[doi 10.1016%2fj.finel.2004.10.007] yao weian; hui wang -- virtual boundary element integral method...

8/11/2019 [Doi 10.1016%2Fj.finel.2004.10.007] Yao Weian; Hui Wang -- Virtual Boundary Element Integral Method for 2-D Pi

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Finite Elements in Analysis and Design 41 (2005) 875891

www.elsevier.com/locate/finel

Virtual boundary element integral method for2-D piezoelectric media

Yao Weian, Hui Wang

State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics,

Dalian University of Technology, Dalian 116023, PR China

Received 27 February 2004; accepted 12 October 2004

Available online 24 January 2005

Abstract

This paper presents a virtual boundary element integral method for 2-D piezoelectric media, which is based

on the fundamental solutions of 2-D piezoelectric media, the principle of superposition and the basic idea of the

virtual boundary element method for elasticity. Besides all the advantages of the conventional boundary element

method (BEM) over domain discretization methods, this method avoids the computation of singular integral on the

boundary by introducing the virtual boundary. In the end, several numerical examples are given to demonstrate theperformance of this method, and the results show that they agree well with the exact solutions. 2005 Elsevier B.V. All rights reserved.

Keywords:Virtual boundary element integral method; Piezoelectric media; Fundamental solution; Principle of superposition;

Singular integral

1. Introduction

Since the piezoelectric effect was found by J. Curie and P. Curie in 1880, piezoelectric media have

been widely used. Because of excellent coupling characteristic between electric effect and mechanicaldeformation, piezoelectric media have been preferred to the core components of transducers sensors andactuators and have been widely used in smart structures and micro-electro-mechanical systems (MEMS).

The direct piezoelectric effect has been used to create pressure sensors, microphones, accelerometersand so on. The indirect piezoelectric effect can be employed to create piezoelectric bending elements

Corresponding author.

E-mail addresses: [email protected](Y. Weian), [email protected](H. Wang).

0168-874X/$ - see front matter 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.finel.2004.10.007
http://www.elsevier.com/locate/finelmailto:[email protected]:hnwhprotect%20LY1extunderscore%[email protected]:hnwhprotect%20LY1extunderscore%[email protected]:[email protected]://www.elsevier.com/locate/finel


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876 Y. Weian, H. Wang /Finite Elements in Analysis and Design 41 (2005) 875891

(bimorphs), ultrasonic rotary motors, etc. Because intelligent structures or smart structures have attractedmore attention[1,2], the accurate analysis of piezoelectric media is more and more important. How-ever, because of its complex coupled basic governing equations, it is difficult to get the exact solutions.

Numerical methods are usually adopted.At present, there have been increasing efforts in the research on modeling piezoelectric media using

the finite element method (FEM), the finite difference method (FDM) and the boundary element method

(BEM) based on the 2-D and 3-D piezoelectric theory. Since the proposed method in this paper involvesonly the boundary integrals, it can be viewed as one of general BEMs. As a result, only developments

corresponding to BEM in this field are briefly reviewed in this section. For example, Lee and Jiang [3]firstly developed the BEM formulation of piezoelectricity using the method of weighted residuals andthey also derived the fundamental solution for plane piezoelectric media using double Fourier transform

technology. Lu and Mahrenholtz [4] deriveda variational boundaryintegralequationfor the sameproblem.Meng and Du[5]derived 2-D fundamental solutions for isotropic piezoelectric media subjected to the

generalized unit loads. Lee[6]also developed a boundary element method for electro-elastic interactionin piezoelectricity. Ding et al. [7]obtained fundamental solutions for transverse isotropic piezoelectricmedia by using potential theory and constructing a kind of harmonic function for 3-D piezoelectric

media. Ding et al.[8,9]also derived 2-D fundamental solutions using the Fourier transformation and theboundary integral equation by utilizing the reciprocal theorem of works. In addition, Qin[10]developeda new type of boundary element formulation for crack problems of piezoelectric materials, based on the

dislocation theory. Liu and Fan[11]developed an advanced BEM for accurate computation of thin filmsand coatings, avoiding the computation of the nearly singular integrals.

