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Closed-form solution for a piezoelectric strip with two collinear cracks normal to the strip boundaries
European Journal of Mechanics A/Solids 21 (2002) 981–989
Closed-form solution for a piezoelectric strip with two collinear cracks normal to the strip boundaries Xian-Fang Li a,b a College of Mathematics and Physics, Hunan Normal University, Changsha, Hunan 410081, PR China b School of Aerospace and Materials Engineering, National University of Defence Technology, Changsha, Hunan, PR China
Received 2 November 2001; revised and accepted 21 June 2002
Abstract The problem of two collinear cracks of equal length and normal to the strip boundaries in an infinitely long piezoelectric strip of finite width is analyzed. By using the Fourier series method, the mixed boundary value problem is reduced to triple series equations, which are then transformed to a singular integral equation. For four combined cases of uniform antiplane shear and uniform inplane electric loading at infinity, the solution is obtained in closed-form, and explicit expressions for the electroelastic field are determined. The formulae for calculating the intensity factors of the electroelastic field and the energy release rate at the inner and outer crack tips are given, respectively. Some special cases for the electroelastic field intensity factors and the energy release rate of the present results are discussed. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
1. Introduction Piezoelectric materials have been used widely in technology such as transducers, actuators, sensors, etc. due to the intrinsic coupling characteristics between electric and elastic behaviors. Recently, crack problems in brittle piezoelectric materials have been investigated intensively under a variety of electric and elastic boundary conditions. So far many papers have been devoted to the analysis of the electroelastic field of a cracked piezoelectric material subjected to applied electromechanical loading by means of different approaches including analytic, approximate, and numerical schemes. Due to the importance in applications and the simplicity in mathematics, a class of crack problems in an infinite piezoelectric material (or composite) subjected to static antiplane shear and inplane electric loading have been studied and closed-form solutions have been obtained by numerous researchers (Pak, 1990; Li et al., 1990; Zhang and Tong, 1996; Zhong and Meguid, 1997a, 1997b; Hou and Mei, 1998; Deng and Meguid, 1999; etc.). In particular, for a cracked piezoelectric strip subjected to static antiplane shear and inplane electric loading, Shindo et al. (1997) studied the case of a central crack parallel to the strip boundaries. Furthermore, Shin et al. (2000), and Kwon and Lee (2000) considered the case of an eccentric crack in a piezoelectric strip and in a rectangular piezoelectric body, respectively. Narita and Shindo (1999) analyzed a similar problem of an interface crack of piezoelectric and orthotropic layers. In these works, the intensity factors of electroelastic field and the energy release rate at the crack tips are determined numerically via solving an integral equation by means of numerical schemes. Recently, Li and Duan (2001) reexamined a piezoelectric strip with a central crack parallel to the strip boundaries and obtained a closed-form solution. In addition, for a piezoelectric strip containing a central crack normal to the strip boundaries, by employing the Fourier transform method and solving a resulting Fredholm integral equation, Shindo et al. (1996) obtained the numerical intensity factors and the numerical energy release rate.
E-mail address: [email protected] (X.-F. Li). 0997-7538/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 0 9 9 7 - 7 5 3 8 ( 0 2 ) 0 1 2 4 1 - X
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As we know, an analytic solution in closed-form has some advantages over numerical and approximate solutions, so that in many cases, analytical solutions in closed-form are desired for accurate analysis and design. Moreover, analytical solution can serve as a benchmark for the purpose of judging the accuracy and efficiency of various numerical and approximate methods. However, owing to the mathematical complexity, certain practical problems of complicated configurations are only solved with recourse to numerical schemes, and it is difficult to obtain their analytical solutions in closed-form. This paper is concerned with the electroelastic problem of two collinear cracks normal to the strip boundaries and embedded symmetrically in a piezoelectric strip. Under four combined cases of remote uniform antiplane shear and remote uniform inplane electric loading, by using the Fourier series method, the associated boundary value problem is reduced to triple series equations, which are further transformed to a singular integral equation with Cauchy kernel. With the aid of a well-known solution, all the quantities of concern including the distribution of the stress and the electric displacement, the crack sliding displacement, the intensity factors of the electroelastic field, and the energy release rate are determined analytically. The electric field is always uniform in the entire piezoelectric strip for all the four combined cases. Finally, several special cases are discussed for the intensity factors and the energy release rate.
