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Wave propagation in one-dimensional solid–fluid quasi-periodic and aperiodic phononic crystals
Physica B 407 (2012) 324–329
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Wave propagation in one-dimensional solid–fluid quasi-periodic and aperiodic phononic crystals A-Li Chen a,n, Yue-Sheng Wang a, Chuanzeng Zhang b a b
Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, China Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 July 2011 Received in revised form 10 October 2011 Accepted 24 October 2011 Available online 31 October 2011
The propagation of the elastic waves in one-dimensional (1D) solid–fluid quasi-periodic phononic crystals is studied by employing the concept of the localization factor, which is calculated by the transfer matrix method. The solid–fluid interaction effect at the interfaces between the solid and the fluid components is considered. For comparison, the periodic systems and aperiodic Thue–Morse sequence are also analyzed in this paper. The splitting phenomenon of the pass bands and bandgaps are discussed for these 1D solid–fluid systems. At last the influences of the material impedance ratios on the band structures of the 1D solid–fluid quasi-periodic phononic crystals arranged as Fibonacci sequence are discussed. & 2011 Elsevier B.V. All rights reserved.
Keywords: Elastic waves Solid–fluid interaction Phononic crystals Quasi-periodic Aperiodic Bandgaps
1. Introduction Analog to the traditional quasicrystals, the quasi-periodic photonic/phononic crystals (PNCs) are those which have no translational symmetry but have long-range order. Shechtman et al. [1] found the quasicrystals experimentally for the first time in 1984. And in the same year, Levine and Steinhardt [2] confirmed the existence of the quasicrystal by analytically computing the diffraction pattern of an ideal quasicrystal, which showed the five-fold symmetry. Lifshitz [3] discussed the symmetry of quasi-periodic crystals. Steurer and Haibach [4] studied the periodic average structures (PASs) of particular quasicrystals. A review on quasicrystals was presented by Steurer [5], in which various quasicrystal structures were discussed. More details about periodic and quasi-periodic structures can be found in the monograph written by Albuquerque and Cottam [6]. Since Kushwaha et al. [7] proposed the concept of phononic crystals, some researchers focused their interest to the one-dimensional (1D) quasi-periodic PNCs [8–15]. Luo [8] studied the propagation of the acoustic waves in 1D quasi-periodic PNCs made up of water and air layers using the transfer matrix method. Cao et al. [9] analyzed the bandgap characteristics of 1D quasi-periodic PNCs using the eigen-mode matching theory. Aynaou et al. [10]
n
Corresponding author. Tel.: þ86 10 51682713. E-mail address: [email protected] (A.-L. Chen).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.10.041
investigated the propagation and localization of acoustic waves in the phononic bandgap structures made of slender tube loops pasted together with slender tubes of finite length according to a Fibonacci sequence. Sesion Jr. et al. [11] calculated the transmission spectrum of the 1D piezoelectric quasi-periodic PNCs. King and Cox [12] studied experimentally and theoretically the 1D periodically and quasi-periodically modulated waveguides. Chen et al. [13] discussed the engineering of the bandgaps using the combination of periodic and Fibonacci quasi-periodic 1D composite thin plates. The researches mentioned above mostly used the transmission spectrum to discuss the propagation properties of the 1D quasi-periodic PNCs. Introducing a well defined localization factor, the present authors have studied the localization phenomenon of the 1D solid–solid quasi-periodic PNCs arranged as a Fibonacci sequence and 2D PNCs with a Fibonacci sequence in one direction [14,15]. The models in most of these studies are composed of two materials, which are all solid states or fluid states. Boudouti et al. [16] reported a review on the layered structures alternated by solid and liquid layers. Quasi-periodic structures were also discussed in that paper. solid–fluid layered structures may exhibit different features from the solid–solid or fluid–fluid ones [16]. In the present paper the localization factor will be introduced to describe the propagation of the 1D solid–fluid quasi-periodic PNCs arranged as the Fibonacci sequence. The solid–fluid interaction will be considered to derive the transfer matrix. Both normal and oblique incidences are considered. For comparison,
A.-L. Chen et al. / Physica B 407 (2012) 324–329
the localization factors are calculated for the periodic and aperiodic systems [17]. The band splitting phenomenon is discussed, and the influences of the different material combinations on the band structures are analyzed.
