Moving crack in a piezoelectric ceramic strip under anti-plane shear loading

Pergamon

Mechanics Research Communications, Vol. 27, No. 3, pp. 327-332, 2000 Copyright © 2000 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413100IS-see front matter

PII: S0093-6413(00)00099-9

MOVING CRACK IN A PIEZOELECTRIC CERAMIC STRIP UNDER ANTI-PLANE SHEAR LOADING ,long Ho Kwon, Kang Yong Lee" and Soon Man Kwon I~nt of Mechanical l~.n~neefing, Yonsei University, Seoul 120749, Korea * Corresponding author, Professor, Phone & Fax. : 82-2-361-2813, E-mail : fracture~yonsei.ac.kr

(Received 27 August 1999; accepted for print 27 April 2000)

Introduction The electro-mecl~aical behaviors of the piezoelectric materials have been studied by many researchers. However, dynamic crack propagating problem of the piezoelectric material in the viewpoint of theomical fracture mechanics has been less performed. Li and Magata [1,2] considered the anti-plane semi-infmite crack propa~ting problem in an unbounded hexagonal piezoelectric medium, and presented the solutions under the electrode and vacuum bounda~ conditions for the crack surface. They also reported that an acoustic surface wave may occur for certain orientations of piezoelectric materials, and the speed of a surface wave influences the crack propagation. Chen and Yu [3] investigated the anti-plane Yoffe's crack problem in an unbounded piezoelectric medium, and reported that the stress and electric displacement on the crack plane are independent of the crack speed under the electrically impermeable crack surface condition. Recently, Chen et al. [4] studied the problem of a finite Griffith crack moving along the imerface of two dissimilar piezoelectric half planes, and showed that the stress and electric displacement intensity factors are dependent on the crack speed under the electrically impermeable condition at crack surface. In the piezoelectric fracture problems, how to impose the electrical boundary conditions on the crack surface is commversial. Although the impermeable crack condition has been widely used in the previous works, the results under impermeable condition show ann-physical s i n g u ~ t y around the crack

and disagree with the experimental results [5,6]. Recently, Gao and Fan [7] presented that the normal components of electric displacement and the tangential components of electric field should be continuous across the crack surface became the real cracks in piezoelectric media are filled with vacuum or air. We investigate the crack problem of an infinitely long piezoelectric ceramic strip, which is transversely isotropic with hexagonal symmetry, containing a Griffith crack moving with constant velocity. The combined out-of-plane mechanical and in-plane electrical loads are appfied, and the continuous crack boundary condition [7] is considered. Fourier transforms are used to reduce the problem to two pairs of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. The freld intensity factors and energy release rate are determined, and numerical results for PZT-SH ceramic are obtained and discussed. The present solutions show the differences with those of the previous works.

Problem Formulation Consider a piezoelectric ceramic strip containing a Griffith crack moving with constant velocity subjected to the mechanical and electrical loads as shown in FIG. 1. A crack of length 2a is located at the center of thickness 2h and parallel to the edges of strip. The Cartesian coordinate (X, Y,Z) is fixed for reference. The piezoelectric medium is considered to be transversely isotropic with hexagonal symmeUy, which has an L~otropie basal plane of XY-plane and a poling direction of Z-axis. 327

328

J . H . K W O N , K. Y. LEE and S. M. K W O N

The piezoelectric boundmy value problem is simplified in the case of out-of-plane mechanical displacement and in-plane electricfields such that Ux = Ur = 0 , Uz = w ( X , ]I, t ) ,

(I)

E x = E x ( X , Y, t) , E y = E y ( X , Y, t) , E z = 0 .

In this case, the dynamic anti-plane governing equations of the piezoelectricity are simplified to

cq2

c , v 2 w ( X . Y. t ) + elsv2¢(X. ]i. t ) = p - ~ els~zZw(X, Y,t) - % ~2~(X,

where

v 2

= ~ a2 z" + - ~ a z

w(X, ]7. t) .

(2-a) (2-b)

Y,t) = 0 ,

is the two-dim~sional Laplacian operator, and

~(X, Y,t)

b the electric

potential. Also, ca, els, En, and p are the elastic stiffn¢~ of piezoelectric ~ measured in a constant electric field, the piezoelectric constant, the dielectric pcrmittivity me~ured at a constant strain, and the mass demity of the piezoelectric raaterial, respectively. For the problem of a crack moving with constant velocity v along the X direction, it is convenient to introduce a Galilcan tnu~onnalion such as x= X-vt,

y=

Y,

(3)

z= Z ,

(x, y,z) is the lranslating coordinate system attached to the moving crack. In the transformed coordinate system, Eqs. (2) become independent of the time variable t, and t l ~ may be rewritten as where

b~ ~_2 w ( x , y ) +

ax

v'{~(x,y)-

~ az w(x.y)

