Band gap structures for viscoelastic phononic crystals based on numerical and experimental investigation

Applied Acoustics 106 (2016) 93–104

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Band gap structures for viscoelastic phononic crystals based on numerical and experimental investigation Xingyi Zhu, Sheng Zhong, Hongduo Zhao ⇑ Key Laboratory of Road and Traffic Engineering of Ministry of Education, Tongji University, Shanghai 200092, PR China

a r t i c l e

i n f o

Article history: Received 14 August 2015 Received in revised form 22 December 2015 Accepted 4 January 2016

Keywords: Phononic crystal Viscoelasticity Band gaps Defect modes Boundary element method

a b s t r a c t Many materials used as phononic crystals (PCs) are viscoelastic one. It is believed that viscosity results in damping to attenuate wave propagation, which may help to tune the defect modes or band gaps of viscoelastic phononic crystals. To investigate above phenomenon, firstly, we have extended the application of boundary element method (BEM) to the study of viscoelastic phononic crystals with and without a point defect. A new developed BEM within the framework of Bloch theory can easily deal with viscoelastic phononic crystals with arbitrary shapes of the scatterers. Experimental methods have been put forward based on the self-made viscoelastic phononic crystals. Verified by the experimental results, systematic comprehensive parametric studies on the band structure of viscoelastic phononic crystals with varying factors (final–initial value ratio, relaxation time, volume fraction of scatterers, shapes of scatterers) have been discussed by the numerical simulation. To further address the possibility to change the defect modes, the band structure of viscoelastic phononic crystals with a point defect has been studied based on the numerical and experimental methods. From present research work, it can be found that by adjusting the two viscous parameters combined with considering the effect of volume fraction and shapes, a wider and lower initial forbidden frequency or lower and higher quality factor resonant frequency can be obtained. Ó 2016 Published by Elsevier Ltd.

1. Introduction The phononic crystal (PC) is a typical periodic structure. Because of its periodicity, there exist band gaps within which the propagation of wave will be forbidden. Based on this special characteristic, many applicants can be designed, such as wave filters, waveguides, noise barriers, and lenses [1–5]. Therefore, the phononic crystal has been extensively investigated recently by experimental, analytical, and numerical methods [6–9]. Among above researches, the host matrix and scatterers are usually considered as elastic materials. Actually, many materials used as phononic crystals are viscoelastic one, for example, epoxy, rubber, silicon rubber, and many other polymer composites. Besides, most materials behave as elastic bodies at room temperature. However, at high temperature, they present apparent viscoelastic properties [10–12]. Viscoelastic materials possess viscous and elastic properties simultaneously. In the frequency domain, the material parameters are complex numbers and frequency-dependent. In the low frequency ranges, the elastic

⇑ Corresponding author. E-mail address: [email protected] (H.D. Zhao). http://dx.doi.org/10.1016/j.apacoust.2016.01.007 0003-682X/Ó 2016 Published by Elsevier Ltd.

moduli are much smaller, while the elastic moduli become larger in the high frequency ranges. Therefore, it may help to lower the initial forbidden frequency and widen the band gaps [13,14]. Riese and Wegdam [15] believed that viscoelasticity would promote the transverse coupling of neighbouring scatterers, which leads to the wider absolute acoustic band gaps compared with those without viscosity. Psarobas [16] investigated the effect of viscoelastic losses in a high-density rubber–air phononic crystal by the multiple scattering method. In his study, the rubber was modeled as the Kelvin– Voigt system, and the sharp peaks and dips in the resonant states of scatterers were washed out because of the viscous properties. Merheb et al. [17,18] developed the finite difference time domain method to investigate the rubber/air phononic crystals. They found out that the viscoelasticity would attenuate transmission over wider frequency ranges, which results in a lower initial frequency. Liu et al. [19] used the Kelvin–Voigt model based on the fractional derivative method to evaluate dispersion and dissipation phenomenon in the viscoelastic phononic crystals. They noticed that the band gaps are widened and the attenuation is enhanced. Zhao and Wei [20,21] observed the influence of viscosity on band gaps of 1D and 2D phononic crystals by means of plane wave expansion method. They thought the viscosity causes all wave bands shifting

