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Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate
International Journal of Engineering Science 41 (2003) 1405–1422 www.elsevier.com/locate/ijengsci
Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate X.-F. Li a
a,b,*
, G.J. Tang
b
College of Mathematics and Physics, Hunan Normal University, Changsha, Hunan 410081, China b School of Aerospace and Materials Engineering, National University of Defence Technology, Changsha, Hunan 410073, China Received 5 June 2002; received in revised form 16 October 2002; accepted 6 November 2002
Abstract The problem of an anti-plane interface crack in a layered piezoelectric plate composed of two bonded dissimilar piezoelectric ceramic layers subjected to applied voltage is considered. It is assumed that the crack is either impermeable or permeable. An integral transform technique is employed to reduce the problem considered to dual integral equations, then to a Fredholm integral equation by introducing an auxiliary function. Field intensity factors and energy release rate are obtained in explicit form in terms of the auxiliary function. In particular, by solving analytically a resulting singular integral equation, they are determined explicitly in terms of given electromechanical loadings for the case of two bonded layers of equal thickness. Some numerical results are presented graphically to show the influence of the geometric parameters on the field intensity factors and the energy release rate. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Piezoelectric plate; Interface crack; Electroelastic analysis; Field intensity factors; Energy release rate
1. Introduction Piezoelectric materials have been used widely in technology such as transducers, actuators, sensors, etc., due to their intrinsic coupling characteristics between electric and elastic behaviors. Great progress on the study of electroelastic field disturbed by cracks in a piezoelectric material has been made in recent years. Considerable researches in this area are mainly focused on *
Corresponding author. E-mail addresses: xfl[email protected], [email protected] (X.-F. Li).
0020-7225/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0020-7225(03)00038-7
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determining the electroelastic field for a cracked piezoelectric material subjected to various electromechanical loadings and many analytical solutions have been derived in theory. In piezoelectric composite materials, cracks have been found and they usually propagate along the interface. Therefore analyzing electric and elastic behaviors of an interface crack is a prerequisite for studying the failure of piezoelectric composite materials. The investigation of the problem involving interface cracks between two bonded dissimilar piezoelectric materials has recently attracted increasing attention. So far many papers have been devoted to the analysis of electroelastic field of an interface crack such as Suo et al. [31], Boem and Atluri [1], Chen et al. [2], Qin and Yu [22], Hou and Mei [14], Deng and Meguid [5], Wang and Han [32], Ru [23], Govorukha and Lobada [11], Gao and Wang [9], etc. The dynamic behavior of interface cracks in a piezoelectric composite structure has been studied by Meguid and Wang [18], Li et al. [15], Wang [33], etc. The above-mentioned works are mainly related to internal cracks embedded in an infinite piezoelectric sheet. However, some practical situations in engineering applications should be posed with a layered structure having at least two layers. As a result, it is crucial to determine the electric and elastic behaviors disturbed by an interface crack in a layered piezoelectric composite material, in particular in the close vicinity of the crack tip. Along this line, Narita and Shindo [19,20] considered an anti-plane crack located at the interface between a piezoelectric material and an orthotropic elastic medium, and determined the stress intensity factor via solving numerically a singular integral equation by use of the Gauss–Chebyshev integration formula. Nonlinear fracture of piezoelectric materials with an interface crack has been treated [25]. Recently, transient response of a crack in a piezoelectric bimaterial and in a multilayered piezoelectric material under anti-plane mechanical and in-plane electric impacts has been studied by Wang et al. [34,35] and Gu et al. [13], respectively. For a piezoelectric laminate containing multiple interfacial collinear cracks subjected to steady-state anti-plane mechanical and in-plane electric loadings, the dynamic behavior of electroelastic field has been analyzed by Zhao and Meguid [38]. For two arc–curved interface cracks between a piezoelectric fiber and an elastic matrix, the dynamic behavior under a steady-state anti-plane shear wave has been dealt with by Shindo et al. [26,27]. As compared to a lot of theoretical works, experimental results on fracture of a cracked piezoelectric material are very limited [3,8,21,24,28,36]. This paper is concerned with the static problem of an anti-plane shear crack situated at the interface of a layered piezoelectric plate consisting of two bonded dissimilar piezoelectric layers. Within the framework of the reigning linear piezoelectricity theory, the associated mixed boundary value problem, by using an integral transform technique, is reduced to a Fredholm integral equation of the second kind. Under the assumptions of permeable and impermeable cracks, field intensity factors and energy release rate are determined numerically, respectively. Moreover, for the special case of two bonded layers of equal thickness, all the field intensity factors and energy release rate can be obtained analytically. In contrast to numerical results, analytical solution serves a benchmark and provides us with an approach for justifying accuracy of certain numerical results.
2. Formulation of the problem Consider a piezoelectric plate made of two bonded dissimilar piezoelectric layers of thicknesses h1 and h2 , whose interface is along the xz-plane, and whose poling directions are oriented in the
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
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Fig. 1. Schematic of an interface crack in a layered piezoelectric plate made of two bonded dissimilar layers.
z-axis, as shown in Fig. 1, in which the z-axis is not depicted. Plate dimensions along the x- and zaxes are infinite, and a crack of length 2a is located at the interface and penetrates through the piezoelectric plate along the z-axis. From the viewpoint of experiment, it is much simpler to measure or impose applied voltage between the two surfaces of the layered piezoelectric plate than the charge. Moreover, in certain practical applications of piezoelectric devices such as actuators or sensors, piezoelectric layers are sandwiched between exterior media. Consequently, it is assumed that the plate surfaces are rigidly clamped, or have a relative displacement along the z-axis. Thus, the plate geometry and electromechanical loadings at the plate surfaces are independent of z; so the electric and elastic boundary conditions at the plate surfaces can be stated as /ð1Þ ðx; h1 Þ /ð2Þ ðx; h2 Þ ¼ V0 ;
wð1Þ ðx; h1 Þ wð2Þ ðx; h2 Þ ¼ W0 ;
1 < x < 1:
ð1Þ
Here wðjÞ ðx; yÞ and /ðjÞ ðx; yÞ represent the out-of-plane displacements and the in-plane electric potentials, and the subscripts j ¼ 1 and j ¼ 2 specify the quantities in the upper and lower layers, respectively. It is easily seen that for the problem in question, the in-plane elastic deformation is related neither to electric field nor to anti-plane deformation with respect to the xy-plane, and anti-plane deformation is coupled with electric field. Therefore, for the problem stated above, wðjÞ ðx; yÞ and /ðjÞ ðx; yÞ satisfy the basic governing differential equations for anti-plane piezoelectricity, in the absence of body forces and free charges, c44ðjÞ r2 wðjÞ þ e15ðjÞ r2 /ðjÞ ¼ 0;
e15ðjÞ r2 wðjÞ e11ðjÞ r2 /ðjÞ ¼ 0;
ð2Þ
where c44ðjÞ , e11ðjÞ , and e15ðjÞ are the elastic stiffness measured in a constant electric field, the dielectric permittivity measured at a uniform strain, the piezoelectric constant, respectively, and r2 stands for the two-dimensional Laplacian operator. Once functions wðjÞ and /ðjÞ are determined from given boundary conditions, then the components of anti-plane shear stress, strain, in-plane electric displacement, and electric field in each piezoelectric layer are obtainable in terms of the following equations: szxðjÞ ¼ c44ðjÞ czxðjÞ e15ðjÞ ExðjÞ ;
szyðjÞ ¼ c44ðjÞ czyðjÞ e15ðjÞ EyðjÞ ;
ð3Þ
DxðjÞ ¼ e15ðjÞ czxðjÞ þ e11ðjÞ ExðjÞ ;
