Constant moving crack in a piezoelectric block: anti-plane problem

Mechanics of Materials 33 (2001) 649±657

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Constant moving crack in a piezoelectric block: anti-plane problem Soon Man Kwon, Kang Yong Lee * Department of Mechanical Engineering, College of Engineering, Yonsei University, Seoul 120-749, Republic of Korea Received 10 May 2001

Abstract The problem of an anti-plane Grith moving crack in a rectangular piezoelectric ceramic block is solved by integral transform technique under the condition of continuous permeable electric crack faces. The far-®eld anti-plane shear mechanical and in-plane electrical loads are applied to the piezoelectric block. It is expressed as a Fredholm integral equation of the second kind. Expressions for the dynamic ®eld intensity factors and the dynamic energy release rate are obtained. The dynamic stress intensity factor and the dynamic energy release rate depend on the crack propagation speed. Based on the criterion of maximum energy release rate, the propagation orientation of a crack moving at constant speed is predicted. Numerical results for PZT-5H piezoelectric ceramic are also presented by graphical representations. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Moving crack; Piezoelectric material; Field intensity factors; Dynamic energy release rate; Crack branch

1. Introduction The study of fracture of piezoceramics is receiving considerable attention due to extensive use of piezoelectric materials in adaptive (smart) structures. Commercially available piezoceramics are rather brittle and susceptible to fracture. A thorough understanding of crack extension is imperative to the development of reliable piezoceramic actuators for advanced engineering applications. The increasing attention to the study of crack problems in piezoelectric materials in the last decade has led to a lot of signi®cant works. Especially, the in¯uence of the crack moving speed on the crack tip ®eld was a popular subject in classical elastodynamics. However, the results cannot be directly applied to piezoceramics due to intrinsic coupling between mechanical and electrical ®elds. The Yo€e crack problem in a piezoelectric material was ®rst investigated by Chen and Yu (1997). The result showed that the moving speed of the crack had no in¯uence on the intensities of the stress and electric displacement. Chen et al. (1998) studied an interface Yo€e crack problem, and showed that the stress and electric displacement intensity factors are dependent on the crack speed. But the above Yo€e crack researches have drawbacks. The impermeable boundary condition on the crack face was used. With this

*

Corresponding author. Fax: +82-2-2123-2813. E-mail address: [email protected] (K.Y. Lee).

0167-6636/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 0 1 ) 0 0 0 8 2 - 5

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S.M. Kwon, K.Y. Lee / Mechanics of Materials 33 (2001) 649±657

condition enforced, the electric displacement intensity factor depends on the electric load, and the energy release rate is always negative in the presence of electric loading only, irrespective of its sign. This contradicts available experimental observations (Tobin and Pak, 1993; Park and Sun, 1995); in reality, the crack may have some small electrical conductivity. Thus, the zero charge equation of electrostatics in the impermeable crack boundary condition must be replaced by the equation of the continuity of electric charge (Jackson, 1975). Particularly for an anti-plane problem, since no opening displacement exists, the crack surfaces can be in perfect contact. Accordingly, the permeable condition will be enforced in the current study, i.e., both the components of electric displacement and the electric ®eld are continuous across the crack faces. In this paper, we consider the problem for the Yo€e-type permeable crack in a rectangular piezoelectric ceramic block under the combined anti-plane shear and in-plane electrical loadings. Fourier transforms and Fourier series are used to obtain a pair of dual integral equations, which are then expressed into a Fredholm integral equation of the second kind. Numerical results for the dynamic energy release rate are shown graphically for PZT-5H piezoelectric ceramic.

2. Problem statements and method of solution Consider a Grith crack of length 2a moving in a rectangular piezoelectric block at a constant speed v under the remote loads as shown in Fig. 1. During the movement of crack, the length remains unchanged. The Cartesian coordinate system …X ; Y ; Z† is set at the center of the initial crack for reference. The piezoelectric material with the poling axis Z occupies the region, and is thick enough in the Z-direction to allow a state of anti-plane shear. For the steady state solution, we have uX ˆ uY ˆ 0; xˆX

vt;

uZ …X ; Y ; t† ˆ w…x; y†;

U…X ; Y ; t† ˆ /…x; y†;

y ˆ Y;

…1† …2†

where uk …k ˆ X ; Y ; Z†; U and …x; y† are the displacement components, the electric potential, and the translating coordinate system attached to the moving crack, respectively.