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toward lower frequencies. However, the shift amplitude is different for different wave bands. Hussein and Frazier [22] adopted the state-space method to analysis the band structure of viscously damped phononic crystals, they found out that the optical branch is more sensitive to the stiffness-proportional damping, while the acoustical branch is more sensitive to the mass-proportional damping. Based on the above investigations, it can be concluded that viscoelasticity can contribute to widen band gaps and lower the initial forbidden frequency. Therefore, in this paper, we try to discuss the practical design of viscoelastic phononic crystals to get a wider band gap and a lower initial frequency. Besides, to the best of the authors’ knowledge, defect modes for viscoelastic PCs have not yet been researched so far. To analyze the defect modes of PCs, a supercell system has to be established, which results in a largesize computational consumption. To solve this problem, a new boundary element method (BEM) considering the viscoelastic effect is developed to simulate the wave propagation behavior in the viscoelastic phononic crystals with or without defects. Compared with the conventional numerical method, such as plan wave expansion method [23], multiple scattering method [24], finite difference time domain method [25], and wavelet method [26], BEM has some special advantages. It automatically satisfies radiation conditions that are inherent to the scattering problems [27,28], besides, its dimensionality reduction for linear problems offers a higher efficiency and lower storage. Recently, Li et al. [29–31] gave a conventional BEM to research the band structures of solid/solid and solid/liquid phononic crystals. However, in their papers, they only considered phononic crystals as an elastic body, based on the ideas they have developed, we try to extend BEM to the study of viscoelastic phononic crystals. In this paper, a BEM for 2D viscoelastic phononic crystals is developed. By means of the Fourier transformation method, the constitutive relation for an isotropic linear viscoelastic media can be easily transferred from time-domain to frequency-domain. Then, eigen equations which can be further adopted to simulate the viscoelastic phononic crystals are obtained. The threeparameter model and an 8-element generalized Maxwell solid model are used to model the viscoelastic behavior of host matrix. Then, the experimental investigation has been carried out on the self-made viscoelastic phononic crystals with or without defects. The effects of final–initial value ratio, relaxation time, volume fraction of scatterers, and shapes of scatterers are discussed. The localization phenomenon for viscoelastic PCs with a point defect is also researched. Results show that viscoelasticity not only can attenuate transmission over wider range, but also can tune the defect mode. Furthermore, viscous parameters (final–initial value ratio and relaxation time) are two major factors affect the band gaps, and combined with other two parameters (volume fraction and shape of the scatterer), a wider and lower initial forbidden frequency or lower and higher quality factor resonant frequency can be obtained. 2. Methods and models 2.1. Boundary element method for 2D viscoelastic problems Suppose the periodic array of homogeneous and isotropic elastic scatterers are embedded in the linear viscoelastic host materials, see Fig. 1. For an isotropic linear viscoelastic media, the constitutive relation can be given as [32]

Z

rij ðx; y; tÞ ¼

t

1

Gijkl ðt  sÞ

dekl ðx; y; tÞ ds ds

ð1Þ

where rij and eij are the stress tensors and the strain tensors, respectively. Gijkl is the relaxation function which can be written in terms of two time-dependent Lame coefficients (kðtÞ and l(t))

  Gijkl ðtÞ ¼ kðtÞdij dkl þ lðtÞ dik djl þ dil djk

ð2Þ

where dij is the Kronecker delta. Substituting Eq. (2) into Eq. (1), the following relationships can be obtained:

Z

rij ðx; tÞ ¼

t

1



kðt  sÞ

dekk ðx; tÞ dij ds þ ds

Z

t

1

2lðt  sÞ

deij ðx; tÞ dij ds ds

ð3Þ

For the harmonic wave motion, the strain and stress can be written in a harmonic function of time, i.e.,

eij ðx; sÞ ¼ eij ðxÞeixs ; rij ðx; sÞ ¼ rij ðxÞeixs

ð4Þ

deij ðx; sÞ ¼ ixeij ðxÞeixs ds where x is the circular frequency. Therefore, based on all above equations, Eq. (3) can be finally written as

