Quadrupolar topological behavior of elastic waves in nonsymmorphic two-dimensional square lattices

We investigate a higher-order topological behavior in elastic lattices characterized by nonsymmorphic symmetries. In the theoretical spring-mass lattice, altering the vertex mass allows for fine tuning of the topological features within the band gap. We analyze the quadrupole topological behavior in square lattices with nonsymmorphic symmetries using nested Wannier bands. Beyond second-order topological metamaterials, a single-phase topological configuration promotes energy localization at the corners due to a nonzero relative quadrupole moment. Our findings are validated through experimental observations of higher-order topological corner states, which show excellent agreement with simulated results and theoretical predictions. Additionally, the elastic lattices in the self-similar system exhibit fractal higher-order topological behaviors, revealing numerous topological edge and corner states. The self-similar lattice also demonstrates enhanced energy localization, with the number of topological states showing a linear correlation to the corner dimension. This study provides insights into elastic higher-order topological insulators and inspires innovative strategies for simulating topological elastic materials.