- Email: [email protected]
Electroelastic analysis of a piezoelectric half-plane with finite surface electrodes
International Journal of Engineering Science 42 (2004) 1603–1619 www.elsevier.com/locate/ijengsci
Electroelastic analysis of a piezoelectric half-plane with finite surface electrodes Zhen-Bang Kuang *, Zhi-Dong Zhou, Kun-Li Zhou Department of Engineering Mechanics, Shanghai JiaoTong University, Shanghai 200240, PR China Received 10 August 2003; received in revised form 12 February 2004; accepted 4 March 2004 Available online 28 July 2004 (Communicated by S.-A. ZHOU)
Abstract In this paper the problem of the electroelastic fields of piezoelectric ceramic in a half-plane with thin collinear surface electrodes is formulated and analyzed. Using the standard analytic continuation method, a mixed boundary value problem in plane is obtained and then reduced to a standard Hilbert problem. The closed form expressions of electroelastic pffiffi fields are obtained. Some special cases are discussed. The electroelastic filed has singularity with 1= r at the vicinity of the electrode ends. The numerical examples for two surface electrodes are studied to show the practical distributions of the electroelastic fields graphically. 2004 Elsevier Ltd. All rights reserved. Keywords: Piezoelectric ceramic; Surface electrodes; Electroelastic behavior; Hilbert problem
1. Introduction Owing to the large coupling between electric and mechanical fields, piezoelectric ceramics are widely used as smart materials such as transducers, sensors and actuators. In these cases, the thin electrodes are usually attached at the surface or interface of electroceramic blocks [1–8]. Hence it is important to research the electroelastic field produced by the thin electrodes in piezoelectric apparatuses. Shindo et al. [2] studied the static behavior of electroelastic field at the vicinity of single surface electrode by Fourier transformation method in a transverse isotropic material. Ru [3] studied the case of thin electrodes embedded at the interface of two bonded dissimilar *
Corresponding author. Tel.: +86-21-54743067; fax: +86-21-54740344. E-mail address: [email protected] (Z.-B. Kuang).
0020-7225/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2004.03.004
1604
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
piezoelectric half-planes. Ye and He [6] analyzed the electric field concentrations of the edges of a pair of parallel electrodes in a transverse isotropic material when the sizes of electrodes are far smaller than that of the thickness. They found the stresses at the edge of electrodes are very high and can make the ceramics failure. Li and Tang [7] studied the piezoelectric layer with electrodes at the top surface and bottom surface by the conformal mapping technique. Bent and Hagood [8] discussed in detail the properties of a piezoelectric fiber composites with interdigitated electrodes. Zhou et al. studied a single surface electrode for soft and rigid cases [9] by conformal mapping method and got pretty closed solutions. In this paper, the problem of thin collinear electrodes attached at the surface of the half-plane in a general piezoelectric ceramic are formulated and discussed. According to the principle of standard analytic continuation method, the problem is continued to the whole plane and then is reduced to a mixed boundary problem which can be reduced to a Hilbert problem. The closed form solutions of electroelastic filed are obtained, which show the electric and stress fields exhibit inverse square root singularity at the vicinity of the electrode ends. The field concentration is abated when the depth (x2 ) increases. Finally some special examples are given. The distributions of the electric field and stress are useful in engineering application. 2. Basic equations and general solution In a fixed rectangular coordinate system (x1 ; x2 ; x3 ), all of the field variables depend on x1 ; x2 only for a generalized plane piezoelectric problem. Following Suo et al. [10] and Kuang and Ma [11], the general solution in this case can be given by the linear combination of four complex analytical functions u ¼ 2Re½AfðzÞ;
/ ¼ 2Re½BfðzÞ; T
u ¼ ½u1 ; u2 ; u3 ; u ;
/ ¼ ½/1 ; /2 ; /3 ; /4 T ; T
fðzÞ ¼ ½f1 ðz1 Þ; f2 ðz2 Þ; f3 ðz3 Þ; f4 ðz4 Þ ;
ð1Þ
zk ¼ x1 þ pk x2
ðk ¼ 1–4Þ;
where Re stands for the real part of a function; ui ði ¼ 1; 2; 3Þ; u and /i are the displacement components, electric potential and the components of the generalized stress function, respectively; A and B are 4 · 4 complex matrices related to the material constants, expressed as A ¼ ½a1 ; a2 ; a3 ; a4 ;
B ¼ ½b1 ; b2 ; b3 ; b4 :
ð2Þ
The eigenvalues pk and eigenvectors ak are determined by the following equations: ½Q þ ðR þ RT Þp þ Tp2 a ¼ 0
ð3Þ
where
QE Q¼ T e11 QEik ¼ ci1k1 ;
e11 ; j11
RE R¼ T e21
REik ¼ ci1k2 ;
e21 ; j12
TikE ¼ ci2k2 ;
TE T¼ T e22
ðeij Þs ¼ eijs ;
e22 ; j22
ð4Þ
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1605
where cijkl are the elastic stiffnesses under constant electric field, eijs are the piezoelectric constants and jij are the permittivities under constant strain field. The eigenvectors bk can be obtained from the following equations: bk ¼ ðRT þ pk TÞak ¼
1 ðQ þ pk RÞak : pk
ð5Þ
The generalized stresses are represented as t1 ¼ ½r11 ; r12 ; r13 ; D1 T ¼ ½/1;2 ; /2;2 ; /3;2 ; /4;2 T ; T
ð6Þ
T
t2 ¼ ½r21 ; r22 ; r23 ; D2 ¼ ½/1;1 ; /2;1 ; /3;1 ; /4;1 ;
where rij ; Di are stresses and electric displacements, respectively. After the normalization for the eigenvectors A and B, there exist the following relations [11,12]:
BT T B
AT T A
I A A ¼ 0 B B
0 ; I
A A B B
BT T B
AT T A
I 0 ¼ 0 I
ð7Þ
3. Statement and solution procedure of the problem Consider thin collinear electrodes attached at the surface of piezoelectric ceramic as shown in Fig. 1. Let the left and right ends of the i-th electrode be ai and bi ði ¼ 1; 2; . . . ; nÞ, respectively. It is assumed that the total electric charge or potential on the i-th electrode is qi or Vi respectively. Then, the mixed boundary conditions are given by
Fig. 1. A piezoelectric ceramic with collinear surface electrodes.
