Elastic wave localization in two-dimensional phononic crystals with one-dimensional quasi-periodicity and random disorder

Acta Mechanica Solida Sinica, Vol. 21, No. 6, December, 2008 Published by AMSS Press, Wuhan, China. DOI: 10.1007/s10338-008-0862-x

ISSN 0894-9166

ELASTIC WAVE LOCALIZATION IN TWO-DIMENSIONAL PHONONIC CRYSTALS WITH ONE-DIMENSIONAL QUASI-PERIODICITY AND RANDOM DISORDER  Ali Chen1

Yuesheng Wang1

Guilan Yu2

Yafang Guo1

Zhengdao Wang1

1

( Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, China) (2 School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China)

Received 22 June 2008; revision received 19 October 2008

ABSTRACT The band structures of both in-plane and anti-plane elastic waves propagating in two-dimensional ordered and disordered (in one direction) phononic crystals are studied in this paper. The localization of wave propagation due to random disorder is discussed by introducing the concept of the localization factor that is calculated by the plane-wave-based transfer-matrix method. By treating the quasi-periodicity as the deviation from the periodicity in a special way, two kinds of quasi phononic crystal that has quasi-periodicity (Fibonacci sequence) in one direction and translational symmetry in the other direction are considered and the band structures are characterized by using localization factors. The results show that the localization factor is an effective parameter in characterizing the band gaps of two-dimensional perfect, randomly disordered and quasi-periodic phononic crystals. Band structures of the phononic crystals can be tuned by different random disorder or changing quasi-periodic parameters. The quasi phononic crystals exhibit more band gaps with narrower width than the ordered and randomly disordered systems.

KEY WORDS phononic crystal, quasi phononic crystal, disorder, localization factors, plane-wavebased transfer-matrix method, periodic average structure

I. INTRODUCTION [1]

In 1993, Kushwaha proposed the concept of ‘phononic crystals’ (PNCs), artificial periodic elastic/acoustic structures that exhibit so-called ‘phononic band gaps’[2] , which are in analogy to the photonic crystals (PTCs) simultaneously proposed by Yablonovitch[3] and John[4] in 1987. Since then a great deal of works have been devoted to the study of PNCs due to their unique physical properties and many promising applications such as thermal barriers, noise suppression, acoustic filters, waveguides, new transducers, etc. So far, several methods have been developed to calculate the band gaps of the PNCs, for instance, the transfer matrix method[5] , plane-wave-expansion method (PWE)[6–8] , finite-different time-domain method (FDTD)[9–11] , multi-scattering theory (MST)[12] and wavelet-based method[13–15] and etc. A lot of results based on these methods have been reported for the band gaps of the perfectly ordered PNCs or those with defects[16–18] . However in practical cases, disorder, usually caused by randomly distributed material defaults or manufacture defects during manufacture, is very common. This  

Corresponding author. Tel: +86-10-51688417, E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No.10632020).

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may lead to the localization, e.g. the well-known Anderson localization of electron waves in disordered systems[19] . Since the pioneer works of Anderson[19] , localization phenomenon in randomly disordered systems has attracted considerable attention, e.g. localization of acoustic waves and electromagnetic waves in media with properties fluctuating randomly[20–22] , and vibration localization of nearly periodic engineering structures such as beams, bars, plates, etc[23, 24] . In recent years, the band structures and localization phenomenon of electromagnetic waves in disordered PTCs have also been studied[25–27] . However researches on randomly disordered PNCs are very limited, especially on two-dimensional PNCs. The importance of this topic has been stressed in Refs.[28] and [29] in which the band gaps of one-dimensional ordered, randomly disordered and quasi-periodic phononic crystals were calculated using the well-defined localization factor[30] . Similarly, the localization phenomenon of the elastic waves propagating in one-dimensional randomly disordered piezoelectric composite structures was studied in Refs.[31] and [32]. Another topic that is now receiving increasing interest is the wave propagating in quasi-periodic PNCs. As we know, a quasi-periodic system is of the case between ordered and disordered systems, and thus the waves therein should have both propagating and localizing modes[33] . Several authors have reported results on this topic. Peng et al.[34] calculated the diffraction spectrum of the electromagnetic waves propagating in a one-dimensional k-component Fibonacci structure. King et al.[35] studied the acoustic band gaps in one-dimensional quasi-periodically modulated waveguides experimentally. Steurer et al.[36] and Velasco et al.[37] analyzed various kinds of quasi PNCs and PTCs. Sesion Jr et al.[38] discussed the acoustic transmission in piezoelectric Fibonacci PNCs. It is noted that most of the above-mentioned studies are based on the transmission spectra. In this paper, instead of calculating the transmitted waves, we used the well-defined localization factor to characterize the band structures and localization phenomenon of two-dimensional randomly disordered and quasi-periodic PNCs as we did in Refs.[28] and [29]. A two-dimensional PNC with the quasi periodicity of Fibonacci sequence in one direction was also considered. We developed a way to calculate the localization factors based on the plane-wave-based transfer-matrix method that has been used to study the response and dispersion curves for two-dimensional ordered PTCs[39] .

