# Plane and Surface Acoustic Waves Manipulation by Three-Dimensional Composite Phononic Pillars with 3D Bandgap and Defect Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Phononic Crystal and Theoretical Background

#### 2.1. Unit Cell Structure

^{®}. The PnC structure was designed in such a fashion that made it periodic in all three directions.

#### 2.2. Theory and Mathematical Framework

#### 2.3. Model Validation

## 3. Results and Discussion

#### 3.1. Plane Wave Propagation and Defect Analysis

#### 3.2. SAW Propagation and Defect Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Pennec, Y.; Djafari-Rouhani, B.; Vasseur, J.O.; Khelif, A.; Deymier, P.A. Tunable filtering and demultiplexing in phononic crystals with hollow cylinders. Phys. Rev. E Stat. Nonlin. Soft Matter Phys.
**2004**, 69, 046608. [Google Scholar] [CrossRef] [PubMed][Green Version] - Qiu, C.Y.; Liu, Z.Y.; Mei, J.; Shi, J. Mode-selecting acoustic filter by using resonant tunneling of two-dimensional double phononic crystals. Appl. Phys. Lett.
**2005**, 87, 104101. [Google Scholar] [CrossRef] - Kumar, S.; Lee, H.P. The present and future role of acoustic metamaterials for architectural and urban noise mitigations. In Proceedings of Acoustics; Multidisciplinary Digital Publishing Institute: Basel, Switzerland, 2019; pp. 590–607. [Google Scholar]
- Ang, L.Y.L.; Koh, Y.K.; Lee, H.P. Plate-type acoustic metamaterials: Experimental evaluation of a modular large-scale design for low-frequency noise control. In Proceedings of Acoustics; Multidisciplinary Digital Publishing Institute: Basel, Switzerland, 2019; pp. 354–368. [Google Scholar]
- Zhang, X.; Liu, Z. Superlenses to overcome the diffraction limit. Nat. Mater.
**2008**, 7, 435–441. [Google Scholar] [CrossRef] [PubMed] - De Ponti, J.M.; Colombi, A.; Ardito, R.; Braghin, F.; Corigliano, A.; Craster, R.V. Graded metasurface for enhanced sensing and energy harvesting. arXiv
**2019**, arXiv:1907.09297. [Google Scholar] - Jin, Y.B.; Fernez, N.; Pennec, Y.; Bonello, B.; Moiseyenko, R.P.; Hemon, S.; Pan, Y.D.; Djafari-Rouhani, B. Tunable waveguide and cavity in a phononic crystal plate by controlling whispering-gallery modes in hollow pillars. Phys. Rev. B
**2016**, 93, 054109. [Google Scholar] [CrossRef][Green Version] - Lim, C.W.; Reddy, J.N.; Carrera, E.; Xu, X.; Zhou, Z. Surface elastic waves whispering gallery modes based subwavelength tunable waveguide and cavity modes of the phononic crystals. Mech. Adv. Mater. Struct.
**2020**, 27, 1053–1064. [Google Scholar] [CrossRef] - Craster, R.V.; Guenneau, S. Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking; Springer Science & Business Media: Berlin, Germany, 2012; Volume 166. [Google Scholar]
- Cummer, S.A.; Schurig, D. One path to acoustic cloaking. New J. Phys.
**2007**, 9, 45. [Google Scholar] [CrossRef] - Ning, L.; Wang, Y.-Z.; Wang, Y.-S. Active control cloak of the elastic wave metamaterial. Int. J. Solids Struct.
**2020**, 202, 126–135. [Google Scholar] [CrossRef] - Lim, C.W. Analytical modeling and computation on topological properties of protected interface state of 1-d phononic crystal in elastic media. J. Mech. Mater. Struct.
**2020**, 15, 15–35. [Google Scholar] [CrossRef] - Zhou, W.; Lim, C.W. Topological edge modeling and localization of protected interface modes in 1D phononic crystals for longitudinal and bending elastic waves. Int. J. Mech. Sci.
**2019**, 159, 359–372. [Google Scholar] [CrossRef] - Zhou, W.; Su, Y.; Chen, W.; Lim, C.W. Voltage-controlled quantum valley Hall effect in dielectric membrane-type acoustic metamaterials. Int. J. Mech. Sci.
