Archive of Applied Mechanics 72 (2002) 160±170 Ó Springer-Verlag 2002 DOI 10.1007/s00419-001-0199-0

Moving eccentric crack in a piezoelectric strip bonded to elastic materials S.-M. Kwon, H.-S. Choi, K.-Y. Lee

160

Summary The steady-state of a propagation eccentric crack in a piezoelectric ceramic strip bonded between two elastic materials under combined anti-plane mechanical shear and inplane electrical loadings is considered in this paper. The analysis based on the integral transform approach is conducted on the permeable crack condition. Field intensity factors and energy release rate are obtained in terms of a Fredholm integral equation of the second kind. It is shown for this geometry that the crack propagation speed has in¯uence on the dynamic energy release rate. The initial crack branching angle for a PZT-5H piezoceramic structure is predicted by the maximum energy release rate criterion. Keywords Moving Crack, Piezoelectric Material, Field Intensity Factors, Dynamic Energy Release Rate, Crack Branch

1 Introduction The in¯uence of the crack propagation speed on the crack tip ®eld was a popular subject in classical elastodynamics. The ®rst solution of a dynamic steady-state crack problem was given by Yoffe, [1], for a crack of constant length, moving with constant velocity in an in®nite solid. Thus, the crack had one leading edge, where the process is tearing, and one trailing edge, where the process is healing. Based on the criterion of maximum circumferential stress ahead of the crack tip, it was concluded that there is a critical velocity of about 0.6 times the shear wave velocity, at which the crack tends to curve and branch out. At lower velocities, the initial growth is expected to occur along the line of the crack. The ®rst solution for a mode III steady-state problem was given in [2], where a semi-in®nite crack under crack face loading was considered. The same problem was considered in [3] for the in-plane case. The results are analogous, and showed that the upper limit for mode III crack velocities, as expected, is the S-wave velocity. The increasing attention to the study of crack problems in piezoelectric materials has led to signi®cant works in the past decade. However, to the authors' knowledge, few elastodynamical researches have been conducted for piezoelectric materials. The mode III steady-state crack problem in an unbounded piezoelectric material was ®rst investigated in [4]. The result implied that the moving speed of the crack had no in¯uence on the stress intensities and electric displacement. In [5] an interface Yoffe crack problem was studied, and it was shown that the stress and electric displacement intensity factors are dependent on the crack speed. The above early efforts were based on the impermeable-crack assumption: the crack was treated as an impermeable slit and, thus, the electric ®eld inside the crack was assumed to be zero. Though this assumption can simplify some analysis and is shown to be valid for the problem of a nonslender hole, it may lead to erroneous results for crack problems, [6±8]. In reality, the crack can be considered to be a slit ®lled with air or vacuum, i.e. it may have some small electrical conductivity. Thus, the zero-charge equation of electrostatics in the boundary condition for an impermeable crack must be replaced by the equation of the continuity of electric charge, [9]. Particularly, since no opening displacement exists for an anti-plane problem, the crack surfaces can be in perfect contact. Accordingly, the permeable crack model is enforced in the current Received 23 January 2001; accepted for publication 18 October 2001 S.-M. Kwon (&), H.-S. Choi, K.-Y. Lee Department of Mechanical Engineering, Yonsei University, Seoul 120-749, Korea Fax: +82-2-2123-2813 e-mail: soonmankwon@hanmail.net

study, i.e. both the components of the electric displacement and the electric ®eld will be continuous across the crack faces. In this paper, we consider the problem for the steady-state moving eccentric crack in an in®nitely long piezoelectric ceramic strip sandwiched between two elastic materials under the combined anti-plane mechanical shear and in-plane electrical loadings. By using the integral transform techniques, the problem is reduced to the solution of a Fredholm integral equation of the second kind, which is obtained from two pairs of dual integral equations. The two crack intensity factors and energy release rate are derived by using the solution of a Fredholm integral equation. The branching direction is determined as the angle at which the energy release is maximized. Numerical results for the dynamic energy release rate are shown graphically for PZT-5H piezoelectric structures.

