piezomagnetic phononic crystals

International Journal of Solids and Structures 45 (2008) 4203–4210

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International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr

Wave band gaps in two-dimensional piezoelectric/piezomagnetic phononic crystals Yi-Ze Wang a, Feng-Ming Li a,b,*, Wen-Hu Huang a, Xiaoai Jiang c, Yue-Sheng Wang d, Kikuo Kishimoto b a

School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, PR China Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo 152-8552, Japan c Advanced Material Systems Application Lab, GE Global Research, One Research Circle, KW-C282, Niskayuna, NY 12309, USA d Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, PR China b

a r t i c l e

i n f o

Article history: Received 16 November 2007 Received in revised form 28 January 2008 Available online 10 March 2008

Keywords: Phononic crystal Piezoelectricity Piezomagneticity Plane-wave expansion method Band gap

a b s t r a c t In this paper, the elastic wave propagation in phononic crystals with piezoelectric and piezomagnetic inclusions is investigated taking the magneto-electro-elastic coupling into account. The electric and magnetic fields are approximated as quasi-static. The band structures of three kinds of piezoelectric/piezomagnetic phononic crystals—CoFe2O4/quartz, BaTiO3/CoFe2O4 and BaTiO3–CoFe2O4/polymer periodic composites are calculated using the plane-wave expansion method. The piezoelectric and piezomagnetic effects on the band structures are analyzed. The numerical results show that in CoFe2O4/quartz structures, only one narrow band gap exists along the C–X direction for the coupling of xy-mode and z-mode for the filling fraction f being 0.4; while in BaTiO3/CoFe2O4 composites, only one narrow band gap exists along the C–X direction forxy-mode and no band gap exists for z-mode as the filling friction f is 0.5. Moreover, for the new type of magneto-electroelastic phononic crystal—BaTiO3–CoFe2O4/polymer periodic composite, the band gap characteristics are more superior in the whole considered frequency regions due to the big contrast of the material properties in the two constituents and the effects of the piezoelectricity and piezomagneticity on the band gap structures are remarkable. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The propagation of elastic waves in periodic composite media (called phononic crystals) has received much attention in recent years (Sigalas and Economou, 1993; Kushwaha et al., 1993; Tanaka and Tamura, 1998). The properties of frequency band gaps in the phononic crystals have been studied in both theoretical predictions and experiments (Vasseur et al., 2001; Liu et al., 2005; Wu et al., 2005a,b; Zhang et al., 2006). The advantages of such crystals may lead to the development of acoustic devices such as transducers, filters and vibration isolation technology. Many smart structures such as the piezoelectric or piezomagnetic composites are periodically made up of two or more distinct constituents. Compared with the purely elastic phononic crystals, they have superior electric or magnetic effects and show some new acoustic properties. These smart composites can really be considered as new kinds of phononic crystals. Therefore, more and more people have been investigating the new kinds of intelligent phononic crystals. Wilm et al. (2002) applied the plane-wave expansion method to analyze the 1–3 piezoelectric composites taking the piezoelectricity, acoustic losses and electrical excitation conditions into account. Hou et al. (2004) used the same method to study the two-dimen* Corresponding author. Address: School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, PR China. Tel.: +86 451 86414479. E-mail address: [email protected] (F.-M. Li). 0020-7683/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2008.03.001