DyðjÞ ¼ e15ðjÞ czyðjÞ þ e11ðjÞ EyðjÞ ;
ð4Þ
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czxðjÞ ¼ owðjÞ =ox; ExðjÞ ¼ o/ðjÞ =ox;
czyðjÞ ¼ owðjÞ =oy;
ð5Þ
EyðjÞ ¼ o/ðjÞ =oy:
ð6Þ
In order to obtain the desired electroelastic field, appropriate boundary conditions at the crack surfaces must be furnished. Clearly, elastic boundary conditions at the crack surfaces are szyð1Þ ðx; 0þ Þ ¼ szyð2Þ ðx; 0 Þ ¼ 0;
jxj < a:
ð7Þ
For a hole in an infinite piezoelectric material, electric boundary conditions at the rim of the hole have been exactly analyzed by Dunn [6], Zhang and Tong [37], and Sosa and Khutoryansky [30]. However, for a slit-like crack, two types of electric boundary conditions at the crack surfaces prevail, the permeable and impermeable assumptions. For the present study, they are stated as /ð1Þ ðx; 0þ Þ ¼ /ð2Þ ðx; 0 Þ;
Dyð1Þ ðx; 0þ Þ ¼ Dyð2Þ ðx; 0 Þ;
jxj < a
ð8Þ
for a permeable interface crack, or Dyð1Þ ðx; 0þ Þ ¼ Dyð2Þ ðx; 0 Þ ¼ 0;
jxj < a
ð9Þ
for an impermeable interface crack. In addition, perfect bonding exists between two dissimilar piezoelectric layers; so electric and elastic field should fulfill the continuity conditions along the bonded interface wð1Þ ðx; 0þ Þ ¼ wð2Þ ðx; 0 Þ;
szyð1Þ ðx; 0þ Þ ¼ szyð2Þ ðx; 0 Þ;
jxj > a;
ð10Þ
/ð1Þ ðx; 0þ Þ ¼ /ð2Þ ðx; 0 Þ;
Dyð1Þ ðx; 0þ Þ ¼ Dyð2Þ ðx; 0 Þ;
jxj > a:
ð11Þ
3. Solution of the problem Due to the symmetry of the problem under consideration, it is sufficient to analyze the righthalf portion, x P 0. Therefore in the following we confine our attention to the region x P 0. To solve the problem stated above, it is convenient to employ an integral transform technique to reduce the associated mixed boundary value problem to dual integral equations. For this purpose, by use of the Fourier cosine transform, it is easy to obtain that an appropriate solution of Eq. (2) can be expressed as the following integrals: Z 1 ½Aj ðnÞ coshðynÞ þ Bj ðnÞ sinhðynÞ cosðxnÞ dn þ aj y; ð12Þ wðjÞ ðx; yÞ ¼ 0
/ðjÞ ðx; yÞ ¼
Z
1
½Cj ðnÞ coshðynÞ þ Dj ðnÞ sinhðynÞ cosðxnÞ dn þ bj y; 0
ð13Þ
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
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where x P 0, 0 6 y 6 h1 for j ¼ 1, h2 6 y 6 0 for j ¼ 2, Aj ðnÞ; . . . ; Dj ðnÞ ðj ¼ 1; 2Þ are unknown functions to be determined from the conditions given above, and aj and bj are to be determined by use of the boundary conditions (1) and the interface continuity conditions, which are given in the Appendix. With the aid of constitutive equations (3) and (4), from (12) and (13) it is a simple matter to obtain the expressions for the components of the stress, strain, electric displacement and electric field in terms of Aj ðnÞ; . . . ; Dj ðnÞ ðj ¼ 1; 2Þ. For instance, we have Z 1 n½ðc44ðjÞ Aj þ e15ðjÞ Cj Þ sinhðynÞ þ ðc44ðjÞ Bj þ e15ðjÞ Dj Þ coshðynÞ cosðxnÞ dn þ s0 ; szyðjÞ ¼ 0
ð14Þ DyðjÞ ¼
Z
1
n½ðe15ðjÞ Aj e11ðjÞ Cj Þ sinhðynÞ þ ðe15ðjÞ Bj e11ðjÞ Dj Þ coshðynÞ cosðxnÞ dn þ D0 0
ð15Þ with s0 ¼
c0 ðc44ð1Þ h1 þ c44ð2Þ h2 C12 Þ E0 ðe15ð1Þ h1 þ e15ð2Þ h2 C12 Þ ; D
ð16Þ
c0 ðe15ð1Þ h1 þ e15ð2Þ h2 C12 Þ þ E0 ðe11ð1Þ h1 þ e11ð2Þ h2 C12 Þ ; D
ð17Þ
D0 ¼
where c0 , E0 , C12 , and D are defined in the appendix. Our objective is to obtain electroelastic field, in particular in the vicinity of a crack tip. Substituting the expressions for szyðjÞ and DyðjÞ from (12) and (13) into (1) yields four algebraic equations for Aj ðnÞ; . . . ; Dj ðnÞ, from which one gets B1 ðnÞ ¼ A1 ðnÞ cothðh1 nÞ; B2 ðnÞ ¼ A2 ðnÞ cothðh2 nÞ;
D1 ðnÞ ¼ C1 ðnÞ cothðh1 nÞ; D2 ðnÞ ¼ C2 ðnÞ cothðh2 nÞ:
ð18Þ ð19Þ
Using the above results, in conjunction with the interface continuity conditions one finds A1 ðnÞ ¼
½H12 ðnÞ þ m22 A m12 C 2H12 ðnÞ; ½H12 ðnÞ þ m11 ½H12 ðnÞ þ m22 m12 m21
ð20Þ
C1 ðnÞ ¼
½H12 ðnÞ þ m11 C m21 A 2H12 ðnÞ; ½H12 ðnÞ þ m11 ½H12 ðnÞ þ m22 m12 m21
ð21Þ
where 2AðnÞ ¼ A1 ðnÞ A2 ðnÞ, 2CðnÞ ¼ C1 ðnÞ C2 ðnÞ, H12 ðnÞ ¼ tanhðh1 nÞ cothðh2 nÞ, mij Õs are given in the appendix. The functions AðnÞ and CðnÞ are new unknown functions, which may be determined via the remaining boundary conditions. In what follows two cases of electric boundary conditions at the crack surfaces will be considered, respectively.
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3.1. The permeable case In this case, employing the electric boundary conditions for a permeable crack yields CðnÞ ¼ 0. Furthermore, substituting the above into the remaining boundary conditions, (7) and the first of (10), results in dual integral equations for AðnÞ: Z 1 ngðnÞ cothðh2 nÞAðnÞ cosðxnÞ dn ¼ s0 ; 0 6 x < a; ð22Þ 0
Z
1
AðnÞ cosðxnÞ dn ¼ 0;
x > a;
ð23Þ
0
where gðnÞ ¼
2c44ð1Þ ½H12 ðnÞ þ m22 2e15ð1Þ m21 : ½H12 ðnÞ þ m11 ½H12 ðnÞ þ m22 m12 m21
ð24Þ
Due to the complicated form of gðnÞ, it seems unlikely that a closed-form solution of dual integral equations (22) and (23) is obtainable except for a special case when h1 ¼ h2 . For the latter, a closed-form solution can be derived analytically, which is given in the next section. Next, dual integral equations (22) and (23) can be transformed into a Fredholm integral equation of the second kind. To achieve this, following Sneddon [29], we choose AðnÞ given by Z a wðuÞJ0 ðunÞ du; ð25Þ AðnÞ ¼ 0
where J0 ð Þ denotes the Bessel function of the first kind of order zero, and wðuÞ is an unknown auxiliary function. Recalling the result [12] Z 1 J0 ðunÞ cosðxnÞ dn ¼ 0; x > u; ð26Þ 0
inserting (25) into (23) indicates that Eq. (23) willpbe automatically satisfied. On the other hand, ffiffiffiffiffiffiffiffiffiffiffiffiffi we substitute (25) into (22), then multiply by 2=ðp t2 x2 Þ, and integrate with x between 0 and t. Making use of the identity 2 J0 ðtnÞ ¼ p
Z
t 0
cosðxnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi dx; t2 x2
we find that Eq. (22) then becomes a Fredholm integral equation of the second kind, Z a Kðx; u; h1 ; h2 ÞwðuÞ du ¼ xs0 ; 0 6 x < a; gwðxÞ þ g 0
with
ð27Þ
ð28Þ
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
Kðx; u; h1 ; h2 Þ ¼ x
Z
1
0
g¼
gðnÞ cothðh2 nÞ 1 nJ0 ðunÞJ0 ðxnÞ dn; g
1411
ð29Þ
c44ð1Þ þ c44ð2Þ C12 c44ð2Þ e11ð2Þ þ e215ð2Þ ; c44 e11 þ e215 2
ð30Þ
where c44 , e11 , and e15 represent the mean values of the corresponding material constants in the upper and lower layers, and gðnÞ ! g as n ! 1. For the purpose of numerical computation, we introduce the following notations: x x ¼ ; a
u u¼ ; a
h1 h1 ¼ ; a
h2 h2 ¼ ; a
gwðxÞ ; wðxÞ ¼ a
ð31Þ
and hence Eq. (28) is rewritten in a normalized form as wðxÞ þ
Z
1
Kðx; uÞwð uÞ d u ¼ xs0 ;
ð32Þ
0 6 x < 1;
0
where Kðx; uÞ ¼ Kðx; u; h1 ; h2 Þ:
ð33Þ
Taking (16) into account, in addition to c0 and E0 , s0 is related to the material constants and the crack geometry as well. 3.2. The impermeable case In a similar manner, employing the electric and elastic boundary conditions for an impermeable crack yields the system of dual integral equations for AðnÞ and CðnÞ, which may be written in the following compact form Z 1 nMðnÞXðnÞ cothðh2 nÞ cosðxnÞ dn ¼ T0 ; 0 6 x < a; ð34Þ 0
Z
1
XðnÞ cosðxnÞ dn ¼ 0;
ð35Þ
x > a;
0
where T
XðnÞ ¼ ðAðnÞ; CðnÞÞ ;
T
T0 ¼ ðs0 ; D0 Þ ;
MðnÞ ¼
g11 ðnÞ g12 ðnÞ
g12 ðnÞ ; g22 ðnÞ
ð36Þ
in which the superscript T stands for the transpose of a matrix, and the elements gij ðnÞ in MðnÞ are given in the Appendix.
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By representing XðnÞ in terms of a new unknown WðuÞ as Z a XðnÞ ¼ WðuÞJ0 ðunÞ du;
ð37Þ
0
proceeding as before, according to the notations in (31) dual integral equations for XðnÞ then become a Fredholm integral equations in a normalized form Z 1 Wð uÞ þ Kðx; uÞM1 Wð uÞ d u ¼ xT0 ; 0 6 x < 1 ð38Þ 0
with
MWðuÞ Wð uÞ ¼ ; a K ij ðx; uÞ ¼ gij x
Z
M¼ 1
0
"
g11 g12
g12 ; g22 #
Kðx; uÞ ¼
K 11 ðx; uÞ K 12 ðx; uÞ
gij ðnÞ cothð h2 nÞ 1 nJ0 ðunÞJ0 ðxnÞ dn; gij
K 12 ðx; uÞ ; K 22 ðx; uÞ
i; j ¼ 1; 2:
ð39Þ
ð40Þ
In fact, gij ¼ gij ðnÞ as n ! 1. 4. Closed-form solution for a central interface crack In this section, we will derive a closed-form solution for a crack lying at the interface between two dissimilar piezoelectric layers of equal thickness, i.e. h1 ¼ h2 ¼ h. In this case, H12 ðnÞ ¼ 1. 4.1. The permeable case Dual integral equations (22) and (23) for a permeable crack reduce to Z 1 AðnÞ cosðxnÞ dn ¼ 0; x > a;
ð41Þ
0
Z
1
nAðnÞ cothðhnÞ cosðxnÞ dn ¼ s0 ;
g
0 6 x < a:
ð42Þ
0
Owing to the special form of dual integral equations, a closed-form solution is obtainable. To this end, as in Li and Duan [16], we choose AðnÞ in terms of w0 ðxÞ: Z a 2 w0 ðsÞ sinðsnÞ ds; ð43Þ AðnÞ ¼ pn 0 instead of (25), where 2wðxÞ is the crack displacement jump across the crack surfaces, i.e. 2wðxÞ ¼ wð1Þ ðx; 0þ Þ wð2Þ ðx; 0 Þ, the prime denotes differentiation with respect to the argument. Taking into account wðxÞ ¼ 0 for jxj P a, Eq. (41) will be fulfilled identically.
