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The fundamental solutions for transversely isotropic
piezoelectricity and boundary element method
Haojiang Dinga,*, Jian Liangb
aDepartment of Civil Engineering, Zhejiang University, 310027, Hangzhou, People's Republic of ChinabDepartment of Mechanics, Zhejiang University, 310027, Hangzhou, People's Republic of China
Received 18 February 1997; accepted 12 November 1998
Abstract
In this paper, we rst supplement two groups of simplied general solution based on previous work. Those results
are in terms of harmonic functions and t for the cases of multiple eigenvalues. Then by trial and error, we obtain
the fundamental solutions for three cases for a piezoelectric innite media by giving the expressions of harmonic
functions. Finally, by use of those solutions, we implement a boundary element method program to perform
numerical calculations. The numerical results agree well with the analytical ones. # 1999 Elsevier Science Ltd. All
rights reserved.
Keywords: Piezoelectricity; Boundary element method; Harmonic functions
1. Introduction
For the equilibrium problem of transversely isotro-
pic media, Hu [2] obtained his solution by introducing
two potential functions F and c to express the displa-
cements. For the case of s16s2 (s1 and s2 are the eigen-
values of transversely isotropic materials), Hu [3] later
gave a general solution in terms of three harmonic
functions, and has solved a series of problems. Pan
and Chou [4] got the fundamental solutions for the in-
nite media for the cases of s16s2 and s1=s2 by apply-
ing the expressions of three potential functions. For
the equilibrium problem of piezoelectricity, Chen [5]
and Chen and Lin [6] expressed the innite body
Green's functions and their rst and second derivatives
as the contour integrals over the unit circle by using
the triple Fourier transform. Dunn [7] gave an explicit
solution for the Green's functions for an innite trans-
versely isotropic piezoelectric solid by taking the
Radon transform, coordinator transformation and
evaluation of residues in sequence. Lee and Jiang [8]
obtained a fundamental solution for an innite plane
by using the double Fourier transform. Lee and Jiang
[8] proposed a boundary element formulation based on
a weighted residual statement, giving a fundamental
solution for planar piezoelectric media. Lu and
Mahrenholtz [9] extended the variational boundary el-
ement model to piezoelectricity based on a modied
functional with six kinds of independent variables. But
none of them has performed numerical calculations.
Wang and Zheng [10] obtained a general solution in
terms of four harmonic functions in the case of distinct
eigenvalues. Ding et al. [1] have studied the general
solution systematically for transversely isotropic piezo-
electricity, and in the case of distinct eigenvalues, their
results are simplied and reduce to the Wang's sol-
ution [10], but the simplied general solutions for the
other two cases of multiple eigenvalues have not
appeared yet in the literature. In this paper, based on
the work of Ding et al. [1], we rst supplement these
results which are in terms of four harmonic functions.
Computers and Structures 71 (1999) 447455
0045-7949/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.
P I I : S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 2 3 7 - 5
* Corresponding author.
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As an application, we then obtain the fundamental sol-
ution of piezoelectric innite media for three cases by
giving three groups of expressions of the harmonic
functions. Finally we use those fundamental solutions
to complement a boundary element method programas well as perform numerical calculations. The numeri-
cal results agree well with the analytical ones.
2. The simplied general solutions
The constitutive relations of the transversely isotro-
pic piezoelectricity are
sx c11du
dx c12
dv
dy c13
dw
dz e31
df
dz
sy c12du
dx c11
dv
dy c13
dw
dz e31
df
dz
sz c13du
dx c13
dv
dy c33
dw
dz e33
df
dz
tyz c44
dv
dz
dw
dy
e15
df
dy
txz c44dudz
dw
dx
e15df
dx
txy c66
du
dy
dv
dx
Dx e15
du
dz
dw
dx
e11
df
dx
Dy e15dv
dz
dw
dy e11
df
dy
Dz e31du
dx e31
dv
dy e33
dw
dz e33
df
dz1
where si (tij), Di, u(n,w ) and f are the components of
stress, electric displacement, mechanical displacement
and electric potential, respectively; cij, eij and eij are the
elastic, piezoelectric and dielectric constants, respect-
ively. Ding et al. [1] have obtained the general sol-
utions for the basic equations of transversely isotropic
piezoelectricity as follows:
u dc0dy
m1L m2 d2
dz2
d 2Fdxdz
v dc0dx
m1L m2
d2
dz2
d 2F
dydz
w
c11e11L2
m3L
d 2
dz2 c44e33
d4
dz4
F
f
c11e15L
2 m4Ld 2
dz2 c44e33
d 4
dz4
F 2
where
L d2
dx 2
d 2
dy23
and the functions c0 and F satisfy the following
equations:2L
d 2
dz20
3c0 0 4
P3i1
2L
d 2
dz2i
3F 0 5
where zi=siz (i=0,1,2,3), and eigenvalues s20=c66/c44,
s 2i (i = 1,2,3) are the three roots of the following
equation
as6
bs4
cs2
d 0 6
where the coecients a,b,c,d are combinations of ma-
terial constants. Ding et al. [1] pointed out that F
admits the presentation
case a, when s1,s2,s3 are distinct: F F1 F2 F3 7
case b, when s1 T s2 s3:F F1 F2 z2F3 8
case c, when s1 s2 s3:F F1 z1F2 z21F3 9
where Fi (i=1,2,3) satisfy the following equations, re-spectively,2L
d 2
dz2i
3Fi 0, i 1,2,3 10
It will be very easy to nd a particular solution of Fi.
