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Electroelastic stress in an electrostrictive material with charged surface electrodes
International Journal of Engineering Science 48 (2010) 2066–2080
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Electroelastic stress in an electrostrictive material with charged surface electrodes Quan Jiang a,b,*, Cun-Fa Gao a, Zhen-Bang Kuang c a
College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China College of Civil Engineering, Nantong University, Nantong 226007, China c Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200240, China b
a r t i c l e
i n f o
Article history: Received 4 July 2008 Received in revised form 8 August 2009 Accepted 16 April 2010 Available online 8 May 2010 Communicated by M. Kachanov Keywords: Electrostictive materials Surface electrodes Field singularities
a b s t r a c t Field singularities of collinear electrodes on the surface of a half-infinite electrostrictive solid are studied in terms of the complex variable method. Firstly, the general solutions for the potential functions of electric fields and stresses are derived for the case of collinear electrodes based on analytical continuation method. Then, explicit results are given for the cases of one and two electrodes, respectively. Finally, numerical calculations are made to discuss the effects of applied electric loading on the field singularities near the electrodes. It is found that even at the low level of applied electric loading, considerable high stresses will be induced around the tip of electrodes. Moreover, for the cases of multiple electrodes, the distribution of electric fields and stresses is more complicated than the case of a single electrode. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction As a necessary carrier of electric loading, electrodes are usually used in smart structures and devices, which are in general made of piezoelectric or electrostrictive materials. In these structures and devices, electrodes can be placed in three cases: internal electrodes (which are inserted into a homogeneous material), interface electrodes (which are placed between two materials), and surface electrodes (which are located on the surface of the material). For all the cases, the high field variables may be induced around the electrode edges due to the different mechanical and electrical behavior of electrodes and matrix materials. The nonuniform local field results in crack initiation and crack growth and finally leads to the failure of the device. Thus, with increasingly wide applications of smart structures and devices made of piezoelectric/electrostrictive materials, the problems of local fields near electrodes have received much attention in the recent decade. In fact, some important results have been presented for the above three cases of electrode distributing. For the case of internal electrodes, Ye and He [35] studied the electric field and stresses concentrations at the edge of parallel electrodes in piezoelectric ceramics; Lucato et al. [16] investigated the constraint-induced crack initiation at electrode edges in piezoelectric ceramics; Chen and Chue [3] made the fracture mechanics analysis of a composite piezoelectric strip with an internal semi-infinite electrode; Wang and Sun [30] derived the intensity factors for some common piezoelectric fracture mechanics specimens with conducting cracks or electrodes; Gao et al. [4] addressed the generalized two-dimensional problem of a conductive crack and electrode in piezoelectric materials using dielectric breakdown model; Yang et al. [34] gave * Corresponding author at: College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China. Tel.: +86 25 84895115; fax: +86 25 84891422. E-mail address: [email protected] (Q. Jiang). 0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2010.04.008
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the electrical characteristics of central driving type piezoelectric transformers with different electrode distributing. For the cases of interfacial electrodes, Ru[22,23] studied the fracture problems of a soft electrode in piezoelectric multilayer materials or at the interface of two piezoelectric half-planes by means of complex potentials; Shindo et al. [25] and Narita et al. [20] also analyzed the electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators or between two piezoelectric half-planes by using experiment and finite element simulation, respectively; Hausler et al. [8], Hausler and Balke [7] studied the generalized two-dimensional problems of collinear and periodic electrode-ceramic interfacial cracks in piezoelectric bimaterials and gave the structure of singular fields at the tip of electrode; Beom et al. [1] made the analysis of a thin electrode layer between two dissimilar piezoelectric solids, and Narita et al. [19] analyzed the electroelastic field concentrations and polarization switching induced by circular electrode at the interface of piezoelectric disk composites. On the other hand, a lot of works for the cases of surface electrodes are also made. For example, Shindo et al. [24] conducted the electroelastic analysis of piezoelectric ceramics with surface electrodes; He and Ye [9], Li and Duan [14], and Wang [29] studied the concentration of electroelastic field near electrodes on a piezoelectric layer, respectively. Yang [32] addressed the electromechanical interaction of linear piezoelectric materials with a surface electrode; Kuang et al. [12] derived the analytical solutions for the electroelastic field of a piezoelectric half-plane with finite surface electrodes; Zhang and Shen [36] made the three-dimensional analysis for rectangular 1–3 piezoelectric fiber-reinforced composite laminates with the interdigitated electrodes under electromechanical loadings; Li and Lee [15] studied the elastic response of a piezoelectric ceramic plate with two penny-shaped electrodes at the opposite surfaces under an electric load, and Malits [17] investigated the periodic anti-plane problem of a piezoelectric layer with electrodes by using the dual equations method. However, it should be noted that all the works cited above are for the cases of piezoelectric materials. For electrostrictive materials with electrodes, only a few of works can be found for the study of electroelastic field concentrations near the electrodes due to the difficulty involved in the constitutive equations of electrostrictive materials. An earlier work on studying electroelastic fields in a designing cofired multilayer electrostrictive actuator with electrodes was given by Winzer et al. [31]. Then, Gong [5] derived the stresses near the end of an internal electrode in multilayer electrostrictive ceramic actuators, and Yang and Suo [33], Hao et al. [6] made the numerical calculation on the electroelastic fields for the design of ectrostrictive ceramic multilayer actuators, respectively. Recently, Beom et al. [2] presented some theoretical results for an internal electrode in an electrostrictive material under the small scale saturation condition with the use of complex function theory. To the authors’ knowledge, however, no work can be found for the cases of collinear charged electrodes on the surface of an electrostrictive material, although the similar problems have been well studied for the cases of piezoelectric materials [24,25,32]. It is therefore the purpose of this work to study the field singularities of collinear electrodes on the surface of a half-infinite electrostrictive solid in terms of the complex variable method. Below is the plan of the present work: following the introduction, we will outline the basic equations based on the work of Stratton [28] and Landau and Lifshitz [13] in Section 2], and then give the solutions of Knops [10] and Smith and Warren [26,27] for the case of two-dimensional deformation in Section 3. Derived in Section 4 is the general solution for the potential functions of electric fields and stresses for the case of collinear electrodes, and explicit results are given for special cases of one and two electrodes in Section 5. Numerical calculations are made in Section 6 to discuss the effects of applied electric loading on the field singularities near the electrodes, and this work is finally concluded in Section 7. 2. Basic equations of electrostrictive materials The constitutive equations of electrostriction is nonlinear since they contain the square terms of electric field. Stratton [28], Landau and Lifshitz [13] obtained the electrostrictive stress induced by electric field in electrostrictive media. Pao [21] and Kuang [11] also discussed the electrostrictive force induced by electric field in other ways. These results will be outlined as follows. The constitutive equations of electrostrictive materials for the isothermal and isotropic case are
1 a1 Ei Ej þ a2 Ek Ek dij ; 2 Di ¼ dij þ a1 eij þ a2 ekk dij Ej ;
rij ¼ 2leij þ kekk dij
ð1Þ
where rij,eij,Di and Ej are stress, strain, electric displacement and electric field intensity, respectively. a1, a2 are two independent electrostrictive coefficients in isotropic materials. is the permittivity at the state without strain. dij is Kronecker delta. k and l are Lame constants which can be expressed by Young’s modulus E and Poisson ratio m in the form k = E m/ [(1 + m)(1 2m)],l = E/[2(1 + m)]. The equilibrium equation of electro-elastic material is
rij;j þ fie ¼ 0;
ð2Þ
where fie is the body force induced by the electric fields, and it reads:
fie ¼ rM ij;j ¼
1 2
Ei Ej Ek Ek dij
; ;j
ð3Þ
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where
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rMij is called Maxwell stress. Substituting Eq. (3) into equilibrium Eq. (2), we get:
re ij;j ¼ 0; re ij ¼ rij þ rMij :
ð4Þ
e ij the pseudo total stress. With the help of Eqs. (1) and (3), it is easily got: We call r
re ij ¼ 2leij þ kekk dij þ
a1 þ a2 Ei Ej Ek Ek dij : 2 2
ð5Þ
If there is no free charge in the medium, the electric displacement satisfies:
Di;i ¼ 0:
ð6Þ
On the interface of two medium (Fig. 1), in general case (there is no surface charge), the electric interface conditions are:
uþ ¼ u ; Dþn ¼ Dn ;
ð7Þ
where u and Dn are the electric potential and the normal electric displacement on the interface, and the superscripts (+,) refer the variables in the material I or II. The traction induced by the electric fields on the interface of dielectric media I and II can be written as [28,13]:
X ei ¼ rMþ rM nj ; ij ij
ð8Þ
where nj is the outward normal vector on the interface of the material I. The mechanical interface conditions on the interface are
uþi ¼ ui ; rþij rij nj ¼ X mi þ X ei ¼ X mi rMþ rM nj ; ij ij
ð9Þ ð10Þ
m where u i is the displacements on the interface. X i is the given mechanical traction on interface, and X i is the total interface traction. Eq. (10) can also be changed to:
ei ¼ r e þij nj ¼ r e ij nj ; X
ð11Þ
e i is named as pseudo total interface traction. where X If the material II degenerated to enviroment, it should be noted that the boundary conditions Eqs. (7), (9) and (11) are still valid, but in this case, the functions with superscript () will be some given values. 3. Complex variable solution for 2D problems It is obvious that the electrostrictive problem is nonlinear from Eq. (1). So the exact solution of the problem is hard to be given. In general, since the dielectric constant satisfies the condition akeijdij for small strain, we can uncouple the electric problem from the whole problem, that is, the electric field may be calculated firstly based on the theory of electrostatics. And then the stress field can be given with the help of the known electric field, which is similar to solving the problems of thermal stress.
x2
Dielectric material II
B
n ds Dielectric material I
Di
X
m
Di ij ij
A 0
x1 Fig. 1. The boundary condition of mechanical and electrical fields.