1606
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
r2i ¼ 0;
D2 ¼ 0;
x2 ¼ 0;
z 2 L0 ¼ ½1; a1 þ ½b1 ; a2 þ þ ½bn ; þ1;
r2i ¼ 0;
E1 ¼ 0;
x2 ¼ 0;
z 2 L00 ¼ ½a1 ; b1 þ þ ½an ; bn ;
rij ! 0;
Dj ! 0;
jzj ! 1;
i ¼ 1; 2; 3;
ð8Þ
j ¼ 1; 2;
where L0 þ L00 ¼ L. Analogous to elastic contact problem [13], using the Gauss law, along the surface electrodes the supplement boundary conditions can be written as Z D i ¼ 1; 2; . . . ; n ð9aÞ 2 ðx1 Þdx1 ¼ qi ; L00i
or ui ¼ V i ;
i ¼ 1; 2; . . . n and
n Z X i¼1
L00i
D 2 ðx1 Þ dx1 ¼ Q ¼
n X
qi ;
ð9bÞ
i¼1
where Q is the total charge on all electrodes. If Q ¼ 0, then the supplement conditions can be expressed by pure potentials. If Q 6¼ 0, then Q is included in the solution. Eqs. (9a) and (9b) are equivalent. The superscript ‘)’ indicates the limit value taken from the lower half-plane. The boundary conditions (8) and (9a) or (9b) guarantee that the discussed problem has an unique solution. Let FðzÞ ¼ f 0 ðzÞ, following Eq. (8)1 , the generalized stresses along all electrode-free part L0 can be written as BF ðx1 Þ þ BF ðx1 Þ ¼ 0;
x 2 L0 ;
ð10Þ
which can be rewritten as [13] BF ðx1 Þ þ BFþ ðx1 Þ ¼ 0;
ð11Þ
where the superscript ‘+’ indicates the limit value taken from the upper half-plane. Now a new function of complex variable z is defined as B1 BFðzÞ z 2 S þ ð12Þ hðzÞ ¼ FðzÞ z 2 S The function hðzÞ is analytic in the whole plane except at the segment L00 . Using Eqs. (1) and (6), Eq. (8)2 can be written as BF ðx1 Þ þ BFþ ðx1 Þ ¼ g;
T 00 g ¼ ð0; 0; 0; D 2 ðx1 ÞÞ ; x 2 L ;
ð13Þ
where D 2 ðx1 Þ is unknown. Substituting Eq. (12) into Eq. (13), we get hþ ðx1 Þ h ðx1 Þ ¼ B1 g; 1 hþ a ðx1 Þ ha ðx1 Þ ¼ Ba4 D2 ðx1 Þ;
a ¼ 1; . . . ; 4;
ð14Þ
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1607
which is the standard Hilbert problem. The solution of Eq. (14) is B1 hðzÞ ¼ 2pi
Z L00
gðx1 Þ dx1 and Fa ðza Þ ¼ ha ðza Þ when z 2 S : x1 z
ð15Þ
It is known that the electric displacement can be expressed in the form as [13,14] P ðzÞ D2 ðzÞ ¼ Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðz ai Þðz bi Þ
z 2 S;
ð16Þ
i¼1
where P ðzÞ is a polynomial of degree ðn 1Þ and we select the Branch z ! 1 along axis x, P ðzÞ ¼ iðcn1 zn1 þ þ c1 z þ c0 Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz ai Þðz bi Þ ! x when
ð17Þ
in which ci ði ¼ 1; 2; . . . ; n 1Þ are n unknown complex numbers which should be determined by Eq. (9a) or (9b). Substituting Eqs. (16) and (17) into Eq. (15) and noting that D takes its value at S , we get [13] P ðza Þ Fa ðza Þ ¼ B1 a4 Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 ðza ai Þðza bi Þ
ð18Þ
i¼1
fa ðza Þ ¼
B1 a4
Z
P ðza Þdza iC 1 B ; n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Q 2 a4 2 ðza ai Þðza bi Þ
ð19Þ
i¼1
where C can be taken as a real constant due to that its imaginary part does not infect the potential (see Eq. (27)). The expression of fa ðza Þ may not be expressed by elementary functions except some special cases, so in some cases using Eq. (9a) is more convenient. Using E1 ¼ 0 on L00 shown in Eq. (8)2 , we have AFðx1 Þ þ AFðx1 Þ ¼ ð; ; ; 0Þ
ð20Þ
in which ‘*’ denotes some unknown quantities which are not concerned here. Eq. (20) can be rewritten as
1608
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
iP ðx1 Þ A4a B1 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ðx1 ai Þðbi x1 Þ ðx1 aj Þðx1 bj Þ j¼1 j6¼i
iP ðx1 Þ A4a B1 a4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0; ðx1 aj Þðx1 bj Þ ðx1 ai Þðbi x1 Þ
x1 2 L00i ;
ð21Þ
j¼1 j6¼i
pffiffiffiffiffiffiffi where i ¼ 1. According to [3], [13] and [14] and that H44 ¼ iA4a B1 a4 is real, so Eq. (21) can be reduced to x1 2 L00i :
P ðx1 Þ þ P ðx1 Þ ¼ 0;
ð22Þ
So we can get the relation of ci ci ci ¼ 0;
ð23Þ
which means that ci are all the real constants. The generalized stresses can be expressed as t2b ¼ Re
4 X a¼1
P ðza Þ 1 Bba Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ba4 : ðza ai Þðza bi Þ
ð24Þ
i¼1
The generalized displacements are ub ¼ 2Re½Aba fa ðza Þ:
ð25Þ
It is obvious that t2b exhibit inverse square root singularity at the vicinity of electrode ends. Substituting Eq. (24) into (9a) we get Z
bi
t24 ðx1 Þdx1 ¼
Z
ai
bi
D 2 ðx1 Þdx1 ai
¼
Z ai
bi
P ðx1 Þdx1 Re Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx1 aj Þðx1 bj Þ j¼1
¼
Z
bi ai
iP ðx1 Þdx1 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qi : Q ðx1 ai Þðbi x1 Þðx1 aj Þðx1 bj Þ
ð26Þ
j¼1 j6¼i
There are n equations to determine n unknowns c0 ; c1 ; . . . ; cn1 , so the problem is solved. Or substituting Eq. (25) into (9b) we get
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
ðu44 Þi ¼ ui ¼ H44 Re
Z
bi ai
Pn
Z
i¼1 ai
iP ðx1 Þdx1 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ H44 C ¼ Vi ; Q ðx1 ai Þðx1 bi Þ j¼1
bi
1609
P ðx1 Þdx1 Re Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Q: ðx1 ai Þðx1 bi Þ
ð27Þ
i¼1
