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Periodic cracks in a functionally graded piezoelectric layer bonded to a piezoelectric half-plane
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Theoretical and Applied Fracture Mechanics 49 (2008) 313–320 www.elsevier.com/locate/tafmec
Periodic cracks in a functionally graded piezoelectric layer bonded to a piezoelectric half-plane S.H. Ding a, X. Li b,* b
a Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China Department of Mathematics and Computer Science, Ningxia University, Yinchuan 750021, China
Available online 16 February 2008
Abstract Studied is the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material. The properties of the functionally graded piezoelectric strip, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. The effects of the periodic crack spacing, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Functionally graded piezoelectric material; Periodic group cracks; Impermeable; Permeable
1. Introduction Piezoelectric materials (PMs) have been widely used as transducers, sensors and actuators because of their intrinsic electro-mechanical coupling behavior. Piezoelectric bimorphs are a special type of piezoelectric devices that consist of two long and thin piezoelectric elements. The stress peaks will be induced at the interfaces to cause failure such as cracking or debonding. The principal disadvantage of piezoelectric materials is that the bonding agent may crack at low temperature and creep at high temperature. These drawbacks may lead to lifetime limitation and restrict the use of piezoelectric actuator that require high reliability. To improve the reliability and durability problems arising largely from high residual and thermal stress, poor interfacial bonding strength, the functionally graded piezoelectric materials (FGPMs) as a new class of advanced composites have been developed. The concept of fracture mechanics on a finite
*
Corresponding author. E-mail address: [email protected] (X. Li).
0167-8442/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2008.02.002
crack in a strip of functionally graded piezoelectric material (FGPM) has been applied in [1]. They found that the singular stresses and electrical displacements at the tip of the crack in the FGPM carry the same forms as those in a homogeneous piezoelectric material (PM) but the magnitudes of the intensity factors are dependent on the gradient of the FGPM properties. This discover makes us to research the problem of FGPM by using the methods which investigates the problem of piezoelectric material. The dynamic anti-plane problem for a functionally graded piezoelectric strip containing a central crack vertical to the boundary is considered [2]. Both the impermeable and permeable cases are considered. They employed integral transforms and dislocation density functions to reduce the problem to Cauchy singular integral equations. The propagation problem of an anti-plane moving crack in a function graded piezoelectric strip is solved [3]. Investigated in [4] is the anti-plane crack and collinear crack problems in FGPM. A class of functional forms has been assumed to describe the mechanical and the electrical properties of the medium. For the permeable crack, the stress and the electrical displacement intensity factors depend only on the applied mechanical loads. The piezoelectric effect has
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no effect on the stress intensity factors. A variety of different crack problems in functionally graded piezoelectric medium have been studied [5–10]. As described in the foregoing, although a variety of challenging issues related to certain crack problems in the functionally graded piezoelectric materials have been addressed, one of the important mechanics problems here is the determination of the length parameter describing the surface crack periodicity. To date, the problem of periodic array of cracks [11,12] in the isotropic and anisotropic material is studied. The anti-plane shear problem for periodic cracking in functionally graded coatings has been considered [13–15]. Fracture by periodic cracking under electrical–mechanical loading has drawn increasing attention in both the academic and industrial communities. Reference can be made to the contributions presented in [16–19]. In the paper, we considered the problem of a FGPM strip containing a periodical array of parallel cracks bonded to a homogeneous piezoelectric material. The focus of this paper is the development of a mathematical model to predict the length scale for the spacing of transverse cracks that form in a functionally graded piezoelectric medium subjected to a coupled electro-mechanical external loading condition. The crack surface condition is assumed to be electrically impermeable or permeable. Integral transform techniques are used to obtain a system of singular integral equations, which are then solved numerically. Numerical results are presented for the stress intensity factor, the energy release rate and the energy density factor. 2. Formulation of the problem Fig. 1 shows a functionally graded piezoelectric strip and a piezoelectric material perfectly bonded together along x = h. The functionally graded piezoelectric strip is assumed to contain periodic cracks perpendicular to the interface. The crack of length and the x-coordinate of the crack center are defined as 2a0 = b a and d = (b + a)/2, respectively. All materials exhibit transversely isotropic behavior and poled in the z-direction, and the anti-plane
mechanical field and inplane electric field are coupled. The constitutive equation can be written as owk o/ þ e15k ðxÞ k ; ox ox owk o/k þ e15k ðxÞ ; syzk ¼ c44k ðxÞ oy oy owk o/ D:xk ¼ e15k ðxÞ e11k ðxÞ k ; ox ox owk o/k e11k ðxÞ ; Dyk ¼ e15k ðxÞ oy oy
sxzk ¼ c44k ðxÞ
ð1Þ
ð2Þ
where sizk, wk, Dik and /k (i = x, y, k = 1, 2) are the shear stresses, anti-plane displacements, inplane electrical displacements and electric potentials, respectively, while subscripts k = 1,2 refer to the FGPM strip 1 and the homogeneous PM 2. The variations of material constants c44k(x), e15k(x) and e11k(x) called the shear modulus, piezoelectric constants, and dielectric constants, respectively, are assumed in the following exponential forms; c441 ðxÞ ¼ c0 expðbxÞ;
e151 ðxÞ ¼ e0 expðbxÞ;
e111 ðxÞ ¼ e0 expðbxÞ; 0 < x < h; c442 ðxÞ ¼ c0 expðbhÞ; e152 ðxÞ ¼ e0 expðbhÞ;
ð3Þ
e112 ðxÞ ¼ e0 expðbhÞ; x > h;
ð4Þ
where b is called nonhomogeneous parameters. The constants c0, e0, and e0 are the material properties at x = 0. The static equilibrium equation and Maxwell’s equation under electro-static condition are given as osxzk osyzk þ ¼ 0; ox oy
oDxk oDyk þ ¼ 0; k ¼ 1; 2: ox oy
ð5Þ
By separating the homogeneous solution through an appropriate superposition, the problem may be reduced to a perturbation solution in which self-equilibration crack surface tractions are the only nonzero external loads. For the problem described in Fig. 1, it has to be solved under the following mixed boundary conditions: w1 ðh; yÞ ¼ w2 ðh; yÞ; /1 ðh; yÞ ¼ /2 ðh; yÞ; sxz1 ðh; yÞ ¼ sxz2 ðh; yÞ; Dx1 ðh; yÞ ¼ Dx2 ðh; yÞ; sxz1 ð0; yÞ ¼ 0; Dx1 ð0; yÞ ¼ 0; w2 ðx; 0Þ ¼ 0; /2 ðx; 0Þ ¼ 0; w1 ðx; cÞ ¼ 0; /1 ðx; cÞ ¼ 0; w2 ðx; cÞ ¼ 0;
/2 ðx; cÞ ¼ 0;
ð6Þ
and the boundary conditions on the crack surfaces can be written as syz1 ðx; 0Þ ¼ sðxÞ; w1 ðx; 0Þ ¼ 0;
Dy1 ðx; 0Þ ¼ DðxÞ; a < x < b;
ð7Þ
/1 ðx; 0Þ ¼ 0; 0 < x < a; b < x < h;
ð8Þ
for the impermeable case, and syz1 ðx; 0Þ ¼ sðxÞ;
Fig. 1. The geometry of periodically cracked FGPM strip bonded to a homogeneous PM.