2. Statement of the problem Consider an infinitely long piezoelectric strip of width 2h occupying −h x h, −∞ < y < ∞, containing two collinear cracks separated each other by distance 2a and situated at a |x| b, y = 0 (0 a < b h), as shown in Fig. 1. Here it is assumed that Cartesian coordinates x, y, z are the principal axes of the material symmetry while the z-axis, which is not depicted, is oriented in the poling direction of the piezoelectric strip. The crack fronts are assumed to be parallel to the z-axis and the crack surfaces are perpendicular to the strip boundaries. Assume that the piezoelectric strip is subjected to four combined cases of uniform electromechanical loading at infinity, i.e., case 1:
τzy (x, y) → τ0 ,
Dy (x, y) → D0 ,
y → ±∞;
(1)
case 2:
γzy (x, y) → γ0 ,
Ey (x, y) → E0 ,
y → ±∞;
(2)
case 3:
τzy (x, y) → τ0 ,
Ey (x, y) → E0 ,
y → ±∞;
(3)
case 4:
γzy (x, y) → γ0 ,
Dy (x, y) → D0 ,
y → ±∞;
(4)
where τ0 (γ0 ) and D0 (E0 ) are constants. Hence, for the piezoelectric strip in question, the antiplane deformation and the inplane electric field are coupled, which are decoupled from the inplane deformation of the strip. Consequently, the elastic displacement components u, v along the x- and y-axes, respectively, and the electric field component Ez along the z-axis vanish, and then there are only nonvanishing the out-of-plane displacement w(x, y) and the potential φ(x, y), which give the antiplane strain and the inplane electric field as follows: γzx =
∂w , ∂x
γzy =
∂w , ∂y
Fig. 1. Geometry and coordinates of a strip with two collinear cracks.
(5)
Ex = −
∂φ , ∂x
X.-F. Li / European Journal of Mechanics A/Solids 21 (2002) 981–989
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∂φ . ∂y
(6)
Ey = −
These quantities are related to the antiplane shear stress and the inplane electric displacement by the following constitutive equations: τzx = c44 γzx − e15 Ex ,
τzy = c44 γzy − e15 Ey ,
(7)
Dx = e15 γzx + ε11 Ex ,
Dy = e15 γzy + ε11 Ey ,
(8)
where c44 , ε11 , and e15 are the elastic stiffness, dielectric permittivity, piezoelectric constant of the piezoelectric strip, respectively. From these equations, using the equilibrium equations of forces and electric displacements, in the absence of body forces and free charges, yields the basic governing differential equations for antiplane piezoelectricity, c44 ∇ 2 w + e15 ∇ 2 φ = 0,
e15 ∇ 2 w − ε11 ∇ 2 φ = 0,
(9)
where ∇ 2 represents the two-dimensional Laplacian operator. The mixed boundary conditions relevant to the problem stated above can be expressed below: τzy (x, 0+ ) = τzy (x, 0− ) = 0, Dy (x, 0+ ) = Dy (x, 0− ),
a < |x| < b,
(10)
Ex (x, 0+ ) = Ex (x, 0− ),
a < |x| < b,
(11)
w(x, 0+ ) = w(x, 0− ),
τzy (x, 0+ ) = τzy (x, 0− ),
|x| < a, b < |x| < h,
(12)
φ(x, 0+ ) = φ(x, 0− ),
Dy (x, 0+ ) = Dy (x, 0− ),
|x| < a, b < |x| < h,
(13)
Dx (±h, y) = 0,
τzx (±h, y) = 0,
−∞ < y < ∞.