There are n þ 1 equations to determine the n þ 1 unknowns c0 ; c1 ; . . . ; cn1 and C. It is also known that from electricity theory one of Vi can take arbitrary value, especially zero. When z ! 1, Eqs. (15) and (18)1 are respectively reduced to Z 1 D lim F4 ðz4 Þ ¼ B44 2 ðx1 Þdx1 =2piz4 ; z!1
L00
lim F4 ðz4 Þ ¼ iB1 44 cn1 =2z4 :
z!1
Therefore we have Z 1 1 Q cn1 ¼ D : 2 ðx1 Þdx1 ¼ ðQÞ ¼ pi L00 p p
ð28Þ
It is also noted that if Ei is determined by using Eq. (26), then the potentials can be determined as follows: Let u1 ¼ V0 at electrode 1 ða1 6 x 6 b1 ; y ¼ 0Þ, then ui at electrode i ðai 6 x 6 bi ; y ¼ 0Þ can be determined by Z ai E1 dx1 ¼ Vi ; i ¼ 2; 3; . . . ; n: ð29Þ ui ¼ ui1 þ bi1
So the relations between Vi and qi are obtained. It means that Eqs. (9a) and (9b) are equivalent. 4. Some special cases 4.1. When the collinear electrodes are degenerated into a single surface electrode with charge q and a1 ¼ a, b1 ¼ a, then using Eq. (28) the function Fa ðza Þ is P ðza Þ qi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ffi ¼ B1 Fa ðza Þ ¼ B1 a4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 2 2 2 ðza a Þ 2p ðz2a a2 Þ Integrating Eq. (30) we have ( ) qffiffiffiffiffiffiffiffiffiffiffiffiffiffi iq fa ðza Þ ¼ ln za : þ z2a a2 þ ln b B1 a4 ; 2p
ð30Þ
ð31Þ
1610
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
where b is a real constant. Let u ¼ V0 on the electrode, then we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q 2 2 ¼ V0 : u ¼ 2RefA4a fa ðza Þg ¼ Re iA4a Ba4 lnðx1 i a x1 Þ þ ln b p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Because H44 ¼ iA4a B1 a2 x21 is a, we get a4 is real and the modulus of x1 i q 1 H44 ln ba ¼ V0 or b ¼ exp p a
V0 p : qH44
ð32Þ
So in this case the electric potential u is qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qH44 1 Re ln za þ z2a a2 u¼ þ V0 : a p
ð33Þ
Usually H44 < 0. Now the normalized stresses can be expressed as 4 X q 1 t2b ¼ Im Bba pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 a4 p ðz2a a2 Þ a¼1
t1b
4 X q pa ¼ Im Bba pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 a4 p ðz2a a2 Þ a¼1
ð34Þ
The above equations are identical with Ref. [9]. For the dielectric without piezoelectric effect then Q, R, T each has one component, Q44 ¼ e11 , R44 ¼ e13 , T44 ¼ e33 . The eigenvalue equation is reduced to one equation, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 e33 : ð35Þ ðe11 þ 2p4 e13 þ p4 e33 Þa4 ¼ 0; p4 ¼ p ¼ e13 þ i e11 e33 e213 Thus A, B each only has one component, B44 ¼ ðe12 þ p4 e33 Þ, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A44 ¼ i e11 e33 e213 a44 and H44 ¼ ðe11 e33 e213 Þ1=2 < 0. The results become " # iq pffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 F44 ðz4 Þ ¼ 44 ; 2p z24 a2
ð36Þ
" # ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r q z4 z 4 2 ffi Re ln þ 1 ð37Þ uðz4 Þ ¼ V0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a p e11 e33 e213 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 For isotropic medium, eij ¼ edij , u ¼ V0 peq Re ln½az þ ðaz Þ 1, which is identical with the result in the usual dielectric theory.
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1611
Fig. 2. A piezoelectric ceramic with two surface electrodes.
4.2. Two surface electrodes with same configuration are considered as shown in Fig. 2. 4.2.1. Case 1: The electrodes 1 and 2 are located at a1 ¼ 0:03 m, b1 ¼ 0:01 m, a2 ¼ 0:01 m, b2 ¼ 0:03 m and the charges on them are q and q respectively. From Eq. (28) it is obvious that c1 ¼ 0. c0 can be obtained by R 1 numerical method and c0 ¼ 1:1469q. Obviously if let u1 ¼ V0 on electrode 1, then u2 ¼ V0 1 E1 dx1 on the electrode 2. The problem is solved. By numerical integration we get u2 ¼ V0 þ 1:0707q. 4.2.2. Case 2: The electric charge of each electrode is q and their ends are located at a1 ¼ 0:03 m, b1 ¼ 0:01 m, a2 ¼ 0:01 m, b2 ¼ 0:03 m, respectively. PZT-5H ceramic with poling axis ox2 is taken as an example. The material constants of which are listed in Table 1. The function of Eqs. (17) and (28) are respectively reduced to P ðzÞ ¼ iðc1 z þ c0 Þ;
2 c1 ¼ q: p
ð38Þ
Due to the symmetry of the distribution of electric charge, so c0 ¼ 0. Table 1 Elastic stiffnesses, piezoelectric coefficients and dielectric constants of PZT-5Ha c11 (N/m2 )
c13 (N/m2 )
c12 (N/m2 )
c22 (N/m2 )
c55 (N/m2 )
e21 (C/m2 )
e22 (C/m2 )
e16 (C/m2 )
j11 (C/Vm)
j22 (C/Vm)
12.6 · 1010
5.5 · 1010
5.2 · 1010
11.7 · 1010
3.53 · 1010
)6.5
23.3
17.0
1.51 · 108
1.30 · 108
a
The constitutive equations are: ri ¼ cij ej eij Ej , Di ¼ eij ej þ jij Ej .
1612
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
4.2.3. Numerical results: Figs. 3–6 give the distributions of normalized electric displacement ðD1 =q; D2 =qÞ and normalized stresses ðr12 =q; r22 =qÞ with x1 =a along axis ox1 at different x2 =a
Fig. 3. Distributions of electric displacement D2 =q with x1 =a axis ðx2 =a ¼ 1; 0:1; 0:01Þ. (a) Case 1; (b) Case 2.