The above discussion has clearly demonstrated the accuracy and efficiency of the piezoelectric BEM

for piezoelectric media. However, BEMs all inevitably come across the problem of the computation ofsingular integrals, which are caused when the source point lies on the integral element or close to the

integral element. Moreover, extra quadrature equations are needed to compute qualities at internal pointswithin the domain. In order to avoid these problems, Sun et al. [12] developed the virtual boundaryelement integral and collocation points method, which is similar to the fundamental method (MFS)[13],

and applied this method to solve many engineering problems successfully. The basic idea of this method[12]is stated as follows.

Suppose that the interesting domain is placed into an infinite domain. If a virtual boundary is selected

at certain locations within this infinite domain, which is usually different from the real boundary ofthe interesting domain, and there are some fictitious or virtual loads applied on this virtual boundary,according to the principle of superposition, the qualities at every point in the interesting domain or on

its boundary can be expressed as the linear combination of fundamental solutions whose coefficients are

those fictitious loads. If these qualities at points on the real boundary satisfy corresponding boundaryconditions, the fictitious source loads can be determined. Further, the qualities at arbitrary points in theinteresting domain or on its boundary can be obtained. Evidently, the singular integral is avoided owing tothe proper distance between the virtual boundary and the real boundary and no extra quadrature equations

are needed to compute qualities at internal points within the domain. At the same time, this method iseasily implemented due to its simple theory. Additionally, this method also has the same advantages asconventional BEM, such as elegant and economic computation. Surely, there are some disadvantages of

this method, such as the treatment of inhomogeneous and non-linear problems, requiring knowledge ofa suitable fundamental solution, and so on. These also exist in conventional BEM and need to be further

throughtout.
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Y. Weian, H. Wang /Finite Elements in Analysis and Design 41 (2005) 875891 877

In this paper, suppose that there are distributed fictitious loads actingon the virtual or fictitious boundarywithin the infinite domain. As a result, the virtual boundary element integral method is developed for planepiezoelectric problems. In Section 2 the fundamental equations for 2-D piezoelectric media are introduced,

and in Section 3 the paper derives the basic computing formulations by using the fundamental solutionsdeveloped in Ref. [8]. Several numerical examples are employed in Section 4 to verify the performance ofthe method proposed in this paper, and the results show good agreement with the exact solutions. Finally,

the paper is summarized in Section 5.

2. Fundamental equations for 2-D piezoelectric media

Let us consider the transverse isotropic linear piezoelectric material, such as PZT-4 ceramic. It occupiesa domain in R2 which has a piecewise smooth boundary shown inFig. 1.

Suppose that thexyplane is isotropic plane and thez-direction is the polarization direction of piezo-electric media. If thex zplane is chosen as the study of plane electromechanical phenomena, the basicgoverning equations can be given as follows.

The equilibrium equations and Gausss Law of Electrostatics can be written as

x

x+

xz

z+ fx = 0,

xz

x+

z

z+ fz= 0,

Dx

x+

Dz

z= , (1)

where x , z and xz denote the stress components, fx and fz are the body force components, Dx and

Dzare the electric displacement components in thex- andz-directions, respectively,is the free electric

volume charge.

The straindisplacement relations are

x =u

x, z =

w

z, xz =

u

z+

w

x, Ex =

x, Ez=

z, (2)

wherex ,zand xz are the strain components,ExandEzare the electric fields in thex- andz-directions,respectively,u and w are the elastic displacement components, and is the electric potential.

1

2

1

2

N

N

x

z

o

Fig. 1. Discretization of the virtual boundary and collocation points on the real one.


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The elastic constitutive equations for plane-strain response can be expressed as[14]

xzxz

=c11 c13 0

c13 c33 00 0 c44

xzxz

0 e31

0 e33e15 0

ExEz

, (3)

DxDz

=

0 0 e15e31 e33 0

xzxz

+

11 0

0 33

ExEz

, (4)

where c11, c13, c33, c44 denote elastic modulus measured at zero or constant electric field; e15, e31, e33denote piezoelectric constants, and 11, 33denote the dielectric permittivity measured at zero or constantstrain.

The boundaryconditions onhave mechanics conditions and electric conditions. Mechanics conditions

can be divided into the given displacement boundary conditions

u = u, w= w on u (5)

and the given surface traction boundary conditions

x nx + xz nz= tx , xz nx + znz= tz on t. (6)

Electric conditions can be divided into the prescribed surface charge density boundary conditions

Dx nx + Dznz = on (7)

and the prescribed electric potential boundary conditions

= on , (8)

where u, w, , tx , tz and are specified values on the boundary, and nx and nz are components of the

unit outward normal vector, respectively. Note thatt u= = ,t u= = 0.

