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Acoustic beam controlling in water by the design of phononic crystal
Extreme Mechanics Letters xxx (xxxx) xxx
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Acoustic beam controlling in water by the design of phononic crystal X.K. Han, Z. Zhang
∗
State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China
article
info
Article history: Received 9 September 2019 Received in revised form 6 November 2019 Accepted 7 November 2019 Available online xxxx Keywords: Phononic crystal Line-defects Self-collimation-based acoustic waves Beam splitter
a b s t r a c t Three-component phononic crystals line-defects composed of aluminum, rubber and water were introduced to the aluminum pillars arrayed in water for controlling of the acoustic beam. The propagation properties of self-collimation-based acoustic waves were investigated by finite element method in the designed PC structure. Results reveal that the line-defects can lead to two splitting wave beams. The beam splitter PC structure can be controlled by adjusting the radii of the rods and the coating layer in the line defects. When the acoustic speed of the coating layer changes, the splitting propagation of the acoustic wave can be changed. The output acoustic wave can be altered by controlling the phase difference of the two incident sound sources. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction As a periodic artificial structure, phononic crystals (PCs) are well known due to the unique frequency characteristics on acoustic or elastic wave propagating [1–8]. Besides the remarkable feature band gap (BG) [9,10], the pass band properties like sonic focusing [11,12], self-collimation effect [13,14] and negative refraction [15,16], have also attracted considerable interests and attentions. Equi-frequency contours (EFCs) of the certain band are efficient for detailed description on the response of the PC structure for the incident wave with a certain frequency [17–21]. When contour lines are flat at some frequencies, self-collimation occurs. Self-collimation acoustic beam can propagate in PCs with almost no broadening. This new feature has no direct relation with the band gap or the defect waveguides [22–25]. Due to the energy propagation along the vertical direction of the contour lines, the direction of the acoustic wave propagation is normal to the contour lines. Self-collimation in PCs is investigated in both theory and experiment [26–28]. The unidirectional propagation of the self-collimation wave can limit the application of the acoustic devices. So, it is necessary to control the self-collimation waves. Guo et al. [29] designed 2D PC directional waveguide to study acoustic beam splitting in the PC nano-structure with no defect in low GHz range. Xie et al. [30] designed efficient sharp bending structure for controlling of surface acoustic waves based on the prism-shaped surface PC. The combination of two same prismshaped PCs can form an interface and it leads to the splitting propagation of the surface acoustic waves. Lee et al. [31] designed ∗ Corresponding author. E-mail address: [email protected] (Z. Zhang).
an elastic plate which is composed of elastic PC prisms and found the triple beam splitting in experimental observation. Kaya et al. [32] investigated a Mach–Zehnder interferometer in a 2D PC by splitting and recombining self-collimated beams. Li et al. [33] arranged circular rods and square rods line defects in PCs and found that the self-collimated beams can be split by controlling of the radii and the orientation. As mentioned above, the beam splitter can be controlled by line defects for phononic crystals with two kind of materials. When the defects PCs consist of three components with rods and coatings, both the rods and the coatings can affect the characteristics of the beam splitter. It is interesting to know whether new features can be found for the three components with rods and coatings when the line defects are introduced. This is the motivation of current work. 2. Model and method As shown in Fig. 1(a), two-dimensional PCs composed of aluminum pillars arranged in a square lattice in water are discussed. The acoustic longitudinal speed and the density of aluminum pillar are ca = 2730 m/s and ρa = 6782 kg/m3 . The acoustic longitudinal speed and the density of water are cw = 1490 m/s and ρw = 1000 kg/m3 . The lattice constant a is 0.01 m and the radii of the pillars are R1 = 0.39a. The splitting structure is composed of 21×21 unit cells. Perfectly matched layers (PMLs) are placed along the sides of the structure to restrain artificial back reflections √ in Fig. 1(b). The length of each cell along x-direction is t = 2a. As shown in Fig. 1(b), unit cells are periodically arranged. The unit cells in adjacent columns are stagger. So, the x-axis in the structure corresponds to the Γ M direction in the PC unit cell. Acoustic beam splitting occurs when a line defect
https://doi.org/10.1016/j.eml.2019.100602 2352-4316/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: X.K. Han and Z. Zhang, Acoustic beam controlling in water by the design of phononic crystal, Extreme Mechanics Letters (2019) 100602, https://doi.org/10.1016/j.eml.2019.100602.
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Fig. 1. The phononic crystal structures. (a) The unit cell (b) beam splitting structure.
Fig. 2. The equi-frequency contours of the first band.
Fig. 3. Simulation results for acoustic beam propagation at different frequencies. (a) f = 70 500 Hz (b) f = 60 000 Hz.
is placed in the PC model. Line defects are formed by threecomponent PCs line-defects composed of aluminum, rubber and water. The material parameters for the rubber are chosen as
follows: the density ρr = 1300 kg/m3 and the longitudinal speed cr = 200 m/s. Each defect pillar has a radius R2 for aluminum
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Fig. 4. Variation of the acoustic field distributions for different aluminum pillars radii R2 : (a) R2 = 0.418a (b) R2 = 0.416a (c) R2 = 0.412a (d) R2 = 0.406a (e) R2 = 0.4a (f) R2 = 0.394a.
Fig. 5. Effects of the R2 : (a) sound intensity of the B-boundary (b) the reflection efficiency.
pillar and a radius R3 for the coating layer. As shown in Fig. 1(b), the line defects are placed along the Γ X direction. The wave equations in the water can be described as: 1 ∂ 2 p(r , t) c2
∂ 2t
= ∇ 2 p(r , t)
(1)
where c is the acoustic longitudinal speed, p(r , t) = p(r)e−iωt is the sound pressure. The sound pressure field p(r) can be described as [34–36] by the Bloch theorem: p(r) = pk (r)e(ik ·r )
(2)
where k = (kx , ky ) describes the wave vector, pk (r) represents a vector function which is periodic.