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619 1.0
1613
x2/a=-1
0.8
x2/a=-0.1
0.6
x2/a=-0.01
0.4
D1/q(m-2)
0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -4
-3
-2
0
-1
x1/a
(a)
1
2
3
4
Case 1
0.8
x2/a=-1 x2/a=-0.1
0.6
x2/a=-0.01
0.4
D1/q(m-2)
0.2 0.0 -0.2 -0.4 -0.6 -0.8 -4
(b)
-3
-2
-1
0
1
2
3
4
x /a 1
Case 2
Fig. 4. Distributions of electric displacement D1 =q with x1 =a axis ðx2 =a ¼ 1; 0:1; 0:01Þ. (a) Case 1; (b) Case 2.
values ðx2 =a ¼ 1; 0:1; 0:01Þ, respectively. As shown in these figures, we can see that in case 1, the stress r22 and electric displacement D2 are anti-symmetric to x2 =a, but r21 and D1 are
1614
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
Fig. 5. Distributions of stress r12 =q with x1 =a axis ðx2 =a ¼ 1; 0:1; 0:01Þ. (a) Case 1; (b) Case 2.
symmetric to x2 =a. In case 2, the stress r12 and electric displacement D1 are anti-symmetric to x2 =a, but r21 and D1 are symmetric to x2 =a. From Eq. (29) and the anti-symmetry of E1 related to x2 =a
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1615
Fig. 6. Distributions of stress r22 =q with x1 =a axis ðx2 =a ¼ 1; 0:1; 0:01Þ. (a) Case 1; (b) Case 2.
in case 2, the electric potentials on two electrodes are the same. But in case 1 the potentials on two electrodes are different because E1 is symmetric to axis x2 . When x2 =a approaches to zero, the
1616
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
electric and mechanical fields are singular (with inverse square root singularity) at the vicinity of the electrode ends. The distribution behaviors of electroelastic fields at the place between two electrodes (1 < x1 =a < 1) and at the place outside the electrodes (x1 =a < 3 or x1 =a > 3) are different. This is shown that the effects of multiple electrodes are obvious. Comparing the values of two cases we can see that the region between two electrodes and x2 is not large, then the electroelastic fields in case 1 are larger than that in case 2.
Fig. 7. Distributions of stresses in depth direction ðx1 =a ¼ 3; 2; 1; 0:5; 0Þ. (a) r22 =q; (b) r12 =q.
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1617
The calculations also show that the values of electric displacement D2 =q on jx2 =aj < 4, x2 =a ¼ 10 are less than 6 103 /m2 and 5:8 102 /m2 for case 1 and 2 respectively. It is shown that at the remote area from the electrodes, the values of D2 =q approach to zero in both cases. But the speed approaching zero in case 1 is faster than that in case 2. The further study shows that the potential in case 1 is finite at infinity but the material in case 2 approaches )const. ln r as r ! 1.
Fig. 8. Distributions of electric displacements in depth direction ðx1 =a ¼ 3; 2; 1; 0:5; 0Þ. (a) D2 =q; (b) D1 =q.
1618
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
The behaviors of normalized stresses and normalized electric displacements along x2 =a axis at different x1 =a values shown in Figs. 7 and 8 respectively are only calculated for case 2. As expected, stresses r22 and r12 vanish on the electrodes and free surface. The electric displacement D2 vanishes on the free surface and has a finite value on the electrodes. The electric field E1 vanishes on the surface electrodes and takes nonzero value on the free surface (which is not shown by figures). At the vicinity of the electrode ends (x1 =a ¼ 3; 1; 1; 3, x2 =a ¼ 0), the electric and mechanical fields are concentration, which will induce depoling and even breakdown. The high electric and mechanical fields near the electrode ends are declined quickly when the depth ðx2 =aÞ increases. At a finite distance from the surface D2 ðE2 Þ approaches a lower even value, D1 and r12 approach zero, but r22 varies still finite. All these variables approach zero when jzj ! 1. The situation for case 1 can be discussed in a similar manner. 5. Conclusions The problem of the half-plane with thin collinear surface electrodes of piezoelectric ceramic is formulated and solved in a closed form. Some special cases with numerical results are given. For these examples, the distributions of electric and mechanical fields are given graphically. These figures show that the electroelastic fields of multi-electrodes are different with that for one electrode. It is also shown that there are significant differences for the case with zero total charge on all electrodes and the case with Q 6¼ 0. It is found that electric and mechanical fields are singular at the vicinity of the electrode ends. These results are important for the function and failure to the piezoelectric apparatuses, especially for the interdigitated electrodes.
Acknowledgement This work was supported by the National Natural Science Foundation through Grant No. 10132010.
References [1] R.D. Mindlin, Coupled piezoelectric vibration of quartz plates, Int. J. Solids Struct. 10 (1974) 453–459. [2] Y. Shindo, F. Narita, H. Sosa, Electroelectric analysis of piezoelectric ceramics with surface electrodes, Int. J. Eng. Sci. 36 (1998) 1001–1009. [3] C.Q. Ru, Exact solution for finite electrode layers embedded at the interface of two piezoelectric half-planes, J. Mech. Phys. Solids 48 (2000) 693–708. [4] N. Bugdayci, D.B. Bogy, A two-dimensional theory for piezoelectric layers used in electro-mechanical transducers I derivation, Int. J. Solid Struct. 17 (1981) 1159–1178. [5] M. Yoshida, F. Narita, Y. Shindo, M. Karaiwa, K. Horiguchi, Electroelastic field concentration by circular electrodes in piezoelectric ceramics, Smart Mater. Struct. 12 (2003) 972–978. [6] R.Q. Ye, L.H. He, Electric field and stresses concentrations at the edge of parallel electrodes in piezoelectric ceramics, Int. J. Solids Struct. 38 (2001) 6943–6951. [7] X.F. Li, G.J. Tang, Electroelastic analysis for a piezoelectric layer with surface electrodes, Mech. Res. Commun. 30 (2003) 345–351.
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1619
[8] A.A. Bent, N.W. Hagood, Piezoelectric Fiber composites with interdigitated electrodes, J. Intell. Mater. Sys. Struct. 8 (1997) 903–919. [9] K.-L. Zhou, Z.-D. Zhou, Z.-B. Kuang, Surface electrode problem in piezoelectric materials, International Symposium on Macro-, Meso-, Micro- and Nano-Mechanics of Materials (MM2003), December 8–10, 2003 HKUST, Clear Water Bay, Hong Kong. [10] Z. Suo, C.M. Kuo, D.M. Barnett, J.R. Willis, Fracture mechanics for piezoelectric ceramics, J. Mech. Phys. Solids 40 (1992) 739–765. [11] Z.B. Kuang, F.S. Ma, Crack Tip Fields, Xi’an Jiaotong University Press, Xi’an, 2002 (in Chinese). [12] T.C.T. Ting, Anisotropic Elasticity, Theory and Application, Oxford University Press, Oxford, 1996. [13] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, The Netherlands, 1963. [14] Z.Y. Huang, Z.B. Kuang, A mixed electric boundary value problem for a two-dimensional piezoelectric crack, Int. J. Solids Struct. 40 (2003) 1433–1453.
