Fig. 1. Schematic diagram of phononic crystal plate structure, its unit cell and corresponding Brillouin zone: (a) The phononic crystal plate structure with triangular through-holes; (b) the unit cell of the phononic crystal plate; (c) the first Brillouin zone and irreducible Brillouin zone (red region) of the lattice.

Fig. 2. Band structure and displacement field distribution (DFD) in z-direction at eigenstates of the phononic crystal plate with triangular holes with different rotation angle: (a) , modes are below modes. The left DFD demonstrates mode shape of shear horizontal mode corresponding to the points, while the right group of DFDs illustrate the mode shapes of and modes. The color of the points on the dispersion curves corresponds to the polarization index; (b) , a double Dirac cone is formed, and the right DFDs show eigenstates distributions of and modes; (c) , modes are above modes.

Fig. 3. Effect of lattice parameters on eigenfrequencies of the dipole modes and quadrupole modes at the center Γ point of Brillouin zone: (a) The effect of the rotation angle θ of the triangular holes; (b) the effect of the side length l of the triangular holes.

Fig. 4. (a) Supercell composed of 5 TTCs and 5 TNCs; (b) the band structure of the supercell in (a), red and blue dots represent the edge states, A and B dots represent the pseudospin states in , and the gray circles represent the shear horizontal modes; (c) the DFDs in z-directiona corresponding to points A and B in (b), the enlarged figure shows the mechanical energy flux direction.

Fig. 5. Phononic crystal plate composed of TTCs and TNCs with different waveguide channels. (a) The black dashed line represents the edge formed by TTCs and TTCs, and the blue dashed line represents the edge formed by TTCs and TNCs. Three excitation points A, B and C are set at red points, and the DFDs under the vibration excitation in z-direction are shown in below. (b) The blue dashed line represents the zigzag edge formed by TTCs and TNCs.

Fig. 6. Defective phononic crystal plate composed of TTCs and TNCs: (a) The zigzag channel with missing triangular holes (the red point is the excitation position), and the DFDs under the z-direction vibration excitation are shown in below; (b) the zigzag channel with disordered triangular holes

Fig. 7. One-way propagation of elastic wave is realized by using multi-point excitation strategy, and the corresponding DFDs clearly show the one-way propagation phenomenon of elastic wave based on the topological protected edge states: (a) The pseudospin + state is generated by the strategy; (b) the pseudospin － state is generated by the strategy.

Fig. 8. Multichannel waveguide switch based on topologically protected edge states: (a) The phononic crystal plate with an excitation point (blue point) on the left side and four waveguide channels (blue dashed lines) which formed by different types of crystals, in which the blue or red circular arrow indicates the pseudospin direction of elastic wave propagating along the channel from the excitation point; (b) the DFDs when the excitation point is the left blue point and the enlarged figure shows the mechanical energy flux direction; (c) the phononic crystal plate with an excitation point (red point) on the upper side; (d) the DFDs when the excitation point is the upper red point.