Dirac degeneracy and elastic topological valley modes induced by local resonant states

In elastic systems, the current technique to produce Dirac degeneracy is based on Bragg scattering eigenstates, which, however, suffers from operating only at relatively high frequency determined by lattice constant. Here, an elastic metamaterial plate that presents an analog to the quantum valley Hall effect (QVHE) is proposed to achieve Dirac degeneracy with local resonant states, enabling us to tune the operating frequency without altering the lattice. By introducing resonator pair into a hexagonal lattice, a local resonance states-induced Dirac cone is produced right below the local resonant band gap caused by the resonator pair. After gapping the Dirac cone with unequal masses of the resonator pair, a new local resonant band gap supporting topological edge modes immune to backscattering appears. This band gap is formed by mixing effective negative mass effect and Bragg scattering effect due to large virtual mass. These ideas are demonstrated by numerical simulation, as well as validated by the experiment on flexural wave in a textured plate. The proposed design provides a new degree of freedom to control elastic topological mode and paves the way to explore subwavelength elastic topological interface states.

Figure 1

The principle scheme of local resonance states induced Dirac degeneracy. (a) The traditional elastic valley Hall insulator. (b) A resonator pair is introduced into the classic valley Hall insulator. (c) The obtained local resonant valley Hall insulator.

Figure 2

Theoretical analysis based on discrete hexagonal lattice. (a) Discrete hexagonal mass-spring lattice with local resonators. (b) The first Brillouin zone (BZ). (c) The dispersion diagram for the case where the two resonators are the same (dotted lines) and that for the traditional hexagonal phononic lattice without resonators (red solid lines). (d) The dispersion diagram for the case where the masses of the two resonators are slightly different from each other. (e) The Berry curvatures for the first and second branches shown in (d).

Figure 3

(a) The designed continuous hexagonal elastic lattice. (b) The dispersion structure for the case the two resonators are the same. (c) The dispersion structure for the case the two resonators are slightly different from each other. (d) The out-of-plane effective mass density.

Figure 4

The dispersion curves for (a) $h_1=h_2=2.0\mathrm{mm}$ and (b) $h_1=h_2=4.05\mathrm{mm}$.

Figure 5

Numerical analysis of the local resonant topological interface state. (a) The FE model of the strip supercell. (b) Band structure of the strip, where the green solid line represents the interface branch. (c) Eigenvector corresponding to the interface mode evaluated at $f=2500\mathrm{Hz}$. (d–f) The displacement fields at $f=2500\mathrm{Hz}$ for the bulk state, straight-line interface state and Z-shape interface state, respectively. (g) Simulated displacement amplitude profile of a cutline shown in (e), where position “0” corresponds to the resonators closest to the interface. (h) Calculated transmission spectra of the three states under study.

Figure 6

Experimental demonstration of the local resonant topological interface state. (a) The fabricated lattice sample and the experimental setup. (b) The measured frequency-response functions (FRF) of the displacements at points A and B. (c–e) The measured root mean squared distributions of the velocity field at 1500, 2045, and 2500 Hz, respectively.

Figure 7

Realization of double Dirac cone at desired frequency based on local resonant effect. (a) Schematic of the lattice with local resonators. (b) The first BZs of the original unit cell and the supercell. The calculated dispersion diagrams for (c) discrete lattice model and (d) solid continuous lattice model.

Figure 8

(left) The measured frequency-response functions. (right) Band structure of the strip supercell calculated by using the real stiffness of the microstructure beam retrieved from the experimental results. Good agreements between the calculations and experiments are observed.

Figure 9

The calculated displacement fields with effect of material damping for Z-shape interface route at $f=2500\mathrm{Hz}$.