2 Problem statement Consider a Grif®th crack of length 2a moving eccentrically at a constant speed v in an in®nitely long piezoelectric strip sandwiched between two elastic materials with thick…e† nesses, h1 and 1, respectively. The structure is subjected to the combined mechanical and electric loads as shown in Fig. 1. A set of cartesian coordinates (X; Y; Z) is attached to the center of the crack for reference purposes. The piezoelectric ceramic strip poled with Z-axis occupies the region( 1 < X < 1; h2  Y  h1 † with 2h ˆ h1 ‡ h2 , and is thick enough in the Z-direction to allow a state of anti-plane shear. The uniform far-®eld shear stress, s1 , and uniform electric displacement, D0 , are applied. For convenience, we assume that the piezoelectric strip consists of upper (Y  0, thickness h1 ) and lower (Y  0, thickness h2 ) regions. The piezoelectric boundary value problem is simpli®ed under the out-of-plane displacement and the in-plane electric ®elds in the form …p†

…p†

…p†

uXi ˆ uYi ˆ 0; EXi ˆ EXi …X; Y; t†; …e†

…e†

uXj ˆ uYj ˆ 0;

…p†

uZi ˆ wi …X; Y; t† ; EYi ˆ EYi …X; Y; t†; …e†

…e†

uZj ˆ wj …X; Y; t†;

…1† EZi ˆ 0 ;

…2†

… j ˆ 1; 2† ;

…3†

where uk and Ek …k ˆ X; Y; Z† are displacement and electric ®eld components, respectively. Superscripts p and e stand for piezoelectric layer and two elastic regions, respectively. Subscripts i and j …i; j ˆ 1; 2† stand for upper and lower regions of piezoelectric layer, and upper and lower elastic materials bonded to a piezoelectric strip, respectively. For the problem of a crack moving at a constant velocity v along the X-direction, it is convenient to introduce a Galilean transformation such as

xˆX

vt;

yˆY ;

…4†

where …x; y† is the translating coordinate system attached to the moving crack.

Fig. 1. A piezoelectric ceramic strip with an eccentric moving crack bonded to elastic materials: de®nition of geometry and loading

161

In the transformed coordinate system, the dynamic anti-plane governing equations for the piezoelectric and elastic materials can be simpli®ed to the following forms

162

a2

o2 …p† o2 …p† w …x; y† ‡ w …x; y† ˆ 0; ox2 i oy2 i

…a  1† ;

…5†

b2j

o2 …e† o2 …e† w …x; y† ‡ w …x; y† ˆ 0; j ox2 oy2 j

…bj  1† ;

…6†

r2 wi …x; y† ˆ 0 ;

…7†

where

e15 …p† w ; d11 i

wi  /i

c44j ˆ

s c44j ˆ ˆ ; CT qj …e†

v Mˆ ; CT

Cj

…e† c44j

CTj

qj ˆ

;

l

v !2 u u M ; bj ˆ t1 Cj v u …e† r u c44j l …e† CT ˆ ; CTj ˆ t …e† ; q q

p a ˆ 1 M2 ;

…e† qj

q

;

…8†

j

l ˆ c44 ‡

e215

d11

;

…e†

and /i ; c44 ; d11 ; e15 and c44j are the electric potential, the elastic shear modulus measured in a constant electric ®eld, the dielectric permittivity measured at a constant strain, the piezoelectric …e† …e† constant, and the shear moduli of elastic materials, respectively. Also, wi ; CT ; CTj ; l; q and qj are the Bleustein function, [10], the speed of the piezoelectrically stiffened bulk shear wave, the shear wave velocities of the elastic materials, the piezoelectrically stiffened elastic constant, the piezoelectric material density and the elastic material densities, respectively. …p† …e† In terms of the independent variables wi ; wi and wj , the constitutive relations can be written as follows: …p†

…p†

skzi ˆ lwi;k ‡ e15 wi;k ; Dki ˆ …p†

d11 wi;k ;

…e†

…e†

…e†

skzj ˆ c44j wj ;k ; …k ˆ x; y†; …i; j ˆ 1; 2† ;