3. Virtual boundary element integral method

Let us first extend the domain to the infinite domain, in which we select a virtual boundary as

shown inFig. 1.For the sake of simplicity, the notations ui (i =1, 2, 3) and i (i =1, 2, 3, 4, 5) areintroduced, which representu,w,and x , z, xz , Dx , Dz, respectively. In addition, assume that thereareLpointsX0k (k = 1, 2, . . . , L )within the domain, at which there are x- andz-direction point loadsand point chargeFk1, Fk2, Fk3, respectively.

Suppose that there are distributed unknown virtual loads (X) = [1(X),2(X

),3(X)] on the

virtual boundary , where 1,2,3 denote x- and z-direction loads and charge at random point X,

respectively. If these virtual loads are viewed as source loads, in the absence of body forces and free

electric volume charge, according to the principle of superposition and the fundamental solutions, whichare analytically free space solutions of the governing differential equation under the action of point source,

we can express the displacements, electric potential, stress and electric displacements at random pointX


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in domain as

ui (X) =

3j=1

uij(X,X)j(X

) d + L

k=1

3j=1

uij(X,X0k )Fkj (i= 1, 2, 3), (9)

i (X) =

3

j=1

ij(X,X)j(X

)

d + L

k=1

3j=1

ij(X,X0k )Fkj (i= 1, 2, 3, 4, 5), (10)

where the expressions of the fundamental solutionsuij (i = 1, 2, 3; j= 1, 2, 3)and ij (i = 1, 2, 3, 4, 5;

j = 1, 2, 3)can be seen in the Appendix. For example, u12 (X, X)denotes displacementu at the point

Xwhen there is azdirection unit load at pointX

.X R2

,X

R2

,X0k R

2

. At the same time, we cansee that Eqs. (9) and (10) satisfy all governing equations within the domain .

Now the virtual boundary is discretized by n elements and the total number of nodes is N. Supposethat there are m nodal points on the random element p. So the value of the distributed loads and charge

at the arbitrary point on the elementp can be expressed as

j() =

ml=1

Nl ()lj (j= 1, 2, 3), (11)

where Nl () (l =1, 2, . . . , m ) are the interpolation shape functions, is the dimensionless coordinatedefined same as in BEM, andljis the value of loads and charge at the lth nodal point.

Substituting Eq. (11) into Eqs. (9) and (10), we can obtain

ui (X) =

np=1

p

3

j=1

uij(X,X)j(X

)

d + L

k=1

3j=1

uij(X,X0k )Fkj

=

np=1

ml=1

3j=1

p

uij(X,X())Nl () d

lj +

Lk=1

3j=1

uij(X, X0k )Fkj (i= 1, 2, 3)

(12)

i (X) =

np=1

p

3

j=1

ij(X,X)j(X

)

d + L

k=1

3j=1

ij(X,X0k )Fkj

=

np=1

ml=1

3j=1

p

ij(X,X())Nl () d

lj +

Lk=1

3j=1

ij(X,X0k )Fkj

(i= 1, 2, 3, 4, 5). (13)


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In order to solve unknown nodal values ljon the virtual boundary, we select Ncollocation points

Xq (q = 1, 2, . . . , N )on the real boundary and let the arbitrary field pointXin Eqs. (12) and (13) reachthe collocation pointXq (q= 1, 2, . . . , N )in turn. At the same time, we define

hpqijl =

p

uij(Xq , X())Nl () d

, (14a)

gpqijl =

p

ij(Xq , X())Nl () d

. (14b)

They can easily be computed by using the standard Gaussian quadrature approach. Substituting

Eq. (14) into Eqs. (12) and (13), Eqs. (12) and (13) can be rewritten in a simple way:

ui (Xq ) =

np=1

ml=1

3j=1

hpqijl

lj +

Lk=1

3j=1

uij(Xq , X0k )Fkj (i= 1, 2, 3), (15)

i (Xq ) =

np=1

mj=1

3k=1

gpqkij

jk +

Lk=1

3j=1

ij(Xq , X0k )Fkj (i= 1, 2, 3, 4, 5). (16)

