# Low Frequency Attenuation Characteristics of Two-Dimensional Hollow Scatterer Locally Resonant Phonon Crystals

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Method

**u**is the displacement vector;

**r**represents the position vector and $\omega $ is the angular frequency. Due to the periodicity of the structure, only the band gap characteristics of one unit cell needed to be calculated. According to Bloch’s theorem, periodic boundary conditions were applied to the four boundaries of the unit cell and the displacement field is expressed below:

_{n}represents the lattice vector. By substituting Equation (2) into Equation (1), the following characteristic equation can be obtained [3]:

**K**and

**M**are the global stiffness matrix and mass matrix of the lattice element, respectively. By changing the value of k in the first Brillouin zone and solving for the eigenvalue or frequency ω problem using the FEM, the dispersion relation and eigenmode could be obtained. To obtain the propagation modes of waves in various directions [0–360°], the wave vector k needed to vary over the entire first irreducible Brillouin zone, as shown in Figure 1c (i.e., triangular XMΓ). Γ, X and M were highly symmetric points in the Brillouin zone.

_{in}is the displacement acceleration excitation applied on the left side of the structure, and d

_{out}is the displacement acceleration excitation collected on the right side.

## 3. Simulation and Analysis

#### 3.1. Two-Component Phononic Crystal Plate

#### 3.2. Calculation of Band Gap Characteristics

#### 3.3. Analysis of Band Gap Formation Mechanism

_{i}is the coefficient, M

_{i}is determined by the vibration part of the scatterer or connecting plate; and K is the stiffness coefficient of the equivalent spring.

_{max}is the maximum vibration displacement, and U, W and x

_{max}are considered dimensionless due to regularization and normalization. By calculating the strain energy W and the maximum vibration displacement x

_{max}using the FEM, the equivalent stiffness K of the spring can be obtained based on the above equation and substituted into Equation (4) to calculate the equivalent frequency f, and then the energy band structure can be verified. In the energy band structure shown in Figure 5d, points B and D are located in the 7th and 9th frequency bands, respectively. Taking points B and D as examples, based on the corresponding vibration modes in Figure 6B,D, the frequencies at points B and D were calculated using the equation (expressed in f1) and compared with the characteristic frequencies calculated by the FEM (expressed in f2). The results are shown in Table 2. The table shows that the frequencies at points B and D calculated by the two methods were basically consistent, and the error mainly came from the neglect of the equivalent spring mass in the model, the selection of δ and so on.

