# Ultra-Wide Bandgap in Two-Dimensional Metamaterial Embedded with Acoustic Black Hole Structures
## 3 Results and Discussion  
Figure 3 illustrates the phononic dispersion relations and the eigenstates of the lattice. Figure 3a shows the band structures of the lattice made of Polycarbonate, and it has two complete bandgaps from 5 kHz to 15 kHz, and from 18 kHz to 22 kHz. The width of the bandgap can be expressed using the gap-mid gap ratio, a non-dimensional parameter that avoids frequency dependence [29]  
\[BG\%=\frac{2\big{(}f_{top}-f_{bot}\big{)}}{f_{top}+f_{bot}}\% \tag{3}\]  
where \(f_{top}\) and \(f_{bot}\) are the bounding frequency of the bandgap. The gap-mid ratios of the first two bandgaps are 98% and 20%. As the material changes to epoxy resin, the first bandgap width increases significantly, and ranges from 6.4 kHz to 19.3 kHz with a gap-mid gap ratio equal to 100% shown in Figure 3b. The second bandgap ranges from 23.5 kHz to 27.5 kHz. Compared with Figure 3a, when the ABH structure is removed in the hexagonal lattice edge, the band structures in Figure 3c have no bandgaps within 25 kHz. This result shows that the traditional hexagonal cell structure can't attenuate the incident wave. On the contrary, if a designed structure is introduced into the traditional hexagonal lattice, it can be given a certain attenuation ability to waves and vibrations. Figure 4 illustrates the schematic diagram and Bloch modes of the simple lattice for Figure 3b. Due to the introduction of ABH structure at each edge of the hexagonal lattice, the stiffness of each edge of hexagonal lattice decreases, resulting in a local lumped mass at the junction of the three edges. It can be simplified as the form of lumped masses vibrating on springs, and the first six eigenstates: the rotational eigenstates S1 (717 Hz) and S2 (6442 Hz), and the transverse eigenstates S3 (6436 Hz), S4 (6436 Hz), S5 (19,365 Hz), and S6 (19,365 Hz), as shown in Figure 4d. The two lumped masses of State S1 rotate in opposite directions, and the rotation direction of two adjacent lumped masses in State S2 is the same. States S3 and S4 are the same eigenstates, and the phase difference of two adjacent masses is \(\pi\). Five ABHs of each lattice are mainly subjected to tensile and compressive stress. Similarly, States S5 and S6 belong to the same kind eigenstate, but the phase difference of  
Figure 2: Schematic diagram of the proposed 2D lattice. (**a**,**b**) The geometry of the 2D lattice embedded with ABHs at the middle of each hexagonal edge. The curve of ABH is driven by y(x) = 61.797x\({}^{2}\) + 0.0015 m. (**c**) The red triangle shows the first irreducible Brillouin zone of the 2D lattice. (**d**,**e**) Periodic array composed of hexagonal unit cells and the supercell. two adjacent masses is 0. Four ABHs of each lattice are mainly subjected to the shear stress, and one ABH structure is subjected to tensile and compressive stress. Figure 4: Schematic diagram and eigenstates of the proposed 2D lattice. (**a**) The simplified lattice of supercell. It can be simplified as the form of lumped masses vibrating on springs: the rotational eigenstates (**b**) and transverse mode (**c**). (**d**) The eigenstates S1–S6 in Figure 3b. Figure 3: Band structures of the lattice. The materials in (**a,b**) are polycarbonate ([PERSON]’s modulus E = 2.5 GPa, Poisson’s ratio \(\
u\) = 0.35, and density \(\varrho\) = 1100 kg/m\({}^{3}\)) and epoxy resin ([PERSON]’s modulus E = 4.35 GPa, [PERSON]’s ratio \(\
u\) = 0.3, and density \(\varrho\) = 1180 kg/m\({}^{3}\)), respectively. (**c**) The band structures of the reference hexagonal lattice. It is made of polycarbonate. Unit cells are depicted at the top right corner in the subfigure. (**d**) The associated displacement field distributions for the degenerated eigenstates at the \(\Gamma\) point. Due to the introduction of ABH structures at the edge of the hexagonal lattice, the width of the edge is driven by the nonlinear power-law profile, resulting in the geometric nonlinearity and the non-uniformity of modal distribution.