wave at an angle y0 with the phase velocity c0, ky ¼ oc1 0 a1 sin y0 . The normalized displacement and stress components can be calculated by vx ¼
2. The models and the method
@j @c þ , @x @Z
vx ¼ @j=@x þ @c=@y,
vy ¼ @j=@y@c=@x
ð1Þ
Then the governing equations for wave motion in the solid layers can be written as 2
r j¼
c2 L
j€ , r
2
€ c ¼ c2 T c
ð2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi where r ¼ @ =@x þ @ =@y ; and cL ¼ ðl þ2mÞ=r and cT ¼ m=r are the longitudinal and shear wave speeds, respectively. For convenience, we define the normalized local coordinates xj ¼xj/a1 and Zj ¼yj/a1, where a1 is the thickness of material A, and j ¼1, 2 represent sub-cells A and B, respectively. Then the general normalized harmonic solutions to Eq. (2) are given by 2
2
2
2
2
j1 ðx1 , Z1 ,tÞ ¼ ½A1 expðiqL1 x1 Þ þ A2 expðiqL1 x1 Þexpðiky Z1 iotÞ c1 ðx1 , Z1 ,tÞ ¼ ½B11 expðiqT1 x1 Þ þ B21 expðiqT1 x1 Þexpðiky Z1 iotÞ ð3Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where 0 r x1 r z1 ¼ a1 ¼ 1; i2 ¼ 1; qL1 ¼ ðoa1 =cL1 Þ2 ky and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 qT1 ¼ ðoa1 =cT1 Þ2 ky ; o is the circular frequency; A1, A2, B11
x1 x2
A B A A B 0
@2 f
þ
ð4Þ
c2 ðx2 , Z2 ,tÞ ¼ ½B12 expðiqL2 x2 Þ þ B22 expðiqL2 x2 Þexpðiky Z2 iotÞ ð5Þ where B12 and B22 are unknown coefficients; 0 r x2 r z2 ¼ a2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 =a1 ; qL2 ¼ ðoa1 =cf Þ2 ky and cf ¼ C 11f =r is the sound velocity of the fluid. For the solid layers, considering the ideal solid–fluid interaction at the interfaces, we have tyx 9x ¼ 0 ¼ tyx 9x ¼ a1 ¼ 0. And then 1 1 the four unknown coefficients in Eq. (3) can be reduced into two (here we use B11 and B21). The interface continuity conditions between the solid and the fluid layers are vx 9solid ¼ vx 9fluid ,
sx 9solid ¼ sx 9fluid
ð6Þ
T
We define V ¼ fvx , sx g as the state vector. Then at the left and right sides of each sub-cell in the kth unit cell, we have VðkÞ ¼ fvx , sx gTxj ¼ 0 ¼ Mj fB1j ,B2j gT , jL
VðkÞ ¼ fvx , sx gTxj ¼ aj ¼ Nj fB1j ,B2j gT jR ð7Þ
where the elements of the matrices Mj and Nj are given in Appendix. The above two state vectors have the following relation ðkÞ 0 ðkÞ VðkÞ ¼ Nj M1 j V jL 9Tj V jL jR
ð8Þ
where T0j ¼ Nj M1 is the 2 2 transfer matrix of the jth sub-cell. j By considering the continuity of the displacements and stresses at the interfaces between the two sub-cells and between the two unit cells, i.e. Eq. (6), the relationship between the state vectors of the kth and the (k 1)th unit cells can be obtained as ðk1Þ VðkÞ 2R ¼ Tk V 2R
2.5
ð9Þ
n = 72 n = 305 n = 1292 n = 5473
2.0 localization factor
o1
@j @c @Z @x
For the fluid layers there is only one displacement potential c2 of which the general normalized harmonic solution is given by
and B21 are the unknown coefficients to be determined; and ky is the y-component of the normalized wave vector. For an incident
y1 y2
vy ¼
! ! @2 f @2 f @2 c m þ þ 2 , 2 2 @x@Z @Z2 @x @x ! @2 j @2 c @2 c tyx ¼ m 2 þ @x@Z @Z2 @x2
sx ¼ l Consider a 1D solid–fluid quasi-periodic PNC arranged as the Fibonacci sequence, which can be obtained by repeating operations of the concurrent substitution rules: A-AB and B-A [18] as shown in Fig. 1, where A and B are sub-layerspmade up of different ffiffiffi materials with the length ratio a2 =a1 ¼ ð1 þ 5Þ=2. Besides, other three systems are also discussed: the periodic average structure of the above quasi-periodic PNC with AB as the unit cell, the periodic system with the fourth generation of the Fibonacci sequence (ABAAB) as the supercell, and Thue–Morse sequence, which is an aperiodic sequence obeying the concurrent substitution rules: A-AB and B-BA [19]. In this paper the localization factor [14] is introduced to describe the band structures for elastic waves propagating in an arbitrary direction in the above layered systems. The transfer matrix method is used to calculate the localization factor by considering the continuity relationships of the solid–fluid interaction at the interfaces. We introduce two displacement potentials, j and c, which are related to the displacements by [20]
325
1.5
1.0
0.5
Elastic waves 0.0
a1
a2
0
1
2
3
4
L1 Fig. 1. Schematic diagram of the 1D quasi-periodic PNC arranged as a Fibonacci sequence.
Fig. 2. Influence of the layer number on the calculated localization factors.
5
326
A.-L. Chen et al. / Physica B 407 (2012) 324–329 0
exponent g1 is the localization factor. The wave amplitude can be estimated roughly by the localization factor. For instance, when the waves propagate through the whole structure with n unit cells, the amplitude attenuation coefficient is approximately e ng1.
where Tk ¼T0 2T1 is the 2 2 transfer matrix between the two consecutive unit cells. The localization factor, similar to the Lyapunov exponent, applied to characterize the spatial evolution of a nearly periodic system is the average exponential rate of growth or decay of the wave amplitudes [21]. For a 2m 2m transfer matrix, the expression for calculating the localization factor of the system was given by Wolf et al. [22] as
gm ¼ lim
1
n-1 n
n X
ðkÞ
ln:v^ 2R,m :
3. Numerical examples and discussions First, we consider the elastic wave propagating normally (ky ¼ 0) along the 1D quasi-periodic PNCs, which is made up of Al (Aluminum) and water arranged as the Fibonacci sequence. Fig. 2 shows the influence of the number (2n) of the layers on the calculated localization factors varying with the normalized frequency OL1 ¼ oa1/cL1. The quasi-periodic PNCs with 144, 610, 2584
ð10Þ
k¼1
ðkÞ where the vector v^ 2R,m is obtained by the Gram–Schmidt orthonormalization procedure. For details, we refer to Ref. [23]. In this ðkÞ paper m ¼1, then v^ 2R,m ¼ vðkÞ 2R . So the only one positive Lyapunov
1.0
0.8
Transmission coefficient
Transmission coefficient
1.0
0.6 0.4 0.2
0.8 0.6 0.4 0.2 0.0
0.0
0
1
2
3
5
4
0
1
2
L1
3
4
5
L1
Fig. 3. Transmission spectra of a 34-layered finite system: (a) results from the localization factor; and (b) results from the transfer matrix method directly.