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
1413
Using the following result [12] Z
1
0
pu cothðanÞ cosðunÞ dn ¼ ln 2 sinh ; n 2a
ð44Þ
cothðhnÞ sinðsnÞ sinðxnÞ 1 tanhðbsÞ þ tanhðbxÞ ; dn ¼ ln n 2 tanhðbsÞ tanhðbxÞ
ð45Þ
p : 2h
ð46Þ
we find Z
1
0
where b¼
With (45) in hand, substituting (43) into (42) yields Z tanhðbsÞ tanhðbxÞ 1 a 0 ds ¼ s0 x ; jxj < a: w ðsÞ ln p 0 tanhðbsÞ þ tanhðbxÞ g
ð47Þ
The resulting integral equation is a weakly singular integral equation with logarithmic kernel. With the help of the known results [7], the solution of (47) is found to be s0 1 coshðbxÞ wðxÞ ¼ cos ; jxj < a: ð48Þ bg coshðbaÞ Especially, for h ! 1, i.e. b ! 0, a simple expression for the half of the crack displacement jump across the crack surfaces wðxÞ ¼
s0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 ; g
jxj < a
ð49Þ
follows immediately from (48). Furthermore, a direct evaluation leads to the distribution of the stress as follows: s0 sinhðbxÞ szyðjÞ ðx; 0Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; sinh2 ðbxÞ sinh2 ðbaÞ
jxj > a; j ¼ 1; 2:
ð50Þ
In addition, the electric displacement and electric field along the crack plane may be obtained similarly. After some algebra, the final results are respectively 2 DyðjÞ ðx; 0Þ ¼
3
e15ð1Þ þ e15ð2Þ C12 6 sinhðbxÞ 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15s0 þ D0 ; c44ð1Þ þ c44ð2Þ C12 sinh2 ðbxÞ sinh2 ðbaÞ
ð51Þ
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ExðjÞ ðx; 0Þ ¼ 0
ð52Þ
for jxj > a, j ¼ 1; 2. The above expressions give the exact distribution of electroelastic field along the interface. It indicates that both stress and electric displacement near the crack tips exhibit an inverse squareroot singularity. 4.2. The impermeable case T
In this case, the system of dual integral equations for XðnÞ ¼ ðAðnÞ; CðnÞÞ can be written in the following compact form: Z
1
nXðnÞ cothðhnÞ cosðxnÞ dn ¼ M1 T0 ;
0 6 x < a;
ð53Þ
0
Z
1
XðnÞ cosðxnÞ dn ¼ 0;
ð54Þ
x > a:
0
As compared to dual integral equations (41) and (42), one finds that they are the same in structure. So an entirely analogous procedure yields M1 T0 1 coshðbxÞ cos UðxÞ ¼ ; coshðbaÞ b
jxj < a;
ð55Þ
where UðxÞ denotes ðwðxÞ; /ðxÞÞT , defined by wðxÞ ¼
wð1Þ ðx; 0þ Þ wð2Þ ðx; 0 Þ ; 2
/ðxÞ ¼
/ð1Þ ðx; 0þ Þ /ð2Þ ðx; 0 Þ : 2
ð56Þ
As a consequence, the tangential components jumps of strain and electric field across the crack, denoted as czx ðxÞ and Ex ðxÞ, respectively, can be evaluated as M1 T0 sinhðbxÞ ðczx ðxÞ; Ex ðxÞÞT ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; sinh2 ðbaÞ sinh2 ðbxÞ
jxj < a;
ð57Þ
where czx ðxÞ ¼
czxð1Þ ðx; 0þ Þ czxð2Þ ðx; 0 Þ ; 2
Ex ðxÞ ¼
Exð1Þ ðx; 0þ Þ Exð2Þ ðx; 0 Þ : 2
ð58Þ
Further, the normal components of stress and electric displacement along the interface can be obtained to be
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
szyðjÞ ðx; 0Þ DyðjÞ ðx; 0Þ
T0 sinhðbxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; sinh2 ðbxÞ sinh2 ðbaÞ
jxj > a:
1415
ð59Þ
Unlike the normal components of stress and electric displacement, the normal components of strain and electric field along the interface have two different values, respectively, depending upon the directions from the upper and lower layers, due to their discontinuity at the interface, i.e. czyð1Þ ðx; 0þ Þ 6¼ czyð2Þ ðx; 0 Þ and Eyð1Þ ðx; 0þ Þ 6¼ Eyð2Þ ðx; 0 Þ. A comparison reveals that stress, strain, and electric displacement near the crack tips have the singularity of an inverse square-root, while electric field near the crack tips is dominated by the same singularity for an impermeable crack, but uniform for a permeable crack. 5. Field intensity factors and energy release rate In analyzing crack problems in piezoelectric materials, the energy release rate, or J -integral, defined as the change of energy of a cracked piezoelectric material for an infinitesimal crack extension, is a significant fracture criterion [4,10,31]. Assume that under applied electromechanical loading the crack tip advances along the crack plane from x ¼ a to x ¼ a þ d (0 < d a). The energy release rate per unit length during this process is given by GIII
1 ¼ lim d!0 d
Z
aþd
TTy ðxÞUða þ d xÞ dx;
ð60Þ
a
where Ty ðxÞ ¼ ðszy ðx; 0Þ; Dy ðx; 0ÞÞT is that ahead of the crack tip prior to extension, UðxÞ ¼ T ðwðxÞ; /ðxÞÞ is that at the crack surfaces posterior to extension. To obtain GIII , it is expedient to determine first field intensity factors near the crack tips. Prior to the presentation of general results, we consider a piezoelectric plate made of two layers of equal thickness. Based on analytical expressions obtained above for electroelastic field, field intensity factors can be calculated immediately as s D T ; KIII Þ ðKIII
¼
e15ð1Þ þ e15ð2Þ C12 1; c44ð1Þ þ c44ð2Þ C12
T
pffiffiffiffiffiffi Y s0 pa;
pffiffiffiffiffiffi c E T ; KIII Þ ¼ ðg1 ; 0ÞT Y s0 pa ðKIII
ð61Þ ð62Þ
for a permeable interface crack, and pffiffiffiffiffiffi s D T ; KIII Þ ¼ T0 Y pa; ðKIII
pffiffiffiffiffiffi c E T ðKIII ; KIII Þ ¼ M1 T0 Y pa
ð63Þ
for an impermeable crack, with rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanhðbaÞ ; Y ¼ ba
ð64Þ
1416
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where field intensity factors are defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s D T ðKIII ; KIII Þ ¼ limþ 2pðx aÞðszy ðx; 0Þ; Dy ðx; 0ÞÞT ;
ð65Þ
x!a
T
c E ðKIII ; KIII Þ ¼ lim x!a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2pða xÞðczx ðx; 0Þ; Ex ðx; 0ÞÞ :
ð66Þ
Furthermore, upon knowing field intensity factors, evaluating the integral (60) the energy release rate yields GIII ¼
pffiffiffiffiffiffi 1 ½Y s0 pa 2 2g
ð67Þ
for a permeable interface crack, and pffiffiffiffiffiffi GIII ¼ 12TT0 M1 T½Y s0 pa 2
ð68Þ
for an impermeable crack. These provide us with analytic formulae for calculating the required field intensity factors and energy release rate. We in turn consider an eccentric interface crack, i.e. the upper and lower piezoelectric layers are not of equal thickness. In this case, a governing integral equation for the problem under discussion can be solved via some numerical schemes. According to ordinary treatments, Fredholm integral equations (32) and (38) may be discretized by use of some known integration formulae and then converted to a finite system of algebraic equations. Once the auxiliary functions wðxÞ and WðuÞ are determined from Eqs. (32) and (38), the asymptotic electroelastic field in the vicinity of a crack tip is obtainable. For example, to get the distribution of szyð1Þ ðx; 0þ Þ near the right crack tip, say, inserting (25) into (14), after neglecting certain lower-order terms, yields the asymptotic stress field ahead of the right crack tip as follows: s KIII szyðjÞ ða þ r; 0Þ ¼ pffiffiffiffiffiffiffi ; 2pr
j ¼ 1; 2;
ð69Þ
where pffiffiffiffiffiffi s KIII ¼ wð1Þ pa:
ð70Þ
Similarly, we have 0
w ða rÞ ¼
c KIII
rffiffiffiffiffi 2r ; p
Ex ðxÞ ¼ 0;
D KIII DyðjÞ ða þ r; 0Þ ¼ pffiffiffiffiffiffiffi ; 2pr
j ¼ 1; 2;
ð71Þ
where c ¼ KIII
wð1Þ pffiffiffiffiffiffi pa; g
D ¼ KIII
e15ð1Þ þ e15ð2Þ C12 pffiffiffiffiffiffi wð1Þ pa: c44ð1Þ þ c44ð2Þ C12
ð72Þ
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
1417
Thus, the energy release rate is easily obtained as GIII ¼
pa ½wð1Þ 2 2g
ð73Þ
for a permeable interface crack. On the other hand, a similar evaluation can result in field intensity factors as pffiffiffiffiffiffi s D T ðKIII ; KIII Þ ¼ Wð1Þ pa;
pffiffiffiffiffiffi c E T ðKIII ; KIII Þ ¼ M1 Wð1Þ pa
ð74Þ
and the energy release rate as GIII ¼
pa T Wð1Þ M1 Wð1Þ 2
ð75Þ
for an impermeable interface crack. Strictly speaking, the kernels of Eqs. (32) and (38) depend on the bimaterial constants and the ratio of thickness of the upper and the lower layers to the crack length, and so do field intensity factors. However, if the upper and lower piezoelectric layers are with the same materials, we have m11 ¼ m22 ¼ 1, m12 ¼ m21 ¼ 0, g ¼ c44 , gij ðnÞ=gij ¼ 2=ð1 þ H 12 ðnÞÞ. Under such circumstances, the kernels of Eqs. (32) and (38) are independent of the material constants, and so the stress intensity factor is independent of material properties for both the permeable and impermeable cracks, which has been treated by an approximate technique in Li [17]. As a check, if taking h1 ! 1, h2 ! 1, the kernels of Eqs. (32) and (38) vanish and the closedform solutions for an interface crack between two bonded dissimilar piezoelectric half-planes can be easily obtained, which are in accordance with the results (67) and (68), respectively, when h ! 1.
