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Lamb waves in two-dimensional phononic crystal plate with anisotropic inclusions
Ultrasonics 51 (2011) 602–605
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Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Lamb waves in two-dimensional phononic crystal plate with anisotropic inclusions Yuanwei Yao a,⇑, Fugen Wu b, Zhilin Hou c, Zhang Xin a a
Department of Physics, Guangdong University of Technology, Guangzhou 510006, China Experiment and Educational Center, Guangdong University of Technology, Guangzhou 510006,China c Department of Physics, South China University of Technology, Guangzhou 510640, China b
a r t i c l e
i n f o
Article history: Received 9 October 2010 Received in revised form 28 December 2010 Accepted 30 December 2010 Available online 7 January 2011 Keywords: Lamb wave Band gap Anisotropy
a b s t r a c t An analysis is given to the band structure of the two-dimensional phononic crystal plate constituted of a square array of elastic anisotropic, circular Pb cylinders embedded in elastic isotropic epoxy. The numerical results show that the band gap can be tuned by rotating the anisotropic material orientation. It is found that the influence of anisotropy on band gap of Lamb wave is clearly different from that on the band gap of bulk waves. The thickness of the system under study is a sensitive parameter to affect the influence of anisotropic materials on the normalized gap width. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction In the past two decades, phononic band gap materials, called phononic crystals, have inspired great interest of researchers for their rich physics and potential applications. The key motivation behind the proposal of phononic crystals is the possibility of modifying the propagation of acoustic or elastic waves by creating phononic band gaps in the band structure of the synthetic periodic structures. Since the phononic band gap is a major feature of a phononic crystal, the techniques for enlarging and tuning frequency band gap have become an important research topic in the field of physics. Several numerical methods, such as the plane-waveexpansion method (PWE) [1,2], the multi-scattering theory (MST) [3,4], and the finite-difference time-domain [5,6], have been developed and extensively used to analyze the frequency band structure. The earlier theoretical and experimental studies have demonstrated that the bandwidth of the forbidden band depended on the contrast between the physical characteristics (mass density and elastic moduli) of both materials and the filling fraction of inclusions [7,8]. In addition to the factors above, the band gap can be enlarged and tuned by decreasing the symmetry of the system [9,10]. Anisotropic materials have different physical properties in different directions relative to the crystal orientation of the materials. Some interesting phenomena have been observed and studied by introducing anisotropic materials into infinite or semi-infinite phononic crystal. Hou et al. [11] have studied the 2D anisotropic ⇑ Corresponding author. Tel./fax: +86 20 39322265. E-mail address: [email protected] (Y. Yao). 0041-624X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2010.12.016
phononic crystal composed of parallel Pb (Lead) cylinders embedded in the uniform matrix (Poly-vinyl-chloride). They have found that the band gap structure varied with the rotating angle of these parallel cylinders. The band structure of the systems with higher filling fractions were more sensitive to the anisotropic material orientation than the lower filling fraction systems. Wilm et al. [12] used a plane-wave-expansion method to compute the band structure of an anisotropic infinite square array of parallel quartz rods embedded in an epoxy matrix and found that the band structure of out-of-plane was changed with the values of the wave-vector component parallel to the rods. Wu et al. [13] have investigated the property of the surface acoustic wave propagating in the system composed of elastic anisotropy materials. They have found that the mode exchanged suddenly around the sharp bend area where some of the crossing over of the dispersion curves was indeed sharp bends. More recently, theoretical and experimental studies of the band structure of Lamb waves in a finite system have been reported [14–19]. The numerical results have revealed that the band structure of Lamb wave in an two-dimensional finite phononic crystal was entirely different from that of the infinite phononic crystal with same geometry and composition [20]. The anisotropy materials have also been introduced into finite phononic crystal plate by the authors [19,20], but they have not presented relationship between anisotropic materials and band structure of phononic crystal plate. In this paper, band gap characteristic of the Lamb waves with different rotating angle of anisotropic inclusions is calculated and discussed by using the supercell plane-wave-expansion method. The different influence of anisotropy on the band gap of twodimensional phononic crystal plate, as is compared with those of infinite phononic crystal, is also analyzed.