Dy1 ðx; 0Þ ¼ Dc ðx; 0Þ ¼ DðxÞ;
a < x < b; w1 ðx; 0Þ ¼ 0; 0 < x < a; b < x < h;
ð9Þ ð10Þ
/1 ðx; 0Þ ¼ 0; 0 < x < h;
ð11Þ
S.H. Ding, X. Li / Theoretical and Applied Fracture Mechanics 49 (2008) 313–320
for the permeable case, where Dc(x,0) denotes the electric displacement of the space of the crack itself.
kðx; uÞ ¼ F 1 ðx; uÞ þ F 2 ðx; uÞ þ F 3 ðx; uÞ; Z 1 1 i m2 expðm1 cÞ m1 expðm2 cÞ þ jaj F 1 ðx; uÞ ¼ 2 1 a expðm1 cÞ expðm2 cÞ
3. Singular integral equations
expðiaðu xÞÞ da;
Consider the electrically impermeable case. By using Fourier transform method, we can obtain 8 w1 ðx; yÞ > > > R1 > 1 > ¼ 2p ½A ðaÞ expðm1 yÞ þ A2 ðaÞexpðm2 yÞ expðiaxÞda > > 1 1 > > 1 > P > > > þ ½C 1k ðck Þ expðp1k xÞ þ C 2k ðck Þ expðp2k xÞ sinðck yÞ; < k¼1
F 2 ðx; uÞ ¼
1 pX ½F 2j expðcj ðu þ xÞÞ; c j¼1
F 3 ðx; uÞ ¼
1 þ Oðu þ xÞ; uþx
F 2j ¼
ð12Þ
> /1 ðx; yÞ > > R1 > > 1 > ¼ 2p ½B ðaÞ expðm1 yÞ þ B2 ðaÞexpðm2 yÞ expðiaxÞda > 1 1 > > > 1 P > > > þ ½D1k ðck Þ expðp1k xÞ þ D2k ðck Þ expðp2k xÞ sinðck yÞ; :
> > : /2 ðx; yÞ ¼
k¼1 1 P
þ
ð13Þ
k¼1
where m1 ¼ m2 ¼
The integral equations (16) and (17) will be solved for different crack types, that is, for the internal crack (a > 0) and surface crack (a = 0) problem. It is to be point out that for the internal crack, the function g1(u) and g2(u) must fulfill the single-valuedness as Z b Z b g1 ðuÞ du ¼ g2 ðuÞ du ¼ 0: ð19Þ a
ð14Þ
The solutions of undetermined function Ai(a), Bi(a), Cik(ck), Dik(ck), E2k(ck) and F2k(ck) (i = 1, 2) depends on the mechanical and electrical conditions of crack surfaces. Define two dislocation function g1(x) and g2(x) ( ow1 ðx;0Þ ; a < x < b; ox g1 ðxÞ ¼ ; 0; x < a; x > b; ( o/1 ðx;0Þ ; a < x < b; ox : ð15Þ g2 ðxÞ ¼ 0; x < a; x > b; Substituting Eqs. (12)–(15) into (6)–(8), from Eq. (7) it follows that: Z 1 b 1 þ kðx; uÞ g1 ðuÞ c0 syz1 ðx; 0Þ ¼ sðxÞ ¼ expðbxÞ p a ux 1 þ e0 þ kðx; uÞ g2 ðuÞ du; ð16Þ ux Z 1 b 1 þ kðx; uÞ g1 ðuÞ Dy1 ðx; 0Þ ¼ DðxÞ ¼ expðbxÞ e0 p a ux 1 e0 þ kðx; uÞ g2 ðuÞ du; ð17Þ ux where
expðp2j xÞc2j ½expðp1j ðu hÞÞ þ expðp1j h p2j uÞ: kj p1j D2 ð18Þ
F 2k ðck Þ expðck xÞ sinðck yÞ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b a2 þ iba; p1k ¼ kk ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b b2 p2k ¼ þ kk ; k ¼ þ c2k ; ck ¼ kp=c: 2 4
expðp1j xÞc2j ½p2j ðp1j þ cj Þ expðp1j ðu hÞÞ kj p1j D2 þ p1j ðp2j þ cj Þ expðp2j ðu hÞÞ
k¼1
8 1 P > > < w2 ðx; yÞ ¼ E2k ðck Þ expðck xÞ sinðck yÞ;
315
a
For the internal crack problem and surface crack problem, we define the following normalized quantities: bþa bþa bþa bþa rþ ; u¼ sþ ; 2 2 2 2 f 1 ðsÞ ¼ g1 ðuÞ; f 2 ðsÞ ¼ g2 ðuÞ;
ð20Þ
Kðr; sÞ ¼ kðx; uÞ; sðrÞ ¼ syz1 ðx; 0Þ expðbxÞ; DðrÞ ¼ Dy1 ðx; 0Þ expðbxÞ;
ð21Þ
x¼
then (16) and (17) become Z 1 1 1 þ Kðr; sÞ f1 ðsÞ c0 sðrÞ ¼ p 1 sr 1 þe0 þ Kðr; sÞ f2 ðsÞ ds; sr Z 1 1 1 þ Kðr; sÞ f1 ðsÞ e0 DðrÞ ¼ p 1 sr 1 e0 þ Kðr; sÞ f2 ðsÞ ds: sr
ð22Þ
ð23Þ
Eqs. (22) and (23) are the singular integral equation of the first kind. The fundamental solution for anpinternal pffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi crack and surface crack are 1= 1 s2 and 1= 1 s, respectively. f1(s) and f2(s) may be expressed as F 1 ðsÞ f1 ðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð1 þ sÞð1 sÞ for internal crack, and
F 2 ðsÞ f 2 ðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð1 þ sÞð1 sÞ
ð24Þ
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S.H. Ding, X. Li / Theoretical and Applied Fracture Mechanics 49 (2008) 313–320
F 1 ðsÞ f1 ðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi ; 1s
F 2 ðsÞ f 2 ðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi ; 1s
ð25Þ
for surface crack, and expanding F1(s) and F2(s) in forms of Chebyshev polynomials F 1 ðsÞ ¼
1 X
An T n ðsÞ;
F 2 ðsÞ ¼
n¼0
1 X
Bn T n ðsÞ:
ð26Þ
n¼0
