Stress intensification near an elliptical crack border

theoretical and applied fracture mechanics ELSEVIER

Theoretical and Applied Fracture Mechanics 22 (1995) 229-237

Stress intensification near an elliptical crack border Z.K. Wang *, S.H. Huang Department of Engineering Mechanics, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China

Abstract Analyzed in this work is the stress intensification around an elliptical crack in a piezoelectric medium with transverse isotropy. The electroelastic field equations in three dimensions are expressed in terms of the displacement components and electric potential. Two potential functions are introduced to obtain the solution for an elliptical crack subjected to uniform tractions and electric disturbance. The plane of transverse isotropy is parallel to the crack. Complex interactions of the local stresses and electric displacements are found, the variations of which around the crack border are expressed in terms of the stress intensity factor. Numerical results are obtained for four different peizo-ceramic materials; they are PZT-6B, PZT-2, BaTiO 3 and PZT-5H. Enhancement of piezoelectric properties is shown to reduce the crack border stress intensification.

1. Introduction Piezoelectric materials have a wide range of applications owing to their unique interplay between mechanical deformation and electric current. They are used as transducers and sensors in many fields of engineering associated with sonar projectors, fluid monitors, pulse generators and surface acoustic wave devices. An inherent weakness of piezoelectric ceramics lies in their brittle mechanical behavior. During the course of operation where mechanical and electrical effects are interchanged, they are vulnerable to cracking initiating from pre-existing defects such as small voids, inclusions and other discontinuities. The singular character of crack tip stresses and displacements has been studied [1,2] for piezo-

electric materials under in-plane extension and antiplane shear. A three-dimensional eigenfunction expansion was made in [3] to determine the singular character of a crack situated in the plane of transversely isotropic piezoelectric material. More general considerations of plane problems of piezoelectric material with defects have also been made [4]. Considered in [5] is the geometry of a penny-shaped crack in an axially symmetric material with transverse isotropy. This work is extended to the non-axisymmetric case of an elliptical crack, whereby the electroelastic equations are solved by the introduction of two potential functions.

2. Equations of electroelasticity

* Corresponding author.

The theory of electroelasticity requires that the stresses tr/j and electric displacements D i

0167-8442/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0167-8442(94)00061-1

230

Z.K. Wang, S.H. Huang / Theoretical and Applied Fracture Mechanics 22 (1995) 229-237

satisfy the following

Orij,j "~- 0

1

Di.i = O,

and

( 1)

where i, j = 1, 2, 3. Differentiations with respect to the coordinate variables x, y and z are inferred by the comma notation. The first and second expression in Eqs. (1) are, respectively, the equation of equilibrium in the absence of body forces and Maxwell's equation with free charges. The mechanical and electrical effects are coupled via the constitutive relations O'ij = Cijklff, kl --

e l¢ijE k

02 U

2(Cll

32 U

_

_

_

1

32 U -~- C44

c12) 0x2 + Cll 0y2

_

_

Oz2

02u

02w

+ ~-(C'I + C12) ~"-~ q- (C13 -{'-(-"44) 0y0~ Z oxoy a2q~

+(els +e31 )

=0

ayaz

(6)

02W 02W ] 02W (?44 OX----" ~- jr_ - 7 ] -}- C33 az 2

(2)

02u

02v )

-1- -OyOz q- ( C13 -{- C44) -OXOZ

and (3)

D i = eiklC.kl q- d i k E k .

(02(~

1

2W

(4)

Eij = ~(Igi, j -k lgj,i) ,

while the electric field E i is expressible in terms of the electric potential qb:

In Eqs. (2) and (3), c~jkl, eij k and dij stand for the elastic, piezoelectric and dielectric constants of the material. There are in general 45 independent constants: 21 for elasticity, 18 for piezoelectricity and 6 for dielectricity. For a transversely isotropic medium, this number reduces to 10:5 for elasticity, 3 for piezoelectricity and 2 for dielectricity. Referring to a system of Cartesian coordinates (x, y, z) with xy being the plane of transverse isotropy, Eqs. (4) and (5) may be inserted into Eqs. (2) and (3) and the results can then be put into Eqs. (1). This gives a system of four equations solving for u, v, w and ¢b. They are given in component form as follows: a2u 1 - 2 + f(Cllcu -Ox

02/2

32/,/

02(J~

02W t

02W

e,~ 7x~ + T 2 I + e~ 0~2 02/,/ 02U ) --]- - -

(5)

E i = -~l),i.