2.5
2.5 PAS
2.0
quasi
2.0
1.5
1.5
1.0
1.0
0.5
0.5 0.0
0.0
0
1
2
3
4
5
0
1
2
L1
3 L1
4
5
2.5
2.5 2.0
2.0
ABAAB
1.5
1.5
1.0
1.0
0.5
0.5
Thue-Morse
0.0
0.0
0
1
2
3 L1
4
5
0
1
2
3 L1
4
5
ky = 0
Fig. 4. Band structures characterized by the localization factors for the four systems in the case of normal propagation of waves: (a) PAS of the Fibonacci sequence; (b) Fibonacci sequence; (c) periodic PNC with the supercell ABAAB and (d) Thue–Morse sequence.
A.-L. Chen et al. / Physica B 407 (2012) 324–329
and 10946 layers (i.e., the 11th, 14th, 17th and 20th generation of the Fibonacci sequence) are considered, respectively. It can be seen that the curves for these four systems are similar with each other. The values of the localization factor decrease with the number of layers increasing. The intervals with localization factors much bigger than zero are known as the stop bands or bandgaps. It is noted that almost all bandgaps are split into two parts. This is a distinguishing feature of the quasi-periodic systems. Numerical tests show that the values of the localization factor are almost unchanged for the Fibonacci sequences from the 20th generation up (i.e. nZ5473). Therefore it can be concluded that the 20th generation of the Fibonacci sequence is enough to show the most important properties of the 1D quasi-periodic PNCs. So in the following discussion we will take the calculation model as the 20th generation of the Fibonacci sequence (n¼5473) with Al and water without special statement. For the 20th generation of the Fibonacci sequence there are some frequency intervals in which the localization factors are almost zero. These intervals are the pass bands. Most pass bands shown in Fig. 2 are very narrow except the one in the interval of OL1A(3.0, 3.3). In order to further demonstrate the efficiency of the localization factor in characterizing the band structures for the 1D solid–fluid quasi-periodic PNCs, Fig. 3 shows the transmission spectrum for a
327
34-layered finite system calculated by the localization factors (Fig. 3a) and by the transfer matrix method directly (Fig. 3b), respectively. A good agreement is shown between pass/stop bands characterized by the localization factor and the transmission response. The only difference is that the values of the transmission coefficients in the pass bands calculated by the localization factor are bigger than those calculated by the transfer matrix method directly. This is understood by considering the fact that the localization factor describes the wave that propagates with the weakest attenuation. Therefore, the localization factor not only can characterize the band structures but also can give a rough estimation for wave transmission through a finite layered structure. Fig. 4 presents the localization factors versus the normalized frequency for waves propagating normally in the four systems addressed in Section 2: the Fibonacci sequence (Fig. 4b), its PAS (Fig. 4a), the periodic PNC with the supercell ABAAB (Fig. 4c) and the Thue–Morse sequence (Fig. 4d). Compared with its PAS, the Fibonacci sequence exhibits clearly splitting phenomenon in both pass bands and bandgaps. For periodic PNC with the supercell ABAAB splitting phenomenon is shown in the pass bands but not in most bandgaps. The pass bands and the bandgaps are all split into multiple narrower bands as shown in Fig. 4d for the aperiodic Thue–Morse sequence. It should be noticed that the bandgaps, (1.396, 1.781) and (4.564,
quasi PAS 1.0E-4 0.010 0.020 0.040 0.080 0.17 0.33 0.67 1. 3 2. 7 5. 4 6. 7 8. 0 9. 4 11 13
B
L1
3
A D
2
C F
1
1.0E-4 0.010 0.020 0.040 0.080 0.17 0.33 0.67 1. 3 2. 7 5. 4 6. 7 8. 0 9. 4 11
4 B 3
A
L1
4
D
2
C F
1
E
E
H G 0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
0.0
2.0
0.2
0.4
0.6
0.8
1.2
1.0 ky
1.4
1.6
2.0
1.8
ABAAB
B A
L1
3
D
2
C F
1
Thue-Morse 4 B 3
A
L1
4
1.0E-4 0.010 0.020 0.040 0.080 0.17 0.33 0.67 1.3 2.7 4.0 5.4 6.7 8.0 9.4 12
D
2
C F
1
E
1.0E-4 0.010 0.020 0.080 0.17 0.33 0.67 1.3 2.7 4.0 5.4 6.7 8.0 9.4 10
E
H G 0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
Fig. 5. Localization factors varying with the normalized frequency and ky for the four systems: (a) PAS of the Fibonacci sequence; (b) Fibonacci sequence; (c) periodic PNC with the supercell ABAAB and (d) Thue–Morse sequence.
328
A.-L. Chen et al. / Physica B 407 (2012) 324–329
4.949), in the PAS (Fig. 4a) are split into two or more smaller bandgaps in the other three systems. Also the two pass bands, (2.958, 3.156) and (3.178, 3.354), in the PAS (Fig. 4a) are split into multiple smaller bands in the other three cases. It should be mentioned that the random disordered PNCs do not exhibit the band splitting phenomenon [24]. Compared with other three systems, the Thue–Morse sequence (Fig. 4d) is the best candidate for the band splitting function. The localization factors of the Thue–Morse sequence are the smallest, which means that the wave attenuation in such a sequence is the weakest. Fig. 5 shows the localization factors varying with the normalized frequency (OL1) and wave vector ky for waves propagating obliquely through the four systems stated above. Clear bandgaps and pass bands are shown in Fig. 5a and c for the periodic systems, but not in Fig. 5b and d for the quasi-periodic and aperiodic systems because of the wave localization [24,25]. Splitting phenomenon as shown in Fig. 4 can be observed in Fig 5b, c and d. This feature can be applied in the design of acoustic filters. Unlike the solid–solid systems [24], many pass bands and bandgaps in the solid–fluid systems vary only very slightly with ky, except for those in very low frequency regions or the near crossover points AH. Furthermore one may observe a dark oblique line passing through
the points E and (0, 0) (excluding the very low frequency region) in each figure, meaning a strong attenuation for the wave modes in this dark region. This does not appear in the band structures for the solid–solid PNCs [24] and quasi PNC [14]. At last we discuss the influences of the material combinations on the band structures of the Fibonacci sequence. We choose four material combinations: Al–Hg (Hydrargyrum), Al–Water, Au (Aurum)–Hg and Au–Water. The acoustic impedance ratios for these four cases are ZAl/ZHg ¼0.8729, ZAl/ZWater ¼11.56, ZAu/ZHg ¼ 3.320 and ZAu/ZWater ¼ 43.97. The localization factors versus the normalized frequency and ky are shown in Fig. 6 for these four systems. Al–Hg system with a small acoustic impedance ratio exhibits very broad pass bands with narrow bandgaps (Fig. 6a). With the increase of the impedance ratio, the bandgaps become wider. For Au–Water system (Fig. 6c) only a few narrow pass bands appear. A dark curved zone can be seen in each figure (especially Fig. 6a) for OL1 o2. The zones in all the cases are similar in shape. The localization factors in these zones are relatively big, implying that the wave propagation is forbidden therein. This feature is different from the solid–solid systems as discussed in Refs. [14,15]. It should be an intrinsic property for a solid–fluid system.