…9†

…e†

where skzi ; skzj and Dki are the stress and electric displacement components, respectively. The boundary conditions with the permeable crack model are written as …p†

syzi …x; 0† ˆ 0; …p†

…0  x < a† ; …p†

w1 …x; 0† ˆ w2 …x; 0†;

…10†

…a  x < 1† ;

…11†

Dy1 …x; 0‡ † ˆ Dy2 …x; 0 †;

…0  x < a† ;

…12†

Ex1 …x; 0‡ † ˆ Ex2 …x; 0 †;

…0  x < a† ;

…13†

/1 …x; 0† ˆ /2 …x; 0†; …p†

…p†

…a  x < 1† ;

…14†

syz1 …x; 0† ˆ syz2 …x; 0†;

…a  x < 1† ;

…15†

Dy1 …x; 0† ˆ Dy2 …x; 0†;

…a  x < 1† ;

…16†

…p†

…e†

syz1 …x; h1 † ˆ syz1 …x; h1 † ;

…17†

…p†

…e†

syz2 …x; h2 † ˆ syz2 …x; h2 † ; …p†

…18†

…e†

w1 …x; h1 † ˆ w1 …x; h1 † ; …p†

…19†

…e†

w2 …x; h2 † ˆ w2 …x; h2 † ; …e†

…20†

…e†

syz1 …x; h3 † ˆ syz2 …x; 1† ˆ s1 ;

…21†

Dy1 …x; h1 † ˆ Dy2 …x; h2 † ˆ D0 :

…22† 163

3 Solution Applying Fourier cosine transforms to Eqs. (5)±(7), we can obtain the results as follows: …p† wi …x; y†

2 ˆ p

2 wi …x; y† ˆ p …e† w1 …x; y†

…e† w2 …x; y†

Z1 ‰A1i …s†esay ‡ A2i …s†e

say

Šcos…sx†ds ‡ a0 y ;

…23†

0

Z1

‰B1i …s†esy ‡ B2i …s†e

sy

Šcos…sx†ds

b0 y ;

…24†

 cos…sx†ds ‡ c0 y ‡ e01 ;

…25†

0

2 ˆ p 2 ˆ p

Z1

 C1 …s†esb1 y ‡ C2 …s†e

sb1 y

0

Z1

D1 …s†esb2 y cos…sx†ds ‡ d0 y ‡ e02 ;

…26†

0

where Aji …s†; Bji …s†; Ci …s† and Di …s† …i; j ˆ 1; 2† are the unknowns to be solved, and a0 ; b0 ; c0 ; d0 and e0i …i ˆ 1; 2† are real constants, which will be determined from the far ®eld and interface loading conditions. The corresponding stress, electric displacement, electric potential and electric ®eld components are obtained in the form

…p† syzi …x; y†

2 ˆ p

Z1

s‰laA1i …s†esay

…e† syz2 …x; y†

…e†

2c b ˆ 441 1 p …e†

2c b ˆ 442 2 p

Dyi …x; y† ˆ

say

‡ e15 B1i …s†esy

e15 B2i …s†e

sy

Š

0

 cos…sx†ds ‡ la0 …e† syz1 …x; y†

laA2i …s†e

2d11 p

Z1

e15 b0 ;

 s C1 …s†esb1 y

…27†

C2 …s†e

sb1 y

 …e† cos…sx†ds ‡ c441 c0 ;

…28†

0

Z1

…29†

0

Z1 0

…e†

sD1 …s†esb2 y cos…sx†ds ‡ c442 c0 ;

s‰B1i …s†esy

B2i …s†e

sy

Šcos…sx†ds ‡ d11 b0 ;

…30†

e15 2 /i …x; y† ˆ d11 p ‡

164

2 p

Z1 ‰A1i …s†esay ‡ A2i …s†e

Šcos…sx†ds

0

Z1

‰B1i …s†esy ‡ B2i …s†e

sy

 Šcos…sx†ds ‡

0

e15 2 Exi …x; y† ˆ d11 p ‡

say

2 p

Z1

s‰A1i …s†esay ‡ A2i …s†e

say

e15 a0 d11

 b0 y ;