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sional piezoelectric phononic crystals and discussed the effects of filling fraction on the band gaps. Wu et al. (2005a,b) investigated the propagation of surface acoustic and bulk waves in two-dimensional periodic piezoelectric structures. Khelif et al. (2006) studied the acoustic wave propagation in a phononic crystal slab consisting of piezoelectric inclusions and drew some important conclusions. Li and Wang (2005), Li et al. (2007) studied the propagation and localization of SH-waves in disordered periodic layered piezoelectric composites. Benchabane et al. (2006) found a complete surface acoustic wave band gap in piezoelectric phononic crystals in their experiment and the band gap is well in agreement with theoretical calculations. Sesion et al. (2007) studied the acoustic–phonon transmission spectra in piezoelectric AlN/GaN Fibonacci phononic crystals using a theoretical model beyond the elastic continuum approach. To our knowledge, however, no work on elastic wave propagation in piezoelectric/piezomagnetic phononic crystals has been reported. In this paper, three kinds of piezoelectric/piezomagnetic phononic crystals, i.e. CoFe2O4/quartz, BaTiO3/CoFe2O4 and BaTiO3–CoFe2O4/polymer periodic composites with the electromagnetic field are investigated taking the piezoelectric and piezomagnetic effects into account. From the numerical results, it can be seen that the BaTiO3–CoFe2O4/ polymer composites have complete band gaps for both xy-mode and z-mode. The effects of the piezoelectric, piezomagnetic and elastic constants on the band gap characteristics are analyzed. 2. Equations of wave motion For general cases, we consider the anisotropic two-dimensional piezoelectric/piezomagnetic phononic crystals with square lattice where the cylinder inclusions with radius r are embedded periodically in the matrix material. The lattice constant of the square lattice is a. Fig. 1 shows the two-dimensional phononic crystals and the corresponding first irreducible Brillouin zone. The constitutive equations of a composite material with piezoelectric and piezomagnetic phases are given by (Liu et al., 2001; Tian and Gabbert, 2005; Rojas-Dı´az et al., 2008) rij ¼ cijmn emn  enij En  qnij Hn ; ð1Þ

Di ¼ eimn emn þ 2in En þ kin Hn ; Bi ¼ qimn emn þ kin En þ Cin Hn ;

ði; j; m; n ¼ 1; 2; 3Þ;

where rij is elastic stress tensor, Di electric displacement vector, Bi magnetic induction, emn elastic strain tensor, En electric field, Hn magnetic field, cijmn(r) elastic constants, enij(r) piezoelectric constants, qnij(r) piezomagnetic constants, 2in(r) dielectric constants, Cin(r) magnetic permeability, kin(r) electromagnetic constants which are all position-dependent material constants and r=(x, y) the position vector. In Eq. (1) the summation convention is employed. Because the phononic crystals are homogeneous along the z axis, these constants are independent of the coordinate z. The elastic strain tensor emn can be expressed as the following forms: emn ¼

1 ðum;n þ un;m Þ; 2

ð2Þ

where um={ux,uy,uz}T is the elastic displacement vector. By the quasi-static approximation, the electric field En and magnetic field Hn can be expressed as: En ¼ 

ou ¼ u;n ; oxn

Hn ¼ 

o/ ¼ /;n ; oxn

ð3Þ

where u and / are the electric potential and magnetic potential.

a

b

z y

ky

M

2r

x 2π/a

a

kx Γ

X

2π /a Fig. 1. (a) Square lattice two-dimensional phononic crystals consisting of cylindrical inclusion embedded in matrix. (b) The corresponding first irreducible Brillouin zone.

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The differential equations of motion in the absence of body forces are given by: €i ; rij;j ¼ qu

Di;i ¼ 0;

Bi;i ¼ 0;

ð4Þ

where q(r) is the mass density of the materials and it is dependent on position, and dot denotes differentiation with respect to the time. Based on the periodicity of material constants and Bloch–Floquet theorem, the plane-wave expansion method is used to obtain the following generalized eigenvalue equation: x2 RU ¼ QU;