**2020**, 172, 105368. [Google Scholar] [CrossRef] - Zhou, W.J.; Chen, W.Q.; Lim, C.W. Surface effect on the propagation of flexural waves in periodic nano-beam and the size-dependent topological properties. Compos. Struct.
**2019**, 216, 427–435. [Google Scholar] [CrossRef] - Zhou, W.J.; Chen, W.Q.; Chen, Z.Y.; Lim, C.W. Actively controllable flexural wave band gaps in beam-type acoustic metamaterials with shunted piezoelectric patches. Eur. J. Mech. Solid
**2019**, 77, 103807. [Google Scholar] [CrossRef] - Brule, S.; Javelaud, E.H.; Enoch, S.; Guenneau, S. Experiments on seismic metamaterials: Molding surface waves. Phys. Rev. Lett.
**2014**, 112, 133901. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lim, C.W. Wide Rayleigh waves bandgap engineered metabarriers for seismic shielding of civil infrastructures. J. Eng. Mech.
**2020**, in press. [Google Scholar] - Lim, C.W. Elastic waves propagation in thin plate metamaterials and evidence of low frequency pseudo and local resonance bandgaps. Phys. Lett. A
**2019**, 383, 2789–2796. [Google Scholar] [CrossRef] - Lim, C.W.; Reddy, J.N. Built-up structural steel sections as seismic metamaterials for surface wave attenuation with low frequency wide bandgap in layered soil medium. Eng. Struct.
**2019**, 188, 440–451. [Google Scholar] [CrossRef] - Wu, T.; Lim, C.W. Forest trees as naturally available seismic metamaterials: Low frequency Rayleigh waves with extremely wide bandgaps. Int. J. Struct. Stab. Dyn.
**2020**, 2043014. [Google Scholar] [CrossRef] - Zhou, W.J.; Wu, B.; Du, Q.J.; Huang, G.L.; Lu, C.F.; Chen, W.Q. Actively tunable transverse waves in soft membrane-type acoustic metamaterials. J. Appl. Phys.
**2018**, 123, 165304. [Google Scholar] [CrossRef] - Huang, H.H.; Sun, C.T.; Huang, G.L. On the negative effective mass density in acoustic metamaterials. Int. J. Eng. Sci.
**2009**, 47, 610–617. [Google Scholar] [CrossRef] - Park, J.; Park, B.; Kim, D.; Park, J. Determination of effective mass density and modulus for resonant metamaterials. J. Acoust. Soc. Am.
**2012**, 132, 2793–2799. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ren, X.; Das, R.; Tran, P.; Ngo, T.D.; Xie, Y.M. Auxetic metamaterials and structures: A review. Smart Mater. Struct.
**2018**, 27, 023001. [Google Scholar] [CrossRef] - Yu, K.; Fang, N.X.; Huang, G.; Wang, Q. Magnetoactive Acoustic Metamaterials. Adv. Mater.
**2018**, 30, 1706348. [Google Scholar] [CrossRef] [PubMed] - Yu, X.; Zhou, J.; Liang, H.; Jiang, Z.; Wu, L. Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review. Prog. Mater. Sci.
**2018**, 94, 114–173. [Google Scholar] [CrossRef] - An, N.; Domel, A.G.; Zhou, J.; Rafsanjani, A.; Bertoldi, K. Programmable Hierarchical Kirigami. Adv. Funct. Mater.
**2020**, 30, 1906711. [Google Scholar] [CrossRef] - Barnhart, M.V.; Xu, X.C.; Chen, Y.Y.; Zhang, S.; Song, J.Z.; Huang, G.L. Experimental demonstration of a dissipative multi-resonator metamaterial for broadband elastic wave attenuation. J. Sound Vib.
**2019**, 438, 1–12. [Google Scholar] [CrossRef] - Lim, C.W. Dissipative multiresonant pillared and trampoline metamaterials with amplified local resonance bandgaps and broadband vibration attenuation. J. Vib. Acoust.
**2020**, 142, 061012. [Google Scholar] [CrossRef] - Lim, C.W.; Li, J.T.H.; Zhao, Z. Lightweight architected lattice phononic crystals with broadband and multiband vibration mitigation characteristics. Extrem. Mech. Lett.
**2020**, 41, 100994. [Google Scholar] [CrossRef] - Ang, L.Y.L.; Koh, Y.K.; Lee, H.P. Acoustic Metamaterials: A Potential for Cabin Noise Control in Automobiles and Armored Vehicles. Int. J. Appl. Mech.