It follows that a system of linear algebraic equations for the unknown coefficients An and Bn. The stress intensity factor (SIF) and the electrically displacement intensity factor (EDIF) can be expressed in different forms as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 3 ðbÞ ¼ limþ 2ðx bÞsyz1 ðx; 0Þ x!b pffiffiffiffiffi pffiffiffiffiffi ð27Þ ¼ c0 expðbbÞ a0 F 1 ð1Þ e0 expðbbÞ a0 F 2 ð1Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 3 ðaÞ ¼ lim 2ða xÞsyz1 ðx; 0Þ x!a pffiffiffiffiffi pffiffiffiffiffi ¼ c0 expðbaÞ a0 F 1 ð1Þ þ e0 expðbaÞ a0 F 2 ð1Þ; ð28Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ limþ 2ðx bÞDy1 ðx; 0Þ x!b pffiffiffiffiffi pffiffiffiffiffi ¼ e0 expðbbÞ a0 F 1 ð1Þ þ e0 expðbbÞ a0 F 2 ð1Þ; ð29Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K D3 ðaÞ ¼ lim 2ða xÞDy1 ðx; 0Þ x!a pffiffiffiffiffi pffiffiffiffiffi ¼ e0 expðbaÞ a0 F 1 ð1Þ e0 expðbaÞ a0 F 2 ð1Þ;
K D3 ðbÞ
ð30Þ for internal crack problem, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 3 ðbÞ ¼ limþ 2ðx bÞsyz1 ðx; 0Þ x!b pffiffiffi pffiffiffi ¼ c0 expðbbÞ bF 1 ð1Þ e0 expðbbÞ bF 2 ð1Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K D3 ðbÞ ¼ limþ 2ðx bÞDy1 ðx; 0Þ x!b pffiffiffi pffiffiffi ¼ e0 expðbbÞ bF 1 ð1Þ þ e0 expðbbÞ bF 2 ð1Þ;
ð31Þ
1 ¼ expðbxÞ p Dy1 ðx; 0Þ ¼ DðxÞ 1 ¼ expðbxÞ p
Z
b a
Z
b a
1 þ kðx; uÞ g1 ðuÞ du; c0 ux
ð32Þ
1 þ kðx; uÞ g1 ðuÞ du: e0 ux
ð33Þ
Compared to the internal crack problem, the surface crack problem is more practical. So, all analyses and results presented in the section are presented for a = 0 (i.e., the surface crack problem). For the anti-plane problem, the energy release rate (ERR) can be expressed directly in terms of stress intensity factor and electric displacement intensity factor as [20] " # 1 e0 K 23 ðbÞ þ 2e0 K 3 ðbÞK D3 ðbÞ c0 ðK D3 ðbÞÞ2 GðbÞ ¼ ; 2 expðbbÞ c0 e0 þ e20 ð39Þ for the impermeable case, and GðbÞ ¼
K 23 ðbÞ ; 2c0 expðbbÞ
ð40Þ
for the permeable case. Recently, some studies have shown the superiority of the energy density factor (EDF) in analyzing the fracture behavior of the piezoelectric structure [21,22]. For the anti-plane problem, the energy density factor can be expressed directly in terms of stress intensity factor and electric displacement intensity factor as " # 2 1 e0 K 23 ðbÞ e0 K 3 ðbÞK D3 ðbÞ þ 2c0 ðK D3 ðbÞÞ SðbÞ ¼ ; 8 expðbbÞ c0 e0 þ e20
for the impermeable case, and SðbÞ ¼
K 23 ðbÞ ; 8c0 expðbbÞ
ð42Þ
for the permeable case. It can be found that energy density factor is always positive. For the pure mechanical case, energy density factor is equivalent to the traditional definition of energy release rate [13]. 4. Results and discussion
as
for internal crack problem, and
ð38Þ
ð34Þ
The SIF and EDIF can be expressed in different forms pffiffiffiffiffi K 3 ðbÞ ¼ c0 expðbbÞ a0 F 1 ð1Þ; pffiffiffiffiffi K 3 ðaÞ ¼ c0 expðbaÞ a0 F 1 ð1Þ; pffiffiffiffiffi K D3 ðbÞ ¼ e0 expðbbÞ a0 F 1 ð1Þ; pffiffiffiffiffi K D3 ðaÞ ¼ e0 expðbaÞ a0 F 1 ð1Þ;
for surface crack problem. We find that e0 K D3 ¼ K 3 : c0
ð37Þ
ð41Þ
for surface crack problem. For the electrically permeable case, the corresponding stress and electric displacement can be expressed as syz1 ðx; 0Þ ¼ sðxÞ
pffiffiffi K 3 ðbÞ ¼ c0 expðbbÞ bF 1 ð1Þ; pffiffiffi K D3 ðbÞ ¼ e0 expðbbÞ bF 1 ð1Þ;
ð35Þ ð36Þ
In the following discussions, take BaTiO3 as the base material. The material properties at x = 0 can be found in [19]. The loading combination parameter k is determined as k = D0e0/(s0e0). For convenience, the stress intensity factor, the energy release rate andpthe ffiffiffi energy density factor are normalized as k 0 ¼ s0 h, G0 ¼ 4s20 h=c0 and S 0 ¼ s20 h=c0 , respectively. If c/h ? 1, it is known that the numerical results will approach the general results of a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric mate-
S.H. Ding, X. Li / Theoretical and Applied Fracture Mechanics 49 (2008) 313–320
Fig. 2. Variations of the normalized SIF with b/h for a FGPM strip bonded to a homogeneous PM with only a crack (k = 0, c/h = 30).
