Band Gap Structure of Acoustic Wave in Hexagonal Phononic Crystals with Polyethylene Matrix

Band Gap Structure of Acoustic Wave in Hexagonal Phononic Crystals with Polyethylene Matrix

Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 79 (2014) 612 – 616 37th National Conference on Theoretical and Applied...

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Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 79 (2014) 612 – 616

37th National Conference on Theoretical and Applied Mechanics (37th NCTAM 2013) & The 1st International Conference on Mechanics (1st ICM)

Band gap structure of acoustic wave in hexagonal phononic crystals with polyethylene matrix Yu-Lin Tsai* and Tungyang Chen Department of Civil Engineering, National Cheng Kung University, Tainan 70101, Taiwan

Abstract Phononic crystals have received some attention in the last decade due to their intrinsic property of band gaps. A band gap is a frequency range that acoustic waves cannot propagate within the periodic structure. In this paper, we propose a two-dimensional hexagonal phononic crystal that is composed of lead cylinders embedded in a polyethylene matrix. Using the finite element method, the dispersion diagram is calculated to identify the existence of band gaps. In addition, when a coating layer is introduced between the inclusion and the matrix, we find that the bandwidth and values of the gaps can be adjusted. In our demonstration, the band gap appear to be within the audible range with frequency 15400~18000Hz. Thus, the present proposition may have potential applications in blocking high-frequency noise. © 2013 Published The Authors. Published byThis Elsevier © 2014 by Elsevier Ltd. is an Ltd. open access article under the CC BY-NC-ND license Selection and peer-review under responsibility of the National Tsing Hua University, Department of Power Mechanical (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the National Tsing Hua University, Department of Power Mechanical Engineering. Engineering Keywords: hexagonal phononic crystals, polyethylene, lead cylinder, band gap.

Nomenclature ui x a k

displacement location vector length of lattice wave vector

* Corresponding author. Tel.: 886-6-2354326; fax: 886-6-2354326. E-mail address: [email protected]

1877-7058 © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the National Tsing Hua University, Department of Power Mechanical Engineering doi:10.1016/j.proeng.2014.06.387

Yu-Lin Tsai and Tungyang Chen / Procedia Engineering 79 (2014) 612 – 616

t c

time speed of acoustic wave (343 m/s)

1. Introduction Metamaterials are important in many applications, particularly in electromagnetic wave [1-4], acoustic wave [5-8] and thermo conductivity [9-10]. For acoustic waves, Kushwaha et al. proposed that elastic wave also could exhibit band gap effect based on the concept of photonic crystals [11]. A band gap represents a frequency range that acoustic waves cannot propagate within a periodic structure. With proper design of band structures, many applications of phononic crystals were proposed accordingly, such as elastic wave filters and waveguides. A famous example is an art sculpture in Spain, that exhibits a sound attenuation effect [12]. Liu et al. (2000) designed a sonic material which is composed by a lead ball, coated with silicone rubber and embedded in an epoxy matrix. They showed that the complete band gap could be achieved at 400Hz~1100Hz [13]. Based on one dimensional springmass system, Yao et al. [14] found that negative density effect could be identified at band gaps frequency, and that one-dimensional mass-spring-mass resonant lattice system can result in more specific band gaps with high frequencies [15-16]. But the frequency interval was still too narrow for practical applications. Later, multi-resonator acoustic metamaterials were proposed by Huang and Sun [17] to improve the narrow range of gaps. In addition, two-dimension multi-resonant structures and their effective mass density was studied [18] and extended to continuum medium [19-20]. In this work, we adopt polyethylene as the background matrix material and investigate the dispersion relation of the periodic lead cylinders. We find that it is possible to attain higher values and at the same time wider range of band gap. We mention that polyethylene is one of plastic materials which is usually used in the manufacture of plastic bottles. Recycling of the plastic bottles is an important environmental issue, and thus, from this environmental perspective, the usage of polyethylene could also be beneficial to our environment. 2. Method Base on Bloch’s theorem, wave propagation in the periodic structure can be characterized by analyzing all wave vectors k through the first Brillouin zone. For the periodic structure, the periodic boundary conditions in the unit cell have to satisfy ui x  a , t e ik˜a ui x, t (1) where ui is the displacement in the i direction, i=1, 2, x= x1 , x2 is the position vector, a= a1 , a 2 is the length of lattice, and k= k1, k 2 is the wave vector. In this work, we examine the hexagonal phononic crystals structures, Fig 1(a). A finite element method based on the software COMSOL Multiphysics is used to calculate the eigenfrequencies for each different wave vectors along the reduced Brillouin zone, * o 0 o . o * . Here, the top 10 eigen-frequencies, normalized by a 2Sc , are used to obtain the dispersion diagram, where c is the sound speed 343 m/s.