The periodic boundary conditions are obtained by Eq. (2): p (r + a) = p (r ) e(ik ·a)
(3)
where a represents the spatial vector. Band structures of the models are calculated using the ComsolMultiphysics which is an efficient software to calculate the PC dispersion relations and eigenmodes. Due to the Bloch theorem, periodic boundary conditions are adopted on the interfaces between the adjacent unit cells. Single unit cell is used for calculation due to the periodicity of the structure. When wave vectors are swept on Brillouin zone, eigenvectors and the corresponding eigenfrequencies can be obtained. Then, band structures are obtained.
Please cite this article as: X.K. Han and Z. Zhang, Acoustic beam controlling in water by the design of phononic crystal, Extreme Mechanics Letters (2019) 100602, https://doi.org/10.1016/j.eml.2019.100602.
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Fig. 6. Effects of the R3 : (a) sound intensity of the B-boundary (b) the reflection efficiency.
3. Results and discussions The equi-frequency contours calculated in the first Brillouin zone for the first band are shown in Fig. 2. It can be seen that flat equi-frequency exits at frequency of 70 500 Hz. The flat equifrequency means that self-collimated propagation of the wave occurs without diffraction along Γ –M direction. To validate the self-collimated propagation phenomenon, propagations of two wave beams with different frequencies are selected for comparison. The incident acoustic beam at frequency of 70 500 Hz propagates with no mirror diffraction along the Γ –M direction in Fig. 3(a). The phenomenon above is called as selfcollimation. For comparison, a case with frequency of 60 000 Hz is used. Divergent phenomenon occurs when the acoustic beam propagates, as shown in Fig. 3(b). No self-collimation exits for the other frequencies in the first band along Γ –M direction. 3.1. Effects of the parameters of the defect PCs on splitting For the beam splitting structure proposed in Fig. 1(b), linedefects can lead to the splitting of acoustic beams into two outgoing wave beams: one beam is the transmitted beam which propagates along the original direction and another beam is the reflected beam which is perpendicular to the incident acoustic beam. The signal at boundary C indicates the transmission and the signal at boundary B indicates the reflection. To control the beam splitter, the radii of the rods and the coating layers in line defects can be adjusted. When the radius of the defect PC coating layer R3 is 0.42a, the acoustic field distributions for different R2 are shown in Fig. 4. A majority of the wave energy of the incident acoustic beam can propagate through the defects when R2 = 0.418a in Fig. 4(a). The wave energy of the reflected beam can be increased to be higher than the transmitted beam when R2 is decreased, as shown in Fig. 4(b, c). Total reflection occurs when R2 is 0.406a in Fig. 4(d). However, the transmitted beam changes to be stronger again as shown in Fig. 4(e) when R2 decreases continuously. The transmitted beam and the incident beam are stagger in wave propagation path. For the beam spilling structure, only reflected beam exists when R2 is smaller than 0.394a as shown in Fig. 4(f). To study the effects of the R2 intuitively, sound intensity I of the B-boundary of the structure in Fig. 1(b) and reflection efficiency curves are given in Fig. 5. Sound intensity of the reflected beam increases when R2 decreases from 0.418a to 0.406a and then it decreases when R2 decreases to 0.4a in Fig. 5(a). The sound intensity remains constant when R2 is further decreased to be smaller than 0.394a. The reflection efficiency is defined as the ratio of the sound power in B-boundary and the incident
Fig. 7. The acoustic field distribution when R3 = 0.344a.
energy. The reflection efficiency is close to be 1 when R2 is smaller than 0.396a in Fig. 5(b) which indicates that total reflection can be realized. The reflection efficiency fluctuates with R2 . This means that the transmission and the reflection acoustic beams can fluctuate with R2 , which is observed in Fig. 4. When R2 is fixed to be 0.31a, the efficiency of the beam splitter can be altered by adjusting R3 . As shown in Fig. 6(a), the sound intensity grows rapidly when R3 increases from 0.312a to 0.322a. Then, the sound intensity becomes decreased when R3 is further increased to be higher than 0.332a. The reflection efficiencies are also calculated in Fig. 6(b) for different selections of R3 . The reflection efficiency increases initially and it turns to be stabilized at about 0.95 after two periods of fluctuations. Total reflection exits when R3 is larger than 0.35a. For the sound intensity in Fig. 6(a), two peaks exist when R3 is 0.344a which indicates that two reflected acoustic beams reach B-boundary. From the acoustic field distribution in Fig. 7, when the incident acoustic wave propagates to the defects, part of wave is reflected to the B-boundary by the defects. The rest part of the incident wave propagates along the defects. Then it splits into two beams pointing to B and C boundaries, respectively. Three beams of acoustic wave can be split by the proposed structure. When the ratio R2 /R3 is 0.942, the second reflected wave is in the left half of the B-boundary in Fig. 8(a), which indicates that the incident wave propagates downward along the defects PCs before the second reflection in Fig. 8(c). However,
Please cite this article as: X.K. Han and Z. Zhang, Acoustic beam controlling in water by the design of phononic crystal, Extreme Mechanics Letters (2019) 100602, https://doi.org/10.1016/j.eml.2019.100602.
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Fig. 8. Sound intensities of the B and C boundaries: (a) R2 /R3 = 0.942 (b) R2 /R3 = 0.897 The acoustic field distributions: (c) R2 /R3 = 0.942 (d) R2 /R3 = 0.897.
Fig. 9. Sound intensities of the B and C boundaries: (a) cr = 200 m/s (b) cr = 400 m/s (c) cr = 600 m/s; The acoustic field distributions when R2 = 0.458a: (d) cr = 200 m/s (e) cr = 400 m/s (f) cr = 600 m/s.