rial with only a crack. Fig. 2 shows a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material with only a surface crack under only the shear loading (in the numerical analysis, c/h = 30). The general feature of these curves is that, for the same crack length, K3(b)/k0 increases with the increasing of bh. As expected, K3(b)/k0 increases with the increasing of b/h. Fig. 3 plots the variation of the normalized stress intensity factor for the periodically edge-cracked piezoelectric materials with b/h. In this case, the FGPM strip is reduced to a homogeneous piezoelectric strip. It can be found that K3(b)/k0 increases with increasing c/h for the same crack length. Clearly, multiple cracking has a great tendency to reduce the field intensity factor. K3(b)/k0 increases with increasing b/h, and it means that the crack is easier to propagate as the crack length increases when b = 0. Figs. 4 and 5 show the effect c/h on the normalized stress intensity factor for periodic surface cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material with bh = 0.5 and bh = 0.5. For bh > 0, K3(b)/k0 increases with increasing c/h for the same crack
Fig. 3. Variations of the normalized SIF with b/h for a FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.0).
317
Fig. 4. Variations of the normalized SIF with b/h for a FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
Fig. 5. Variations of the normalized SIF with b/h for a FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
length. For bh < 0, K3(b)/k0 first increases, goes through a peak value and then decreases. We also found that single crack gives the biggest stress intensity factor. Figs. 6 and 7 show the normalized stress intensity factor for periodic surface cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material with bh = 0.5 and bh = 0.5. It can be found that K3(b)/k0 always decreases with decreasing c/h and approaches zero as c/h ? 0. It can be seen that the influence of the crack span c/h on the stress intensity factor is quite insignificant for a very small crack length (such as b/h = 0.1). Figs. 8 and 9 show the effect bh on the normalized stress intensity factor for periodic surface cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material. It can be seen that K3(b)/k0 increases with increasing bh, and it means that the decrease of bh could impede the crack extension. We also found that the effect of bh on the stress intensity factor is not obvious for small values of relative crack length (such as b/h = 0.1).
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Fig. 6. Variations of the normalized SIF with c/h for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
Fig. 7. Variations of the normalized SIF with c/h for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
Fig. 8. Variations of the normalized SIF with bh for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, b/h = 0.5).
Figs. 10 and 11 show the comparison of the normalized stress intensity factor and the energy release rate between
Fig. 9. Variations of the normalized SIF with bh for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, c/h = 0.5).
Fig. 10. Comparison of the normalized SIF between permeable and impermeable case (k = 0, c/h = 0.5).
the permeable case and impermeable case under only the shear loading. It can be found that K3(b)/k0 and G(b)/G0 increase with increasing b/h and bh for both the permeable case and impermeable case. The stress intensity factor and the energy release rate for the permeable case is greater than that for the impermeable case. Figs. 12 and 13 show the normalized energy density factor for periodic surface cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material with bh = 0.5 and bh = 0.5. Since the material properties at the different crack tip are variable, the decrease or increase of K3(b)/k0 does not always decide the crack propagation. It can be found from Figs. 12 and 13 that the energy density factor always increases with the increasing of crack length b/h, and it means that the crack is easier to propagate as the crack length increases. Fig. 14 shows the effects of the parameter k on the energy density factor for bh = 0.5 and c/h = 0.5. It can be found that, as the absolute value of k increases, S(b)/
S.H. Ding, X. Li / Theoretical and Applied Fracture Mechanics 49 (2008) 313–320
Fig. 11. Comparison of the normalized ERR between permeable and impermeable case (k = 0, c/h = 0.5).
319
Fig. 14. Effect of k on EDF for bh = 0.5 and c/h = 0.5 (for impermeable case).
the positive electric load, and this means that the crack is more likely to propagate under the negative electric loading than under the positive one. 5. Conclusion
Fig. 12. Variations of the normalized EDF with b/h for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
The reliability of piezoelectric materials is of great importance due to their abundant applications in the smart systems and structures. The coupled electro-mechanical behavior poses a challenge in obtaining close form solutions that describe the field quantities. The fracture behavior of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material has been considered. The singular integral equations for the impermeable and the permeable case are obtained by using integral transforms. The obtained results show that stress intensity factors can be released significantly by increasing crack density (decreasing crack spacing). The crack is easier to propagate under the negative electric loading, and the decrease of bh could impede the crack extension. Acknowledgements The project is supported by the National Natural Science Foundation of China (10661009) and the Ningxia Natural Science Foundation (NZ0604). References
Fig. 13. Variations of the normalized EDF with b/h for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
S0 always increase. However, the energy density factor under the negative electric load is larger than that under
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[15] B.L. Wang, Y.W. Mai, A periodic array of cracks in functionally graded materials subjected to thermomechanical loading, Int. J. Eng. Sci. 43 (2005) 432–446. [16] T.H. Hao, Periodical collinear air containing cracks in a piezoelectric material, Int. J. Fract. 112 (2001) 197–204. [17] J. Chen, Z.X. Liu, On the dynamic behavior of a functionally graded piezoelectric strip with periodic cracks vertical to the boundary, Int. J. Solids Struct. 42 (2005) 3133–3146. [18] B.L. Wang, Y.W. Mai, Multiple surface cracks in a piezoelectric layer bonded to an elastic substrate under transient electromechanical loads, Mech. Mater. 39 (2007) 564–579. [19] S.H. Ding, X. Li, The periodic crack problem in bonded piezoelectric materials, Acta Mech. Solida Sin. 20 (2007) 171–179. [20] K.Q. Hu, Z. Zhong, B. Jin, Anti-plane shear crack in a functionally gradient piezoelectric layer bonded to dissimilar half spaces, Int. J. Mech. Sci. 47 (2005) 82–93. [21] G.C. Sih, J.Z. Zuo, Multiscale behavior of crack initiation and growth in piezoelectric ceramics, Theor. Appl. Fract. Mech. 34 (2000) 123–141. [22] Z.J. Zuo, G.C. Sih, Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics, Theor. Appl. Fract. Mech. 34 (2000) 17–33.
