# Band gap structures for 2D phononic crystals with composite scatterer

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## Abstract

We investigated the band gap structures in two-dimensional phononic crystals with composite scatterer. The composite scatterers are composed of two materials (Bragg scattering type) or three materials (locally resonance type). The finite element method is used to calculate the band gap structure, eigenmodes and transmission spectrum. The variation of the location and width of band gap are also investigated as a function of material ratio in the scatterer. We have found that the change trends the widest band gap of the two phononic crystals are different as the material ratio changing. In addition to this, there are three complete band gaps at most for the Bragg-scattering-type phononic crystals in the first six bands; however, the locally resonance-type phononic crystals exist only two complete band gap at most in the first six bands. The gap-tuning effect can be controlled by the material ratio in the scatterer.

## 1 Introduction

At present, the research on 2D crystals is mainly divided into two branches: photonic crystals and phononic crystals. Photonic crystals are artificial materials with a permittivity, which is a periodic function of the position, with a period comparable to the wavelength of light. Composite materials with elastic coefficients, which are periodic functions of the position, are named phononic crystals. The research on photonic crystals focuses on electromagnetic waves and in the case of phononic crystals, elastic waves, and so they have different practical applications [1, 2, 3]. In the past few years, much work has been devoted to the study of the propagation of elastic or acoustic waves in phononic crystals, which are periodic structures composed of two or more materials [4, 5]. The most striking features of phononic crystals are frequency bands (band gaps) within which sound and vibration are barred in any direction [6, 7, 8]. The existence of a band gap has endowed phononic crystals with great prospects in vibration and noise reduction applications as well as in phononic functional devices, such as acoustic filters, waveguides, focusing, acoustic wave transducers, etc. [9, 10, 11]. The major mechanism proposed for the formation of such a band gap is the Bragg scattering principle [12]. Phononic crystals can be generally defined into three types according to their dimension of periodicity: one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) phononic crystals [13, 14]. Several methods have been used to study the acoustic or elastic band gap structures, such as, the finite element method (FEM) [15], the plane-wave expansion method (PWE) [16], the finite difference time domain method (FDTD) [17], the multiple-scattering method (MST) [18], the variational method [19], and the perturbative approach [20]. FEM is the most widely used method for calculating the band structures because it has the wide applicability, good convergence and a lot of mature commercial software, though it has some errors in the high frequency ranges, the result can be improved by refining the mesh. The band gap properties of phononic crystal can be controlled by the lattice-filling factor, the constituent material contrast between the scatterer and matrix, and the lattice symmetry, as well as the scatterer shape.

The present work is motivated by a similar work in photonic crystals and phononic crystals. Oral et al. investigated theoretically the propagation of acoustic waves in two-dimensional sonic crystals with elliptic rods and found that the band gap can be changed by tuning the rotation angles of the scatterers and the filling fraction [11]. Khelif et al. discovered that the thickness-to-lattice pitch ratio plays a crucial role in the opening of complete band gaps [21]. A homogeneous plate with periodic truncated cones and tapered surface was investigated by Chen and Zhang, respectively, and the result shows that the relative bandwith is enlarged by changing the value of the semiangle [22, 23]. The arrangement of the stubs was found to influence the band gaps in the study of the multi-stub locally resonant phononic crystal plate by Wang et al. [24]. Jiang et al. found that acoustic band gaps are affected by the distance between two adjacent square stubs in the investigation of the multi-stub phononic crystal plate [25]. Goffaux and Wu et al. have researched how the band gaps are affected by the rotation angle of the square scattering from a different point of view, respectively [26, 27]. In the literature, the band gaps of various two-dimensional phononic crystals structures were investigated and the scatterers of these structures mostly consisted of a homogeneous material.

In the paper, the propagation of elastic or acoustic wave in two-dimensional periodic arrays of composite scatterers embedded in matrix is considered. We have investigated the effects of material ratio in the scatterer on the phononic band gap.