Fig. 1. Illustration of the unit cell and Brillouin zone of a unit cell with (a) a non-coated cylinder and (b)ġwith a coated cylinder.

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Lead cylinders are placed as hexagonal array in the background matrix material, polyethylene. The lattice length is 10cm. We discuss the band gap structure by selecting various size of lead cylinders. As suggested by Huang and Sun, they showed that the range of the gap could be significantly increased by an insertion of a compliant layer [16]. Here, in our investigation, we also consider a coated cylinder, embedding a fluid-like material between the inclusion, lead, and the matrix, polyethylene, Fig 1(b). We also discuss the band structure for the latter coated cylinder configuration. 3. Result and discussion Here we consider the periodic array of the unit cell in Fig.1, in which the geometric configurations and the constituent materials for the cell are fixed. We vary the filling fraction (size) of the cylinder to discuss its effect on the eigenfrequency and its corresponding eigenmode of acoustic waves in phononic crystals. Two different configurations (one without coating and the other with coating) with two different sizes of cylindrical leads, one 1cm and the other 2cm, are examined. The first case, we consider a periodic array containing non-coated lead cylinders. The second one is that the cylinder is coated by a fluid-like material with 1cm coating thickness. Also, to understand the effect of coating thickness, we also consider the case of 2 cm coating thickness. 3.1. Periodic lead cylinders without coating For the first case, the dispersion diagram is shown in Fig 2. It is seen that there is nearly no band gap for 1cm cylindrical lead. When the diameter of the cylindrical lead is increased, it is seen that the complete gaps appear at 4000~5000Hz and 6800~8200Hz (Fig 2(b)). As suggested by the concept of multi-resonator acoustic metamaterial, the gaps can behave differently by considering more inner mass-spring system or adding more compliant layer between the cylinder and the matrix. Thus, we will also consider embed a fluid-like coating material between the lead cylinder and polyethylene to discuss the coating effect of the dispersion relation.

Fig. 2. The dispersion diagram of hexagonal phononic crystals without coating for (a) 1cm radius lead cylinder and (b) 2cm radius lead cylinder.

3.2. Periodic lead cylinders with 1cm coating To increase range and frequency of complete gaps, we construct a model with a matrix-based material (polyethylene) reinforced by a coated cylinder. It is seen in Fig 3(a) that there are no complete gaps, though the dispersion structure is different with Fig 2(a). However, in comparison of Fig 3(b) and Fig 2(b), band gaps at 5100~8000Hz and 8500~10000Hz are observed for the case with 1cm coating, much wider and higher compared to the non-coated hexagonal crystals.

Yu-Lin Tsai and Tungyang Chen / Procedia Engineering 79 (2014) 612 – 616

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Fig. 3. The dispersion diagraph of hexagonal phononic crystals with 1cm coating for (a) 1cm radius lead cylinder and (b) 2cm radius lead cylinder.

3.3. Periodic lead cylinders with 2cm coating As motivated by the idea of multi-resonator acoustic metamaterials, a spring with very compliant property can increase the bandwidth of complete gap. Here, we find that the thickness of coating layer will also influence the range of the gap. Figure 4 illustrates for the dispersion diagram of a periodic array of lead cylinders with 2cm coating. Compared with Fig 4(a) and Fig 3(a), the band gaps of 1cm radius lead cylinders hexagonal phononic array is much larger than that of case 1 and case 2, where the complete gap appears at 7200~8500Hz. Fig 4(b) shows that the complete gaps appear at 6800~13000Hz and 15400~18000Hz, and its range and frequency are also wider and higher than that of the 1cm coating. Since, human can hear the acoustic frequency at 20~20000Hz, therefore, our model could be of use in blocking high frequency noise.