=

0

(4-a) ,

(4-b)

e_~,,, w(x,y)} = 0 ,

whem ~--1 i ~

,

C~ =

c ±e_J~ 44" ~11

=

C"~

,

V2

=

a2

a2

,

(5)

c r is the shear wave velocity and ~ is the piezoelectric stiffened elastic constant. In this case, the fundamental relations become r,i (x, y ) = c14~%i(x, y ) -- elsEi(x, y ) , E~(x, y) = - ~ (x, y), ~ ,

D i ( x , y ) = gl.syzi(x, y ) "b e u E i ( x , y ) ,

(6) (7)

wl~re r~.(x.y),

r=(x.y), E;(x,y) and D~(x.y) ( i = x . y ) are the components of the stress, strain, electric field and electric displacement vectors, respectively. Because of the symmetry in geometry and loading, it is sufficiem to consider only the q u a ~ r space resion defined by x > 0 and y > 0 . According to the electrically continuous crack surface condition of G-~ and Fan [7], the following boundary conditions am considered, r,~(x.0) = 0 . w(x.0) = 0 .

(Case 1)

D,(x.0 +) = D,(x.O-) .

E~(x.O ÷) = E,(x,O-)

r,y(x,h) = r0 , D y ( x , h ) = Do ,

(0
~(x,0) = 0

(Case 2)

7 , y ( x , h ) = 70 , E y ( x , h ) = Eo ,

(8) (9) (10-a)

M O V I N G C R A C K IN P I E Z O E L E C T R I C C E R A M I C S

(Ca.~ 3)

rzr(x,h) = r o , Ey(x,h) = E o ,

(Case 4)

329

~,,y(x,h) --- 7'o , Dy(x,h) = D o ,

(10-b)

where to, to, Eo, and Do are a uniform shear stress, uniform shear strain, uniform electric field, and

uniform electric displacement, respectively. Solution Procedure Applying Fourier cosine transforms to Eqs. (4), the solutions are formed as follows,

w(x, y) = zt fo (Al(s)exp(sfly) + A 2 ( s ) e x p ( - sBy)} cos(sx) ds + aoy , g

(ll-a)

'*

tb(x,y ) ='~11w(x,y ) +'~ff fo {Bl(s )exp(sY) + Bz(s ) e x p ( - sy) } cos(sx) d s - boY ,

(ll=b)

where Ai(s) and Bj(s) ( i = 1 , 2 ) are the unknowns to be solved. The real constant a0 and b0 will be determined from the edge loading conditions. The field components can be obtained from a simple p ~ , and then applying the bo~m~_~/ conditions, the unknown functions and real constants are dctermiI~d as follows, (Case 1)

ao = r° + ~~D °

(Case 3)

ao=

¢:'44

, bo = Do

r0 + elsEo c~ ' bo=

A](s) = e x p ( - 2 s B h ) A 2 ( s )

(Case 2)

~11

,

~ E o + e-l~-ro c~ e~l

ao-- ~'o , bo = E o + ~ ' o ,

(Case 4)

Bl(s) = exp(-2sh)B2(s)

(12-a)

/3. ao=7o , bo -- 'E- n' u '

.

(12-b) (13)

By Eq. (13), the two mixed boundary conditions of Eqs. (8) and (9) lead to two simultaneous &~al integral equations in the forms,

fo~s["~B(1-exp(-2sBh))A2(s)+els{1-exp(-2sh)}B2(s)]cos(sx)ds

fo~{1+exp(-2sBh)}A2(s)cos(sx)

ds = 0

= -~Co

(O~x
(14--b)

(a
fo~s['-~-(l +exp(-2sZh)}Az(s)+{l +exp(-2sh))B2(s)]sin(sx)ds = 0 [ ~'11

(O
fo~[ e-£t~-{l+ e x p ( - 2 s B h ) ) A z ( s ) + ( l +exp(-2sh)}Bz(s)]cos(sx)ds = 0

( a ~ x
[ Ell

where

(14-a)

co = c-'-~ao- elsbo .

9

(15-a) (15.-b) (16)

Equations (15) give the following relation,

Bz(s) = - el__.£ 1+ e x v ( - 2 s B h ) Az(s )

(17)

~11 l + e x p ( - 2 s h )

Using Eq. (17), the set of two simultaneous dual integral equations is reduced to the following forms,

fo°~s['d-~a{1-exp(-2s#h)} -e-~sen tanh(sh){1-exp(-2sh))]A2(s)cos(sx)ds=-~co fo~{l+exp(-2sBh)}Az(s)cos(sx)ds=O

( a ~ x
(0
330

J . H . K W O N , K. Y. LEE and S. M. K W O N

Equations (18) can be solved by the method of Copson [8], and the solution is found as follow, Az(s)

(19)

2 ~e1~B-e~s {l+exp(-2s/YO}

where ]0( ) is the zorn-order Bessel function of the first kind. Inserting Eq. (19) into Eqs. (18), the new function ~ ( $ ) should .~tisfy the Fredhoim integral equation of the second kind in the form, !