02(~ ]

+e,~ 0,---V+--9-~ I +e~3-7=0

The elastic strains % are related to the mechanical displacements u i as

+ (e15 + e31) 3xOz

3yOz

[ 02(~) 32(/) t 02(J) =0. - ",~I ~-gr + T I - d.,.~~ 7

3. Potential

function

formulation

Potential function formulation is well known for solving the system of equations in the classical theory of elasticity. Let the four unknowns u, v, w and q~ be expressed in terms of two potential functions 0(x, y, z) and )¢(x, y, z) such that u-

aq,

ax

00

Ox

Oy '

0x

v = Oy + -OX -

'

q 2 ) - Oy - 2 + c447z: w=klsz,

02U 02W + ~ { c . + c~2) ~ + (cl3 + c,4)

q~

ke~-z'

(7)

1

ax3z

02~

+(el5 +e31)

ayOz

=0

where k 1 and k 2 are constants. Substituting Eqs. (7) into (6), there results four equations:

321//

321// 02q, a x - - y + - - +2 a j O - f T = 0 a y

j=1,2,3

(8)

Z.IC Wang, S.H. Huang/ Theoreticaland Applied FractureMechanics 22 (1995) 229-237 and 021#4

021#4 0x---g-+ --Oy 2 +/~4 0"~ = 0

¢44 + (613 + C44)k1 + (el5 + e31)k2

A2= A3=

c33k 1 + e33k 2

e15 + e31 + elsk I -

dnk

(10)

2 '

2Can ~4

Cll -- C12

Nontrivial solutions of 1#j(j = 1, 2, 3) prevail only if the conditions (11)

a I =A2=A 3 =h

+ c33c44dll W c11e23

(c13 + c44)2d33 -Jr-(c24 --}-c11c33)d331//c11, J

(14) D = - (c44e23 + c11c33d33)/c11.

Cll

c13 + Can + c44kl + e15k 2 ' e33k 1 _ d33k 2

e31)¢44¢11

¢13 '}- c44

(9)

with X(X, y, z ) = 1#4(x, y, z). The quantities hi(i = 1, 2, 3, 4) are given by /~l---

els(el5 +

+

321#4

are satisfied so that ¢44 + (C13 + ¢44)kl + (e,5 + e3,)k2

The three roots of Eq. (13) are denoted by At (j = 1, 2, 3), where A1 is assumed to be a positive real number, and A2 and A3 are either positive real numbers or a pair of complex conjugate roots with a positive real part• Corresponding to At, there prevail the constants klj and k2j to be determined from Eq. (12). Hence, the solution has been reduced to finding the four functions 1#j (j = 1, 2, 3, 4) governed by 321#i

321#i 32 I//i 3x-----~ + - - 0 y2 +A i - 3 z2 = 0

i=1,2,3,4.

Now, the quasi-harmonic operator 32 32 32 32 32 g~2 = 3 7 + - + 3y2 - - +2 - - +3y -- 2 ai ~-7"2 0X

Cll

=a.

(12)

el5 + e31 + el5k 1 - d l l k 2

The constants k I and k 2 can be eliminated giving a cubic equation in A: AA3 +BA2 + CA + D = 0

(13)

in which A, B, C and D are given by A =e25

+c44d11,

B = (2e25c13 - c44e21 + 2e15e31c13 - 2elsclle33 +dnc23 + 2c13cand11 - c33Clldll

-c44Clld33)//c11, r C = ](el5 + e31)2c33 -- 2(c13 + c44)(e15 + e31)e33

32:2

321#i

321#i 321#i 3x-----T+--@T+ 3z~--.2= 0

i=1,2,3,4.

(17)

The variables z i are related to z as zi = Siz,

~i = 1//82

i = 1, 2, 3, 4.

(18)

The four potential functions 1#j are regarded as harmonic with reference to the coordinates (x, y, zj). Under the above consideration, Eqs. (7) expressed in terms of 1#j (j = 1, 2, 3, 4) take the forms 3 u = g(1#1 3 U=

31#4 + 1#2 + 1#3) -

3--7'

01//4 , 1#1 + 1#2 -{- 1#3) -1- - -3X

(19)

els(e15 + e31)c44c33 c13 --[-can

32

can be introduced so that Eqs. (8) can be written as

c13 --t-c44 + cank 1 + elsk 2 e33k 1 -- d33k 2

(15)

(16)

c33k 1 + e33k 2

=

231

W = kll

+ k12

31#3 + k13 3z '