That means Fi may be obtained one by one. So Eq.
(2), the general solutions, must be simplied for each
case, respectively. For case a, Ding et al. [1] gave the
following simplied form:
u 3i1
dcidx
dc0dy
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v 3i1
dcidy
dc0dx
wm 3i1
aim dcidzi
m 1,2 11
where w1 represents the mechanical displacement com-
ponent w, w2 represents the electric potential, and the
coecients aim are given by
ai1 c11e11 m3s
2i c44e33s
4i
m1si m2s3i
ai2 c11e15 m4s
2i c44e33s
4i
m1si m2s3i
i 1,2,3 12
and the functions ci (i=0,1,2,3) satisfy2L
d2
dz2i
3ci 0 i 0,1,2,3 13
For cases b and c, the simplied results can be
obtained by substituting Eqs. (8) and (9), respectively,
into Eq. (2). After a series of work having been done,
the explicit forms ofci are found one by one for some
particular load cases to obtain the solutions. The next
section we describe the details.
3. The fundamental solutions for innite media
3.1. Solutions to the problem of combination of point
force P in z direction and point charge Q
This is an axisymmetric problem; assume c0=0 and
case a
ci Ai signz lnRi sijzj i 1,2,3 14
case b
ci signzBi lnRi sijzj i 1,2
c3 B31
R215
case c
c1 signzC1 lnR1 S1jzj
ci Ci
R1
i 2,3 16
where
Ri
x 2 y2 siz
2
qi 1,2,3
and Ai, Bi, Ci (i=1,2,3) are undetermined coecients.
Substituting Eqs. (14)(16) into the corresponding ex-
pressions of simplied general solutions yields the dis-placements and electric potential. Hence, by utilizing
the constitutive relations, the representations of the
stress components can be obtained. For case a, those
components are expressed as
u signz3i1
Aix
RiRi sijzj
v signz
3
i1
Aiy
RiRi sijzj
wm 3i1
aimAi
Ri
sx c11 c12 signz
3i1
Ai
41
RiRi sijzj
x 2
R3i Ri sijzj
x 2
R2
i Ri sijzj2 5
3i1
xiAizi
R3i
sy c11 c12 signz
3i1
Ai
41
RiRi sijzj
y2
R3i Ri sijzj
y2
R2i Ri sijzj
2 53i1
xiAizi
R3i
txy 2c66xy signz
3i1
Ai
41
R3i Ri sijzj
1
R2i Ri sijzj2
5
txm 3i1
o im AixR3i
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tym 3i1
o imAiy
R3i
sm 3i1
WimAiziR3i
17
where m = 1,2, s1,s2,tx1,tx2,ty1 and ty2 represent, re-
spectively, sz,Dz,txz,Dx,tyz and Dy; the coecients oimand Wim (i=1,2,3) are given by
o i1 c44si ai1 e15ai2, o i2 e15si ai1 e11ai2
Wi1 c33ai1 e33ai2si c13,
Wi2 e33ai1 e33ai2si e3118
Continuity of displacements u, v and stresses sx,sy,txyacross the interface z=0 implies that
3i1
Ai 0 19
Consider the equilibrium conditions of a layer between
two planes of z=2e, yielding
I
I I
I szx,y,e szx,y, e dx dy P 0 20I
I
II
Dzx,y,e Dzx,y, e
dx dy Q
0 21
Substituting Eq. (17) into Eqs. (20) and (21) yields
4p3i1
Wi1Ai P 0 22
4p3i1
Wi2Ai Q 0 23
The constants Ai(i=1,2,3) can be determined by Eqs.