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3.1. Electric field in the medium The complex electric potential x(z) is in the form:
xðzÞ ¼ uðx1 ; x2 Þ þ iAðx1 ; x2 Þ;
ð12Þ
where z = x1 + ix2 in cartesian coordinate. x(z) is analytic function of z in the whole plane, and the electric potential u(x1,x2) has the relationship :
Ei ¼
ouðx1 ; x2 Þ : oxi
ð13Þ
Noting that A(x1,x2) is the supplementary function of u(x1,x2), which is defined as
ou oA ou oA ¼ ; ¼ : ox1 ox1 ox2 ox2
ð14Þ
So we have:
E1 þ iE2 ¼ x0 ðzÞ;
ð15Þ 0
where the symbol (—) denote the conjugate operation to the function, and x (z) = dx(z)/d z. Let z = t on the boundary, the electric boundary can be written as
xðtÞ þ xðtÞ ¼ 2u on L0 ; x0 ðtÞðnx þ iny Þ ¼ 2iDn on L00 ; 0
ð16Þ
00
where L and L stand for the boundary with the given electric potential u and given electric displacement Dn , respectively. Once these conditions are given, the complex potential x(z) can be solved by using complex variable method. 3.2. Stress field in the medium Introducing the pseudo total stress function U as
re 11 ¼
o2 U ; ox22
re 22 ¼
o2 U ; ox21
re 12 ¼
o2 U : ox1 ox2
ð17Þ
Then Eq. (4) is automatically satisfied. The compatible relation can be expressed by
r4 U ¼ 16
i o4 U o4 h ¼ 4j 2 2 x0 ðzÞx0 ðzÞ ; 2 2 oz oz oz oz
ð18Þ
where j = (1 2m)(a1 + 2a2)/[2(1 m)]. Therefore, the function U can be expressed by two analytic functions g(z) and /(z) in the following form:
U¼
i 1 1h gðzÞ þ gðzÞ þ z/ðzÞ þ z /ðzÞ þ j xðzÞ xðzÞ: 2 4
ð19Þ
The pseudo total stresses are
h
i
re 22 þ re 11 ¼ jx0 ðzÞx0 ðzÞ þ 2 /0 ðzÞ þ /0 ðzÞ ;
ð20Þ
re 22 re 11 þ 2i re 12 ¼ jx00 ðzÞxðzÞ þ 2½z/00 ðzÞ þ w0 ðzÞ; 0
where w(z) = g (z). The displacement for plane strain is
2 l ðu1 þ iu2 Þ ¼ ð3 4 mÞ/ðzÞ
j 2
a xðzÞx0 ðzÞ z/0 ðzÞ wðzÞ þ XðzÞ; 2
ð21Þ
where a = a1 . In the plane stress case, the constants m,E,a1 and a2 should change to be m/(1 + m), [E(1 + 2m)/(1 + m)2], a1 and (1 2m)a2/ (1 m), respectively. Moreover, using Eqs. (11) and (17), the traction can be expressed as
Z e 1 þ iP e2 ¼ i i P A
B
B e 1 þ iX e 2 ds ¼ 2 oU : X oz A
ð22Þ
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Substituting Eq. (19) into Eq. (22), we get:
iB h e1 þ iP e 2 ¼ /ðtÞ þ t/0 ðtÞ þ wðtÞ þ j xðtÞx0 ðtÞ ; i P 2 A
ð23Þ
along the given boundary. For the infinite plate, the complex potentials x(z),/(z) andw(z) take the form:
xðzÞ ¼
Q ln z þ C3 z þ xðzÞ; 2p
ð24Þ
and [18]
e1 þ iP e2 P ln z þ C1 z þ /ðzÞ; 8ð1 mÞ e 1 iP e2 ð3 4mÞ P wðzÞ ¼ ln z þ C2 z þ wðzÞ; 8ð1 mÞ
/ðzÞ ¼
ð25Þ
where x(z),/(z) and w(z) are complex analytic function in the discussing region, respectively; and C3, C1 and C2 are related to the electric fields and the tractions at infinity, respectively. 4. General solution for collinear electrodes As shown in Fig. 2, an electrostrictive material is in the lower half-plane S while the upper half-plane S+ is filled with air. The soft electrodes are coated along the surface of the material and are denoted by L0k # ½ak ; bk ; k ¼ 1; 2; . . . ; n. The union of P 0 P 00 electrodes is represented by L0 ¼ Lk , while the other part of the x1 axis by L00 ¼ Lk ; k ¼ 0; 1; 2; . . . ; n. It is assumed that each of the electrodes contains the given charge qi and has the electric potential Vk. And there are no electrical field and mechanical stress at infinity. 4.1. Functions of the electric field In this case, the electric boundary conditions Eq. (16) reads:
x01 ðtÞ þ x01 ðtÞ ¼ 0; x02 ðtÞ þ x02 ðtÞ ¼ 0 on L0 ;
ð26Þ
and
x01 ðtÞ þ x01 ðtÞ ¼ x02 ðtÞ þ x02 ðtÞ; h
1 x01 ðtÞ x01 ðtÞ
i
h i ¼ 2 x02 ðtÞ x02 ðtÞ ;
on L00 ;
ð27Þ
where 1 and 2 are the dielectric constants of the air and material, respectively; x1(z) and x2(z) stand for the electric potentials in S+ and S, respectively; and moreover x1(1) = x2(1) = 0.
x2
air ( ε 1) L"0 a1
L"1
L’1 b1
L’2 a2
L"2 b2
L’3 a3
b3
L"3 ......
L"n−1
L’n an
electrostrictive material (ε 2)
Fig. 2. Collinear electrodes on the surface of half-plane.
L"n bn
x1
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Introduce two auxiliary functions H1 and H2 as follows:
(
H1 ðzÞ ¼ and
( H2 ðzÞ ¼
x01 ðzÞ x02 ðzÞ in Sþ x02 ðzÞ x01 ðzÞ in S
1 x01 ðzÞ þ 2 x02 ðzÞ 2 x02 ðzÞ þ 1 x01 ðzÞ
ð28Þ
in Sþ
ð29Þ
in S
It is found that these two functions H1 and H2 are analytic in S+ and S, respectively. Moreover, they decrease to zero at infinity. And from the above two equations, we have:
Hþ1 ðtÞ H1 ðtÞ ¼ 0;
on L00 ;
Hþ2 ðtÞ H2 ðtÞ ¼ 0;
ð30Þ
00
where t takes the value of z on L . And the superscript ‘‘” denotes the variable approaches to x1 axis from the region x2 < 0, 00 ‘‘+” from x2 > 0. Eq. (30) indicates that H1 and H2 are continuous from S+ to S on the boundary L . On the other hand, from Eqs. (28) and (29), we have:
x01 ðzÞ ¼ ð1 þ 2 Þ1 ½H2 ðzÞ þ 2 H1 ðzÞ; x01 ðzÞ ¼ ð1 þ 2 Þ1 ½H2 ðzÞ 2 H1 ðzÞ; x02 ðzÞ ¼ ð1 þ 2 Þ1 ½H2 ðzÞ þ 1 H1 ðzÞ; x02 ðzÞ ¼ ð1 þ 2 Þ1 ½H2 ðzÞ 1 H1 ðzÞ;
ð31Þ
Substituting Eq. (31) into Eq. (26), we have:
Hþ2 ðtÞ þ 2 Hþ1 ðtÞ þ H2 ðtÞ 2 H1 ðtÞ ¼ 0;
on L0 :
H2 ðtÞ þ 1 H1 ðtÞ þ Hþ2 ðtÞ 1 Hþ1 ðtÞ ¼ 0;
ð32Þ
Using Eq. (32), we obtain:
Hþ1 ðtÞ H1 ðtÞ ¼ 0; Hþ2 ðtÞ þ H2 ðtÞ ¼ 0;
on L0 :
ð33Þ
According to Liouville theorem, we get:
H1 ðzÞ ¼ 0;
ð34Þ
which is obtained by using the first equation in Eqs. (30) and (32), and the condition H1(1) = 0. On the other hand, from Eqs. (24) and (29), it is obvious that H2(z) = 1/z + O(1/z2)jz?1. The function H2(z) can be solved by the second equation in Eq. (33). The result is
H2 ðzÞ ¼ PðzÞXðzÞ;
ð35Þ
where
XðzÞ ¼ PðzÞ ¼
n Y
1
1
ðz ak Þ2 ðz bk Þ2 ;
k¼1 n X
ð36Þ
c k zk ;
ð37Þ
k¼1
where ck are the complex constants to be found. Finally, from Eq. (31), the electric potential in the whole region reads:
x01 ðzÞ ¼ x02 ðzÞ ¼ ð1 þ 2 Þ1
n1 X k¼1
c k zk
n Y
1
1
ðz ak Þ2 ðz bk Þ2 :
ð38Þ
k¼1
In the above equation, cn = 0 is considered due to H2(1) = 0, and c0,c1,c2, . . ., cn1 can be determined by the electric charge qk on the electrodes, as shown in Appendix A. 4.2. Functions of the stress field In this case, Eq. (23) can be reduced to:
i
Z
þ j rM12 þ irM22 dt ¼ /ðtÞ þ t/0 ðtÞ þ wðtÞ þ x2 ðtÞx02 ðtÞ on L0 þ L00 : 2
ð39Þ
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Since the dielectric constant of the air is much smaller than that of the medium, i.e., 1 2, we can let the Maxwell stress
rMij 0 on the surface of the material, that is, Eq. (39) reads: /ðtÞ þ t/0 ðtÞ þ wðtÞ þ
j 2
x2 ðtÞx02 ðtÞ ¼ 0 on L0 þ L00 ;
ð40Þ
where functions /(z),w(z) are analytic in S. In the following, the analytic extension of /(z) are made according to the following definition:
z 2 Sþ ;
/ðzÞ ¼ z/0 ðzÞ wðzÞ f ðzÞ
ð41Þ
where
f ðzÞ ¼
j 2
x2 ðzÞx02 ðzÞ:
ð42Þ
Therefore, /(z) is analytic function in the whole plane. From Eq. (41), we have:
wðzÞ ¼ /ðzÞ z/0 ðzÞ f ðzÞ z 2 Sþ :
ð43Þ
Thus, the pseudo stresses in Eq. (20) can be expressed only by /(z) with the help of Eq. (43), as
h
i
re 22 þ re 11 ¼ jx02 ðzÞx02 ðzÞ þ 2 /0 ðzÞ þ /0 ðzÞ ;
re 22 re 11 þ 2i re 12 ¼ jx002 ðzÞx2 ðzÞ jx02 ðzÞx02 ðzÞ jx2 ðzÞx002 ðzÞ þ 2 ðz zÞ/00 ðzÞ /0 ðzÞ /0 ðzÞ ;
ð44Þ
where the unknown function /(z) can be obtained from solving the standard Hilbert problem, which is given in Appendix B. Moreover, it is known that from Appendix B that /(z) can be expressed by x02 ðzÞ. Substituting the solution of /(z) into Eq. (44), we have:
h
i
re 22 þ re 11 ¼ jx02 ðzÞx02 ðzÞ jV 1 x002 ðzÞ þ x002 ðzÞ ; h
i
re 22 re 11 þ 2i re 12 ¼ jx002 ðzÞ x2 ðzÞ þ V 1 jx02 ðzÞx02 ðzÞ þ j½V 1 x2 ðzÞx002 ðzÞ þ jV 1 ðz zÞx000 ðzÞ;
ð45Þ
where it is assumed that all the electrodes have the same voltage V1. It is shown from Eq. (45) that the stress fields can be obtained once the electric potential x2(z) is solved.