If Eqs. (15) and (16) satisfy the given boundary conditions at every collocation point Xq (q =

1, 2, . . . , N ), we can obtain a linear system of equations in matrix form

HA = Y, (17)

whereA = [11 12

13

N1

N2

N3 ]

T is an unknown vector of dimension 3N,Y represents a3Ndimensional vector which is formed according to the given boundary conditions at collocation points

Xq (q = 1, 2, . . . , N )on the real boundary and Hstands for an influence matrix of dimensions 3N 3N.Solving Eq. (17), we can obtain the unknown vector A, that is to say that the nodal values of distributed

fictitious loads on the virtual boundary are determined. Further, according to Eq. (11), we can know the

approximate distribution of fictitious loads on the virtual boundary.

Finally, substituting virtual distributed loads obtained into Eqs. (15) and (16), we can obtain numericalsolutions of the displacement, electric potential, stress and electric displacement components at arbitrarypoints within the domain and on its boundary.

It should be mentioned that the geometry of the virtual boundary can be arbitrary in theory. In order to

be convenient in the process of computation, it can be determined by defining the similar ratio between thevirtual boundary and the real one[12].However, due to the interference of singularity of the fundamentalsolution, the accuracy of the result can degrade when the distance between the virtual boundary and the

real boundary becomes very close[12].The study in Ref.[12]indicated that, generally, for the boundaryof inner domain, the similar ratio can be selected as 1.23.5; for the boundary of external domain, the

similar ratio can be selected as 0.60.85.


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4. Numerical examples

To verify the method presented in the previous section, several numerical examples are considered and

the calculated results are compared with analytical solutions. In these cases, PZT-4 ceramics is specifiedand its mechanical and electric constants are:

modulus of elasticity measured at constant electric fields

c11= 12.6, c12= 7.48, c13= 7.43, c33= 11.5, c44= 2.56 (1010 N m2),

piezoelectric constants

e15= 12.7, e31= 5.2, e33= 15.1(C m2),

dielectric constants measured at constant strain

11= 730 0, 33= 635 0 in which 0= 8.85419 1012(C V1m1).

Example 1. Consider an infinite piezoelectric plane filled with transversely isotropic piezoelectric ma-terial and containing a circular hole (Fig. 2). Mechanical loads0 = 10 parallel to theozaxis and appliedat remote distances from the hole induce deformations and electric fields in the whole region. In the

calculation, the radius of the hole is 1 and a total of 20 constant elements are employed. The similar ratioof the virtual boundary and the real boundary is 0.6 (see Ref.[12]). The corresponding exact solution canbe found in Ref.[15].

x

z

r

Fig. 2. A circular hole in an infinite piezoelectric medium.


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1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Radius r

Stress

EXACTVBEM

Fig. 3. Distribution of radius stress versus r radiusralong the line = 0 subjected to remote mechanical load.

1 2 3 4 5 6 7 8 9 1010

12

14

16

18

20

22

24

26

28

EXACTVBEM

Stress

Radius r

Fig. 4. Distribution of hoop stress versus radiusralong the line = 0 subjected to remote mechanical load.

Figs. 3and 4 describe the distributions ofr and when radius rapproaches infinity along the line = 0, respectively. Results show that the maximum value of stress on the rim of the hole is found to be26.92 (the exact solution gives 27.21).

Figs. 5and6depict the distribution of and D on the rim of the circular hole, respectively. Wecan observe that the maximum value of occurs at = 0

, while D reaches its maximum at about

= 65.

Example 2. Consider the shear deformation of a 1.01.0 mm piezoelectric strip. The strip is subjected toa uniform pressp = 5 N/mm2 and an applied voltageV0 = 1000 V as shown inFig. 7. The corresponding


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0 10 20 30 40 50 60 70 80 9015

10

5

0

5

10

15

20

25

30

Angle

Stress

VBEMEXACT

Fig. 5. Distribution of hoop stress versus angle on the rim of the circular hole subjected to remote mechanical load.