## 2 Model of the unit cell

There are two structures considered in the paper, structure A and structure B, as shown in Fig. 1a, b, respectively. Structure A is composed of three regions 1, 2 and 3, and structure B is composed of four regions 1, 2, 3 and 4. The materials labeled 1, 2, 3 and 4 are epoxy resin, aluminum and plumbum, silicone rubber, respectively. The materials are specified and the choice of the materials refers to literature [28]. Both the structures are manufactured by bonding or 3D printing. Figure 1 shows the geometric parameters of the both structures, the lattice constant is *a* = 20 mm, the internal side length of the epoxy resin is *b* = 16 mm, the thickness of plumbum defined as *c* is variable, the external side length of the silicone rubber *d* = 18 mm. The material parameters are chosen as follows: the elastic modulus, *E*_{1} = 4.35GPa, *E*_{2} = 77.6GPa, *E*_{3} = 40.8GPa, *E*_{4} = 0.1175 MPa, the Poisson’s ratio, *ν*_{1} = 0.368, *ν*_{2} = 0.352, *ν*_{3} = 0.369, *ν*_{4} = 0.469, the density, *ρ*_{1} = 1180 kg/m^{3}, *ρ*_{2} = 2730 kg/m^{3}, *ρ*_{3} = 11,600 kg/m^{3}, *ρ*_{4} = 1300 kg/m^{3}.

**K**] and [

**M**] are the stiffness matrix and the mass matrix, respectively.

**U**is the displacement matrix of unit cell,

*ω*is the angular frequency. Periodic boundary conditions were used for the interfaces between the nearest unit cell according to the Bloch-Floquet theory; the displacement matrix met the following description [29]:

*i*is the imaginary unit,

**r**is the position vector,

**a**is the lattice constant vector,

**k**is the wave vector. To combine Eqs. (1) and (2), we can obtain the relation between the wave vector

**k**and angular frequency

*ω*. By varying the reduced wave vector

**k**in the first irreducible Brillouin zone along the path

*M*–

*Γ*–

*X*–

*M*, as shown in Fig. 1c, the eigenfrequencies and corresponding eigenvectors are obtained. From this, we plot the dispersion relationship.

## 3 Numerical results and discussion

### 3.1 Band structures and transmission spectra

Figure 2 shows the dispersion relationships of structure A when *c* is equal to 0, 8, 16 mm, respectively. In the paper, we only calculate the first six dispersion curves. There exists one complete band gap between the third and fourth dispersion curves when *c* = 0 mm and the band gap extends from 64.66 to 107.8 KHz. There are two complete band gaps when *c* = 8 mm at 40.439–60.719 and 65.833–68 KHz, respectively. It also exhibits two complete band gaps when *c* = 16 mm at 31.02–46.56 and 51.52–51.88 KHz, respectively. There are two points that should be noticed. First, the frequency of the band gap becomes lower when the thickness of plumbum (material ratio) *c* increases. Second, an additional band gap is formed between the fourth and fifth dispersion curves with the increasing *c*.

Figure 3 shows the dispersion relationships of structure B when *c* is equal to 0, 8, 16 mm, respectively. There is only one complete band gap when *c* = 0 mm at 0.904–2.285 KHz. There are two complete band gaps when *c* = 8 mm at 0.708–2.156 and 0.565–0.591 KHz. There is also one complete band gap when *c* = 16 mm from 0.445 to 2.124 KHz. It is the same with structure A in that the frequency of band gap becomes lower when the thickness of plumbum (material ratio) *c* increases. However, it is different when compared with structure A in that an additional band gap is formed between the second and third dispersion curves with the increasing *c*.

In order to verify the validity of band gap structure, we calculated the transmission spectra of 5 × 5 unit cells for structure A and structure B when *c* = 8 mm, as shown in Fig. 4. The location and width of band gap agree well between the transmission spectra and dispersion relationship.