II. PROBLEM STATEMENT AND PLANE-WAVE-BASED TRANSFER-MATRIX METHOD Consider a two-dimensional solid-solid PNC with lattice constant a as shown in Fig.1(a). Rectangular rods with wide length 2l are embedded in the host material and arranged in the square lattice. The filling fraction of the rods is F = (2l/a)2 . The disordered system we considered in this paper is generated by perturbing the filling fraction F of the above ordered structure randomly in x-direction, see Fig.2(a). If the perturbation of the filling fraction F results in a Fibonacci sequence, then we have a special system with quasi-periodicity in x-direction, which is named as a quasi-periodic phononic crystal (QPNC),

Fig. 1. The schematic structure of the 2D PNC (a), the first Brillouin zone (b) and the calculated unit cells along the high-symmetry lines of the first Brillouin Zone for Γ X direction (c) and M Γ direction (d).

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Fig. 2. The schematic structure for the 2D PNC with the filling fraction randomly disordered in x-direction (a), QPNC with the lattice constant (b) and QPNC with the filling fraction (c) arranged in the Fibonacci sequence in x-direction.

see Fig.2(c). Another kind of the QPNC, which is generated by arranging the lattice constant in a Fibonacci sequence while keeping the filling fraction F unchanged, is also considered, see Fig.2(b). It should be noted that the disordered and quasi-periodic PNCs we considered are perfectly periodic in the y-direction. Therefore the structures can be viewed as an array of different sub-layers, each of which is a one-dimensional PNC layered in the y-direction. The following analysis will start with the perfectly periodic system shown in Fig.1(a). We consider the wave with frequency ω and wave vector k propagating through the structure. The plane-wave-based transfer-matrix method[39] is employed. We first cut the unit cell of the structure into n slices. It is noticed that different cutting-directions should be chosen to calculate the responses along different high-symmetry lines of the first Brillouin Zone (BZ) (see Fig.1(b)). The calculated unit cells along Γ X and M Γ directions are shown in Figs.1(c) and (d), respectively. For the in-plane wave propagating in the two-dimensional isotropic media, we have the equilibrium and constitutive equations −ρω 2 u1 = [(λ + 2μ)u1,1 + λu2,2 ],1 + σ21,2 −ρω 2 u2 = (μu1,2 + μu2,1 ),1 + σ22,2 σ21 = μu1,2 + μu2,1

(1)

σ22 = λu1,1 + (λ + 2μ)u2,2 where uj (j = 1, 2) is the displacement component; σij (i, j = 1, 2) is the stress component; ρ is the mass density; λ and μ are Lam´e constants. Due to the periodicity of the system along the y-direction, the elastic parameters in each slice can be written in Fourier series  f (y) = fG eiGy (2) G

where i2 = −1; f (y) represents the material parameters ρ(y), λ(y) and μ(y); G is the y-component of the reciprocal lattice; fG is the Fourier coefficient and its values can be found in Ref.[40]. According to the Bloch’s theorem and the periodicity in the y-direction, the displacement field can be expressed as  u(x, y, t) = ei(ky y−ωt) uky +G (G)eiGy eiβx (3) G

where u can be u1 or u2 ; ky and β are the y- and x-components of the wave vector, respectively. Substituting Eqs.(2) and (3) into Eqs.(1), we can obtain the following eigenvalue problem ⎤⎡ ⎡ ⎡ ⎤ ⎤ ⎤⎡ u1 0 A1 −I 0 A 0 0 0 u1 ⎢ B 1 0 0 −I ⎥ ⎢ u2 ⎥ ⎢ 0 B 0 0 ⎥ ⎢ u2 ⎥ ⎥⎢ ⎢ ⎢ ⎥ ⎥ ⎥⎢ (4) ⎣ 0 −C −I 0 ⎦ ⎣ iσ 21 ⎦ = β ⎣ C 1 0 0 0 ⎦ ⎣ iσ 21 ⎦ −D 0 0 −I 0 D1 0 0 iσ 22 iσ 22