Electroelastic analysis of a piezoelectric half-plane with finite surface electrodes Zhen-Bang Kuang *, Zhi-Dong Zhou, Kun-Li Zhou Department of Engineering Mechanics, Shanghai JiaoTong University, Shanghai 200240, PR China Received 10 August 2003; received in revised form 12 February 2004; accepted 4 March 2004 Available online 28 July 2004 (Communicated by S.-A. ZHOU)
Abstract In this paper the problem of the electroelastic fields of piezoelectric ceramic in a half-plane with thin collinear surface electrodes is formulated and analyzed. Using the standard analytic continuation method, a mixed boundary value problem in plane is obtained and then reduced to a standard Hilbert problem. The closed form expressions of electroelastic pffiffi fields are obtained. Some special cases are discussed. The electroelastic filed has singularity with 1= r at the vicinity of the electrode ends. The numerical examples for two surface electrodes are studied to show the practical distributions of the electroelastic fields graphically. 2004 Elsevier Ltd. All rights reserved. Keywords: Piezoelectric ceramic; Surface electrodes; Electroelastic behavior; Hilbert problem
1. Introduction Owing to the large coupling between electric and mechanical fields, piezoelectric ceramics are widely used as smart materials such as transducers, sensors and actuators. In these cases, the thin electrodes are usually attached at the surface or interface of electroceramic blocks [1–8]. Hence it is important to research the electroelastic field produced by the thin electrodes in piezoelectric apparatuses. Shindo et al. [2] studied the static behavior of electroelastic field at the vicinity of single surface electrode by Fourier transformation method in a transverse isotropic material. Ru [3] studied the case of thin electrodes embedded at the interface of two bonded dissimilar *
Corresponding author. Tel.: +86-21-54743067; fax: +86-21-54740344. E-mail address: [email protected] (Z.-B. Kuang).
0020-7225/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2004.03.004
1604
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
piezoelectric half-planes. Ye and He [6] analyzed the electric field concentrations of the edges of a pair of parallel electrodes in a transverse isotropic material when the sizes of electrodes are far smaller than that of the thickness. They found the stresses at the edge of electrodes are very high and can make the ceramics failure. Li and Tang [7] studied the piezoelectric layer with electrodes at the top surface and bottom surface by the conformal mapping technique. Bent and Hagood [8] discussed in detail the properties of a piezoelectric fiber composites with interdigitated electrodes. Zhou et al. studied a single surface electrode for soft and rigid cases [9] by conformal mapping method and got pretty closed solutions. In this paper, the problem of thin collinear electrodes attached at the surface of the half-plane in a general piezoelectric ceramic are formulated and discussed. According to the principle of standard analytic continuation method, the problem is continued to the whole plane and then is reduced to a mixed boundary problem which can be reduced to a Hilbert problem. The closed form solutions of electroelastic filed are obtained, which show the electric and stress fields exhibit inverse square root singularity at the vicinity of the electrode ends. The field concentration is abated when the depth (x2 ) increases. Finally some special examples are given. The distributions of the electric field and stress are useful in engineering application. 2. Basic equations and general solution In a fixed rectangular coordinate system (x1 ; x2 ; x3 ), all of the field variables depend on x1 ; x2 only for a generalized plane piezoelectric problem. Following Suo et al. [10] and Kuang and Ma [11], the general solution in this case can be given by the linear combination of four complex analytical functions u ¼ 2Re½AfðzÞ;
/ ¼ 2Re½BfðzÞ; T
u ¼ ½u1 ; u2 ; u3 ; u ;
/ ¼ ½/1 ; /2 ; /3 ; /4 T ; T
fðzÞ ¼ ½f1 ðz1 Þ; f2 ðz2 Þ; f3 ðz3 Þ; f4 ðz4 Þ ;
ð1Þ
zk ¼ x1 þ pk x2
ðk ¼ 1–4Þ;
where Re stands for the real part of a function; ui ði ¼ 1; 2; 3Þ; u and /i are the displacement components, electric potential and the components of the generalized stress function, respectively; A and B are 4 · 4 complex matrices related to the material constants, expressed as A ¼ ½a1 ; a2 ; a3 ; a4 ;
B ¼ ½b1 ; b2 ; b3 ; b4 :
ð2Þ
The eigenvalues pk and eigenvectors ak are determined by the following equations: ½Q þ ðR þ RT Þp þ Tp2 a ¼ 0
ð3Þ
where
QE Q¼ T e11 QEik ¼ ci1k1 ;
e11 ; j11
RE R¼ T e21
REik ¼ ci1k2 ;
e21 ; j12
TikE ¼ ci2k2 ;
TE T¼ T e22
ðeij Þs ¼ eijs ;
e22 ; j22
ð4Þ
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1605
where cijkl are the elastic stiffnesses under constant electric field, eijs are the piezoelectric constants and jij are the permittivities under constant strain field. The eigenvectors bk can be obtained from the following equations: bk ¼ ðRT þ pk TÞak ¼
1 ðQ þ pk RÞak : pk
ð5Þ
The generalized stresses are represented as t1 ¼ ½r11 ; r12 ; r13 ; D1 T ¼ ½/1;2 ; /2;2 ; /3;2 ; /4;2 T ; T
ð6Þ
T
t2 ¼ ½r21 ; r22 ; r23 ; D2 ¼ ½/1;1 ; /2;1 ; /3;1 ; /4;1 ;
where rij ; Di are stresses and electric displacements, respectively. After the normalization for the eigenvectors A and B, there exist the following relations [11,12]:
BT T B
AT T A
I A A ¼ 0 B B
0 ; I
A A B B
BT T B
AT T A
I 0 ¼ 0 I
ð7Þ
3. Statement and solution procedure of the problem Consider thin collinear electrodes attached at the surface of piezoelectric ceramic as shown in Fig. 1. Let the left and right ends of the i-th electrode be ai and bi ði ¼ 1; 2; . . . ; nÞ, respectively. It is assumed that the total electric charge or potential on the i-th electrode is qi or Vi respectively. Then, the mixed boundary conditions are given by
Fig. 1. A piezoelectric ceramic with collinear surface electrodes.
