Green's functions for generalized 2D problems in piezoelectric media with an elliptic hole

Mechm-dcs Research Communications, Vol. 25, No. 6, pp. 685-693, 1998 Copyright © 1998 Elsevier Science Ltd Printed in the USA. All fights reserved 0093-6413/98 $19.00 + .00

Pergamon

PII S0093-6413(98)00088-3

GREEN'S FUNCTIONS FOR GENERALIZED 2D PROBLEMS IN P I E Z O E L E C T R I C MEDIA WITH AN ELLIPTIC HOLE

Cun-Fa GAO and Wei-Xun FAN Department of Aircraft, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, P.R.China

(Received 13 January 1998; accepted for print 15 July 1998) Introduction

The Green's functions play an important role in linear elasticity. For example, they can be used to construct many analytical solutions of practical problems. More importantly they are used as the fundamental solutions in the boundary element method (BEM) to solve some complicated problems. With increasingly-wide application of piezoelectric materials in the engineering, the study on the Green s functions of piezoelectric solids has received much interest. Recent development includes the works of Benveniste [1], Chen [2], Chen and Lin [3], Lee and Jiang [4], Dunn [5], Dunn and Wienecke [6], Akamatsu and Tanuma [7], Ding and Chert [8], Ding et al [9], and Gao and Fan[10]. To the authors' knowledge, however, these earlier studies were focused on a piezoelectric material without hole. In fact, it is more important to study the Green's functions of a infinite piezoelectric medium containing an elliptic hole. Since the presence of the hole has already been taken into account in the special Green s functions which satisfy the boundary conditions on the hole surface (zone of the high stress concentration), it is not necessary to consider any more these boundary conditions in BEM analyses, and therefore more accurate results can be obtained. More recently, Liu et al [11], and Lu and Williams [12] presented the Green's functions of a infinite two-dimensional piezoelectric material with an elliptic hole, respectively. In their analysis, however, the hole is assumed to be impermeable, i.e., the electric field inside the hole is neglected, so their results are not valid to the case when the hole is very slender or it degenerates into a crack [ 13,14]. In the present work, the Green's functions, which satisfy exact boundary conditions, for the 2D problem of a piezoelectric material containing an elliptic hole are studied based on the use of the Stroh s formalism combined with the method of analytical continuations. When the hole becomes a crack, the field intensity factors are obtained. It is shown that the impermeable boundary condition is not valid to the crack problem in piezoelectric media.

The Stroh's formalism

Consider a piezoelectric solid in a fixed rectangular coordinate system x~ 0=1,2,3). Assuming that all the field variables of the solid depend on x L and x 2 only, then a general solution for the twodimensional problem can be expressed as It 51

685

686

C.-F. GAO and W.-X FAN

.: 2Re[AS(=)],

: 2Re["I(:)]

(l)

where

.: [,,,,,,2.u,.+]', f(~) = [J](ZI),J2(Z2),S](z3)aS4(z4)] / ,

Zk :

(k:]-4)

)(,-t- p k x 2 ,

and Re stands for taking the real part; T denotes a transpose: u, (j = 1-3),(pand ¢, (k = 1 - 4 ) represent the displacement components, electric potential and generalized stress functions, respectively; fk (zk) (k=l~4) are complex potentials: A and B are the elastic matrices only related to the material constants; p~ (k=l~4) are the complex eigenvalues, which can be determined through solving the eigenvalue problem l~sl. In this paper, we assume that p~ are distinct. For this case, A and B are nonsingular, and there is the following orthogonality relation:

3

/

where the overbar indicates complex conjugate; I is a 4 x 4 unit matrix. The force and electric displacement boundary conditions can be written as

Bf +Bf = --

~tds.

t=(t,,t2,t3,D,,

)

1

(3)

where t~ (j=l~3) are the rectangular-coordinate components of force; D,, denotes the normal component of electric displacement; s stands for the arc-length Eqn (3) can be reduced to

where M = B i ~ , A = B J. For the sake of explicitness, eqn (4) can be expanded into 4

4

/:i

]:i

fk(z,)+ Z MksSs(zj)=~2 A,,tsds,

(k:l-4)

(5)

Once fk (zk) is obtained for the given boundary conditions, the field solutions can be determined by ~sl =-~bs,2, aj2 =4bi,i

(j=1,2,3)

Ol = -~b4.2, D2 = 04.1 , El = -(,°1, E2 = -~o:

(6) (7)

where a comma means partial differentiation; cr, D and E are stress, electric displacement and electric field, respectively.