Theoretical and Applied Fracture Mechanics 49 (2008) 313–320 www.elsevier.com/locate/tafmec
Periodic cracks in a functionally graded piezoelectric layer bonded to a piezoelectric half-plane S.H. Ding a, X. Li b,* b
a Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China Department of Mathematics and Computer Science, Ningxia University, Yinchuan 750021, China
Available online 16 February 2008
Abstract Studied is the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material. The properties of the functionally graded piezoelectric strip, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. The effects of the periodic crack spacing, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Functionally graded piezoelectric material; Periodic group cracks; Impermeable; Permeable
1. Introduction Piezoelectric materials (PMs) have been widely used as transducers, sensors and actuators because of their intrinsic electro-mechanical coupling behavior. Piezoelectric bimorphs are a special type of piezoelectric devices that consist of two long and thin piezoelectric elements. The stress peaks will be induced at the interfaces to cause failure such as cracking or debonding. The principal disadvantage of piezoelectric materials is that the bonding agent may crack at low temperature and creep at high temperature. These drawbacks may lead to lifetime limitation and restrict the use of piezoelectric actuator that require high reliability. To improve the reliability and durability problems arising largely from high residual and thermal stress, poor interfacial bonding strength, the functionally graded piezoelectric materials (FGPMs) as a new class of advanced composites have been developed. The concept of fracture mechanics on a finite
*
Corresponding author. E-mail address: [email protected] (X. Li).
0167-8442/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2008.02.002
crack in a strip of functionally graded piezoelectric material (FGPM) has been applied in [1]. They found that the singular stresses and electrical displacements at the tip of the crack in the FGPM carry the same forms as those in a homogeneous piezoelectric material (PM) but the magnitudes of the intensity factors are dependent on the gradient of the FGPM properties. This discover makes us to research the problem of FGPM by using the methods which investigates the problem of piezoelectric material. The dynamic anti-plane problem for a functionally graded piezoelectric strip containing a central crack vertical to the boundary is considered [2]. Both the impermeable and permeable cases are considered. They employed integral transforms and dislocation density functions to reduce the problem to Cauchy singular integral equations. The propagation problem of an anti-plane moving crack in a function graded piezoelectric strip is solved [3]. Investigated in [4] is the anti-plane crack and collinear crack problems in FGPM. A class of functional forms has been assumed to describe the mechanical and the electrical properties of the medium. For the permeable crack, the stress and the electrical displacement intensity factors depend only on the applied mechanical loads. The piezoelectric effect has
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no effect on the stress intensity factors. A variety of different crack problems in functionally graded piezoelectric medium have been studied [5–10]. As described in the foregoing, although a variety of challenging issues related to certain crack problems in the functionally graded piezoelectric materials have been addressed, one of the important mechanics problems here is the determination of the length parameter describing the surface crack periodicity. To date, the problem of periodic array of cracks [11,12] in the isotropic and anisotropic material is studied. The anti-plane shear problem for periodic cracking in functionally graded coatings has been considered [13–15]. Fracture by periodic cracking under electrical–mechanical loading has drawn increasing attention in both the academic and industrial communities. Reference can be made to the contributions presented in [16–19]. In the paper, we considered the problem of a FGPM strip containing a periodical array of parallel cracks bonded to a homogeneous piezoelectric material. The focus of this paper is the development of a mathematical model to predict the length scale for the spacing of transverse cracks that form in a functionally graded piezoelectric medium subjected to a coupled electro-mechanical external loading condition. The crack surface condition is assumed to be electrically impermeable or permeable. Integral transform techniques are used to obtain a system of singular integral equations, which are then solved numerically. Numerical results are presented for the stress intensity factor, the energy release rate and the energy density factor. 2. Formulation of the problem Fig. 1 shows a functionally graded piezoelectric strip and a piezoelectric material perfectly bonded together along x = h. The functionally graded piezoelectric strip is assumed to contain periodic cracks perpendicular to the interface. The crack of length and the x-coordinate of the crack center are defined as 2a0 = b a and d = (b + a)/2, respectively. All materials exhibit transversely isotropic behavior and poled in the z-direction, and the anti-plane