Z

 kðnÞeixn dn ekk ðxÞdij 1 Z 1  þ ix 2lðgÞeixn dn eij ðxÞ

rij ðxÞ ¼ ix

1

1

ð5Þ

R1 It can be observed that the terms ix 1 kðnÞeixn dn and R1 ix 1 lðnÞeixn dn are just the mathematical Fourier transformation formulations employed to transfer the problem from timedomain to frequency-domain. Then, if we introduce the frequency-dependent Lame constants kðxÞ and l(x) into Eq. (5), almost the same constitutive relation as for linear elastic problems can be obtained, i.e.,

rij ðx; xÞ ¼ kðxÞdij ekk ðx; xÞdij þ 2lðxÞeij ðx; xÞ

ð6Þ

The only difference between the elastic and viscoelastic constitutive relation is that kðxÞ and l(x) are complex-valued frequency dependent functions. Based on the constitutive relations for linear viscoelastic problems (see Eq. (6)), the fundamental solutions for 2D anti-plane viscoelastic problems are

U3 ¼

  1 xr ; K0 i 2plðxÞ c2

P3 ¼ 

  i x @r xr K1 i 2p c2 @n c2

ð7Þ

where K0(z) and K1(z) are the modified Bessel functions of order 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and 1, respectively. c2 ¼ lðxÞ=q is the transverse wave speed, it is also dependent on the frequency; and q is the density, pffiffiffiffiffiffiffi r = |x  y|, and i ¼ 1. Because of the periodicity of the phononic crystals, we only need to calculate the band gaps among the unit cell, see Fig. 1. The boundary integral equations corresponding to the anti-plane time-harmonic problems for matrix and scatterers can be given respectively as follows:

Z m ckl ðyÞum Pm 3 ðx; y; xÞu3 ðx; xÞdSðxÞ 3 ðy; xÞ þ si Z m Um 8y 2 Si ðxÞ; i ¼ 1; 2; 3; 4 ¼ 3 ðx; y; xÞp3 ðx; xÞdSðxÞ

ð8Þ

si

Z Ps3 ðx; y; xÞus3 ðx; xÞdSðxÞ ckl ðyÞus3 ðy; xÞ þ s0 Z ¼ U s3 ðx; y; xÞps3 ðx; xÞdSðxÞ 8y 2 S0 ðxÞ s0

ð9Þ

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Fig. 1. 2D viscoelastic phononic crystals with square lattice.

where ckl is the geometry-dependent parameter, which can be determined as 12 dkl for the smooth boundary. u3 and p3 are the displacements and tractions, respectively. Subscripts m and s indicate matrix and scatterer. S(x) is the boundary of the domain, see Fig. 1. If Eqs. (8) and (9) are discretized by N constant elements, we can finally obtain a linear algebraic system based on two N  N matrices for matrix domain (H) and scatterer domain (G): m Hm ij USj ¼ Gij PSj

ð10Þ

Hsi0 U0S0 ¼ Gsi0 P0S0

ð11Þ

2.2. Linear viscoelastic model To describe the creep/recovery and stress relaxation behavior of viscoelastic materials, the simplest model is the three-parameter solid model, see Fig. 2. We can think it as composed of a spring in series with a Kelvin–Voigt element. Under a constant load, this model can be used to simulate two types of deformations, namely, the spontaneous elastic deformation and delayed elastic deformation (reversible creep). The differential constitutive relation of the three-parameter solid model can be written as

r þ p1 r_ ¼ q0 e þ q1 e_

where USj and PSj are the displacements and tractions of the matrix at the boundary Sj , while U0S0 and P0S0 are the displacements and tractions of the scatterers at the boundary S0. USj and PSj can be written as T

T

USj ¼ ½ US0 US1 US2 US3 US4  ; PSj ¼ ½ PS0 PS1 PS2 PS3 PS4  ; Hij and Gij are two N  N matrices, which are in the form of

Hij ¼

1 dij þ 2

Z P3 dS;

