Uncovering higher-order topological states in fractal mechanical metamaterials

Abstract

Recently, researchers have incorporated topological phases into mechanical metamaterials to facilitate defect-immune elastic wave and vibration manipulation. The topological mechanical metamaterials developed thus far have achieved extraordinary wave control capabilities through the construction of robust elastic waveguides at the boundaries and interfaces of 1D, 2D, and 3D periodic mechanical lattices. Given the overwhelming focus of previous research on traditional integer-dimensional mechanical architectures, an unexplored opportunity exists to investigate the emergence of topological phases in fractal mechanical metamaterials, which have a non-integer dimension and exhibit self-similarity across multiple scales. This research addresses the unexplored opportunity and advances the state of the art through the synthesis of a 1.89D fractal mechanical metamaterial that harnesses higher-order topological phases to enable multifaceted elastic wave and vibration control. The proposed fractal topological mechanical metamaterial is a thin plate with embedded torsional spring-mass resonators that are arranged into the pattern of a 1.89D Sierpiński carpet. A numerical eigenfrequency analysis uncovers coexisting topological corner and edge states that trap wave energy at the myriad corner and edge interfaces available in the 1.89D fractal. The outcomes from this study provide insight into the attainment of higher-order topological states in fractal metamaterials that localize elastic waves and vibrations across various locations and frequencies, opening the door for future research of topological phases in mechanical metamaterials with fractal architectures.