**2016**, 8, 1650072. [Google Scholar] [CrossRef][Green Version] - Vyas, N.S.; Lim, C.W. A Novel Application of Multi-Resonant Dissipative Elastic Metahousing For Bearings. SAGE J. Vib. Control
**2020**. under review. [Google Scholar] - Liu, Y.; Du, J.; Cheng, L. Bandgap formation under temperature-induced quasi-periodicity in an acoustic duct with flexible walls. J. Sound Vib.
**2020**, 486, 115615. [Google Scholar] [CrossRef] - Huang, Z.; Zhao, S.; Su, M.; Yang, Q.; Li, Z.; Cai, Z.; Zhao, H.; Hu, X.; Zhou, H.; Li, F.; et al. Bioinspired Patterned Bubbles for Broad and Low-Frequency Acoustic Blocking. ACS Appl. Mater. Interfaces
**2020**, 12, 1757–1764. [Google Scholar] [CrossRef] [PubMed] - Allam, A.; Sabra, K.; Erturk, A. 3D-Printed Gradient-Index Phononic Crystal Lens for Underwater Acoustic Wave Focusing. Phys. Rev. Appl.
**2020**, 13, 064064. [Google Scholar] [CrossRef] - Lim, C.W. From photonic crystals to seismic metamaterials: A review via phononic crystals and acoustic metamaterials. Appl. Mech. Rev.
**2020**. under review. [Google Scholar] - 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] - Meseguer, F.; Holgado, M.; Caballero, D.; Benaches, N.; Sanchez-Dehesa, J.; Lopez, C.; Llinares, J. Rayleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal. Phys. Rev. B
**1999**, 59, 12169–12172. [Google Scholar] [CrossRef][Green Version] - 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, 1–7. [Google Scholar] [CrossRef][Green Version] - Williams, E.G.; Roux, P.; Rupin, M.; Kuperman, W.A. Theory of multiresonant metamaterials forA0Lamb waves. Phys. Rev. B
**2015**, 91, 104307. [Google Scholar] [CrossRef][Green Version] - Oudich, M.; Djafari-Rouhani, B.; Bonello, B.; Pennec, Y.; Sarry, F. Phononic Crystal Made of Multilayered Ridges on a Substrate for Rayleigh Waves Manipulation. Crystals
**2017**, 7, 372. [Google Scholar] [CrossRef][Green Version] - Oudich, M.; Djafari-Rouhani, B.; Bonello, B.; Pennec, Y.; Hemaidia, S.; Sarry, F.; Beyssen, D. Rayleigh Waves in Phononic Crystal Made of Multilayered Pillars: Confined Modes, Fano Resonances, and Acoustically Induced Transparency. Phys. Rev. Appl.
**2018**, 9, 034013. [Google Scholar] [CrossRef] - Hussein, M.I.; Leamy, M.J.; Ruzzene, M. Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook. Appl. Mech. Rev.
**2014**, 66, 040802. [Google Scholar] [CrossRef] - Zhou, X.; Assouar, M.B.; Oudich, M. Acoustic superfocusing by solid phononic crystals. Appl. Phys. Lett.
**2014**, 105, 233506. [Google Scholar] [CrossRef][Green Version] - Siddiqi, M.; Lee, J. Wide acoustic bandgap solid disk-shaped phononic crystal anchoring boundaries for enhancing quality factor in AlN-on-Si MEMS resonators. Micromachines
**2018**, 9, 413. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zhu, J.; Chen, Y.; Zhu, X.; Garcia-Vidal, F.J.; Yin, X.; Zhang, W.; Zhang, X. Acoustic rainbow trapping. Sci. Rep.
**2013**, 3, 1728. [Google Scholar] [CrossRef][Green Version] - Bilal, O.R.; Hussein, M.I. Trampoline metamaterial: Local resonance enhancement by springboards. Appl. Phys. Lett.
**2013**, 103, 111901. [Google Scholar] [CrossRef][Green Version] - Achaoui, Y.; Laude, V.; Benchabane, S.; Khelif, A. Local resonances in phononic crystals and in random arrangements of pillars on a surface. J. Appl. Phys.
**2013**, 114, 104503. [Google Scholar] [CrossRef] - Pennec, Y.; Djafari Rouhani, B.; Larabi, H.; Akjouj, A.; Gillet, J.N.; Vasseur, J.O.; Thabet, G. Phonon transport and waveguiding in a phononic crystal made up of cylindrical dots on a thin homogeneous plate. Phys. Rev. B
**2009**, 80, 144302. [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] - Wu, T.-C.; Wu, T.-T.; Hsu, J.