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Analysis of piezoelectric plates with a hole using nature boundaryintegral equation and domain decomposition

Xing-Yuan Miao, Guo-Qing Lin

School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China

a r t i c l e i n f o

Article history:Received 5 May 2013Accepted 18 November 2013Available online 22 December 2013

Keywords:Piezoelectric mediumBoundary integration equationHarmonic analysisDefectsDomain decomposition

a b s t r a c t

In this paper, the plane problems of piezoelectricity are studied by using nature boundary integralequation and domain decomposition. A general displacement solution in terms of three potentialfunctions is adopted to solve exterior boundary value problems of piezoelectricity, and three mappingrelations corresponding to three potential functions are proposed for domain decomposition. Bysymbolic matrix inversion and derivation calculus, each potential function is governed by harmonicsecond-order partial differential equation in transformed domain with prescribed boundary condition.Therefore, three classic harmonic problems equivalent to the original plane piezoelectricity areestablished. Two cases of boundary conditions are considered, in which the displacement and electricpotential are prescribed or the traction and electric displacement are given on the boundary. Allproblems considered are equivalent to three independent harmonic problems, which are solved by usingnature boundary integration method proposed by Feng and Yu. A piezoelectric plate with a circular holeis analyzed as numerical examples. The results show that the proposed method is valid for thepiezoelectric plates with holes. The proposed method has potential applications to analyze multi-eldcoupling problems.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The study on coupled elds involved with piezoelectricity,electrostriction and magnetostriction has aroused many researchers'interests, and the plane problems of piezoelectricity have beenwidely and deeply investigated in the past years. Sosa and Castro[16] rst studied the concentrated loads acted on the boundary of apiezoelectric half plane and extended the complex potential functionmethod proposed by Lekhnitskii to analyze electroelastic problems,who also investigated the plane problems in piezoelectric mediumwith elliptic hole [15,17]. Ding et al. [3] derived the general solutionof plane problems of piezoelectric medium expressed by harmonicfunctions. Rajapakse [14] investigated the upper half plane problemof piezoelectric medium by means of the Fourier transform. Further-more, Benveniste [1], Chung and Ting [2], Dunn andWienecke [7], Luet al. [11], Pan and Yuan [13], Pan and Tonon [12] and Hu et al. [8]studied the Green's functions for a series of numerical computationsof plane problems. Li et al. [10] analyzed the piezoelectric planeproblem with xed electrodes. As for computing methods, Yu andZhao [22] and Wu and Yu [19] applied the natural boundary integralequation in many numerical experiments based on natural boundaryreduction. Natural BIE has unique superiorities in multiple boundaryvalue problems. In recent years, the natural BIE has been applied to

solve a series of exterior problems for continuous or discrete domainin two-dimensional space. It can be referred to Wu and Yu [19], Duan Yu [5], Yu [21], Du and Yu [6], Yu and Zhao [22], Huang et al. [9]for more details.

In this paper, a coupling problem is decomposed into threesuccinct harmonic problems. First, the piezoelectric plane problemwith specied displacement and electric potential boundary con-ditions will be investigated. The general solutions of displacementand electric potential are expressed by three harmonic functions.Then, we separate the boundary conditions into three mappingregions, and eventually the original problem would be trans-formed into harmonic problem in three different mapping regions.Second, we discuss piezoelectric plane problem of the second kindwith specied distributed force and electric displacement bound-ary conditions. The general solutions of the stress and electricdisplacement are denoted by three harmonic functions. Hence, theequivalent three harmonic problems have been obtained by meansof separating the boundary conditions as well.

2. General solution to plane piezoelectricity

In this section, a general solution to plane piezoelectricity usingdisplacement potential functions will be briey introduced, whichis rst proposed by Ding et al. [4]. Considering the piezoelectricmedium occupying a plane without body force and electric charge,

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/enganabound

Engineering Analysis with Boundary Elements

0955-7997/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.enganabound.2013.11.012

n Corresponding author: Tel.: +86 13971016840; fax: +86 027 87543538.E-mail address: lig57@hust.edu.cn (G.-Q. Li).