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where I is an N × N unit matrix and AGG = ρG−G ω 2 − (ky + G)(ky + G )c11G−G , A1GG = (ky + G)c12G−G BGG = ρG−G ω 2 − (ky + G)(ky + G )c44G−G , B1GG = (ky + G)c44G−G CGG = (ky + G )c44G−G ,

C1GG = c44G−G



DGG = (ky + G )c12G−G ,

D1GG = c11G−G

with G and G taking the values of (M, . . . , M )× 2π/a that include totally N = 2M + 1 terms. Equation (4) gives a scalar equation about β for a given frequency ω. Solving this equation, 4N values of β and the corresponding 4N eigen-vectors can be obtained. Generally, the eigenvalue set η(β) of Eq.(4) consists of two parts: η = η1 + η2 , where η1 contains all of the real positive β and the complex ones with positive imaginary parts corresponding to the rightforward subset η(βR ); and η2 contains all of the real negative β and the complex ones with negative imaginary parts corresponding to the left-forward subset η(βL ), respectively. So, the solution of Eq.(4) can be written as ⎡ j ⎤ ⎡ ⎤ u1 E α (y) 0 0 0





j j ⎢ uj ⎥ ⎢ 0 E (y) 0 0 ER 0 ⎥ AjR α ⎢ 2 ⎥ ⎢ ⎥ UR UL β (x) (5) ⎢ j ⎥=⎢ ⎥ 0 EL ⎣ iσ 21 ⎦ ⎣ 0 0 E α (y) 0 ⎦ τ j2R τ j2L AjL β (x) 0 0 0 E α (y) iσ j22 L where j is the slice label; E α (y) is an N × N diagonal matrix; E R β (x) and E β (x) are 2N × 2N diagonal matrices; U and τ 2 are 2N × 2N eigen-vector matrices; AR andAL are 2N × 1 amplitude vectors corresponding to the right- and left-forward waves.  (j) −j −j −j T Then we take the state vectors at the left and right sides of the j-th slice as V L = u−j 1 u2 iσ 21 iσ 22  (j) +j +j +j T and V R = u+j where the superscript +/−j and the subscript R/L denote the 1 u2 iσ 21 iσ 22 right/left boundary of the j-th slice. The boundary conditions of the interface between j-th and (j +1)-th slices are (j) (j+1) VR =VL (6)

Substituting the solutions of the j-th and (j + 1)-th slices (which have the form of Eq.(5)) into Eq.(6), and comparing the coefficients of each mode eiαy , we get −(j+1)

j +j j+1 U jR A+j R + U L AL = U R AR

τ j2R A+j R

+

τ j2L A+j L

=

−(j+1) τ j+1 2R AR

−(j+1)

+ U j+1 L AL +

−(j+1) τ j+1 2L AL

(7) (8)

To solve Eqs.(7) and (8), we define the reflection matrix Rj+ , transmission matrix T j and general reflection matrix Rj− as j +j A+j L = R+ AR −(j+1) AR

and

=T

j

A+j R

j R Rj− = E L β (hj )R+ E β (hj )

where hj is the thickness of the j-th slice. Then Eqs.(7) and (8) can be written as j j+1



j+1 Tj UR U R + U j+1 −U jL L R− = j+1 j+1 Rj+ τ j2R τ j+1 −τ j2L 2R + τ 2L R−

(9) (10) (11)

(12)

Note that the general reflection matrix Rout − = 0 in the outputting slice equals zero, and that the amplitude of the incident wave A1R on the left-most substrate is known. So the reflection and transmission matrix of each slice can be obtained by repeatedly using Eq.(12). At last the transfer matrix of the u-th unit cell C u can be obtained as 1 C u = X n ER β (hn ) · · · X

(13)

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where X n is the transfer matrix of the n-th slice of the u-th unit cell. It should be noticed that the above derivation is applicable not only for the ordered PNCs but also the disordered and quasi-periodic ones. Although no representative unit cell exists because of the disorder in the latter case, we still define the unit cell disordered from the original one of the periodic system as the ‘unit cell’. However, the transfer matrices of the ‘unit cells’ are different from each other because of the difference between the ‘unit cells’.