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Y. Yao et al. / Ultrasonics 51 (2011) 602–605
U j ðr; zÞ ¼
2. Formulations
XX Gjj
The considered system with thickness h is shown in Fig. 1a. The anisotropic cylindrical fillers (Pb) are embedded in the host material (Epoxy) in square lattice array with lattice constant a. The phononic crystal coordinate system(global coordinate system) and the materials coordinate system (local coordinate system) are taken to be x, y, z and x0 , y0 , z0 , respectively. For simplicity, the z-axis and z0 axis are kept parallel with the direction of thickness. The elastic stiffness is presented by components cmnrelated to the global coordinate axes and by component c0 mn related to the material coordinate axes. From the theory of transformation, the stiffness transformation law is
½c ¼ M½c0 M t
ð1Þ
where M is the Bond stress transformation matrix, Mt is its transpose. If the rotation angle is h, the stiffness matrix c has the form
0
c011 b B c0 þ b B 12 B 0 B c12 ½c ¼ B B0 B B @0 a
1
c012 þ b c012
0
0
a
c011 c012
c012 c011
0
0
a
0
0
0
0
0
c044
0
0
0
0 0
0 0
c044 0
0 c044 þ b
b
a
C C C C C C C C A
a ¼ dðsin4hÞ=4; b ¼ dð1 cos4hÞ=4:
ð2Þ
In the absence of an external force, the equation of lamb waves motion for displacement vector can be written as 2
q
tial U i tialt
2
¼ ðC ijkl U k;l Þ; j
ð3Þ
and the stress-free boundary conditions on two free surface is
T zj ¼ C zjkl U k;l ¼ 0
ð4Þ
where i, j, k, l = x, y, z, respectively, Tzj is the stress vector, Ui is the elastic displacement vector, q is the mass density and cijkl is the elastic stiffness. The stiffness constant cijkl can be described as cmn in abbreviated subscript notation. The supercell plane wave method have been successfully used in two-dimensional phononic crystal plate [19,21]. The basic idea of the method is that two imaginary ‘‘vacuum’’ layers are added on the outer surfaces of the system to satisfy the stress-free boundary condition automatically. This is more effective to search roots of plate waves because it does not require the boundary conditions on the free surface explicitly. The supercell of the studied system is shown in Fig. 1b. based on the idea of the supercell plane-waveexpansion method. The displacement can be expressed as:
(a)
y
ð5Þ
and elastic parameters are expressed as
nðr; zÞ ¼
XX Gjj
nGjj ;Gz eiGjj r eiGz z
ð6Þ
Gz
where, Gallel = (Gx, Gy) with Gx = 2nxp/a, Gy = 2nyp/a, Gz = 2np/ ty(n = 0, ±1, ±2, . . . , ±1) with ty = 2tv + h. Substitution of Eqs. (5) and (6) into Eq. (3) yields a general eigenvalue equation about the frequency x
0
0
1
1 B12 B13 B11 qGG0 0 0 G;G0 G;G0 G;G0 B C C 0 B 21 22 23 2B qGG0 0 x @0 AU G ¼ @ BG;G0 BG;G0 BG;G0 C AU G0 31 32 33 0 0 qGG0 BG;G0 BG;G0 BG;G0
where U G0 ¼ ðU xG0 ; U yG0 ; U zG0 ÞT and qGG0 is the diagonal matrix. An detail expression of the sub-matrix BijG;G0 (i, j = 1, 2, 3) can be found in Ref. [21].
In this section, we use the supercell plane wave method to calculate the band structure of the phononic crystal plates illustrated in Fig. 1. In the system, the single crystal Pb belonging to the cubic crystal system with c011 ¼ 5:03 1010 dyn=cm2 ; c012 ¼ 3. 93 1010 dyn=cm2 ; c044 ¼ 1:40 1010 dyn=cm2 , and qpb = 11,340 kg/m3 is selected as the material of the cylinder. The isotropic material epoxy with c011 ¼ 0:758 1010 dyn=cm2 ; c012 ¼ 0. 442 1010 dyn=cm2 ; c044 ¼ 0:158 1010 dyn=cm2 and qepoxy = 1180 kg/m3 is chosen as the host material. To ensure the converge of the numerical calculation, 1125 plane waves(15 15 5) are used. Fig. 2 shows the band structures of the system with h = 0.8a, f = p(r/ a)2 = 0.55 for two different rotating angles h = 0 and h = 25. It can be seen that no absolute band gap occurs in Fig. 2a, but an absolute band gap exists in Fig. 2b for a rotating angle of 25°. The difference between the two band structures demonstrates that the band gap of Lamb waves can be tuned by rotating anisotropic materials. The complete evolution of the normalized gap width Dx/xg of the lowest absolute band gap with the h values is shown in Fig. 3, where xg is the mid-gap value. In the figure, only the numerical results with h 6 45° are reported for the symmetric property. We presents the complete evolution for the case of two-dimensional bulk wave phononic crystal in panel (a) of Fig. 3. As one can see, the normalized gap width, which is zero when h < 10°, begins to increase progressively with the increase of rotating angle when h = 10°. It reaches the maximum value at h = 25° and decreases gradually when h > 25°. The normalized gap width shows a nonlinear dependence on the angle of the anisotropic material orienta-
(b)
y
tv
X
h
θ
tv
z y x
x r
a
ð7Þ
3. Numerical results and discussion
where a, b are defined as
d ¼ c011 c012 2c044 ;
U Gjj ;Gz eiðkjj þGjj Þr eiGz z
Gz
h
Fig. 1. Cross-section of a two-dimensional phononic crystal plate. (a) A projection in the xy-plane. (b) An unit cell of the system and a projection in xz-plane.