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intensities of the transmitted beam and reflected beam, as shown in Fig. 9(c). 3.2. Acoustic beam controlling by splitting structure for two sound sources
Fig. 10. The acoustic field distribution for two sound sources when R2 = 0.4a and R3 = 0.42a.
the second reflected wave is in the right half of the B-boundary in Fig. 8(b) when the ratio R2 /R3 is 0.897. Incident beam propagates upward along the defects before the second reflection from Fig. 8(d). When the ratio R2 /R3 is 0.897, the intensity of the transmitted beam in C boundary is much smaller which indicates that a majority of the wave energy of the incident acoustic beam is reflected. The propagation of the incident wave can be affected by the acoustic longitudinal speed of the defect coating layer. Three cases with different acoustic longitudinal speeds, cr = 200 m/s, 400 m/s and 600 m/s, are selected to study the effects of the acoustic longitudinal speed. R3 is fixed to be 0.64a. As shown in Fig. 9, when R2 is 0.458a, the transmitted beam intensity in 200 m/s is much lower than 400 m/s and it almost keeps constant when cr increases from 400 m/s to 600 m/s. The intensity of the reflected beam is higher than the transmitted beam intensity when cr is 200 m/s. This means the reflected acoustic beam can be obviously found in this case. The sound intensities of the transmitted beam and reflected beam are almost equal when R2 slightly decreases to 0.457a, as shown in Fig. 9(a). When the acoustic longitudinal speed is increased to 400 m/s, R2 need to be decreased to 0.446a to realize equal sound intensities of the transmitted beam and reflected beam, as shown in Fig. 9(b). When the acoustic longitudinal speed is further increased to 600 m/s, R2 need to be decreased to 0.42a to realize equal sound
By adjusting the radii of the defects composed of three-component PCs, the splitting structure can control the propagation of the incident acoustic wave generated by one sound source. When two sources are given, it is also necessary to control the outgoing wave. For the splitting structure with R2 = 0.4a and R3 = 0.42a, two sound sources send sound waves to the defect three-component PCs. The first sound source S1 is placed in the left of the structure. The second sound source S2 is at the upper side of the structure. As shown in Fig. 10, two same intensity wave beams are ejected to B-boundary and C-boundary, respectively. Both the two acoustic beams can be reflected by the introduced defects composed of three-component PCs. The sound intensity curves corresponding to one sound source and two sound sources are shown in Fig. 11. The sound intensities are shown in Fig. 11(a) when only S1 exits. It can be seen that the sound intensity of the transmitted acoustic beam is higher than the reflected acoustic beam. However, the wave beams reaching B-boundary/C-boundary are the superposition of both the reflected wave beam and the transmitted beam from S1 and S2. As shown in Fig. 11(b), the sound intensities reaching B and C boundaries are equal when S1 and S2 exist. When S1 keeps constant, the intensities of the outgoing beams in B and C boundaries are adjusted by changing of the phase position of S2. As shown in Fig. 12, when θ decreases from 0 to – π /2, the intensity of the reflected beam in B-boundary tends to be larger and the intensity of the transmitted beam in C-boundary becomes smaller. The wave beam reaching the B-boundary becomes weaker when θ increases from 0 to π /2. Instead, the wave beam reaching C-boundary becomes increased when θ increases. As shown in Fig. 13(a), when the phase position of S2 is selected to be θ = π /2, a majority of the wave energy of the two incident acoustic beams propagates along x-direction (horizontal direction) when the two incident beams meet at the middle of the defect PCs. The structure behaves like a polarization device. By the change of the phase difference of the two incident waves, the intensities of the acoustic beams along x-direction and y-direction (vertical direction) can be controlled. 4. Conclusions In this work, three-component phononic crystals line-defects composed of aluminum, rubber and water are introduced to
Fig. 11. Sound intensities of the B and C boundaries: (a) one sound source (b) two sound sources.
Please cite this article as: X.K. Han and Z. Zhang, Acoustic beam controlling in water by the design of phononic crystal, Extreme Mechanics Letters (2019) 100602, https://doi.org/10.1016/j.eml.2019.100602.
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Fig. 12. Variation of sound intensities with different phase of S2: (a) B-boundary (b) C-boundary.
Fig. 13. The acoustic field distributions for two sound sources: (a) θ = π /2 (b) θ = −π /2.
the aluminum pillars arrayed in water to control the acoustic beam. The propagation properties of self-collimation-based acoustic waves are investigated by finite element method in the designed PC structure. Results reveal that the line-defects can lead to two splitting wave beams. In certain conditions, the wave can be split to three beams. This is the new feature of the proposed splitting structure. The rod radius and the coating layer radius in the line defects are two key factors for the efficiency of the beam splitter. When the acoustic speed of the coating layer changes, acoustic wave propagation is influenced. The proposed structure can be altered like a polarization device by controlling the phase difference of the two incident sound sources. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 11572074). References [1] M. Maldovan, Sound and heat revolutions in phononics, Nature 503 (2013) 209–217.
[2] J.J. Chen, K.W. Zhang, J. Gao, J.C. Cheng, Stopbands for lower-order Lamb waves in one-dimensional composite thin plates, Phys. Rev. B 73 (2006) 094307. [3] R. Zhu, G.L. Huang, H.H. Huang, C.T. Sun, Experimental and numerical study of guided wave propagation in a thin metamaterial plate, Phys. Lett. A 375 (2011) 2863–2867. [4] Z. Zhang, X.K. Han, G.M. Ji, Mechanism for controlling the band gap and the flat band in three-component phononic crystals, J. Phys. Chem. Solids 123 (2018) 235–241. [5] X.K. Han, Z. Zhang, Topological optimization of phononic crystal thin plate, Sci. Rep. 9 (2019) 8331. [6] A. Arreola-Lucas, J.A. Franco-Villafañe, G. Báez, R.A. Méndez-Sánchez, Inplane vibrations of a rectangular plate: Plane wave expansion modelling and experiment, J. Sound Vib. 342 (2015) 168–176. [7] Y. Lai, X.D. Zhang, Z.Q. Zhang, Large sonic band gaps in 12-fold quasicrystals, J. Appl. Phys. 91 (2002) 6191–6193. [8] J. Mei, G.C. Ma, M. Yang, Z.Y. Yang, W.J. Wen, P. Sheng, Dark acoustic metamaterials as super absorbers for low-frequency sound, Nature Commun. 3 (2012) 756. [9] Z. Zhang, X.K. Han, A new hybrid phononic crystal in low frequencies, Phys. Lett. A 380 (2016) 3766–3772. [10] M. Hirsekorn, Small-size sonic crystals with strong attenuation bands in the audible frequency range, Appl. Phys. Lett. 84 (2004) 3364–3366. [11] X.W. Xue, P. Li, F. Jin, Refraction behavior investigation and focusing control of phononic crystals under external magnetic fields, Ultrasonics 92 (2019) 261–266. [12] M. Kurosu, D. Hatanaka, K. Onomitsu, H. Yamaguchi, On-chip emporal focusing of elastic waves in a phononic crystal waveguide, Nature Commun. 9 (2018) 1331. [13] X.F. Yu, S.H. Fan, Bands and splitters for self-collimated beams in photonic crystals, Appl. Phys. Lett. 83 (2003) 3251–3253. [14] D.W. Prather, S.Y. Shi, J. Murakowski, G.J. Schneider, A. Sharkawy, C.H. Chen, B.L. Miao, R. Martin, J. Phys. D: Appl. Phys. 40 (2007) 2635–2651.