The Green s functions

Consider an infinite piezoelectric solid with an elliptic hole L (see Fig.l). The piezoelectric solid is subjected to a line three (qlo,qz~,q3o) and a free line-charge q40 at an arbitrary point z o = xlo + iX2o. Additionally, the hole is assumed to be free of force and external charge.

PIEZOELECTRIC MEDIUM WITH ELLIPTIC HOLE

x2 !

687

qo

L zoY" - -

Fig.1

__

_J-

Xl

A piezoelectric solid with an elliptic hole

The complex potential inside the hole Let the electric potential ~oo(z) inside the hole be ~o(z) = 2 Re/o (z)

(8)

where fo(Z) is an analytic function. Then inside L, the electric field components (E°,E °) and electric displacement components (D°,D ° ) can be expressed as: E ° = -2 Re F o(z),

E ° = 2 Im F o(z)

D o = - 2 ~ o ReFo(z ),

D° = 2 c olmFo(z )

(9) (10)

where Im indicates the imaginary part; ~0 is the dielectric constant of air; Fo(z ) = dfo(z)/dz. Using Gauss' law, one has

~D°ds = ~D°d~, - D°dx2

(11)

Inserting eqn (10) into the right side ofeqn (11) gives

~D°ds = 2L"o Im/0(z )

(12)

Noting that the exterior of the ellipse L can be mapped onto the exterior of the unit circle y in the ~'o plane by Z((o)= Ro(g"o+mo~'o'),

Ro =(a+b)/2,

mo=(a-b)/(a+b )

then fo(Z) can be expressed inside L in the form of the Faber series as [)6]

L(~-o):

± .,,o(~-o,,+ m~-~,, )

(l,)

n:l

where fo(~'o)=/o[Z(Co)] ; a:' are complex coefficients to be determined.

The complex potential in the solid The complex potential in the solid can be written as fk(~_k)=qk ln(z k - zk0) * fk0(Zk)

(k=l~4)

(14)

where zko = x~o + pkX2o ; Ao(zk) is a holomorphic function in the zk -plane (z k -plane is obtained from z-plane by the affine transformation: z k = xj + pkx2 ) up to infinity; qk is a complex constant to be determined. To find qk, substituting eqn (14) into eqns (1)1 and (3)1, and then considering the force equilibrium condition and the single-valued conditions of displacement and electric potential, one can obtain 4~ Re[otq] = 0, 4~ Re[iBq] : qo (15)

688

C.-F. GAO and W.-X FAN where

qo=(qio,q20,q30,q40) ~ , q=(ql,q2,q3,q4) 7 Using eqn (2), one has from eqn (15) that 1

q=-2~ASqo

(16)

On the hole surface, the continuous conditions of force, electric displacement and electric potential require

, :

(o,o,o, ~,,o),

On

L

(i

7)

4

2Re~+A4,S}+(z+)=2Refo(z ) ,

On L

(18)

Inserting eqn (17) into eqn (5), and then using eqn ( [ 2) gives 4

]s<(zk)+ E M,]fs(z,) = 2a'oAk, lmfo(Z)+

(k=l~4)

(19)

7=1

Let us now introduce the lbllowing mapping function z~ ( 5 , ) :

:+(5+)= R,(5+ +,~+5;) + R~ =(~-++~)/2+ °,+ =(~++,b)i(a-+,b)