Engineering Analysis with Boundary Elements 40 (2014) 7177

the governing equations for this piezoelectric problem are given asfollows:

Dfu;w;gT 0; 1where

D

c11 2

x21c44 2x23 c13c44

2x1x3

e15e31 2x1x3c13c44 2x1x3 c44

2x21

c33 2x23 e152x21

e33 2x23e15e31 2x1x3 e15

2x21

e33 2x23

11 2

x2133 2x23

2666664

3777775: 2

In the above equations, u and w are the mechanical displacementcomponents along x1 and x3 directions, is the electric potential,D represents a differential matrix operator. Cij; eij; ij are the elastic,piezoelectric and dielectric constants, respectively.

Based on the derivation in Ding's work [4], the fundamentalsolution to Eq. (1) is expressed in terms of three quasi-harmonicfunctions:

2

y2k1

2

y2k3

! k 0; 3

where subscript k 1;2;3, which will be always implicit impliedin this paper, and k are potential functions, yk1 x1kx3,yk3 kx3 and sk k ik are the three roots of the followingequation:

as6bs4cs2d 0; 4where

a c44e233c3333;b c33c4411e15e31233c11c33c244c13c442

e332c44e15c11e332c13c44e15e31;c c44c1133e15e31211c11c33c244c13c442

e152c11e33c44e152c13c44e15e31;d c11e215c4411: 5

It has been studied on sk by Ding et al. [4] and by [18] that, theeigenvalue sk has a positive imaginary part, i.e. k40, and sk canalso be a pure imaginary number. For simplicity, we only considerthe sk as imaginary in this paper. In the rest of this paper, yk1 x1,yk3 yk skx3 will be always established.

Therefore, the displacement components and electric potentialcan be obtained as

u 3

k 1a1k

kx1

; w 3

k 1a2k

kyk

; 3

k 1a3k

kyk

; 6

where

a1k 1;

a2k c1111m3s2kc4433s4k

m1m2s2k sk;

a3k c11e15m4s2kc44e33s4k

m1m2s2k sk; 7

in which

m1 c13c4411e15e31e15;m2 c13c4433e15e31e33;m3 c1133c4411e15e312;m4 c11e33c13e15e31c44e31: 8

Applying the constitutive equations of the plane piezoelectricity tothe displacement solution, the stress components and electric

displacement can be obtained by

s11 3

k 1b11

2 kx23

; s33 3

k 1b12

2 kx21

; s13 3

k 1b13

2 kx1x3

;

9

D1 3

k 1b14

2 kx21

; D3 3

k 1b15

2 ky2k

; 10

where

b11 C11a1kC13a2ke31a3k;b12 C13a1kC33a2ke33a3k;b13 C44a1ka2ke15a3k;b14 e15a1ka2k11a3k;b15 e31a1ke33a2k33a3k:

9>>>>>>=>>>>>>;

11

In summary, there are three harmonic functions k in transformeddomain kx1; yk satisfying the governing equation of planepiezoelectricity, and all physical elds can be expressed in termsof the functions k. In order to obtain the solution of k fulllingthe prescribed boundary equations, the equations on x1; x3will be separated into three independent boundary conditions onthe transformed boundary kx1; yk, which will be presented inthe following sections.

3. Plane piezoelectricity with displacement and electricpotential conditions

In this section, the displacement and electric potential bound-ary conditions will be spliced into three independent conditions intransformed domain.