ð5Þ

fuxG ; uyG ; uzG ; uG ; /G gT

is the generalized displacement vector, R and Q the 5N  5N matrices which are the functions where U ¼ of k, G, G0 , the material constants and the filling fraction f = pr2/a2. The elements of matrices R and Q for CoFe2O4/quartz composites are listed in Appendix A. For detailed deducing process of Eq. (5), one can refer to the work by Kushwaha et al. (1994). It should be noted that the z direction is along the poling axis, and the symmetries for quartz, BaTiO3 and CoFe2O4 used in the calculation are Trig.32, Uniaxial k and Uniaxial k, respectively. For the anisotropic periodic structure, i.e. CoFe2O4/quartz, the xy-mode and z-mode are coupled. But for the other two kinds of piezoelectric/piezomagnetic phononic crystals, i.e. BaTiO3/CoFe2O4 and BaTiO3–CoFe2O4/polymer, the xy-mode and z-mode are decoupled. From the eigenvalue equation of the decoupled cases, we can observe that the piezoelectric and piezomagnetic effects only influence the z-mode just like the piezoelectric phononic crystals investigated by Hou et al. (2004) which is simplified from the coupled case. If the piezoelectric and piezomagnetic effects are ignored, the eigenvalue equation, Eq. (5), will become 3N  3N dimensional. By solving the eigenvalue equation for x as a function of the wave vector k in the first Brillouin zone, the band structures can be built. 3. Numerical examples and discussions In this section, numerical simulations of band structures for CoFe2O4/quartz, BaTiO3/CoFe2O4 and BaTiO3–CoFe2O4/polymer have been conducted. The material constants used in the calculation are mainly from Auld (1973), Ramirez et al. (2006), Shane Fazzio (2006), Annigeri et al. (2006) and Atul Daga et al. (2008) and listed in Table 1. Along the C–M–X–C path indicated in Fig. 1(b), Fig. 2 shows the band structures of the CoFe2O4/quartz phononic crystals in the first Brillouin zone for the symmetry axis. The filling fraction f = pr2/a2 is taken as 0.4 (corresponding to r/a = 0.357) for the normalized frequency xa/2pct. In Fig. 2, we find one narrow band gap along the C–X direction between the third and forth band. But there is no one band gap along other directions, i.e. X–M and M–C directions. Consequently, it is implied that complete band gaps are impossibly generated in this kind of piezoelectric/piezomagnetic phononic crystal. This is mainly due to the contrasts of the material constants between quartz and CoFe2O4 being not remarkable. BaTiO3/CoFe2O4 is a typical piezoelectric/piezomagnetic composite investigated by many works (Nan, 1994; Sih and Chen, 2003; Jiang and Pan, 2004; Zheng et al., 2007). In this paper, we also calculate the band structures of the BaTiO3/CoFe2O4 phononic crystals. The results are presented in Fig. 3(a) and (b) for xy-mode and z-mode, respectively. The filling fraction is taken as 0.5. The generalized eigenvalue equation of this kind of periodic structure becomes simpler, which can be found in Appendix B.

Table 1 Material constants of quartz, BaTiO3, CoFe2O4, BaTiO3—CoFe2O4 and polymer Material constants

Quartz

BaTiO3

CoFe2O4

BaTiO3–CoFe2O4

Polymer

q(103kg/m3) c11 (GPa) c12 (GPa) c14 (GPa) c22 (GPa) c44 (GPa) c55 (GPa) c66 (GPa) e11 (C/m2) e41 (C/m2) e15 (C/m2) e25 (C/m2) 211(109C2/Nm2) 222(109C2/Nm2) q15 (N/Am) C11 (106N s2/C2) k11 (109N s/VC) k33( 109N s/VC)

2.64867 86.7997 7.0362 18.0612 86.7997 58.2231 58.2231 39.8817 0.1719 0.0390 0 0.0390 0.039136 0.039136 0 5 0 0

5.8 166 77 0 166 43 43 44.5 0 0 11.6 0 11.2 11.2 0 5 0 0

5.3 286 173 0 286 45.3 45.3 56.5 0 0 0 0 0.08 0.08 550 590 0 0

5.73 166 77 0 166 43 43 44.5 0 0 11.6 0 11.2 11.2 550 5 0.005 0.003

1.15 7.8 4.7 0 7.8 1.6 1.6 1.55 0 0 0 0 0.0398 0.0398 0 5 0 0

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2

1.6

1.2

0.8

0.4

0

Γ

X

Γ

M

Fig. 2. Band structures of two-dimensional phononic crystals with CoFe2O4 cylinders in quartz matrix. The vertical coordinates, i.e. the normalized frequency xa/2pct, are plotted against the reduced wave vector along the C–M–X–C path of the first irreducible Brillouin zone for the coupling of xy-mode and z-mode. The filling fraction f is 0.4.

a

b

2

2

1.6

1.6

1.2

1.2

0.8

0.8

0.4

0.4

0

Γ

X

M

0

Γ

Γ

X

M

Γ

Fig. 3. Band structures of two-dimensional phononic crystal with BaTiO3 cylinders in CoFe2O4 matrix. (a) xy-mode, (b) z-mode. The filling fraction f is 0.5.