(19), (22) and (23):
A1
PW22 W32 QW21 W31aDa
A2
PW32 W12 QW31 W11
aDa
A3 A1 A2
PW12 W22 QW11 W21aDa
Da 4p
W11 W31W22 W32 W21 W31W12
W32
24
For the other two cases, the undetermined constants
can be derived by the similar method:
B1 QW41 PW42aDb B2 QW41 PW42aDb
B3
QW11 W21 PW12 W22aDb
Db 4pW12W41 W41W22 W11W42 W21W4225
and
C1 0, C2 QW51 PW52aDc
C3 QW41 PW42aDc
Dc 4pW42W51 W41W5226
where
W41
c33a21 a41 e33a22 a42
s2
W42
e33a21 a41 e33a22 a42
s2
W51 c332a41 a51 e332a42 a52s1W52
e332a41 a51 e332a42 a52
s1 27
3.2. Solution to the problem of point force T in the x
direction
For case a, assume
c0 D0y
R0 s0jzj
ci Dix
Ri sijzji 1,2,3 28
for case b, assume
c0 H0y
R0 s0jzj
ci Hix
Ri sijzji 1,2
c3 signz H3x
R2R2 s2jzj29
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for case c, assume
c0 G0y
R0 s0jzj
c1 G1x
R1 s1jzj
ci signzGix
R1R1 s1jzji 2,3 30
where Di, Hi, Gi (i= 0,1,2,3) are undetermined con-stants. The components of displacement and stress can
be obtained by utilizing the simplied general solutions
and constitutive relations. For case a, they are:
u D0
41
R0 s0jzj
y2
R0R0 s0jzj2
5
3i1
Di
41
Ri sijzj
x 2
RiRi sijzj2
5
v D0xy
R0R0 s0jzj2
xy3i1
Di
RiRi sijzj2
wm signzx3i1
aimDi
RiRi sijzj
sx c11 c12D0x
41
R0R0 s0jzj2
2y2
R20R0 s0jzj3
y2
R30R0 s0jzj2
5
x3i1
Di
@xi
R3i c11 c12
43
RiRi sijzj2
2x
2
R2i Ri sijzj3 x
2
R3i Ri sijzj2
5A
sy c11 c12D0x
41
R0R0 s0jzj2
2y2
R20R0 s0jzj3
y2
R30R0 s0jzj25
x3i1
Di
@xi
R3i c11 c12
41
RiRi sijzj2
2y2
R2i Ri sijzj3
y2
R3i Ri sijzj2
5A
txy c66D0y
41
R30
2
R0R0 s0jzj2
4x 2
R20R0 s0jzj3
2x 2
R30R0 s0jzj2
5
2c66y3i1
Di
41
RiRi sijzj2
2x 2
R2i Ri sijzj3
x 2
R3i Ri sijzj2
5
txm o 0m signzD04 1
R0R0 s0jzj
y2
R30R0 s0jzj
y2
R20R0 s0jzj2
5
signz3i1
o imDi
41
RiRi sijzj
x 2
R3i Ri sijzj
x 2
R2i Ri sijzj2
5
tym o 0m signzD0xy
41
R30R0 s0jzj
1
R20R0 s0jzj2
5 signzxy
3i1
o imDi
41
R3i Ri sijzj
1
R2i Ri sijzj2
5
sm 3i1
Wim DixR3i
31
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where
o01 c44s0 o02 e15s0
xi c13ai1 e31ai2si c12 i 1,2,3 32
As before, the continuity of the components wm, txmand tym on z=0 yields
3i1
aimDi 0 m 1,2 33
o0mD0 3i1
o imDi 0 m 1,2 34
and the equilibrium condition givesII
II
txzx,y,h txzx,y, h
dx dy T
0 35
Substituting the 7th equation of Eq. (31) into Eq. (35)
yields
2pc44s0D0 2p3i1
o i1Di T 0 36
Eqs. (34) can be further reduced to a simple one with
the expression ofoim substituted:
3i0
siDi 0 37
Now the undetermined coecients Di (i=0,1,2,3) can
be derived from Eqs. (33), (36) and (37):
D0 Ta4pc44s0
D1 a21a32 a31a22TaDd
D2 a31a12 a11a32TaDd
D3 a11a22 a21a12TaDd
Dd 4pc44
s1a21a32 a31a22 s2a31a12
a11a32 s3a11a22 a21a12X 38
By the similar method, we have
H0 Ta4pc44s0
H1 a22a41 a21a42TaDh
H2 a11a42 a12a41TaDh
H3 a11a22 a12a21TaDh
Dh 4pc44