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[15] H.L. He, C.Y. Qiu, L.P. Ye, X.X. Cai, X.Y. Fan, M.Z. Ke, F. Zhang, Z.Y. Liu, Nature 560 (2018) 61–64. [16] Z.L. Hou, H.Q. Ni, B. Assouar, PT-symmetry for elastic negative refraction, Phys. Rev. Appl. 10 (2018) 044071. [17] J.H. Wen, D.L. Yu, L. Cai, X.S. Wen, Acoustic directional radiation operating at the pass band frequency in two-dimensional phononic crystals, J. Phys. D: Appl. Phys. 42 (2009) 115417. [18] P.S. Ma, H.W. Kim, J.H. Oh, Y.Y. Kim, Mode separation of a single-frequency bi-modal elastic wave pulse by a phononic crystal, Appl. Phys. Lett. 99 (2011) 201906. [19] A. Sukhovich, L. Jing, J.H. Page, Negative refraction and focusing of ultrasound in two-dimensional phononic crystal, Phys. Rev. B 77 (2008) 014301. [20] J.F. Zhao, R. Marchal, B. Bonello, O. Boyko, Efficient focalization of antisymmetric Lamb waves in gradient-index phononic crystal plates, Appl. Phys. Lett. 101 (2012) 261905. [21] E.M. Hamham, N. Jiménez, R. Picó, V.J. Sánchez-Morcillo, L.M. García-Raffi, Nonlinear self-collimated sound beams in sonic crystals, Phys. Rev. B 92 (2015) 054302. [22] I. Pérez-Arjona, V.J. Sánchez-Morcillo, J. Redondo, V. Espinosa, K. Staliunas, Theoretical prediction of the nondiffractive propagation of sonic waves through periodic acoustic media, Phys. Rev. B 75 (2007) 014304. [23] D.W. Prather, S.Y. Shi, J. Murakowski, G.J. Schneider, A. Sharkawy, C.H. Chen, B.L. Miao, R. Martin, Self-collimation in photonic crystal structures: a new paradigm for applications and device development, J. Phys. D: Appl. Phys. 40 (2007) 2635–2651. [24] S.Y. Shi, A. Sharkawy, C.H. Chen, D.M. Pustai, D.W. Prather, Dispersionbased beam splitter in photonic crystals, Opt. Lett. 29 (2004) 617–619. [25] X.F. Yu, S.H. Fan, Bends and splitters for self-collimated beams in photonic crystals, Appl. Phys. Lett. 83 (2003) 3251–3253.
[26] C.Y. Chiang, P.G. Luan, Imaging off-plane shear waves with a twodimensional phononic crystal lens, J. Phys.: Condens. Matter 22 (2010) 055405. [27] A. Cicek, O.A. Kaya, B. Ulug, Acoustic beam splitting in a sonic crystal around a directional band gap, Chin. Phys. B 22 (2013) 114301. [28] T. Zhang, Y. Cheng, J.Z. Guo, J.Y. Xu, X.J. Liu, Acoustic logic gates and boolean operation based on self-collimating acoustic beams, Appl. Phys. Lett. 106 (2015) 113503. [29] Y.N. Guo, D. Brick, M. Großmann, M. Hettich, T. Dekorsy, Acoustic beam splitting at low GHz frequencies in a defect-free phononic crystal, Appl. Phys. Lett. 110 (2017) 031904. [30] S. Xie, S.L. Ouyang, Z.J. He, X.Y. Wang, K. Deng, H.P. Zhao, Bending and splitting of spoof surface acoustic waves through structured rigid surface, Results Phys. 8 (2018) 52–56. [31] H. Lee, J.H. Oh, Y.Y. Kim, Multiple beam splitting in elastic phononic crystal plates, Ultrasonics 56 (2015) 178–182. [32] O.A. Kaya, A. Cicek, A. Salman, B. Ulug, Acoustic Mach–Zehnder interferometer utilizing self-collimated beams in a two-dimensional phononic crystal, Sensors Actuators B 203 (2014) 197–203. [33] J. Li, F.G. Wu, H.L. Zhong, et al., Acoustic beam splitting in two-dimensional phononic crystals using self-collimation effect, J. Appl. Phys. 118 (2015) 144903. [34] Z. Zhang, G.M. Ji, X.K. Han, Optimization design of a novel zigzag lattice phononic crystal with holes, Internat. J. Modern Phys. B 33 (2019) 1950124. [35] Z. Zhang, X.K. Han, G.M. Ji, The band gap controlling by geometrical symmetry design in hybrid phononic crystal, Internat. J. Modern Phys. B 32 (2018) 1850034. [36] Y.L. Huang, J. Li, W.Q. Chen, R.H. Bao, Tunable bandgaps in soft phononic plates with spring-mass-like resonators, Int. J. Mech. Sci. 151 (2019) 300–313.