Al-Hg 1.0E-4 0.010 0.020 0.080 0.17 0.33 0.67 1.3 2.7 4.0 5.4 6.7 7.6 8.7
L1
3
2
4
1.0E-4 0.010 0.020 0.040 0.080 0.17 0.33 0.67 1.3 2.7 5.4 6.7 8.0 9.4 11
3 L1
4
Al-Water
2 1
1
0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
4
0.0
L1
1
0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
Au-water 1.0E-4 0.010 0.020 0.080 0.17 0.33 0.67 1. 3 2. 7 4. 0 5. 4 6. 7 8. 0 9. 4 11 12
3 L1
1.0E-4 0.010 0.020 0.080 0.17 0.33 0.67 1.3 2.7 4.0 5.4 6.7 8.0 8.6 10
2
0.4
4
Au-Hg
3
0.2
2
1
0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
Fig. 6. Influence of the material combinations on the band structures of the Fibonacci sequence: (a) Al–Hg system; (b) Al–Water system; (c) Au–Hg system and (d) Au–Water system.
A.-L. Chen et al. / Physica B 407 (2012) 324–329
2
4. Conclusions
it 5 expðiqL1 z1 Þ
1. The localization factor is an efficient parameter to describe the elastic wave propagating in one dimensional solid–fluid quasiperiodic and aperiodic PNCs. 2. Compared with their PAS, the quasi-periodic PNCs exhibit clearly splitting phenomenon in both pass bands and bandgaps. Splitting phenomenon is shown in the pass bands but not in most bandgaps for periodic PNCs with the supercell ABAAB. For the aperiodic, Thue–Morse sequence, the pass bands and the bandgaps are all split into multiple narrower bands. The Thue–Morse sequence is the best candidate for the band splitting function. 3. The material combinations have influences on the band structures. With the increase of the acoustic impedance mismatch, the bandgaps become wider. 4. Unlike the solid–solid systems, many pass bands and bandgaps in the solid–fluid systems vary only very slightly with the y-component of the normalized wave vector, except for those in very low frequency regions or near crossover points. And there exists a zone in which the wave attenuation is very strong.
2expðiqT1 z1 Þþ expð2iqT1 z1 iqL1 z1 Þt 1 t4 =t 6
" M2 ¼
The authors are grateful for the support by the National Science Foundation under Grant nos. 10902012 and 10632020, the Fundamental Research Funds for the Central Universities under Grant no. 2011JBM272, and the German Research Foundation (DFG) under Grant no. ZH15/11-1.
and Nj are given by 3 it 5 t 3 ½2 expðiqT1 z1 þ iqL1 z1 Þ 7 5 expð2iqT1 z1 Þ1t 1 t4 =t 6
2
2
C 11f ðq2L2 þ ky Þ
,
iqL2 expðiqL2 z2 Þ
iqL2 expðiqL2 z2 Þ
2 C 11f ðq2L2 þ ky ÞexpðiqL2 z2 Þ
2 C 11f ðq2L2 þ ky ÞexpðiqL2 z2 Þ
# ,
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[15] [16] [17] [18] [19] [20]
Appendix
#
where j¼ 1 and 2 stand for the solid and fluid layers, respectively; 2 2 t1 ¼ 2mkyqL1; t 2 ¼ mðq2T1 ky Þ; t 3 ¼ ðl þ2mÞq2L1 þ lky ; t4 ¼2mkyqT1; t5 ¼ (kyt1 þqL1t2)/t2 and t 6 ¼ ½expð2iqT1 z1 Þ1t 2 .
[12] [13] [14]
Acknowledgments
5, expðiqL1 z1 Þt 3 ½expðiqL1 z1 Þ 2 expðiqT1 z1 Þþ expð2iqT1 z1 þ iqL1 z1 Þt1 t 4 =t6
iqL2
iqL2 C 11f ðq2L2 þ ky Þ
" N2 ¼
3
it 5 expðiqL1 z1 Þ
N1 ¼ 4 expðiqL1 z1 Þt3 þ½expðiqL1 z1 Þ
The propagation of the elastic waves in one-dimensional solid–fluid quasi-periodic PNCs arranged as the Fibonacci sequence is studied by employing the concept of the localization factor. For comparison, the periodical average structure of the quasi-periodic PNCs, the PNCs taking the fourth generation of the Fibonacci sequence as the supercell and the aperiodic Thue– Morse sequence, are also considered. The splitting of the pass bands and bandgaps are discussed for these four systems. At last the influences of the material combinations on the band structures of the 1D quasi-periodic PNCs arranged as the Fibonacci sequence are discussed. The present results show:
The elements of the matrices Mj 2 it5 6 M1 ¼ 4 t 3 þ ½2 expðiqT1 z1 iqL1 z1 Þ expð2iqT1 z1 Þ1t 1 t 4 =t 6
329
[21] [22] [23] [24] [25]
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Physica B journal homepage: www.elsevier.com/locate/physb
Wave propagation in one-dimensional solid–fluid quasi-periodic and aperiodic phononic crystals A-Li Chen a,n, Yue-Sheng Wang a, Chuanzeng Zhang b a b
Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, China Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 July 2011 Received in revised form 10 October 2011 Accepted 24 October 2011 Available online 31 October 2011
The propagation of the elastic waves in one-dimensional (1D) solid–fluid quasi-periodic phononic crystals is studied by employing the concept of the localization factor, which is calculated by the transfer matrix method. The solid–fluid interaction effect at the interfaces between the solid and the fluid components is considered. For comparison, the periodic systems and aperiodic Thue–Morse sequence are also analyzed in this paper. The splitting phenomenon of the pass bands and bandgaps are discussed for these 1D solid–fluid systems. At last the influences of the material impedance ratios on the band structures of the 1D solid–fluid quasi-periodic phononic crystals arranged as Fibonacci sequence are discussed. & 2011 Elsevier B.V. All rights reserved.