s1a22a41 a21a42 s2a11a42
a12a41 a12a21 a11a22
39
and
G0 Ta4pc44s0
G1 a41a52 a42a51TaDg
G2 a11a52 a12a51TaDg
G3 a12a41 a42a11TaDg
Dg 4pc44s1a12a51 a42a51 a11a52 a41a52 40
4. Boundary integral formulation
Letting S be the boundary of piezoelectric materialdomain, the boundary conditions of the basic
equations of piezoelectricity are given by
sijnj "t i on St
ui "u i on Su
and
Dini "o on So
f "
f on Sf 41
where ti represents the surface traction, o is the surface
charge, ni is the unit outward normal vector, and the
overbar indicates a prescribed value. The boundary
sets satisfy
St Su So Sf S,St Su So Sf X 42
Based on the extended Somigliana equation, the
boundary integral formulation is obtained:
Cu S
UtdS
S
Tu dS
V
Ub dV 43
where C is the coecient matrix, the general displace-
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ment u, surface traction t and body force b are
u
PTTR
u
v
w
f
QUUS, t
PTTR
txtytz
o
QUUS, b
PTTR
fxfyfzq
QUUS 44
and the two matrices composed by fundamental sol-
utions are
U
PTTR
u11 u12 u
13 f
1
u21 u22 u
23 f
2
u31 u32 u
33 f
3
u41 u42 u
43 f
4
QUUS,
T
P
TTRt11 t
12 t
13 o
1
t21 t22 t
23 o
2
t31 t32 t
33 o
3
t41 t42 t43 o4
Q
UUS
45
where u ij and t
ij (i,j=1,2,3) are, respectively, displace-
ment components and surface traction at a eld point
x in the xj coordinate directions due to a unit load act-
ing in one of the xi directions at a source point x on
the boundary, u 4j and t
4j (j= 1,2,3) are, respectively,
displacement components and surface traction at x in
the xj coordinate directions due to a unit electric
charge at x, f i and o
i (i= 1,2,3) are, respectively,
electric potential and surface charge at x due to a unit
load acting at x in one of the xi directions, and f
4 and
o
4 are, respectively, electric potential and surfacecharge at x due to a unit electric charge at x. In the
absence of body forces, Eq. (43) can be rewritten in
the form
Cxxxuxxx
S
Uxxx,xtx dS
S
Txxx,xux dS 46
If the boundary is in discretization with eight-node iso-
parameteric quadratic elements, Eq. (46) becomes
Cixxxuixxx ePS
8
i1
1
1
1
1
UtiNijJj dz dZ
ePS
8i1
11
11
TuiNijJj dz dZ 47
where Ni is shape function, J is Jacobian matrix, uiand ti represent, respectively, the general displacement
and surface traction at discrete grid, and Ci is a 4 4
coecient matrix, which by applying rigid-body
motion and constant electric potential, can be derived
from the basic equations and Eq. (47)
for nite media
Cixxx ePS
8i1
11
11
TNijJj dz dZ 48
for innite media
Cixxx I4 ePS
8i1
11
11
TNijJj dz dZ 49
where I4 is a 44 unit matrix. By utilizing above for-
mulae and fundamental solutions, we have performed
some numerical calculations. Table 1 gives the values
of material constants in our numerical examples.