Please cite this article as: X.K. Han and Z. Zhang, Acoustic beam controlling in water by the design of phononic crystal, Extreme Mechanics Letters (2019) 100602, https://doi.org/10.1016/j.eml.2019.100602.
Contents lists available at ScienceDirect
Extreme Mechanics Letters journal homepage: www.elsevier.com/locate/eml
Acoustic beam controlling in water by the design of phononic crystal X.K. Han, Z. Zhang
∗
State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China
article
info
Article history: Received 9 September 2019 Received in revised form 6 November 2019 Accepted 7 November 2019 Available online xxxx Keywords: Phononic crystal Line-defects Self-collimation-based acoustic waves Beam splitter
a b s t r a c t Three-component phononic crystals line-defects composed of aluminum, rubber and water were introduced to the aluminum pillars arrayed in water for controlling of the acoustic beam. The propagation properties of self-collimation-based acoustic waves were investigated by finite element method in the designed PC structure. Results reveal that the line-defects can lead to two splitting wave beams. The beam splitter PC structure can be controlled by adjusting the radii of the rods and the coating layer in the line defects. When the acoustic speed of the coating layer changes, the splitting propagation of the acoustic wave can be changed. The output acoustic wave can be altered by controlling the phase difference of the two incident sound sources. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction As a periodic artificial structure, phononic crystals (PCs) are well known due to the unique frequency characteristics on acoustic or elastic wave propagating [1–8]. Besides the remarkable feature band gap (BG) [9,10], the pass band properties like sonic focusing [11,12], self-collimation effect [13,14] and negative refraction [15,16], have also attracted considerable interests and attentions. Equi-frequency contours (EFCs) of the certain band are efficient for detailed description on the response of the PC structure for the incident wave with a certain frequency [17–21]. When contour lines are flat at some frequencies, self-collimation occurs. Self-collimation acoustic beam can propagate in PCs with almost no broadening. This new feature has no direct relation with the band gap or the defect waveguides [22–25]. Due to the energy propagation along the vertical direction of the contour lines, the direction of the acoustic wave propagation is normal to the contour lines. Self-collimation in PCs is investigated in both theory and experiment [26–28]. The unidirectional propagation of the self-collimation wave can limit the application of the acoustic devices. So, it is necessary to control the self-collimation waves. Guo et al. [29] designed 2D PC directional waveguide to study acoustic beam splitting in the PC nano-structure with no defect in low GHz range. Xie et al. [30] designed efficient sharp bending structure for controlling of surface acoustic waves based on the prism-shaped surface PC. The combination of two same prismshaped PCs can form an interface and it leads to the splitting propagation of the surface acoustic waves. Lee et al. [31] designed ∗ Corresponding author. E-mail address: [email protected] (Z. Zhang).
an elastic plate which is composed of elastic PC prisms and found the triple beam splitting in experimental observation. Kaya et al. [32] investigated a Mach–Zehnder interferometer in a 2D PC by splitting and recombining self-collimated beams. Li et al. [33] arranged circular rods and square rods line defects in PCs and found that the self-collimated beams can be split by controlling of the radii and the orientation. As mentioned above, the beam splitter can be controlled by line defects for phononic crystals with two kind of materials. When the defects PCs consist of three components with rods and coatings, both the rods and the coatings can affect the characteristics of the beam splitter. It is interesting to know whether new features can be found for the three components with rods and coatings when the line defects are introduced. This is the motivation of current work. 2. Model and method As shown in Fig. 1(a), two-dimensional PCs composed of aluminum pillars arranged in a square lattice in water are discussed. The acoustic longitudinal speed and the density of aluminum pillar are ca = 2730 m/s and ρa = 6782 kg/m3 . The acoustic longitudinal speed and the density of water are cw = 1490 m/s and ρw = 1000 kg/m3 . The lattice constant a is 0.01 m and the radii of the pillars are R1 = 0.39a. The splitting structure is composed of 21×21 unit cells. Perfectly matched layers (PMLs) are placed along the sides of the structure to restrain artificial back reflections √ in Fig. 1(b). The length of each cell along x-direction is t = 2a. As shown in Fig. 1(b), unit cells are periodically arranged. The unit cells in adjacent columns are stagger. So, the x-axis in the structure corresponds to the Γ M direction in the PC unit cell. Acoustic beam splitting occurs when a line defect
https://doi.org/10.1016/j.eml.2019.100602 2352-4316/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: X.K. Han and Z. Zhang, Acoustic beam controlling in water by the design of phononic crystal, Extreme Mechanics Letters (2019) 100602, https://doi.org/10.1016/j.eml.2019.100602.
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Fig. 1. The phononic crystal structures. (a) The unit cell (b) beam splitting structure.
Fig. 2. The equi-frequency contours of the first band.
Fig. 3. Simulation results for acoustic beam propagation at different frequencies. (a) f = 70 500 Hz (b) f = 60 000 Hz.
is placed in the PC model. Line defects are formed by threecomponent PCs line-defects composed of aluminum, rubber and water. The material parameters for the rubber are chosen as
follows: the density ρr = 1300 kg/m3 and the longitudinal speed cr = 200 m/s. Each defect pillar has a radius R2 for aluminum
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Fig. 4. Variation of the acoustic field distributions for different aluminum pillars radii R2 : (a) R2 = 0.418a (b) R2 = 0.416a (c) R2 = 0.412a (d) R2 = 0.406a (e) R2 = 0.4a (f) R2 = 0.394a.
Fig. 5. Effects of the R2 : (a) sound intensity of the B-boundary (b) the reflection efficiency.
pillar and a radius R3 for the coating layer. As shown in Fig. 1(b), the line defects are placed along the Γ X direction. The wave equations in the water can be described as: 1 ∂ 2 p(r , t) c2
∂ 2t
= ∇ 2 p(r , t)
(1)
where c is the acoustic longitudinal speed, p(r , t) = p(r)e−iωt is the sound pressure. The sound pressure field p(r) can be described as [34–36] by the Bloch theorem: p(r) = pk (r)e(ik ·r )
(2)
where k = (kx , ky ) describes the wave vector, pk (r) represents a vector function which is periodic.
