Elastic Wave Propagation Research on Two-Dimensional Photonic Crystal Structures of W-Silicone Rubber

the analyzed methods calculate the band structures. Some calculation methods have as transfer matrix Abstract The two-dimensional phononic crystal structures of W-silicone rubber are designed here for the low-frequency vibration absorption by applying both the plane wave expansion method and finite element method. The band structures of phononic crystal are calculated and some special phenomena in the band structures are discussed, such as the Dirac point and the degenerate state near the Dirac point and so on. Moreover, the main deformations of different modes for the structure are given, which indicated that the stress concentration is easy to occur in joint parts of the matrix and scatter. Furthermore, the influence of different parameters on the band gaps of the phononic crystal is discussed by finite element method, including the density ratio and the Young’s modulus ratio. At last, the defect state of two-dimensional phononic crystal is visualized which verifies that acoustic wave can propagate along point defect or line defect of the system. The results will provide certain theoretical guidance for engineering application. method, plane wave expansion method, time-domain finite difference method, multiple scattering method and finite element method, etc. [23-26].


Introduction
Phononic crystal are periodic function materials composed of two or more than two different mechanical properties materials.
The most important characteristic of these structures is the existence of phononic band gaps [1][2][3], which mean some specific frequency ranges of elastic wave propagation can't pass through the structures. Therefore, the phononic crystal structures have a broad application prospect, for example, vibration isolation, vibration acoustic, noise reduction, isolator waveguide filter device and so on [4][5][6][7][8][9]. Researches on phononic crystal structures are mainly concentrated on the propagation performance of elastic wave in different phononic crystal structures. For example, Sorokin et al. [10] considered these phenomena in periodic plates and cylindrical shells with and without heavy fluid loading. Romeo et al. [11,12] present general three-coupled phononic crystal structures through the transfer matrix method. Li et al. [13] introduced a microscale cavity for the two-dimensional phononic crystal. Shan et al. [14] analyzed the propagation of elastic waves in porous phononic crystal structures. Baravelli et al [15] designed a beamlike assembly with a stiff external frame and an internal resonating lattice, which was characterized by high frequency band gaps and tuned vibration attenuation at low frequencies. Zhang et al. [16] discussed the band gaps and damping characteristics of thin plate phononic crystals. The application of complex phononic crystal structures are also reported in some fields, for example, Huang et al. [17] showed its application in semi-active control and health monitoring of intelligent structures. Huang et al. [18] analyzed the electromagnetic wave absorption characteristics of aluminum gradient phononic crystal structures. Li et al. [19] researched the low frequency vibration isolation characteristics of local resonant phononic crystal plates. Wan et al. [20] proposed a low frequency noise attenuation system based on phononic crystal structures and discussed the effects of structure parameters on the band gaps.
Zou et al. [21] showed the decoupling of two coupled phononic crystal waveguides M. Zubtsov et al. [22] designed a normal incidence phononic crystal sensor and analyzed its characteristics.
Another important researches for phononic crystal structures are the analyzed methods to calculate the band structures. Some calculation methods have been developed, such as transfer matrix

Model Introduction
Since the phononic crystals have characteristics of periodicity, one-unit cell can represent the entire structure to be analyzed for ideal phononic crystals [32]. As shown in Figure 1(a), this two-dimensional square lattice unit cell is composed osf circular scatters and square matrix, and the material of scatters and the matrix are made of tungsten and silicone rubber, respectively.
The parameters are including lattice constant a and scattered radius r, and scattered fill rate is Then according to Bloch's theorem, the periodic boundary conditions are applied to the all boundaries of the unit cell, as shown in

Introduction of the Plane Wave Expansion Method
The plane wave expansion method is one of the most h G is the parameter of Fourier series, G is reciprocal vector and can be defined as follows for the two-dimension crystal Here, 1 n and 2 n are integers, 1 b and 2 b means the basic vectors for G .
The parameters for the ( ) h G in Eq (1) are calculated by Here, S means the area of the unit cell.
satisfied the following equation, where f is the filling ratio of the scatter in the unit cell, A h and B h are the corresponding parameters for the scatter and matrix, respectively.
Based on the boundary condition of the periodicity, the integral of the second term for Eq.5 becomes zero, that is where ( ) P G is the structure function and relates to the shape of the scatters?
For the cycle scatter where 1 J is the first order and type of Bessel function, and G is the model of the reciprocal vector.
Then, based on Eqs. (4) and (6), the parameter ( ) h G is rewritten as follows The displacements are expressed in the following form by the Bloch theorem where k is the Bloch wave vector, confined within the first Brillouin zone (see Figure 1 Finally, the eigenvalue equation can be obtained by further Then, the band structures of W-silicone rubber phononic crystal structures are obtained by MATLAB based on the obtained eigenvalue equation (13). In this paper, 961 plane wave numbers are selected.