1606
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
r2i ¼ 0;
D2 ¼ 0;
x2 ¼ 0;
z 2 L0 ¼ ½1; a1 þ ½b1 ; a2 þ þ ½bn ; þ1;
r2i ¼ 0;
E1 ¼ 0;
x2 ¼ 0;
z 2 L00 ¼ ½a1 ; b1 þ þ ½an ; bn ;
rij ! 0;
Dj ! 0;
jzj ! 1;
i ¼ 1; 2; 3;
ð8Þ
j ¼ 1; 2;
where L0 þ L00 ¼ L. Analogous to elastic contact problem [13], using the Gauss law, along the surface electrodes the supplement boundary conditions can be written as Z D i ¼ 1; 2; . . . ; n ð9aÞ 2 ðx1 Þdx1 ¼ qi ; L00i
or ui ¼ V i ;
i ¼ 1; 2; . . . n and
n Z X i¼1
L00i
D 2 ðx1 Þ dx1 ¼ Q ¼
n X
qi ;
ð9bÞ
i¼1
where Q is the total charge on all electrodes. If Q ¼ 0, then the supplement conditions can be expressed by pure potentials. If Q 6¼ 0, then Q is included in the solution. Eqs. (9a) and (9b) are equivalent. The superscript ‘)’ indicates the limit value taken from the lower half-plane. The boundary conditions (8) and (9a) or (9b) guarantee that the discussed problem has an unique solution. Let FðzÞ ¼ f 0 ðzÞ, following Eq. (8)1 , the generalized stresses along all electrode-free part L0 can be written as BF ðx1 Þ þ BF ðx1 Þ ¼ 0;
x 2 L0 ;
ð10Þ
which can be rewritten as [13] BF ðx1 Þ þ BFþ ðx1 Þ ¼ 0;
ð11Þ
where the superscript ‘+’ indicates the limit value taken from the upper half-plane. Now a new function of complex variable z is defined as B1 BFðzÞ z 2 S þ ð12Þ hðzÞ ¼ FðzÞ z 2 S The function hðzÞ is analytic in the whole plane except at the segment L00 . Using Eqs. (1) and (6), Eq. (8)2 can be written as BF ðx1 Þ þ BFþ ðx1 Þ ¼ g;
T 00 g ¼ ð0; 0; 0; D 2 ðx1 ÞÞ ; x 2 L ;
ð13Þ
where D 2 ðx1 Þ is unknown. Substituting Eq. (12) into Eq. (13), we get hþ ðx1 Þ h ðx1 Þ ¼ B1 g; 1 hþ a ðx1 Þ ha ðx1 Þ ¼ Ba4 D2 ðx1 Þ;
a ¼ 1; . . . ; 4;
ð14Þ
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1607
which is the standard Hilbert problem. The solution of Eq. (14) is B1 hðzÞ ¼ 2pi
Z L00
gðx1 Þ dx1 and Fa ðza Þ ¼ ha ðza Þ when z 2 S : x1 z
ð15Þ
It is known that the electric displacement can be expressed in the form as [13,14] P ðzÞ D2 ðzÞ ¼ Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðz ai Þðz bi Þ
z 2 S;
ð16Þ
i¼1
where P ðzÞ is a polynomial of degree ðn 1Þ and we select the Branch z ! 1 along axis x, P ðzÞ ¼ iðcn1 zn1 þ þ c1 z þ c0 Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz ai Þðz bi Þ ! x when
ð17Þ
in which ci ði ¼ 1; 2; . . . ; n 1Þ are n unknown complex numbers which should be determined by Eq. (9a) or (9b). Substituting Eqs. (16) and (17) into Eq. (15) and noting that D takes its value at S , we get [13] P ðza Þ Fa ðza Þ ¼ B1 a4 Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 ðza ai Þðza bi Þ
ð18Þ
i¼1
fa ðza Þ ¼
B1 a4
Z
P ðza Þdza iC 1 B ; n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Q 2 a4 2 ðza ai Þðza bi Þ
ð19Þ
i¼1
where C can be taken as a real constant due to that its imaginary part does not infect the potential (see Eq. (27)). The expression of fa ðza Þ may not be expressed by elementary functions except some special cases, so in some cases using Eq. (9a) is more convenient. Using E1 ¼ 0 on L00 shown in Eq. (8)2 , we have AFðx1 Þ þ AFðx1 Þ ¼ ð; ; ; 0Þ
ð20Þ
in which ‘*’ denotes some unknown quantities which are not concerned here. Eq. (20) can be rewritten as
1608
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
iP ðx1 Þ A4a B1 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ðx1 ai Þðbi x1 Þ ðx1 aj Þðx1 bj Þ j¼1 j6¼i
iP ðx1 Þ A4a B1 a4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0; ðx1 aj Þðx1 bj Þ ðx1 ai Þðbi x1 Þ
x1 2 L00i ;
ð21Þ
j¼1 j6¼i
pffiffiffiffiffiffiffi where i ¼ 1. According to [3], [13] and [14] and that H44 ¼ iA4a B1 a4 is real, so Eq. (21) can be reduced to x1 2 L00i :
P ðx1 Þ þ P ðx1 Þ ¼ 0;
ð22Þ
So we can get the relation of ci ci ci ¼ 0;
ð23Þ
which means that ci are all the real constants. The generalized stresses can be expressed as t2b ¼ Re
4 X a¼1
P ðza Þ 1 Bba Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ba4 : ðza ai Þðza bi Þ
ð24Þ
i¼1
The generalized displacements are ub ¼ 2Re½Aba fa ðza Þ:
ð25Þ
It is obvious that t2b exhibit inverse square root singularity at the vicinity of electrode ends. Substituting Eq. (24) into (9a) we get Z
bi
t24 ðx1 Þdx1 ¼
Z
ai
bi
D 2 ðx1 Þdx1 ai
¼
Z ai
bi
P ðx1 Þdx1 Re Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx1 aj Þðx1 bj Þ j¼1
¼
Z
bi ai
iP ðx1 Þdx1 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qi : Q ðx1 ai Þðbi x1 Þðx1 aj Þðx1 bj Þ
ð26Þ
j¼1 j6¼i
There are n equations to determine n unknowns c0 ; c1 ; . . . ; cn1 , so the problem is solved. Or substituting Eq. (25) into (9b) we get
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
ðu44 Þi ¼ ui ¼ H44 Re
Z
bi ai
Pn
Z
i¼1 ai
iP ðx1 Þdx1 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ H44 C ¼ Vi ; Q ðx1 ai Þðx1 bi Þ j¼1
bi
1609
P ðx1 Þdx1 Re Q n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Q: ðx1 ai Þðx1 bi Þ
ð27Þ
i¼1