…31†

Šsin…sx†ds

0

Z1

s‰B1i …s†esy ‡ B2i …s†e

sy

Šsin…sx†ds :

…32†

0

By applying the far-®eld and interface loading conditions, Eqs. (17)±(22), the constants a0 ; b0 ; c0 ; d0 and e0i are evaluated as follows:

a0 ˆ

d11 s1 ‡ e15 D0 ; c44 d11 ‡ e215

e01 ˆ

e15 D0 s1 ‡ ld11 l

b0 ˆ !

s1 …e†

c441

D0 ; d11

h1 ;

c0 ˆ

s1 …e† c441

e02 ˆ

;

d0 ˆ

e15 D0 s1 ‡ ld11 l

s1 …e†

c442

;

s1 …e†

c442

!

…33†

h2 ;

It is convenient to use the following de®nitions:

A11 …s†

A12 …s† ˆ A21 …s†

A22 …s†  D…s† ;

…34†

B11 …s†

B12 …s† ˆ B21 …s†

B22 …s†  E…s† :

…35†

Using Eqs. (34), (35) and the two mixed boundary conditions Eqs. (10)±(14), we obtain the following two simultaneous dual integral equations:

Z1 sf …s†D…s† cos…sx†ds ˆ

p s1 ; 2 c

D…s† cos…sx†ds ˆ 0;

…a  x < 1† ;

0

…0  x < a† ;

…36†

Z1 …37†

0

 Z1  e15 s D…s† ‡ E…s† sin…sx†ds ˆ 0; d11

…0  x < a† ;

…38†

…a  x < 1† ;

…39†

0

Z1 0

 e15 D…s† ‡ E…s† cos…sx†ds ˆ 0; d11

where

 1 f …s† ˆ laQ…s† c c ˆ la

e215 ; d11

 e215 R…s† ; d11

…40† …41†

f12 e 2sah1 †… f22 f21 e 2sah2 † ; f11 f22 f12 f21 e 2sa…h1 ‡h2 †

1 e 2sh1 1 e 2sh2 R…s† ˆ ; 1 e 2s…h1 ‡h2 †

f11 ; f12 ˆ l a  c441 b1 k ; f21 ; f22 ˆ l a  c442 b2 ; Q…s† ˆ

kˆ

… f11

tanh‰sb1 …h3

h1 † Š ˆ

…e†

tanh…sb1 h1 † :

…42† …43† …44† …45† 165

From Eqs. (38) and (39), we can ®nd the following relation:

e15 D…s† : d11

E…s† ˆ

…46†

To solve Eqs. (36) and (37), let D…s† be expressed by another function X…n† in the form

p s 1 a2 D…s† ˆ 2 c

Z 1 p nX…n†J0 …san†dn ;

…47†

0

where J0 …san† is the zero-order Bessel function of the ®rst kind. Putting Eq. (47) into Eqs. (36) and (37), we can ®nd that the auxiliary function X…n† is given by a Fredholm integral equation of the second kind in the form

Z1 X…n† ‡

L…n; g†X…g†dg ˆ

p n ;

…48†

0

where 1 p Z h  s  L…n; g† ˆ ng s f a

i 1 J0 …sg†J0 …sn†ds :

…49†

0

Equation (48) can be degenerated to some special problems. The corresponding static solution is obtained by letting v ˆ 0. In what follows, the problem is reduced to the moving eccentric crack problem in a piezoelectric strip bonded to elastic half planes by letting h3 ! 1 . In cases of v ˆ 0, h3 ! 1 and the same elastic material properties, it is reduced to the result given in [11]. Also, it degenerates to the result of [12], if we neglect both the elastic materials and eccentricity.