mechanical field and inplane electric field are coupled. The constitutive equation can be written as owk o/ þ e15k ðxÞ k ; ox ox owk o/k þ e15k ðxÞ ; syzk ¼ c44k ðxÞ oy oy owk o/ D:xk ¼ e15k ðxÞ e11k ðxÞ k ; ox ox owk o/k e11k ðxÞ ; Dyk ¼ e15k ðxÞ oy oy
sxzk ¼ c44k ðxÞ
ð1Þ
ð2Þ
where sizk, wk, Dik and /k (i = x, y, k = 1, 2) are the shear stresses, anti-plane displacements, inplane electrical displacements and electric potentials, respectively, while subscripts k = 1,2 refer to the FGPM strip 1 and the homogeneous PM 2. The variations of material constants c44k(x), e15k(x) and e11k(x) called the shear modulus, piezoelectric constants, and dielectric constants, respectively, are assumed in the following exponential forms; c441 ðxÞ ¼ c0 expðbxÞ;
e151 ðxÞ ¼ e0 expðbxÞ;
e111 ðxÞ ¼ e0 expðbxÞ; 0 < x < h; c442 ðxÞ ¼ c0 expðbhÞ; e152 ðxÞ ¼ e0 expðbhÞ;
ð3Þ
e112 ðxÞ ¼ e0 expðbhÞ; x > h;
ð4Þ
where b is called nonhomogeneous parameters. The constants c0, e0, and e0 are the material properties at x = 0. The static equilibrium equation and Maxwell’s equation under electro-static condition are given as osxzk osyzk þ ¼ 0; ox oy
oDxk oDyk þ ¼ 0; k ¼ 1; 2: ox oy
ð5Þ
By separating the homogeneous solution through an appropriate superposition, the problem may be reduced to a perturbation solution in which self-equilibration crack surface tractions are the only nonzero external loads. For the problem described in Fig. 1, it has to be solved under the following mixed boundary conditions: w1 ðh; yÞ ¼ w2 ðh; yÞ; /1 ðh; yÞ ¼ /2 ðh; yÞ; sxz1 ðh; yÞ ¼ sxz2 ðh; yÞ; Dx1 ðh; yÞ ¼ Dx2 ðh; yÞ; sxz1 ð0; yÞ ¼ 0; Dx1 ð0; yÞ ¼ 0; w2 ðx; 0Þ ¼ 0; /2 ðx; 0Þ ¼ 0; w1 ðx; cÞ ¼ 0; /1 ðx; cÞ ¼ 0; w2 ðx; cÞ ¼ 0;
/2 ðx; cÞ ¼ 0;
ð6Þ
and the boundary conditions on the crack surfaces can be written as syz1 ðx; 0Þ ¼ sðxÞ; w1 ðx; 0Þ ¼ 0;
Dy1 ðx; 0Þ ¼ DðxÞ; a < x < b;
ð7Þ
/1 ðx; 0Þ ¼ 0; 0 < x < a; b < x < h;
ð8Þ
for the impermeable case, and syz1 ðx; 0Þ ¼ sðxÞ;
Fig. 1. The geometry of periodically cracked FGPM strip bonded to a homogeneous PM.
Dy1 ðx; 0Þ ¼ Dc ðx; 0Þ ¼ DðxÞ;
a < x < b; w1 ðx; 0Þ ¼ 0; 0 < x < a; b < x < h;
ð9Þ ð10Þ
/1 ðx; 0Þ ¼ 0; 0 < x < h;
ð11Þ
S.H. Ding, X. Li / Theoretical and Applied Fracture Mechanics 49 (2008) 313–320
for the permeable case, where Dc(x,0) denotes the electric displacement of the space of the crack itself.
kðx; uÞ ¼ F 1 ðx; uÞ þ F 2 ðx; uÞ þ F 3 ðx; uÞ; Z 1 1 i m2 expðm1 cÞ m1 expðm2 cÞ þ jaj F 1 ðx; uÞ ¼ 2 1 a expðm1 cÞ expðm2 cÞ
3. Singular integral equations
expðiaðu xÞÞ da;
Consider the electrically impermeable case. By using Fourier transform method, we can obtain 8 w1 ðx; yÞ > > > R1 > 1 > ¼ 2p ½A ðaÞ expðm1 yÞ þ A2 ðaÞexpðm2 yÞ expðiaxÞda > > 1 1 > > 1 > P > > > þ ½C 1k ðck Þ expðp1k xÞ þ C 2k ðck Þ expðp2k xÞ sinðck yÞ; < k¼1
F 2 ðx; uÞ ¼
1 pX ½F 2j expðcj ðu þ xÞÞ; c j¼1
F 3 ðx; uÞ ¼
1 þ Oðu þ xÞ; uþx
F 2j ¼
ð12Þ
> /1 ðx; yÞ > > R1 > > 1 > ¼ 2p ½B ðaÞ expðm1 yÞ þ B2 ðaÞexpðm2 yÞ expðiaxÞda > 1 1 > > > 1 P > > > þ ½D1k ðck Þ expðp1k xÞ þ D2k ðck Þ expðp2k xÞ sinðck yÞ; :
> > : /2 ðx; yÞ ¼
k¼1 1 P
þ
ð13Þ
k¼1
where m1 ¼ m2 ¼
The integral equations (16) and (17) will be solved for different crack types, that is, for the internal crack (a > 0) and surface crack (a = 0) problem. It is to be point out that for the internal crack, the function g1(u) and g2(u) must fulfill the single-valuedness as Z b Z b g1 ðuÞ du ¼ g2 ðuÞ du ¼ 0: ð19Þ a
ð14Þ
The solutions of undetermined function Ai(a), Bi(a), Cik(ck), Dik(ck), E2k(ck) and F2k(ck) (i = 1, 2) depends on the mechanical and electrical conditions of crack surfaces. Define two dislocation function g1(x) and g2(x) ( ow1 ðx;0Þ ; a < x < b; ox g1 ðxÞ ¼ ; 0; x < a; x > b; ( o/1 ðx;0Þ ; a < x < b; ox : ð15Þ g2 ðxÞ ¼ 0; x < a; x > b; Substituting Eqs. (12)–(15) into (6)–(8), from Eq. (7) it follows that: Z 1 b 1 þ kðx; uÞ g1 ðuÞ c0 syz1 ðx; 0Þ ¼ sðxÞ ¼ expðbxÞ p a ux 1 þ e0 þ kðx; uÞ g2 ðuÞ du; ð16Þ ux Z 1 b 1 þ kðx; uÞ g1 ðuÞ Dy1 ðx; 0Þ ¼ DðxÞ ¼ expðbxÞ e0 p a ux 1 e0 þ kðx; uÞ g2 ðuÞ du; ð17Þ ux where
expðp2j xÞc2j ½expðp1j ðu hÞÞ þ expðp1j h p2j uÞ: kj p1j D2 ð18Þ
F 2k ðck Þ expðck xÞ sinðck yÞ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b a2 þ iba; p1k ¼ kk ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b b2 p2k ¼ þ kk ; k ¼ þ c2k ; ck ¼ kp=c: 2 4
expðp1j xÞc2j ½p2j ðp1j þ cj Þ expðp1j ðu hÞÞ kj p1j D2 þ p1j ðp2j þ cj Þ expðp2j ðu hÞÞ
k¼1
8 1 P > > < w2 ðx; yÞ ¼ E2k ðck Þ expðck xÞ sinðck yÞ;
315
a
For the internal crack problem and surface crack problem, we define the following normalized quantities: bþa bþa bþa bþa rþ ; u¼ sþ ; 2 2 2 2 f 1 ðsÞ ¼ g1 ðuÞ; f 2 ðsÞ ¼ g2 ðuÞ;
ð20Þ
Kðr; sÞ ¼ kðx; uÞ; sðrÞ ¼ syz1 ðx; 0Þ expðbxÞ; DðrÞ ¼ Dy1 ðx; 0Þ expðbxÞ;
ð21Þ
x¼