Gij ¼

U 3 dS

ð12Þ

sj

Because of the continuity at the interface between the scatterers and the matrix, we have

U0S0 ¼ US0 ;

P0S0 ¼ PS0

ð13Þ

In a periodic structure, the stress, displacements, and their normal derivatives should obey the Bloch–Floquet periodical condition, which is

/ðx þ aÞ ¼ eika /ðxÞ

ð14Þ

where a is the lattice vector, and k = (kx, ky) is the Bloch wave vector in the first irreducible Brillouin zones. Therefore, based on the boundary conditions of Eqs. (13) and (14), Eqs. (10) and (11) can be combined in a more compact form as

AX þ nBX ¼ 0

g1 E1 þ E2

; q0 ¼

E1 E2 g E2 ; q1 ¼ 1 E1 þ E2 E1 þ E2

ð15Þ

T where X ¼ ½ US1 US2 TS1 TS2 US0 TS0  . Since we only need to figure out the band structures along the first irreducible Brillouin zones, A and B can be given along the boundaries of the first irreducible Brillouin zones MC; CX; XM separately, see Appendix A [29–31,33]. In Eq. (15), n ¼ eikx a or n ¼ eiky a , which should be on the unit circle, therefore, we save the wave vectors (kx, ky) for the given x when |n| = 1. Based on the given x and calculated wave vectors (kx, ky), we can finally obtain the band structures of the viscoelastic phononic crystals.

ð17Þ

where g1 is the viscosity of the dashpot component, and E1, E2 are the elastic parameters of the spring components, r_ ¼ ddtr ; e_ ¼ ddte. if t ! 0; E0 ¼ E2 is the initial elastic modulus of the viscoelastic mateE2 rials, while if t ! 1; E1 ¼ EE11þE is the final elastic modulus of the 2 viscoelastic materials, and

Z

sj

p1 ¼

ð16Þ

sE ¼ E1gþE1 2 is the relaxation time.

In the frequency domain, the elastic moduli of viscoelastic material are frequency-dependent. Based on Eq. (16), the complex Young’s modulus of the three-parameter solid model can be obtained as

EðxÞ ¼

q0 þ q1 ix E1 þ E0 s2E x2 ðE0  E1 ÞsE x ¼ þ i 1 þ p1 ix 1 þ s2E x2 1 þ s2E x2

ð18Þ

Similarly, the complex shear modulus can be expressed as

lðxÞ ¼

l1 þ l0 s2l x2 ðl1  l0 Þsl x þ i 1 þ s2l x2 1 þ s2l x2

ð19Þ

The real part of the complex elastic modulus, namely, the storage modulus, reflects the frequency-dependent characteristic, while the imaginary part, namely, the loss modulus, reflects the dispersive characteristic of the materials. Therefore, we only need to consider the real part of the complex elastic modulus when investigating the wave propagation behavior in the viscoelastic phononic crystals [20,21]. Suppose a = E1/E0 is the final–initial value ratios, Fig. 3 plots frequency-dependent curves of shear

E1

E2

η1 Fig. 2. The three-parameter solid model.

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Fig. 3. Frequency-dependent curves of shear modulus under different a and relaxation time.

modulus under different a and relaxation time. If E1 = E0 or l1 = l0, the materials only present elastic behavior. On the contrary, the materials become more viscous with the decrease of a. 2.3. Experimental method Four types of viscoelastic phononic crystals are fabricated, .i.e., silicon rubber matrix/air scatterers and silicon rubber matrix/lead scatterers with or without a point defect (the radius of the point defect is 10 mm). A square array of 17  8 parallel 42 cm long air/lead cylinders is embedded in the silicon rubber matrix. The lattice parameter is 12 mm and the radius of the cylinder is 4 mm. The relaxation modulus of silicon rubber is obtained based on the dynamic mechanical analysis tests [17], and an 8-element generalized Maxwell solid model is used to fit the modulus data, see Fig. 4. The reason to use this model is that it can fit the test results better. The fitted viscoelastic parameters Ei and si (si = gi/Ei) listed in Table 1 are used in the numerical simulation. For an 8-element generalized Maxwell solid model, the dynamic storage modulus E0 (x) can be written as