0 10 20 30 40 50 60 70 80 903

2

1

0

1

2

3

EXACTVBEM

Electricdisplacement1.0

E+9

Angle

Fig. 6. Distribution of hoop electric displacement Dversus angle on the rim of the circular hole subjected to remote mechanical

load.

boundary conditions are:

u(x= 0, z) = 0, tz(x = 0, z=0) = 0, (x= 0, z) = V0, w(x= 0, z = 0) = 0;

tx (x= L,z) = 0, tz(x = L,z) = 0, (x= L,z) = V0;

tx (x,z = h) = 0, ,z(x,z = h) = 0, tz(x,z = h) = p.

In this example, suppose the length of they-direction is much smaller than other directions, so we can

view this problem as a plane stress problem. In calculation, the similar ratio of the virtual boundary and


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x

z

L

2h

p=5N/mm2

+V0 V0

Fig. 7. Piezoelectric strip subjected to a uniform stress and an applied voltage.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

VBEMEXACT

x (mm)

Displacement1.0

E+8(m)

Fig. 8. Variation of displacementu along the bottom edge of the piezoelectric strip.

the real boundary is 2.0 (see Ref.[12]), a total of 36 linear elements are used on the virtual boundary and36 corresponding points are distributed on the real boundary. Both displacements and electric potential

on the bottom edge of the PZT-4 strip calculated using the method presented in this paper are compared


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

0

2

4

6

8

10

Dis

placeme

nt

1.0E

+7(m

)

x (mm)

VBEMEXACT

Fig. 9. Variation of displacementw along the bottom edge of the piezoelectric strip.

0 0.1 0.2 0.3 0.4 0.5 0.6 11000

800

600

400

0

200

400

600

800

1000

200

0.90.80.7

x (mm)

Electricpotential(V)

VBEMEXACT

Fig. 10. Variation of displacement along the bottom edge of the piezoelectric strip.

with the exact solution given in literature [16]. Figs. 810 all show the good match between the calculatedsolution and the exact one.

Example 3. Consider another piezoelectric strip in bending which has the same geometry as Example 2.The boundary conditions vary as shown in Fig. 11.A voltage is applied to the top and bottom surfaces and

the right side is subjected to a linearly varying stress. We set V0=1000 V,0=5 N/mm2, 1=20 N/mm

3
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L

x

z

+V0

V0

2h 0+ 1z

Fig. 11. Piezoelectric strip subjected to a linearly varying stress and an applied voltage.

0

1

2

3

z (mm)

Dis

placement1.0

E+7(m)

VBEMEXACT

3.5

2.5

1.5

0.50.5 0.4 0.3 0.2 0.1 0.1 0.2 0.4 0.50.3

Fig. 12. Variation of displacementu along the right-side edge of the piezoelectric strip.

and the corresponding boundary conditions are:

u(x= 0, z) = 0, tz(x = 0, z=0) = 0, ,x (x = 0, z) = 0, w(x= 0, z = 0) = 0;

tx (x= L,z) = 0+ 1z, tz(x= L,z) = 0, ,x (x= L,z) = 0;

tx (x,z = h) = 0, tz(x,z = h) = 0, (x,z = h) = V0.

In the calculation, the similar ratio of the virtual boundary and the real boundary also is 2.0(see Ref.[12]), same as Example 2, and total 44 linear elements are used on the virtual boundary and

44 corresponding points are distributed on the real boundary. Both calculated displacements are given inFigs. 12and13,and calculated electric potentials are shown inFig. 14.As a result, they also match the

exact solution given in Ref.[16].


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0

1

2

z (mm)

Displacement1.0

E+7(m

)

VBEMEXACT

1

2

3

4

50.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5

Fig. 13. Variation of displacementw along the right-side edge of the piezoelectric strip.

z (mm)

Electric

potential(V)

VBEM

EXACT

1000

1000

800

800

600

600

400

400

200

200

0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5

0

Fig. 14. Variation of displacement along the right-side edge of the piezoelectric strip.

5. Conclusion

In this paper, the virtual boundary element integral method is presented to analyze 2-D piezoelectricmedia. Compared to the conventional BEM, the proposed method avoids the singular integrals appearing

in the conventional BEM and does not require extra integral equations to calculate corresponding qualitiesat points in the interior of the domain. In addition, from the derivation of computing formulas it can beseen that this method is very simple and is easier to be implemented. Finally, the numerical solutions

matching the analytical ones given in the literatures demonstrate the good performance of this method.So the virtual boundary element integral method presented in this paper can provide a quick and effective

analysis for solving the coupled electrical and mechanical piezoelectric problems.