(20)

which transforms the exterior of the ellipse Lk in the z~ -plane into the exterior of a unit circle ,v in ~+ -plane. Then, eqn (14) can be expressed as

],(5,)= +, ~n[:,(¢~)-:,o(¢,o)] + ];o(5,1

(21)

where ] , ( 5 , ) = f~ [z, (5,)] ; J'k*~(5,) is holomorphic outside 7+ ; 5,o is the conformal point of z .... and ]5,0]>1 Using eqn (20), one has

5J<5,0 J

(22)

Considering that the second term on the right side of eqn (22) is a holomorphic function outside y , eqn (21) can be rewritten as

L(5,)=q,~n(5,

5,<,)+];o(5~),

5~r

.....

(23)

where J-,'o(5+) is a new holomorphic Ihnction outside y . Furthermore, extend L ( 5 , ) from outside y into inside y by way of

]*(Sk) =

M,J,(5~ )

5, ~Y,,,

(24)

/=1

Then, eqn (23) can be written after extension as

] k ( 5 k ) = qk In(5, - 5 , o ) - ~

M,/q~-ln(St: t - ~'joso)+ ],o(Sk) j~l

where ]+o(5+) is a hoiomorphic function in the whole 5, -plane except on y .

(25)

PIEZOELECTRIC M E D I U M WITH ELLIPTIC HOLE

689

On the hole surface, g-o = g-, = cr = e m . and eqn (19) together with eqns (24) and (13) leads to j~k+(t7)- j~k-(o-) = 2e-oA,4 I m ~ a,°,(cr" + m~cr-")

(26)

.fko(~',)= igoA,4 ~ ( a , ° 4 - a,~ ~"~k"

(27)

From eqn (26), one can obtain [171 n= i

Substituting eqn (27) into eqn (25) produces fk(g-k) = q, ln(g', - (ko) -- E M'sqJ In(g-7,' - (io) + i'oAk, j=l

a°rno' - a,° g'kk"

(28)

~=

On the hole surface, eqn (28) becomes 4

__

- -

f.',(o-)=q, ln(tr-g-,o)- X M,jqjln(o--i-g-)o)

+.

~

0

n

0

tcoA,4~,(a,,rao -a~,, cr

j=l

n

(29)

n:l

in which n

.

tn(cY-(,o) = ln(-g-,o)- Z / ( ~ _ ) ,

.

.

1

.

ln(ty-'-g-so) = ln(-(jo ) - ,,~-in

1

(30)

n = l /'l \ ~ k 0 /

To determine a'.~ in eqn (29), substituting eqns (13) and (29) together with eqn (30) into eqn (18), and then equating the coefficients of the same power tr " on both sides ofeqn (18), one obtains 0 • a,,m on (1 -'goPo) + a,,(I + iCoPo)= c°

(31)

where 4 4 A4rAr4, c,°, = 1 - - ~ A a ~ M ,

PO = r=l

1 ~ ...... qy(jo - - - ~ A 4 , q r ( , o

?1 r=I j = l

(32)

n r=]

Eqn (31) and its conjugal equation results in o

. o • ~ n o-,, - moC,, + l~o(Cnp o - moCnpo)

a~) =

(33)

A n

where

A.= (I-m~")(l + e2poP~o)- 2~o(I+ m~")Impo

(34)

Defining fl,, = a,,m o o ,, _ a o , one has from eqns (33) and (28) that =

n

0

0

2n

•

--

2,

(.) A n

)~,(g-k) = q, ln(g-k-g-,o)-~Mkj~ln(g-~k'-d,~o)+Aj.~Vfk

(36)

j=l

where v f , = i ~ o Z P,,(, n= I

On the other hand, eqn (36) can be expressed in the form of matrices as

(37)

690

C.-F. GAO and W.-X FAN 4

:(z) : Fo(Z)q - Z Fj(Z)B "' al tq

(38)