6. Numerical results In this section, the dependence of field intensity factors and energy release rate upon the crack geometry is examined. Prior to the presentation of numerical results, a comparison of the norpffiffiffiffiffiffi s =s0 pa, between the analytic and numerical results for a central malized stress intensity factor, KIII interface crack inpaffiffiffiffiffi piezoelectric plate is shown in Fig. 2, where the solid line denotes the analytical ffi s solution, KIII =s0 pa ¼ Y , calculated from (64), and the dashed line stands for the numerical solution via solving Eqs. (32) or (38) corresponding to the permeable or impermeable cases. It indicates that the numerical procedure provides a very accurate result. In the following numerical examples, we choose PZT-5H as the upper piezoelectric layer and PZT-6B as the lower layer. The relevant material constants are listed in Table 1. For the case of clamped boundaries, i.e. W0 ¼ 0, electric field E0 ¼ V0 =ðh1 þ h2 Þ gives pffiffiffiffiffiffi rise to the stress concens =E0 pa, with the ratio h2 =h1 for tration, and the variations of the stress intensity factor, KIII h1 =a ¼ 1; 2 are presented in Fig. 3. Here the negative stress field induced by E0 represents only the direction opposite to the z-axis, and so a negative electric loading E0 will cause a positive stress field. Moreover, like the stress field near the crack tips, the electric displacement in the vicinity of
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Fig. 2. Comparison of the results obtained from analytic and numerical solutions for a central interface crack.
Table 1 The relevant material constants 10
2
c44 (10 N/m ) e15 (C/m2 ) e11 (1010 C/Vm)
PZT-5H
PZT-6B
3.53 17 151
2.71 4.6 36
the crack tips is also dominated by a square-root singularity, and the curves of the corresponding intensity factor with the ratio h2 =h1 for h1 =a ¼ 1; 2 are plotted in Fig. 4. It is observed that the intensity factors of stress and electric displacement decrease monotonically in magnitude and
Fig. 3. Stress intensity factors versus h2 =h1 when h1 =a ¼ 1; 2 for a permeable and an impermeable crack.
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
1419
Fig. 4. Electric-displacement intensity factors versus h2 =h1 when h1 =a ¼ 1; 2 for a permeable and an impermeable crack.
approaches gradually a constant, respectively, with increasing the ratio h2 =h1 for a permeable interface crack, while for an impermeable interface crack, they increase in magnitude as h2 =h1 is small, reaching a peak, and then decrease as h2 =h1 is raised. Furthermore, for special values of h1 =a ¼ 1 and h2 =h1 ¼ 1; 2, the dependence of energy release rate GIII upon applied electric loadings E0 , E0 ¼ V =ðh1 þ h2 Þ, is displayed in Fig. 5 for a permeable interface crack and an impermeable crack of length 2a ¼ 0:02 m. It is found that in the case of clamped boundaries, the energy release rate GIII is always positive for a permeable interface crack, and is always negative for an impermeable interface crack, irrespective of the direction of applied
Fig. 5. The dependence of energy release rates on applied electric loading for a permeable or an impermeable crack in a piezoelectric plate.
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electric loadings, which coincide with those for an anti-plane shear hole in an infinite piezoelectric plane by Zhang and Tong [37]. Since negative energy release rates, or negative crack driving-force, lack clear physical interpretation, electric boundary conditions for the former are more suitable than those for the latter.
Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant No. 10272043.
Appendix Some constants and functions are defined below m11 ¼
c44ð1Þ e11ð2Þ þ e15ð1Þ e15ð2Þ ; c44ð2Þ e11ð2Þ þ e215ð2Þ
m12 ¼
e15ð1Þ e11ð2Þ e11ð1Þ e15ð2Þ ; c44ð2Þ e11ð2Þ þ e215ð2Þ
m21 ¼
c44ð1Þ e15ð2Þ e15ð1Þ c44ð2Þ ; c44ð2Þ e11ð2Þ þ e215ð2Þ
m22 ¼
e11ð1Þ c44ð2Þ þ e15ð1Þ e15ð2Þ c44ð2Þ e11ð2Þ þ e215ð2Þ
a1 ¼
c0 ðh1 þ h2 m22 Þ þ E0 h2 m12 ; D
a2 ¼
c0 ðh1 m11 þ h2 C2 Þ E0 h1 m12 ; D
b1 ¼
c0 h2 m21 þ E0 ðh1 þ h2 m11 Þ ; D
b2 ¼
c0 h1 m21 E0 ðh1 m22 þ h2 C12 Þ ; D
D¼
ðh1 þ h2 m11 Þðh1 þ h2 m22 Þ h22 m12 m21 ; h1 þ h2
C12 ¼
c44ð1Þ e11ð1Þ þ e215ð1Þ c44ð2Þ e11ð2Þ þ e215ð2Þ
;
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
c0 ¼
W0 ; h1 þ h2
E0 ¼
V0 ; h1 þ h2
g11 ¼
c44ð1Þ þ c44ð2Þ C12 c44ð2Þ e11ð2Þ þ e215ð2Þ ; c44 e11 þ e215 2
g12 ¼
e15ð1Þ þ e15ð2Þ C12 c44ð2Þ e11ð2Þ þ e215ð2Þ ; c44 e11 þ e215 2
g22 ¼
e11ð1Þ þ e11ð2Þ C12 c44ð2Þ e11ð2Þ þ e215ð2Þ ; 2 c44 e11 þ e215
g11 ðnÞ ¼
2c44ð1Þ H 12 ðnÞ þ 2c44ð2Þ C12 ; ½H 12 ðnÞ þ m11 ½H 12 ðnÞ þ m22 m12 m21
g12 ðnÞ ¼
2e15ð1Þ H 12 ðnÞ þ 2e15ð2Þ C12 ; ½H 12 ðnÞ þ m11 ½H 12 ðnÞ þ m22 m12 m21
g22 ðnÞ ¼
2e11ð1Þ H 12 ðnÞ þ 2e11ð2Þ C12 ; ½H 12 ðnÞ þ m11 ½H 12 ðnÞ þ m22 m12 m21
h1 nÞ cothð h2 nÞ. where H 12 ðnÞ ¼ tanhð References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
H.G. Boem, S.N. Atluri, Int. J. Fract. 75 (1996) 163. Z.T. Chen, S. Yu, B.L. Karihaloo, Int. J. Fract. 86 (1997) L9. J. Cheng, C. Qian, M. Zhao, S.W.R. Lee, T.Y. Zhang, Smart. Mater. Struct. 39 (2000) 2477. C. Dascalu, G.A. Maugin, Int. J. Eng. Sci. 32 (1994) 755. W. Deng, S.A. Meguid, Acta Mech. 137 (1999) 161. M.L. Dunn, Eng. Fract. Mech. 48 (1994) 25. R. Estrada, R.P. Kanwal, IMA J. Appl. Math. 45 (1989) 133. R. Fu, C.-F. Qian, T.-Y. Zhang, Appl. Phys. Lett. 76 (2000) 126. C.-F. Gao, M.-Z. Wang, Int. J. Solids Struct. 38 (2001) 5323. H. Gao, T.-Y. Zhang, P. Tong, J. Mech. Phys. Solids 45 (1997) 491. V.B. Govorukha, V.V. Lobada, Acta Mech. 140 (2000) 233. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1980. B. Gu, S. Wang, S. Yu, D. Gross, Eng. Fract. Mech. 69 (2002) 565. M.S. Hou, F.L. Mei, Chin. Sci. Bull. 43 (1998) 216. X.-F. Li, T.-Y. Fan, X.-F. Wu, Int. J. Eng. Sci. 38 (2000) 1219. X.-F. Li, X.-Y. Duan, Mech. Res. Comm. 28 (2001) 703. X.-F. Li, Int. J. Solids Struct. 39 (2002) 1097. S.A. Meguid, X.D. Wang, Int. J. Fract. 91 (1998) 391.