-C. Waveguiding and frequency selection of Lamb waves in a plate with a periodic stubbed surface. Phys. Rev. B
**2009**, 79, 104306. [Google Scholar] [CrossRef] - Wu, T.T.; Huang, Z.G.; Tsai, T.C.; Wu, T.C. Evidence of complete band gap and resonances in a plate with periodic stubbed surface. Appl. Phys. Lett.
**2008**, 93, 98–101. [Google Scholar] [CrossRef] - Oudich, M.; Senesi, M.; Assouar, M.B.; Ruzenne, M.; Sun, J.-H.; Vincent, B.; Hou, Z.; Wu, T.-T. Experimental evidence of locally resonant sonic band gap in two-dimensional phononic stubbed plates. Phys. Rev. B
**2011**, 84, 165136. [Google Scholar] [CrossRef] - Oudich, M.; Zhou, X.; Assouar, M.B. General analytical approach for sound transmission loss analysis through a thick metamaterial plate. J. Appl. Phys.
**2014**, 116, 193509. [Google Scholar] [CrossRef][Green Version] - Jin, Y.; Pennec, Y.; Pan, Y.; Djafari-Rouhani, B. Phononic crystal plate with hollow pillars connected by thin bars. J. Phys. D Appl. Phys.
**2016**, 50, 035301. [Google Scholar] [CrossRef] - Jin, Y.; Pennec, Y.; Pan, Y.; Djafari-Rouhani, B. Phononic Crystal Plate with Hollow Pillars Actively Controlled by Fluid Filling. Crystals
**2016**, 6, 64. [Google Scholar] [CrossRef][Green Version] - Achaoui, Y.; Khelif, A.; Benchabane, S.; Robert, L.; Laude, V. Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars. Phys. Rev. B
**2011**, 83, 1–5. [Google Scholar] [CrossRef][Green Version] - Robillard, J.F.; Devos, A.; Roch-Jeune, I. Time-resolved vibrations of two-dimensional hypersonic phononic crystals. Phys. Rev. B
**2007**, 76, 092301. [Google Scholar] [CrossRef] - Giannetti, C.; Revaz, B.; Banfi, F.; Montagnese, M.; Ferrini, G.; Cilento, F.; Maccalli, S.; Vavassori, P.; Oliviero, G.; Bontempi, E.; et al. Thermomechanical behavior of surface acoustic waves in ordered arrays of nanodisks studied by near-infrared pump-probe diffraction experiments. Phys. Rev. B
**2007**, 76, 125413. [Google Scholar] [CrossRef][Green Version] - Yudistira, D.; Boes, A.; Graczykowski, B.; Alzina, F.; Yeo, L.Y.; Sotomayor Torres, C.M.; Mitchell, A. Nanoscale pillar hypersonic surface phononic crystals. Phys. Rev. B
**2016**, 94, 094304. [Google Scholar] [CrossRef][Green Version] - Colombi, A.; Roux, P.; Guenneau, S.; Gueguen, P.; Craster, R.V. Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances. Sci. Rep.
**2016**, 6, 19238. [Google Scholar] [CrossRef][Green Version] - De Ponti, J.M.; Colombi, A.; Ardito, R.; Braghin, F.; Corigliano, A.; Craster, R.V. Graded elastic metasurface for enhanced energy harvesting. New J. Phys.
**2020**, 22, 013013. [Google Scholar] [CrossRef] - Chaplain, G.J.; De Ponti, J.M.; Colombi, A.; Fuentes-Dominguez, R.; Dryburg, P.; Pieris, D.; Smith, R.J.; Clare, A.; Clark, M.; Craster, R.V. Tailored elastic surface to body wave Umklapp conversion. Nat. Commun.
**2020**, 11, 3267. [Google Scholar] [CrossRef] [PubMed] - D’Alessandro, L.; Belloni, E.; Ardito, R.; Braghin, F.; Corigliano, A. Mechanical low-frequency filter via modes separation in 3D periodic structures. Appl. Phys. Lett.
**2017**, 111, 231902. [Google Scholar] [CrossRef] - Lim, C.W. Ultrawide 3D phononic bandgap metastructures as broadband low frequency filter. Sci. Rep.
**2020**. submitted. [Google Scholar] - Lim, C.W. Ultrawide bandgap by 3D monolithic mechanical metastructure for vibration and noise control. Arch. Civ. Mech. Eng.
**2020**. submitted. [Google Scholar] - Maurel, A.; Marigo, J.-J.; Pham, K.; Guenneau, S. Conversion of Love waves in a forest of trees. Phys. Rev. B
**2018**, 98, 134311. [Google Scholar] [CrossRef][Green Version] - Hofstadter, D.R. Energy-Levels and Wave-Functions of Bloch Electrons in Rational and Irrational Magnetic-Fields. Phys. Rev. B
**1976**, 14, 2239–2249. [Google Scholar] [CrossRef]