232

Z.K. Wang, S.H. Huang / Theoretical and Applied Fracture Mechanics 22 (1995) 229-237

001

502

00 3

J¢

c]) = k 21TzZ + k 22~ -Z +k23 ~ The constants k~l and k21 are associated with /~/. The roots A 2 and A 3 are determined as a pair of complex conjugate numbers with the corresponding complex conjugates k~2 and kl3 and k22 and k23. It follows that 02 and 03 must be a pair of complex conjugate functions. Substituting Eqs. (19) into Eqs. (2) and using ~- instead of ~r for the shear stresses, the six stress components are found:

=

Cll OX2 + C12~ 2

020i

- ( C , l - c,2) ~:~- + (c,3kli + e3,~2i) a ~ , C12~

+ C11~

O',. =C13 ~

+ (e33klj -- d33k2j) ~-

020i

+ (c.kli + e~k~) az ~ .

020~ 0204 . = [C44(1 + k u ) +el5k2j] -OyOz - - - -]-C44 OZOX' Tvz (20) a20j 0204 'rzx = [c44(1 + k l j ) + elsk2j ] OZO~---X--C44blyOz '

-

cl2 ) 2

5204 ~ty2

5204 ] OX2 '

while Eqs. (3) yield the electric displacements

Dy=[els(l+klj)

The summation convention applies to Eqs. (20) and (21).

Let a transversely isotropic piezoelectric medium contain an elliptical crack with semi-axes a and b such that a > b as shown in Fig. 1. The crack lies in the xy-plane, being parallel to the plane of isotropy, while the z-axis is directed normal to the crack.

4.1. Boundary conditions Equal and opposite uniform pressure p and electric displacement q are applied to the upper and lower crack surfaces such that

(01 + 02 + 03)

D~= [e,.0 +k,~)-dHk~j] 0%

02tOJ az 2 •

4. Elliptical crack loaded mechanically and electrically

+ --0y2 (01 + 02 + 03)

020~ +(c33k u +e33k2j) az 2 ,

1

"02 32 ) D~ =e31 ~Ox + --~y2 (01 + 0 2 + 0 3 )

(01 + 02 -t- ~3)

0204

-(c1~- c~,) ~ -

Fig. 1. Schematic of elliptical crack.

(I//I + 4/'2 4-' 0q)

(J204

O'y =

A"

@ = - p,

Dz = q

x 2 y2 ~-~-+~-~<1,

z=0. (22)

0~04

azOx

e 15 Oy(}Z '

On the plane z = 0, the conditions

dllk2j] 020J ayaz

(1204 e~5 azax '

w=0,

(21)

q~=0

~',.:=r=, = 0

X2 y 2 ~-2+~-~>1

z=0,

Ixl <0% lYf <0% z = O

(23) (24)

Z.I( Wang, S.H. Huang / Theoretical and Applied Fracture Mechanics 22 (1995) 229-237

233

are to be enforced. All disturbances vanish as ~/X2 ..{_y 2 + Z 2 ...+ oo. That is

This implies that the functions 1]/1, 1[/2 and ~/3 a r e not independent, i.e.

~rx = % = cr~ = r~y = ryz = r~. = D . = Dy =D~ = 0.

nl~l

(25) Making use of Eqs. (16), % and D z in Eqs. (20) and (21) may be written as 2 ~/1

O'z

=Al-~-72v. 1 +A2

~2~02 0z--~ + A 3

Note that B~ are given by Bj = c44 + canklj + elsk2j,

B2

(34) satisfying

v?q,,(x,y,z,)

A j = --C13 + c33kljS 2 + e33k2jS 2,

while j = 1, 2, 3. (27)

Without loss of generality, ~04 may be set to zero on account of symmetry. Eqs. (22) give 0201

~}202

0203

X2

72

(28) while Eqs. (23) and (24) yield the respective equations kllSl ~

k1232002 +k13S3003 = 01 + 022 Og3 002

0~3

satisfy ~721]12(x,Y,Z2) = O,

~72~3(x,Y,Z3) = O.

(37)

Substituting Eqs. (34) and (36) into Eqs. (28), it is found on the crack plane for z = 0 that

(38)

02~t3 z=0 = Mlq - Nip OZ2 M1N 2 - MzN 1 '

where A1B3

M2 =A 3 - --, B1

C1B 2

N 1=C 2 - - , N B1

72

C1B3

2=C 3 - Bl

(39)

(29)

and 02 - - ( B l t b 1+ 8202 + B303) = 0 OyOz Ob2 - - ( B I 0 1 + B202 + B303) = 0 OxOz

(36)

qJ3(x,y,z3)

A1B2

1

a-~- + ~ - > 1, z = O.