### 3.2 Modal analysis and discussion

To understand the mechanism of the band gap, the vibration eigenmodes at the band gap edges were calculated for both structures when *c* = 8 mm, as shown in Fig. 5. The direction and length of the arrows indicate the displacement direction and size of the medium at the starting point of the arrow, respectively. The colors indicate total displacement of the medium. The different modal displacements corresponding to the modes (I–IV and I′–IV′) as labeled in Figs. 2b and 3b. Mode I shows that the cell unit boundary is slightly stretched to open the first band gap, while the cell unit boundary of mode II is compressed leading to the termination of the first band gap. The energy distribution and movement of mode III and mode III are almost same because they are at the same dispersion curve with different wave vector. The cell unit boundary of mode IV is obviously stretched to close the second band gap. The energy is concentrated at the left part of the structure A for mode I–IV. Because the interaction between cell units plays a leading role for the band gap formation of structure A, the band gap of structure is defined as Bragg scattering mechanism. For the mode I′ the energy is concentrated at the core composed of aluminum and plumbum, the unit cell produces translational resonance to open the first band gap. For mode II′ the unit cell produces rotational resonance to close the first band gap. Because II′ and III′ are located in the same dispersion curve, the energy distribution and movement of mode III′ is almost same with mode II′ to open the second band gap. For mode IV′ the unit cell produces translational resonance and energy is distributed in the rubber coating, which may cause finish of the second band gap. From mode I′–IV′ we can find that the resonance characteristic of the scatterer is the main reason for forming band gap of structure B, therefore the band gap of structure B is the locally resonance type.

### 3.3 Band gap map

In the end, as mentioned above, the material thickness (material ratio) of phononic crystal is significant in band gap control. We further investigate the band gap structure in the range of 0 ≤ *c* ≤ 16 mm. The phononic band gap map extracted from the calculation of band gap is shown in Fig. 6, in which the line with “×” represents the band gap between the second and third bands; the dot line represents the band gap between the third and fourth bands; the line with “☆” represents the band gap between the fourth and fifth bands; the solid line represents the band gap between the fifth and sixth bands. It is noteworthy to point out that the band gaps are quite sensitive to the variation of the thickness of plumbum *c* for the both structures. Figure 6a shows there are three band gaps at most for structure A in the whole range of *c*. The first band gap is only continuous as the varying of *c*. The upper and lower boundaries of the first band gap monotonously reduce; however, the upper boundary decays faster than the lower boundary and so the band width reduces. The physical mechanism of the first band gap changing can be explained below. The effective elastic modulus of scatterer decreases gradually with the increase *c*, which leads to the decrease of the wave velocity and affects the upper boundary of the band gap. However, the effective density of scatterer becomes bigger with the increase *c*, which leads to reduce the resonant frequency of rigid body, and make it further affect lower boundary of band gap. Figure 6b shows that structure B has two band gaps at most with the increase *c*. The second band gap is the main band gap. The upper boundary and lower boundary of the second band gap are almost linearly attenuated, and the upper boundary attenuates slower than the lower boundary, so the band width enlarges. According to the simplified model of local resonant type phononic crystals [28], the mass of local oscillator affects the initial frequency and cutoff frequency of band gap with different effect degree. As the increase of *c*, the mass of local oscillator of structure B becomes bigger, which lead to reduce the initial frequency and cutoff frequency of band gap.

## 4 Conclusions

In conclusion, we have investigated the band gap structure in the phononic crystals with composite scatterer by using the finite element method. We found that the location and the width of the band gap are sensitive to the ratio of materials in the scatterer. For Bragg scattering type phononic crystal can produce three band gaps at most in the first six bands. The first band gap is the widest, and its width decreases with the thickness of plumbum increasing. For locally resonant-type phononic crystal, however, there are two band gaps at most in the first six bands. The second band gap is main band gap and its width becomes bigger with the thickness of plumbum increasing. The material 2 and material 3 can be manufactured into blocks of the same size to realize the interchange of materials. The method of adjusting band gap can overcome the defect that once the structure is developed, the band gap cannot be changed. The results provided in this paper will provide important guidance in the band gap tuning, and a new method for designing novel acoustic devices, for example, acoustic transducers and acoustic filters.

## Notes

### Acknowledgements

We gratefully acknowledge financial support from the National Natural Science Foundation of China (no. 51775403), and the work is supported by the Fundamental Research Funds for the Central Universities (Grant no. JB180404).

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