(14)
It is noted that electrically permeable boundary conditions at the crack surfaces are assumed in the present study. The reason is that the upper and lower crack surfaces maintain contact with each other for antiplane shear deformation. In addition, if considering a crack to be a very thin hole, the fact that electric displacement normal component and the electric field tangent component are continuous across the crack surfaces leads to the above assumption (Dunn, 1994; Zhang and Tong, 1996; Sosa and Khutoryansky, 1996; Shindo et al., 1997). Due to the symmetry of the problem, it is sufficient to analyze the right-half portion of the upper strip, i.e., 0 x h, 0 y < ∞. Therefore in the following we restrict our attention to this region and the elastic and electric fields in the remaining section can be directly given by symmetry.
3. Closed-form solution The above mixed boundary-value problem can be solved by employing many techniques. For example, for the case when a = 0, using the Fourier transform method, Shindo et al. (1996) converted it to a Fredholm integral equation, and then obtained the numerical solution. Here, it is convenient to utilize the Fourier series method, which can transform the problem to a singular integral equation admitting its solution in closed-form. To this end, taking (14) into account, it is easily verified that an appropriate solution to Eqs. (9) can be expressed in terms of the following Fourier series: w(x, y) = φ(x, y) =
∞ n=1 ∞
An e−2nβy cos(2nβx) + A0 y,
0 x h, 0 y < ∞,
(15)
Bn e−2nβy cos(2nβx) + B0 y,
0 x h, 0 y < ∞,
(16)
n=1
where β=
π , 2h
(17)
and An and Bn (n 0) are unknown coefficients to be determined from given boundary conditions. By virtue of the constitutive equations (7) and (8), it is not difficult to obtain the series expressions for the components of the antiplane stress and the inplane electric displacement. For example, we have
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τzy (x, y) = −2β Dy (x, y) = −2β
∞ n=1 ∞
n(c44 An + e15 Bn ) e−2nβy cos(2nβx) + c44 A0 + e15 B0 ,
(18)
n(e15 An − ε11 Bn ) e−2nβy cos(2nβx) + e15 A0 − ε11 B0 .
(19)
n=1
Hence, from the remote boundary conditions we get case 1: case 2: case 3:
ε τ + e15 D0 A0 = 11 0 , 2 c44 ε11 + e15
e τ − c44 D0 B0 = 15 0 ; 2 c44 ε11 + e15
(20)
A0 = γ0 , τ + e15 E0 , A0 = 0 c44
B0 = −E0 ;
(21)
B0 = −E0 ;
(22)
e γ − D0 B0 = 15 0 . ε11
(23)
A0 = γ0 ,
case 4:
Furthermore, in view of (15), (16), (18) and (19), using the boundary conditions (10) to (13) yields unknown coefficients Bn = 0 (n 1) and the system of simultaneous triple series equations for An , ∞ n=1 ∞
An cos(2nβx) = 0, nAn cos(2nβx) =
n=1
x a, b x h,
P , 2βc44
(24)
a < x < b,
(25)
where P = c44 A0 + e15 B0 , or case 1:
P = τ0 ;
(26)
case 2:
P = c44 γ0 − e15 E0 ;
(27)
case 3:
P = τ0 ; e2 P = c44 + 15 γ0 − ε11
(28)
case 4:
e15 D0 . ε11
(29)
To solve the resulting triple series equations for An , we denote g(x) =
∂w(x, 0) , ∂x
(30)
which allows us to express An in terms of g(x) An = −
1 hnβ
b g(u) sin(2nβu) du.
(31)
a
Upon substituting (31) into (25), since ∞
sin(2nβu) cos(2nβx) = −
n=1
1 sin(2βu) , 2 cos(2βu) − cos(2βx)
(32)
we obtain a singular integral equation for g(x) of the form 1 h
b a
sin(2βu)g(u) P , du = cos(2βu) − cos(2βx) c44
a < x < b.
(33)
We now introduce new variables: u1 = cos(2βu),
x1 = cos(2βx),
a1 = cos(2βa),
b1 = cos(2βb).
(34)
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Taking into account β = π/2h, then Eq. (33) can be rewritten as the following form 1 π
a1 b1
where
G(u1 ) P du1 = , u1 − x1 c44
b1 < x < a1 ,
(35)
G(u1 ) = G cos(2βu) ≡ g(u).