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F. Narita, Y. Shindo, Theor. Appl. Fract. Mech. 30 (1998) 119. F. Narita, Y. Shindo, Int. J. Fract. 98 (1999) 87. S.B. Park, C.T. Sun, J. Am. Ceram. Soc. 78 (1995) 1475. Q.-H. Qin, S. Yu, Int. J. Solids Struct. 34 (1997) 581. C. Ru, J. Appl. Mech. 67 (2000) 255. G.A. Schneider, V. Heyer, J. Europ. Ceram. Soc. 19 (1999) 1299. S. Shen, T. Nishioka, Z.-B. Kuang, Z. Liu, Mech. Mater. 32 (2000) 57. Y. Shindo, K. Tanaka, F. Narita, Acta Mech. 120 (1997) 31. Y. Shindo, K. Minamida, F. Narita, Smart. Mater. Struct. 11 (2002) 534. Y. Shindo, H. Murakami, K. Horiguchi, F. Narita, J. Am. Ceram. Soc. 85 (2002) 1243. I.N. Sneddon, The Use of Integral Transforms, McGraw Hill, New York, 1972. H. Sosa, N.M. Khutoryansky, Int. J. Solids Struct. 33 (1996) 3399. Z. Suo, C.-M. Kuo, D.M. Barnett, J.R. Willis, J. Mech. Phys. Solids 40 (1992) 739. T.C. Wang, X.L. Han, Int. J. Fract. 98 (1999) 15. X.D. Wang, Int. J. Solids Struct. 38 (2001) 815. B.L. Wang, J.C. Han, S.Y. Du, Eng. Fract. Mech. 61 (1998) 607. B.L. Wang, J.C. Han, S.Y. Du, Int. J. Solids Struct. 38 (2000) 5219. H. Wang, R.N. Singh, J. Appl. Phys. 81 (1997) 7471. T.-Y. Zhang, P. Tong, Int. J. Solids Struct. 33 (1996) 343. X. Zhao, S.A. Meguid, Int. J. Solids Struct. 39 (2002) 2477.
Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate X.-F. Li a
a,b,*
, G.J. Tang
b
College of Mathematics and Physics, Hunan Normal University, Changsha, Hunan 410081, China b School of Aerospace and Materials Engineering, National University of Defence Technology, Changsha, Hunan 410073, China Received 5 June 2002; received in revised form 16 October 2002; accepted 6 November 2002
Abstract The problem of an anti-plane interface crack in a layered piezoelectric plate composed of two bonded dissimilar piezoelectric ceramic layers subjected to applied voltage is considered. It is assumed that the crack is either impermeable or permeable. An integral transform technique is employed to reduce the problem considered to dual integral equations, then to a Fredholm integral equation by introducing an auxiliary function. Field intensity factors and energy release rate are obtained in explicit form in terms of the auxiliary function. In particular, by solving analytically a resulting singular integral equation, they are determined explicitly in terms of given electromechanical loadings for the case of two bonded layers of equal thickness. Some numerical results are presented graphically to show the influence of the geometric parameters on the field intensity factors and the energy release rate. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Piezoelectric plate; Interface crack; Electroelastic analysis; Field intensity factors; Energy release rate
1. Introduction Piezoelectric materials have been used widely in technology such as transducers, actuators, sensors, etc., due to their intrinsic coupling characteristics between electric and elastic behaviors. Great progress on the study of electroelastic field disturbed by cracks in a piezoelectric material has been made in recent years. Considerable researches in this area are mainly focused on *
Corresponding author. E-mail addresses: xfl[email protected], [email protected] (X.-F. Li).
0020-7225/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0020-7225(03)00038-7
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determining the electroelastic field for a cracked piezoelectric material subjected to various electromechanical loadings and many analytical solutions have been derived in theory. In piezoelectric composite materials, cracks have been found and they usually propagate along the interface. Therefore analyzing electric and elastic behaviors of an interface crack is a prerequisite for studying the failure of piezoelectric composite materials. The investigation of the problem involving interface cracks between two bonded dissimilar piezoelectric materials has recently attracted increasing attention. So far many papers have been devoted to the analysis of electroelastic field of an interface crack such as Suo et al. [31], Boem and Atluri [1], Chen et al. [2], Qin and Yu [22], Hou and Mei [14], Deng and Meguid [5], Wang and Han [32], Ru [23], Govorukha and Lobada [11], Gao and Wang [9], etc. The dynamic behavior of interface cracks in a piezoelectric composite structure has been studied by Meguid and Wang [18], Li et al. [15], Wang [33], etc. The above-mentioned works are mainly related to internal cracks embedded in an infinite piezoelectric sheet. However, some practical situations in engineering applications should be posed with a layered structure having at least two layers. As a result, it is crucial to determine the electric and elastic behaviors disturbed by an interface crack in a layered piezoelectric composite material, in particular in the close vicinity of the crack tip. Along this line, Narita and Shindo [19,20] considered an anti-plane crack located at the interface between a piezoelectric material and an orthotropic elastic medium, and determined the stress intensity factor via solving numerically a singular integral equation by use of the Gauss–Chebyshev integration formula. Nonlinear fracture of piezoelectric materials with an interface crack has been treated [25]. Recently, transient response of a crack in a piezoelectric bimaterial and in a multilayered piezoelectric material under anti-plane mechanical and in-plane electric impacts has been studied by Wang et al. [34,35] and Gu et al. [13], respectively. For a piezoelectric laminate containing multiple interfacial collinear cracks subjected to steady-state anti-plane mechanical and in-plane electric loadings, the dynamic behavior of electroelastic field has been analyzed by Zhao and Meguid [38]. For two arc–curved interface cracks between a piezoelectric fiber and an elastic matrix, the dynamic behavior under a steady-state anti-plane shear wave has been dealt with by Shindo et al. [26,27]. As compared to a lot of theoretical works, experimental results on fracture of a cracked piezoelectric material are very limited [3,8,21,24,28,36]. This paper is concerned with the static problem of an anti-plane shear crack situated at the interface of a layered piezoelectric plate consisting of two bonded dissimilar piezoelectric layers. Within the framework of the reigning linear piezoelectricity theory, the associated mixed boundary value problem, by using an integral transform technique, is reduced to a Fredholm integral equation of the second kind. Under the assumptions of permeable and impermeable cracks, field intensity factors and energy release rate are determined numerically, respectively. Moreover, for the special case of two bonded layers of equal thickness, all the field intensity factors and energy release rate can be obtained analytically. In contrast to numerical results, analytical solution serves a benchmark and provides us with an approach for justifying accuracy of certain numerical results.
2. Formulation of the problem Consider a piezoelectric plate made of two bonded dissimilar piezoelectric layers of thicknesses h1 and h2 , whose interface is along the xz-plane, and whose poling directions are oriented in the
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
1407
Fig. 1. Schematic of an interface crack in a layered piezoelectric plate made of two bonded dissimilar layers.
z-axis, as shown in Fig. 1, in which the z-axis is not depicted. Plate dimensions along the x- and zaxes are infinite, and a crack of length 2a is located at the interface and penetrates through the piezoelectric plate along the z-axis. From the viewpoint of experiment, it is much simpler to measure or impose applied voltage between the two surfaces of the layered piezoelectric plate than the charge. Moreover, in certain practical applications of piezoelectric devices such as actuators or sensors, piezoelectric layers are sandwiched between exterior media. Consequently, it is assumed that the plate surfaces are rigidly clamped, or have a relative displacement along the z-axis. Thus, the plate geometry and electromechanical loadings at the plate surfaces are independent of z; so the electric and elastic boundary conditions at the plate surfaces can be stated as /ð1Þ ðx; h1 Þ /ð2Þ ðx; h2 Þ ¼ V0 ;
wð1Þ ðx; h1 Þ wð2Þ ðx; h2 Þ ¼ W0 ;
1 < x < 1:
ð1Þ