Keywords: Elastic waves Solid–fluid interaction Phononic crystals Quasi-periodic Aperiodic Bandgaps
1. Introduction Analog to the traditional quasicrystals, the quasi-periodic photonic/phononic crystals (PNCs) are those which have no translational symmetry but have long-range order. Shechtman et al. [1] found the quasicrystals experimentally for the first time in 1984. And in the same year, Levine and Steinhardt [2] confirmed the existence of the quasicrystal by analytically computing the diffraction pattern of an ideal quasicrystal, which showed the five-fold symmetry. Lifshitz [3] discussed the symmetry of quasi-periodic crystals. Steurer and Haibach [4] studied the periodic average structures (PASs) of particular quasicrystals. A review on quasicrystals was presented by Steurer [5], in which various quasicrystal structures were discussed. More details about periodic and quasi-periodic structures can be found in the monograph written by Albuquerque and Cottam [6]. Since Kushwaha et al. [7] proposed the concept of phononic crystals, some researchers focused their interest to the one-dimensional (1D) quasi-periodic PNCs [8–15]. Luo [8] studied the propagation of the acoustic waves in 1D quasi-periodic PNCs made up of water and air layers using the transfer matrix method. Cao et al. [9] analyzed the bandgap characteristics of 1D quasi-periodic PNCs using the eigen-mode matching theory. Aynaou et al. [10]
n
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investigated the propagation and localization of acoustic waves in the phononic bandgap structures made of slender tube loops pasted together with slender tubes of finite length according to a Fibonacci sequence. Sesion Jr. et al. [11] calculated the transmission spectrum of the 1D piezoelectric quasi-periodic PNCs. King and Cox [12] studied experimentally and theoretically the 1D periodically and quasi-periodically modulated waveguides. Chen et al. [13] discussed the engineering of the bandgaps using the combination of periodic and Fibonacci quasi-periodic 1D composite thin plates. The researches mentioned above mostly used the transmission spectrum to discuss the propagation properties of the 1D quasi-periodic PNCs. Introducing a well defined localization factor, the present authors have studied the localization phenomenon of the 1D solid–solid quasi-periodic PNCs arranged as a Fibonacci sequence and 2D PNCs with a Fibonacci sequence in one direction [14,15]. The models in most of these studies are composed of two materials, which are all solid states or fluid states. Boudouti et al. [16] reported a review on the layered structures alternated by solid and liquid layers. Quasi-periodic structures were also discussed in that paper. solid–fluid layered structures may exhibit different features from the solid–solid or fluid–fluid ones [16]. In the present paper the localization factor will be introduced to describe the propagation of the 1D solid–fluid quasi-periodic PNCs arranged as the Fibonacci sequence. The solid–fluid interaction will be considered to derive the transfer matrix. Both normal and oblique incidences are considered. For comparison,
A.-L. Chen et al. / Physica B 407 (2012) 324–329
the localization factors are calculated for the periodic and aperiodic systems [17]. The band splitting phenomenon is discussed, and the influences of the different material combinations on the band structures are analyzed.
wave at an angle y0 with the phase velocity c0, ky ¼ oc1 0 a1 sin y0 . The normalized displacement and stress components can be calculated by vx ¼
2. The models and the method
@j @c þ , @x @Z
vx ¼ @j=@x þ @c=@y,
vy ¼ @j=@y@c=@x
ð1Þ
Then the governing equations for wave motion in the solid layers can be written as 2
r j¼
c2 L
j€ , r
2
€ c ¼ c2 T c
ð2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi where r ¼ @ =@x þ @ =@y ; and cL ¼ ðl þ2mÞ=r and cT ¼ m=r are the longitudinal and shear wave speeds, respectively. For convenience, we define the normalized local coordinates xj ¼xj/a1 and Zj ¼yj/a1, where a1 is the thickness of material A, and j ¼1, 2 represent sub-cells A and B, respectively. Then the general normalized harmonic solutions to Eq. (2) are given by 2
2
2
2
2
j1 ðx1 , Z1 ,tÞ ¼ ½A1 expðiqL1 x1 Þ þ A2 expðiqL1 x1 Þexpðiky Z1 iotÞ c1 ðx1 , Z1 ,tÞ ¼ ½B11 expðiqT1 x1 Þ þ B21 expðiqT1 x1 Þexpðiky Z1 iotÞ ð3Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where 0 r x1 r z1 ¼ a1 ¼ 1; i2 ¼ 1; qL1 ¼ ðoa1 =cL1 Þ2 ky and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 qT1 ¼ ðoa1 =cT1 Þ2 ky ; o is the circular frequency; A1, A2, B11
x1 x2
A B A A B 0
@2 f
þ
ð4Þ
c2 ðx2 , Z2 ,tÞ ¼ ½B12 expðiqL2 x2 Þ þ B22 expðiqL2 x2 Þexpðiky Z2 iotÞ ð5Þ where B12 and B22 are unknown coefficients; 0 r x2 r z2 ¼ a2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 =a1 ; qL2 ¼ ðoa1 =cf Þ2 ky and cf ¼ C 11f =r is the sound velocity of the fluid. For the solid layers, considering the ideal solid–fluid interaction at the interfaces, we have tyx 9x ¼ 0 ¼ tyx 9x ¼ a1 ¼ 0. And then 1 1 the four unknown coefficients in Eq. (3) can be reduced into two (here we use B11 and B21). The interface continuity conditions between the solid and the fluid layers are vx 9solid ¼ vx 9fluid ,