E 0 ð xÞ ¼

n X Ei x2 s2i 1 þ x2 s2i i

ð20Þ

Therefore, the storage modulus curve for silicon rubber can be plotted as shown in Fig. 5. To avoid the disturbance from other sound resource and reflection from other obstacles, the experiments are preceded in the anechoic room. The experimental set-up can be found in Fig. 6. To produce compression waves (P waves), the speaker is put into a box, and the size of radiation outlet should be larger than the size of phononic crystals, and the outlet should be close to the PC as much as possible. 3. Wave propagation in viscoelastic phononic crystals without defects First, we check out if the viscoelastic phononic crystals will help to widen the band gaps and lower the initial forbidden frequency. Here, we consider cylindrical or elliptical elastic Aurum scatterers (qAu ¼ 19; 500 kg=m3 ; cAu t ¼ 1239 m=s) embedded in the viscoelastic Epoxy matrix (qEp ¼ 1180 kg=m3 ; cEp t ¼ 1160 m=s) in a square lattice, the volume fraction of the scatters f is 0.1. From Fig. 7, it can be clearly observed that for the calculations including Ep the viscoelastic effect (a ¼ EEp 1 =E0 ¼ 0:1), the gaps are widened significantly. Especially for the elliptical Au–Ep systems, almost

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97

Fig. 4. Experimental and fitted relaxation modulus of silicon rubber.

3.1. Effect of the final–initial value ratio

Table 1 Values of viscoelastic parameters Ei and si for silicon rubber. i

Ei (Pa)

si (s)

1 2 3 4 5 6 7 8

4.20E+06 2.94E+06 9.00E+06 2.41E+06 1.87E+06 1.31E+06 1.15E+06 2.13E+06

5.84E08 3.51E07 4.32E09 2.28E06 1.68E05 2.83E04 0.00854 1.89E+07

Fig. 5. Storage modulus curve for silicon rubber.

twice width as large as the band gaps without considering the viscosity. As for the cylindrical Au–Ep systems, the first band gaps are widened by 1.2 times, while the second band gaps are widened by 2 times. Besides, the initial forbidden frequency is also decreased from 0.33 to 0.14 and from 0.24 to 0.09 for elliptical and cylindrical Au–Ep systems respectively. Since the modulus of host viscoelastic materials varies with the frequency, the dispersion becomes more evident in the relatively higher frequencies, and less distinct in the relatively lower frequencies, which results in a wider gap and a lower initial forbidden frequency.

Consider circular elastic Aurum scatterers embedded in the viscoelastic Epoxy matrix. The volume fraction of the scatterers is f = 0.3. The developed numerical results based on BEM are compared with the results from plane wave expansion method (PWE). It can be observed form Fig. 8(a) that the BEM results match quite well with the PWE results when a = 1 (elastic case). Fig. 8 gives the details of band structures with the variation of final– Ep initial value ratios a ¼ EEp 1 =E0 . It is noted that the shape of the band structures changes little, which indicates that the viscoelastic characteristic of materials would not affect the band structures essentially. Fig. 9 gives the detail information (upper frequency, lower frequency, and band width) of the complete band gaps (CBG, M–C–X–M) and the directional band gaps (DBG, MC; CX; XM) of Aurum scatterers/Epoxy matrix phononic crystals. It is shown that the band gaps are widened and the initial forbidden frequencies are lowered when the matrix is more viscous no matter for CBG or for DBG. The smaller the final–initial value ratio is, the stronger the viscosity of host matrix is. Therefore, the initial frequency of CBG decreases from 2052 Hz to 774 Hz when a changes from a = 1 (elastic case) to a = 0.1, almost decreases 2.7 times compared with the elastic case. As for DBG, the initial forbidden frequencies decrease 2.65, 2.73, and 2.65 times (when a = 0.1) for MC; CX, and XM directions respectively compared with the elastic case. Except for getting the lower initial forbidden frequency, the band gap is also gradually widened with the decrease of a, which results from the dispersion of epoxy. The gaps have been widened from 2230 Hz to 2342 Hz when matrix becomes more viscous.