Fig. 1. Rectangular piezoelectric material with a moving Grith crack.

S.M. Kwon, K.Y. Lee / Mechanics of Materials 33 (2001) 649±657

651

In the transformed coordinate system, the dynamic anti-plane governing equations for piezoelectric materials can be simpli®ed to the following forms: a2

o2 w…x; y† o2 w…x; y† ‡ ˆ0 ox2 oy 2

…a 6 1†;

…3†

r2 w…x; y† ˆ 0; where aˆ

…4†

q 2 1 …v=CT † ;

w/

e15 w=d11 ;

CT ˆ

p l=q;

l ˆ c44 ‡ e215 =d11 ;

…5†

and c44 ; d11 , e15 and / are the elastic shear modulus measured in a constant electric ®eld, the dielectric permittivity measured at a constant strain, the piezoelectric constant, and the electric potential, respectively. Also, w; CT ; l and q are the Bleustein function (Bleustein, 1968), the speed of the piezoelectrically sti€ened bulk shear wave, the piezoelectrically sti€ened elastic constant, and the material density, respectively. In terms of the independent variables w and w, the constitutive relations can be written as follows (Li and Mataga, 1996): skz ˆ lw;k ‡ e15 w;k ;

Dk ˆ

d11 w;k

…k ˆ x; y†;

…6†

where skz and Dk …k ˆ x; y† are the stress and electric displacement components, respectively. The boundary conditions with the electric continuous permeable condition on the Yo€e-type crack faces are written as syz …x; h† ˆ s0 ;

Dy …x; h† ˆ D0

…0 6 x 6 b†;

…7†

sxz …b; y† ˆ 0

…0 6 y 6 h†;

…8†

Dx …b; y† ˆ 0

…0 6 y 6 h†;

…9†

Dy …x; 0‡ † ˆ Dy …x; 0 †

…0 6 x < a†;

…10†

Ex …x; 0‡ † ˆ Ex …x; 0 †

…0 6 x < a†;

…11†

/…x; 0† ˆ 0

…a < x 6 b†;

syz …x; 0† ˆ 0

…12†

…0 6 x < a†;

w…x; 0† ˆ 0

…13†

…a < x 6 b†;

…14†

where the geometric symmetry is considered and only a quarter plane is taken into account. Also s0 and D0 are the applied shear traction and electric displacement, respectively. The solutions of the displacement component w…x; y† and the Bleustein function w…x; y† of Eqs. (3) and (4) are given in terms of the following Fourier cosine transforms and Fourier sine series, respectively: Z 1 X 2 1 cosh‰as…h y†Š cos…sx† ds ‡ w…x; y† ˆ A1 …s† B1 …n† cosh … bx=ah† sin … by=h† ‡ a0 y; …15a† p 0 cosh…ash† nˆ0 w…x; y† ˆ

2 p

Z

1 0

A2 …s†

1 X cosh‰s…h y†Š cos…sx† ds ‡ B2 …n† cosh … bx=h† sin … by=h† cosh…sh† nˆ0

b0 y;

…15b†

where Ai …s† and Bi …n† …i ˆ 1; 2† are the unknowns to be solved, and a0 and b0 are the real constants which are determined from the far-®eld loading conditions. The parameter b is given by b ˆ …2n ‡ 1†p=2.

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S.M. Kwon, K.Y. Lee / Mechanics of Materials 33 (2001) 649±657

By applying the far-®eld loading conditions of Eq. (7), the constants a0 and b0 are determined as follows: a0 ˆ …s0 ‡ e15 D0 =d11 †=l;

b0 ˆ D0 =d11 :

…16†

From Eqs. (8) and (9), we have the following relations: Z 1 4 s B1 …n† ˆ A1 …s† sin…sb† ds; pah sinh…bb=ah† 0 s2 ‡ …b=ah†2 Z 1 4 s B2 …n† ˆ A2 …s† sin…sb† ds: ph sinh…bb=h† 0 s2 ‡ …b=h†2 Eqs. (10)±(14) give two pairs of dual integral equations in the forms: Z 2 1 s‰la tanh…ash†A1 …s† ‡ e15 tanh…sh†A2 …s†Š cos…sx† ds p 0 Z 1 1 X 4…b=h† sA1 …s† sin…sb† cosh…bx=ah† ‡l ds 2 pah sinh…bb=ah† s2 ‡ …b=ah† 0 nˆ0 Z 1 1 X 4…b=h† sA2 …s† sin…sb† cosh…bx=h† ‡ e15 ds ‡ c0 ˆ 0; 2 ph sinh…bb=h† s2 ‡ …b=h† 0 nˆ0 Z 1 A1 …s† cos…sx† ds ˆ 0; a < x 6 b; 0

