Multi-dimensional wave manipulation with 3D mechanical higher-order topological insulators

High-order topological insulators (HOTIs) have emerged as a promising platform for controlling wave propagation and trapping due to their unique topological properties. In this study, we investigate the multi-dimensional manipulation of elastic waves in two types of 3D HOTIs induced by the Dirac hierarchy. These two types of 3D HOTIs are characterized by molecule-zigzag hinge configurations and armchair hinge configurations, both derived from a 3D stack-layered structure featuring an eightfold-degenerate Dirac cone (DC). By gradually introducing alternating interlayer couplings in the vertical direction, varying the intercell and intracell couplings in the horizontal plane, and breaking mirror symmetry, we lift the eightfold-degenerate DC and construct the 3D mechanical HOTI with molecule-zigzag hinge configurations. Remarkably, the realization of 3D mechanical HOTIs with armchair hinge configurations does not require breaking mirror symmetry. Both of these 3D mechanical HOTIs exhibit intriguing hierarchical topological states, including surface states, hinge states, and corner states. Our findings provide new insights into the manipulation of elastic waves and offer potential applications in robust waveguiding, vibration signal processing and energy harvesting.