#### 3.4. Effect of Geometric Parameters on the Band Gap of Phonon Crystals

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Brillouin, L. Propagation in Periodic Structures: Electric Filters and Crystal Lattices; Dover Publications: Mineola, NY, USA, 1953. [Google Scholar]
- Maldovan, M. Sound and heat revolutions in phononics. Nature
**2013**, 503, 209–217. [Google Scholar] [CrossRef] [PubMed] - Wang, Y.F.; Wang, Y.Z.; Wu, B.; Chen, W.; Wang, Y.S. Tunable and active phononic crystals and metamaterials. Appl. Mech. Rev.
**2020**, 72, 040801. [Google Scholar] [CrossRef] - Li, H.; Zhao, G.; Zhu, M.; Guo, J.; Wang, C. Robust Large-Sized Photochromic Photonic Crystal Film for Smart Decoration and Anti-Counterfeiting. ACS Appl. Mater. Interfaces
**2022**, 14, 14618–14629. [Google Scholar] [CrossRef] [PubMed] - Yadav, A.; Gerislioglu, B.; Ahmadivand, A.; Kaushik, A.; Cheng, G.J.; Ouyang, Z.; Wang, Q.; Yadav, V.S.; Mishra, Y.M.; Wu, Y.; et al. Controlled self-assembly of plasmon-based photonic nanocrystals for high performance photonic technologies. Nano Today
**2021**, 37, 101072. [Google Scholar] [CrossRef] - Kushwaha, M.S.; Halevi, P.; Dobrzynski, L.; Djafari-Rouhani, B. Acoustic band structure of periodic elastic composites. Phys. Rev. Lett.
**1993**, 71, 2022. [Google Scholar] [CrossRef] - Khelif, A.; Achaoui, Y.; Benchabane, S.; Laude, V.; Aoubiza, B. Locally resonant surface acoustic wave band gaps in a two-dimensional phononic crystal of pillars on a surface. Phys. Rev. B
**2010**, 81, 214303. [Google Scholar] [CrossRef] - Zhou, X.Z.; Wang, Y.S.; Zhang, C. Effects of material parameters on elastic band gaps of two-dimensional solid phononic crystals. J. Appl. Phys.
**2009**, 106, 014903. [Google Scholar] [CrossRef] - Reinke, C.M.; Su, M.F.; Olsson, R.H., III; El-Kady, I. Realization of optimal bandgaps in solid-solid, solid-air, and hybrid solid-air-solid phononic crystal slabs. Appl. Phys. Lett.
**2011**, 98, 061912. [Google Scholar] [CrossRef] - Liu, Z.; Zhang, X.; Mao, Y.; Zhu, Y.Y.; Yang, Z.; Chan, C.T.; Sheng, P. Locally resonant sonic materials. Science
**2000**, 289, 1734–1736. [Google Scholar] [CrossRef] - Pennec, Y.; Djafari-Rouhani, B.; Larabi, H.; Vasseur, J.O.; Hladky-Hennion, A.C. Low-frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate. Phys. Rev. B
**2008**, 78, 104105. [Google Scholar] [CrossRef] - Hsu, J.C. Local resonances-induced low-frequency band gaps in two-dimensional phononic crystal slabs with periodic stepped resonators. J. Phys. D Appl. Phys.
**2011**, 44, 055401. [Google Scholar] [CrossRef] - Oudich, M.; Li, Y.; Assouar, B.M.; Hou, Z. A sonic band gap based on the locally resonant phononic plates with stubs. New J. Phys.
**2010**, 12, 083049. [Google Scholar] [CrossRef] - Yang, Z.; Mei, J.; Yang, M.; Chan, N.H.; Sheng, P. Membrane-type acoustic metamaterial with negative dynamic mass. Phys. Rev. Lett.
**2008**, 101, 204301. [Google Scholar] [CrossRef] [PubMed] - Ding, H.X.; Shen, Z.H.; Ni, X.W.; Zhu, X.F. Multi-splitting and self-similarity of band gap structures in quasi-periodic plates of Cantor series. Appl. Phys. Lett.
**2012**, 100, 083501. [Google Scholar] [CrossRef] - Huang, H.; Huo, S.; Chen, J. Subwavelength elastic topological negative refraction in ternary locally resonant phononic crystals. Int. J. Mech. Sci.
**2021**, 198, 106391. [Google Scholar] [CrossRef] - Badreddine Assouar, M.; Oudich, M. Enlargement of a locally resonant sonic band gap by using double-sides stubbed phononic plates. Appl. Phys. Lett.
**2012**, 100, 123506. [Google Scholar] [CrossRef] - Deng, J.; Guasch, O.; Zheng, L. Reconstructed Gaussian basis to characterize flexural wave collimation in plates with periodic arrays of annular acoustic black holes. Int. J. Mech. Sci.
**2021**, 194, 106179. [Google Scholar] [CrossRef] - Li, Y.; Chen, T.; Wang, X.; Xi, Y.; Liang, Q. Enlargement of locally resonant sonic band gap by using composite plate-type acoustic metamaterial. Phys. Lett. A
**2015**, 379, 412–416. [Google Scholar] [CrossRef] - Pennec, Y.; Vasseur, J.O.; Djafari-Rouhani, B.; Dobrzyński, L.; Deymier, P.A. Two-dimensional phononic crystals: Examples and applications. Surf. Sci. Rep.
**2010**, 65, 229–291. [Google Scholar] [CrossRef] - Zhao, D.G.; Li, Y.; Zhu, X.F. Broadband lamb wave trapping in cellular metamaterial plates with multiple local resonances. Sci. Rep.
**2015**, 5, 9376. [Google Scholar] [CrossRef] - Zhu, H.F.; Sun, X.W.; Song, T.; Wen, X.D.; Liu, X.X.; Feng, J.S.; Liu, Z.J. Tunable characteristics of low-frequency bandgaps in two-dimensional multivibrator phononic crystal plates under prestrain. Sci. Rep.
**2021**, 11, 8389. [Google Scholar] [CrossRef] [PubMed] - Wang, X.P.; Jiang, P.; Song, A.L. Low-frequency and tuning characteristic of band gap in a symmetrical double-sided locally resonant phononic crystal plate with slit structure. Int. J. Mod. Phys. B
**2016**, 30, 1650203. [Google Scholar] [CrossRef] - Yu, K.; Chen, T.; Wang, X. Band gaps in the low-frequency range based on the two-dimensional phononic crystal plates composed of rubber matrix with periodic steel stubs. Phys. B Condens. Matter
**2013**, 416, 12–16. [Google Scholar] [CrossRef] - Zhang, Z.; Han, X.K. A new hybrid phononic crystal in low frequencies. Phys. Lett. A
**2016**, 380, 3766–3772. [Google Scholar] [CrossRef] - Li, S.; Chen, T.; Wang, X.; Li, Y.; Chen, W. Expansion of lower-frequency locally resonant band gaps using a double-sided stubbed composite phononic crystals plate with composite stubs. Phys. Lett. A
**2016**, 380, 2167–2172. [Google Scholar] [CrossRef] - Li, L.; Gang, X.; Sun, Z.; Zhang, X.; Zhang, F. Design of phononic crystals plate and application in vehicle sound insulation. Adv. Eng. Softw.
**2018**, 125, 19–26. [Google Scholar] [CrossRef] - Li, L.; Wen, T.; Han, J.; Yang, P. Two component phononic crystal integrated bandgap algorithm and regularity of bandgap parameter. J. Sci. Technol. Eng.
**2015**, 15, 123–127. [Google Scholar] - Zhu, X.; Xu, T.; Liu, S.; Cheng, J. Study of acoustic wave behavior in silicon-based one-dimensional phononic-crystal plates using harmony response analysis. J. Appl. Phys.
**2009**, 106, 104901. [Google Scholar] - Zhu, X.; Zou, X.-Y.; Liang, B.; Cheng, J. One-way mode transmission in one-dimensional phononic crystal plates. J. Appl. Phys.
**2010**, 108, 124909. [Google Scholar]