@(~e)+ fo L(~e'~/)~(~)d~/ = ~ ,

(20)

where

Extending the traditional concept of stress intensity factor to other field variables [9], the field intensity factors are defined and determined in the forms,

KV(v) ffi l i m ~ r ~ x - a ) r,y(x,O) = z--.a

el13

z c0vr~'~aO(1)

E44~ 11~ - - ~15

_

K~(V) = Km •--- l ~ ¢ ' - ~ x - a )

~

els(1-#)

(23) '

c0'/'~a~(1),

r,,(x, 0) = c0Vr'~~a~(1) ,

K ° ( v ) •ffi VLm'l"~x--a) D,(x,O) =

el~Ell

c0Vr-~a~(1) ,

(24) (25) (26)

where the (v) denotes the quantities for the crack moving with constant velocity, and g r ( v ) , g e ( v ) , K~(v), and K ° ( v ) are the strain intensity factor, electric field intensity factor, stress intensity factor, and electric displacmnem intensity factor, respectively. Park and Sun [11,12] suggested to use only the mechanical energy release rate term, considering a fracture criterion for the piezoelectric materials, because the fracture is a mechanical process and hence controlled only by the mechanical part of energy. Therefore, from Eqs. (23) and (25), the mechanical enelgy release rate for the crack moving with constant velocity becomes

G(v)(~ = .J~2Kr(v)K~(v) = ~'a2 c"~eit~-elZsell~Co2~2(1) .

(27)

The field imensity factors and mechanical energy release rate for an infinite piezoelectric material are obtained easily fmm Eqs. (23)-(27) since ~(1)--'1 as h--,oo in Eq. (20). Discussions

Chen and Yu [3] presented that the moving speed of the crack has no influence on the stress and electric displacemora intensity factors in an unbounded piezoelectric material. However, in the presem solution under the electrically continuous crack surface condition, other intensity factors except stress intensity factor of Eqs. (23)~(26) depend on B which is determined by the normalized crack speed v/cr. This difference is because they used the electrically impermeable crack surface condition which contradicts the experimental observation. Also, Shindo et al. [10] reported that the electric field intensity factor does not exist in the static crack problem in a piezoelectric ceramic strip with a kind of electrically

M O V I N G C R A C K IN P I E Z O E L E C T R I C C E R A M I C S

331

continuous crack ~rface condition However, in this moving crack problem, the electric field intemity factor e ~ t s and depends on the crack speed. In case of B = 1, the electric field intensity factor is vanished, and other field intensity factors are ~me as the corresponding terms of the static problem [10]. The PZT-SH piezoelectric ceramic strip is considered for the nmnerical analyses, and material properties are given in TABLE I. TABLE 1. Material propc~ies of PZT-5H c~m-,ic Ela~c ~ffnem Piezoeleclric congant

c~ els

3.53×101° (N/m2) 17.0 (C/mz)

Dielectric pennittivity Cri~cal energy release rate

•n Go,

151.0× 10- 10 ( C~Vra) 5.0

The solution ~ ( g ) of Eq. (20) can be mnnerically calculated by using Cmussian quadralnre. The normalized energy release rate G(v)Cm/G~(v) (t~ with the variance of the nornlalized crack length a/h is shown for various nonmli~d velocity V/Cr in FIG. 2. G(v)¢m/G,(v) (~ increa~ with the increuse of a/h and v/cr. Also, G(v)(~/G~(v) (~ approaches to unity with the decrease of a/h for all v/cr. FIG. 3 shows the mechanical energy release rate G(v)(~/G~, with a variance of the norm~li~d crack speed v/cr for the various normalized crack length a/h in Cases 1 and 3. G(v)(m/G~, increases with the increase of a/h and v/cr. FIGs. 4 and 5 show the mechanical energy release rate G(v)(m/G~, with the variance of the applied uniform electric field E0(Ca~ 2) and uniform electric displacement D0(Case 4) under a uniform shear strain 70 for varions v/cr, respectively. The mechanical energy release rate can have a minimnm value with the variation of electrical loadings, and iuc~ses with the increase of v/cr in Cases 2 and 4. The similar results with FIGs. 2 - 5 are obtained under other loading cases, and the numerical solutions of static case for v/cr=O agree with the results of Shindo et al. [10].