It is assumed that the original conditions for static elasto-electric are given as follows:

uj u0x1; x3; wj w0x1; x3; j 0x1; x3: 12Substituting Eq. (6) into Eq. (12) yields the boundary conditions forthe displacement and electric potential, that is

3

k 1a1k

kx1

u0x1; x3;

3

k 1a2k

kyk

w0x1; x3;

3

k 1a3k

kyk

0x1; x3: 13

Considering a simply connected region surrounded by one smoothcurve, which can be dened by the following t-parametric equa-tions:

x1 f 1t; x3 f 3t; 14For purpose of separating the boundary conditions into thetransformed domains, the coordinate transformation relation isused. Then, Eq. (13) can be rewritten with respect to t-parameteras follows:

3

k 1

a1kf 1

kt

u0x1; x3;

3

k 1

a2kskf

3

kt

w0x1; x3;

3

k 1

a3kskf

3

kt

0x1; x3: 15

Rewriting Eq. (15) into matrix form leads to

At kt

Ut; 16

X.-Y. Miao, G.-Q. Li / Engineering Analysis with Boundary Elements 40 (2014) 717772

where A1k a1k=f 1, A2k a2k=skf 3, A3k a3k=skf 3 and Uu0;w0;0T . Therefore, kt

A1tUt k; 17

where A1 is the inverse matrix of A and will be obtainedby symbolic matrix inversion operation. The partial differential k=t has connection with the normal differential k=nk,which is

knk

kt

tnk

f 1f 3skkf 3sk cos k f 1 sin k

; 18

where k is the direction angle shown in Fig. 1.

cos dx1ds

n2; sin dx3ds

n1;

cos k dyk1dsk

nk2; sin k dyk3dsk

nk1: 19

Finally, three equivalent problems used to describe the originalproblem are established by

k 0 on k; knk

f 1f 3skkf 3sk cos k f 1 sin k

on k:

9>=>; k 1;2;3 20

It should be noticed that each of the equivalent problems isdened in a certain x1; yk coordinate system. During the symbolicmatrix operations on the right side of the second equation of Eq.(17), the prescribed boundary conditions shown in Eq. (13) shouldbe simultaneously replaced by u0x1; yk, w0x1; yk and 0x1; ykfor k 1;2;3 respectively. Obviously, the equivalent problemsshown in Eq. (20) are three classic harmonic problems withNeumann conditions.

4. Plane piezoelectricity with traction and electricdisplacement conditions

In this section, the traction and electric displacement boundaryconditions will be studied using the same approach as the lastsection.

At rst, applying the parameter equation of the boundaryshown in Eq. (14) to Eqs. (9) and (10) yields the stress componentsand electric displacement, these are

s11 3

k 1bk11

2 kt2

; s33 3

k 1bk12

2 kt2

; s13 3

k 1bk13

2 kt2

;

21

D1 3

k 1bk14

2 kt2

; D3 3

k 1bk15

2 kt2

; 22

where

bk11 C11a1kf 21

C13a2kskf 23

e31a3kskf 23

;

bk12 C13a1kf 21

C33a2kskf 23

e33a3kskf 23

;

bk13 C44a1ka2kskf 1f 3

e15a3k

skf 1f 3;

bk14 e15a1ka2kskf 1f 3

11a3k

skf 1f 3;

bk15 e31a1kf 21

e33a2kskf 23

33a3kskf 23

:

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

k 1;2;3 23

Next, the stress and electric displacement boundary conditionswill be separated into three independent conditions. It is assumedthat the original conditions for static elastoelectric are given asfollows:

pj p0x1; x3; qj q0x1; x3; dj d0x1; x3; 24

where p, q, d represent the normal traction, the tangentialtraction and the electric displacement, respectively, and they arewritten by

s11n1s13n3 p;s13n1s33n3 q;D1n1D3n3 d; 25

where n1;n2 represent the direction cosines on the boundary.Substituting Eqs. (21) and (22) together with Eq. (25) into

Eq. (24) yields

3

k 1tk12 kt2

p0x1; x3;

3

k 1tk22 kt2

q0x1; x3;

3

k 1tk32 kt2

d0x1; x3; 26

where

tk1 n21bk11n22bk122n1n2bk13;tk2 n1n2bk12bk11n21n22bk13;tk3 n1bk14n2bk15:

9>>>=>>>;

k 1;2;3 27

Rewriting Eq. (26) into matrix form leads to

Tt2 kt2

Pt; 28

Fig. 1. Direction angle in two domains.

