Enabling higher-order topological states in nested fractal elastic metamaterials

• A rich hierarchy of topological states is achieved in fractal mechanical lattices • Knowledge of fractal topological states in mechanical structures is established • A nested fractal metamaterial enables exotic elastic wave control capabilities • Experimental realization of topological states in fractal mechanical systems • Fractal states have superb confinement, scalable quantities, and flexible locations Topological metamaterials facilitate the defect-immune manipulation of elastic waves in mechanical systems. Thus far, research on topological metamaterials has concentrated on confining elastic waves at the boundaries and interfaces of mechanical lattices with integer-dimensional periodicity. On the other hand, topological metamaterial architectures with fractal dimensionality represent an uncharted domain that has the potential to unlock elastic wave phenomena not attainable in periodic mechanical lattices. In this work, we introduce a topological elastic metamaterial that harnesses the complex boundaries and interfaces of nested fractal mechanical lattices to enable a rich collection of higher-order topological states. Theoretical modeling of a spring-mass lattice patterned into a Sierpiński carpet is utilized to establish a deep understanding of topological corner and edge states in fractal mechanical systems. A systematic design framework is pioneered that facilitates the efficient realization of fractal topological states in discrete mechanical lattices. This framework is extended to continuous structures by creating a fractal topological metamaterial composed of a thin plate with nested fractal lattices of embedded mechanical resonators. Numerical and experimental investigations of the proposed metamaterial uncover an abundance of well-confined fractal topological states that have predictable frequencies, scalable quantities, and flexible spatial locations. The presented outcomes broaden the design space for topological metamaterials, paving the way for mechanical wave control devices with unprecedented functionalities.