4.1. Example 1
A piezoelectric column under uniaxial tension or
uniform electric displacement. Its size is aab. Two
Table 1
Material properties for PZT-4 and PZT-5H ceramics
PZT-4 PZT-5H
Elastic constants (N/m2) c11=12.61010 c11=12.610
10
c12=7.781010 c12=5.51010
c13=7.431010 c13=5.310
10
c33=11.51010 c33=11.710
10
c44=2.561010 c44=3.5310
10
Piezoelectric constants (C/m2) e31=5.2 e31=6.5
e33=15.1 e33=23.3
e15=12.7 e15=17.0
Dielectric constants (C/Vm) e11=6.463109 e11=1.5110
8
e33=5.622109 e33=1.3010
8
Eigenvalues s0=0.970261 s0=1.002828
s1=1.104405 s1=1.154188
s2=1.046767 s2=1.035093
+0.363468i +0.406737i
s3=1.046767 s3=1.0350930.363468i 0.406737i
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load cases are considered. Its boundary conditions are
given by
when z 2ba2:
txz tyz 0
sz p, Dz 0 load case 1
sz 0, Dz D0 load case 2
when z 0:
f 0
when x 2aa2:
sx txz Dx 0
when y 2aa2:
sy tyz Dy 0 50
If we assume
c0 0, ci
1
2ai
x2
y2
2z2i
i 1,2,3 51
the analytical results can be obtained by substituting
Eq. (51) into Eq. (11):
u s13 d31d33ak33 px
w
s33 d233ak33
pz
f d33ak33 pz 52
where [sij] (i,j=1,2,3) is the matrix of exibility factors,
the inverse matrix of elasticity constants [cij] (i,j =1,2,3), and
d31 e31s11 s12 e33s13
d33 2e31s13 e33s33
k33 e33 2e231s11 s12 4e33e31s13 e
233s33 53
In numerical calculation, we use six eight-node
quadratic elements, and set p=100N/m2, D0=1010 C/
m2. Because of the linear relationship, the comparison
between the analytical results and numerical ones isonly needed at an arbitrary node, say at node (a/2,a/
2,b/2). Table 2 lists those results. The ``decoupled''
items mean analytical results of a transversely isotropic
column, whose elastic constants and boundary con-
ditions are the same as PZT-4 or PZT-5H columns.
Table 2 shows that the absolute values of coupledcases are less than those of decoupled cases. In fact, by
further study of the constants listed in Table 1, it can
be veried that,
s11 s12 b 0, s13`0
d31`0, d33 b 0, k33 b 0 54
By this relation, Eq. (52) shows clearly that the piezo-
electric eects always cause the absolute values to be
less in the load case of uniaxial tension.
4.2. Example 2
Piezoelectric innity with spheroidal cavity. Assume
there are load cases applying at the innity: (1) uniax-
ial tension sIz , (2) all-around tension sr = sy = sI
r ;
and (3) uniform electric displacement Dz=DI
z . We use
242 nodes and 80 quadratic elements to perform the
numerical calculation. The stress and induction con-
centration at cavity surface are tabulated in Table 3.
The analytical results of PZT4 material are cited
directly from Kogan et al. [12], while the decoupled
results are calculated by Chen's solutions [11] and for
PZT-4 material.If we apply the load cases 1 and 3 only, the circum-
ferential stress component s ez at the equator and spy at
the poles are expressed by linear combination of our
numerical results
sez 1X870sIz 0X025D
Iz a10
10
sp
y 0X91sIz 0X127D
Iz a10
10 for PZT-4
material55
and
sez 1X9258sIz 0X0167D
Iz a10
10
sp
y 0X652sIz 0X0696D
Iz a10
10 for PZT-5H
material56
Assume only DIz is changeable in Eqs. (55) and (56).
If DIz =0, Eqs. (55) and (56) yield
sez
b
s
p
y 57
The position of the maximum principal stress locatesat the equator, but when DIz increases to a large
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enough amount, Eqs. (55) and (56) yieldsez `spy 58It can be concluded from Eqs. (57) and (58) that chan-
ging the electric loads can make the position of the
maximum principal stress move along the cavity sur-face.
5. Conclusions
The fundamental solutions for transversely isotropic
piezoelectricity are obtained in closed forms. Those
solutions are expressed very concisely by assuming the
expressions of the harmonic functions.
Through those numerical examples, the boundary el-
ement method and the fundamental solutions are veri-
ed. By comparing with the results of coupled
problems and decoupled problems, this paper shows
clearly that the piezo-electric constants have great in-
uence over the mechanical components.
Acknowledgements
This work was supported by the National Natural
Science Foundation of China.
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Table 2
Piezoelectric column
U/a (1010) w/b (1010) f/b
Load case Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
PZT-4 Numerical 1.160 0.589 4.115 1.32 1.321 0.00425
Analytical 1.164 0.559 4.117 1.321 1.321 0.00425
Decoupled 2.997 8.220
PZT-5H Numerical 0.784 0.291 3.557 0.716 0.7163 0.00227
Analytical 0.785 0.291 3.558 0.716 0.7162 0.00227
Decoupled 1.703 5.816
Table 3
Stress and induction concentration
Load case Position PZT-5H PZT-4 Ref.[12] Ref.[11]
1: sz/sI
z r=a,z=0 1.926 1.870 1.88 2.01
2: sr/sI
r r=0,z=a 2.431 2.885 2.75a 2.35
3: Dz/DI
z r=a,z=0 1.509 1.510 1.51
a
We get the value of 2.887 after recalculating the Kogan et al. [12] solution.
H. Ding, J. Liang / Computers and Structures 71 (1999) 447455 455