(36)
Eq. (35) is a standard singular integral equation with Cauchy kernel. By using the techniques described in Muskhelisvili (1953), the solution to Eq. (35) is found to be u1 − b1 a1 − u1 P C , (37) − +√ G(u1 ) = 2c44 a1 − u1 u1 − b1 (a1 − u1 )(u1 − b1 ) where C is an arbitrary constant, which can be determined for the problem under consideration by the requirement that b g(u) du = 0.
(38)
a
After some manipulations, we get C=
P 2 cos (βa) + cos2 (βb) − 2χ cos2 (βb) , c44
(39)
χ=
Π(c, k) , K(k)
(40)
with
and then P (u1 + 1) − (b1 + 1)χ , √ c44 (a1 − u1 )(u1 − b1 ) where K(k) and Π(c, k) are the complete elliptical integrals of the first and third kinds, respectively, i.e., G(u1 ) =
π/2
K(k) = 0
with
k=
π/2
1
Π(c, k) =
dθ,
1 − k 2 sin2 (θ)
tan2 (βb) − tan2 (βa) , tan(βb)
0
c=
1
dθ,
(41)
(42)
[1 + c sin2 (θ)] 1 − k 2 sin2 (θ)
tan2 (βa) − tan2 (βb) . sec2 (βb)
(43)
Now substituting (41) into (31), then into (18) and (19), we obtain the explicit expression for the antiplane shear stress and the electric displacement along the crack line in terms of the original variables as follows: τzy (x, 0) = Dy (x, 0) =
P [sec2 (βb) − χ sec2 (βx)]
cos(βb)
[tan2 (βa) − tan2 (βx)][tan2 (βb) − tan2 (βx)] cos(βa)
,
2 cos(βb) c44 ε11 + e15 B0 , − c44 c44 [tan2 (βa) − tan2 (βx)][tan2 (βb) − tan2 (βx)] cos(βa)
e15 P [sec2 (βb) − χ sec2 (βx)]
(44) (45)
for 0 < x < a, and τzy (x, 0) =
P [χ sec2 (βx) − sec2 (βb)]
cos(βb)
[tan2 (βx) − tan2 (βa)][tan2 (βx) − tan2 (βb)] cos(βa)
,
2 cos(βb) c44 ε11 + e15 B0 , − c44 c44 [tan2 (βx) − tan2 (βa)][tan2 (βx) − tan2 (βb)] cos(βa) for b < x < h, where P is given by (26)–(29), respectively, for four cases under consideration.
Dy (x, 0) =
e15 P [χ sec2 (βx) − sec2 (βb)]
(46) (47)
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As expected, the stress and electric displacement exhibit a usual square-root singularity near the crack tips. In addition, clearly, the electric field is always uniform in the entire piezoelectric strip, namely Ey = −B0 . On the other hand, the crack sliding displacement can be determined in closed-form as well. With the solution (41) in hand, integrating according to the original variables u(u1 = cos(2βu)) between the limits x and b (a x b) yields the crack sliding displacement as w(x, 0) =
cos(βb) P χF (ψ, k) − Π(ψ, c, k) , c44 β tan(βb) cos(βa)
where F (ψ, k) and Π(ψ, c, k) are the incomplete elliptical integrals of the first and the third kinds, respectively, with tan2 (βb) − tan2 (βx) −1 , ψ = sin tan2 (βb) − tan2 (βa)
(48)
(49)
k and c being given in (43). It is readily seen that the crack sliding displacement is independent of the applied electric loading, the permittivity and the piezoelectric constant for the cases 1 and 3, i.e., the applied stress is prescribed. Otherwise, the crack sliding displacement depends on the applied electric field and the piezoelectric constant, and not on the permittivity for the case 2, i.e., the remote strain and the remote electric field are prescribed. For the case 4 when the remote strain and the electric displacement are prescribed, the crack sliding displacement depends on the applied electric loading, and three material constants c44 , ε11 , and e15 .