+VfB-'i,

i=1

where F 0 ( g ) : drag[ [n(~", - ~',0 ), l n(~2 - ~20 ), l n(~3 - ~"30 ), In(Ca - ~"40 )]

F,(Z) = dia~ln((i"- ~jo),ln((~ 1 - ~/o),ln(ff3' - ~lo),ln((?,' - ~ojo)]

i, : diag[ l,O,O,O] , 12 : diag[O,l.O,O], 13 = diag[ O,O,l,O] , 14 : diag[O.O,O,l] Vf : diag[Vf,,Vf2, V/3, Vf4] , /4 = (0,0,0,1)' The final solutions of the Green's functions can be obtained by substituting eqn (38) together with eqn (16) into eqn (1) as u= l lm[aFo(Z)hT~qo + --Im~{AE,(Z)B l 4 zr

t

~

7/"

¢j=llm{BFo(Z)A'}qo+

I-BIiA--7 }qo +2Re[AVfB

i=1

1-Im~'4 {BF,(Z)B '--BI,A-' 7/"

}qo +2 Re[BVfB

I/4]

(39)

'i4]

(40)

/=1

If the hole is not very slender (e.g. b / a > 10-2 ), the electric field inside the hole can be neglected [13], i.e. *:o can be assumed to be zero. In this case, eqns (39) and (40) become u = I lm{AFo(Z) Ar}qo

1 4 +--ImE[AFz(Z) B- t--aliA-r Iqo /r

i:11

q)=llm~aFo(Z)h:,~q ° +--ImZIBF,(Z)B I 4 ~r t l. :r

(41)

1

,--BI,A-r tqo

j:~ t

!

(42)

which are consistent with the results of Lu and Williams [12] using a semi-inverse approach.

The field intensity factor

When the elliptic hole degenerates into a crack along the x t axis. letting mo = 1. one has from eqns (34) and (35) that -

A, : _4~olmp~ ,

0

fl = time. e o lm Po

(43)

Inserting eqn (43) into eqn (37) results in V~/] = - T ~ 1 ~(Imc,°,~"~" lmpo

(44)

,,~1

From eqn (32)2, one has lmc: = - l m l • 2 A 4 , M , j q , ( j n

= 1 lm F/

o + 2 A4rq,¢,•'- ~ A4rq~rr;' + 254rq,~; o

Lr:l/= I

~2

Lr=l i=1

r=l

"44,M,jq,(:, + ~ A4,q,(: 2 r~l

Letting ,5.t be the Kronecker delta, then one has

\r=l

r=

(45)

PIEZOELECTRIC

MEDIUM

4

4

A4rqr~rr~

WITH

HOLE

691

4

= ~ ~ A4r (~rl)ql~llg' I=l r=l

r=l

= ZZA4, kJ=l

~'

]=l l=l kr:l __

4

Mrj = ~-'~ArtBo

qt(2

A,jBst

/=lr=l

Substituting eqn (46) and the identity

ELLIPTIC

I

into eqn (45) gives

I=1 1

4

4

4

. . . .

n

Imc,°, = - I m [ ~ Z ( Z A4rArj]B#q'(#o n LJ=i/=lkr= I ) 1

4

=-'~

n j=l

4

4

4

,'l

A4rArj]nflql~l; ]

+ZZ(Z ]=ll=lkr=l

J

)

~lm(~A,,Ar, qr,,,q,<;',o + ,,,q,c,d ] 4

4

/=i

. . . .

\r=l

#~

n

(47)

/

./L

Inserting eqn (47) into eqn (44), and then using

one can obtain

Vfk

=

Ir

(±

(

(

)

lm A..A~j B#qtln l-g~-ol~"~kI +Bjtqtln 1lmpo j=l 1=1 \r=l )k

)l

('~kI

(48)

J

Vfk

Eqn (48) shows that for the crack problem, is independent of c,,. This implies that the field solutions in piezoelectric media are not related to c,,.