5. Special cases: one or two electrodes 5.1. Single electrode on the surface As shown in Fig. 3, we consider a special case for a single electrode with the length 2a and charge q0, situated at [ a, + a] on the surface of the electrostrictive medium. In this case, the function H2 can be solved with Eq. (35), and the result is
c0 H2 ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; z2 a2
ð46Þ
where c0 may be given as
c0 ¼
q0
p
by using Eq. (A-3) in Appendix A. Therefore, by considering Eq. (38), the electric potential in the medium is
K
e ; x02 ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
ð47Þ
z a
where Ke = q0/[p(1 + 2)]. From Eq. (47), we have
"
x2 ðzÞ ¼ K e ln
z þ a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi# z2 1 þ V 1; a2
where V1 is the electric potential on electrode. With x2(z), the stress in the medium can solved by using Eq. (45).
ð48Þ
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x2
Electrode
−a
x1
a
Electrostrictive medium
Fig. 3. Single electrode with the length 2a and charge q0 on the surface.
5.2. Double collinear electrodes For the case of two collinear electrodes, as shown in Fig. 4, where it is assumed that each electrode has the same charge q0. From Eq. (35), we have:
c1 z þ c0 H2 ðzÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 z ða þ bÞ2 z2 ðb aÞ2
ð49Þ
Considering the geometric symmetry and the nature of the function H2(z), the constant c1,c0 may be given as
c1 ¼
2q0
p
; c0 ¼ 0;
ð50Þ
by using Eq. (A-3) in Appendix A. Therefore,the electric potential in the medium can be given by Eq. (38) as
K z
x02 ðzÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; z2 ða þ bÞ2
ð51Þ
z2 ðb aÞ2
where Ke = 2q0/[p(1 + 2)].
x2 Electrode −(b+a)
−(b−a)
Electrode b−a
b+a
x1
Electrostrictive medium
Fig. 4. Double electrodes with the length 2a and charge q0 on the surface.
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From Eq. (51), we have:
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 z2 ða þ bÞ2 þ z2 ðb aÞ2 5 þ V 1; pffiffiffiffiffiffi x2 ðzÞ ¼ K e ln 4 2 ab
ð52Þ
where V1 is the electric potential on the left electrode. The V2 on the right electrode can be calculated out by Eq. (A-6). Since x2(z) is odd function, we get V2 = V1. With x2(z) in Eq. (52), one also gets the results of stress fields by using Eq. (45).
6. Numerical examples In this section, we give some numerical results of electric fields and stresses near the electrodes. Since all the fields are singular at the tip of electrodes, only the field variables around the electrodes are shown in Figs. 5–10, where we take V1 = 100Ke, and the electric and stress fields are normalized by the constants j,Ke and a. Given in Fig. 5(a) is the electric fields induced by the single electrode along x1 in depth x2 = 0.01a and x2 = a, respectively. In Fig. 5(a), the curves 1, 2 and 3, 4 are electric fields distributing along x1, while in Fig. 5(b) they stand for the electric fields distributing in depth at x1 = a and x1 = a. It is shown that the electric field E1 is dissymmetry about axis x2, and E2 is symmetry. The electric fields decrease with the distance from the electrode and have large value near the points ± a, which is also shown in Eq. (47) with singularity r1/2 at the points.
Fig. 5. Electric fields of single electrode, where, for Fig. (a), 1, 2 and 3, 4 are electric fields distributing in depth x2 = 0.01a and x2 = a along x1, while in Fig. (b), 1, 2 and 3, 4 are electric fields distributing in depth at x1 = a and x1 = a.
Fig. 6. Stress fields of single electrode, where Figs. (a), (b), (c) and (d) show the stress in depth of x2 = 0.05a, x2 = 0.5a, x2 = a and x2 = 5a, respectively.
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Fig. 7. Electric fields of double electrodes, where for Figs. (a), (b) and (c), 1, 2 and 3, 4 are the electric fields distribute along x1 in depth of x2 = 0.01a and x2 = 0.01a with the geometric distributions of the electrodes b = 1.5a, b = a and b = 6a, while in Fig. (d), 1, 2 and 3, 4 are the electric fields distributing along x2 at x1 = b + a and x1 = b a with b = 2a.
Fig. 8. Stress distributions along x1 in different depth with b = 2a for double electrodes, where Figs. (a), (b), (c) and (d) show the stress fields distributing in depth of x2 = 0.05a,x2 = 0.5a, x2 = a and x2 = 5a, respectively.
Shown in Fig. 6 are the stress fields induced by the single electrode along x1 in depth of x2 = 0.05a,x2 = 0.5a,x2 = a and x2 = 5a, respectively. It is revealed that stresses r22 and r11 are symmetry about axis x2, while r12 are dissymmetry about axis x2. It can also be found that the stresses are large near the points ± a, which are induced by the singular electric field. The stresses degenerate along with the depth. When depth increases to x2 = a, the stress field is much smaller than that of x2 = 0.05. In general, the electrostrictive constants are negative, which shows that the normalized factors of stress are negative too. Therefore, the stress r22 approaching to the electrode is compressive, that is the electrode would be attracted to the material. Furthermore, The stress r11 under the electrode is tensile. For the materials BaTiO3 having the properties of high brittleness
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Fig. 9. Stress distributions along x1 in depth of x2 = 0.05a with different distance between two electrodes, b = 1.5a,b = a,b = 4a and b = 6a in Figs. (a), (b), (c) and (d), respectively.
Fig. 10. Stress fields distribution along x1 in depth of x2 = a with different distance between two electrodes, b = 1.5a,b = a,b = 4a and b = 6a in Figs. (a), (b), (c) and (d), respectively.
or low strength, the electrostrictive constants a1 amd a2 of such materials would be around 108 N/V2. When the electric potential of electrode is approximately 10 KV, and the electrode has the length of 103 m in magnitude, the stress r11 at about 0.01a ahead the tip of the electrode would be larger than 100 MPa in magnitude. This means that the failure would occur on the surface of electrode near the ends for the tensile stress r11. Given in Fig. 7(a)–(c) are the electric fields induced by the double electrodes along x1 in depth x2 = 0.01a and x2 = 0.01a with the geometric distributions of the electrodes b = 1.5a, b = a and b = 6a, respectively. And in Fig. 7(d), the electric fields distributing along x2 at x1 = b + a and x1 = b a with b = 2a are shown. It is shown that the electric field E1 is dissymmetry about axis x2, and E2 is symmetry, which are similar to the case of a single electrode. The electric fields decrease with the distance and have large value near the electrode ends. The interaction of the electrodes decreases as their distance
Q. Jiang et al. / International Journal of Engineering Science 48 (2010) 2066–2080
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2(b a) increase, as shown in Fig. 7(a)–(c). Furthermore, in Fig. 7(c) (b = 6a), the interaction of the electrodes can be neglected so that we can only consider the total electric field, just giving the sum of ones induced by two single electrodes situated on [(b + a),(b a)] and [b a,b + a]. In other words, the electric field near the electrodes, when they are far from each other, can be calculated as the single electrode problem. In Fig. 8(a)–(d), it is shown that the stress distributions under the double electrodes along x1 in different depth. Similar to the single electrode, the stresses r22 and r11 are symmetry about axis x2,r12 are dissymmetry about axis x2. It can be seen from Fig. 8 that the stresses are very large near the electrode ends. Shown in Figs. 9 and 10 are the stress distributions along x1 in depth of x2 = 0.05a and x2 = a with different distance between two electrodes. It is shown that the interaction of the electrodes reduces as the distance between each other increases. For example, when b = 4a, i.e., the distance of the electrodes is 2a, the stress fields are nearly the same as those in the case of a single electrode loading. 7. Conclusions This work has presented the general solution for the electroelastic fields of collinear electrodes on the surface of a halfinfinite electrostrictive material, and closed form solution for two special cases of one and two electrodes. Numerical results are also given to discuss the distribution of the electroelastic fields near the electrodes. Several conclusions can be obtained as follows: (1) The stress fields in an electrostrictive material can be finally expressed by the electrical potential. Thus, once the electrical potential is obtained, one can easily determine the electroelastic fields from Eq. (45) of the present work. (2) It is found that the electic fields have the singularity of r1/2, while the stress fields have the singularity of r1 at the tips of the electrodes. Thus, the elastic singularity is stronger than that in the linear problem of traditional materials. (3) When the electric loading is applied, the stress r11 below the surface of the electrodes is tensile, while r11 ahead of the electrode ends is compressed. Therefore, the failure may occur firstly from the surface of the electrode. (4) When the distance between the two adjacent ends of two different electrodes is much larger than their lengthes, the interactions between them can be neglected, that is, the case of multiple electrodes can be treated as the case of a single electrode.
