Quadrupole topological behavior of elastic waves in two-dimensional square lattices with nonsymmorphic symmetries

We investigate a novel 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 bandgap. 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 non-zero 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 new insights into elastic higher-order topological insulators and inspires innovative strategies for simulating topological elastic materials.