Valley-protected topological interface states in metastructures with internal geometric rotation

Topological mechanical metamaterials have garnered significant attention for their ability to exhibit robust, defect-immune wave propagation and mechanical behaviors, inspired by topological protection mechanisms in condensed matter physics. In this study, we present a novel mechanical metamaterial design that introduces rotational geometric parameters within the unit cell to explicitly break spatial inversion symmetry. By precisely tuning the rotation angle of structural elements in the unit cell, we induce asymmetric valley states with opposite Berry curvatures, thereby realizing the valley Hall effect in a mechanical framework. This purely geometric approach avoids reliance on material composition gradients or external fields, offering intrinsic control over valley polarization through structural design alone. Numerical simulations and mechanical analyses demonstrate that the proposed metamaterial supports topologically protected interface states at the boundary between regions of distinct valley topologies. These interface states exhibit unidirectional propagation, confirming their topological protection. This work provides a universal geometric strategy to engineer topological phenomena in structural systems. The realized topologically protected interface states hold promise for applications in high-precision sensors, energy harvesting devices, and vibration isolation systems.