The periodic boundary conditions are obtained by Eq. (2): p (r + a) = p (r ) e(ik ·a)
(3)
where a represents the spatial vector. Band structures of the models are calculated using the ComsolMultiphysics which is an efficient software to calculate the PC dispersion relations and eigenmodes. Due to the Bloch theorem, periodic boundary conditions are adopted on the interfaces between the adjacent unit cells. Single unit cell is used for calculation due to the periodicity of the structure. When wave vectors are swept on Brillouin zone, eigenvectors and the corresponding eigenfrequencies can be obtained. Then, band structures are obtained.
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Fig. 6. Effects of the R3 : (a) sound intensity of the B-boundary (b) the reflection efficiency.
3. Results and discussions The equi-frequency contours calculated in the first Brillouin zone for the first band are shown in Fig. 2. It can be seen that flat equi-frequency exits at frequency of 70 500 Hz. The flat equifrequency means that self-collimated propagation of the wave occurs without diffraction along Γ –M direction. To validate the self-collimated propagation phenomenon, propagations of two wave beams with different frequencies are selected for comparison. The incident acoustic beam at frequency of 70 500 Hz propagates with no mirror diffraction along the Γ –M direction in Fig. 3(a). The phenomenon above is called as selfcollimation. For comparison, a case with frequency of 60 000 Hz is used. Divergent phenomenon occurs when the acoustic beam propagates, as shown in Fig. 3(b). No self-collimation exits for the other frequencies in the first band along Γ –M direction. 3.1. Effects of the parameters of the defect PCs on splitting For the beam splitting structure proposed in Fig. 1(b), linedefects can lead to the splitting of acoustic beams into two outgoing wave beams: one beam is the transmitted beam which propagates along the original direction and another beam is the reflected beam which is perpendicular to the incident acoustic beam. The signal at boundary C indicates the transmission and the signal at boundary B indicates the reflection. To control the beam splitter, the radii of the rods and the coating layers in line defects can be adjusted. When the radius of the defect PC coating layer R3 is 0.42a, the acoustic field distributions for different R2 are shown in Fig. 4. A majority of the wave energy of the incident acoustic beam can propagate through the defects when R2 = 0.418a in Fig. 4(a). The wave energy of the reflected beam can be increased to be higher than the transmitted beam when R2 is decreased, as shown in Fig. 4(b, c). Total reflection occurs when R2 is 0.406a in Fig. 4(d). However, the transmitted beam changes to be stronger again as shown in Fig. 4(e) when R2 decreases continuously. The transmitted beam and the incident beam are stagger in wave propagation path. For the beam spilling structure, only reflected beam exists when R2 is smaller than 0.394a as shown in Fig. 4(f). To study the effects of the R2 intuitively, sound intensity I of the B-boundary of the structure in Fig. 1(b) and reflection efficiency curves are given in Fig. 5. Sound intensity of the reflected beam increases when R2 decreases from 0.418a to 0.406a and then it decreases when R2 decreases to 0.4a in Fig. 5(a). The sound intensity remains constant when R2 is further decreased to be smaller than 0.394a. The reflection efficiency is defined as the ratio of the sound power in B-boundary and the incident
Fig. 7. The acoustic field distribution when R3 = 0.344a.
energy. The reflection efficiency is close to be 1 when R2 is smaller than 0.396a in Fig. 5(b) which indicates that total reflection can be realized. The reflection efficiency fluctuates with R2 . This means that the transmission and the reflection acoustic beams can fluctuate with R2 , which is observed in Fig. 4. When R2 is fixed to be 0.31a, the efficiency of the beam splitter can be altered by adjusting R3 . As shown in Fig. 6(a), the sound intensity grows rapidly when R3 increases from 0.312a to 0.322a. Then, the sound intensity becomes decreased when R3 is further increased to be higher than 0.332a. The reflection efficiencies are also calculated in Fig. 6(b) for different selections of R3 . The reflection efficiency increases initially and it turns to be stabilized at about 0.95 after two periods of fluctuations. Total reflection exits when R3 is larger than 0.35a. For the sound intensity in Fig. 6(a), two peaks exist when R3 is 0.344a which indicates that two reflected acoustic beams reach B-boundary. From the acoustic field distribution in Fig. 7, when the incident acoustic wave propagates to the defects, part of wave is reflected to the B-boundary by the defects. The rest part of the incident wave propagates along the defects. Then it splits into two beams pointing to B and C boundaries, respectively. Three beams of acoustic wave can be split by the proposed structure. When the ratio R2 /R3 is 0.942, the second reflected wave is in the left half of the B-boundary in Fig. 8(a), which indicates that the incident wave propagates downward along the defects PCs before the second reflection in Fig. 8(c). However,
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Fig. 8. Sound intensities of the B and C boundaries: (a) R2 /R3 = 0.942 (b) R2 /R3 = 0.897 The acoustic field distributions: (c) R2 /R3 = 0.942 (d) R2 /R3 = 0.897.
Fig. 9. Sound intensities of the B and C boundaries: (a) cr = 200 m/s (b) cr = 400 m/s (c) cr = 600 m/s; The acoustic field distributions when R2 = 0.458a: (d) cr = 200 m/s (e) cr = 400 m/s (f) cr = 600 m/s.