III. LOCALIZATION FACTOR We introduce the concept of the localization factor, which is used to describe the vibration localization phenomenon in nearly periodic engineering structures, to study the plane wave propagation and localization in randomly disordered PNCs. This factor is defined as the smallest positive Lyapunov exponent[30] . As we know, the Lyapunov exponent is the average exponential rate of convergence or divergence between two neighboring phase orbits in the phase space and is considered as a measure of chaos in dynamic systems[41] . The localization factor, a similar concept applied to characterize the spatial evolution of a nearly periodic system is the average exponential rate of growth or decay of the wave amplitudes[42] . If the dimension of the transfer matrices is 2m × 2m, there are m pairs of Lyapunov exponents that may be arranged as followings[43] , γ1 ≥ γ2 ≥ · · · ≥ γm ≥ γm+1 (= −γm ) ≥ γm+2 (= −γm−1 ) ≥ · · · ≥ γ2m (= −γ1 )

(14)

The smallest positive Lyapunov exponent γm (or equivalently −γm+1 ) is the localization factor, because γm represents the wave that has potentially the least amount of decay, propagates longest in all waves and has the farthest distance of energy propagation in the structure. The expression for calculating the localization factor of the system was given by Wolf[41] n

1

(k) ln Vˆ R,m n→∞ n

γm = lim

(15)

k=1

(k) where the vector Vˆ R,m is obtained by following the procedure below. Consider a 2m×2m transfer matrix. In order to calculate the m-th Lyapunov exponent, m orthogonal (0) (0) (0) unit vectors of 2m-dimension, u1 , u2 , · · · , um , are chosen as the initial state vectors. At the k-th iteration, we have (k)

VR,p = C k u(k−1) p

(k = 1, 2, · · · , n;

p = 1, 2, · · · , m)

(16)

where the matrix C k is the transfer matrix of the k-th unit cell and can be obtained by Eq.(13). It is (k) noted that the vectors V R,p (p = 1, 2, · · · , m) are usually not orthogonal. Therefore the Gram-Schmidt

(k) orthonormalization procedure[43] will be applied to produce m orthogonal unit vectors, Vˆ R,p . In this paper, the Lyapunov exponents, calculated by the plane-wave-based transfer-matrix method from which the transfer matrixes are subsequently obtained from the outputting slice to the inputting slice which is in reverse order to the ordinal transfer matrix method[32] , are all negative and arranged in a descending order. So, in order to obtain the localization factor, we just need to calculate the first Lyapunov exponent, i.e., the highest negative Lyapunov exponent, which has an opposite sign with the localization factor. Therefore, the calculation is simplified without the Gram-Schmidt orthonormalization procedure.

IV. NUMERICAL EXAMPLES AND DISCUSSION In this section we consider the phononic crystals of a square lattice made of plumbum (scattering material) and epoxy (matrix material). Detailed calculation is performed for ordered, randomly disordered and quasi-periodic PNCs. It is known that the BZ is no longer applicable to the non-periodic systems. In the following analysis, if we mention the BZ for the disordered or quasi-periodic PNCs, we refer it to the corresponding ordered systems.

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4.1. Ordered and Randomly Disordered Phononic Crystals For the ordered PNC shown in Fig.1(a), we take the filling fraction F0 = 0.5 in calculations; and for the disordered PNC shown in Fig.2(a), we consider that the filling fraction F that varies randomly in x-direction in the uniformly distributed interval, √ √ F ∈ [F0 (1 − 3δ), F0 (1 + 3δ)] (17) where δ is defined as the disorderd degree of the system. Introducing a standard uniformly distributed random variable, t ∈ (0, 1), F is expressed as √ F = F0 [1 + 3δ(2t − 1)] (18) First, in order to testify the correctness of our method, the localization factors (Figs.3(c) and 4(c)), the transmission coefficients (Figs.3(a) and 4(a)) and the dispersion curves (Figs.3(b) and 4(b)) along the high-symmetry lines of the first BZ, i.e. Γ X (Fig.3) and M Γ (Fig.4) directions, are calculated for the in-plane wave modes by using the present method, the eigenmode match theory[40] and the plane wave expansion method, respectively. In the figures the circular frequency and the wave number are normalized as ω ¯ = ωa/(2πcT 2 ) and k¯y = ky a/π, respectively, where cT 2 is the velocity of the transverse wave propagating in the matrix. According to the definition of the localization factor, if its value is zero, the corresponding frequency intervals are known as passbands; otherwise if its value is positive, the intervals are known as stopbands or band gaps. It can be seen that the band gaps, the frequency intervals of (0.52, 0.70), (0.71, 0.845)

Fig. 3. The band gaps in Γ X direction predicted by the transmitted coefficients (a), dispersion curves (b) and localization factors (c), respectively.

Fig. 4. The band gaps in M Γ direction predicted by the transmitted coefficients (a), dispersion curves (b) and localization factors (c), respectively.