sx 9solid ¼ sx 9fluid
ð6Þ
T
We define V ¼ fvx , sx g as the state vector. Then at the left and right sides of each sub-cell in the kth unit cell, we have VðkÞ ¼ fvx , sx gTxj ¼ 0 ¼ Mj fB1j ,B2j gT , jL
VðkÞ ¼ fvx , sx gTxj ¼ aj ¼ Nj fB1j ,B2j gT jR ð7Þ
where the elements of the matrices Mj and Nj are given in Appendix. The above two state vectors have the following relation ðkÞ 0 ðkÞ VðkÞ ¼ Nj M1 j V jL 9Tj V jL jR
ð8Þ
where T0j ¼ Nj M1 is the 2 2 transfer matrix of the jth sub-cell. j By considering the continuity of the displacements and stresses at the interfaces between the two sub-cells and between the two unit cells, i.e. Eq. (6), the relationship between the state vectors of the kth and the (k 1)th unit cells can be obtained as ðk1Þ VðkÞ 2R ¼ Tk V 2R
2.5
ð9Þ
n = 72 n = 305 n = 1292 n = 5473
2.0 localization factor
o1
@j @c @Z @x
For the fluid layers there is only one displacement potential c2 of which the general normalized harmonic solution is given by
and B21 are the unknown coefficients to be determined; and ky is the y-component of the normalized wave vector. For an incident
y1 y2
vy ¼
! ! @2 f @2 f @2 c m þ þ 2 , 2 2 @x@Z @Z2 @x @x ! @2 j @2 c @2 c tyx ¼ m 2 þ @x@Z @Z2 @x2
sx ¼ l Consider a 1D solid–fluid quasi-periodic PNC arranged as the Fibonacci sequence, which can be obtained by repeating operations of the concurrent substitution rules: A-AB and B-A [18] as shown in Fig. 1, where A and B are sub-layerspmade up of different ffiffiffi materials with the length ratio a2 =a1 ¼ ð1 þ 5Þ=2. Besides, other three systems are also discussed: the periodic average structure of the above quasi-periodic PNC with AB as the unit cell, the periodic system with the fourth generation of the Fibonacci sequence (ABAAB) as the supercell, and Thue–Morse sequence, which is an aperiodic sequence obeying the concurrent substitution rules: A-AB and B-BA [19]. In this paper the localization factor [14] is introduced to describe the band structures for elastic waves propagating in an arbitrary direction in the above layered systems. The transfer matrix method is used to calculate the localization factor by considering the continuity relationships of the solid–fluid interaction at the interfaces. We introduce two displacement potentials, j and c, which are related to the displacements by [20]
325
1.5
1.0
0.5
Elastic waves 0.0
a1
a2
0
1
2
3
4
L1 Fig. 1. Schematic diagram of the 1D quasi-periodic PNC arranged as a Fibonacci sequence.
Fig. 2. Influence of the layer number on the calculated localization factors.
5
326
A.-L. Chen et al. / Physica B 407 (2012) 324–329 0
exponent g1 is the localization factor. The wave amplitude can be estimated roughly by the localization factor. For instance, when the waves propagate through the whole structure with n unit cells, the amplitude attenuation coefficient is approximately e ng1.
where Tk ¼T0 2T1 is the 2 2 transfer matrix between the two consecutive unit cells. The localization factor, similar to the Lyapunov exponent, applied to characterize the spatial evolution of a nearly periodic system is the average exponential rate of growth or decay of the wave amplitudes [21]. For a 2m 2m transfer matrix, the expression for calculating the localization factor of the system was given by Wolf et al. [22] as
gm ¼ lim
1
n-1 n
n X
ðkÞ
ln:v^ 2R,m :
3. Numerical examples and discussions First, we consider the elastic wave propagating normally (ky ¼ 0) along the 1D quasi-periodic PNCs, which is made up of Al (Aluminum) and water arranged as the Fibonacci sequence. Fig. 2 shows the influence of the number (2n) of the layers on the calculated localization factors varying with the normalized frequency OL1 ¼ oa1/cL1. The quasi-periodic PNCs with 144, 610, 2584
ð10Þ
k¼1
ðkÞ where the vector v^ 2R,m is obtained by the Gram–Schmidt orthonormalization procedure. For details, we refer to Ref. [23]. In this ðkÞ paper m ¼1, then v^ 2R,m ¼ vðkÞ 2R . So the only one positive Lyapunov
1.0
0.8
Transmission coefficient
Transmission coefficient
1.0
0.6 0.4 0.2
0.8 0.6 0.4 0.2 0.0
0.0
0
1
2
3
5
4
0
1
2
L1
3
4
5
L1
Fig. 3. Transmission spectra of a 34-layered finite system: (a) results from the localization factor; and (b) results from the transfer matrix method directly.
2.5
2.5 PAS
2.0
quasi
2.0
1.5
1.5
1.0
1.0
0.5
0.5 0.0
0.0
0
1
2
3
4
5
0
1
2
L1
3 L1
4
5
2.5
2.5 2.0
2.0
ABAAB
1.5
1.5
1.0
1.0
0.5
0.5
Thue-Morse
0.0
0.0
0
1
2
3 L1
4
5
0
1
2
3 L1
4
5
ky = 0
Fig. 4. Band structures characterized by the localization factors for the four systems in the case of normal propagation of waves: (a) PAS of the Fibonacci sequence; (b) Fibonacci sequence; (c) periodic PNC with the supercell ABAAB and (d) Thue–Morse sequence.