3.2. Effect of relaxation time Relaxation time is another important parameter to determine the level of viscosity of the materials, which therefore further affects the band gaps. Fig. 10 gives the variation of band gaps with the relaxation time. It can be observed that the initial frequency of the first CBG increases with the increase of the relaxation time, and the values of initial frequencies for viscoelastic materials (a < 1.0) are all less than the values for elastic materials (a = 1.0), see Fig. 10(a). This situation is quite reasonable, since the

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Fig. 6. Sketch of tests for (a) transmission frequency spectrum of phononic crystals and (b) defect modes of phononic crystals with defects.

1.1 1.0

Second band gap for viscoelasc media

0.9 9

Second band gap for elasc media

0.8 0.7

First band gap for viscoelasc media

ωa/2πct

0.6

First band gap for elasc media

0.5

Consider Epoxy matrix as viscoelastic medium(α =0.1) Consider Epoxy matrix as an elastic medium

0.4 0.3 0.2 0.1 1 0.0

M

Γ

X

M

(a) Cylindrical Aurum scatters ( f =0.1, τ =0.0003 )

1.2

1.0

0.8

ωa/2πct

0.6

Consider Epoxy matrix as an elastic medium Consider Epoxy matrix as a viscoelastic medium(α =0.1)

Band gap for 0.4 viscoelasc media

Band gap for elasc media

0.2

0.0

M

Γ

X

M

(b) Elliptical Aurum scatters ( f=0.1, τ =0.0003 ) Fig. 7. The band gaps with and without viscosity.

storage elastic modulus (i.e., Re(E)) of viscoelastic materials varies with the frequency and the values are much lower at low frequency than at high frequency, see Fig. 3. Besides, the values of

storage elastic moduli are all smaller than the elastic moduli of elastic materials under different relaxation time. Therefore, for viscoelastic case, the initial frequency of band gaps would shift

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Fig. 8. Band structures of Au/Ep viscoelastic phononic crystals with the variation of final–initial value ratios.

toward the lower frequency compared with the elastic materials. If the relaxation time is very short, the body exhibits properties of a viscous fluid. On the contrary, if the relaxation time is large enough, the materials are more like an elastic body. However, the response of most materials is a combination of viscous fluidity and elastic solidity. The smaller the value of relaxation time is, the shorter the time required for completing the same percentage of the total quantity of stress relaxation. Hence, when the relaxation time is quite small, the materials would get a smaller storage elastic modulus, but, the decrease of storage elastic moduli is not evident, see Fig. 3(c). Based on this fact, the structure gets a lower initial frequency when the relaxation time is relatively small (s = 1E  5), however, the corresponding band gap width is still narrow and even much narrower than the elastic case. From Fig. 10(b), we can also find out that for Au/Ep viscoelastic phononic crystals, only when the relaxation time is in the range of s = 3E – 40.01, the viscous characteristic can help to widen the band gap width. The main reason is that at this range, a best combination of lower and upper frequencies of first CBG can be achieved, which can help to widen the band gaps of PCs.

3.3. Effect of volume fraction of scatterers The effects of volume fraction (f) of scatterers are considered here. From Fig. 11, it can be concluded that at any volume fractions, with the decrease of a, the band gap width becomes wider, at the meanwhile, the initial forbidden frequency of the first CBG shifts to a lower value. Fig. 12 gives the variation of volume fractions under different final–initial value ratios. The volume fraction of the scatterers has a significant effect on the band gaps of phononic crystals. It can be observed that the band gap width and initial frequency of CBG do not vary monotonically with the increase of volume fraction. Based on Fig. 11, for the Epoxy–Aurum system, there exists