Acknowledgements This work is supported by National Natural Scientific Foundation of China (No. 10972103, No. 10902055); Post-Doctor Foundation of China and Jiangsu Province ( No. 20070411046, No. 0701018C); Natural Scientific Foundation of Jiangsu Province (No. BK 2006544). Appendix A The constants in Eq. (37) c0, c1, c2, . . ., cn1 can be determined by the electric charge qk on the electrodes. Assuming all the qk are known on the boundary L0k , we have:
Z
bk
ak
D2 ðtÞdt þ
Z
ak
bk
Dþ2 ðtÞdt ¼ qk :
ðA-1Þ
With the help of Eqs. (15) and (31), we get:
Z
bk
ak
H2 ðtÞ Hþ2 ðtÞ dt ¼ 2iqk ;
ðA-2Þ
and the above equation can be written as
Z
bk
ak
H2 ðtÞdt ¼ iqk ;
ðA-3Þ
by considering the second condition of Eq. (33). Since there are n integral equations of Eq. (A-3) on the boundary L0k ; 1 ¼ 1; 2; 3 . . . n, the constants ck can be determined. Therefore, the electric functions x01 ðzÞ and x02 ðzÞ can be given with the relationship of Eq. (31). The complex electric potential x1(z) and x2(z) will be:
x1 ðzÞ ¼
Z
x01 ðzÞdz þ c1 ; x2 ðzÞ ¼
Z
x02 ðzÞdz þ c2 ;
where the constants c1,c2 are real numbers in general.
ðA-4Þ
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If one of the electric potential on the electrodes is given, for example, it is Vk on L0k , we have:
c2 ¼ V k
1 2
Z
x02 ðzÞdz þ
Z
x02 ðzÞdz
ðA-5Þ
; z¼t
in the electrostrictive materials, where the relationship 2V k ¼ x2 ðtÞ þ x2 ðtÞ is used. It is noted that the constants c1 can also be given with the same method. For the problem of this paper, x1(z) = x2(z), x01 ðzÞ ¼ x02 ðzÞ and c1 = c2. The electric potential Vk+1 of next electrode situated on L0kþ1 must be:
V kþ1 ¼ V k
Z
akþ1
E1 dt ¼ V k þ
bk
1 2
Z
akþ1 bk
x02 ðtÞ þ x02 ðtÞ dt;
ðA-6Þ
with the help of Eq. (13). Using the same method, all the other electric potential can be solved. It should be emphasized that the electric potential on the electrodes are not independent. If the potential on one of the electrode is given, the others must be specified. Furthermore, if all the potentials on the electrodes is known, the charges qk are not random, they are dependent on each other. Appendix B The boundary-value problem of /(z) is studied in this section. Firstly, substituting Eq. (43) into Eq. (40), we get:
/þ ðtÞ / ðtÞ ¼ f ðtÞ f ðtÞ;
ðB-1Þ
where z = t and z ¼ t are used on the axis x1. When examining the nature of x02 ðtÞ, we have the relationship:
x02 ðtÞ ¼ x02 ðtÞ; x2 ðtÞ ¼ x2 ðtÞ; on the boundary
L00k ,
ðB-2Þ
and
x02 ðtÞ þ x02 ðtÞ ¼ 0; x2 ðtÞ þ x2 ðtÞ ¼ 2V k ;
ðB-3Þ
on the boundary L0k . Thus, we can get:
/þ ðtÞ / ðtÞ ¼ 0 on L00k :
ðB-4Þ
+
Therefore, /(z) is continuous from S to S on the boundary
/þ ðtÞ / ðtÞ ¼
jh 2
L00k .
For
L0k ,
we have:
i
x2 ðtÞx02 ðtÞ x2 ðtÞx02 ðtÞ ¼ jV k x02 ðtÞ;
ðB-5Þ
where the relationships of Eq. (B-3) and t ¼ t are used. Consequently, the function /(z) is
/ðzÞ ¼
1 2pi
Z
jV k x02 ðtÞ
L0k
dt : tz
ðB-6Þ
From Eq. (38), the functions x2(t) and x02 ðtÞ are analytical in the whole region except the singular points ak, bk(k = 1,2, . . ., n) on the axis x1. Consider a clockwise contour integration Hk(z) on Kk approaching to the boundary L0k very nearly:
Hk ðzÞ ¼
1 2pi
I Kk
jV k x02 ðtÞ
dt ; tz
which is shown in Fig. 11.
Fig. 11. The contour integration on Kk approach to the boundary L0k .
ðB-7Þ
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Using the relationship of the first equation in (B-3), the cauchy integration Hk(z) is
Hk ðzÞ ¼ jV k x02 ðzÞ þ jV k
n nh i o X ðbm Þ mÞ Gða m ðzÞ þ Gm ðzÞ ð1 dkm Þ ;
ðB-8Þ
m¼1 ðam Þ mÞ 0 where dkm is Kronecker delta, Gm ðzÞ; Gðb m ðzÞ are the cauchy principal value of the function x2 ðtÞ at the singular points. In 0 þ 0 0 addition, it is obviously that ½x2 ðtÞ ¼ ½x2 ðtÞ on L . We have:
/ðzÞ ¼
n 1X Hk ðzÞ: 2 k¼1
ðB-9Þ
If Vk(k = 1,2, . . ., n) have the value on L0k , the sum of the functions /(z) can be given as
/ðzÞ ¼
jV 1 2
x02 ðtÞ:
ðB-10Þ
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Suo, Fracture mechanics for the design of ceramic multilayer actuators, Journal of the Mechanics and Physics of Solids 44 (1) (1996) 23–48. [7] C. Hausler, H. Balke, Full form of the near tip field for the interface crack between a piezoelectric material and a thin electrode, Functionally Graded Materials VIII (492) (2005) 261–266. [8] C. Hausler, C.F. Gao, H. Balke, Collinear and periodic electrode-ceramic interfacial cracks in piezoelectric bimaterials, ASME, Journal of Applied Mechanics-Transactions 71 (4) (2004) 486–492. [9] L.H. He, R.Q. Ye, Concentration of electric field near electrodes on piezoelectric layer, Theoretical and Applied Fracture Mechanics 33 (2) (2000) 101– 106. [10] R.J. Knops, Two-dimensional electrostriction, Quart Journal of Mechanics and Applied Mathematics 16 (3) (1963) 377–388. [11] Z.B. Kuang, Nonlinear Mechanics of Continuous Media, Shanghai Jiaotong University Press, Shanghai, 2002. in Chinese. [12] Z.B. Kuang, Z.D. Zhou, K.L. Zhou, Electroelastic analysis of a piezoelectric half-plane with finite surface electrodes, International Journal of Engineering Science 42 (15–16) (2004) 1603–1619. [13] L.D. Landau, E.M. Lifshitz, Electodynamics of Continous Media, Pergamon Press, Oxford, 1960. [14] X.F. Li, X.Y. Duan, Electroelastic analysis of a piezoelectric layer with electrodes, International Journal of Fracture 111 (4) (2001) L73–L78. [15] X.F. Li, K.Y. Lee, Elastic response of a piezoelectric ceramic plate with two penny-shaped electrodes at the opposite surfaces under an electric load, Journal of Intelligent Material Systems and Structures 18 (4) (2007) 743–751. [16] S.L.D.E. Lucato, D.C. Lupascu, M. Kamlah, et al, Constraint-induced crack initiation at electrode edges in piezoelectric ceramics, Acta Materialia 49 (14) (2001) 2751–2759. [17] P. Malits, Dual equations method for the periodic antiplane problem of a piezoelectric layer with electrodes, Zeitschrift Fur Angewandte Mathematik Und Physik 58 (3) (2007) 475–487. [18] N.I. Muskhelishvili, Some Basic Problem in the Mathematical Theory of Elasticity, Nordhoff, Netherlads, 1954. [19] F. Narita, Y. Shindo, M. Mikami, Electroelastic field concentrations and polarization switching induced by circular electrode at the interface of piezoelectric disk composites, European Journal of Mechanics A – Solids 26 (3) (2007) 394–404. [20] F. Narita, M. Yoshida, Y. Shindo, Electroelastic effect induced by electrode embedded at the interface of two piezoelectric half-planes, Mechanics of Materials 36 (10) (2004) 999–1006. [21] Pao, Y.H., 1978. Electromagnetic force in deformable continua. In: Mechanics Today, Printed in Great Britain by Pitman. [22] C.Q. Ru, Electrode-ceramic interfacial cracks in piezoelectric multilayer materials, ASME, Journal of Applied Mechanics-Transactions 67 (2) (2000) 255– 261. [23] C.Q. Ru, Exact solution for finite electrode layers embedded at the interface of two piezoelectric half-planes, Journal of the Mechanics and Physics of Solids 48 (4) (2000) 693–708. [24] Y. Shindo, F. Narita, H. Sosa, Electroelastic analysis of piezoelectric ceramics with surface electrodes, International Journal of Engineering Science 36 (9) (1998) 1001–1009. [25] Y. Shindo, M. Yoshida, F. Narita, Electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators: experiment and finite element simulation, Journal of the Mechanics and Physics of Solids 52 (5) (2004) 1109–1124. [26] T.E. Smith, W.E. Warren, Some problems in two-dimensional electrostriction, Journal of Mathematical Physics 45 (1) (1966) 45–51. [27] T.E. Smith, W.E. Warren, Some problems in two-dimensional electrostriction (corrigenda), Journal of Mathematical Physics 47 (1) (1968) 109–110. [28] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. [29] B.L. Wang, Circular surface electrode on a piezoelectric layer, Journal of Applied Physics 95 (8) (2004) 4267–4274. [30] B.L. Wang, Y.G. Sun, Intensity factors for some common piezoelectric fracture mechanics specimens with conducting cracks or electrodes, International Journal of Engineering Science 42 (11–12) (2004) 1129–1153. [31] S.R. Winzer, N. Shankar, A.P. Ritter, Designing cofired multilayer electrostrictive actuators for reliability, Journal of the American Ceramic Society 72 (12) (1989) 2246–2257. [32] F.Q. Yang, Electromechanical interaction of linear piezoelectric materials with a surface electrode, Journal of Materials Science 39 (8) (2004) 2811– 2820. [33] W. Yang, Z. Suo, Cracking in ceramic actuators caused by electrostriction, Journal of the Mechanics and Physics of Solids 42 (4) (1994) 649– 663.