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Fig. 5. The transmitted coefficients (a) and localization factors (the dashes line in (b)) of the disordered PNC with the disordered degree of 0.05. The solid line in (b) is for the corresponding perfect one (δ = 0).

(in Fig.3) and (0.48, 0.70) (in Fig.4), predicted by the localization factors are in good agreement with those shown by both the transmission coefficients and the dispersion curves. Similar results, although not presented here, are also shown for the anti-plane wave modes. Therefore, it can be concluded that the localization factor is an effective parameter to describe the band gaps of two-dimensional PNCs. Next we consider the band structures of the two-dimensional randomly disordered PNC with the filling fraction having a disorder degree of δ = 0.05. The band structures characterized by the transmitted coefficients and localization factors in Γ X direction are calculated and shown in Figs.5(a) and (b) (the dashed line), respectively. It is observed that the localization factor becomes positive in the passbands for the corresponding ordered system (Fig.3) when δ is nonzero. This behavior is the so-called localization of elastic waves which can be found in the frequency intervals (0.695, 0.715) and (0.845, 0.910) as shown in Fig.5(b). Comparison between Fig.5(b) (disordered PNC) and Fig.3(a) (ordered PNC) shows that the transmitted coefficients of the disordered PNC (Fig.5(b)) in these two frequency intervals decrease greatly, which is due to the wave localization behavior. As discussed above, it can be seen that both localization factor and transmitted coefficient predict the same wave propagation and localization behaviors. This implies that the localization factor is an effective and efficient parameter to describe the localization behavior of the two-dimensional randomly disordered PNCs. To show the effects of k¯y on the band structures, we illustrate the localization factors varying with the normalized frequency ω ¯ and k¯y in the gray-scale maps for both ordered PNC (Figs.6(a) and 7(a)) and disordered PNC with δ = 0.05 (Figs.6(b) and 7(b)) for oblique propagation of the in-plane (Fig.6) and anti-plane (Fig.7) waves. Similar structures of the gray-scale maps are shown for both ordered (Figs.6(a) and 7(a)) and disordered (Figs.6(b) and 7(b)) systems. But the localization factors for the disordered system are of small positive values in the corresponding frequency regions where it vanishes

Fig. 6. Gray-scale map of the localization factors for the oblique propagation of the in-plane wave modes in the ordered PNC (a) and disordered PNC with the disorder degree of 0.05 (b).

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Fig. 7. Gray-scale map of the localization factors for the oblique propagation of the anti-plane wave modes in the ordered PNC (a) and disordered PNC with the disorder degree of 0.05 (b).

¯y = 0.2 parallel to Γ X direction predicted by the dispersion curves (a) and localization Fig. 8. Band gaps along the line of k factors (b), respectively.

for the ordered system. This localization phenomenon becomes more and more pronounced with increase of the frequency ω ¯ . Particularly notice that a band gap appears first as the frequency increases from zero for k¯y > 0. This behavior was also shown in the one-dimensional layered PNCs[28] , which is due to the total reflection of the waves at the interfaces. The values of the localization factors shown in Figs.6 and 7 in this paper are generally smaller than those shown in Figs.3 and 7 of Ref.[28]. The localization factors in any directions represented by the lines parallel to Γ X direction in the reciprocal space can be obtained from Figs.6 and 7. For instance, considering the directions shown by the dashed line of k¯y = 0.2 in Fig.1(b), the localization factors corresponding to this line are obtained by first sectioning the surface of the localization factor versus ω ¯ and k¯y with the plane of k¯y = 0.2 as shown by the solid line in Fig.7(a) and then projecting the sectioned curves on the plane of ω ¯ . The band structures presented by dispersion curves and the localization factors are shown in Figs.8(a) and (b), respectively. It can be seen that the localization factor predicts the same band gaps as the dispersion curve does. 4.2. Quasi-periodic Phononic Crystals Two kinds of two-dimensional QPNCs shown in Figs.2(b) and (c) will be considered. We first consider the one shown in Fig.2(b). It is generated by arranging the lattice parameters in the x-direction, b and c, in the one-dimensional Fibonacci sequence with b/c = τ (τ is the golden ratio), while the scatterer size and the lattice constant in y-direction unchanged. As we know, a periodic average structure (PAS) can be defined for each quasi-periodic structure[44] ; and for the Fibonacci sequence, its PAS has the lattice constant of apas = (3 − τ )c[36] . In calculations, we take the lattice constant of the QPNC in y-direction as apas . Then the filling fractions of the sub-layers with different lattice constants, b and c,