4. Intensity factors and energy release rate Of particular interest is the crack-tip field from the point of view of fracture mechanics, and for this purpose it is desirable to determine the intensity factors of the electroelastic field, which characterizes the singular field near the crack tips. For the problem considered, the intensity factors can be evaluated straightforwardly from the above expressions for electroelastic field. According to the definition of the intensity factors at the inner and outer crack tips, i.e.,
2π(a − x)q(x, 0), Kout = lim 2π(x − b)q(x, 0), (50) Kinn = lim x→a −
x→b+
where q(x, 0) stands for the stress or the electric displacement, respectively, we arrive at τ = PY , τ = PY , Kinn Kout a b e15 e D D P Ya , Kout = 15 P Yb , Kinn = c44 c44
with
cos2 (πa/(2h)) − χ cos2 (πb/(2h)) πa
, 2h sin(πa/(2h)) cos2 (πa/(2h)) − cos2 (πb/(2h)) πb cos(πb/(2h))[χ − 1]
Yb = 2h cot , 2 2h cos (πa/(2h)) − cos2 (πb/(2h))
Ya =
(51) (52)
2h tan
(53)
(54)
where P is given by (26)–(29), χ = Π(c, k)/K(k), K(k) and Π(c, k) being defined by (42), respectively. On the other hand, from the viewpoint of Griffith’s energy balance used in classical linear elastic fracture mechanics, the energy release rate, defined as the change of the energy of a cracked medium for an infinitesimal crack extension, is also a significant fracture criterion for the study of failure analysis of piezoelectric materials (Suo et al., 1992; Dascalu and Maugin, 1994; Gao et al., 1997). Assume that under applied electromechanical loadings the crack tip advances along the crack plane from x = b to x = b + δ (δ b). The energy release rate at the outer crack tip x = b per unit length during this process is given by Pak (1990) 1 Gout = lim 2δ δ→0
δ
τzy (r, 0) · 2w(δ − r) + Dy (x, 0) · 2φ(δ − r) dr,
(55)
0
where r is the distance from the crack tip x = b. Due to the electrically permeable assumption, the contribution of the second part in the integrand to the energy release rate Gout vanishes. As a consequence, the energy release rate reduces to the mechanical strain energy release rate proposed by Park and Sun (1995), which is in accordance with their experimental data.
X.-F. Li / European Journal of Mechanics A/Solids 21 (2002) 981–989
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Using the results obtained above, the energy release rate at the outer crack tip can be evaluated in closed-form as P 2 h cot(πb/(2h)) cos2 (πb/(2h)) [χ − 1]2 . c44 cos2 (πa/(2h)) − cos2 (πb/(2h)) The same route as the above can result in the energy release rate at the inner crack tip Gout =
(56)
P 2 2h[cos2 (πa/(2h)) − χ cos2 (πb/(2h))]2 . (57) c44 sin(πa/ h)[cos2 (πa/(2h)) − cos2 (πb/(2h))] It is readily seen that 1 τ 2 (58) K , G= 2c44 in agreement with that obtained by Zhang and Tong (1996), Shindo et al. (1997), Hou and Mei (1998). Obviously, from the expressions for P given by (26) to (29), it is concluded that for the case when the remote stress is prescribed (cases 1 and 3), the intensity factors of the stress and the electric displacement, and the energy release rate are independent of the remote electric loading, while for the case when the remote strain is prescribed (cases 2 and 4), the remote electric loading has a remarkable effect on the intensity factors of the stress and the electric displacement, and the energy release rate. Ginn =
5. Special cases 5.1. A piezoelectric strip with a central crack First, we consider an infinitely long piezoelectric strip of width 2h with a central crack of length 2b normal to the strip boundaries. In this case, the constant a in the above expressions should be chosen to be 0, which gives πb , (59) χ − 1 = tan2 2h and we find the intensity factors of the stress and the electric displacement at the crack tip x = b as πb τ , (60) K = P 2h tan 2h πb e P 2h tan , (61) K D = 15 c44 2h and the energy release rate at the crack tip x = b as πb P 2h tan . (62) G= c44 2h It is pointed out that the numerical intensity factors of the stress and the electric displacement at the crack tip x = b have been determined by Shindo et al. (1996) via the numerical solution of a Fredholm integral equation. Nevertheless, here the explicit expressions for these intensity factors and the energy release rate are obtained. 5.2. A piezoelectric strip with two collinear edge cracks Secondly, consider two collinear edge cracks perpendicular to the strip boundaries and distributed symmetrically in a piezoelectric strip. For this case, taking b as the width of the strip, i.e., b = h, the intensity factors of the stress and the electric displacement at the crack tip x = a are πa τ , (63) K = P 2h cot 2h e15 P πa D , (64) K = 2h cot c44 2h and the energy release rate at the crack tip x = a is πa P 2h G= cot . c44 2h
(65)
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5.3. An infinite piezoelectric medium with two collinear cracks Finally, we consider an infinite piezoelectric medium with two collinear internal cracks of equal length b − a, which can be taken as the limiting case of the present study as h → ∞. In fact, it is readily proven from (42) that c
S(c, k) − K(k) , (66) Π(c, k) = K(k) + c + k2 where S(c, k) =
π/2 1 − k 2 sin2 (θ) 0
1 + c sin2 (θ)
dθ.