(35)

and

M 1=A 2 - --, B1

k21SlO~l+k22S2~z2+k2383~z3=O; x2

qlz(X,y,z2)

= o,

02~02 z = N z p - M z q OZ2 =0 M1N2 - MeN1 '

~-+~
0201 0202 0203 c, 0z-~-+ c2 0z---~-+ c3 0232 - q

x,Y,Zl)

(26)

provided that

Cj = - e 3 1 + e33kljS 2 -- d33k2jS 2

(33)

B3

t~l( X , Y , Z l ) = -- -~l t~2( x , Y , Z l ) -- g ~ 3 (

02~//1 021~2 02~t3 D z = C , Oz---y1+ C 2 az--7- + C 3 az--'~

A1 Oz + A2 0z + A3 0z - = - P

j = 1, 2, 3.

The required potential functions are

02~t3

0z 2

(32)

[ z=O = - (B21]/2 + B 3 ~ 3 ) I z=O.

4.2. Elliptical coordinates

Ixl <%171 <°~, lz = 0,

(30)

a2(a 2 -- b 2 ) x 2 = (a 2 + ~ ) ( a 2 + r/)(a 2 + ~), b2(b 2 - a 2 ) y 2 = (b 2 + ~ ) ( b 2 + rl)(b 2 + ~'),

which gives ( Blab1 + B2~02 + B3~//3) [ z=0 -- 0.

It is expedient to introduce the ellipsoidal coordinates (~, r/, ~'), which are related to the Cartesian coordinates (x, y, z) as

(31)

a2bez 2 = ~:'q~',

(40)

234

Z.K. Wang, S.H. Huang / Theoretical and Applied Fracture Mechanics 22 (1995) 229-237

while sc, rl and ~" correspond to the roots v of the ellipsoidal equation

y2 + l'

X2

- -

a 2 + l:

Z2

+ b-7--~ +---

1=0

l'

(41)

Note that z = 0 and ~ = 0 correspond to points inside the ellipse ( x : / a 2 ) + ( y 2 / b 2 ) < 1, while z = 0 and r I = 0 to points outside the ellipse. According to [6], the solutions of Eqs. (35) and (37) satisfying Eqs. (38) are given by

01(x,y,zl)

( B2H2 B1

+

.

.

J~

B1

y2

Zl

2 ~,

×

v[Q( U1 )

k d v.~

X2 -a 2-+ +u k

y2

z2

b ~ -+- -u-a.- + - - t'- k

1

l

k=2,3,

142)

The crack border stress intensity factor K 1 can be found by referring to a system of local polar coordinates (r, 0) as indicated in Fig. 1 as provided in [6], i.e. x=a

where Q(t'k) = vk( az + v~)(b: + t,k)

(43)

and H~ (k = 2, 3) stand for ab 2 ( q M 2 _ p N 2 ) H2= 2E(k)(MIN2_M2N,

) '

ab2( p N 1 - qM l ) H3=

~

a2_b 2

~

(45)

Differentiation of Eqs. (42) involves the quantities 1 [~[ l J = 2J~i [

j = 1, 2, 3

x2 ~

- - + - - - l Y 2 z2 + b2+t'J

cos 4~+r cos 0 cos/3,

y = b sin 4~ + r cos 0 sin 3, (49) z = r sin 0, where 4~ is the angle in the parametric equations of the ellipse and 13 is the angle between the outer normal of the elliptical crack border and x-axis. The elliptical coordinates are related to (r, 0) by the relations

(44)

2 E ( k )( M, N2 - M 2 N , )

The complete elliptic integral of the second kind is denoted by E ( k ) w i t h the argument k =

(48)

4.3. Stress intensity factor

~

= ~g k

where u, snu, cnu, dnu are all elliptic functions, and E ( u ) is the elliptic integral of the second kind. With j = 2 and j = 3 in Eqs. (46) and making use of Eqs. (37) with the recognition that if ~ = 0 , then u 2 = u 3 = ' r r / 2 and hence E ( u l ) = E ( u 2 ) = E ( k ) and snu i c n u J d n u j = O , H 2 and H3 in Eqs. (44) are determined. Since

for z = 0 and zj = 0, it is seen from Eqs. (42) that the boundary conditions in Eqs. (29) are satisfied.