The Band Structure of a Unit Cell
Choose the scattered radius of the single cell structure as 4mm and the lattice constant as 10mm, and material parameters are shown in Table 1. The band structure of unit cell for the twodimensional W-silicone rubber phononic crystal is obtained with two different methods, which shown in Figure 3. The pink curve is the result simulated by finite element method (FEM), while the blue one is the calculation by the plane wave expansion method (PWE).
It is easy to find that the range of the first complete band gap calculated by PWE is 382Hz-997Hz, the bandwidth is 615Hz and its center frequency is 689.5Hz. Meanwhile, the range of the same gap by the FEM is from 396.9Hz to 1009.4Hz, the bandwidth is 639.5Hz and its center frequency is 703.15Hz. Therefore, the results of these two methods basically agree with each other, which verify the correctness of the calculation by PWE. Figure 3

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It is proved that the above phononic crystal has the same range of the band gaps between the unit cell and the super cell. The research on the unit cell can be extended to the super cell structure, which is extremely beneficial for simplifying the calculation process. To analyze the relationship between frequency and wave vector more intuitively, (Figure 6) gives the three-dimensional map of the first energy band of the phononic crystal and its two-dimensional cloud image. Moreover, the frequency of wave vector is 0 at the position of the higher symmetric point τ, and has a larger frequency rang in the XM direction. The deformation trend of the W-silicone rubber phononic crystal is analyzed in different modes to guide the engineering practice.    290 Figures 7a & 7b show the change trend of total displacements of modes A and B (corresponding to mode A and mode B in Figure   3), respectively. It is found that mode A mainly undergoes torsional deformation and mode B is shear deformation. Moreover, the deformation in the center and four corner positions of the unit cell is small for the mode A, however, the one at the junction of the scattered and the matrix is large. For mode B, the deformation of the four corner positions is large and that of the scattered is relatively uniform. Therefore, the connection between the scattered and the matrix is a stress concentration area for torsional deformation, and the effect of shear deformation on the scattered is not obvious.
The connection part of the scattered and the matrix needs to be reinforced in engineering applications (Figure7).

Effects of Material Parameters on Bandgap
Another effect factor for the two-dimensional phonon band gaps is material parameters, here, 10 different scatter materials are selected in Table 2

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The defect state of phononic crystals have important application in the design of acoustic devices. When a defect exists in a periodic structure, an elastic wave of a specific frequency will be localized within the defect or propagate along the defect to form an acoustic waveguide. Phononic crystal defect characteristics have the advantages of high transmission rate, low loss, and good stability and so on, which could meet the requirements of various acoustic frequency selection, filtering, and waveguide devices for transmission performance. Therefore, the defect state phenomenon of the two-dimensional phononic crystal is investigated and visualized here, showing its intuitive image. (Figure 10) is a crosssectional view of the supercell structure of the two-dimensional W-silicone rubber phononic crystal, which all contains 5*10 primitive cells. Figure 10(a) is a complete structure without defects.

Conclusion
This paper calculates the band structure of two-dimensional W-silicone rubber phononic crystals and analyzes the special phenomena appearing in the structure. The influence of material parameters on the band structure are discussed and the defect state phenomenon of the phononic crystal structure are analyzed. The following conclusions are obtained for two-dimensional W-silicone rubber periodic structural composites: a. The phonon crystal structure composed of W-silicone rubber has band gaps in the mid-low frequency ranges. The stress concentration is easily occurred at the junction of the scatter and the matrix.
b. As the density ratio increases, the starting frequency of the first complete band gap gradually increases in the twodimensional W-silicone rubber phononic crystal, but the cut-off frequency changes little and the bandwidth gradually decreases. By adjusting the density of the material, the band gap characteristics of the phononic crystal structure will be applied in the low frequency vibration absorber. Moreover, there is no clear relationship between the first complete band gap and the Young's modulus ratio.
c. Acoustic waves can propagate forward along point or line defects.