Here wðjÞ ðx; yÞ and /ðjÞ ðx; yÞ represent the out-of-plane displacements and the in-plane electric potentials, and the subscripts j ¼ 1 and j ¼ 2 specify the quantities in the upper and lower layers, respectively. It is easily seen that for the problem in question, the in-plane elastic deformation is related neither to electric field nor to anti-plane deformation with respect to the xy-plane, and anti-plane deformation is coupled with electric field. Therefore, for the problem stated above, wðjÞ ðx; yÞ and /ðjÞ ðx; yÞ satisfy the basic governing differential equations for anti-plane piezoelectricity, in the absence of body forces and free charges, c44ðjÞ r2 wðjÞ þ e15ðjÞ r2 /ðjÞ ¼ 0;
e15ðjÞ r2 wðjÞ e11ðjÞ r2 /ðjÞ ¼ 0;
ð2Þ
where c44ðjÞ , e11ðjÞ , and e15ðjÞ are the elastic stiffness measured in a constant electric field, the dielectric permittivity measured at a uniform strain, the piezoelectric constant, respectively, and r2 stands for the two-dimensional Laplacian operator. Once functions wðjÞ and /ðjÞ are determined from given boundary conditions, then the components of anti-plane shear stress, strain, in-plane electric displacement, and electric field in each piezoelectric layer are obtainable in terms of the following equations: szxðjÞ ¼ c44ðjÞ czxðjÞ e15ðjÞ ExðjÞ ;
szyðjÞ ¼ c44ðjÞ czyðjÞ e15ðjÞ EyðjÞ ;
ð3Þ
DxðjÞ ¼ e15ðjÞ czxðjÞ þ e11ðjÞ ExðjÞ ;
DyðjÞ ¼ e15ðjÞ czyðjÞ þ e11ðjÞ EyðjÞ ;
ð4Þ
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X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
czxðjÞ ¼ owðjÞ =ox; ExðjÞ ¼ o/ðjÞ =ox;
czyðjÞ ¼ owðjÞ =oy;
ð5Þ
EyðjÞ ¼ o/ðjÞ =oy:
ð6Þ
In order to obtain the desired electroelastic field, appropriate boundary conditions at the crack surfaces must be furnished. Clearly, elastic boundary conditions at the crack surfaces are szyð1Þ ðx; 0þ Þ ¼ szyð2Þ ðx; 0 Þ ¼ 0;
jxj < a:
ð7Þ
For a hole in an infinite piezoelectric material, electric boundary conditions at the rim of the hole have been exactly analyzed by Dunn [6], Zhang and Tong [37], and Sosa and Khutoryansky [30]. However, for a slit-like crack, two types of electric boundary conditions at the crack surfaces prevail, the permeable and impermeable assumptions. For the present study, they are stated as /ð1Þ ðx; 0þ Þ ¼ /ð2Þ ðx; 0 Þ;
Dyð1Þ ðx; 0þ Þ ¼ Dyð2Þ ðx; 0 Þ;
jxj < a
ð8Þ
for a permeable interface crack, or Dyð1Þ ðx; 0þ Þ ¼ Dyð2Þ ðx; 0 Þ ¼ 0;
jxj < a
ð9Þ
for an impermeable interface crack. In addition, perfect bonding exists between two dissimilar piezoelectric layers; so electric and elastic field should fulfill the continuity conditions along the bonded interface wð1Þ ðx; 0þ Þ ¼ wð2Þ ðx; 0 Þ;
szyð1Þ ðx; 0þ Þ ¼ szyð2Þ ðx; 0 Þ;
jxj > a;
ð10Þ
/ð1Þ ðx; 0þ Þ ¼ /ð2Þ ðx; 0 Þ;
Dyð1Þ ðx; 0þ Þ ¼ Dyð2Þ ðx; 0 Þ;
jxj > a:
ð11Þ
3. Solution of the problem Due to the symmetry of the problem under consideration, it is sufficient to analyze the righthalf portion, x P 0. Therefore in the following we confine our attention to the region x P 0. To solve the problem stated above, it is convenient to employ an integral transform technique to reduce the associated mixed boundary value problem to dual integral equations. For this purpose, by use of the Fourier cosine transform, it is easy to obtain that an appropriate solution of Eq. (2) can be expressed as the following integrals: Z 1 ½Aj ðnÞ coshðynÞ þ Bj ðnÞ sinhðynÞ cosðxnÞ dn þ aj y; ð12Þ wðjÞ ðx; yÞ ¼ 0
/ðjÞ ðx; yÞ ¼
Z
1
½Cj ðnÞ coshðynÞ þ Dj ðnÞ sinhðynÞ cosðxnÞ dn þ bj y; 0
ð13Þ
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
1409
where x P 0, 0 6 y 6 h1 for j ¼ 1, h2 6 y 6 0 for j ¼ 2, Aj ðnÞ; . . . ; Dj ðnÞ ðj ¼ 1; 2Þ are unknown functions to be determined from the conditions given above, and aj and bj are to be determined by use of the boundary conditions (1) and the interface continuity conditions, which are given in the Appendix. With the aid of constitutive equations (3) and (4), from (12) and (13) it is a simple matter to obtain the expressions for the components of the stress, strain, electric displacement and electric field in terms of Aj ðnÞ; . . . ; Dj ðnÞ ðj ¼ 1; 2Þ. For instance, we have Z 1 n½ðc44ðjÞ Aj þ e15ðjÞ Cj Þ sinhðynÞ þ ðc44ðjÞ Bj þ e15ðjÞ Dj Þ coshðynÞ cosðxnÞ dn þ s0 ; szyðjÞ ¼ 0
ð14Þ DyðjÞ ¼
Z
1
n½ðe15ðjÞ Aj e11ðjÞ Cj Þ sinhðynÞ þ ðe15ðjÞ Bj e11ðjÞ Dj Þ coshðynÞ cosðxnÞ dn þ D0 0
ð15Þ with s0 ¼
c0 ðc44ð1Þ h1 þ c44ð2Þ h2 C12 Þ E0 ðe15ð1Þ h1 þ e15ð2Þ h2 C12 Þ ; D
ð16Þ
c0 ðe15ð1Þ h1 þ e15ð2Þ h2 C12 Þ þ E0 ðe11ð1Þ h1 þ e11ð2Þ h2 C12 Þ ; D
ð17Þ
D0 ¼
where c0 , E0 , C12 , and D are defined in the appendix. Our objective is to obtain electroelastic field, in particular in the vicinity of a crack tip. Substituting the expressions for szyðjÞ and DyðjÞ from (12) and (13) into (1) yields four algebraic equations for Aj ðnÞ; . . . ; Dj ðnÞ, from which one gets B1 ðnÞ ¼ A1 ðnÞ cothðh1 nÞ; B2 ðnÞ ¼ A2 ðnÞ cothðh2 nÞ;
D1 ðnÞ ¼ C1 ðnÞ cothðh1 nÞ; D2 ðnÞ ¼ C2 ðnÞ cothðh2 nÞ:
ð18Þ ð19Þ
Using the above results, in conjunction with the interface continuity conditions one finds A1 ðnÞ ¼
½H12 ðnÞ þ m22 A m12 C 2H12 ðnÞ; ½H12 ðnÞ þ m11 ½H12 ðnÞ þ m22 m12 m21
ð20Þ
C1 ðnÞ ¼
½H12 ðnÞ þ m11 C m21 A 2H12 ðnÞ; ½H12 ðnÞ þ m11 ½H12 ðnÞ þ m22 m12 m21
ð21Þ
where 2AðnÞ ¼ A1 ðnÞ A2 ðnÞ, 2CðnÞ ¼ C1 ðnÞ C2 ðnÞ, H12 ðnÞ ¼ tanhðh1 nÞ cothðh2 nÞ, mij Õs are given in the appendix. The functions AðnÞ and CðnÞ are new unknown functions, which may be determined via the remaining boundary conditions. In what follows two cases of electric boundary conditions at the crack surfaces will be considered, respectively.
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3.1. The permeable case In this case, employing the electric boundary conditions for a permeable crack yields CðnÞ ¼ 0. Furthermore, substituting the above into the remaining boundary conditions, (7) and the first of (10), results in dual integral equations for AðnÞ: Z 1 ngðnÞ cothðh2 nÞAðnÞ cosðxnÞ dn ¼ s0 ; 0 6 x < a; ð22Þ 0
Z
1
AðnÞ cosðxnÞ dn ¼ 0;
x > a;
ð23Þ
0
where gðnÞ ¼
2c44ð1Þ ½H12 ðnÞ þ m22 2e15ð1Þ m21 : ½H12 ðnÞ þ m11 ½H12 ðnÞ þ m22 m12 m21
ð24Þ
Due to the complicated form of gðnÞ, it seems unlikely that a closed-form solution of dual integral equations (22) and (23) is obtainable except for a special case when h1 ¼ h2 . For the latter, a closed-form solution can be derived analytically, which is given in the next section. Next, dual integral equations (22) and (23) can be transformed into a Fredholm integral equation of the second kind. To achieve this, following Sneddon [29], we choose AðnÞ given by Z a wðuÞJ0 ðunÞ du; ð25Þ AðnÞ ¼ 0
where J0 ð Þ denotes the Bessel function of the first kind of order zero, and wðuÞ is an unknown auxiliary function. Recalling the result [12] Z 1 J0 ðunÞ cosðxnÞ dn ¼ 0; x > u; ð26Þ 0
inserting (25) into (23) indicates that Eq. (23) willpbe automatically satisfied. On the other hand, ffiffiffiffiffiffiffiffiffiffiffiffiffi we substitute (25) into (22), then multiply by 2=ðp t2 x2 Þ, and integrate with x between 0 and t. Making use of the identity 2 J0 ðtnÞ ¼ p
Z
t 0
cosðxnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi dx; t2 x2
we find that Eq. (22) then becomes a Fredholm integral equation of the second kind, Z a Kðx; u; h1 ; h2 ÞwðuÞ du ¼ xs0 ; 0 6 x < a; gwðxÞ þ g 0
with
ð27Þ
ð28Þ
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
Kðx; u; h1 ; h2 Þ ¼ x
Z
1
0
g¼
gðnÞ cothðh2 nÞ 1 nJ0 ðunÞJ0 ðxnÞ dn; g
1411
ð29Þ
c44ð1Þ þ c44ð2Þ C12 c44ð2Þ e11ð2Þ þ e215ð2Þ ; c44 e11 þ e215 2
ð30Þ
where c44 , e11 , and e15 represent the mean values of the corresponding material constants in the upper and lower layers, and gðnÞ ! g as n ! 1. For the purpose of numerical computation, we introduce the following notations: x x ¼ ; a
u u¼ ; a
h1 h1 ¼ ; a
h2 h2 ¼ ; a
gwðxÞ ; wðxÞ ¼ a
ð31Þ
and hence Eq. (28) is rewritten in a normalized form as wðxÞ þ
Z
1
Kðx; uÞwð uÞ d u ¼ xs0 ;
ð32Þ
0 6 x < 1;
0
where Kðx; uÞ ¼ Kðx; u; h1 ; h2 Þ:
ð33Þ
Taking (16) into account, in addition to c0 and E0 , s0 is related to the material constants and the crack geometry as well. 3.2. The impermeable case In a similar manner, employing the electric and elastic boundary conditions for an impermeable crack yields the system of dual integral equations for AðnÞ and CðnÞ, which may be written in the following compact form Z 1 nMðnÞXðnÞ cothðh2 nÞ cosðxnÞ dn ¼ T0 ; 0 6 x < a; ð34Þ 0
Z
1
XðnÞ cosðxnÞ dn ¼ 0;
ð35Þ
x > a;
0
where T
XðnÞ ¼ ðAðnÞ; CðnÞÞ ;
T
T0 ¼ ðs0 ; D0 Þ ;
MðnÞ ¼
g11 ðnÞ g12 ðnÞ
g12 ðnÞ ; g22 ðnÞ
ð36Þ
in which the superscript T stands for the transpose of a matrix, and the elements gij ðnÞ in MðnÞ are given in the Appendix.