] - - ,

Ok(x,y,z~)

1

47,

+ t: 1

] d/,l

- - +---1 + b2+v, U1

sn,cnu,] dnuj '

,~lj z. r~ dv--2J i~z, - JJ< v j ~

B3H3) 1 f~[ a 2x2 -

21

ab 2 E(uj)

and - a 2 ~ < ~ ' 4 -b2~
--

whose derivatives are i'2I, 2~_}/2[~_,(a2b2- rlj~j) -a2b2('tlj + ~,)-(a2 +b2)'rlj~j] ~2~ a2b2(~j_Tij)(~)_~j)(a2 4_~2)l/2(b2 A_~2)l/2

t'i

]

dr) v"Q(t;I)

'

(46)

~ = (a 2 sin2q~ +

cos%),/2

cos

,

- 2abr sin ~ rI = (a 2 sin24~ + b 2 cos2&) L/2

(50)

= _ (a 2 sin 2& + b 2 cosZ&). The potential functions in Eqs. (42) may be inserted into Eqs. (26) with the help of Eqs. (49) and (50) to yield O"z = { ( a 2 sin2t~ q- b 2 c0s2(b)1/4}{ ~/2E(k)[A,(B3C 2 - B2C3)

+ BI(A2C 3_ A3C2)+CI(A3B 2_ A2B3)]

}-I

Z.BL Wang, S.H. Huang / Theoretical and Applied Fracture Mechanics 22 (1995) 229-237

235

Table 1 Elastic, piezoelectric and dielectric constants for piezo-ceramics Material

Elastic constants (10 l ° N m -e)

PZT-6B PZT-2 BaTiO 3 PZT-5H

•

Piezoelectric constants (Cm 2)

dielectric constants (10 - l ° F m I)

Cll

C33

C44

C13

C12

e15

431

433

dll

d33

16.8 13.5 15.0 12.6

16.3 11.3 14.6 11.7

2.71 2.22 4.4 2.3

6.0 6.79 6.6 7.95

6.0 6.81 6.6 8.41

4.6 9.8 11.4 17.0

-

7.1 9.0 17.5 23.3

36 87 128 277

34 40 150 301

0.9 1.9 4.35 6.5

p l i e d to E q . (51). T h e r e s u l t w h e n i n t o t h e s t r e s s i n t e n s i t y f a c t o r [6]

. AI[(BeC 3 - B 3 C z ) p + ( A 2 B 3 - A3B2)q]

K 1 = lim ~r~(x,

+ A2[(B3C 1 - B~C3) p + ( A 3 B I - AIB3)q]

substituted

y, 0)

(53)

r---~ 0

1

gives

× ~22 + A3[(B1C2 - B2CI)P

(a 2 sinZq5+ b 2 cos2~b)1/4 + (A1B 2 - A2BI)q]

(51)

cos?

K~ ~ E(k)[AI(B3C2 _ B2C3) +BI(A2C3 •

and

AI[(B2C 3 -B3C2)P + (A2B 3 -A3B2)(I ]

O z = {(a 2 sine* + b e cose~b)l/4} { v~-E(k)[ AI(B3C 2 - B2C3) +A2[(B3C1

+ BI(A2C 3 - A3C2 ) + Ci (A3B2 _ A2B3)] } - i . Vf~a " ( CI[( BeC3- B3Ce)P +( A 2 B 3 - A3B2)q] 1 × ~11 + C2[ ( B 3 C 1 - B I C 3 ) p + ( A 3 B 1 -

X~

1

A3C2) + CI(A3B2 -A2B3) ]

1 BIC3)P + (A3B 1 - A I B 3 ) q ] ~ s 2

+A3[(B1C 2 B2CI)P+(A1B2-A2BI)q]

AIB3)q ]

(54)

.

A similar expression can be defined for the electric d i s p l a c e m e n t i n t e n s i t y f a c t o r

+ C3[(B,C e - BeCI)p K ~ = lim ~ D z ( x ,

y, 0 ) .

(55)

r---~ 0

+ ( A I B 2 - AeB,)q]

cos T .