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Y. Yao et al. / Ultrasonics 51 (2011) 602–605
0.8
(b)
0.8
0.6
0.4
0.4
ω
⁄⁄2 π
0.6
ω
⁄⁄2 π
(a)
0.2
0.0
0.2
Γ
Χ
Γ
Μ
0.0
Γ
Χ
Μ
Γ
Fig. 2. Band structure of the two-dimensional phononic crystal plate with square lattice. The plate consists of pb cylinder and epoxy. The filling fraction and plate thickness are f = 0.55 and h = 0.8a, respectively. (a) For h = 0°. (b) For h = 25°. Shade area shows phononic band gap.
(a)
(b)
0.20
(c)
0.20
0.20
'
C44
0.15
0.10
0.10
0.10
0.05
Δω ⁄ω
0.15
Δω ⁄ω
Δω ⁄ω
'
0.15
0.05
0.00
Rotation angle θ
'
C11
0.05
0.00 0 5 10 15 20 25 30 35 40 45
C12
0 5 10 15 20 25 30 35 40 45
0.00
Rotation Angle θ
0 5 10 15 20 25 30 35 40 45
Rotation Angle θ
Fig. 3. The normalized width of lowest complete band gap as a function of rotation angle. The filling fraction of the studied system is f = 0.55. The thickness of the twodimensional phononic crystal plate is 0.8a. (a) For bulk waves. (b) For lamb waves. (c) For lamb waves with decreased d.
0.20
h0.5 h0.6 h0.7 h0.8
0.16
Δω ⁄ω
tion. Panel (b) of Fig. 3 gives the complete evolution for the twodimensional phononic crystal plate with h = 0.8a, f = 0.55. As can be easily found that the curve in panel (b) is not same as that in panel (a). For example, the rotating angles is 0° instead of 10° when the normalized gap width begins to increase. It is 35° instead of 25° when the normalized gap width reaches maximum value. A comparison of panel (a) with panel(b) shows that the effects of anisotropic materials on the band gap of Lamb waves is different from that on bulk waves. The difference is due to the fact that the mechanism for opening band gap in a phononic crystal plate is different from that in a bulk wave phononic crystal. It is well known that elastic waves in a finite structure can be scattered not only by periodically arranged scatterers but also by the surface of the plate. Panel (c) of Fig. 3 indicates the dependence of the normalized gap width of the lowest absolute band gap on a rotating angle when d is decreased by 0.2 1010 dyn/cm2. Eq. (2) in Section 2 shows that there are many ways to obtain the same decreased value of d because c011 ; c012 and c044 are independent parameters in a cubic crystal system. In the numerical simulation, among the three independent parameters, the value of one parameter is changed and that of the other two is kept the same as that in the panel (b). The three changed parameters are chosen as : c011 ¼ 4:83 1010 dyn=cm2 ; c012 ¼ 4. 13 1010 dyn=cm2 and c044 ¼ 1:5 1010 dyn=cm2 . Thus, the anisotropic factors is 3.11 for the system with c011 changed and the system with c012 changed, but it is 2.73 for the sytem with c044 changed. In the panel, the c011 curve and c012 curve are almost the same, but the c044 curve shows an entirely different behavior. A comparing of panel (c) with panel (b) shows that the behavior of complete evolution in panel (b) is different from that in panel (c). These
0.12 0.08 0.04 0.00
0
5 10 15 20 25 30 35 40 45
Rotation angle θ Fig. 4. The complete evolution of normalized gap width with anisotropic material orientation for four different thicknesses h = 0.5a, h = 0.6a, h = 0.7a, and h = 0.8a. The filling fraction of the phononic crystal plate is f = 0.55.