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intensities of the transmitted beam and reflected beam, as shown in Fig. 9(c). 3.2. Acoustic beam controlling by splitting structure for two sound sources
Fig. 10. The acoustic field distribution for two sound sources when R2 = 0.4a and R3 = 0.42a.
the second reflected wave is in the right half of the B-boundary in Fig. 8(b) when the ratio R2 /R3 is 0.897. Incident beam propagates upward along the defects before the second reflection from Fig. 8(d). When the ratio R2 /R3 is 0.897, the intensity of the transmitted beam in C boundary is much smaller which indicates that a majority of the wave energy of the incident acoustic beam is reflected. The propagation of the incident wave can be affected by the acoustic longitudinal speed of the defect coating layer. Three cases with different acoustic longitudinal speeds, cr = 200 m/s, 400 m/s and 600 m/s, are selected to study the effects of the acoustic longitudinal speed. R3 is fixed to be 0.64a. As shown in Fig. 9, when R2 is 0.458a, the transmitted beam intensity in 200 m/s is much lower than 400 m/s and it almost keeps constant when cr increases from 400 m/s to 600 m/s. The intensity of the reflected beam is higher than the transmitted beam intensity when cr is 200 m/s. This means the reflected acoustic beam can be obviously found in this case. The sound intensities of the transmitted beam and reflected beam are almost equal when R2 slightly decreases to 0.457a, as shown in Fig. 9(a). When the acoustic longitudinal speed is increased to 400 m/s, R2 need to be decreased to 0.446a to realize equal sound intensities of the transmitted beam and reflected beam, as shown in Fig. 9(b). When the acoustic longitudinal speed is further increased to 600 m/s, R2 need to be decreased to 0.42a to realize equal sound
By adjusting the radii of the defects composed of three-component PCs, the splitting structure can control the propagation of the incident acoustic wave generated by one sound source. When two sources are given, it is also necessary to control the outgoing wave. For the splitting structure with R2 = 0.4a and R3 = 0.42a, two sound sources send sound waves to the defect three-component PCs. The first sound source S1 is placed in the left of the structure. The second sound source S2 is at the upper side of the structure. As shown in Fig. 10, two same intensity wave beams are ejected to B-boundary and C-boundary, respectively. Both the two acoustic beams can be reflected by the introduced defects composed of three-component PCs. The sound intensity curves corresponding to one sound source and two sound sources are shown in Fig. 11. The sound intensities are shown in Fig. 11(a) when only S1 exits. It can be seen that the sound intensity of the transmitted acoustic beam is higher than the reflected acoustic beam. However, the wave beams reaching B-boundary/C-boundary are the superposition of both the reflected wave beam and the transmitted beam from S1 and S2. As shown in Fig. 11(b), the sound intensities reaching B and C boundaries are equal when S1 and S2 exist. When S1 keeps constant, the intensities of the outgoing beams in B and C boundaries are adjusted by changing of the phase position of S2. As shown in Fig. 12, when θ decreases from 0 to – π /2, the intensity of the reflected beam in B-boundary tends to be larger and the intensity of the transmitted beam in C-boundary becomes smaller. The wave beam reaching the B-boundary becomes weaker when θ increases from 0 to π /2. Instead, the wave beam reaching C-boundary becomes increased when θ increases. As shown in Fig. 13(a), when the phase position of S2 is selected to be θ = π /2, a majority of the wave energy of the two incident acoustic beams propagates along x-direction (horizontal direction) when the two incident beams meet at the middle of the defect PCs. The structure behaves like a polarization device. By the change of the phase difference of the two incident waves, the intensities of the acoustic beams along x-direction and y-direction (vertical direction) can be controlled. 4. Conclusions In this work, three-component phononic crystals line-defects composed of aluminum, rubber and water are introduced to
Fig. 11. Sound intensities of the B and C boundaries: (a) one sound source (b) two sound sources.
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Fig. 12. Variation of sound intensities with different phase of S2: (a) B-boundary (b) C-boundary.
Fig. 13. The acoustic field distributions for two sound sources: (a) θ = π /2 (b) θ = −π /2.
the aluminum pillars arrayed in water to control the acoustic beam. The propagation properties of self-collimation-based acoustic waves are investigated by finite element method in the designed PC structure. Results reveal that the line-defects can lead to two splitting wave beams. In certain conditions, the wave can be split to three beams. This is the new feature of the proposed splitting structure. The rod radius and the coating layer radius in the line defects are two key factors for the efficiency of the beam splitter. When the acoustic speed of the coating layer changes, acoustic wave propagation is influenced. The proposed structure can be altered like a polarization device by controlling the phase difference of the two incident sound sources. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 11572074). References [1] M. Maldovan, Sound and heat revolutions in phononics, Nature 503 (2013) 209–217.
[2] J.J. Chen, K.W. Zhang, J. Gao, J.C. Cheng, Stopbands for lower-order Lamb waves in one-dimensional composite thin plates, Phys. Rev. B 73 (2006) 094307. [3] R. Zhu, G.L. Huang, H.H. Huang, C.T. Sun, Experimental and numerical study of guided wave propagation in a thin metamaterial plate, Phys. Lett. A 375 (2011) 2863–2867. [4] Z. Zhang, X.K. Han, G.M. Ji, Mechanism for controlling the band gap and the flat band in three-component phononic crystals, J. Phys. Chem. Solids 123 (2018) 235–241. [5] X.K. Han, Z. Zhang, Topological optimization of phononic crystal thin plate, Sci. Rep. 9 (2019) 8331. [6] A. Arreola-Lucas, J.A. Franco-Villafañe, G. Báez, R.A. Méndez-Sánchez, Inplane vibrations of a rectangular plate: Plane wave expansion modelling and experiment, J. Sound Vib. 342 (2015) 168–176. [7] Y. Lai, X.D. Zhang, Z.Q. Zhang, Large sonic band gaps in 12-fold quasicrystals, J. Appl. Phys. 91 (2002) 6191–6193. [8] J. Mei, G.C. Ma, M. Yang, Z.Y. Yang, W.J. Wen, P. Sheng, Dark acoustic metamaterials as super absorbers for low-frequency sound, Nature Commun. 3 (2012) 756. [9] Z. Zhang, X.K. Han, A new hybrid phononic crystal in low frequencies, Phys. Lett. A 380 (2016) 3766–3772. [10] M. Hirsekorn, Small-size sonic crystals with strong attenuation bands in the audible frequency range, Appl. Phys. Lett. 84 (2004) 3364–3366. [11] X.W. Xue, P. Li, F. Jin, Refraction behavior investigation and focusing control of phononic crystals under external magnetic fields, Ultrasonics 92 (2019) 261–266. [12] M. Kurosu, D. Hatanaka, K. Onomitsu, H. Yamaguchi, On-chip emporal focusing of elastic waves in a phononic crystal waveguide, Nature Commun. 9 (2018) 1331. [13] X.F. Yu, S.H. Fan, Bands and splitters for self-collimated beams in photonic crystals, Appl. Phys. Lett. 83 (2003) 3251–3253. [14] D.W. Prather, S.Y. Shi, J. Murakowski, G.J. Schneider, A. Sharkawy, C.H. Chen, B.L. Miao, R. Martin, J. Phys. D: Appl. Phys. 40 (2007) 2635–2651.