**Figure 1.**The phononic crystal (PnC) unit cell structure and finite-length model: (

**a**) 3D, top and plan view; (

**b**) unit cell structure with perfect boundary condition (PBC) applied along all vertical edges and top-bottom; (

**c**) supercell model used for plane wave transmission curve study; (

**d**) unit cell structure utilized for surface acoustic wave (SAW) band structure study; (

**e**) finite-length model with SAW propagation at the surface of semi-infinite half-space can be observed; and (

**f**) supercell model adopted for SAW transmission, attenuation and defect analysis.

**Figure 3.**(

**a**) The 3D band structure of PnC unit cell structure, with irreducible Brillouin zone (IBZ) shown at the inset. (

**b**,

**c**) transmission curve for harmonic excitation applied in the x-y and the x-y-z directions. The BGs are highlighted. (

**d**) The vibration modes involved in the opening and closing of BGs.

**Figure 4.**The defect analysis (

**a**) supercell structure with defect height ${h}_{d}$; (

**b**) frequency response spectrum with defect frequencies for various ${h}_{d}$. At the defect frequency, wave energy is concentrated inside the defect, and wave propagation is isolated for other unit cell structures.

**Figure 6.**SAW band structures for (

**a**) ${h}_{2}=0.2a$; (

**b**) ${h}_{2}=0.4a$; (

**c**) ${h}_{2}=0.5a$ (parent structure); (

**d**) ${h}_{2}=0.6a$; and (

**e**) ${h}_{2}=0.8a$. One can deduce identical vibration modes as reported for the parent structure in Figure 5c.

**Figure 7.**SAW defect analysis for ${h}_{2d}=0.8a$; (

**a**) the schematic diagram in which defect location is highlighted, and the transmission curve is plotted. At defect position (

**b**,

**h**) SAW trapping (

**c**–

**e**,

**f**,

**g**) vibration isolation where partial SAW propagation in neighboring parent cells occur can be seen.

**Figure 8.**SAW defect analysis for ${h}_{2d}=0.2a$: (

**a**) the schematic diagram in which the defect location is highlighted, and the transmission curve is plotted. At defect position (

**b**,

**h**), SAW trapping (

**c**–

**e**,

**f**,

**g**) and vibration isolation with partial SAW propagation in neighboring parent cells can be seen.

Young Modulus E (GPa) | Density ρ (kg/m^{3}) | $\mathbf{Poison}\text{}\mathbf{Ratio}\text{}\mathit{\nu}$ | |
---|---|---|---|

Silicon (Si) | 166 | 2330 | 0.28 |

Tungsten (W) | 400 | 19,270 | 0.3 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Muhammad; Lim, C.W.; Leung, A.Y.T. Plane and Surface Acoustic Waves Manipulation by Three-Dimensional Composite Phononic Pillars with 3D Bandgap and Defect Analysis. *Acoustics* **2021**, *3*, 25-41.
https://doi.org/10.3390/acoustics3010004

**AMA Style**

Muhammad, Lim CW, Leung AYT. Plane and Surface Acoustic Waves Manipulation by Three-Dimensional Composite Phononic Pillars with 3D Bandgap and Defect Analysis. *Acoustics*. 2021; 3(1):25-41.
https://doi.org/10.3390/acoustics3010004

**Chicago/Turabian Style**

Muhammad, C.W. Lim, and Andrew Y. T. Leung. 2021. "Plane and Surface Acoustic Waves Manipulation by Three-Dimensional Composite Phononic Pillars with 3D Bandgap and Defect Analysis" *Acoustics* 3, no. 1: 25-41.
https://doi.org/10.3390/acoustics3010004