There are n þ 1 equations to determine the n þ 1 unknowns c0 ; c1 ; . . . ; cn1 and C. It is also known that from electricity theory one of Vi can take arbitrary value, especially zero. When z ! 1, Eqs. (15) and (18)1 are respectively reduced to Z 1 D lim F4 ðz4 Þ ¼ B44 2 ðx1 Þdx1 =2piz4 ; z!1
L00
lim F4 ðz4 Þ ¼ iB1 44 cn1 =2z4 :
z!1
Therefore we have Z 1 1 Q cn1 ¼ D : 2 ðx1 Þdx1 ¼ ðQÞ ¼ pi L00 p p
ð28Þ
It is also noted that if Ei is determined by using Eq. (26), then the potentials can be determined as follows: Let u1 ¼ V0 at electrode 1 ða1 6 x 6 b1 ; y ¼ 0Þ, then ui at electrode i ðai 6 x 6 bi ; y ¼ 0Þ can be determined by Z ai E1 dx1 ¼ Vi ; i ¼ 2; 3; . . . ; n: ð29Þ ui ¼ ui1 þ bi1
So the relations between Vi and qi are obtained. It means that Eqs. (9a) and (9b) are equivalent. 4. Some special cases 4.1. When the collinear electrodes are degenerated into a single surface electrode with charge q and a1 ¼ a, b1 ¼ a, then using Eq. (28) the function Fa ðza Þ is P ðza Þ qi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ffi ¼ B1 Fa ðza Þ ¼ B1 a4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 2 2 2 ðza a Þ 2p ðz2a a2 Þ Integrating Eq. (30) we have ( ) qffiffiffiffiffiffiffiffiffiffiffiffiffiffi iq fa ðza Þ ¼ ln za : þ z2a a2 þ ln b B1 a4 ; 2p
ð30Þ
ð31Þ
1610
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
where b is a real constant. Let u ¼ V0 on the electrode, then we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q 2 2 ¼ V0 : u ¼ 2RefA4a fa ðza Þg ¼ Re iA4a Ba4 lnðx1 i a x1 Þ þ ln b p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Because H44 ¼ iA4a B1 a2 x21 is a, we get a4 is real and the modulus of x1 i q 1 H44 ln ba ¼ V0 or b ¼ exp p a
V0 p : qH44
ð32Þ
So in this case the electric potential u is qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qH44 1 Re ln za þ z2a a2 u¼ þ V0 : a p
ð33Þ
Usually H44 < 0. Now the normalized stresses can be expressed as 4 X q 1 t2b ¼ Im Bba pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 a4 p ðz2a a2 Þ a¼1
t1b
4 X q pa ¼ Im Bba pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 a4 p ðz2a a2 Þ a¼1
ð34Þ
The above equations are identical with Ref. [9]. For the dielectric without piezoelectric effect then Q, R, T each has one component, Q44 ¼ e11 , R44 ¼ e13 , T44 ¼ e33 . The eigenvalue equation is reduced to one equation, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 e33 : ð35Þ ðe11 þ 2p4 e13 þ p4 e33 Þa4 ¼ 0; p4 ¼ p ¼ e13 þ i e11 e33 e213 Thus A, B each only has one component, B44 ¼ ðe12 þ p4 e33 Þ, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A44 ¼ i e11 e33 e213 a44 and H44 ¼ ðe11 e33 e213 Þ1=2 < 0. The results become " # iq pffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 F44 ðz4 Þ ¼ 44 ; 2p z24 a2
ð36Þ
" # ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r q z4 z 4 2 ffi Re ln þ 1 ð37Þ uðz4 Þ ¼ V0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a p e11 e33 e213 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 For isotropic medium, eij ¼ edij , u ¼ V0 peq Re ln½az þ ðaz Þ 1, which is identical with the result in the usual dielectric theory.
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1611
Fig. 2. A piezoelectric ceramic with two surface electrodes.
4.2. Two surface electrodes with same configuration are considered as shown in Fig. 2. 4.2.1. Case 1: The electrodes 1 and 2 are located at a1 ¼ 0:03 m, b1 ¼ 0:01 m, a2 ¼ 0:01 m, b2 ¼ 0:03 m and the charges on them are q and q respectively. From Eq. (28) it is obvious that c1 ¼ 0. c0 can be obtained by R 1 numerical method and c0 ¼ 1:1469q. Obviously if let u1 ¼ V0 on electrode 1, then u2 ¼ V0 1 E1 dx1 on the electrode 2. The problem is solved. By numerical integration we get u2 ¼ V0 þ 1:0707q. 4.2.2. Case 2: The electric charge of each electrode is q and their ends are located at a1 ¼ 0:03 m, b1 ¼ 0:01 m, a2 ¼ 0:01 m, b2 ¼ 0:03 m, respectively. PZT-5H ceramic with poling axis ox2 is taken as an example. The material constants of which are listed in Table 1. The function of Eqs. (17) and (28) are respectively reduced to P ðzÞ ¼ iðc1 z þ c0 Þ;
2 c1 ¼ q: p
ð38Þ
Due to the symmetry of the distribution of electric charge, so c0 ¼ 0. Table 1 Elastic stiffnesses, piezoelectric coefficients and dielectric constants of PZT-5Ha c11 (N/m2 )
c13 (N/m2 )
c12 (N/m2 )
c22 (N/m2 )
c55 (N/m2 )
e21 (C/m2 )
e22 (C/m2 )
e16 (C/m2 )
j11 (C/Vm)
j22 (C/Vm)
12.6 · 1010
5.5 · 1010
5.2 · 1010
11.7 · 1010
3.53 · 1010
)6.5
23.3
17.0
1.51 · 108
1.30 · 108
a
The constitutive equations are: ri ¼ cij ej eij Ej , Di ¼ eij ej þ jij Ej .
1612
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
4.2.3. Numerical results: Figs. 3–6 give the distributions of normalized electric displacement ðD1 =q; D2 =qÞ and normalized stresses ðr12 =q; r22 =qÞ with x1 =a along axis ox1 at different x2 =a
Fig. 3. Distributions of electric displacement D2 =q with x1 =a axis ðx2 =a ¼ 1; 0:1; 0:01Þ. (a) Case 1; (b) Case 2.