 e15 s A1 …s† ‡ A2 …s† sin…sx† ds ˆ 0; d11 0  Z 1 e15 A1 …s† ‡ A2 …s† cos…sx† ds ˆ 0; d11 0

Z

1

…17† …18†

0 6 x < a;

…19† …20†



0 6 x < a;

…21†

a < x 6 b;

…22†

and c0 ˆ la0

e15 b0 ˆ s0 :

…23†

From Eqs. (21) and (22), we can ®nd the following relation: e15 A2 …s† ˆ A1 …s†; d11

…24†

and let A…s†  cA1 …s†;

c ˆ la

e215 =d11 :

To solve Eqs. (19) and (20), let A…s; p† be expressed by another function X…n† in the form: Z pc0 a2 1 p nX…n†J0 …san† dn; A…s† ˆ 2 0

…25†

…26†

where J0 …san† stands for the zero-order Bessel function of the ®rst kind. Putting Eq. (26) into Eqs. (19) and (20), we can ®nd that the auxiliary function X…n† is given by a Fredholm integral equation of the second kind in the form:

S.M. Kwon, K.Y. Lee / Mechanics of Materials 33 (2001) 649±657

Z X…n† ‡

1

0

where L1 …n; g† ˆ

X…g†‰L1 …n; g†

p Z ng

1 0

L2 …n; g† ‡ L3 …n; g†Š dg ˆ

s‰ f …s=a†

p n;

1ŠJ0 …sn†J0 …sg† ds;

653

…27†

…28a†

L2 …n; g† ˆ

1 l p X ~ ba† ~ pbh~2 ‰coth…bh= ng ac nˆ0

~ ~ 1ŠI0 …bhn=a†I 0 …bhg=a†;

…28b†

L3 …n; g† ˆ

1 e215 p X ~ b† ~ ng pbh~2 ‰coth…bh= cd11 nˆ0

~ 0 …bhg†; ~ 1ŠI0 …bhn†I

…28c†

f …s=a† ˆ h~ ˆ a=h;

1h ~ la tanh…as=h† c

i ~ ; …e215 =d11 † tanh…s=h†

b~ ˆ a=b;

…28d† …28e†

and I0 … † represents the modi®ed zero-order Bessel function of the ®rst kind. The obtained Fredholm integral equation is computed numerically by Gaussian quadrature integration technique.

3. Field 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   K r …v† r h1 h p l sxz ˆ sin ; …29a† …e215 =d11 † sin r1 2 2 c 2pr    r   K r …v† r h1 h syz ˆ p la cos ; …29b† …e215 =d11 † cos r1 2 2 c 2pr     K D …v† h K D …v† h Dx ˆ p sin ; Dy ˆ p cos ; …30† 2 2 2pr 2pr where

q r ˆ …x a†2 ‡ y 2 ;

r1 ˆ

h ˆ tan

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

1



y



; a  ay  h1 ˆ tan 1 : x a x

…31† …32†

Also K r …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 p p …33† K r …v†  lim‡ 2p…x a†syz …x; 0† ˆ s0 paX…1†; x!a

K D …v†  lim‡ x!a

p e15 r e15 s0 p K …v† ˆ paX…1†: 2p…x a†Dy …x; 0† ˆ c c

…34†

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S.M. Kwon, K.Y. Lee / Mechanics of Materials 33 (2001) 649±657

Evaluating the dynamic energy release rate (DERR) G…v† for the anti-plane case (Narita and Shindo, 1998; Pak, 1990), we obtain G…v† ˆ

la‰K r …v†Š2 2c2

2

‰K D …v†Š ‰K r …v†Š2 s20 pa 2 s20 pa X …1† ˆ X2 …1†: ˆ ˆ 2c 2d11 2c 2…la e215 =d11 †