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in x-direction are (2l)2 3 − τ  2l 2 = = 0.854F0 = 0.427 bapas τ apas  2l 2 (2l)2 = (3 − τ ) = 1.382F0 = 0.691 Fc = capas apas

Fb =

(19) (20)

The other QPNC as shown in Fig.2(c) is generated by perturbing the filling fraction F (or equivalently, changing the scatterer size) of the ordered structure with the lattice constant of apas resulting in a Fibonacci sequence in x-direction. Particularly, we consider the QPNC formed by two kinds of sub-layers with Fb = 0.427 and Fc = 0.691 arranged in a Fibonacci sequence in x-direction. Concisely, these two kinds of QPNCs are named as QPNC-A and QPNC-B, respectively. Figure 9 shows the band structures of QPNC-A. The dashed line is for the transmission coefficients and the black solid line for the localization factors. It is seen that the localization factors and the transmitted coefficients predict the same band gaps, e.g. the frequency intervals of (0.295, 0.33) and (0.46, 0.83). For comparison, the results for the perfect PNC (δ = 0) and randomly disordered PNC (δ = 0.05) are also shown in Fig.10 by the dotted and dashed lines, respectively. One distinguishing feature of the QNPC is that it exhibits more band gaps with narrower width than the ordered and randomly disordered systems. Even at very lower frequencies, small band gaps, e.g. the frequency intervals of (0.185, 0.2) and (0.29, 0.33) as shown in Fig.10, appear in the QNPC but not in the ordered and randomly disordered ones. It is also noted that the localization factors in the passbands of the ordered PNCs, e.g. (0.7, 0.71) and (0.84, 0.92), do not vanish for the QPNC. This implies that localization is the inherent feature for QPNCs. However, this localization behavior is generally slighter for QPNCs

Fig. 9. Band structures of the QPNC-A predicted by the transmission coefficients (the dashed line) and localization factors (the solid line), respectively.

Fig. 10. Localization factors of QPNC-A (the solid line), its PAS (the dotted line) and the randomly disordered PNC with the disorder degree of 0.05 (the dashed line).

Fig. 11. Gray-scale map of the localization factors for the oblique propagation of the in-plane wave modes in QPNC-A (a) and QPNC-B (b).

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than for randomly disordered PNCs. Similar feature has been demonstrated for the one-dimensional layered Fibonacci sequence in Ref.[29]. Finally we illustrate the localization factors varying with ω ¯ and k¯y in the grey-scale maps, see Fig.11 for in-plane wave propagating in QPNC-A (Fig.11(a)) and QPNC-B (Fig.11(b)), respectively. The corresponding grey-scale map of their PAS is shown as Fig.6(b). Comparison of these three figures shows that the band structures can be greatly changed when one-dimensional quasi-periodicity is introduced to the perfect two-dimensional PNCs in various ways. The QPNCs involve more band gaps that can be tuned by choosing different quasi-periodic parameters such as lattice constant a, filling friction F , etc. Compared with the results of the one-dimensional QPNC with Fibonacci sequence discussed in Ref.[29], the grey-scale maps (Fig.11) in this paper have absolutely white zones (i.e. passbands) without introducing the translational symmetry.

V. CONCLUSION REMARKS The plane-wave-expansion based transfer matrix method is generalized to calculate the localization factor which is introduced to describe the band structures and localization behaviors of twodimensional perfect phononic crystals and two-dimensional phononic crystals with random disorder or quasi-periodicity (Fibonacci sequence) in one direction and translational symmetry in the other direction. Comparison with other parameters such as the dispersion relations and transmission coefficients, etc. shows The localization factor is an effective parameter in characterizing the band gaps of two-dimensional perfect phononic crystals. It can characterize the band structures not only along the high-symmetry edges but also the inner of the first Brillouin Zone. The localization factor is an effective parameter in characterizing the band gaps and localization behaviors of two-dimensional randomly disordered phononic crystals. Localization phenomenon appears in the random disordered phononic crystals. Band structures of the phononic crystals can be tuned by changing random disorder. The localization factor is also an effective parameter in characterizing the band gaps and localization behaviors of quasi phononic crystals. Localization is the inherent feature for quasi phononic crystals but is generally slighter than that in randomly disordered phononic crystals. The quasi phononic crystals exhibit more band gaps with narrower width than the ordered and randomly disordered systems. For the quasi phononic crystals band gaps may appear in the low frequencies, showing potential application in design of low-frequency filters. Acknowledgement The authors would like to thank Dr. Z.L. Hou from Nacy University in France for helpful discussion and providing the eigenmode match code. We are also grateful to Dr. Z.Y. Li from Chinese Academy of Sciences for helpful discussion on the theory of the plane-wave-based transfer-matrix method.

