologically protected wave propagation are proposed. Based on the Timoshenko beam theory, general solutions of this periodic beam-foundation topological system are obtained. Subsequently, by applying the Bloch theory, the dispersion relation is obtained using the transfer matrix method. Comparing with finite element numerical solutions, excellent agreement is observed with only a minor difference due to the assumptions in the Timoshenko beam theory. Generally, any breaking of spatial symmetry in beam systems can be achieved by changing geometry of beam cross-sections or tuning the material properties of substructures. We herein propose a new approach to break the spatial symmetry, i.e., tuning the elastic stiffnesses of the periodically alternated elastic supports. It is found that beam bending on an infinite continuously distributed elastic foundation leads to a band-crossing point. After tuning the elastic support stiffness in a periodic manner, the band-crossing point is broken and a new bandgap appears due to the breaking of structural symmetry. Further, a mode shape inversion and Zak phase transition are captured. Based on a unit cell analysis, we examine the topologically protected interface modes in an array constructed with periodically arranged unit cells with two corresponding phase states. Furthermore, several numerical examples are used to demonstrate the defect- and disorder-immune properties in this one-dimensional topological system. This new proposed general mechanism can be extended to establish topological phase transitions in other mechanical and dynamic systems.