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[34] Z.P. Yang, L.L. Yang, X.L. Chao, et al, Electrical characteristics of central driving type piezoelectric transformers with different electrode distributing, Sensors and Actuators A – Physical 136 (1) (2007) 341–346. [35] R.Q. Ye, L.H. He, Electric field and stresses concentrations at the edge of parallel electrodes in piezoelectric ceramics, International Journal of Solids and Structures 38 (38–39) (2001) 6941–6951. [36] H.Y. Zhang, Y.P. Shen, Three-dimensional analysis for rectangular 1–3 piezoelectric fiber-reinforced composite laminates with the interdigitated electrodes under electromechanical loadings, Composites Part B – Engineering 37 (7–8) (2006) 603–611.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Electroelastic stress in an electrostrictive material with charged surface electrodes Quan Jiang a,b,*, Cun-Fa Gao a, Zhen-Bang Kuang c a
College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China College of Civil Engineering, Nantong University, Nantong 226007, China c Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200240, China b
a r t i c l e
i n f o
Article history: Received 4 July 2008 Received in revised form 8 August 2009 Accepted 16 April 2010 Available online 8 May 2010 Communicated by M. Kachanov Keywords: Electrostictive materials Surface electrodes Field singularities
a b s t r a c t Field singularities of collinear electrodes on the surface of a half-infinite electrostrictive solid are studied in terms of the complex variable method. Firstly, the general solutions for the potential functions of electric fields and stresses are derived for the case of collinear electrodes based on analytical continuation method. Then, explicit results are given for the cases of one and two electrodes, respectively. Finally, numerical calculations are made to discuss the effects of applied electric loading on the field singularities near the electrodes. It is found that even at the low level of applied electric loading, considerable high stresses will be induced around the tip of electrodes. Moreover, for the cases of multiple electrodes, the distribution of electric fields and stresses is more complicated than the case of a single electrode. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction As a necessary carrier of electric loading, electrodes are usually used in smart structures and devices, which are in general made of piezoelectric or electrostrictive materials. In these structures and devices, electrodes can be placed in three cases: internal electrodes (which are inserted into a homogeneous material), interface electrodes (which are placed between two materials), and surface electrodes (which are located on the surface of the material). For all the cases, the high field variables may be induced around the electrode edges due to the different mechanical and electrical behavior of electrodes and matrix materials. The nonuniform local field results in crack initiation and crack growth and finally leads to the failure of the device. Thus, with increasingly wide applications of smart structures and devices made of piezoelectric/electrostrictive materials, the problems of local fields near electrodes have received much attention in the recent decade. In fact, some important results have been presented for the above three cases of electrode distributing. For the case of internal electrodes, Ye and He [35] studied the electric field and stresses concentrations at the edge of parallel electrodes in piezoelectric ceramics; Lucato et al. [16] investigated the constraint-induced crack initiation at electrode edges in piezoelectric ceramics; Chen and Chue [3] made the fracture mechanics analysis of a composite piezoelectric strip with an internal semi-infinite electrode; Wang and Sun [30] derived the intensity factors for some common piezoelectric fracture mechanics specimens with conducting cracks or electrodes; Gao et al. [4] addressed the generalized two-dimensional problem of a conductive crack and electrode in piezoelectric materials using dielectric breakdown model; Yang et al. [34] gave * Corresponding author at: College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China. Tel.: +86 25 84895115; fax: +86 25 84891422. E-mail address: [email protected] (Q. Jiang). 0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2010.04.008
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the electrical characteristics of central driving type piezoelectric transformers with different electrode distributing. For the cases of interfacial electrodes, Ru[22,23] studied the fracture problems of a soft electrode in piezoelectric multilayer materials or at the interface of two piezoelectric half-planes by means of complex potentials; Shindo et al. [25] and Narita et al. [20] also analyzed the electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators or between two piezoelectric half-planes by using experiment and finite element simulation, respectively; Hausler et al. [8], Hausler and Balke [7] studied the generalized two-dimensional problems of collinear and periodic electrode-ceramic interfacial cracks in piezoelectric bimaterials and gave the structure of singular fields at the tip of electrode; Beom et al. [1] made the analysis of a thin electrode layer between two dissimilar piezoelectric solids, and Narita et al. [19] analyzed the electroelastic field concentrations and polarization switching induced by circular electrode at the interface of piezoelectric disk composites. On the other hand, a lot of works for the cases of surface electrodes are also made. For example, Shindo et al. [24] conducted the electroelastic analysis of piezoelectric ceramics with surface electrodes; He and Ye [9], Li and Duan [14], and Wang [29] studied the concentration of electroelastic field near electrodes on a piezoelectric layer, respectively. Yang [32] addressed the electromechanical interaction of linear piezoelectric materials with a surface electrode; Kuang et al. [12] derived the analytical solutions for the electroelastic field of a piezoelectric half-plane with finite surface electrodes; Zhang and Shen [36] made the three-dimensional analysis for rectangular 1–3 piezoelectric fiber-reinforced composite laminates with the interdigitated electrodes under electromechanical loadings; Li and Lee [15] studied the elastic response of a piezoelectric ceramic plate with two penny-shaped electrodes at the opposite surfaces under an electric load, and Malits [17] investigated the periodic anti-plane problem of a piezoelectric layer with electrodes by using the dual equations method. However, it should be noted that all the works cited above are for the cases of piezoelectric materials. For electrostrictive materials with electrodes, only a few of works can be found for the study of electroelastic field concentrations near the electrodes due to the difficulty involved in the constitutive equations of electrostrictive materials. An earlier work on studying electroelastic fields in a designing cofired multilayer electrostrictive actuator with electrodes was given by Winzer et al. [31]. Then, Gong [5] derived the stresses near the end of an internal electrode in multilayer electrostrictive ceramic actuators, and Yang and Suo [33], Hao et al. [6] made the numerical calculation on the electroelastic fields for the design of ectrostrictive ceramic multilayer actuators, respectively. Recently, Beom et al. [2] presented some theoretical results for an internal electrode in an electrostrictive material under the small scale saturation condition with the use of complex function theory. To the authors’ knowledge, however, no work can be found for the cases of collinear charged electrodes on the surface of an electrostrictive material, although the similar problems have been well studied for the cases of piezoelectric materials [24,25,32]. It is therefore the purpose of this work to study the field singularities of collinear electrodes on the surface of a half-infinite electrostrictive solid in terms of the complex variable method. Below is the plan of the present work: following the introduction, we will outline the basic equations based on the work of Stratton [28] and Landau and Lifshitz [13] in Section 2], and then give the solutions of Knops [10] and Smith and Warren [26,27] for the case of two-dimensional deformation in Section 3. Derived in Section 4 is the general solution for the potential functions of electric fields and stresses for the case of collinear electrodes, and explicit results are given for special cases of one and two electrodes in Section 5. Numerical calculations are made in Section 6 to discuss the effects of applied electric loading on the field singularities near the electrodes, and this work is finally concluded in Section 7. 2. Basic equations of electrostrictive materials The constitutive equations of electrostriction is nonlinear since they contain the square terms of electric field. Stratton [28], Landau and Lifshitz [13] obtained the electrostrictive stress induced by electric field in electrostrictive media. Pao [21] and Kuang [11] also discussed the electrostrictive force induced by electric field in other ways. These results will be outlined as follows. The constitutive equations of electrostrictive materials for the isothermal and isotropic case are