then (16) and (17) become Z 1 1 1 þ Kðr; sÞ f1 ðsÞ c0 sðrÞ ¼ p 1 sr 1 þe0 þ Kðr; sÞ f2 ðsÞ ds; sr Z 1 1 1 þ Kðr; sÞ f1 ðsÞ e0 DðrÞ ¼ p 1 sr 1 e0 þ Kðr; sÞ f2 ðsÞ ds: sr
ð22Þ
ð23Þ
Eqs. (22) and (23) are the singular integral equation of the first kind. The fundamental solution for anpinternal pffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi crack and surface crack are 1= 1 s2 and 1= 1 s, respectively. f1(s) and f2(s) may be expressed as F 1 ðsÞ f1 ðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð1 þ sÞð1 sÞ for internal crack, and
F 2 ðsÞ f 2 ðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð1 þ sÞð1 sÞ
ð24Þ
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S.H. Ding, X. Li / Theoretical and Applied Fracture Mechanics 49 (2008) 313–320
F 1 ðsÞ f1 ðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi ; 1s
F 2 ðsÞ f 2 ðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi ; 1s
ð25Þ
for surface crack, and expanding F1(s) and F2(s) in forms of Chebyshev polynomials F 1 ðsÞ ¼
1 X
An T n ðsÞ;
F 2 ðsÞ ¼
n¼0
1 X
Bn T n ðsÞ:
ð26Þ
n¼0
It follows that a system of linear algebraic equations for the unknown coefficients An and Bn. The stress intensity factor (SIF) and the electrically displacement intensity factor (EDIF) can be expressed in different forms as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 3 ðbÞ ¼ limþ 2ðx bÞsyz1 ðx; 0Þ x!b pffiffiffiffiffi pffiffiffiffiffi ð27Þ ¼ c0 expðbbÞ a0 F 1 ð1Þ e0 expðbbÞ a0 F 2 ð1Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 3 ðaÞ ¼ lim 2ða xÞsyz1 ðx; 0Þ x!a pffiffiffiffiffi pffiffiffiffiffi ¼ c0 expðbaÞ a0 F 1 ð1Þ þ e0 expðbaÞ a0 F 2 ð1Þ; ð28Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ limþ 2ðx bÞDy1 ðx; 0Þ x!b pffiffiffiffiffi pffiffiffiffiffi ¼ e0 expðbbÞ a0 F 1 ð1Þ þ e0 expðbbÞ a0 F 2 ð1Þ; ð29Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K D3 ðaÞ ¼ lim 2ða xÞDy1 ðx; 0Þ x!a pffiffiffiffiffi pffiffiffiffiffi ¼ e0 expðbaÞ a0 F 1 ð1Þ e0 expðbaÞ a0 F 2 ð1Þ;
K D3 ðbÞ
ð30Þ for internal crack problem, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 3 ðbÞ ¼ limþ 2ðx bÞsyz1 ðx; 0Þ x!b pffiffiffi pffiffiffi ¼ c0 expðbbÞ bF 1 ð1Þ e0 expðbbÞ bF 2 ð1Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K D3 ðbÞ ¼ limþ 2ðx bÞDy1 ðx; 0Þ x!b pffiffiffi pffiffiffi ¼ e0 expðbbÞ bF 1 ð1Þ þ e0 expðbbÞ bF 2 ð1Þ;
ð31Þ
1 ¼ expðbxÞ p Dy1 ðx; 0Þ ¼ DðxÞ 1 ¼ expðbxÞ p
Z
b a
Z
b a
1 þ kðx; uÞ g1 ðuÞ du; c0 ux
ð32Þ
1 þ kðx; uÞ g1 ðuÞ du: e0 ux
ð33Þ
Compared to the internal crack problem, the surface crack problem is more practical. So, all analyses and results presented in the section are presented for a = 0 (i.e., the surface crack problem). For the anti-plane problem, the energy release rate (ERR) can be expressed directly in terms of stress intensity factor and electric displacement intensity factor as [20] " # 1 e0 K 23 ðbÞ þ 2e0 K 3 ðbÞK D3 ðbÞ c0 ðK D3 ðbÞÞ2 GðbÞ ¼ ; 2 expðbbÞ c0 e0 þ e20 ð39Þ for the impermeable case, and GðbÞ ¼
K 23 ðbÞ ; 2c0 expðbbÞ
ð40Þ
for the permeable case. Recently, some studies have shown the superiority of the energy density factor (EDF) in analyzing the fracture behavior of the piezoelectric structure [21,22]. For the anti-plane problem, the energy density factor can be expressed directly in terms of stress intensity factor and electric displacement intensity factor as " # 2 1 e0 K 23 ðbÞ e0 K 3 ðbÞK D3 ðbÞ þ 2c0 ðK D3 ðbÞÞ SðbÞ ¼ ; 8 expðbbÞ c0 e0 þ e20
for the impermeable case, and SðbÞ ¼
K 23 ðbÞ ; 8c0 expðbbÞ
ð42Þ
for the permeable case. It can be found that energy density factor is always positive. For the pure mechanical case, energy density factor is equivalent to the traditional definition of energy release rate [13]. 4. Results and discussion
as
for internal crack problem, and
ð38Þ
ð34Þ
The SIF and EDIF can be expressed in different forms pffiffiffiffiffi K 3 ðbÞ ¼ c0 expðbbÞ a0 F 1 ð1Þ; pffiffiffiffiffi K 3 ðaÞ ¼ c0 expðbaÞ a0 F 1 ð1Þ; pffiffiffiffiffi K D3 ðbÞ ¼ e0 expðbbÞ a0 F 1 ð1Þ; pffiffiffiffiffi K D3 ðaÞ ¼ e0 expðbaÞ a0 F 1 ð1Þ;
for surface crack problem. We find that e0 K D3 ¼ K 3 : c0
ð37Þ
ð41Þ
for surface crack problem. For the electrically permeable case, the corresponding stress and electric displacement can be expressed as syz1 ðx; 0Þ ¼ sðxÞ
pffiffiffi K 3 ðbÞ ¼ c0 expðbbÞ bF 1 ð1Þ; pffiffiffi K D3 ðbÞ ¼ e0 expðbbÞ bF 1 ð1Þ;
ð35Þ ð36Þ
In the following discussions, take BaTiO3 as the base material. The material properties at x = 0 can be found in [19]. The loading combination parameter k is determined as k = D0e0/(s0e0). For convenience, the stress intensity factor, the energy release rate andpthe ffiffiffi energy density factor are normalized as k 0 ¼ s0 h, G0 ¼ 4s20 h=c0 and S 0 ¼ s20 h=c0 , respectively. If c/h ? 1, it is known that the numerical results will approach the general results of a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric mate-
S.H. Ding, X. Li / Theoretical and Applied Fracture Mechanics 49 (2008) 313–320
Fig. 2. Variations of the normalized SIF with b/h for a FGPM strip bonded to a homogeneous PM with only a crack (k = 0, c/h = 30).
