# Introducing Obliquely Perforated Phononic Plates for Enhanced Bandgap Efficiency

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## Abstract

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## 1. Introduction

## 2. OPhP Designs and Modal Band Analysis

## 3. Bandgap of Mixed Guided Wave Modes by A-OPhPs

- increasing perforation angle introduces and/or widens fundamental low frequency bandgaps;
- increasing perforation angle shifts higher order bandgaps towards lower frequency levels; and,
- narrowing of a bandgap is generally associated with development of a lower bandgap.

## 4. Bandgap of Asymmetric Wave Modes by S-OPhPs

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Deymier, P. Acoustic Metamaterials and Phononic Crystals; Springer: Berlin, Germany, 2011. [Google Scholar]
- Wu, T.T.; Chen, Y.T.; Sun, J.H.; Lin, S.C.S.; Huang, T.J. Focusing of the lowest antisymmetric Lamb wave in a gradient-index phononic crystal plate. Appl. Phys. Lett.
**2011**, 98, 171911. [Google Scholar] [CrossRef] - Olsson Iii, R.H.; El-Kady, I.F. Microfabricated phononic crystal devices and applications. Meas. Sci. Technol.
**2009**, 20, 012002. [Google Scholar] [CrossRef] - Celli, P.; Gonella, S. Low-frequency spatial wave manipulation via phononic crystals with relaxed cell symmetry. J. Appl. Phys.
**2014**, 115, 103502. [Google Scholar] [CrossRef] - Hedayatrasa, S.; Kersemans, M.; Abhary, K.; Uddin, M.; Van Paepegem, W. Optimization and experimental validation of stiff porous phononic plates for widest complete bandgap of mixed fundamental guided wave modes. Mech. Syst. Signal Process.
**2018**, 98, 786–801. [Google Scholar] [CrossRef] [Green Version] - 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] - Khelif, A.; Adibi, A. Phononic Crystals: Fundamentals and Applications; Springer: Berlin, Germany, 2015. [Google Scholar]
- Craster, R.V.; Guenneau, S. Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking; Springer Science & Business Media: Berlin, Germany, 2012. [Google Scholar]
- Liu, J.; Li, F.; Wu, Y. The slow zero order antisymmetric Lamb mode in phononic crystal plates. Ultrasonics
**2013**, 53, 849–852. [Google Scholar] [CrossRef] [PubMed] - Hedayatrasa, S. Design Optimisation and Validation of Phononic Crystal Plates for Manipulation of Elastodynamic Guided Waves; Springer: Berlin, Germany, 2018. [Google Scholar]
- Shelke, A.; Banerjee, S.; Habib, A.; Rahani, E.K.; Ahmed, R.; Kundu, T. Wave guiding and wave modulation using phononic crystal defects. J. Intell. Mater. Syst. Struct.
**2014**, 25, 1541–1552. [Google Scholar] [CrossRef] - Zhu, H.; Semperlotti, F. A passively tunable acoustic metamaterial lens for damage detection applications. In Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems; International Society for Optics and Photonics: Bellingham, WA, USA, 2014; Volume 9061, p. 906107. [Google Scholar]
- Semperlotti, F.; Zhu, H. Achieving selective interrogation and sub-wavelength resolution in thin plates with embedded metamaterial acoustic lenses. J. Appl. Phys.
**2014**, 116, 054906. [Google Scholar] [CrossRef] [Green Version] - Lin, S.-C.S. Acoustic metamaterials: Tunable gradient-index phononic crystals for acoustic wave manipulation. In Engineering Science and Mechanics; The Pennsylvania State University: State College, PA, USA, 2012. [Google Scholar]
- Wu, T.-T.; Hsu, J.-C.; Sun, J.-H. Phononic plate waves. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2011**, 58, 2146–2161. [Google Scholar] [PubMed] - 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] - Andreassen, E.; Manktelow, K.; Ruzzene, M. Directional bending wave propagation in periodically perforated plates. J. Sound Vib.
