Fig. 1

(a) Schematic of the periodic structure and (b) schematic of the unit cell. The figure shows the corresponding positions of the structural parameters, and the specific parameter values are shown in Table 1. (c) The longitudinal diagram of the unit cell. (d) Within the reciprocal lattice space, the first Brillouin zone is identified, with its fundamental sector highlighted in blue, representing the irreducible portion of the first Brillouin zone (color online)"

Table 1

Geometric parameters Unit: mm"

Fig. 2

(a)-(c) Out-of-plane band structures of (a) α = 0º (named as Unit cell B), (b) α = 30º (named as Unit cell C), and (c) α = -30º (named as Unit cell D). The Dirac point formed by Dirac degeneracy in (a) is marked by a black dot, and the opened out-of-plane bandgaps in (b) and (c) are represented by gray regions (color online)"

Fig. 3

Analysis of modes with topological properties: (a) the connection between the bandwidth and the rotation angle α of the prism, where the solid lines indicate the bandwidths of Cells C and D at the wave vector k; (b) kinetic energy distributions of Unit cells C and D in (a). The color represents the amplitude of the kinetic energy Wk, where the energy distribution of Unit cells C and D reveals the opposite chirality, which indicates a topological phase transition; (c) the top views of the out-of-plane phases of Unit cells C and D in (a), and the in-plane displacement is almost invisible, where the color represents the amplitude of the out-of-plane displacement (color online)"

Fig. 4

Schematic dispersion curves and modes of the supercell. (a) Schematic of the supercell, in which Cells D and C are marked in blue and orange, respectively. (b) Dispersion curves of the supercell, where the blue and gray curves are the topological edge modes. E1/F1 and E2/F2 mark the eigenfrequencies of the edge states at wave vectors k=-0.5 and k=0.5, respectively (color online)"

Fig. 5

Dispersion curves of the supercell and the topological edge eigenstates at a particular point. (a) Schematic of the supercell and topological edge modes, where the green, blue, and orange cells are Cells B, C, and D, respectively. (b) Dispersion curves of the supercell, where the blue curve is the valley-locked edge state, and the shaded zone is the theoretical bandgap. G1 and G2 are the eigenfrequencies at k = -0.8 and k = 0.8, respectively. The right panels show the eigenstate of topological edges at G1 and G2 (color online)"

Fig. 6

At a frequency of 525 Hz, the displacement field characteristics of valley-locked elastic waves are as follows: (a) the schematic of the phononic crystal plate composed of Cells B, C, and D, where the green, blue, and orange cells are Cells B, C, and D, respectively, regions S1 and S2 within the range of Cells B are selected to calculate the transmission rate and are marked in light purple, and the red arrow marks the excitation position; (b) waveguide under impurity, where cells with dark green color in the inserted panel denote the local impurity; (c) waveguide under local disorder; (d) transmittance of valley-locked elastic waves under wide interface, impurity, and local disorders. The shaded area marks the theoretical bandgap (color online)"

Fig. 7

Klein tunneling of elastic waves: (a) a conical dispersion profile associated with the Klein tunneling, defined by an energy barrier of the amplitude V and a lateral dimension of D, where the red and blue lines signify branches with contrary pseudospin orientations, mirroring the properties of a massless relativistic Dirac quasiparticle, two types of composite panels consist of Cells A and B (marked as red and green, respectively), and regions S1 and S2 (labeled as light purple) are the integral areas of calculating transmission; (b) dispersion curves of Cells A and B, where the red shaded area represents the barrier; (c) transmittance of the designed two composite plates and displacement fields corresponding to the frequencies 480 Hz, 575 Hz, and 740 Hz of the star on the transmittance (color online)"

Fig. 8

Experiments and results obtained: (a) the experimental setup; (b) the transmittance obtained by the experiment (color online)"

Fig. 9

Schematics of phononic crystal plates and transmittance: (a) Case 1, Cell B is inserted into Cell A as a potential barrier; (b) Case 2, Cell A is inserted into Cell B as a potential well, where the red, green, blue, and orange cells are Cells A, B, C, and D, respectively. In Case 1 (Case 2), regions S1 and S2 within the range of Cell A (Cell B) are selected to calculate the transmission rate, and are marked in light purple, and the position of the black arrow indicates the excitation; (c) transmittances of Cases 1 and 2, where the frequencies of Points Q1 and Q2 are 525 Hz and 642 Hz, respectively; (d) and (e) displacement fields for Cases 1 and 2 at two frequency points, respectively (color online)"

Fig. 10

Elastic wave splitter: (a) schematic diagram of Case 1 phononic crystal plate, where the red, green, blue, and orange cells are cells A, B, C, and D, respectively. Cell B is inserted into Cell A as a potential barrier. The input port is indicated by the black arrow, denoted as I; the output ports are indicated by blue arrows, denoted as O1 and O2; (b) transmission of outputs O1 and O2 is derived from the simulation study. The light gray shade and the light purple shade represent two frequency intervals, and the three frequency points are marked with stars; (c)-(e) displacement fields under different frequencies obtained from simulation: (c) 575 Hz, (d) 606 Hz, and (e) 640 Hz (color online)"

Fig. 11

Elastic wave splitter: (a) schematic diagram of Case 2 phononic crystal plate, where the red, green, blue, and orange cells are cells A, B, C, and D, respectively. Cell A is inserted into Cell B as a potential barrier. The input port is indicated by the black arrow, denoted as I; the output ports are indicated by blue arrows, denoted as O1 and O2; (b) transmittance of outputs O1 and O2 obtained from simulation, where the light gray shade and the light orange shade represent two frequency intervals, and the two frequency points are marked with stars; (c) displacement field at the frequency of 575 Hz; (e) displacement field at the frequency of 636 Hz (color online)"

Table 2

AND gate"

Table 3

OR gate"

Fig. 12

Schematic diagram of a phononic crystal plate, where the red, green, blue, and orange cells are Cells A, B, C, and D, respectively. Cell A is inserted into Cell B as a potential barrier. The input ports are indicated by the black arrow, denoted as I1 and I2; the output port is indicated by blue arrows, denoted as O (color online)"