…35†

The DERR can be expressed in terms of the DSIF and depends on the resultant stress distribution only generated by mechanical deformation and the electromechanical interaction. Using the polar coordinate system …r; h† de®ned at the right of crack tip (Fig. 1), we obtain the DERR when the crack branching initiates in the form: G…v; h† ˆ

2

la‰K r …v; h†Š2 2c2

‰K D …v; h†Š ˆ G…v†F …h†; 2d11

…36†

where

  1 e215 h laH2 …h† cos2 ; c 2 d11   1 h1 h1 e215 h laR…h† cos h cos ‡ lR…h† sin h sin cos ; H…h† ˆ c 2 2 2 d11 s r 4 r 1 ‡ tan2 h ; h1 ˆ tan 1 …a tan h†: R…h† ˆ ˆ r1 1 ‡ a2 tan2 h F …h† ˆ

…37† …38†

…39†

4. Numerical results and discussion We consider PZT-5H piezoelectric ceramic of which material properties (Pak, 1990) are given as follows: 2

c44 ˆ 3:53  1010 …N=m †;

2

e15 ˆ 17:0 …C=m †;

d11 ˆ 151  10

10

…C=V m†;

where N is the force in newton, C is the charge in coulomb, and V is the electric potential in volt. (1) To illustrate the in¯uence of the crack moving speed on the DERR, a Mach number as the ratio of the crack moving speed to the electroelastic shear wave speed, M ˆ v=CT , is introduced. Eq. (35) shows that the DERR may have the negative values at certain crack propagation speeds, and that it depends on the crack propagation speed and the material properties. Solving Eq. (35) with respect to M, the DERR can be shown to be positive when q 0 < M < Md ˆ c44 …l ‡ e215 =d11 †=l: …40† For a PZT-5H ceramic, Md ˆ 0:936. The DERR, G…v†, will increase rapidly and approaches to positive in®nity in the case when M approaches to Md . Figs. 2 and 3 show the variations of G…v†=G0 versus M for the cases of ®xed a=h and a=b, respectively, in the interval M < Md . Here G0 ˆ G…v ˆ 0† (for the details of G0 , see Kwon and Lee, 2000). The G…v†=G0 values increase with the increase of M for both the ®gures. The geometric e€ects, however, show opposite trends, and the e€ects of a=h on the DERR are greater than those of a=b. (2) The solution of an in®nite piezoelectric strip containing a central crack parallel to the strip edges …b ! 1† can be derived. In this case, the function X…1† is governed by

S.M. Kwon, K.Y. Lee / Mechanics of Materials 33 (2001) 649±657

655

Fig. 2. G…v†=G0 versus M with the variations of a=b.

Fig. 3. G…v†=G0 versus M with the variations of a=h.

Z X…n† ‡

0

1

X…g†L1 …n; g† dg ˆ

p n

…41†

because the functions L2 …n; g† and L3 …n; g† represented by Eqs. (28b) and (28c) are obviously vanished. Eq. (41) is the same as the result of Kwon et al. (2000). But their result of the DERR is not same as the present one since they used the mechanical part only of the DERR. (3) Letting h ! 1 in Eqs. (27), (28a)±(28e), considering ~ lim‰tanh…as=h† ~ h!0

~ 1Š ˆ lim‰tanh…s=h† ~ h!0

1Š ˆ 0;

…42†

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S.M. Kwon, K.Y. Lee / Mechanics of Materials 33 (2001) 649±657

~ ˆ ds; bh~ ˆ s, the solution of a piezoelectric strip with a central crack perpenand assuming limh!0 …ph† ~ dicular to its edges may be also obtained from Eq. (27) in the form: Z 1 p X…n† ‡ X…g†‰L4 …n; g† L5 …n; g†Š dg ˆ n; …43† 0

where

Z

L4 …n; g† ˆ

l p ng ac

L5 …n; g† ˆ

e215 p ng cd11

1

0

Z

~ s‰coth…s=ba† 1

0

~ s‰coth…s=b†

1ŠI0 …sn=a†I0 …sg=a† ds;

…44a†

1ŠI0 …sn†I0 …sg†:

…44b†

(4) The present solutions of the piezoelectric block can be easily degenerated to those of an in®nite material by considering X…1† ! 1 in cases of b ! 1 and h ! 1. The DSIF, DEDIF and DERR in an in®nite piezoelectric body are given as follows: p p e15 s0 pa r D K1 …v† ˆ s0 pa; K1 …v† ˆ ; …45a† la e215 =d11 G1 …v† ˆ

s20 pa : 2…la e215 =d11 †

…45b†

From the above results, it can be observed that the DSIF does not depend on crack speed as like in a purely elastic in®nite material. But, it is noted that the DERR does depend on crack speed v. If e15 ˆ 0, Eq. (45b) is also reduced to the purely elastic result of Freund (1990). Recently, Chen and Yu (1997) presented that M had no in¯uence on the DSIF and DEDIF in an in®nite piezoelectric material. However, in the present solution, it is true for the DSIF only in an in®nite cracked body but not for the DEDIF. On the other hand, when the geometry of the medium is such as the block, the values of ®eld intensity factors of Eqs. (33) and (34) are dependent on both the ®nite geometry and the crack propagation speed M. This di€erence is

Fig. 4. F …h† versus h.

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657

because of their using the electrically impermeable crack face condition which contradicts the experimental observations. (5) To investigate the crack branching of the brittle electroelasticity, we use the criterion of maximum energy release rate. By calculating the extreme values of F …h† of Eq. (37), we can get the critical Mach number, Mc ˆ 0:36 for PZT-5H. At lower Mach numbers M 6 Mc , F …h† monotonically decrease with the increase of h (see Fig. 4). The maximum value of the DERR G…v; h† occurs at the crack axis h ˆ 0°. Therefore, the initial growth is expected to occur along the line of the crack. In case of M > Mc , the function F …h† increases with the increase of h ®rstly and then decreases after it reaches a certain peak value. The corresponding angle hb to a peak value is the branch angle based on the criterion of maximum energy release rate. Therefore, there is a possibility for the crack to branch out toward the orientation hb . Furthermore, the higher the crack moving speed, the larger the branch angle.

5. Conclusions The electroelastic problem of a constant moving crack in a rectangular transversely isotropic piezoelectric ceramic under the combined anti-plane mechanical shear and in-plane electrical loadings is analyzed by the continuous permeable crack condition and integral transform approach. The ®eld intensity factors and energy release rate are derived. In a piezoelectric material with surrounding geometries, the moving speed of the crack have in¯uence on the dynamic stress intensity factor and dynamic energy release rate. The propagation of a crack at the high speed, v > Mc CT , brings about the branch phenomena in the piezoelectric media. Numerical results are shown graphically for PZT-5H piezoelectric ceramic.

References Bleustein, J.L., 1968. A new surface wave in piezoelectric materials. Appl. Phys. Lett. 13, 412±413. Chen, Z.T., Yu, S.W., 1997. Antiplane Yo€e crack problem in piezoelectric materials. Int. J. Fract. 84, L41±L45. Chen, Z.T., Karihaloo, B.L., Yu, S.W., 1998. A Grith crack moving along the interface of two dissimilar piezoelectric materials. Int. J. Fract. 91, 197±203. Freund, L.B., 1990. Dynamic Fracture Mechanics. Cambridge Press, Cambridge. Jackson, J.D., 1975. Classical Electrodynamics. Wiley, New York. Kwon, S.M., Lee, K.Y., 2000. Analysis of stress and electric ®elds in a rectangular piezoelectric body with a center crack under antiplane shear loading. Int. J. Solids Struct. 37, 4859±4869. Kwon, J.H., Lee, K.Y., Kwon, S.M., 2000. Moving crack in a piezoelectric ceramic strip under anti-plane shear loading. Mech. Res. Commun. 27, 327±332. Li, S., Mataga, P.A., 1996. Dynamic crack propagation in piezoelectric materials ± Part I. Electrode solution. J. Mech. Phys. Solids 44, 1799±1830. Narita, F., Shindo, Y., 1998. Dynamic anti-plane shear of a cracked piezoelectric ceramic. Theoret. Appl. Fract. Mech. 29, 169±180. Park, S.B., Sun, C.T., 1995. E€ect of electric ®eld on fracture of piezoelectric ceramics. Int. J. Fract. 70, 203±216. Pak, Y.E., 1990. Crack extension force in a piezoelectric materials. Trans. ASME J. Appl. Mech. 57, 647±653. Tobin, A.G., Pak, Y.E., 1993. E€ect of electric ®eld on fracture behavior of PZT ceramics. Proc. SPIE, Smart Struct. Mater. 1916, 78± 86.