behaviors lead us to draw a conclusion that the band structure of two-dimensional phononic crystal plate will be changed by changing the value of d. If d is kept constant, the system with different anisotropy factor has different complete evolution. The complete evolution of normalized gap as a function of rotating angle for different plate thicknesses is shown in Fig. 4. In the calculation, the plate thicknesses are h = 0.5a, h = 0.6a, h = 0.7a and h = 0.8a, respectively. Seen from the figure, the most striking feature of this graph is that these complete evolutions are not iden-
Y. Yao et al. / Ultrasonics 51 (2011) 602–605
tical. For example, the maximum value of the normalized gap width is 0.069 for h = 0.5a, and it is 0.179 for h = 0.8a. Further more, the different systems reach their maximum value of normalized gap width at different angle values. The h is 25° for h = 0.6a and h = 0.7a while it is 35° for h = 0.8a. Besides the difference mentioned above, it is also found that the normalized band gap for h = 0.5a is closed when h = 30°, but it is still open in the other three systems. Based on these comparison above, it is clear that the anisotropic materials exert different influences on the band structure when the thickness of a phononic crystal plate is varied. 4. Brief summary In conclusion, by using supercell plane-wave-expansion method, we have made a theoretically study of the effects of anisotropic property of inclusions on the band structure of Lamb wave in a finite phononic crystal. It is found that the phononic band gap can be enlarged and reduced by the rotation of the anisotropic inclusions. The influence of anisotropic materials on band gap is quite different in the case of a slab as compared to the case of a two-dimensional infinite structure. If the plate thickness or the value of d is changed, the system will show different complete evolution of normalized gap with the anisotropic material orientation. Acknowledgement This work was supported by the National Natural Science foundation of China under Grant nos. 10747119, 11004039 and 10674032. References [1] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari- Rouhani, Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 (1993) 2022–2025. [2] M.S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, B. Djafari-Rouhani, Theory of acoustic band structure of periodic elastic composites, Phys. Rev. B 49 (1994) 2313–2322. [3] Zhengyou Liu, C.T. Chan, Ping Sheng, Elastic wave scattering by periodic structures of spherical objects: theory and experiment, Phys. Rev. B 62 (2000) 2446–2457. [4] I.E. Psarobas, N. Stefanou, Scattering of elastic waves by periodic arrays of spherical bodies, Phys. Rev. B 62 (2000) 278–291.
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[5] D. Garica-Pablos, M. Sigalas, F.R. Montero de Espinosa, M. Kafesaki, N. Garcia, Theory and experiments on elastic band gaps, Phys. Rev. Lett. 84 (2000) 4349– 4352. [6] Yongjun Cao, Zhilin Hou, Youyan Liu, Finite difference time-domain method for band-structure calculations of two-dimensional phononic crystals, Solid State Commun. 132 (2004) 39–543. [7] M.M. Sigalas, E.N. Economou, Band structure of elastic waves in two dimensional systems, Solid State Commun. 86 (1993) 141–143. [8] J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzynski, M.S. Kushwaha, P. Halevi, Complete acoustic band gaps in periodic fibre reinforced composite materials: the carbon/epoxy composite and some metallic systems, J. Phys.: Condens. Matter 6 (1994) 8759–8770. [9] J.V. Caballero, J. Sánchez-Dehesa, C. Rubio, R. Mártinez-Sala, J.V. Sánchez-Pérez, F. Meseguer, J. Llinares, Large two-dimensional sonic band gaps, Phys. Rev. E 60 (1999) R6316–R6319. [10] Xin Zhang, Zhengyou. Liu, Youyan. Liu, Fugen. Wu, Elastic wave band gaps for three-dimensional phononic crystals with two structural units, Phys. Lett. A 313 (2003) 455–460. [11] Zhilin Hou, Xiujun Fu, Youyan Liu, Acoustic wave in a two-dimensional composite medium with anisotropic inclusions, Phys. Lett. A 317 (2003) 127– 134. [12] M. Wilm, A. Khelif, S. Ballandras, V. Laude, Out-of-plane propagation of elastic waves in two-dimensional phononic band-gap materials, Phys. Rev. E 67 (2003) (065602(1)-065602(4)). [13] Tsung-Tsong Wu, Zi-Gui Huang, S. Lin, Surface and bulk acoustic waves in twodimensional phononic crystal consisting of materials with general anisotropy, Phys. Rev. B 69 (2004) (094301(1)-094301(10)). [14] Jiu-Jiu Chen, Kai-Wen Zhang, Jian Gao, Jian-Chun Cheng, Stopbands for lowerorder Lamb waves in one-dimensional composite thin plates, Phys. Rev. B 73 (2006) (094307(1)-094307(5)). [15] Jin-Chen Hsu, Tsung-Tsong Wu, Phys. Efficient formulation for band-structure calculations of two-dimensional phononic-crystal plates, Phys. Rev. B 74 (2006). 