**2015**, 335, 187–203. [Google Scholar] [CrossRef] [Green Version] - Miniaci, M.; Gliozzi, A.S.; Morvan, B.; Krushynska, A.; Bosia, F.; Scalerandi, M.; Pugno, N.M. Proof of concept for an ultrasensitive technique to detect and localize sources of elastic nonlinearity using phononic crystals. Phys. Rev. Lett.
**2017**, 118, 214301. [Google Scholar] [CrossRef] [PubMed] - Hedayatrasa, S.; Abhary, K.; Uddin, M. Numerical study and topology optimization of 1D periodic bimaterial phononic crystal plates for bandgaps of low order Lamb waves. Ultrasonics
**2015**, 57, 104–124. [Google Scholar] [CrossRef] [PubMed] - Hung, C.H.; Wang, W.S.; Lin, Y.C.; Liu, T.W.; Sun, J.H.; Chen, Y.Y.; Esashi, M.; Wu, T.T. Design and fabrication of an AT-cut quartz phononic Lamb wave resonator. J. Micromech. Microeng.
**2013**, 23, 065025. [Google Scholar] [CrossRef] - Foehr, A.; Bilal, O.R.; Huber, S.D.; Daraio, C. Spiral-based phononic plates: From wave beaming to topological insulators. Phys. Rev. Lett.
**2018**, 120, 205501. [Google Scholar] [CrossRef] [PubMed] - Hedayatrasa, S.; Kersemans, M.; Abhary, K.; Uddin, M.; Guest, J.K.; Van Paepegem, W. Maximizing bandgap width and in-plane stiffness of porous phononic plates for tailoring flexural guided waves: Topology optimization and experimental validation. Mech. Mater.
**2017**, 105, 188–203. [Google Scholar] [CrossRef] - Miniaci, M.; Pal, R.K.; Morvan, B.; Ruzzene, M. Observation of topologically protected helical edge modes in Kagome elastic plates. arXiv
**2017**, arXiv:1710.11556v2 [physics.app-ph]. [Google Scholar] - Miniaci, M.; Mazzotti, M.; Radzieński, M.; Kherraz, N.; Kudela, P.; Ostachowicz, W.; Morvan, B.; Bosia, F.; Pugno, N.M. Experimental observation of a large low-frequency band gap in a polymer waveguide. Front. Mater.
**2018**, 5, 8. [Google Scholar] [CrossRef] - Pourabolghasem, R.; Dehghannasiri, R.; Eftekhar, A.A.; Adibi, A. Waveguiding Effect in the Gigahertz Frequency Range in Pillar-based Phononic-Crystal Slabs. Phys. Rev. Appl.
**2018**, 9, 014013. [Google Scholar] [CrossRef] - Bilal, O.R.; Hussein, M.I. Trampoline metamaterial: Local resonance enhancement by springboards. Appl. Phys. Lett.
**2013**, 103, 111901. [Google Scholar] [CrossRef] - Bilal, O.R.; MHussein, I. Topologically evolved phononic material: Breaking the world record in band gap size. In Photonic and Phononic Properties of Engineered Nanostructures; International Society for Optics and Photonics: Bellingham, WA, USA, 2012; p. 826911. [Google Scholar]
- Halkjær, S.; Sigmund, O.; Jensen, J.S. Maximizing band gaps in plate structures. Struct. Multidiscip. Optim.
**2006**, 32, 263–275. [Google Scholar] [CrossRef] - Hedayatrasa, S.; Abhary, K.; Uddin, M.; Ng, C.T. Optimum design of phononic crystal perforated plate structures for widest bandgap of fundamental guided wave modes and maximized inplane stiffness. Mech. Phys. Solids
**2016**, 89, 31–58. [Google Scholar] [CrossRef] - Hedayatrasa, S.; Abhary, K.; Uddin, M.S.; Guest, J.K. Optimal design of tunable phononic bandgap plates under equibiaxial stretch. Smart Mater. Struct.
**2016**, 25, 055025. [Google Scholar] [CrossRef] - Zhu, H.; Semperlotti, F. Phononic thin plates with embedded acoustic black holes. Phys. Rev. B
**2015**, 91, 104304. [Google Scholar] [CrossRef] - Zhu, H.; Semperlotti, F. Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces. Phys. Rev. Lett.
**2016**, 117, 034302. [Google Scholar] [CrossRef] [PubMed] - Aberg, M.; Gudmundson, P. The usage of standard finite element codes for computation of dispersion relations in materials with periodic microstructure. J. Acoust. Soc. Am.
**1997**, 102, 2007–2013. [Google Scholar] [CrossRef] - Sigmund, O.; Jensen, J.S. Systematic design of phononic band-gap materials and structures by topology optimization. Philos. Trans. R. Soc. Lond. A
**2003**, 361, 1001–1019. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Schematic of proposed obliquely perforated porous phononic crystal plates (OPhPs) produced by oblique perforation at an angle $\theta $ (about y-axis) on a uniform plate and (