4 Intensity factors and energy release rate The singular parts of the stresses and the electric displacements in the neighborhood of the crack tip can be written as  r    2    K T …v† r h1 e15 h p la ; …50† sin sxz ˆ sin 2 d11 r1 2 c 2pr    2   r   K T …v† r h1 e15 h ; …51† cos syz ˆ p la cos 2 d11 r1 2 c 2pr     K D …v† h K D …v† h Dx ˆ p sin ; Dy ˆ p cos ; …52† 2 2 2pr 2pr where

rˆ

q …x a†2 ‡ y2 ;

 y  ; x a  ay  : h1 ˆ tan 1 x a

h ˆ tan

q r1 ˆ …x a†2 ‡ …ay†2 ;

1

…53a† …53b†

Here, K T …v† and K D …v† are the dynamic stress intensity factor (DSIF) and the dynamic electric displacement intensity factor (DEDIF), respectively. These ®eld intensity factors are de®ned and determined as 166

K T …v†  lim‡

p p 2p…x a†syz …x; 0† ˆ s1 paX…1† ;

…54†

K D …v†  lim‡

p e15 T e15 s1 p K …v† ˆ paX…1† : 2p…x a†Dy …x; 0† ˆ c c

…55†

x!a

x!a

Following the results of [13] and [14], we can evaluate the dynamic energy release rate (DERR) G…v† by the aid of Eqs. (50)±(55) in the form

G…v† ˆ ˆ

la‰K T …v†Š2 2c2

‰K D …v†Š2 2d11

‰K T …v†Š2 2 s2 pa s2 paX2 …1† X …1† ˆ 1 X2 …1† ˆ  p1 2c 2c 2 l 1 M2

e215 d11

 :

…56†

The DERR can be expressed in terms of the DSIF and depends only on the resultant stress distribution generated by the mechanical deformation and the electromechanical interaction. To get X…1† in Eqs. (54)±(56), the Fredholm integral equation (48) is solved numerically by Gauss±Lagueere and Gauss±Legendre technique. The present solutions can be easily degenerated to those of an in®nite piezoelectric material, by considering h ! 1. In this case, the DSIF, DEDIF and DERR are given as follows:

p l s0 …1 D0 † pa; c44  T 2 K1 …v† G1 …v† ˆ ; 2c T K1 …v† ˆ

D K1 …v† ˆ

T e15 K1 …v† ; c

…57† …58†

where s0 is the pure mechanical shear stress at zero electrical load. The relation between the applied shear stress s1 and s0 can be found as

s1 ˆ

l s0 …1 c44

D0 † ;

…59†

where

D0

  c44 e15 D0 ˆ : ld11 s0

…60†

Thus, D0 is an electrical-to-mechanical load ratio, which illustrates the in¯uence of the dynamic electric load on the propagation of a crack. Examining Eqs. (57) and (58), the DSIF may have negative values when D0 > 1, but not the DERR. From the above results, it can be also observed that the DSIF does not depend on crack speed as in a purely elastic in®nite material. But it is noted that the DERR as a fracture criterion in piezoelectric materials does depend on crack speed v. If we consider mechanical terms only, i.e. e15 ˆ 0, Eqs. (57) and (58) are reduced to the purely elastic results, [15]. In [4], it was shown that the Mach number, M, had no in¯uence on both the DSIF and the DEDIF in an in®nite piezoelectric material. However, in the present solution, it is true for the DSIF in an in®nite cracked body but not for the DEDIF. This difference, due to using the electrically impermeable

crack face condition, contradicted experimental observations. On the other hand, when the geometry of the medium is such as the strip, the values of ®eld intensity factors of Eqs. (54) and (55) are dependent on both the ®nite geometry and the crack propagation speed v. Solving Eq. (56) with respect to M, the DERR can be shown to be positive when

0 < M < Md ˆ

r   e2 c44 l ‡ d1511 l

:

…61†

5 Numerical results and discussion In this Section, we consider PZT-5H piezoelectric ceramic, sandwitched in nickel and/or aluminum as elastic materials, which material properties are given in Table 1. For the PZT-5H piezoelectric ceramic, Eq. (61) yields Md ˆ 0:936. Figure 2 displays G … G…v†=G1 …v ˆ 0†† versus M for various material combinations in the interval M < Md . Here, G ˆ ‰…c44 =c†X2 …1†Š. It shows that G increases with the increase of M regardless of the material combinations. It is noted that the structure of Ni/PZT-5H/Ni has the lowest values of G among the combinations. In Fig. 3, we can observe that the values of G are smaller if the outer elastic layers are thicker in a structure of Ni/PZT-5H/Ni. Figure 4 shows G …e† versus e=h with the variations of a=h ˆ 0:1; 1:0 and 10.0, for the case of a=h1 ! 0. The trends  of G with the variations of e=h are dependent on the given material combinations and geometries. It also shows that a higher value of a=h gives a lower value of G . This trend is not always true, as shown in the static results, [11], but depends on the material combinations. To investigate the crack branching of the brittle electroelasticity, we use the criterion of maximum energy release rate. Using the polar coordinate system …r; h†, the ®eld intensity factors along the orientation h are

Table 1. Material properties of PZT-5H piezoelectric ceramic and elastic materials Material

q (103 kg/m3)

d11 (10)10 C/Vm)

c44 (1010 N/m2)

e15 (C/m2)

PZT-5H Ni Al

7.6 8.9 2.8

151 ± ±

3.53 8.01 2.65

17.0 ± ±

Fig. 2. G versus M for various material combinations

167

168

Fig. 3. G versus M with the varia…e† tions of a=h1 for a Ni/PZT-5H/Ni structure

Fig. 4. G versus e=h

K T …v; h† ˆ K T …v†H…h†;

h e15 T h K …v† cos ; K D …v; h† ˆ K D …v† cos ˆ c 2 2

…62†

where

  1 h1 h1 e215 h H…h† ˆ lar…h† cos h cos ‡ lR…h† sin h sin cos ; 2 2 d11 c 2 r r 2 r 4 1 ‡ tan h ; tan h1 ˆ a tan h : ˆ r…h† ˆ r1 1 ‡ a2 tan2 h Therefore, the DERR can be found at crack branching in the form

…63†

G…v; h† ˆ

la‰K T …v; h†Š 2c2

2

2

‰K D …v; h†Š ˆ G…v†F…h† ; 2d11

…64†

where

 1 F…h† ˆ laH2 …h† c

 e215 2h : cos d11 2

…65†

The critical Mach number, Mc ˆ 0:36, for PZT-5H ceramic is obtained by calculating the extreme values of F…h†. At lower Mach numbers, M  Mc , F…h† monotonically decreases with the increase of h, Fig. 5, the maximum value of the DERR G…v; h† occurs at the crack axis h ˆ 0 , and the direction of the crack growth is along the crack axis.

Fig. 5. F…h† versus h

Fig. 6. G…v†=s20 pa versus D0 for a Ni/PZT-5H/Ni structure

169

For the case of M > Mc , ®rst the function F…h† increases with the increase of h, and then decreases after it reaches a certain peak value. The angle hb corresponding to a peak value is the branching angle based on the criterion of maximum energy release rate. Furthermore, the higher is the crack speed, the larger is the branching angle. Finally, to illustrate the in¯uence of the dynamic electric load D0 on the propagation of a crack, a plot of G…v; h†=s20 pa versus h for a Ni/PZT-5H/Ni structure is shown in Fig. 6. It is noted that an increase of D0 decreases the normalized DERR.

170

6 Conclusions A Grif®th eccentric crack moving at constant velocity in a transversely isotropic piezoelectric ceramic strip bonded to two elastic materials under combined anti-plane mechanical shear and in-plane electrical displacement loads is analyzed by the permeable crack model and the integral transform approach. The traditional concept of linear elastic fracture mechanics is extended to include the piezoelectric effects, and the results are expressed in terms of the ®eld intensity factors and the dynamic energy release rate. For the case of the piezoelectric material with the surrounding geometries, the crack propagation speed has in¯uence on the dynamic stress intensity factor and dynamic energy release rate. The kinetic energy of the moving crack at high speed can change the propagation orientation of the moving crack. Also, it can be seen that the increase of the electrical-to-mechanical load ratio decreases the dynamic energy release rate. References

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