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X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
By representing XðnÞ in terms of a new unknown WðuÞ as Z a XðnÞ ¼ WðuÞJ0 ðunÞ du;
ð37Þ
0
proceeding as before, according to the notations in (31) dual integral equations for XðnÞ then become a Fredholm integral equations in a normalized form Z 1 Wð uÞ þ Kðx; uÞM1 Wð uÞ d u ¼ xT0 ; 0 6 x < 1 ð38Þ 0
with
MWðuÞ Wð uÞ ¼ ; a K ij ðx; uÞ ¼ gij x
Z
M¼ 1
0
"
g11 g12
g12 ; g22 #
Kðx; uÞ ¼
K 11 ðx; uÞ K 12 ðx; uÞ
gij ðnÞ cothð h2 nÞ 1 nJ0 ðunÞJ0 ðxnÞ dn; gij
K 12 ðx; uÞ ; K 22 ðx; uÞ
i; j ¼ 1; 2:
ð39Þ
ð40Þ
In fact, gij ¼ gij ðnÞ as n ! 1. 4. Closed-form solution for a central interface crack In this section, we will derive a closed-form solution for a crack lying at the interface between two dissimilar piezoelectric layers of equal thickness, i.e. h1 ¼ h2 ¼ h. In this case, H12 ðnÞ ¼ 1. 4.1. The permeable case Dual integral equations (22) and (23) for a permeable crack reduce to Z 1 AðnÞ cosðxnÞ dn ¼ 0; x > a;
ð41Þ
0
Z
1
nAðnÞ cothðhnÞ cosðxnÞ dn ¼ s0 ;
g
0 6 x < a:
ð42Þ
0
Owing to the special form of dual integral equations, a closed-form solution is obtainable. To this end, as in Li and Duan [16], we choose AðnÞ in terms of w0 ðxÞ: Z a 2 w0 ðsÞ sinðsnÞ ds; ð43Þ AðnÞ ¼ pn 0 instead of (25), where 2wðxÞ is the crack displacement jump across the crack surfaces, i.e. 2wðxÞ ¼ wð1Þ ðx; 0þ Þ wð2Þ ðx; 0 Þ, the prime denotes differentiation with respect to the argument. Taking into account wðxÞ ¼ 0 for jxj P a, Eq. (41) will be fulfilled identically.
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
1413
Using the following result [12] Z
1
0
pu cothðanÞ cosðunÞ dn ¼ ln 2 sinh ; n 2a
ð44Þ
cothðhnÞ sinðsnÞ sinðxnÞ 1 tanhðbsÞ þ tanhðbxÞ ; dn ¼ ln n 2 tanhðbsÞ tanhðbxÞ
ð45Þ
p : 2h
ð46Þ
we find Z
1
0
where b¼
With (45) in hand, substituting (43) into (42) yields Z tanhðbsÞ tanhðbxÞ 1 a 0 ds ¼ s0 x ; jxj < a: w ðsÞ ln p 0 tanhðbsÞ þ tanhðbxÞ g
ð47Þ
The resulting integral equation is a weakly singular integral equation with logarithmic kernel. With the help of the known results [7], the solution of (47) is found to be s0 1 coshðbxÞ wðxÞ ¼ cos ; jxj < a: ð48Þ bg coshðbaÞ Especially, for h ! 1, i.e. b ! 0, a simple expression for the half of the crack displacement jump across the crack surfaces wðxÞ ¼
s0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 ; g
jxj < a
ð49Þ
follows immediately from (48). Furthermore, a direct evaluation leads to the distribution of the stress as follows: s0 sinhðbxÞ szyðjÞ ðx; 0Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; sinh2 ðbxÞ sinh2 ðbaÞ
jxj > a; j ¼ 1; 2:
ð50Þ
In addition, the electric displacement and electric field along the crack plane may be obtained similarly. After some algebra, the final results are respectively 2 DyðjÞ ðx; 0Þ ¼
3
e15ð1Þ þ e15ð2Þ C12 6 sinhðbxÞ 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15s0 þ D0 ; c44ð1Þ þ c44ð2Þ C12 sinh2 ðbxÞ sinh2 ðbaÞ
ð51Þ
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X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
ExðjÞ ðx; 0Þ ¼ 0
ð52Þ
for jxj > a, j ¼ 1; 2. The above expressions give the exact distribution of electroelastic field along the interface. It indicates that both stress and electric displacement near the crack tips exhibit an inverse squareroot singularity. 4.2. The impermeable case T
In this case, the system of dual integral equations for XðnÞ ¼ ðAðnÞ; CðnÞÞ can be written in the following compact form: Z
1
nXðnÞ cothðhnÞ cosðxnÞ dn ¼ M1 T0 ;
0 6 x < a;
ð53Þ
0
Z
1
XðnÞ cosðxnÞ dn ¼ 0;
ð54Þ
x > a:
0
As compared to dual integral equations (41) and (42), one finds that they are the same in structure. So an entirely analogous procedure yields M1 T0 1 coshðbxÞ cos UðxÞ ¼ ; coshðbaÞ b
jxj < a;
ð55Þ
where UðxÞ denotes ðwðxÞ; /ðxÞÞT , defined by wðxÞ ¼
wð1Þ ðx; 0þ Þ wð2Þ ðx; 0 Þ ; 2
/ðxÞ ¼
/ð1Þ ðx; 0þ Þ /ð2Þ ðx; 0 Þ : 2
ð56Þ
As a consequence, the tangential components jumps of strain and electric field across the crack, denoted as czx ðxÞ and Ex ðxÞ, respectively, can be evaluated as M1 T0 sinhðbxÞ ðczx ðxÞ; Ex ðxÞÞT ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; sinh2 ðbaÞ sinh2 ðbxÞ
jxj < a;
ð57Þ
where czx ðxÞ ¼
czxð1Þ ðx; 0þ Þ czxð2Þ ðx; 0 Þ ; 2
Ex ðxÞ ¼
Exð1Þ ðx; 0þ Þ Exð2Þ ðx; 0 Þ : 2
ð58Þ
Further, the normal components of stress and electric displacement along the interface can be obtained to be
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
szyðjÞ ðx; 0Þ DyðjÞ ðx; 0Þ
T0 sinhðbxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; sinh2 ðbxÞ sinh2 ðbaÞ
jxj > a:
1415
ð59Þ
Unlike the normal components of stress and electric displacement, the normal components of strain and electric field along the interface have two different values, respectively, depending upon the directions from the upper and lower layers, due to their discontinuity at the interface, i.e. czyð1Þ ðx; 0þ Þ 6¼ czyð2Þ ðx; 0 Þ and Eyð1Þ ðx; 0þ Þ 6¼ Eyð2Þ ðx; 0 Þ. A comparison reveals that stress, strain, and electric displacement near the crack tips have the singularity of an inverse square-root, while electric field near the crack tips is dominated by the same singularity for an impermeable crack, but uniform for a permeable crack. 5. Field intensity factors and energy release rate In analyzing crack problems in piezoelectric materials, the energy release rate, or J -integral, defined as the change of energy of a cracked piezoelectric material for an infinitesimal crack extension, is a significant fracture criterion [4,10,31]. Assume that under applied electromechanical loading the crack tip advances along the crack plane from x ¼ a to x ¼ a þ d (0 < d a). The energy release rate per unit length during this process is given by GIII
1 ¼ lim d!0 d
Z
aþd
TTy ðxÞUða þ d xÞ dx;
ð60Þ
a
where Ty ðxÞ ¼ ðszy ðx; 0Þ; Dy ðx; 0ÞÞT is that ahead of the crack tip prior to extension, UðxÞ ¼ T ðwðxÞ; /ðxÞÞ is that at the crack surfaces posterior to extension. To obtain GIII , it is expedient to determine first field intensity factors near the crack tips. Prior to the presentation of general results, we consider a piezoelectric plate made of two layers of equal thickness. Based on analytical expressions obtained above for electroelastic field, field intensity factors can be calculated immediately as s D T ; KIII Þ ðKIII
¼
e15ð1Þ þ e15ð2Þ C12 1; c44ð1Þ þ c44ð2Þ C12
T
pffiffiffiffiffiffi Y s0 pa;
pffiffiffiffiffiffi c E T ; KIII Þ ¼ ðg1 ; 0ÞT Y s0 pa ðKIII
ð61Þ ð62Þ
for a permeable interface crack, and pffiffiffiffiffiffi s D T ; KIII Þ ¼ T0 Y pa; ðKIII
pffiffiffiffiffiffi c E T ðKIII ; KIII Þ ¼ M1 T0 Y pa
ð63Þ
for an impermeable crack, with rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanhðbaÞ ; Y ¼ ba
ð64Þ
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X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
where field intensity factors are defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s D T ðKIII ; KIII Þ ¼ limþ 2pðx aÞðszy ðx; 0Þ; Dy ðx; 0ÞÞT ;
ð65Þ
x!a
T
c E ðKIII ; KIII Þ ¼ lim x!a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2pða xÞðczx ðx; 0Þ; Ex ðx; 0ÞÞ :
ð66Þ
Furthermore, upon knowing field intensity factors, evaluating the integral (60) the energy release rate yields GIII ¼
pffiffiffiffiffiffi 1 ½Y s0 pa 2 2g
ð67Þ
for a permeable interface crack, and pffiffiffiffiffiffi GIII ¼ 12TT0 M1 T½Y s0 pa 2
ð68Þ