1 a1 Ei Ej þ a2 Ek Ek dij ; 2 Di ¼ dij þ a1 eij þ a2 ekk dij Ej ;
rij ¼ 2leij þ kekk dij
ð1Þ
where rij,eij,Di and Ej are stress, strain, electric displacement and electric field intensity, respectively. a1, a2 are two independent electrostrictive coefficients in isotropic materials. is the permittivity at the state without strain. dij is Kronecker delta. k and l are Lame constants which can be expressed by Young’s modulus E and Poisson ratio m in the form k = E m/ [(1 + m)(1 2m)],l = E/[2(1 + m)]. The equilibrium equation of electro-elastic material is
rij;j þ fie ¼ 0;
ð2Þ
where fie is the body force induced by the electric fields, and it reads:
fie ¼ rM ij;j ¼
1 2
Ei Ej Ek Ek dij
; ;j
ð3Þ
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where
Q. Jiang et al. / International Journal of Engineering Science 48 (2010) 2066–2080
rMij is called Maxwell stress. Substituting Eq. (3) into equilibrium Eq. (2), we get:
re ij;j ¼ 0; re ij ¼ rij þ rMij :
ð4Þ
e ij the pseudo total stress. With the help of Eqs. (1) and (3), it is easily got: We call r
re ij ¼ 2leij þ kekk dij þ
a1 þ a2 Ei Ej Ek Ek dij : 2 2
ð5Þ
If there is no free charge in the medium, the electric displacement satisfies:
Di;i ¼ 0:
ð6Þ
On the interface of two medium (Fig. 1), in general case (there is no surface charge), the electric interface conditions are:
uþ ¼ u ; Dþn ¼ Dn ;
ð7Þ
where u and Dn are the electric potential and the normal electric displacement on the interface, and the superscripts (+,) refer the variables in the material I or II. The traction induced by the electric fields on the interface of dielectric media I and II can be written as [28,13]:
X ei ¼ rMþ rM nj ; ij ij
ð8Þ
where nj is the outward normal vector on the interface of the material I. The mechanical interface conditions on the interface are
uþi ¼ ui ; rþij rij nj ¼ X mi þ X ei ¼ X mi rMþ rM nj ; ij ij
ð9Þ ð10Þ
m where u i is the displacements on the interface. X i is the given mechanical traction on interface, and X i is the total interface traction. Eq. (10) can also be changed to:
ei ¼ r e þij nj ¼ r e ij nj ; X
ð11Þ
e i is named as pseudo total interface traction. where X If the material II degenerated to enviroment, it should be noted that the boundary conditions Eqs. (7), (9) and (11) are still valid, but in this case, the functions with superscript () will be some given values. 3. Complex variable solution for 2D problems It is obvious that the electrostrictive problem is nonlinear from Eq. (1). So the exact solution of the problem is hard to be given. In general, since the dielectric constant satisfies the condition akeijdij for small strain, we can uncouple the electric problem from the whole problem, that is, the electric field may be calculated firstly based on the theory of electrostatics. And then the stress field can be given with the help of the known electric field, which is similar to solving the problems of thermal stress.
x2
Dielectric material II
B
n ds Dielectric material I
Di
X
m
Di ij ij
A 0
x1 Fig. 1. The boundary condition of mechanical and electrical fields.
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3.1. Electric field in the medium The complex electric potential x(z) is in the form:
xðzÞ ¼ uðx1 ; x2 Þ þ iAðx1 ; x2 Þ;
ð12Þ
where z = x1 + ix2 in cartesian coordinate. x(z) is analytic function of z in the whole plane, and the electric potential u(x1,x2) has the relationship :
Ei ¼
ouðx1 ; x2 Þ : oxi
ð13Þ
Noting that A(x1,x2) is the supplementary function of u(x1,x2), which is defined as
ou oA ou oA ¼ ; ¼ : ox1 ox1 ox2 ox2
ð14Þ
So we have:
E1 þ iE2 ¼ x0 ðzÞ;
ð15Þ 0
where the symbol (—) denote the conjugate operation to the function, and x (z) = dx(z)/d z. Let z = t on the boundary, the electric boundary can be written as
xðtÞ þ xðtÞ ¼ 2u on L0 ; x0 ðtÞðnx þ iny Þ ¼ 2iDn on L00 ; 0
ð16Þ
00
where L and L stand for the boundary with the given electric potential u and given electric displacement Dn , respectively. Once these conditions are given, the complex potential x(z) can be solved by using complex variable method. 3.2. Stress field in the medium Introducing the pseudo total stress function U as
re 11 ¼
o2 U ; ox22
re 22 ¼
o2 U ; ox21
re 12 ¼
o2 U : ox1 ox2
ð17Þ
Then Eq. (4) is automatically satisfied. The compatible relation can be expressed by
r4 U ¼ 16
i o4 U o4 h ¼ 4j 2 2 x0 ðzÞx0 ðzÞ ; 2 2 oz oz oz oz
ð18Þ
where j = (1 2m)(a1 + 2a2)/[2(1 m)]. Therefore, the function U can be expressed by two analytic functions g(z) and /(z) in the following form:
U¼
i 1 1h gðzÞ þ gðzÞ þ z/ðzÞ þ z /ðzÞ þ j xðzÞ xðzÞ: 2 4
ð19Þ
The pseudo total stresses are
h
i
re 22 þ re 11 ¼ jx0 ðzÞx0 ðzÞ þ 2 /0 ðzÞ þ /0 ðzÞ ;
ð20Þ
re 22 re 11 þ 2i re 12 ¼ jx00 ðzÞxðzÞ þ 2½z/00 ðzÞ þ w0 ðzÞ; 0
where w(z) = g (z). The displacement for plane strain is
2 l ðu1 þ iu2 Þ ¼ ð3 4 mÞ/ðzÞ
j 2
a xðzÞx0 ðzÞ z/0 ðzÞ wðzÞ þ XðzÞ; 2
ð21Þ
where a = a1 . In the plane stress case, the constants m,E,a1 and a2 should change to be m/(1 + m), [E(1 + 2m)/(1 + m)2], a1 and (1 2m)a2/ (1 m), respectively. Moreover, using Eqs. (11) and (17), the traction can be expressed as
Z e 1 þ iP e2 ¼ i i P A
B
B e 1 þ iX e 2 ds ¼ 2 oU : X oz A
ð22Þ
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Substituting Eq. (19) into Eq. (22), we get:
iB h e1 þ iP e 2 ¼ /ðtÞ þ t/0 ðtÞ þ wðtÞ þ j xðtÞx0 ðtÞ ; i P 2 A
ð23Þ
along the given boundary. For the infinite plate, the complex potentials x(z),/(z) andw(z) take the form:
xðzÞ ¼
Q ln z þ C3 z þ xðzÞ; 2p
ð24Þ
and [18]
e1 þ iP e2 P ln z þ C1 z þ /ðzÞ; 8ð1 mÞ e 1 iP e2 ð3 4mÞ P wðzÞ ¼ ln z þ C2 z þ wðzÞ; 8ð1 mÞ
/ðzÞ ¼
ð25Þ
where x(z),/(z) and w(z) are complex analytic function in the discussing region, respectively; and C3, C1 and C2 are related to the electric fields and the tractions at infinity, respectively. 4. General solution for collinear electrodes As shown in Fig. 2, an electrostrictive material is in the lower half-plane S while the upper half-plane S+ is filled with air. The soft electrodes are coated along the surface of the material and are denoted by L0k # ½ak ; bk ; k ¼ 1; 2; . . . ; n. The union of P 0 P 00 electrodes is represented by L0 ¼ Lk , while the other part of the x1 axis by L00 ¼ Lk ; k ¼ 0; 1; 2; . . . ; n. It is assumed that each of the electrodes contains the given charge qi and has the electric potential Vk. And there are no electrical field and mechanical stress at infinity. 4.1. Functions of the electric field In this case, the electric boundary conditions Eq. (16) reads:
x01 ðtÞ þ x01 ðtÞ ¼ 0; x02 ðtÞ þ x02 ðtÞ ¼ 0 on L0 ;
ð26Þ
and
x01 ðtÞ þ x01 ðtÞ ¼ x02 ðtÞ þ x02 ðtÞ; h
1 x01 ðtÞ x01 ðtÞ
i
h i ¼ 2 x02 ðtÞ x02 ðtÞ ;
on L00 ;
ð27Þ
where 1 and 2 are the dielectric constants of the air and material, respectively; x1(z) and x2(z) stand for the electric potentials in S+ and S, respectively; and moreover x1(1) = x2(1) = 0.
x2
air ( ε 1) L"0 a1
L"1
L’1 b1
L’2 a2
L"2 b2
L’3 a3
b3
L"3 ......
L"n−1
L’n an
electrostrictive material (ε 2)
Fig. 2. Collinear electrodes on the surface of half-plane.