X.-Y. Miao, G.-Q. Li / Engineering Analysis with Boundary Elements 40 (2014) 7177 73

where P p0; q0; d0T . Taking the inverse matrix operation on Eq.(28) leads to

2 kt2

T1tPt; 29

where T1t is the inverse matrix of Tt, which is obtained usingsymbolic inverse matrix operation similar to Eq. (17). In order toobtain the functions k using natural boundary integral equation,the second-order partial differential equation must be degener-ated into a rst-order partial differential equation on the boundaryk. Thus, the integration of Eq. (29) gives

kt

^ k0Ck k; k 1;2;3 30

where ^ k0 RkT 1ik tPit dt, and the unknown constants Ck can

be determined from the following displacement and electricpotential integral conditions (see Appendix for more detail), i.e.Zu ds 0;

Zw ds 0;

Z ds 0: 31

Once the constants Ck are determined, Eq. (30) plays the same roleas that of Eq. (18). The separated boundary conditions similar toEq. (18) can be obtained by

knk

f 1skf 3kskf 3 cos k f 1 sin k

: 32

In the end, a displacement expression which is analogous to Eq.(20) can be obtained as

k 0 on k; knk

f1f

3skk

f 3sk cos k f 1 sin kon k:

9>>=>>; k 1;2;3 33

Comparing Eq. (33) with Eq. (20) shown that the only differ-ence between them is the boundary function k in Eq. (20)replacing by k in Eq. (33), in which constants Ck are involved.

5. Solution procedure using nature boundary integrationequations

From the above sections, the solution to plane piezoelectricitycan be reduced to solving three independent Laplace problemswhich is shown in Eq. (20) or Eq. (33). It is well known that thereare many methods available to solving Eq. (33) or Eq. (20) in theliterature. In this paper, the method called as nature boundaryintegral equation (NBIE) established by Feng and [20] is adopted.Therefore, the solution procedure will be briey introduced asfollows. Without lost of generality, the solution procedure isdesigned for piezoelectric plate with a unit circular hole, whoseresults will be discussed in detail in the next section.

Prior to the solution procedure, the natural boundary integralmethod for harmonic problem in exterior elliptic region will bebriey introduced [22]. Taking problem described by Eq. (20) as anexample, the original domain is a unit cycle and the trans-formed domain k is a elliptic dened by x21y2ks2k 1, and thenal transformed domain is a unit cycle again dened by k 1using elliptic k;kAk coordinates. The transform relationsfrom Cartesian coordinates to elliptic coordinates are

x1 f 0 cosh cos yk f 0 sinh sin : 34In elliptic coordinates, the Laplace operator remains unchanged forthe new domain.

Another important issue is concerned with computationmeshes. It is worth noting that the uniform or equal spacingmeshes cannot be inherited after coordinate transformations. In

another word, if meshes in k are uniform, meshes in k and in are all non-uniform, and vice versa. Fig. 2 shows the uniformmeshes in but non-uniform meshes in k and

k. In fact, the

uniform meshes in k are always needed for NBIE solution asstated by Yu and Zhao [22]. In this case, the interpolationcalculation will be needed, in which piecewise linear basis func-tions Lj on boundary will be used, that is

Lj

N2

mj1; j1rmrj;N2

j1m; jrmrj1;0; else;

8>>>>>>>:

35

which can be found in Yu and Zhao [22] or Yu [20].The seven main steps in solution procedure will be addressed

in the following.

Step 1: Input the problem denition including material con-stants, geometry and load conditions.

Step 2: Solve the eigen value sk and associated matrix At orTt using Eqs. (...