144303(1)-144303(7). [16] R. Sainidou, B. Djafari-Rouhani, J.O. Vasseur, Surface acoustic waves in finite slabs of three-dimensional phononic crystals, Phys. Rev. B 77 (2008) (094304(1)-094304(9)). [17] Xinya Zhang, Ted Jackson, Emmanuel Laford, Pierre Deymier, Jerome Vasseur, Evidence of surface acoustic wave band gaps in the phononic crystals created on thin plates, Appl. Phys. Lett. 88 (2006) (041911(1)-041911(3)). [18] C. Charles, B. Bonello, F. Ganot, Propagation of guided elastic waves in 2D phononic crystals, Ultrasonics 44 (2006) e1209–e1213. [19] J.O. Vasseur, P.A. Deymier, B. Djafari-Rouhani, Y. Pennec, A.-C. HladkyHennion, Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates, Phys. Rev. B 77 (2008) (085415(1)-085415(15)). [20] A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, V. Laude, Complete band gaps in two-dimensional phononic crystal slabs, Phys. Rev. E 74 (2006) (046610(1)046610(5)). [21] Zhilin Hou, Badreddine M. Assouar, Modeling of Lamb wave propagation in plate with two-dimensional phononic crystal layer coated on uniform substrate using plane-wave-expansion method, Phys. Lett. A 372 (2008) 2091–2097.
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Lamb waves in two-dimensional phononic crystal plate with anisotropic inclusions Yuanwei Yao a,⇑, Fugen Wu b, Zhilin Hou c, Zhang Xin a a
Department of Physics, Guangdong University of Technology, Guangzhou 510006, China Experiment and Educational Center, Guangdong University of Technology, Guangzhou 510006,China c Department of Physics, South China University of Technology, Guangzhou 510640, China b
a r t i c l e
i n f o
Article history: Received 9 October 2010 Received in revised form 28 December 2010 Accepted 30 December 2010 Available online 7 January 2011 Keywords: Lamb wave Band gap Anisotropy
a b s t r a c t An analysis is given to the band structure of the two-dimensional phononic crystal plate constituted of a square array of elastic anisotropic, circular Pb cylinders embedded in elastic isotropic epoxy. The numerical results show that the band gap can be tuned by rotating the anisotropic material orientation. It is found that the influence of anisotropy on band gap of Lamb wave is clearly different from that on the band gap of bulk waves. The thickness of the system under study is a sensitive parameter to affect the influence of anisotropic materials on the normalized gap width. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction In the past two decades, phononic band gap materials, called phononic crystals, have inspired great interest of researchers for their rich physics and potential applications. The key motivation behind the proposal of phononic crystals is the possibility of modifying the propagation of acoustic or elastic waves by creating phononic band gaps in the band structure of the synthetic periodic structures. Since the phononic band gap is a major feature of a phononic crystal, the techniques for enlarging and tuning frequency band gap have become an important research topic in the field of physics. Several numerical methods, such as the plane-waveexpansion method (PWE) [1,2], the multi-scattering theory (MST) [3,4], and the finite-difference time-domain [5,6], have been developed and extensively used to analyze the frequency band structure. The earlier theoretical and experimental studies have demonstrated that the bandwidth of the forbidden band depended on the contrast between the physical characteristics (mass density and elastic moduli) of both materials and the filling fraction of inclusions [7,8]. In addition to the factors above, the band gap can be enlarged and tuned by decreasing the symmetry of the system [9,10]. Anisotropic materials have different physical properties in different directions relative to the crystal orientation of the materials. Some interesting phenomena have been observed and studied by introducing anisotropic materials into infinite or semi-infinite phononic crystal. Hou et al. [11] have studied the 2D anisotropic ⇑ Corresponding author. Tel./fax: +86 20 39322265. E-mail address: [email protected] (Y. Yao). 0041-624X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2010.12.016
