Topological mechanics beyond wave dynamics

Topological phases of matter, originally derived from condensed matter physics, have been extensively exploited in photonics and phononics, and recently in mechanical metamaterial systems. Topological mechanics mainly aims at the anomalous edge channels of wave propaga-tion in mechanical metamaterials that has potential applications like a one-way robust wave -guide. However, most studies on topological mechanics focus on wave-based dynamics. Here, a novel class of static topology in one and two dimensions beyond wave dynamics is presented, via the static Rayleigh deformation mode in delicately designed two-and three-dimensional me-chanical metamaterials. It is shown that the topological properties that are usually presented in dynamic systems, such as the topological invariants and bulk-edge correspondence, are also available in the static systems. It is demonstrated analytically and experimentally that for topo-logically protected zero modes, externally applied boundary loads can be fully blocked from penetrating the bulk, showing the effect of deformation shielding. Moreover, high-order topo-logical corner states and valley Hall effects are also explored with the proposed prototypical mechanical metamaterials. This study generalizes the concepts of topological mechanics beyond frequency-based wave dynamics.