Polarized elastic topological states in hexagonal lattices

Robust elastic wave propagation against bending and weak perturbations can be realized at the edges by analyzing the topological states in mechanical systems. However, both longitudinal and transverse wave components of polarized elastic waves complicate their manipulation. Current studies on the topological properties of elastic waves mainly address a single polarization component, even though the diverse polarization characteristics show significant potential for advanced manipulation of elastic waves. In this study, a hexagonal lattice with additional masses is presented to discuss the topological properties, and the out-of-plane and in-plane polarized valley topological states are realized in the lattice system. First, the dynamical model of hexagonal lattice based on the Timoshenko beam theory is proposed to analyze the topological properties, and the out-of-plane polarized and in-plane polarized topological phases are realized by tuning the additional masses. Then the Symplectic solution method is introduced to simplify the calculation of band structures, and the polarized topological properties are obtained by calculating the Berry curvatures and Wannier centers. Subsequently, an interface with a sharp corner is constructed to demonstrate the existence of polarized topological edge states and corner states. Finally, we analyze the out-of-plane polarized and in-plane polarized topological states in the frequency domain and further realize the prediction of the topological corner states according to the polarization of the elastic waves. The proposed structures provide an effective way to tailor the multi-directional elastic wave propagation and realize the selective transmission of the single-polarized elastic waves.