phononic crystal composed of parallel Pb (Lead) cylinders embedded in the uniform matrix (Poly-vinyl-chloride). They have found that the band gap structure varied with the rotating angle of these parallel cylinders. The band structure of the systems with higher filling fractions were more sensitive to the anisotropic material orientation than the lower filling fraction systems. Wilm et al. [12] used a plane-wave-expansion method to compute the band structure of an anisotropic infinite square array of parallel quartz rods embedded in an epoxy matrix and found that the band structure of out-of-plane was changed with the values of the wave-vector component parallel to the rods. Wu et al. [13] have investigated the property of the surface acoustic wave propagating in the system composed of elastic anisotropy materials. They have found that the mode exchanged suddenly around the sharp bend area where some of the crossing over of the dispersion curves was indeed sharp bends. More recently, theoretical and experimental studies of the band structure of Lamb waves in a finite system have been reported [14–19]. The numerical results have revealed that the band structure of Lamb wave in an two-dimensional finite phononic crystal was entirely different from that of the infinite phononic crystal with same geometry and composition [20]. The anisotropy materials have also been introduced into finite phononic crystal plate by the authors [19,20], but they have not presented relationship between anisotropic materials and band structure of phononic crystal plate. In this paper, band gap characteristic of the Lamb waves with different rotating angle of anisotropic inclusions is calculated and discussed by using the supercell plane-wave-expansion method. The different influence of anisotropy on the band gap of twodimensional phononic crystal plate, as is compared with those of infinite phononic crystal, is also analyzed.
603
Y. Yao et al. / Ultrasonics 51 (2011) 602–605
U j ðr; zÞ ¼
2. Formulations
XX Gjj
The considered system with thickness h is shown in Fig. 1a. The anisotropic cylindrical fillers (Pb) are embedded in the host material (Epoxy) in square lattice array with lattice constant a. The phononic crystal coordinate system(global coordinate system) and the materials coordinate system (local coordinate system) are taken to be x, y, z and x0 , y0 , z0 , respectively. For simplicity, the z-axis and z0 axis are kept parallel with the direction of thickness. The elastic stiffness is presented by components cmnrelated to the global coordinate axes and by component c0 mn related to the material coordinate axes. From the theory of transformation, the stiffness transformation law is
½c ¼ M½c0 M t
ð1Þ
where M is the Bond stress transformation matrix, Mt is its transpose. If the rotation angle is h, the stiffness matrix c has the form
0
c011 b B c0 þ b B 12 B 0 B c12 ½c ¼ B B0 B B @0 a
1
c012 þ b c012
0
0
a
c011 c012
c012 c011
0
0
a
0
0
0
0
0
c044
0
0
0
0 0
0 0
c044 0
0 c044 þ b
b
a
C C C C C C C C A
a ¼ dðsin4hÞ=4; b ¼ dð1 cos4hÞ=4:
ð2Þ
In the absence of an external force, the equation of lamb waves motion for displacement vector can be written as 2
q
tial U i tialt
2
¼ ðC ijkl U k;l Þ; j
ð3Þ
and the stress-free boundary conditions on two free surface is
T zj ¼ C zjkl U k;l ¼ 0
ð4Þ
where i, j, k, l = x, y, z, respectively, Tzj is the stress vector, Ui is the elastic displacement vector, q is the mass density and cijkl is the elastic stiffness. The stiffness constant cijkl can be described as cmn in abbreviated subscript notation. The supercell plane wave method have been successfully used in two-dimensional phononic crystal plate [19,21]. The basic idea of the method is that two imaginary ‘‘vacuum’’ layers are added on the outer surfaces of the system to satisfy the stress-free boundary condition automatically. This is more effective to search roots of plate waves because it does not require the boundary conditions on the free surface explicitly. The supercell of the studied system is shown in Fig. 1b. based on the idea of the supercell plane-waveexpansion method. The displacement can be expressed as:
(a)
y
ð5Þ
and elastic parameters are expressed as
nðr; zÞ ¼
XX Gjj
nGjj ;Gz eiGjj r eiGz z
ð6Þ
Gz
where, Gallel = (Gx, Gy) with Gx = 2nxp/a, Gy = 2nyp/a, Gz = 2np/ ty(n = 0, ±1, ±2, . . . , ±1) with ty = 2tv + h. Substitution of Eqs. (5) and (6) into Eq. (3) yields a general eigenvalue equation about the frequency x
0
0
1
1 B12 B13 B11 qGG0 0 0 G;G0 G;G0 G;G0 B C C 0 B 21 22 23 2B qGG0 0 x @0 AU G ¼ @ BG;G0 BG;G0 BG;G0 C AU G0 31 32 33 0 0 qGG0 BG;G0 BG;G0 BG;G0
where U G0 ¼ ðU xG0 ; U yG0 ; U zG0 ÞT and qGG0 is the diagonal matrix. An detail expression of the sub-matrix BijG;G0 (i, j = 1, 2, 3) can be found in Ref. [21].