From Fig. 3, we can observe that there is also one narrow band gap along the C–X direction between the second and third bands for xy-mode. However, due to the similar values of the material constants between BaTiO3 and CoFe2O4, there are no band gaps along any other directions for xy-mode, and no band gaps for z-mode. It should be noted that for any other filling fractions of CoFe2O4/quartz and BaTiO3/CoFe2O4 composites, the band gaps are also quite small and even disappear. Fig. 4 shows the band structures for xy-mode and z-mode in the magneto-electro-elastic phononic crystals—BaTiO3–CoFe2O4/polymer periodic composites. The elements of matrices R and Q in the eigenvalue equation for this case are the same as those in Appendix B. The filling ratio is taken as 0.5.

a

b

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

Γ

X

M

Γ

0

Γ

X

Fig. 4. The same as Fig. 3 but for BaTiO3–CoFe2O4 cylinders in polymer matrix.

M

Γ

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We can see from Fig. 4(a) that there is a complete broad band gap that appears between the third and forth bands for xymode. In Fig. 4(b), however, there are three band gaps of z-mode. These phenomena are mainly due to the remarkable contrast between the material properties of the two constituents. Consequently, it is implied that the band gap characteristics for this kind of magneto-electro-elastic coupling structure are superior to the other two cases discussed above. In order to give a better insight into the effects of piezoelectricity and piezomagneticity on the phononic crystals, we present the band structures for the z-mode of BaTiO3–CoFe2O4/polymer composites with f = 0.5. The reason for considering the zmode is that only z-mode is influenced by the piezoelectric and piezomagnetic phases. The band structures for the cases with and without piezoelectric effect are shown in Fig. 5(a). In Fig. 5(b), the piezomagnetic effect is analyzed. Fig. 5(c) shows the band structures for considering and neglecting all the piezoelectric and piezomagnetic effects. The most noticeable feature in Fig. 5(a) is that the band structures are similar in lower frequency regions. However, this phenomenon changes obviously for higher frequency regions. The rule is valid for the case with and without the piezomagnetic effect, which can be observed in Fig. 5(b). We can also see that for the case considering the piezoelectric or piezomagnetic effect the band gaps become wider than those of the case without the piezoelectric or piezomagnetic effect. It means that the piezoelectricity or the piezomagneticity can enlarge the band gaps. Comparing Fig. 5(c) with Fig. 5(a) and (b), it can be observed that the piezoelectricity and piezomagneticity have more serious effects on the band structures than each of them does. This is because both the piezoelectric and piezomagnetic effects can prohibit the elastic wave propagation in the phononic crystals. Fig. 6 shows the band structures of two-dimensional magneto-electro-elastic phononic crystals—BaTiO3–CoFe2O4/polymer composites with f = 0.5 for z-mode for the cases of the elastic constant c44 of the BaTiO3–CoFe2O4 being 43 GPa as listed in Table 1 and 0.6  43 = 25.8 GPa. From the results, we can clearly observe that the band structures change with different values of c44, especially for higher frequency regions. When the magnitude of the elastic constant c44 becomes smaller, the band gaps become narrower, which implies that the larger the elastic constants, the broader the band gaps.

a

b

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

Γ

X

Γ

M

c

0

Γ

X

M

Γ

3 2.5 2 1.5 1 0.5 0

Γ

X

M

Γ

Fig. 5. Band structures of two-dimensional magneto-electro-elastic phononic crystals—BaTiO3–CoFe2O4/polymer composites with f = 0.5 for z-mode only. (a) Solid lines and hollow dots denote the cases with and without the piezoelectric effect. (b) Solid lines and hollow dots denote the cases with and without the piezomagnetic effect. (c) Solid lines and hollow dots denote the cases with and without all the piezoelectric and piezomagnetic effects.