L"n bn
x1
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2071
Introduce two auxiliary functions H1 and H2 as follows:
(
H1 ðzÞ ¼ and
( H2 ðzÞ ¼
x01 ðzÞ x02 ðzÞ in Sþ x02 ðzÞ x01 ðzÞ in S
1 x01 ðzÞ þ 2 x02 ðzÞ 2 x02 ðzÞ þ 1 x01 ðzÞ
ð28Þ
in Sþ
ð29Þ
in S
It is found that these two functions H1 and H2 are analytic in S+ and S, respectively. Moreover, they decrease to zero at infinity. And from the above two equations, we have:
Hþ1 ðtÞ H1 ðtÞ ¼ 0;
on L00 ;
Hþ2 ðtÞ H2 ðtÞ ¼ 0;
ð30Þ
00
where t takes the value of z on L . And the superscript ‘‘” denotes the variable approaches to x1 axis from the region x2 < 0, 00 ‘‘+” from x2 > 0. Eq. (30) indicates that H1 and H2 are continuous from S+ to S on the boundary L . On the other hand, from Eqs. (28) and (29), we have:
x01 ðzÞ ¼ ð1 þ 2 Þ1 ½H2 ðzÞ þ 2 H1 ðzÞ; x01 ðzÞ ¼ ð1 þ 2 Þ1 ½H2 ðzÞ 2 H1 ðzÞ; x02 ðzÞ ¼ ð1 þ 2 Þ1 ½H2 ðzÞ þ 1 H1 ðzÞ; x02 ðzÞ ¼ ð1 þ 2 Þ1 ½H2 ðzÞ 1 H1 ðzÞ;
ð31Þ
Substituting Eq. (31) into Eq. (26), we have:
Hþ2 ðtÞ þ 2 Hþ1 ðtÞ þ H2 ðtÞ 2 H1 ðtÞ ¼ 0;
on L0 :
H2 ðtÞ þ 1 H1 ðtÞ þ Hþ2 ðtÞ 1 Hþ1 ðtÞ ¼ 0;
ð32Þ
Using Eq. (32), we obtain:
Hþ1 ðtÞ H1 ðtÞ ¼ 0; Hþ2 ðtÞ þ H2 ðtÞ ¼ 0;
on L0 :
ð33Þ
According to Liouville theorem, we get:
H1 ðzÞ ¼ 0;
ð34Þ
which is obtained by using the first equation in Eqs. (30) and (32), and the condition H1(1) = 0. On the other hand, from Eqs. (24) and (29), it is obvious that H2(z) = 1/z + O(1/z2)jz?1. The function H2(z) can be solved by the second equation in Eq. (33). The result is
H2 ðzÞ ¼ PðzÞXðzÞ;
ð35Þ
where
XðzÞ ¼ PðzÞ ¼
n Y
1
1
ðz ak Þ2 ðz bk Þ2 ;
k¼1 n X
ð36Þ
c k zk ;
ð37Þ
k¼1
where ck are the complex constants to be found. Finally, from Eq. (31), the electric potential in the whole region reads:
x01 ðzÞ ¼ x02 ðzÞ ¼ ð1 þ 2 Þ1
n1 X k¼1
c k zk
n Y
1
1
ðz ak Þ2 ðz bk Þ2 :
ð38Þ
k¼1
In the above equation, cn = 0 is considered due to H2(1) = 0, and c0,c1,c2, . . ., cn1 can be determined by the electric charge qk on the electrodes, as shown in Appendix A. 4.2. Functions of the stress field In this case, Eq. (23) can be reduced to:
i
Z
þ j rM12 þ irM22 dt ¼ /ðtÞ þ t/0 ðtÞ þ wðtÞ þ x2 ðtÞx02 ðtÞ on L0 þ L00 : 2
ð39Þ
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Since the dielectric constant of the air is much smaller than that of the medium, i.e., 1 2, we can let the Maxwell stress
rMij 0 on the surface of the material, that is, Eq. (39) reads: /ðtÞ þ t/0 ðtÞ þ wðtÞ þ
j 2
x2 ðtÞx02 ðtÞ ¼ 0 on L0 þ L00 ;
ð40Þ
where functions /(z),w(z) are analytic in S. In the following, the analytic extension of /(z) are made according to the following definition:
z 2 Sþ ;
/ðzÞ ¼ z/0 ðzÞ wðzÞ f ðzÞ
ð41Þ
where
f ðzÞ ¼
j 2
x2 ðzÞx02 ðzÞ:
ð42Þ
Therefore, /(z) is analytic function in the whole plane. From Eq. (41), we have:
wðzÞ ¼ /ðzÞ z/0 ðzÞ f ðzÞ z 2 Sþ :
ð43Þ
Thus, the pseudo stresses in Eq. (20) can be expressed only by /(z) with the help of Eq. (43), as
h
i
re 22 þ re 11 ¼ jx02 ðzÞx02 ðzÞ þ 2 /0 ðzÞ þ /0 ðzÞ ;
re 22 re 11 þ 2i re 12 ¼ jx002 ðzÞx2 ðzÞ jx02 ðzÞx02 ðzÞ jx2 ðzÞx002 ðzÞ þ 2 ðz zÞ/00 ðzÞ /0 ðzÞ /0 ðzÞ ;
ð44Þ
where the unknown function /(z) can be obtained from solving the standard Hilbert problem, which is given in Appendix B. Moreover, it is known that from Appendix B that /(z) can be expressed by x02 ðzÞ. Substituting the solution of /(z) into Eq. (44), we have:
h
i
re 22 þ re 11 ¼ jx02 ðzÞx02 ðzÞ jV 1 x002 ðzÞ þ x002 ðzÞ ; h
i
re 22 re 11 þ 2i re 12 ¼ jx002 ðzÞ x2 ðzÞ þ V 1 jx02 ðzÞx02 ðzÞ þ j½V 1 x2 ðzÞx002 ðzÞ þ jV 1 ðz zÞx000 ðzÞ;
ð45Þ
where it is assumed that all the electrodes have the same voltage V1. It is shown from Eq. (45) that the stress fields can be obtained once the electric potential x2(z) is solved.
5. Special cases: one or two electrodes 5.1. Single electrode on the surface As shown in Fig. 3, we consider a special case for a single electrode with the length 2a and charge q0, situated at [ a, + a] on the surface of the electrostrictive medium. In this case, the function H2 can be solved with Eq. (35), and the result is
c0 H2 ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; z2 a2
ð46Þ
where c0 may be given as
c0 ¼
q0
p
by using Eq. (A-3) in Appendix A. Therefore, by considering Eq. (38), the electric potential in the medium is
K
e ; x02 ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
ð47Þ
z a
where Ke = q0/[p(1 + 2)]. From Eq. (47), we have
"
x2 ðzÞ ¼ K e ln
z þ a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi# z2 1 þ V 1; a2
where V1 is the electric potential on electrode. With x2(z), the stress in the medium can solved by using Eq. (45).
ð48Þ
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x2
Electrode
−a
x1
a
Electrostrictive medium
Fig. 3. Single electrode with the length 2a and charge q0 on the surface.
5.2. Double collinear electrodes For the case of two collinear electrodes, as shown in Fig. 4, where it is assumed that each electrode has the same charge q0. From Eq. (35), we have:
c1 z þ c0 H2 ðzÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 z ða þ bÞ2 z2 ðb aÞ2
ð49Þ
Considering the geometric symmetry and the nature of the function H2(z), the constant c1,c0 may be given as
c1 ¼
2q0
p
; c0 ¼ 0;
ð50Þ
by using Eq. (A-3) in Appendix A. Therefore,the electric potential in the medium can be given by Eq. (38) as
K z
x02 ðzÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; z2 ða þ bÞ2
ð51Þ
z2 ðb aÞ2
where Ke = 2q0/[p(1 + 2)].
x2 Electrode −(b+a)
−(b−a)
Electrode b−a
b+a
x1
Electrostrictive medium
Fig. 4. Double electrodes with the length 2a and charge q0 on the surface.
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From Eq. (51), we have:
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 z2 ða þ bÞ2 þ z2 ðb aÞ2 5 þ V 1; pffiffiffiffiffiffi x2 ðzÞ ¼ K e ln 4 2 ab
ð52Þ
where V1 is the electric potential on the left electrode. The V2 on the right electrode can be calculated out by Eq. (A-6). Since x2(z) is odd function, we get V2 = V1. With x2(z) in Eq. (52), one also gets the results of stress fields by using Eq. (45).
6. Numerical examples In this section, we give some numerical results of electric fields and stresses near the electrodes. Since all the fields are singular at the tip of electrodes, only the field variables around the electrodes are shown in Figs. 5–10, where we take V1 = 100Ke, and the electric and stress fields are normalized by the constants j,Ke and a. Given in Fig. 5(a) is the electric fields induced by the single electrode along x1 in depth x2 = 0.01a and x2 = a, respectively. In Fig. 5(a), the curves 1, 2 and 3, 4 are electric fields distributing along x1, while in Fig. 5(b) they stand for the electric fields distributing in depth at x1 = a and x1 = a. It is shown that the electric field E1 is dissymmetry about axis x2, and E2 is symmetry. The electric fields decrease with the distance from the electrode and have large value near the points ± a, which is also shown in Eq. (47) with singularity r1/2 at the points.
Fig. 5. Electric fields of single electrode, where, for Fig. (a), 1, 2 and 3, 4 are electric fields distributing in depth x2 = 0.01a and x2 = a along x1, while in Fig. (b), 1, 2 and 3, 4 are electric fields distributing in depth at x1 = a and x1 = a.
Fig. 6. Stress fields of single electrode, where Figs. (a), (b), (c) and (d) show the stress in depth of x2 = 0.05a, x2 = 0.5a, x2 = a and x2 = 5a, respectively.
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Fig. 7. Electric fields of double electrodes, where for Figs. (a), (b) and (c), 1, 2 and 3, 4 are the electric fields distribute along x1 in depth of x2 = 0.01a and x2 = 0.01a with the geometric distributions of the electrodes b = 1.5a, b = a and b = 6a, while in Fig. (d), 1, 2 and 3, 4 are the electric fields distributing along x2 at x1 = b + a and x1 = b a with b = 2a.
Fig. 8. Stress distributions along x1 in different depth with b = 2a for double electrodes, where Figs. (a), (b), (c) and (d) show the stress fields distributing in depth of x2 = 0.05a,x2 = 0.5a, x2 = a and x2 = 5a, respectively.