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619 1.0
1613
x2/a=-1
0.8
x2/a=-0.1
0.6
x2/a=-0.01
0.4
D1/q(m-2)
0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -4
-3
-2
0
-1
x1/a
(a)
1
2
3
4
Case 1
0.8
x2/a=-1 x2/a=-0.1
0.6
x2/a=-0.01
0.4
D1/q(m-2)
0.2 0.0 -0.2 -0.4 -0.6 -0.8 -4
(b)
-3
-2
-1
0
1
2
3
4
x /a 1
Case 2
Fig. 4. Distributions of electric displacement D1 =q with x1 =a axis ðx2 =a ¼ 1; 0:1; 0:01Þ. (a) Case 1; (b) Case 2.
values ðx2 =a ¼ 1; 0:1; 0:01Þ, respectively. As shown in these figures, we can see that in case 1, the stress r22 and electric displacement D2 are anti-symmetric to x2 =a, but r21 and D1 are
1614
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
Fig. 5. Distributions of stress r12 =q with x1 =a axis ðx2 =a ¼ 1; 0:1; 0:01Þ. (a) Case 1; (b) Case 2.
symmetric to x2 =a. In case 2, the stress r12 and electric displacement D1 are anti-symmetric to x2 =a, but r21 and D1 are symmetric to x2 =a. From Eq. (29) and the anti-symmetry of E1 related to x2 =a
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1615
Fig. 6. Distributions of stress r22 =q with x1 =a axis ðx2 =a ¼ 1; 0:1; 0:01Þ. (a) Case 1; (b) Case 2.
in case 2, the electric potentials on two electrodes are the same. But in case 1 the potentials on two electrodes are different because E1 is symmetric to axis x2 . When x2 =a approaches to zero, the
1616
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
electric and mechanical fields are singular (with inverse square root singularity) at the vicinity of the electrode ends. The distribution behaviors of electroelastic fields at the place between two electrodes (1 < x1 =a < 1) and at the place outside the electrodes (x1 =a < 3 or x1 =a > 3) are different. This is shown that the effects of multiple electrodes are obvious. Comparing the values of two cases we can see that the region between two electrodes and x2 is not large, then the electroelastic fields in case 1 are larger than that in case 2.
Fig. 7. Distributions of stresses in depth direction ðx1 =a ¼ 3; 2; 1; 0:5; 0Þ. (a) r22 =q; (b) r12 =q.
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1617
The calculations also show that the values of electric displacement D2 =q on jx2 =aj < 4, x2 =a ¼ 10 are less than 6 103 /m2 and 5:8 102 /m2 for case 1 and 2 respectively. It is shown that at the remote area from the electrodes, the values of D2 =q approach to zero in both cases. But the speed approaching zero in case 1 is faster than that in case 2. The further study shows that the potential in case 1 is finite at infinity but the material in case 2 approaches )const. ln r as r ! 1.
Fig. 8. Distributions of electric displacements in depth direction ðx1 =a ¼ 3; 2; 1; 0:5; 0Þ. (a) D2 =q; (b) D1 =q.
1618
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
The behaviors of normalized stresses and normalized electric displacements along x2 =a axis at different x1 =a values shown in Figs. 7 and 8 respectively are only calculated for case 2. As expected, stresses r22 and r12 vanish on the electrodes and free surface. The electric displacement D2 vanishes on the free surface and has a finite value on the electrodes. The electric field E1 vanishes on the surface electrodes and takes nonzero value on the free surface (which is not shown by figures). At the vicinity of the electrode ends (x1 =a ¼ 3; 1; 1; 3, x2 =a ¼ 0), the electric and mechanical fields are concentration, which will induce depoling and even breakdown. The high electric and mechanical fields near the electrode ends are declined quickly when the depth ðx2 =aÞ increases. At a finite distance from the surface D2 ðE2 Þ approaches a lower even value, D1 and r12 approach zero, but r22 varies still finite. All these variables approach zero when jzj ! 1. The situation for case 1 can be discussed in a similar manner. 5. Conclusions The problem of the half-plane with thin collinear surface electrodes of piezoelectric ceramic is formulated and solved in a closed form. Some special cases with numerical results are given. For these examples, the distributions of electric and mechanical fields are given graphically. These figures show that the electroelastic fields of multi-electrodes are different with that for one electrode. It is also shown that there are significant differences for the case with zero total charge on all electrodes and the case with Q 6¼ 0. It is found that electric and mechanical fields are singular at the vicinity of the electrode ends. These results are important for the function and failure to the piezoelectric apparatuses, especially for the interdigitated electrodes.
Acknowledgement This work was supported by the National Natural Science Foundation through Grant No. 10132010.
References [1] R.D. Mindlin, Coupled piezoelectric vibration of quartz plates, Int. J. Solids Struct. 10 (1974) 453–459. [2] Y. Shindo, F. Narita, H. Sosa, Electroelectric analysis of piezoelectric ceramics with surface electrodes, Int. J. Eng. Sci. 36 (1998) 1001–1009. [3] C.Q. Ru, Exact solution for finite electrode layers embedded at the interface of two piezoelectric half-planes, J. Mech. Phys. Solids 48 (2000) 693–708. [4] N. Bugdayci, D.B. Bogy, A two-dimensional theory for piezoelectric layers used in electro-mechanical transducers I derivation, Int. J. Solid Struct. 17 (1981) 1159–1178. [5] M. Yoshida, F. Narita, Y. Shindo, M. Karaiwa, K. Horiguchi, Electroelastic field concentration by circular electrodes in piezoelectric ceramics, Smart Mater. Struct. 12 (2003) 972–978. [6] R.Q. Ye, L.H. He, Electric field and stresses concentrations at the edge of parallel electrodes in piezoelectric ceramics, Int. J. Solids Struct. 38 (2001) 6943–6951. [7] X.F. Li, G.J. Tang, Electroelastic analysis for a piezoelectric layer with surface electrodes, Mech. Res. Commun. 30 (2003) 345–351.
Z.-B. Kuang et al. / International Journal of Engineering Science 42 (2004) 1603–1619
1619
[8] A.A. Bent, N.W. Hagood, Piezoelectric Fiber composites with interdigitated electrodes, J. Intell. Mater. Sys. Struct. 8 (1997) 903–919. [9] K.-L. Zhou, Z.-D. Zhou, Z.-B. Kuang, Surface electrode problem in piezoelectric materials, International Symposium on Macro-, Meso-, Micro- and Nano-Mechanics of Materials (MM2003), December 8–10, 2003 HKUST, Clear Water Bay, Hong Kong. [10] Z. Suo, C.M. Kuo, D.M. Barnett, J.R. Willis, Fracture mechanics for piezoelectric ceramics, J. Mech. Phys. Solids 40 (1992) 739–765. [11] Z.B. Kuang, F.S. Ma, Crack Tip Fields, Xi’an Jiaotong University Press, Xi’an, 2002 (in Chinese). [12] T.C.T. Ting, Anisotropic Elasticity, Theory and Application, Oxford University Press, Oxford, 1996. [13] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, The Netherlands, 1963. [14] Z.Y. Huang, Z.B. Kuang, A mixed electric boundary value problem for a two-dimensional piezoelectric crack, Int. J. Solids Struct. 40 (2003) 1433–1453.