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However, there are some notes needed to say. In the case that body forces cannot be neglected, thismethod does not show evident advantage over other domain discretization methods such as FEM andFDM, due to extra domain cells needed to calculate corresponding domain integrals. Also, in order to

increase the computational accuracy and stability for complicated boundary conditions, we can choosemore collation points on the real boundary and gain least-squares solutions, which is not discussed indetail because it can be easily derivated from this paper. In other words, the method presented in this

paper is an alternative method to analyze piezoelectric media.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 10172021).

Appendix A. The fundamental solution for 2-D piezoelectric media

For the completeness and reference, the fundamental solutions for 2-D piezoelectric media are listed, inwhich the displacement and electric potential components are first given in literature [8] and the stress and

electric displacement components can be derived by substituting the displacement and electric potentialcomponents into governing equations.

Suppose that at point(, )there are unit loads along x - andz-direction and unit charge; then for theplane strain problem, especially under the situation of material eigenrootssi (i= 1, 2, 3)different fromeach other, the expressions of the displacements, electric potential, stresses and electric displacement

components at random field point(x, z)can be shown as follows:

A.1. The displacement and electric potential components [8]

u11=1

De11

3j=1

sj1t1(2j )1ln rj,

u12=1

De12

3j=1

sj2t1(2j )2arctg

x

sj(z ),

u13= 1

De13

3

j=1

sj3t1(2j )3arctg

x

sj(z )

,

(A.1)

u21=1

De11

3j=1

dj1t1(2j )1arctg

x

sj(z ),

u22=1

De12

3j=1

dj2t1(2j )2ln rj,

u23= 1

De13

3j=1

dj3t1(2j )3ln rj,

(A.2)


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u31=1

De11

3j=1

gj1t1(2j )1arctg

x

sj(z ),

u32=1

De12

3j=1

gj2t1(2j )2ln rj,

u33= 1

De13

3j=1

gj3t1(2j )3ln rj.

(A.3)

A.2. The stress and electric displacement components

11=1

De11

3j=1

(c11sj1 c13dj1sj e31gj1sj)t

1(2j )1

x

r2j

,

12=1

De12

3j=1

(c11sj2+ c13dj2sj + e31gj2sj)t

1(2j )2

sj(z )

r2j

,

13= 1

De13

3j=1


1(2j )3

sj(z )

r2j

,

(A.4)

21=1

De11

3j=1

(c13sj1 c33dj1sj e33gj1sj)t

1(2j )1

x

r2j

,

22=1

De12

3j=1


1(2j )2

sj(z )

r2j

,

23= 1

De13

3j=1


1(2j )3

sj(z )

r2j

,

(A.5)

31=1

De11

3j=1

(c44sj1sj+ c44dj1+ e15gj1)t

1(2j )1

sj(z )

r2j

,

32=1

De12

3j=1

(c44sj2sj+ c44dj2+ e15gj2)t

1(2j )2

x

r2j

,

33= 1

De13

3j=1

(c44sj3sj + c44dj3+ e15gj3)t

1(2j )3

x

r2j

,

(A.6)


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41=1

De11

3j=1

(e15sj1sj + e15dj1 11gj1)t

1(2j )1

sj(z )

r2j

,

42=1

De12

3j=1

(e15sj2sj + e15dj2 11gj2)t

1(2j )2

x

r2j

,

43= 1

De13

3j=1

(e15sj3sj+ e15dj3 11gj3)t

1(2j )3

x

r2j

,

(A.7)

51=1

De11

3j=1

(e31sj1 e33dj1sj + 33gj1sj)t

1(2j )1

x

r2j

,

52=1

De12

3j=1

(e31sj2+ e33dj2sj 33gj2sj)t

1(2j )2

sj(z )

r2j

,

53= 1

De13

3j=1

(e31sj3+ e33dj3sj 33gj3sj)t

1(2j )3

sj(z )

r2j

,

(A.8)

whererj =

(x )2 + s2j(z )2,sij, dij, gij, tij andsjare defined in literature[8].

References

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[doi 10.1016%2fj.finel.2004.10.007] yao weian; hui wang -- virtual boundary element integral method...

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