a narrowest band gap and a highest initial frequency when f = 30%. However, when f = 40% and a = 0.1, the structure gets a widest band gap (4856 Hz) and a lowest initial forbidden frequency (630 Hz). In short, the CBG and DBG size can be widened by changing the volume fraction of scatterers combined with adjusting the viscous of the host materials. 3.4. Effect of shapes of scatterers To analyze the effects of shapes of scatterers on the band gap, the lower and upper frequencies of first complete band gaps are plotted in Fig. 13 for different volume fractions when s = 0.0003. For three geometric shapes, the initial frequencies of first CBG are all decreased and the gap width becomes wider with the decrease of a. It can be concluded that viscoelasticity offers an attractive alternative ways for designing lower and wider frequency-forbidden appliance. Besides, we can also further tune the band gaps combined with the volume fractions and shapes of the scatterers. For the hexagon and elliptical scatterers, the values of initial frequency of first CBG are firstly decreased and then increased with the increase of volume fraction, while for the circle scatterers, the situation is more complicated, see Fig. 12(b). However, it can be noted that the circle rods have the lowest initial forbidden frequency among three geometric shapes for the Ep–Au system. Besides, the other important information is that the gap width of circle rods is also larger than the other two shapes. 4. Wave propagation in viscoelastic phononic crystals with a point defect The investigation of the localization phenomena of viscoelastic PC with a point defect is discussed by BEM. We consider the elliptical Aurum scatterer/Epoxy matrix PC firstly. To get a band gap

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Ep

Ep

Fig. 9. Band gaps width (Dxa=2pc2 0 ) and lower and upper frequencies (xa=2pc2 0 ) of CBG and DBG for Au/Ep phononic crystals (f = 0.3) under different final–initial value ratios (LF = lower frequency, UF = upper frequency).

Fig. 10. Variation of relaxation time under different a.

structure of PC with a point defect, a system in which the supercell consists of 5  9 elliptical cylinders is established. The aspect ratio for the elliptical scatterer is 4:3, and the volume fraction is 37.7%. The point defect can be realized by removing an Aurum scatter at the center of the PC. One defect mode appears in the band gap of

PC, and the normalized frequencies with and without viscosity are 0.48 (Fig. 14(a)) and 0.39 (Fig. 14(b)) correspondingly. Compared with the elastic case (Fig. 14(a)), the viscoelastic one has a lower defect mode frequency, which means viscoelasticity may tune the defect mode.

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Fig. 11. Variation of final–initial value ratios under different volume fractions (s = 0.0003 s).

Fig. 12. Variation of volume fractions under different final–initial value ratios (s = 0.0003 s).

Fig. 15 gives the transmission spectrum and band gaps for silicon rubber matrix/air scatterers PC with and without a point defect based on the experimental and numerical results. The numerical simulation considering the viscosity is also implemented, the viscoelastic parameters used in the calculation can be found in Table 1. When the viscosity in silicon rubber is neglected, i.e., a = 1, the first band gap starts from 921 Hz. However, when the viscoelasticity behavior of rubber is counted on, the initial frequency of the first band gap moves to 783 Hz, which matches well with the experimental results. Comparing the band gaps with or without viscosity, the initial frequency slightly shifts to the lower frequency when viscosity is considered. For the PC without defects, transmission decreases with frequency, and this attenuation becomes distinct at relatively higher frequencies, i.e., the transmission ratio is almost close to zero beyond 2.5 kHz. The reason is that for the silicon rubber, the loss factor (which is the ratio of energy dissipated from the system to the energy stored in the system) increases with the frequency, which means more energy dissipated due to the viscoelasticity of host silicon rubber.

For the PC with a point defect (the point defect is realized by enlarging the size of air scatter at the center), two defect modes are found in the band gaps. From test results, these two defect modes have resonant frequencies at 3.6 kHz and 4.2 kHz, which are almost identical to the numerical results. For the second defect mode, the transmission ratio increases 200 times from 0.02 to 0.4 compared to the case of PC without defects. Compared to the first defect mode, the bandwidth of the second mode is much narrower, which means it has a higher quality factor (resonant frequency/bandwidth of half resonant frequency). Fig. 16 gives the effect of the defect size (rd is the radius of the defect) on the resonant frequency of silicon rubber matrix/air scatterer PC. Here, the Epoxy matrix is considered as the viscoelastic media. With the increase of rd, the resonant frequencies for higher and lower modes all decrease correspondingly. The main reason is that with the increase of the defect size, the Mie scattering effect will be weakened while the Bragg scattering effect will be enhanced, meanwhile, the larger defect will destroy the periodicity

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Fig. 13. Lower and upper frequencies of first complete band gaps under different volume fractions and final–initial value ratios.