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3 2.5 2 1.5 1 0.5 0

Γ

X

M

Γ

Fig. 6. Band structures of two-dimensional magneto-electro-elastic phononic crystals—BaTiO3–CoFe2O4/polymer composites with f = 0.5 for z-mode only. Solid lines and hollow dots denote the band structures corresponding to the elastic constant c44 of the BaTiO3–CoFe2O4 being 43 GPa as listed in Table 1 and 0.6  43 = 25.8 GPa, respectively.

4. Conclusions In this paper, the wave propagation in phononic crystals consisting of piezoelectric and piezomagnetic composites is studied using the plane-wave expansion method. The band gaps for CoFe2O4/quartz, BaTiO3/CoFe2O4 and BaTiO3–CoFe2O4/polymer periodic composites are calculated. The piezoelectric and piezomagnetic effects on the band structures are evaluated. From the results, the following conclusions can be drawn: (1) There is only a narrow directional band gap along the C–X direction for the coupling of xy-mode and z-mode in CoFe2O4/quartz periodic structures and for the xy-mode in BaTiO3/CoFe2O4 composites. (2) The two-dimensional phononic crystals—BaTiO3–CoFe2O4/polymer periodic structures have more superior band gap properties for both the xy-mode and z-mode. (3) The piezoelectric and piezomagnetic effects play an important role on the band gap characteristics in the BaTiO3– CoFe2O4/polymer phononic crystals, especially for higher frequency regions. (4) The elastic constants can also influence the band structures, which is similar to the effects of the piezoelectricity and piezomagneticity.

Acknowledgments The authors express gratitude for the supports provided by the National Natural Science Foundation of China under Grant Nos. 10672017 and 10632020 for this research work. Feng-Ming Li also acknowledge the supports provided by the China Postdoctoral Science Foundation, Heilongjiang Province Postdoctoral Science Foundation and Japan Society for the Promotion of Science (JSPS) to perform research work at Tokyo Institute of Technology, Japan. Appendix A The elements of matrices R and Q in Eq. (5) are expressed as: 2 RG;G

0

6 6 6 ¼6 6 6 4 2

Q G;G0

where

3

qG;G0

7 7 7 7; 7 7 5

qG;G0 qG;G

0

0

ðA:1Þ

0 ð1;1Þ

Q G;G0

6 ð2;1Þ 6Q 6 G;G0 6 6 ð3;1Þ ¼ 6 Q G;G0 6 6 ð4;1Þ 6 Q G;G0 4 0

ð1;2Þ

Q G;G0

ð1;3Þ

Q G;G0

ð2;2Þ

Q G;G0

ð2;3Þ

Q G;G0

Q G;G0

ð3;2Þ

Q G;G0

ð3;3Þ

Q G;G0

ð4;2Þ Q G;G0

ð4;3Þ Q G;G0

ð4;4Þ Q G;G0

0

Q G;G0

Q G;G0 Q G;G0

ð5;3Þ

ð1;4Þ ð2;4Þ ð3;4Þ

ð5;4Þ

Q G;G0

0

3

7 7 7 7 ð3;5Þ 7 Q G;G0 7; 7 ð4;5Þ 7 Q G;G0 7 5 ð5;5Þ Q G;G0 0

ðA:2Þ

Y.-Z. Wang et al. / International Journal of Solids and Structures 45 (2008) 4203–4210 ð1;1Þ

4209

0 0 66 Q G;G0 ¼ c11 G;G0 ðk þ GÞ1 ðk þ G Þ1 þ c G;G0 ðk þ GÞ2 ðk þ G Þ2 ;

ðA:3Þ

ð1;2Þ Q G;G0

c12 G;G0 ðk

0

0

þ GÞ2 ðk þ G Þ1 ;

ðA:4Þ

0 0 14 Q G;G0 ¼ c14 G;G0 ðk þ GÞ1 ðk þ G Þ2 þ c G;G0 ðk þ GÞ2 ðk þ G Þ1 ;

ðA:5Þ

ð1;4Þ Q G;G0

0

ðA:6Þ

0

ðA:7Þ

0

þ GÞ1 ðk þ G Þ1 ;

ðA:8Þ

0 0 14 Q G;G0 ¼ c14 G;G0 ðk þ GÞ1 ðk þ G Þ1  c G;G0 ðk þ GÞ2 ðk þ G Þ2 ;