Conclusions A theoretical analysis was performed for a C a ~ t h crack moving with constant velocity in a PZT-SH piezoelectric ceramic strip, with electrically continuous crack surface condition, under the combined out-of-plane mechanical and in-plane electrical loads. Fourier mmsforms were used to reduce the problem to two ~ of dual integral equations, which were then expressed to a Fredholm integral equation of the second kind. The field intensity factors were determined theoretically. Only the mechanical energy release rate was considered in the fracture criterion for the crack moving with constant velocity in a piezoeleclric material. The mechanical energy release rote increases with the increase of the normalized crack length and the normalized crack speed, and can have a minimum value with the variation of electrical loadings. The results in case of zero normalized crock speed are consistent with the previous static solutions.

Acknowledl~ements This work was supported by the Brain Korea 21 Project, and the authors are grateful for the support provided by a grant from the Korea Science & Engineering Foundation (KOSEF) and Safety and Suuctural Inlegrity Research Center at the Sungkyunkwan University.

References [1] [2] [3] [4] [5] [6]

S. S. Z. Z. T. C.

Li Li T. T. Y. F.

and P. A. Magma, J. M~h. Phys. Solids 44(11), 1799 0996) and P. A. Maga~ J. M ~ K Phys. Solids 44(11), 1831 0996) Chen and S. W. Yu, In~ L Fract. 84, L41 (1997) Chen, B. L. ~ o o and S. W. Yu, Int. J. Fract 91, 197 0998) ZImag and P. Tong, lnt. J. Solids Stm~. 33, 343 (1996) Gau and W. X. Fan, Int. J. Solids S m u t 36, 227 (1999)

332 [7] [8] [9] [10] Ill] [12]

J.H. KWON, K. Y. LEE and S. M. KWON C. E. Y. Y. S. S.

F. Crao and W. X. Fan, InL J. Eng. Sci. 37, 347 (1999) T. Copson, Proc. Glasgow Math. Assoc. 5, 19 (1961) E. Pal(, ASME J. Appl. Mock 57, 647 (1990) Shindo, K. Tanaka and F. Narita, Acta Mech. 120, 31 (1997) B. Park and C. T. Sun, J. Am. Ceram. Soc. 78(6), 1475 (1995) B. Park and C. T. Sun, Int. J. FracL 70, 203 (1995) ~'0

® V

~0

or

®

®

®

.=L= (-a,o)

x

(a,0)

@

2h

x

@

@

tftfttf

FIG. 1 A piezoelectricstripwith a moving crack

D O or . . . .

I

I

. . . .

- -

vr~.oo

..... -- " --

V/C'r = 0 2 V/CT=0.4

-

-

-

--

v

. . . .

E o

-

- -

I

12

. . . .

/~

V/Gr=0.8

I

Io 8

=.. /

-

6

--

>

-

Wh : 0.5 = ~ = 1.0 Wh = 1.5 -

,,

}

I,,

,,

I// / s / / '

4

[,

0.5

,,

1

hi

h

,

,

0

r

1

PZT-5H

20

(CaN

I 2)

b (9

/

15

\

~o

V/CT = 0.0 V/Or = 0.2

-

-

--

--

--

-

v~=o4

t

V/CT = 0 B -

-2xl1

,'

I

-

~

-

\

8

O.g

i

i /-

-

..... ---

-

V/Or

= 0.0

V/~V/Or V/Cr ViCT

= = = =

0.2 0.4 0.6 0.8

/

/

/

\

0

/

2x1~

4x105

G(v)
\.,

L

/~

6x10 s

-2x10 ~

Eo (V/m)

FIG. 4

/

(Cue 4) R=0.01(m),=dh-I, 7o---6.2xi04

/

\

/

PZT-SH

\

V/C.r = 0.8

\

I"

0.0

r

,

\ - .....

0.3

]

tt-0.01(m), full-l,7 o = 9 . 5 x l 0 "s

'

""

V/CT FIG. 3 G(u)(U)/G~, vs. u/cr (Cases l&3)

C(v)(~/G=(v) <~ vs. a/h

25

~

L

1.5

a/h

FIG. 2

/Ji

~/n=2.0

2

,*,

:I I

(3

/"

ol

llr

=fn = 0.0

..... ---

b (9

d

lI

:I

P Z T - 6 H ( C a ~ 1 z 3) * = 0.01 (m), % =2.8 x I 0 ' O,~'/m=) -

/

V/Ca = 0.8

f

vs. Eo (Case 2)

0

..I

2 x 1 0a

4 x 1 0 "3

0 x l 0 "s

/.~

8 x 1 0 "s

Do (C/m=) FIG. 5

G(v)(~/G,.,. vs. D O (Case 4)