A.-L. Chen et al. / Physica B 407 (2012) 324–329
and 10946 layers (i.e., the 11th, 14th, 17th and 20th generation of the Fibonacci sequence) are considered, respectively. It can be seen that the curves for these four systems are similar with each other. The values of the localization factor decrease with the number of layers increasing. The intervals with localization factors much bigger than zero are known as the stop bands or bandgaps. It is noted that almost all bandgaps are split into two parts. This is a distinguishing feature of the quasi-periodic systems. Numerical tests show that the values of the localization factor are almost unchanged for the Fibonacci sequences from the 20th generation up (i.e. nZ5473). Therefore it can be concluded that the 20th generation of the Fibonacci sequence is enough to show the most important properties of the 1D quasi-periodic PNCs. So in the following discussion we will take the calculation model as the 20th generation of the Fibonacci sequence (n¼5473) with Al and water without special statement. For the 20th generation of the Fibonacci sequence there are some frequency intervals in which the localization factors are almost zero. These intervals are the pass bands. Most pass bands shown in Fig. 2 are very narrow except the one in the interval of OL1A(3.0, 3.3). In order to further demonstrate the efficiency of the localization factor in characterizing the band structures for the 1D solid–fluid quasi-periodic PNCs, Fig. 3 shows the transmission spectrum for a
327
34-layered finite system calculated by the localization factors (Fig. 3a) and by the transfer matrix method directly (Fig. 3b), respectively. A good agreement is shown between pass/stop bands characterized by the localization factor and the transmission response. The only difference is that the values of the transmission coefficients in the pass bands calculated by the localization factor are bigger than those calculated by the transfer matrix method directly. This is understood by considering the fact that the localization factor describes the wave that propagates with the weakest attenuation. Therefore, the localization factor not only can characterize the band structures but also can give a rough estimation for wave transmission through a finite layered structure. Fig. 4 presents the localization factors versus the normalized frequency for waves propagating normally in the four systems addressed in Section 2: the Fibonacci sequence (Fig. 4b), its PAS (Fig. 4a), the periodic PNC with the supercell ABAAB (Fig. 4c) and the Thue–Morse sequence (Fig. 4d). Compared with its PAS, the Fibonacci sequence exhibits clearly splitting phenomenon in both pass bands and bandgaps. For periodic PNC with the supercell ABAAB splitting phenomenon is shown in the pass bands but not in most bandgaps. The pass bands and the bandgaps are all split into multiple narrower bands as shown in Fig. 4d for the aperiodic Thue–Morse sequence. It should be noticed that the bandgaps, (1.396, 1.781) and (4.564,
quasi PAS 1.0E-4 0.010 0.020 0.040 0.080 0.17 0.33 0.67 1. 3 2. 7 5. 4 6. 7 8. 0 9. 4 11 13
B
L1
3
A D
2
C F
1
1.0E-4 0.010 0.020 0.040 0.080 0.17 0.33 0.67 1. 3 2. 7 5. 4 6. 7 8. 0 9. 4 11
4 B 3
A
L1
4
D
2
C F
1
E
E
H G 0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
0.0
2.0
0.2
0.4
0.6
0.8
1.2
1.0 ky
1.4
1.6
2.0
1.8
ABAAB
B A
L1
3
D
2
C F
1
Thue-Morse 4 B 3
A
L1
4
1.0E-4 0.010 0.020 0.040 0.080 0.17 0.33 0.67 1.3 2.7 4.0 5.4 6.7 8.0 9.4 12
D
2
C F
1
E
1.0E-4 0.010 0.020 0.080 0.17 0.33 0.67 1.3 2.7 4.0 5.4 6.7 8.0 9.4 10
E
H G 0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
Fig. 5. Localization factors varying with the normalized frequency and ky for the four systems: (a) PAS of the Fibonacci sequence; (b) Fibonacci sequence; (c) periodic PNC with the supercell ABAAB and (d) Thue–Morse sequence.
328
A.-L. Chen et al. / Physica B 407 (2012) 324–329
4.949), in the PAS (Fig. 4a) are split into two or more smaller bandgaps in the other three systems. Also the two pass bands, (2.958, 3.156) and (3.178, 3.354), in the PAS (Fig. 4a) are split into multiple smaller bands in the other three cases. It should be mentioned that the random disordered PNCs do not exhibit the band splitting phenomenon [24]. Compared with other three systems, the Thue–Morse sequence (Fig. 4d) is the best candidate for the band splitting function. The localization factors of the Thue–Morse sequence are the smallest, which means that the wave attenuation in such a sequence is the weakest. Fig. 5 shows the localization factors varying with the normalized frequency (OL1) and wave vector ky for waves propagating obliquely through the four systems stated above. Clear bandgaps and pass bands are shown in Fig. 5a and c for the periodic systems, but not in Fig. 5b and d for the quasi-periodic and aperiodic systems because of the wave localization [24,25]. Splitting phenomenon as shown in Fig. 4 can be observed in Fig 5b, c and d. This feature can be applied in the design of acoustic filters. Unlike the solid–solid systems [24], many pass bands and bandgaps in the solid–fluid systems vary only very slightly with ky, except for those in very low frequency regions or the near crossover points AH. Furthermore one may observe a dark oblique line passing through
the points E and (0, 0) (excluding the very low frequency region) in each figure, meaning a strong attenuation for the wave modes in this dark region. This does not appear in the band structures for the solid–solid PNCs [24] and quasi PNC [14]. At last we discuss the influences of the material combinations on the band structures of the Fibonacci sequence. We choose four material combinations: Al–Hg (Hydrargyrum), Al–Water, Au (Aurum)–Hg and Au–Water. The acoustic impedance ratios for these four cases are ZAl/ZHg ¼0.8729, ZAl/ZWater ¼11.56, ZAu/ZHg ¼ 3.320 and ZAu/ZWater ¼ 43.97. The localization factors versus the normalized frequency and ky are shown in Fig. 6 for these four systems. Al–Hg system with a small acoustic impedance ratio exhibits very broad pass bands with narrow bandgaps (Fig. 6a). With the increase of the impedance ratio, the bandgaps become wider. For Au–Water system (Fig. 6c) only a few narrow pass bands appear. A dark curved zone can be seen in each figure (especially Fig. 6a) for OL1 o2. The zones in all the cases are similar in shape. The localization factors in these zones are relatively big, implying that the wave propagation is forbidden therein. This feature is different from the solid–solid systems as discussed in Refs. [14,15]. It should be an intrinsic property for a solid–fluid system.