**Figure 1.**(

**a**,

**b**) are the two-dimensional hollow scatterer phonon crystals unit cell; (

**c**) the first irreducible Brillouin zone.

**Figure 2.**Three contrasting models of the unit cell: (

**a**) a hollow lead cylinder bonded on silicone rubber embedded in the square connecting plate, (

**b**) a hollow lead cylinder embedded in silicone rubber embedded in the square connecting plate, (

**c**) a hollow lead cylinder bonded on a silicon rubber cylinder embedded on four short connecting plates.

**Figure 4.**Two-dimensional two-component phononic crystals plate; (

**a**) models the unit cell; (

**b**) is the band structure (the black line). The blue marker is the complete band gap with a frequency range of 41,139–44,759 Hz.

**Figure 5.**Band structure and transmission loss maps of (

**a**) S1, (

**b**) S2, (

**c**) S3, and (

**d**) hollow scatterer phonon crystals. The points A, B, C, D, E and F are complete band gap boundary points. The bands where points B, C, D, E and F are located are flat bands consistent with the characteristics of local resonance bands.

**Figure 6.**Intrinsic modes at the edge of a partial band gap of hollow scatterer phonon crystals. Corresponding to the points (

**A**–

**F**) in Figure 5d. The vibrations are concentrated in the connecting plate and the substrate, and the band gap formation mechanism is local resonance.

**Figure 7.**(

**a**) Vibration mode; (

**b**) mass spring model. M corresponds to the hollow scatterers and m corresponds to the connecting plate.

**Figure 8.**(

**a**) Effect of the connecting plate width on the first complete band gap. (

**b**) Effect of the connecting plate thickness on the first complete band gap.

**Figure 9.**(

**a**) Effect of the inner radius of the scatterer on the first complete band gap. (

**b**) Effect of the outer radius of the scatterer on the first complete band gap. (

**c**) Effect of the scatterer height on the first complete band gap.

Material | Density ρ (kg/m^{3}) | Young Modulus E (×10^{10} Pa) | Poisson Ratio v |
---|---|---|---|

Lead | 11,600 | 4.08 | 0.42 |

Silicon rubber | 1300 | 1.37 | 0.47 |

Epoxy resin | 1180 | 0.435 | 0.38 |

Band | x_{max} | W | K | f_{1} | f_{2} |
---|---|---|---|---|---|

7th | 4.32 | 8164.96 | 875.02 | 579.44 | 582 |

9th | 7.32 | 8697.08 | 324.62 | 590.62 | 620 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xu, J.; Chen, C. Low Frequency Attenuation Characteristics of Two-Dimensional Hollow Scatterer Locally Resonant Phonon Crystals. *Materials* **2023**, *16*, 3982.
https://doi.org/10.3390/ma16113982

**AMA Style**

Xu J, Chen C. Low Frequency Attenuation Characteristics of Two-Dimensional Hollow Scatterer Locally Resonant Phonon Crystals. *Materials*. 2023; 16(11):3982.
https://doi.org/10.3390/ma16113982

**Chicago/Turabian Style**

Xu, Jingcheng, and Changzheng Chen. 2023. "Low Frequency Attenuation Characteristics of Two-Dimensional Hollow Scatterer Locally Resonant Phonon Crystals" *Materials* 16, no. 11: 3982.
https://doi.org/10.3390/ma16113982