References [1] Kushwaha,M.S., Halevi,P., Martinez,G., Dobrzynski,L. and Djafarirouhani,B., Acoustic band structure of periodic elastic composites. Physical Review Letters, 1993, 71: 2022-2025. [2] Martinez-Sala,R., Sancho,J., Sanchez,J.V., Gomez,V., Llinares,J. and Meseguer,F., Sound-attenuation by sculpture. Nature, 1995, 378: 241-241. [3] Yablonovitch,E., Inhibited spontaneous emission in solid state physics and electronics. Physical Review Letters, 1987, 58: 2059-2062. [4] John,S., Strong localization of photons in certain disordered dielectric superlattices. Physical Review Letters, 1987, 58: 2486-2489. [5] Sigalas,M.M. and Soukoulis,C.M., Elastic-wave propagation through disordered and/or absorptive layered systems. Physical Review B, 1995, 51: 2780-2789. [6] Vasseur,J.O., Djafari-Rouhani,B., Dobrzynski,L. and Deymier,P.A., Acoustic band gaps in fibre composite materials of boron nitride structure. Journal of Physics: Condense Matter, 1997, 9: 7327-7341. [7] Tanaka,Y. and Tamura,S., Two-dimensional phononic crystals: surface acoustic waves. Physica B, 1999, 263-264: 77-80. [8] Wu,T.T., Huang,Z.G., and Lin,S., Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy. Physical Review B, 2004, 69: 094301.

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[9] Vasseur,J.O., Deymier,P.A., Khelif,A., Lambin,Ph., Djafari-Rouhani,B., Akjouj,A., Dobrzynski,L., Fettouhi,N. and Zemmouri,J., Phononic crystal with low filling fraction and absolute acoustic band gap in the audible frequency range: A theoretical and experimental study. Physical Review E, 2002, 65: 056608. [10] Wu,F.G., Liu,Z.Y. and Liu,Y.Y., Splitting and tuning characteristics of the point defect modes in twodimensional phononic crystals. Physical Review E, 2004, 69: 066609. [11] Khelif,A., Deymier,P.A., Djafari-Rouhani,B., Vasseur,J.O. and Dobrzynski,L., Two-dimensional phononic crystal with tunable narrow pass band: Application to a waveguide with selective frequency. Journal of Applied Physics, 2003, 94: 1308-1311. [12] Psarobas,I.E. and Sigalas,M.M., Elastic band gaps in a fcc lattice of mercury spheres in aluminum. Physical Review B, 2002, 66: 052302. [13] Yan,Z.Z. and Wang,Y.S., Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals. Physical Review B, 2006, 74: 1. [14] Yan,Z.Z. and Wang,Y.S., Wavelet-based method for computing elastic band gaps of one-dimensional phononic crystals. Science in China Series G — Physics and Astronomy, 2007, 50: 622-630. [15] Yan,Z.Z., Wang,Y.S. and Zhang,C.Z., Wavelet method for calculating the defect states of two-dimensional phononic crystals. Acta Mechanica Solida Sinica. 2008, 21: 105-109. [16] Torres,M., Montero de Espinosa,F.R., Garc´ıa-Pablos,D. and Garc´ıa,N., Sonic band gaps in finite elastic media: surface states and localization phenomena in linear and point defects. Physical Review Letters, 1999, 82: 3054-3057. [17] Sigalas,M.M., Defect states of acoustic waves in a two-dimensional lattice of solid cylinders. Journal of Applied Mechanics, 1998, 84: 3026-3030. [18] Kafesaki,M., Sigalas,M.M. and Garc´ıa,N., Frequency modulation in the transmittivity of wave guides in elastic-wave band-gap materials. Physical Review Letters, 2000, 85: 4044-4047. [19] Anderson,P.W., Absence of diffusion in certain random lattices. Physical Review, 1958, 109: 1492-1505. [20] Li,F.M., Wang,Y.S. and Chen,A.L., Wave localization in randomly disordered periodic piezoelectric rods. Acta Mechanica Solida Sinica, 2006, 19: 50-57. [21] Zhang,Y.P., Yao,J.Q., Zhang,H.Y., Zheng,Y. and Wang,P., Bandgap extension of disordered 1D ternary photonic crystals. Acta Photonica Sinica, 2005, 34: 1094-1098 (in Chinese). [22] Zhao,Z., Gao,F., Peng,R.W., Cao,L.S., Li,D., Wang,Z., Hao,X.P., Wang,M. and Ferrari,C., Localizationdelocalization transition of photons in one-dimensional random n-mer dielectric systems. Physical Review B, 2007, 75: 165117. [23] Bendiksen,O.O., Localization phenomena in structural dynamics. Chaos, Solitons and Fractals, 2000, 11: 1621-1660. [24] Ariaratnam,S.T. and Xie,W.C., Wave localization in randomly disordered nearly periodic long continuous beams. Journal of Sound and Vibration, 1995, 181: 7-22. [25] Asatryan,A.A., Robinson,P.A., Botten,L.C., McPhedran,R.C., Nicorovici,N.A. and Martijn de Sterke,C., Effects of disorder on wave propagation in two-dimensional photonic crystals. Physical Review E, 1999, 60: 006118. [26] Vinogradov,A.P. and Merzlikin,A.M., Band theory of light localization in one-dimensional disordered systems. Physical Review E, 2004, 70: 026610. [27] Zhang,D.Z., Hu,W., Zhang,Y.L., Li,Z.L., Cheng,B.Y. and Yang,G.Z., Experimental verification of light localization for disordered multilayers in the visible-infrared spectrum. Physical Review B, 1994, 50: 98109814. [28] Chen,A.L. and Wang,Y.S., Study on band gaps of elastic waves propagating in one dimensional disordered phononic crystals. Physica B, 2007, 392: 369-378. [29] Chen,A.L., Wang,Y.S., Guo,Y.F. and Wang,Z.D., Band structures of Fibonacci phononic quasicrystals. Solid State Communications, 2008, 145: 103-108. [30] Xie,W.C., Chaos, buckling mode localization in nonhomogeneous beams on elastic foundations. Chaos, Solitons and Fractals, 1997, 8: 411-431. [31] Li,F.M. and Wang,Y.S., Study on wave localization in disordered periodic layered piezoelectric composite structures. International Journal of Solids and Structures, 2005, 42: 6457-6474. [32] Li,F.M., Wang,Y.S., Hu,C. and Huang,W.H., Wave localization in randomly disordered periodic layered piezoelectric structures. Acta Mechanica Sinica, 2006, 22: 559-567. [33] Aynaou,H., Boudouti,E.H.EI., Djafari-Rouhani,B., Akjouj,A. and Velasco,V.R., Propagation and localization of acoustic waves in Fibonacci phononic circuits. Journal of Physics: Condensed Matter, 2005, 17: 4245-4262. [34] Peng,R.W., Wang,M., Hu,A., Jiang,S.S., Jin,G.J. and Feng,D., Characterization of the diffraction spectra of one-dimensional k-component Fibonacci structures. Physical Review B, 1995, 52: 13310-13316.