(52)

K n o w i n g f r o m E q s . (18) t h a t zj = S i z a n d z = r sin 0, t h e r e r e s u l t s rj = S i r w h i c h m a y b e a p -

P u t t i n g E q s . (52) i n t o (55), it is f o u n d t h a t (a 2 sin2~b+ b 2 cos2~b)1/a K~

E(k)[AI(B3C2 _ B2C3) + BI(A2C3

A3C2) + CI(A3B2 _ A2B3)] 1

"~(CI[(B2C3-B3C2)p+(A2B3-A3B2)q]~

Table 2 Values of Sj (j = 1, 2, 3) for piezo-ceramics Material PZT-6B PZT-2 BaTiO 3 PZT-5H

1 +C2[( B3CI - BIC3)P + (A3B1 -- AIB3)q] ~22

Parameter Sj in Eqs. (18) Si

$2

0.505 0.626 0.849 0.735

1.02 - 0.475i 0.738 - 0.225i 0.694- 0.410i 0.625 - 0.545i

$3 1.02 + 0.475i 0.738 + 0.225i 0.694 + 0.410i 0.625 + 0.545i

+C3[(BIC2-B2CI)p+(AIB2-A2BI)q]

.

(56)

Coupling between the mechanical and electrical e f f e c t is r e f l e c t e d by t h e p r e s e n c e o f b o t h p a n d q.

236

Z.K. Wang, S.H. Huang / Theoretical and Applied Fracture Mechanics 22 (1995) 229-237

5. Discussion of results 1.2 a/b

SS = 10

6

1,0

o >.

0.~

I ,=

O.E

i

0.4

z 0.2

t 0o

I

I 30 °

I

I 60 °

_1 900

Angle @

Fig. 2. Variations of normalized stress intensity factor with parametric angle for PZT-6B.

a/b = 10 1,6

Numerical values of K t in Eqs. (54) will be obtained for four different piezoelectric materials, namely PZT-6B, PZT-2, BaTiO 3 and PZT5H. Their corresponding elastic, piezoelectric and dielectric constants are listed in Table 1. Once the values of Aj (j = 1, 2, 3) are found from Eqs. (13), Eqs. (18) can be applied to obtain Sj (j = 1, 2, 3). They are given in Table 2 for the four different piezo-ceramics. Displayed graphically in Figs. 2-5 inclusive are plots of the normalized stress intensity factor K~ in Eq. (54) with the parametric angle ~b for the four different peizo-ceramics. As the angle ~h defined in Fig. 1 increases form 0 ° to 90 °, K~ tends to increase with the lowest value at the border intersecting the major axis to a maximum at the minor axis. The rate of increase becomes more pronounced for larger ratios of a/b corresponding to narrower elliptical cracks. This trend is the same as those in [6] for mechanical loads. The influence of piezoelectricity can be seen from the four sets of curves in Figs. 2-5. Amplitudes of K~ are the smallest for PZT-5H in Fig. 5, whereas its piezoelectric constants et5 and e33 and dielectric constants dl~ and d33 in Table 1 are largest.

1.4 ~'K 0.8

1.2

c

m

0.6 =

1,0

/

~

7 0.4

0.8

!

N

z

0.2

0.6

O '4100

Z i

1 300

1

I 60 °

I

90 °

Angle ¢,

Fig. 3. Variations of normalized stress intensity factor with parametric angle for PZT-2.

0o

310 o

6() o

"~0 o

Angle @

Fig. 4. Variations of normalized stress intensity factor with parametric angle for BaTiO3.

Z.K Wang, S.H. Huang / Theoretical and Applied Fracture Mechanics 22 (1995) 229-237

amplitude of mechanical and electrical load. The coefficients of the inverse square root singular stress and electric displacement field are found. They both varied around the crack border and increase from a minimum at the major axis to a maximum at the minor axis. The numerical data for four different piezoelectric materials suggest a reduction of the local stress intensification as the piezoelectric and dielectric constants are increased while the elastic constants remained nearly the same.

1.0

0.8

0.6

i ~

a/b = 10

0.4

°/,

i z

237

0.2

Acknowledgements 1

06

30 °

'

e~)o

90 °

Angle ¢

This work was supported by the National Science Foundation of the People's Republic of China.

Fig. 5. Variations of normalized stress intensity factor with parametric angle for PZT-5H.

References The elastic constants are about the same as the others. Except for the data in Fig. 3 for PZT-2, the curves in Figs. 2, 4 and 5 tend to suggest a reduction in K~ with increasing values of the piezoelectric constants.

6. Conclusions A formation based on potential functions is presented for solving the problem of an elliptical crack in a transversely isotropic piezoelectric medium. Coupling of the mechanical and electrical effect is reflected by the presence of both the

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