In this section, we use the supercell plane wave method to calculate the band structure of the phononic crystal plates illustrated in Fig. 1. In the system, the single crystal Pb belonging to the cubic crystal system with c011 ¼ 5:03 1010 dyn=cm2 ; c012 ¼ 3. 93 1010 dyn=cm2 ; c044 ¼ 1:40 1010 dyn=cm2 , and qpb = 11,340 kg/m3 is selected as the material of the cylinder. The isotropic material epoxy with c011 ¼ 0:758 1010 dyn=cm2 ; c012 ¼ 0. 442 1010 dyn=cm2 ; c044 ¼ 0:158 1010 dyn=cm2 and qepoxy = 1180 kg/m3 is chosen as the host material. To ensure the converge of the numerical calculation, 1125 plane waves(15 15 5) are used. Fig. 2 shows the band structures of the system with h = 0.8a, f = p(r/ a)2 = 0.55 for two different rotating angles h = 0 and h = 25. It can be seen that no absolute band gap occurs in Fig. 2a, but an absolute band gap exists in Fig. 2b for a rotating angle of 25°. The difference between the two band structures demonstrates that the band gap of Lamb waves can be tuned by rotating anisotropic materials. The complete evolution of the normalized gap width Dx/xg of the lowest absolute band gap with the h values is shown in Fig. 3, where xg is the mid-gap value. In the figure, only the numerical results with h 6 45° are reported for the symmetric property. We presents the complete evolution for the case of two-dimensional bulk wave phononic crystal in panel (a) of Fig. 3. As one can see, the normalized gap width, which is zero when h < 10°, begins to increase progressively with the increase of rotating angle when h = 10°. It reaches the maximum value at h = 25° and decreases gradually when h > 25°. The normalized gap width shows a nonlinear dependence on the angle of the anisotropic material orienta-
(b)
y
tv
X
h
θ
tv
z y x
x r
a
ð7Þ
3. Numerical results and discussion
where a, b are defined as
d ¼ c011 c012 2c044 ;
U Gjj ;Gz eiðkjj þGjj Þr eiGz z
Gz
h
Fig. 1. Cross-section of a two-dimensional phononic crystal plate. (a) A projection in the xy-plane. (b) An unit cell of the system and a projection in xz-plane.
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Y. Yao et al. / Ultrasonics 51 (2011) 602–605
0.8
(b)
0.8
0.6
0.4
0.4
ω
⁄⁄2 π
0.6
ω
⁄⁄2 π
(a)
0.2
0.0
0.2
Γ
Χ
Γ
Μ
0.0
Γ
Χ
Μ
Γ
Fig. 2. Band structure of the two-dimensional phononic crystal plate with square lattice. The plate consists of pb cylinder and epoxy. The filling fraction and plate thickness are f = 0.55 and h = 0.8a, respectively. (a) For h = 0°. (b) For h = 25°. Shade area shows phononic band gap.
(a)
(b)
0.20
(c)
0.20
0.20
'
C44
0.15
0.10
0.10
0.10
0.05
Δω ⁄ω
0.15
Δω ⁄ω
Δω ⁄ω
'
0.15
0.05
0.00
Rotation angle θ
'
C11
0.05
0.00 0 5 10 15 20 25 30 35 40 45
C12
0 5 10 15 20 25 30 35 40 45
0.00
Rotation Angle θ
0 5 10 15 20 25 30 35 40 45
Rotation Angle θ
Fig. 3. The normalized width of lowest complete band gap as a function of rotation angle. The filling fraction of the studied system is f = 0.55. The thickness of the twodimensional phononic crystal plate is 0.8a. (a) For bulk waves. (b) For lamb waves. (c) For lamb waves with decreased d.