Fig. 4. The dispersion diagram of hexagonal phononic crystals with 1cm coating for (a) 1cm radius lead cylinder and (b) 2cm radius lead cylinder.

4. Conclusion In summary, a dispersion curve of a hexagonal phononic array made of lead cylinder and polyethylene matrix is calculated by using finite element methods. In our demonstrations, we find that when the size of the lead cylinder (large filling fraction) increases, the band gap will have a larger range. In addition, the range of complete gaps can be further increased by inserting a compliant coating layer between the lead cylinder and the polyrthylene matrix. In

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particular, a thick coating layer will increase the frequency range to a higher value. We also find that that the complete gaps of the frequency appear to be within the audible range. This suggests that the present design may be useful to block noise in practical applications. Acknowledgements The financial support from the National Science Council of Taiwan under the grant NSC 99-2221-E-006-070MY3 is acknowledged. References [1] D.R. Smith, J.B. Pendry, M.C.K. Wiltshire, Metamaterials and negtive refractive index, Science 305 (2004) 788-792. [2] J.B. Pendry, D. Schurig, D.R. Smith, Controlloing electromagnetic fields, Science 312 (2006) 1780-1782. [3] U. Leonhardt, Optical conformal mapping, Science 312 (2006) 1777-1780. [4] D. Schurig, J.J. Mock, B.J. Justice, S.A. Cummer, Metamaterial electromagnetic cloak at microwave frequencies, Science 314 (2006) 977-980. [5] S.A. Cummer, D. Schurig, One path to acoustic cloaking, New J. Phys. 9 (2007) 45. [6] H. Chen, C.T. Chan, Acoustic cloaking and transformation acoustics, J. Phys. D: Appl. Phys. 43 (2010) 113001. [7] D. Torrent, J. Sanchez-Dehesa, Acoustic metamaterials for new two dimensional sonic devices, New J. Phys. 9 (2007) 323. [8] H. Chen, C.T. Chan, Acoustic cloaking in three dimensions using acoustic metamaterials, Appl. Phys. Lett. 91 (2007) 183518. [9] S. Guenneau, C. Amra, D. Veynante, Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Exp. 20 (2012) 8207-8218. [10] R. Schittny, M. Kadic, S. Guenneau, M. Wegener, Experiments on transformation thermodynamics: molding the flow of heat, Phys. Rev. Lett. 110 (2013) 195901. [11] M.S. Kushwaha, P. Halevi, L. Dobrzynsky, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 (1993) 2022. [12] R. MartǮnez-Sala, J. SǢncho, J.V. Sanchez, V. GǴmez, J. Llinares, Sound attenuation by sculpture, Nature 378 (1995) 241. [13] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Locally resonant sonic materials, Science 289 (2000) 1734-1736. [14] S. Yao, X. Zhou, G. Hu, Experimental study on negative mass effective mass in a 1D mass-spring system, New J. Phys. 10 (2008) 043020. [15] H.H. Huang, C.T. Sun, G.L. Huang, On the negative effective mass density in acoustic metamaterials, Int. J. Eng. Sci. 47 (2009) 610-617. [16] H.H. Huang, C.T. Sun, Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density, New J. Phys. 11 (2009) 013003. [17] G.L. Huang, C.T. Sun, Band gaps in a multiresonator acoustic metamaterial, J. Vib. Acoust. 132 (2010) 031003. [18] H.H. Huang, C.T. Sun, G.L. Huang, Locally resonant acoustic metamaterials with 2D anisotropic effective mass density, Philos. Mag. 91 (2011) 981-996. [19] R. Zhu, H.H. Huang, G.L. Huang, C.T. Sun, Microstructure continuum modeling of an elastic metamaterial, Int. J. Eng. Sci. 49 (2011) 14471485. [20] A.P. Liu, R. Zhu, X.N. Liu, G.K. Hu, G.L. Huang, Multi-displacement microstructure continuum modeling of anisotropic elasticmetamaterials, Wave Motion 49 (2012) 411-426.