Al-Hg 1.0E-4 0.010 0.020 0.080 0.17 0.33 0.67 1.3 2.7 4.0 5.4 6.7 7.6 8.7
L1
3
2
4
1.0E-4 0.010 0.020 0.040 0.080 0.17 0.33 0.67 1.3 2.7 5.4 6.7 8.0 9.4 11
3 L1
4
Al-Water
2 1
1
0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
4
0.0
L1
1
0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
Au-water 1.0E-4 0.010 0.020 0.080 0.17 0.33 0.67 1. 3 2. 7 4. 0 5. 4 6. 7 8. 0 9. 4 11 12
3 L1
1.0E-4 0.010 0.020 0.080 0.17 0.33 0.67 1.3 2.7 4.0 5.4 6.7 8.0 8.6 10
2
0.4
4
Au-Hg
3
0.2
2
1
0.0
0.2
0.4
0.6
0.8
1.0 ky
1.2
1.4
1.6
1.8
2.0
Fig. 6. Influence of the material combinations on the band structures of the Fibonacci sequence: (a) Al–Hg system; (b) Al–Water system; (c) Au–Hg system and (d) Au–Water system.
A.-L. Chen et al. / Physica B 407 (2012) 324–329
2
4. Conclusions
it 5 expðiqL1 z1 Þ
1. The localization factor is an efficient parameter to describe the elastic wave propagating in one dimensional solid–fluid quasiperiodic and aperiodic PNCs. 2. Compared with their PAS, the quasi-periodic PNCs exhibit clearly splitting phenomenon in both pass bands and bandgaps. Splitting phenomenon is shown in the pass bands but not in most bandgaps for periodic PNCs with the supercell ABAAB. For the aperiodic, Thue–Morse sequence, the pass bands and the bandgaps are all split into multiple narrower bands. The Thue–Morse sequence is the best candidate for the band splitting function. 3. The material combinations have influences on the band structures. With the increase of the acoustic impedance mismatch, the bandgaps become wider. 4. Unlike the solid–solid systems, many pass bands and bandgaps in the solid–fluid systems vary only very slightly with the y-component of the normalized wave vector, except for those in very low frequency regions or near crossover points. And there exists a zone in which the wave attenuation is very strong.
2expðiqT1 z1 Þþ expð2iqT1 z1 iqL1 z1 Þt 1 t4 =t 6
" M2 ¼
The authors are grateful for the support by the National Science Foundation under Grant nos. 10902012 and 10632020, the Fundamental Research Funds for the Central Universities under Grant no. 2011JBM272, and the German Research Foundation (DFG) under Grant no. ZH15/11-1.
and Nj are given by 3 it 5 t 3 ½2 expðiqT1 z1 þ iqL1 z1 Þ 7 5 expð2iqT1 z1 Þ1t 1 t4 =t 6
2
2
C 11f ðq2L2 þ ky Þ
,
iqL2 expðiqL2 z2 Þ
iqL2 expðiqL2 z2 Þ
2 C 11f ðq2L2 þ ky ÞexpðiqL2 z2 Þ
2 C 11f ðq2L2 þ ky ÞexpðiqL2 z2 Þ
# ,
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[15] [16] [17] [18] [19] [20]
Appendix
#
where j¼ 1 and 2 stand for the solid and fluid layers, respectively; 2 2 t1 ¼ 2mkyqL1; t 2 ¼ mðq2T1 ky Þ; t 3 ¼ ðl þ2mÞq2L1 þ lky ; t4 ¼2mkyqT1; t5 ¼ (kyt1 þqL1t2)/t2 and t 6 ¼ ½expð2iqT1 z1 Þ1t 2 .
[12] [13] [14]
Acknowledgments
5, expðiqL1 z1 Þt 3 ½expðiqL1 z1 Þ 2 expðiqT1 z1 Þþ expð2iqT1 z1 þ iqL1 z1 Þt1 t 4 =t6
iqL2
iqL2 C 11f ðq2L2 þ ky Þ
" N2 ¼
3
it 5 expðiqL1 z1 Þ
N1 ¼ 4 expðiqL1 z1 Þt3 þ½expðiqL1 z1 Þ
The propagation of the elastic waves in one-dimensional solid–fluid quasi-periodic PNCs arranged as the Fibonacci sequence is studied by employing the concept of the localization factor. For comparison, the periodical average structure of the quasi-periodic PNCs, the PNCs taking the fourth generation of the Fibonacci sequence as the supercell and the aperiodic Thue– Morse sequence, are also considered. The splitting of the pass bands and bandgaps are discussed for these four systems. At last the influences of the material combinations on the band structures of the 1D quasi-periodic PNCs arranged as the Fibonacci sequence are discussed. The present results show:
The elements of the matrices Mj 2 it5 6 M1 ¼ 4 t 3 þ ½2 expðiqT1 z1 iqL1 z1 Þ expð2iqT1 z1 Þ1t 1 t 4 =t 6
329
[21] [22] [23] [24] [25]
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