for an impermeable crack. These provide us with analytic formulae for calculating the required field intensity factors and energy release rate. We in turn consider an eccentric interface crack, i.e. the upper and lower piezoelectric layers are not of equal thickness. In this case, a governing integral equation for the problem under discussion can be solved via some numerical schemes. According to ordinary treatments, Fredholm integral equations (32) and (38) may be discretized by use of some known integration formulae and then converted to a finite system of algebraic equations. Once the auxiliary functions wðxÞ and WðuÞ are determined from Eqs. (32) and (38), the asymptotic electroelastic field in the vicinity of a crack tip is obtainable. For example, to get the distribution of szyð1Þ ðx; 0þ Þ near the right crack tip, say, inserting (25) into (14), after neglecting certain lower-order terms, yields the asymptotic stress field ahead of the right crack tip as follows: s KIII szyðjÞ ða þ r; 0Þ ¼ pffiffiffiffiffiffiffi ; 2pr
j ¼ 1; 2;
ð69Þ
where pffiffiffiffiffiffi s KIII ¼ wð1Þ pa:
ð70Þ
Similarly, we have 0
w ða rÞ ¼
c KIII
rffiffiffiffiffi 2r ; p
Ex ðxÞ ¼ 0;
D KIII DyðjÞ ða þ r; 0Þ ¼ pffiffiffiffiffiffiffi ; 2pr
j ¼ 1; 2;
ð71Þ
where c ¼ KIII
wð1Þ pffiffiffiffiffiffi pa; g
D ¼ KIII
e15ð1Þ þ e15ð2Þ C12 pffiffiffiffiffiffi wð1Þ pa: c44ð1Þ þ c44ð2Þ C12
ð72Þ
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
1417
Thus, the energy release rate is easily obtained as GIII ¼
pa ½wð1Þ 2 2g
ð73Þ
for a permeable interface crack. On the other hand, a similar evaluation can result in field intensity factors as pffiffiffiffiffiffi s D T ðKIII ; KIII Þ ¼ Wð1Þ pa;
pffiffiffiffiffiffi c E T ðKIII ; KIII Þ ¼ M1 Wð1Þ pa
ð74Þ
and the energy release rate as GIII ¼
pa T Wð1Þ M1 Wð1Þ 2
ð75Þ
for an impermeable interface crack. Strictly speaking, the kernels of Eqs. (32) and (38) depend on the bimaterial constants and the ratio of thickness of the upper and the lower layers to the crack length, and so do field intensity factors. However, if the upper and lower piezoelectric layers are with the same materials, we have m11 ¼ m22 ¼ 1, m12 ¼ m21 ¼ 0, g ¼ c44 , gij ðnÞ=gij ¼ 2=ð1 þ H 12 ðnÞÞ. Under such circumstances, the kernels of Eqs. (32) and (38) are independent of the material constants, and so the stress intensity factor is independent of material properties for both the permeable and impermeable cracks, which has been treated by an approximate technique in Li [17]. As a check, if taking h1 ! 1, h2 ! 1, the kernels of Eqs. (32) and (38) vanish and the closedform solutions for an interface crack between two bonded dissimilar piezoelectric half-planes can be easily obtained, which are in accordance with the results (67) and (68), respectively, when h ! 1.
6. Numerical results In this section, the dependence of field intensity factors and energy release rate upon the crack geometry is examined. Prior to the presentation of numerical results, a comparison of the norpffiffiffiffiffiffi s =s0 pa, between the analytic and numerical results for a central malized stress intensity factor, KIII interface crack inpaffiffiffiffiffi piezoelectric plate is shown in Fig. 2, where the solid line denotes the analytical ffi s solution, KIII =s0 pa ¼ Y , calculated from (64), and the dashed line stands for the numerical solution via solving Eqs. (32) or (38) corresponding to the permeable or impermeable cases. It indicates that the numerical procedure provides a very accurate result. In the following numerical examples, we choose PZT-5H as the upper piezoelectric layer and PZT-6B as the lower layer. The relevant material constants are listed in Table 1. For the case of clamped boundaries, i.e. W0 ¼ 0, electric field E0 ¼ V0 =ðh1 þ h2 Þ gives pffiffiffiffiffiffi rise to the stress concens =E0 pa, with the ratio h2 =h1 for tration, and the variations of the stress intensity factor, KIII h1 =a ¼ 1; 2 are presented in Fig. 3. Here the negative stress field induced by E0 represents only the direction opposite to the z-axis, and so a negative electric loading E0 will cause a positive stress field. Moreover, like the stress field near the crack tips, the electric displacement in the vicinity of
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X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
Fig. 2. Comparison of the results obtained from analytic and numerical solutions for a central interface crack.
Table 1 The relevant material constants 10
2
c44 (10 N/m ) e15 (C/m2 ) e11 (1010 C/Vm)
PZT-5H
PZT-6B
3.53 17 151
2.71 4.6 36
the crack tips is also dominated by a square-root singularity, and the curves of the corresponding intensity factor with the ratio h2 =h1 for h1 =a ¼ 1; 2 are plotted in Fig. 4. It is observed that the intensity factors of stress and electric displacement decrease monotonically in magnitude and
Fig. 3. Stress intensity factors versus h2 =h1 when h1 =a ¼ 1; 2 for a permeable and an impermeable crack.
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
1419
Fig. 4. Electric-displacement intensity factors versus h2 =h1 when h1 =a ¼ 1; 2 for a permeable and an impermeable crack.
approaches gradually a constant, respectively, with increasing the ratio h2 =h1 for a permeable interface crack, while for an impermeable interface crack, they increase in magnitude as h2 =h1 is small, reaching a peak, and then decrease as h2 =h1 is raised. Furthermore, for special values of h1 =a ¼ 1 and h2 =h1 ¼ 1; 2, the dependence of energy release rate GIII upon applied electric loadings E0 , E0 ¼ V =ðh1 þ h2 Þ, is displayed in Fig. 5 for a permeable interface crack and an impermeable crack of length 2a ¼ 0:02 m. It is found that in the case of clamped boundaries, the energy release rate GIII is always positive for a permeable interface crack, and is always negative for an impermeable interface crack, irrespective of the direction of applied
Fig. 5. The dependence of energy release rates on applied electric loading for a permeable or an impermeable crack in a piezoelectric plate.
1420
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
electric loadings, which coincide with those for an anti-plane shear hole in an infinite piezoelectric plane by Zhang and Tong [37]. Since negative energy release rates, or negative crack driving-force, lack clear physical interpretation, electric boundary conditions for the former are more suitable than those for the latter.
Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant No. 10272043.
Appendix Some constants and functions are defined below m11 ¼
c44ð1Þ e11ð2Þ þ e15ð1Þ e15ð2Þ ; c44ð2Þ e11ð2Þ þ e215ð2Þ
m12 ¼
e15ð1Þ e11ð2Þ e11ð1Þ e15ð2Þ ; c44ð2Þ e11ð2Þ þ e215ð2Þ
m21 ¼
c44ð1Þ e15ð2Þ e15ð1Þ c44ð2Þ ; c44ð2Þ e11ð2Þ þ e215ð2Þ
m22 ¼
e11ð1Þ c44ð2Þ þ e15ð1Þ e15ð2Þ c44ð2Þ e11ð2Þ þ e215ð2Þ
a1 ¼
c0 ðh1 þ h2 m22 Þ þ E0 h2 m12 ; D
a2 ¼
c0 ðh1 m11 þ h2 C2 Þ E0 h1 m12 ; D
b1 ¼
c0 h2 m21 þ E0 ðh1 þ h2 m11 Þ ; D
b2 ¼
c0 h1 m21 E0 ðh1 m22 þ h2 C12 Þ ; D
D¼
ðh1 þ h2 m11 Þðh1 þ h2 m22 Þ h22 m12 m21 ; h1 þ h2
C12 ¼
c44ð1Þ e11ð1Þ þ e215ð1Þ c44ð2Þ e11ð2Þ þ e215ð2Þ
;
X.-F. Li, G.J. Tang / International Journal of Engineering Science 41 (2003) 1405–1422
c0 ¼
W0 ; h1 þ h2
E0 ¼
V0 ; h1 þ h2
g11 ¼
c44ð1Þ þ c44ð2Þ C12 c44ð2Þ e11ð2Þ þ e215ð2Þ ; c44 e11 þ e215 2
g12 ¼
e15ð1Þ þ e15ð2Þ C12 c44ð2Þ e11ð2Þ þ e215ð2Þ ; c44 e11 þ e215 2
g22 ¼
e11ð1Þ þ e11ð2Þ C12 c44ð2Þ e11ð2Þ þ e215ð2Þ ; 2 c44 e11 þ e215
g11 ðnÞ ¼
2c44ð1Þ H 12 ðnÞ þ 2c44ð2Þ C12 ; ½H 12 ðnÞ þ m11 ½H 12 ðnÞ þ m22 m12 m21
g12 ðnÞ ¼
2e15ð1Þ H 12 ðnÞ þ 2e15ð2Þ C12 ; ½H 12 ðnÞ þ m11 ½H 12 ðnÞ þ m22 m12 m21
g22 ðnÞ ¼
2e11ð1Þ H 12 ðnÞ þ 2e11ð2Þ C12 ; ½H 12 ðnÞ þ m11 ½H 12 ðnÞ þ m22 m12 m21
h1 nÞ cothð h2 nÞ. where H 12 ðnÞ ¼ tanhð References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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