**b**) selected unit-cell and relevant irreducible Brillouin zone, and cross section of (

**c**) an asymmetric OPhP (A-OPhP) and (

**d**) a symmetric OPhP (S-PhP) lattice, respectively, with asymmetric and symmetric through-the-thickness design.

**Figure 2.**Perforation profiles chosen to study the bandgap efficiency of OPhPs, (

**a**) prescribed topologies with arbitrary perforation profile (${r}_{2}=0.9a$ and different ${r}_{1}$), and optimized topologies with (

**b**) maximized complete bandgap of mixed guided waves and (

**c**) maximized bandgap of asymmetric guided wave modes [10].

**Figure 3.**Total relative bandgap width (RBW) of mixed guided wave modes versus perforation angle calculated for (

**a**,

**b**) prescribed topologies PT1-PT4 and (

**c**,

**d**) optimized topologies CT1-CT4, (

**a**,

**c**) partial bandgap in $\mathsf{\Gamma}\mathsf{{\rm X}}$ and (

**b**,

**d**) complete bandgap in $\mathsf{\Gamma}\mathsf{{\rm X}}\mathsf{{\rm M}}\mathsf{\Gamma}\mathsf{{\rm N}}\mathsf{{\rm M}}$.

**Figure 4.**Variation of frequency ranges corresponding to the bandgaps of mixed guided wave modes opened within the first 10 modal branches versus perforation angle $0\xb0\le \theta \le 60\xb0$, (

**a**) partial bandgap of prescribed topology PT2 and (

**b**) partial bandgap of optimized topology CT1 in $\mathsf{\Gamma}\mathsf{{\rm X}}$.

**Figure 5.**Total RBW of guided wave modes versus perforation angle calculated for prescribed topologies PT1-PT4, (

**a**,

**b**) asymmetric wave modes and (

**c**,

**d**) symmetric wave modes, (

**a**,

**c**) partial bandgap in $\mathsf{\Gamma}\mathsf{{\rm X}}$, and (

**b**,

**d**) complete bandgap in $\mathsf{\Gamma}\mathsf{{\rm X}}\mathsf{{\rm M}}\mathsf{\Gamma}\mathsf{{\rm N}}\mathsf{{\rm M}}$.

**Figure 6.**(

**a**,

**b**) Total RBW of asymmetric guided wave modes versus perforation angle calculated for optimized topologies AT1-AT4, and (

**c**,

**d**) corresponding bandgap frequency ranges for topology AT3, (

**a**,

**c**) partial bandgap in $\mathsf{\Gamma}\mathsf{{\rm X}}$ and (

**b**,

**d**) complete bandgap in $\mathsf{\Gamma}\mathsf{{\rm X}}\mathsf{{\rm M}}\mathsf{\Gamma}\mathsf{{\rm N}}\mathsf{{\rm M}}$.

**Figure 7.**Modal band structure of S-OPhP of topology AT3 at perforation angles (

**a**) $30\xb0$ and (

**c**) $0\xb0$, and (

**b**) transmission spectrum in a finite phononic structure for both perforation angles.

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**MDPI and ACS Style**

Hedayatrasa, S.; Kersemans, M.; Abhary, K.; Van Paepegem, W.
Introducing Obliquely Perforated Phononic Plates for Enhanced Bandgap Efficiency. *Materials* **2018**, *11*, 1309.
https://doi.org/10.3390/ma11081309

**AMA Style**

Hedayatrasa S, Kersemans M, Abhary K, Van Paepegem W.
Introducing Obliquely Perforated Phononic Plates for Enhanced Bandgap Efficiency. *Materials*. 2018; 11(8):1309.
https://doi.org/10.3390/ma11081309

**Chicago/Turabian Style**

Hedayatrasa, Saeid, Mathias Kersemans, Kazem Abhary, and Wim Van Paepegem.
2018. "Introducing Obliquely Perforated Phononic Plates for Enhanced Bandgap Efficiency" *Materials* 11, no. 8: 1309.
https://doi.org/10.3390/ma11081309