(67)
Thus, a direct evaluation from (51) and (52) as h → ∞ leads to the intensity factors of the stress and the electric displacement at the crack tips x = a and x = b as follows: τ = P √πa Kinn
λ − (a/b)2
, a/b 1 − (a/b)2
λ − (a/b)2 D = e15 P √πa
, Kinn c44 a/b 1 − (a/b)2
√ τ = P πb 1 − λ Kout , 1 − (a/b)2 √ D = e15 P πb 1 − λ Kout , c44 1 − (a/b)2
(68) (69)
respectively, and the energy release rate at the crack tips x = a and x = b as Ginn = where λ=
P 2 π(b2 λ − a 2 )2 , 2c44 a(b2 − a 2 )
E( 1 − (a/b)2 )
, K( 1 − (a/b)2 )
Gout =
P 2 πb3 (1 − λ)2 , 2c44 b2 − a 2
(70)
(71)
K(·) and E(·) being the complete elliptical integrals of the first, and the second kinds, respectively. As a check, if imposing the requirement that the piezoelectric constant e15 = 0, the piezoelectric strip then reduces to a purely elastic strip. The well-known results for the latter (see, e.g., Tada et al. (1985)) can be recovered from the results given in this section by setting e15 = 0.
6. Conclusions An infinitely long piezoelectric strip of finite width with two collinear cracks of equal length is considered. The cracks are assumed to be perpendicular to the strip boundaries and distributed symmetrically in the strip. Explicit expressions for the electroelastic field are obtained for four cases of uniform electromechanical loadings at infinity. The results indicate that the electric field is always uniform in the entire strip, and the stress and the electric displacement in the vicinity of the crack tips are dominated by a term behaving like r −1/2 , r being the distance from the crack tips. The intensity factors of the stress and the electric displacement, and the energy release rate are determined in closed-form. In particular, for the case when the applied stress is prescribed, the singular electroelastic field is independent of the applied electric loading, while for the case when the applied strain is prescribed, the applied electric loading can give rise to the singular electroelastic field.