Shown in Fig. 6 are the stress fields induced by the single electrode along x1 in depth of x2 = 0.05a,x2 = 0.5a,x2 = a and x2 = 5a, respectively. It is revealed that stresses r22 and r11 are symmetry about axis x2, while r12 are dissymmetry about axis x2. It can also be found that the stresses are large near the points ± a, which are induced by the singular electric field. The stresses degenerate along with the depth. When depth increases to x2 = a, the stress field is much smaller than that of x2 = 0.05. In general, the electrostrictive constants are negative, which shows that the normalized factors of stress are negative too. Therefore, the stress r22 approaching to the electrode is compressive, that is the electrode would be attracted to the material. Furthermore, The stress r11 under the electrode is tensile. For the materials BaTiO3 having the properties of high brittleness
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Fig. 9. Stress distributions along x1 in depth of x2 = 0.05a with different distance between two electrodes, b = 1.5a,b = a,b = 4a and b = 6a in Figs. (a), (b), (c) and (d), respectively.
Fig. 10. Stress fields distribution along x1 in depth of x2 = a with different distance between two electrodes, b = 1.5a,b = a,b = 4a and b = 6a in Figs. (a), (b), (c) and (d), respectively.
or low strength, the electrostrictive constants a1 amd a2 of such materials would be around 108 N/V2. When the electric potential of electrode is approximately 10 KV, and the electrode has the length of 103 m in magnitude, the stress r11 at about 0.01a ahead the tip of the electrode would be larger than 100 MPa in magnitude. This means that the failure would occur on the surface of electrode near the ends for the tensile stress r11. Given in Fig. 7(a)–(c) are the electric fields induced by the double electrodes along x1 in depth x2 = 0.01a and x2 = 0.01a with the geometric distributions of the electrodes b = 1.5a, b = a and b = 6a, respectively. And in Fig. 7(d), the electric fields distributing along x2 at x1 = b + a and x1 = b a with b = 2a are shown. It is shown that the electric field E1 is dissymmetry about axis x2, and E2 is symmetry, which are similar to the case of a single electrode. The electric fields decrease with the distance and have large value near the electrode ends. The interaction of the electrodes decreases as their distance
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2077
2(b a) increase, as shown in Fig. 7(a)–(c). Furthermore, in Fig. 7(c) (b = 6a), the interaction of the electrodes can be neglected so that we can only consider the total electric field, just giving the sum of ones induced by two single electrodes situated on [(b + a),(b a)] and [b a,b + a]. In other words, the electric field near the electrodes, when they are far from each other, can be calculated as the single electrode problem. In Fig. 8(a)–(d), it is shown that the stress distributions under the double electrodes along x1 in different depth. Similar to the single electrode, the stresses r22 and r11 are symmetry about axis x2,r12 are dissymmetry about axis x2. It can be seen from Fig. 8 that the stresses are very large near the electrode ends. Shown in Figs. 9 and 10 are the stress distributions along x1 in depth of x2 = 0.05a and x2 = a with different distance between two electrodes. It is shown that the interaction of the electrodes reduces as the distance between each other increases. For example, when b = 4a, i.e., the distance of the electrodes is 2a, the stress fields are nearly the same as those in the case of a single electrode loading. 7. Conclusions This work has presented the general solution for the electroelastic fields of collinear electrodes on the surface of a halfinfinite electrostrictive material, and closed form solution for two special cases of one and two electrodes. Numerical results are also given to discuss the distribution of the electroelastic fields near the electrodes. Several conclusions can be obtained as follows: (1) The stress fields in an electrostrictive material can be finally expressed by the electrical potential. Thus, once the electrical potential is obtained, one can easily determine the electroelastic fields from Eq. (45) of the present work. (2) It is found that the electic fields have the singularity of r1/2, while the stress fields have the singularity of r1 at the tips of the electrodes. Thus, the elastic singularity is stronger than that in the linear problem of traditional materials. (3) When the electric loading is applied, the stress r11 below the surface of the electrodes is tensile, while r11 ahead of the electrode ends is compressed. Therefore, the failure may occur firstly from the surface of the electrode. (4) When the distance between the two adjacent ends of two different electrodes is much larger than their lengthes, the interactions between them can be neglected, that is, the case of multiple electrodes can be treated as the case of a single electrode.
Acknowledgements This work is supported by National Natural Scientific Foundation of China (No. 10972103, No. 10902055); Post-Doctor Foundation of China and Jiangsu Province ( No. 20070411046, No. 0701018C); Natural Scientific Foundation of Jiangsu Province (No. BK 2006544). Appendix A The constants in Eq. (37) c0, c1, c2, . . ., cn1 can be determined by the electric charge qk on the electrodes. Assuming all the qk are known on the boundary L0k , we have:
Z
bk
ak
D2 ðtÞdt þ
Z
ak
bk
Dþ2 ðtÞdt ¼ qk :
ðA-1Þ
With the help of Eqs. (15) and (31), we get:
Z
bk
ak
H2 ðtÞ Hþ2 ðtÞ dt ¼ 2iqk ;
ðA-2Þ
and the above equation can be written as
Z
bk
ak
H2 ðtÞdt ¼ iqk ;
ðA-3Þ
by considering the second condition of Eq. (33). Since there are n integral equations of Eq. (A-3) on the boundary L0k ; 1 ¼ 1; 2; 3 . . . n, the constants ck can be determined. Therefore, the electric functions x01 ðzÞ and x02 ðzÞ can be given with the relationship of Eq. (31). The complex electric potential x1(z) and x2(z) will be:
x1 ðzÞ ¼
Z
x01 ðzÞdz þ c1 ; x2 ðzÞ ¼
Z
x02 ðzÞdz þ c2 ;
where the constants c1,c2 are real numbers in general.
ðA-4Þ
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If one of the electric potential on the electrodes is given, for example, it is Vk on L0k , we have:
c2 ¼ V k
1 2
Z
x02 ðzÞdz þ
Z
x02 ðzÞdz
ðA-5Þ
; z¼t
in the electrostrictive materials, where the relationship 2V k ¼ x2 ðtÞ þ x2 ðtÞ is used. It is noted that the constants c1 can also be given with the same method. For the problem of this paper, x1(z) = x2(z), x01 ðzÞ ¼ x02 ðzÞ and c1 = c2. The electric potential Vk+1 of next electrode situated on L0kþ1 must be:
V kþ1 ¼ V k
Z
akþ1
E1 dt ¼ V k þ
bk
1 2
Z
akþ1 bk
x02 ðtÞ þ x02 ðtÞ dt;
ðA-6Þ
with the help of Eq. (13). Using the same method, all the other electric potential can be solved. It should be emphasized that the electric potential on the electrodes are not independent. If the potential on one of the electrode is given, the others must be specified. Furthermore, if all the potentials on the electrodes is known, the charges qk are not random, they are dependent on each other. Appendix B The boundary-value problem of /(z) is studied in this section. Firstly, substituting Eq. (43) into Eq. (40), we get:
/þ ðtÞ / ðtÞ ¼ f ðtÞ f ðtÞ;
ðB-1Þ
where z = t and z ¼ t are used on the axis x1. When examining the nature of x02 ðtÞ, we have the relationship:
x02 ðtÞ ¼ x02 ðtÞ; x2 ðtÞ ¼ x2 ðtÞ; on the boundary
L00k ,
ðB-2Þ
and
x02 ðtÞ þ x02 ðtÞ ¼ 0; x2 ðtÞ þ x2 ðtÞ ¼ 2V k ;
ðB-3Þ
on the boundary L0k . Thus, we can get:
/þ ðtÞ / ðtÞ ¼ 0 on L00k :
ðB-4Þ
+
Therefore, /(z) is continuous from S to S on the boundary
/þ ðtÞ / ðtÞ ¼
jh 2
L00k .
For
L0k ,
we have:
i
x2 ðtÞx02 ðtÞ x2 ðtÞx02 ðtÞ ¼ jV k x02 ðtÞ;
ðB-5Þ
where the relationships of Eq. (B-3) and t ¼ t are used. Consequently, the function /(z) is
/ðzÞ ¼
1 2pi
Z
jV k x02 ðtÞ
L0k
dt : tz
ðB-6Þ
From Eq. (38), the functions x2(t) and x02 ðtÞ are analytical in the whole region except the singular points ak, bk(k = 1,2, . . ., n) on the axis x1. Consider a clockwise contour integration Hk(z) on Kk approaching to the boundary L0k very nearly:
Hk ðzÞ ¼
1 2pi
I Kk
jV k x02 ðtÞ
dt ; tz
which is shown in Fig. 11.
Fig. 11. The contour integration on Kk approach to the boundary L0k .
ðB-7Þ
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Using the relationship of the first equation in (B-3), the cauchy integration Hk(z) is
Hk ðzÞ ¼ jV k x02 ðzÞ þ jV k
n nh i o X ðbm Þ mÞ Gða m ðzÞ þ Gm ðzÞ ð1 dkm Þ ;
ðB-8Þ
m¼1 ðam Þ mÞ 0 where dkm is Kronecker delta, Gm ðzÞ; Gðb m ðzÞ are the cauchy principal value of the function x2 ðtÞ at the singular points. In 0 þ 0 0 addition, it is obviously that ½x2 ðtÞ ¼ ½x2 ðtÞ on L . We have:
/ðzÞ ¼
n 1X Hk ðzÞ: 2 k¼1
ðB-9Þ
If Vk(k = 1,2, . . ., n) have the value on L0k , the sum of the functions /(z) can be given as
/ðzÞ ¼
jV 1 2
x02 ðtÞ:
ðB-10Þ
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