Please cite this article as: X.K. Han and Z. Zhang, Acoustic beam controlling in water by the design of phononic crystal, Extreme Mechanics Letters (2019) 100602, https://doi.org/10.1016/j.eml.2019.100602.
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X.K. Han and Z. Zhang / Extreme Mechanics Letters xxx (xxxx) xxx
[15] H.L. He, C.Y. Qiu, L.P. Ye, X.X. Cai, X.Y. Fan, M.Z. Ke, F. Zhang, Z.Y. Liu, Nature 560 (2018) 61–64. [16] Z.L. Hou, H.Q. Ni, B. Assouar, PT-symmetry for elastic negative refraction, Phys. Rev. Appl. 10 (2018) 044071. [17] J.H. Wen, D.L. Yu, L. Cai, X.S. Wen, Acoustic directional radiation operating at the pass band frequency in two-dimensional phononic crystals, J. Phys. D: Appl. Phys. 42 (2009) 115417. [18] P.S. Ma, H.W. Kim, J.H. Oh, Y.Y. Kim, Mode separation of a single-frequency bi-modal elastic wave pulse by a phononic crystal, Appl. Phys. Lett. 99 (2011) 201906. [19] A. Sukhovich, L. Jing, J.H. Page, Negative refraction and focusing of ultrasound in two-dimensional phononic crystal, Phys. Rev. B 77 (2008) 014301. [20] J.F. Zhao, R. Marchal, B. Bonello, O. Boyko, Efficient focalization of antisymmetric Lamb waves in gradient-index phononic crystal plates, Appl. Phys. Lett. 101 (2012) 261905. [21] E.M. Hamham, N. Jiménez, R. Picó, V.J. Sánchez-Morcillo, L.M. García-Raffi, Nonlinear self-collimated sound beams in sonic crystals, Phys. Rev. B 92 (2015) 054302. [22] I. Pérez-Arjona, V.J. Sánchez-Morcillo, J. Redondo, V. Espinosa, K. Staliunas, Theoretical prediction of the nondiffractive propagation of sonic waves through periodic acoustic media, Phys. Rev. B 75 (2007) 014304. [23] D.W. Prather, S.Y. Shi, J. Murakowski, G.J. Schneider, A. Sharkawy, C.H. Chen, B.L. Miao, R. Martin, Self-collimation in photonic crystal structures: a new paradigm for applications and device development, J. Phys. D: Appl. Phys. 40 (2007) 2635–2651. [24] S.Y. Shi, A. Sharkawy, C.H. Chen, D.M. Pustai, D.W. Prather, Dispersionbased beam splitter in photonic crystals, Opt. Lett. 29 (2004) 617–619. [25] X.F. Yu, S.H. Fan, Bends and splitters for self-collimated beams in photonic crystals, Appl. Phys. Lett. 83 (2003) 3251–3253.
[26] C.Y. Chiang, P.G. Luan, Imaging off-plane shear waves with a twodimensional phononic crystal lens, J. Phys.: Condens. Matter 22 (2010) 055405. [27] A. Cicek, O.A. Kaya, B. Ulug, Acoustic beam splitting in a sonic crystal around a directional band gap, Chin. Phys. B 22 (2013) 114301. [28] T. Zhang, Y. Cheng, J.Z. Guo, J.Y. Xu, X.J. Liu, Acoustic logic gates and boolean operation based on self-collimating acoustic beams, Appl. Phys. Lett. 106 (2015) 113503. [29] Y.N. Guo, D. Brick, M. Großmann, M. Hettich, T. Dekorsy, Acoustic beam splitting at low GHz frequencies in a defect-free phononic crystal, Appl. Phys. Lett. 110 (2017) 031904. [30] S. Xie, S.L. Ouyang, Z.J. He, X.Y. Wang, K. Deng, H.P. Zhao, Bending and splitting of spoof surface acoustic waves through structured rigid surface, Results Phys. 8 (2018) 52–56. [31] H. Lee, J.H. Oh, Y.Y. Kim, Multiple beam splitting in elastic phononic crystal plates, Ultrasonics 56 (2015) 178–182. [32] O.A. Kaya, A. Cicek, A. Salman, B. Ulug, Acoustic Mach–Zehnder interferometer utilizing self-collimated beams in a two-dimensional phononic crystal, Sensors Actuators B 203 (2014) 197–203. [33] J. Li, F.G. Wu, H.L. Zhong, et al., Acoustic beam splitting in two-dimensional phononic crystals using self-collimation effect, J. Appl. Phys. 118 (2015) 144903. [34] Z. Zhang, G.M. Ji, X.K. Han, Optimization design of a novel zigzag lattice phononic crystal with holes, Internat. J. Modern Phys. B 33 (2019) 1950124. [35] Z. Zhang, X.K. Han, G.M. Ji, The band gap controlling by geometrical symmetry design in hybrid phononic crystal, Internat. J. Modern Phys. B 32 (2018) 1850034. [36] Y.L. Huang, J. Li, W.Q. Chen, R.H. Bao, Tunable bandgaps in soft phononic plates with spring-mass-like resonators, Int. J. Mech. Sci. 151 (2019) 300–313.
Please cite this article as: X.K. Han and Z. Zhang, Acoustic beam controlling in water by the design of phononic crystal, Extreme Mechanics Letters (2019) 100602, https://doi.org/10.1016/j.eml.2019.100602.