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Closed-form solution for a piezoelectric strip with two collinear cracks normal to the strip boundaries Xian-Fang Li a,b a College of Mathematics and Physics, Hunan Normal University, Changsha, Hunan 410081, PR China b School of Aerospace and Materials Engineering, National University of Defence Technology, Changsha, Hunan, PR China
Received 2 November 2001; revised and accepted 21 June 2002
Abstract The problem of two collinear cracks of equal length and normal to the strip boundaries in an infinitely long piezoelectric strip of finite width is analyzed. By using the Fourier series method, the mixed boundary value problem is reduced to triple series equations, which are then transformed to a singular integral equation. For four combined cases of uniform antiplane shear and uniform inplane electric loading at infinity, the solution is obtained in closed-form, and explicit expressions for the electroelastic field are determined. The formulae for calculating the intensity factors of the electroelastic field and the energy release rate at the inner and outer crack tips are given, respectively. Some special cases for the electroelastic field intensity factors and the energy release rate of the present results are discussed. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
1. Introduction Piezoelectric materials have been used widely in technology such as transducers, actuators, sensors, etc. due to the intrinsic coupling characteristics between electric and elastic behaviors. Recently, crack problems in brittle piezoelectric materials have been investigated intensively under a variety of electric and elastic boundary conditions. So far many papers have been devoted to the analysis of the electroelastic field of a cracked piezoelectric material subjected to applied electromechanical loading by means of different approaches including analytic, approximate, and numerical schemes. Due to the importance in applications and the simplicity in mathematics, a class of crack problems in an infinite piezoelectric material (or composite) subjected to static antiplane shear and inplane electric loading have been studied and closed-form solutions have been obtained by numerous researchers (Pak, 1990; Li et al., 1990; Zhang and Tong, 1996; Zhong and Meguid, 1997a, 1997b; Hou and Mei, 1998; Deng and Meguid, 1999; etc.). In particular, for a cracked piezoelectric strip subjected to static antiplane shear and inplane electric loading, Shindo et al. (1997) studied the case of a central crack parallel to the strip boundaries. Furthermore, Shin et al. (2000), and Kwon and Lee (2000) considered the case of an eccentric crack in a piezoelectric strip and in a rectangular piezoelectric body, respectively. Narita and Shindo (1999) analyzed a similar problem of an interface crack of piezoelectric and orthotropic layers. In these works, the intensity factors of electroelastic field and the energy release rate at the crack tips are determined numerically via solving an integral equation by means of numerical schemes. Recently, Li and Duan (2001) reexamined a piezoelectric strip with a central crack parallel to the strip boundaries and obtained a closed-form solution. In addition, for a piezoelectric strip containing a central crack normal to the strip boundaries, by employing the Fourier transform method and solving a resulting Fredholm integral equation, Shindo et al. (1996) obtained the numerical intensity factors and the numerical energy release rate.
E-mail address: [email protected] (X.-F. Li). 0997-7538/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 0 9 9 7 - 7 5 3 8 ( 0 2 ) 0 1 2 4 1 - X
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As we know, an analytic solution in closed-form has some advantages over numerical and approximate solutions, so that in many cases, analytical solutions in closed-form are desired for accurate analysis and design. Moreover, analytical solution can serve as a benchmark for the purpose of judging the accuracy and efficiency of various numerical and approximate methods. However, owing to the mathematical complexity, certain practical problems of complicated configurations are only solved with recourse to numerical schemes, and it is difficult to obtain their analytical solutions in closed-form. This paper is concerned with the electroelastic problem of two collinear cracks normal to the strip boundaries and embedded symmetrically in a piezoelectric strip. Under four combined cases of remote uniform antiplane shear and remote uniform inplane electric loading, by using the Fourier series method, the associated boundary value problem is reduced to triple series equations, which are further transformed to a singular integral equation with Cauchy kernel. With the aid of a well-known solution, all the quantities of concern including the distribution of the stress and the electric displacement, the crack sliding displacement, the intensity factors of the electroelastic field, and the energy release rate are determined analytically. The electric field is always uniform in the entire piezoelectric strip for all the four combined cases. Finally, several special cases are discussed for the intensity factors and the energy release rate.
2. Statement of the problem Consider an infinitely long piezoelectric strip of width 2h occupying −h x h, −∞ < y < ∞, containing two collinear cracks separated each other by distance 2a and situated at a |x| b, y = 0 (0 a < b h), as shown in Fig. 1. Here it is assumed that Cartesian coordinates x, y, z are the principal axes of the material symmetry while the z-axis, which is not depicted, is oriented in the poling direction of the piezoelectric strip. The crack fronts are assumed to be parallel to the z-axis and the crack surfaces are perpendicular to the strip boundaries. Assume that the piezoelectric strip is subjected to four combined cases of uniform electromechanical loading at infinity, i.e., case 1:
τzy (x, y) → τ0 ,
Dy (x, y) → D0 ,
y → ±∞;
(1)
case 2:
γzy (x, y) → γ0 ,
Ey (x, y) → E0 ,
y → ±∞;
(2)
case 3:
τzy (x, y) → τ0 ,
Ey (x, y) → E0 ,
y → ±∞;
(3)
case 4:
γzy (x, y) → γ0 ,
Dy (x, y) → D0 ,
y → ±∞;
(4)
where τ0 (γ0 ) and D0 (E0 ) are constants. Hence, for the piezoelectric strip in question, the antiplane deformation and the inplane electric field are coupled, which are decoupled from the inplane deformation of the strip. Consequently, the elastic displacement components u, v along the x- and y-axes, respectively, and the electric field component Ez along the z-axis vanish, and then there are only nonvanishing the out-of-plane displacement w(x, y) and the potential φ(x, y), which give the antiplane strain and the inplane electric field as follows: γzx =
∂w , ∂x
γzy =
∂w , ∂y
Fig. 1. Geometry and coordinates of a strip with two collinear cracks.
