# Progress in Topological Mechanics
## 2 Mechanical Topological Edge Sates
### Analogue Valley Hall Insulators  
A new degree of freedom, valley, has been introduced into the classical wave systems. The QVHE has been theoretically and experimentally demonstrated in photonic crystals [38; 39; 40; 41; 84; 85], sonic crystals [42; 49; 50; 53] and mechanical structures [55; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78]. In particular, topologically valley-polarized states in elastic systems have been proposed via bi-layered lattices [69; 74], hexagonal lattices [75; 70; 73; 76; 77; 78; 79; 88; 89; 90] and triangular lattices [71; 72; 86]. The key to realizing the QVHE is breaking mirror symmetries or inversion symmetries. Now we will discuss some iconic achievements regarding QVHE in mechanical systems.  
In 2017, [PERSON] et al. constructed a continuous elastic plate that consists of hexagonal lattices with magnetic cylinders attached to the sublattice sites, as shown in Figure 3a [76]. The magnetic cylinders act as the localized masses at the nodes used to break the mirror symmetry. Particularly, it refers to reducing the original \(C_{3v}\) symmetry to \(C_{3}\) symmetry in a unit cell. The number of cylinders (masses) is controlled by the parameter \(\gamma\). The masses of the two sites a, b in a unit cell are \(m_{a}=m(1+\gamma)\) and \(m_{b}=m(1-\gamma)\) respectively, as shown in Figure 3b. The value \(\gamma=0\) represents no additional masses, and the Dirac cones appear at the K and \(K^{\prime}\) points in this case. The value \(\gamma>0\) refers to additional masses at site a; on the contrary, the value \(\gamma<0\) refers to additional masses at site b. The degeneracy would be lifted when \(\gamma\
eq 0\). The band structure with the valley Chern numbers labelled at the K and \(K^{\prime}\) points is shown in Figure 3c. In this context, the topological boundary state emerges at a line interface of the elastic plate by setting the value \(\gamma=1\) or \(\gamma=-1\) on the left or right side of the interface, as shown in Figure 3d.  
Figure 2: Analogue spin Hall insulators. (**a**) The elastic plate and the zoomed-in view of the unit cell. The inset presents the high-symmetry points \(\Gamma,K,K^{\prime},M\) of the first irreducible Brillouin zone. [61]. (**b**) The schematic of three types of unit cell with distinct holes [61]. (**c**) Dispersion curve for the plate with blind holes. The colors ranging from blue to red suggest that from pure in-plane modes to pure out-of-plane modes [61]. (**d**) The experimental configuration [63]. (**e**) The edge-state transmission along the domain wall at 7.70 kHz frequency [63]. The color represents the power spectral density, which is calculated from the measured out-of-plane velocity of the plate. (**f**) The experimental setup, in which a piezoelectric actuator is positioned at the center to excite the plate, and out-of-plane velocity is measured point-by-point by an LDV (Polytec OFV 534 laser Doppler vibrometer) mounted on a two-axis linear stage [63].  
In 2019, [PERSON] et al. observed the topological valley transport of elastic waves in numerical simulations based on the bi-layered lattices [74]. The structure is constructed by two layers of snowflake plates connected by a honeycomb array of cylinders, as shown in Figure 3e. In the same year, [PERSON] et al. proposed a programmable elastic valley Hall insulator whose edge states could propagate along arbitrary interface routes by a programmable magnet lifting array [79]. The unit cell consists of two cylindrical cavities with a small arched channel connecting them. It is obvious that to gap the K-point (Dirac cone), the inversion symmetry should first be broken. Hence, magnetic fluid with equal volume as one cavity is injected into the unit cell, and the inversion symmetry is broken by independently adjusting the magnetic fluid distribution in each unit cell (see Figure 3f). The measured displacement field for the Z-shaped interface configurations is shown in Figure 3g.