Valley Hall Elastic Edge States in Locally Resonant Metamaterials

Abstract

This paper presents a locally resonant metamaterial periodically rearranged as a local resonator, that is hexagonal holes arranged in a thin plate replace the elastic local resonator to achieve the quantum valley Hall effect. Due to the C3v symmetry in the primitive hexagonal lattice, one Dirac point emerges at high symmetry points in the Brillouin zone in the sub-wavelength area. Rotating the beam element of the resonator can break the spatial inversion symmetry to lift the Dirac degeneracy and form a new bandgap. Thus, the band inversion is discovered by computing the relationship between the associated bandgap and the rotational parameter. We also confirmed this result by analyzing the vortex chirality and calculating the Chern number. We can discover two kinds of edge states in the projected band obtained by computing the supercell composed of different topological microstructures. Finally, the propagation behavior in various heterostructures at low frequencies was analyzed. It is shown that these valley Hall elastic insulators can guide elastic waves along sharp interfaces and are immune to backscattering from defects or disorder. By utilizing elastic resonators, a simple reconfigurable topological elastic metamaterial is realized in the sub-wavelength area.

Sub-wavelength topological boundary states and rainbow trapping of local-resonance phononic crystal plate

“…In our work, the height difference of VPCs is 0.3 (−0.3) mm, and the corresponding C v is about ∓ 0.28 (± 0.28). The strong SIS breaking will cause C v to deviate from its theoretical value in VPCs [34,39], so the our obtained C v deviates from the theoretical value 1/2 (−1/2).…”

“…Valley phononic crystals (VPCs) are a kind of artificial periodic structures based on the analogy of the quantum valley Hall effect, whose realization relies on the design of the spatial structure [22]. As for VPCs, realizing their topological boundary states requires breaking the SIS by introducing differences in mass or stiffness [23][24][25][26][27][28][29][30] and rotation elements [31][32][33][34][35][36]. Vila et al selected additional mass blocks on the hexagonal frame to realize the propagation mode of topological protection of elastic waves along non-trivial interfaces [23].…”

Tailoring Structure‐Borne Sound through Bandgap Engineering in Phononic Crystals and Metamaterials: A Comprehensive Review

“…These valley states are proven to yield opposite 1/2 pseudospins, leading to robust edge states that are immune to certain defects. Local resonance structures have been used to break the SIS in lattices formed by mechanical beams [164,165,[273][274][275][276][277][278][279][280][281] (see example in Figure 6e), and soft materials, [282,283] where the SIS is broken by the variation of resonators in the different valleys. The valley eigenstates have been observed with opposites 1/2 spins, which are characterized by valley-Chern numbers.…”

Local-Resonance-Induced Dual-Band Topological Corner States of Flexural Waves in a Perforated Metaplate

“…Despite the use of resonators in these designs, topological states still exist within the topological band gap opened by the Bragg scattering rather than by the local resonance [29][30][31]. Subsequently, locally resonant topological band gaps, which are formed by opening the local-resonance-created degenerate points, have been demonstrated to support topological edge states [29,[32][33][34][35]. However, the introduction of local resonators is at the cost of complicating the metastructures and imposing difficulties in system modeling, given the imperfect contacts between the resonators and the host structure.…”

“…However, the introduction of local resonators is at the cost of complicating the metastructures and imposing difficulties in system modeling, given the imperfect contacts between the resonators and the host structure. In addition, topological band gaps created by the local resonance [29,[32][33][34][35] typically possess narrow bandwidths, limiting the localization degree of topological waves.…”