Define

k r =(kTo,kD),k r =(kH,k,,kt11) Then, at the right crack tip ( x~ = a ), the field intensity factors can be expressed as

kj(a) = x/~ lim(z k -a)~ cr2j

0=1~4)

(49)

Using eqns (6), (7) and (1), eqn (49) leads to

kj(a)= 2~i2~),~oRef~.(zk -a)~es, aA(z.)i a~,

(50)

In (~. -plane, eqn (50) can be reduced to kja =2

~ ,

k

(51)

k /d(k

=

4

Noting

~.BjkAk4 = 814, one can obtain, by inserting eqn (36) into eqn (51), that k=l

kj(a) = 2

~r Re Z ~ +Bjkqk Z Z ( B , j : M , , ) - ~ - ~ - q#r ' "

Lk=l

bk0

I=1 k=l

i--~/0 j

, (j=l-3)

692

C,IF, GAO and W.-X FAN

=2/~Re[+

B,kqk

= 2,~-X[2Re,~-,

V~k

'--

q~ ]

B/,q, ]

(52)

~ 1-¢,oj

4 ~,_~,t + V °f ; k,,(a) = 2 l f *r ~ [ 2Re /~1

( 1) ]

(53)

Inserting eqns (48) and (32)1 into eqn (53) leads to

ko(a)=-

4~,Iml~-,d4,A,:l|ki(a) \ / i m~ A4rAra j=L ~=1

(54)

r=l

Eqn (54) shows that the intensity tactor of electric displacement depends on the stress intensity factors, while the latter can be obtained by using the following identity

and eqn (52). The result is o,

Similarly, one has at the left tip ( x~ = - a ) that

kj,-a ( ) =-2,--Re__Bt,q, ,/ • 1-

(j=l-3)

(56)

When a pair of self-equilibrium line-loads are applied, respectively, on both faces of the crack, the

solution can be expressed, by using eqns (55) and (15) and the superposition principle, as F

_ qjo ,/a+x~

(j=l-3)

(57)

kj(a) - 2,/7= w-x, When the piezoelectric medium is subjected to uniform loads (o-21,o-~>o-23 , Df ) at infinity, using eqn (57) one can easily obtain that

kz(a) = ~axa(cr~,,o';2,o';~) /

(j:l-3)

(58)

Eqns (54) and (58) show that the singularity of electric displacement is independent of the applied electric loads, while the result based on the impermeable crack assumption is k~ = ,~/~ D~ [18,19]. This means that the impermeable crack assumption may lead to erroneous results. Finally, it should be noted that if" one makes a coordinate transtbrmation: z" = z + a, then eqn (56) can rewritten as tr

4

Z;o - 2a

k,(-a)=-2J--Re~_Bj,q,(I-,~] va

where Zlo is a corresponding point to zm .

~:,

L

V

-'~0

0=1-3) )

(59)

PIEZOELECTRIC MEDIUM WITH ELLIPTIC HOLE

693

Letting a ~ oo in eqn (59), one obtains the solution for a semi-infinite crack which lies along the positive real axis. The result is 4

Bflqt

k,(-a) = -2 2~x ImZ[~/

0=1-3)

(60)

,:,~,/z,o) When the load is applied on the upper crack surface, eqn (60) with eqn (15) gives

kj(-a) =

qj0

0=1_3)

(61)

Eqn (61) shows that in this case, the stress intensity factors are not related to the material constants and electric load. Furthermore, it can also be seen from eqns (54) and (61) that k o is independent of the electric loads. Conclusion We derived the Green's functions of a piezoelectric medium with an elliptic hole using the Stroh's formalism combined with the method of analytical continuations. Since exact boundary conditions at the rim of the hole are used, the Green s functions are still valid to the case when the hole is very slender or it degenerates into a crack. Despite the mathematical complexities inherent to this problem, the present analysis is very straightforward and explicit. This is attributable to a combination of the Stroh's formalism with the Muskhelishvili's theory. This combination makes the Stroh's formalism more powerful and elegant in analyzing the generalized 2D problems of anisotropic media.

References

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