rial with only a crack. Fig. 2 shows a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material with only a surface crack under only the shear loading (in the numerical analysis, c/h = 30). The general feature of these curves is that, for the same crack length, K3(b)/k0 increases with the increasing of bh. As expected, K3(b)/k0 increases with the increasing of b/h. Fig. 3 plots the variation of the normalized stress intensity factor for the periodically edge-cracked piezoelectric materials with b/h. In this case, the FGPM strip is reduced to a homogeneous piezoelectric strip. It can be found that K3(b)/k0 increases with increasing c/h for the same crack length. Clearly, multiple cracking has a great tendency to reduce the field intensity factor. K3(b)/k0 increases with increasing b/h, and it means that the crack is easier to propagate as the crack length increases when b = 0. Figs. 4 and 5 show the effect c/h on the normalized stress intensity factor for periodic surface cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material with bh = 0.5 and bh = 0.5. For bh > 0, K3(b)/k0 increases with increasing c/h for the same crack
Fig. 3. Variations of the normalized SIF with b/h for a FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.0).
317
Fig. 4. Variations of the normalized SIF with b/h for a FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
Fig. 5. Variations of the normalized SIF with b/h for a FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
length. For bh < 0, K3(b)/k0 first increases, goes through a peak value and then decreases. We also found that single crack gives the biggest stress intensity factor. Figs. 6 and 7 show the normalized stress intensity factor for periodic surface cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material with bh = 0.5 and bh = 0.5. It can be found that K3(b)/k0 always decreases with decreasing c/h and approaches zero as c/h ? 0. It can be seen that the influence of the crack span c/h on the stress intensity factor is quite insignificant for a very small crack length (such as b/h = 0.1). Figs. 8 and 9 show the effect bh on the normalized stress intensity factor for periodic surface cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material. It can be seen that K3(b)/k0 increases with increasing bh, and it means that the decrease of bh could impede the crack extension. We also found that the effect of bh on the stress intensity factor is not obvious for small values of relative crack length (such as b/h = 0.1).
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Fig. 6. Variations of the normalized SIF with c/h for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
Fig. 7. Variations of the normalized SIF with c/h for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
Fig. 8. Variations of the normalized SIF with bh for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, b/h = 0.5).
Figs. 10 and 11 show the comparison of the normalized stress intensity factor and the energy release rate between
Fig. 9. Variations of the normalized SIF with bh for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, c/h = 0.5).
Fig. 10. Comparison of the normalized SIF between permeable and impermeable case (k = 0, c/h = 0.5).
the permeable case and impermeable case under only the shear loading. It can be found that K3(b)/k0 and G(b)/G0 increase with increasing b/h and bh for both the permeable case and impermeable case. The stress intensity factor and the energy release rate for the permeable case is greater than that for the impermeable case. Figs. 12 and 13 show the normalized energy density factor for periodic surface cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material with bh = 0.5 and bh = 0.5. Since the material properties at the different crack tip are variable, the decrease or increase of K3(b)/k0 does not always decide the crack propagation. It can be found from Figs. 12 and 13 that the energy density factor always increases with the increasing of crack length b/h, and it means that the crack is easier to propagate as the crack length increases. Fig. 14 shows the effects of the parameter k on the energy density factor for bh = 0.5 and c/h = 0.5. It can be found that, as the absolute value of k increases, S(b)/
S.H. Ding, X. Li / Theoretical and Applied Fracture Mechanics 49 (2008) 313–320
Fig. 11. Comparison of the normalized ERR between permeable and impermeable case (k = 0, c/h = 0.5).
319
Fig. 14. Effect of k on EDF for bh = 0.5 and c/h = 0.5 (for impermeable case).
the positive electric load, and this means that the crack is more likely to propagate under the negative electric loading than under the positive one. 5. Conclusion
Fig. 12. Variations of the normalized EDF with b/h for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
The reliability of piezoelectric materials is of great importance due to their abundant applications in the smart systems and structures. The coupled electro-mechanical behavior poses a challenge in obtaining close form solutions that describe the field quantities. The fracture behavior of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material has been considered. The singular integral equations for the impermeable and the permeable case are obtained by using integral transforms. The obtained results show that stress intensity factors can be released significantly by increasing crack density (decreasing crack spacing). The crack is easier to propagate under the negative electric loading, and the decrease of bh could impede the crack extension. Acknowledgements The project is supported by the National Natural Science Foundation of China (10661009) and the Ningxia Natural Science Foundation (NZ0604). References
Fig. 13. Variations of the normalized EDF with b/h for periodically edgecracked FGPM strip bonded to a homogeneous PM (k = 0, bh = 0.5).
S0 always increase. However, the energy density factor under the negative electric load is larger than that under
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