0.20
h0.5 h0.6 h0.7 h0.8
0.16
Δω ⁄ω
tion. Panel (b) of Fig. 3 gives the complete evolution for the twodimensional phononic crystal plate with h = 0.8a, f = 0.55. As can be easily found that the curve in panel (b) is not same as that in panel (a). For example, the rotating angles is 0° instead of 10° when the normalized gap width begins to increase. It is 35° instead of 25° when the normalized gap width reaches maximum value. A comparison of panel (a) with panel(b) shows that the effects of anisotropic materials on the band gap of Lamb waves is different from that on bulk waves. The difference is due to the fact that the mechanism for opening band gap in a phononic crystal plate is different from that in a bulk wave phononic crystal. It is well known that elastic waves in a finite structure can be scattered not only by periodically arranged scatterers but also by the surface of the plate. Panel (c) of Fig. 3 indicates the dependence of the normalized gap width of the lowest absolute band gap on a rotating angle when d is decreased by 0.2 1010 dyn/cm2. Eq. (2) in Section 2 shows that there are many ways to obtain the same decreased value of d because c011 ; c012 and c044 are independent parameters in a cubic crystal system. In the numerical simulation, among the three independent parameters, the value of one parameter is changed and that of the other two is kept the same as that in the panel (b). The three changed parameters are chosen as : c011 ¼ 4:83 1010 dyn=cm2 ; c012 ¼ 4. 13 1010 dyn=cm2 and c044 ¼ 1:5 1010 dyn=cm2 . Thus, the anisotropic factors is 3.11 for the system with c011 changed and the system with c012 changed, but it is 2.73 for the sytem with c044 changed. In the panel, the c011 curve and c012 curve are almost the same, but the c044 curve shows an entirely different behavior. A comparing of panel (c) with panel (b) shows that the behavior of complete evolution in panel (b) is different from that in panel (c). These
0.12 0.08 0.04 0.00
0
5 10 15 20 25 30 35 40 45
Rotation angle θ Fig. 4. The complete evolution of normalized gap width with anisotropic material orientation for four different thicknesses h = 0.5a, h = 0.6a, h = 0.7a, and h = 0.8a. The filling fraction of the phononic crystal plate is f = 0.55.
behaviors lead us to draw a conclusion that the band structure of two-dimensional phononic crystal plate will be changed by changing the value of d. If d is kept constant, the system with different anisotropy factor has different complete evolution. The complete evolution of normalized gap as a function of rotating angle for different plate thicknesses is shown in Fig. 4. In the calculation, the plate thicknesses are h = 0.5a, h = 0.6a, h = 0.7a and h = 0.8a, respectively. Seen from the figure, the most striking feature of this graph is that these complete evolutions are not iden-
Y. Yao et al. / Ultrasonics 51 (2011) 602–605
tical. For example, the maximum value of the normalized gap width is 0.069 for h = 0.5a, and it is 0.179 for h = 0.8a. Further more, the different systems reach their maximum value of normalized gap width at different angle values. The h is 25° for h = 0.6a and h = 0.7a while it is 35° for h = 0.8a. Besides the difference mentioned above, it is also found that the normalized band gap for h = 0.5a is closed when h = 30°, but it is still open in the other three systems. Based on these comparison above, it is clear that the anisotropic materials exert different influences on the band structure when the thickness of a phononic crystal plate is varied. 4. Brief summary In conclusion, by using supercell plane-wave-expansion method, we have made a theoretically study of the effects of anisotropic property of inclusions on the band structure of Lamb wave in a finite phononic crystal. It is found that the phononic band gap can be enlarged and reduced by the rotation of the anisotropic inclusions. The influence of anisotropic materials on band gap is quite different in the case of a slab as compared to the case of a two-dimensional infinite structure. If the plate thickness or the value of d is changed, the system will show different complete evolution of normalized gap with the anisotropic material orientation. Acknowledgement This work was supported by the National Natural Science foundation of China under Grant nos. 10747119, 11004039 and 10674032. References [1] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari- Rouhani, Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 (1993) 2022–2025. [2] M.S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, B. Djafari-Rouhani, Theory of acoustic band structure of periodic elastic composites, Phys. Rev. B 49 (1994) 2313–2322. [3] Zhengyou Liu, C.T. Chan, Ping Sheng, Elastic wave scattering by periodic structures of spherical objects: theory and experiment, Phys. Rev. B 62 (2000) 2446–2457. [4] I.E. Psarobas, N. Stefanou, Scattering of elastic waves by periodic arrays of spherical bodies, Phys. Rev. B 62 (2000) 278–291.
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