Fig. 14. Defect modes of PC with and without considering viscosity.

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Fig. 15. Transmission spectrum for silicon rubber matrix/air scatterers PC with and without defects (a) experimental results; and (b) numerical results.

Fig. 16. Effect of the defect size on the resonant frequency of silicon rubber matrix/ air scatterer PC.

of PC, and the point defect behaves more like a tunnel rather than an obstacle to transport the elastic wave through the PC.

5. Conclusions Firstly, the previously developed numerical methods cannot simulate the viscoelastic crystals with arbitrary shapes of the scatterers. Here, we developed a new BEM to solve above issue. Then, this developed numerical method is verified by the experimental method. Based on the numerical and experimental methods, we implemented systematic comprehensive parametric studies on the band structure of viscoelastic phononic crystals with varying factors. After that, a strategy can be obtained to tune the band gaps of viscoelastic phononic crystals. Besides, the band structure of viscoelastic phononic crystals with a point defect has also been investigated for the first time. Based on the above research, the following results can be obtained:

(1) The viscoelastic characteristic of materials would not affect the band structures essentially. However, the smaller the final–initial value ratio is, the lower initial frequencies and wider band gaps will be obtained. (2) When the relaxation time is quite small, the materials can reach to a smaller storage elastic modulus, which results in a lower initial frequency. However, the band gap width does not monotonic increase with the increase of relaxation time. An optimum relaxation time exists to widen the band gaps of viscoelastic phononic crystals. (3) The band gap width and initial frequencies of CBG are not varied monotonically with the increase of the volume fraction. The CBG and DBG sizes can be enlarged by changing the volume fractions of scatterers combined with adjusting the viscous of the host materials. (4) The PCs with circle rods have the lowest initial forbidden frequency and largest gap width among the three shapes (circle, ellipse, and hexagon) of scatterers. (5) For the viscoelastic phononic crystals with a point defect, it can be found that the viscosity may tune the defect mode.

Acknowledgements The work described in this paper is supported by the National Natural Science Foundation of China (Nos. 11102104, U1333104), Innovation Program of Shanghai Municipal Education Commission (No. 15ZZ017), and Program for Young Excellent Talents in Tongji University. Appendix A The matrix A and B are given for each edge of MC; CX; XM separately as follows:

"

A¼ " B¼

H0b1

H0b2 þ H0b4

G0b1

0

0

0

H0b3 0

G0b2  G0b4

H0b0

G0b0

0

Hj

Gj

0 G0b3

0 0 0

0

0 0 0

0

#

#

ðA1Þ

104

X.Y. Zhu et al. / Applied Acoustics 106 (2016) 93–104

  for CX, where eiky a ¼ 1; n ¼ eikx a 0 < kx 6 pa .

"

A¼ " B¼

H0b1  H0b3

H0b2

G0b1 þ G0b3

G0b2

H0b0

G0b0

0

0

0

0

Hj

Gj

0

H0b4

0

0

0

G0b4

0 0

0

0

0 0

#

# ðA2Þ

  for XM, where eikx a ¼ 1; n ¼ eiky a 0 < ky 6 pa .

"

A¼ " B¼

H0b1

H0b2

G0b1

G0b2

H0b0

G0b0

0

0

0

0

Hj

Gj

H0b3 0

H0b4 0

G0b3 0

G0b4 0

0 0 0 0

#

#

ðA3Þ

  for MC , where n ¼ eikx a ¼ eiky a 0 < kx ¼ ky 6 pa . In above equations,



T H0ba ¼ H00a H01a H02a H03a H04a

T G0ba ¼ G00a G01a G02a G03a G04a

ðA4Þ

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