· 528 ·

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2008

[35] King,P.D.C. and Cox,T.J., Acoustic band gaps in periodically and quasiperiodically modulated waveguides. Journal of Applied Physics, 2007, 102: 014902. [36] Steurer,W. and Sutter-Widmer,D., Photonic and phononic quasicrystals. Journal of Physics D: Applied Physics, 2007, 40: R229-R247. [37] Velasco,V.R., Perez-Alvarez,R. and Garcia-Moliner,F., Some properties of the elastic waves in quasiregular heterostructures. Journal of Physics: Condensed Matter, 2002, 14: 5933-5957. [38] Sesion Jr,P.D., Albuquerque,E.L., Chesman,C. and Freire,V.N., Acoustic phonon transmission spectra in piezoelectric AlN/GaN Fibonacci phononic crystals. The European Physical Journal B, 2007, 58: 379-387. [39] Li,Z.Y. and Lin,L.L., Photonic band structures solved by a plane-wave-based transfer-matrix method. Physical Review E, 2003, 67: 046607. [40] Hou,Z.L., Kuang,W.M. and Liu,Y.Y., Transmission property anslysis of two-dimensional phononic crystal. Physics Letters A, 2004, 333: 172-180. [41] Wolf,A., Swift,J.B., Swinney,H.L. and Vastano,J.A., Determining Lyapunov exponents from a time series. Physica D, 1985, 16: 285-317. [42] Castanier,M.P. and Pierre,C., Lyapunov exponents and localization phenomena in multi-coupled nearly periodic systems. Journal of Sound and Vibration, 1995, 183: 493-515. [43] Xie,W.C., Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners. Computers and Structures, 1998, 67: 175-189. [44] Sutter-Widmer,D., Deloudi,S. and Steurer,W., Periodic average structures in phononic quasicrystals. Philosophical Magazine, 2007, 87: 3095-3102.