ðA:9Þ

¼

þ GÞ1 ðk þ G Þ2 þ

c66 G;G0 ðk

ð1;3Þ

ð2;1Þ Q G;G0 ð2;2Þ Q G;G0

¼

e11 G;G0 ðk

¼

c12 G;G0 ðk

¼

c22 G;G0 ðk

0

e11 G;G0 ðk

0

c66 G;G0 ðk

0

c66 G;G0 ðk

þ GÞ1 ðk þ G Þ1  þ GÞ2 ðk þ G Þ1 þ þ GÞ2 ðk þ G Þ2 þ

þ GÞ2 ðk þ G Þ2 ; þ GÞ1 ðk þ G Þ2 ;

ð2;3Þ

ð2;4Þ Q G;G0 ð3;1Þ Q G;G0 ð3;2Þ Q G;G0

0

¼ ¼

ð1;3Þ Q G;G0 ;

ðA:11Þ

¼

ð2;3Þ Q G;G0 ;

ðA:12Þ

þ GÞ1 ðk þ G Þ2 

e11 G;G0 ðk

0

e11 G;G0 ðk

þ GÞ2 ðk þ G Þ1 ;

ðA:10Þ

ð3;3Þ

0 0 44 Q G;G0 ¼ c44 G;G0 ðk þ GÞ1 ðk þ G Þ1 þ c G;G0 ðk þ GÞ2 ðk þ G Þ2 ;

ðA:13Þ

ð3;4Þ Q G;G0

ðA:14Þ

ð3;5Þ Q G;G0 ð4;1Þ Q G;G0 ð4;2Þ Q G;G0

0

¼ ¼

q15 G;G0 ðk

¼

ð1;4Þ Q G;G0 ;

ðA:16Þ

¼

ð2;4Þ Q G;G0 ;

ðA:17Þ

ð4;3Þ

þ GÞ1 ðk þ G Þ2 þ 0

þ GÞ1 ðk þ G Þ1 þ

e41 G;G0 ðk

0

e41 G;G0 ðk

q15 G;G0 ðk

þ GÞ2 ðk þ G Þ1 ; 0

þ GÞ2 ðk þ G Þ2 :

ðA:15Þ

ð3;4Þ

Q G;G0 ¼ Q G;G0 ; ð4;4Þ Q G;G0 ð4;5Þ Q G;G0 ð5;3Þ Q G;G0

ðA:18Þ

¼

211 G;G0 ðk

¼

k11 G;G0 ðk

¼

ð3;5Þ Q G;G0 ;

ð5;4Þ

0

211 G;G0 ðk

0

k11 G;G0 ðk

þ GÞ1 ðk þ G Þ1  þ GÞ1 ðk þ G Þ1 

0

þ GÞ2 ðk þ G Þ2 ;

ðA:19Þ

0

þ GÞ2 ðk þ G Þ2 ;

ðA:20Þ ðA:21Þ

ð4;5Þ

Q G;G0 ¼ Q G;G0 ; ð5;5Þ Q G;G0

¼

ðA:22Þ

C11 G;G0 ðk

0

þ GÞ1 ðk þ G Þ1 

C11 G;G0 ðk

0

þ GÞ2 ðk þ G Þ2 :

ðA:23Þ

Appendix B The same as Appendix A, but for ð1;3Þ

ð1;4Þ

ð2;3Þ

ð2;4Þ

ð3;1Þ

ð3;2Þ

ð4;1Þ

ð4;2Þ

Q G;G0 ¼ Q G;G0 ¼ Q G;G0 ¼ Q G;G0 ¼ Q G;G0 ¼ Q G;G0 ¼ Q G;G0 ¼ Q G;G0 ¼ 0;

ðB:1Þ

ð3;4Þ Q G;G0

ðB:2Þ

ð4;3Þ Q G;G0

¼

e15 G;G0 ðk

¼

ð3;4Þ Q G;G0 :

0

þ GÞ1 ðk þ G Þ1 þ

e15 G;G0 ðk

0

þ GÞ2 ðk þ G Þ2 ;

ðB:3Þ

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