(5)
Ex = −
∂φ , ∂x
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∂φ . ∂y
(6)
Ey = −
These quantities are related to the antiplane shear stress and the inplane electric displacement by the following constitutive equations: τzx = c44 γzx − e15 Ex ,
τzy = c44 γzy − e15 Ey ,
(7)
Dx = e15 γzx + ε11 Ex ,
Dy = e15 γzy + ε11 Ey ,
(8)
where c44 , ε11 , and e15 are the elastic stiffness, dielectric permittivity, piezoelectric constant of the piezoelectric strip, respectively. From these equations, using the equilibrium equations of forces and electric displacements, in the absence of body forces and free charges, yields the basic governing differential equations for antiplane piezoelectricity, c44 ∇ 2 w + e15 ∇ 2 φ = 0,
e15 ∇ 2 w − ε11 ∇ 2 φ = 0,
(9)
where ∇ 2 represents the two-dimensional Laplacian operator. The mixed boundary conditions relevant to the problem stated above can be expressed below: τzy (x, 0+ ) = τzy (x, 0− ) = 0, Dy (x, 0+ ) = Dy (x, 0− ),
a < |x| < b,
(10)
Ex (x, 0+ ) = Ex (x, 0− ),
a < |x| < b,
(11)
w(x, 0+ ) = w(x, 0− ),
τzy (x, 0+ ) = τzy (x, 0− ),
|x| < a, b < |x| < h,
(12)
φ(x, 0+ ) = φ(x, 0− ),
Dy (x, 0+ ) = Dy (x, 0− ),
|x| < a, b < |x| < h,
(13)
Dx (±h, y) = 0,
τzx (±h, y) = 0,
−∞ < y < ∞.
(14)
It is noted that electrically permeable boundary conditions at the crack surfaces are assumed in the present study. The reason is that the upper and lower crack surfaces maintain contact with each other for antiplane shear deformation. In addition, if considering a crack to be a very thin hole, the fact that electric displacement normal component and the electric field tangent component are continuous across the crack surfaces leads to the above assumption (Dunn, 1994; Zhang and Tong, 1996; Sosa and Khutoryansky, 1996; Shindo et al., 1997). Due to the symmetry of the problem, it is sufficient to analyze the right-half portion of the upper strip, i.e., 0 x h, 0 y < ∞. Therefore in the following we restrict our attention to this region and the elastic and electric fields in the remaining section can be directly given by symmetry.
3. Closed-form solution The above mixed boundary-value problem can be solved by employing many techniques. For example, for the case when a = 0, using the Fourier transform method, Shindo et al. (1996) converted it to a Fredholm integral equation, and then obtained the numerical solution. Here, it is convenient to utilize the Fourier series method, which can transform the problem to a singular integral equation admitting its solution in closed-form. To this end, taking (14) into account, it is easily verified that an appropriate solution to Eqs. (9) can be expressed in terms of the following Fourier series: w(x, y) = φ(x, y) =
∞ n=1 ∞
An e−2nβy cos(2nβx) + A0 y,
0 x h, 0 y < ∞,
(15)
Bn e−2nβy cos(2nβx) + B0 y,
0 x h, 0 y < ∞,
(16)
n=1
where β=
π , 2h
(17)
and An and Bn (n 0) are unknown coefficients to be determined from given boundary conditions. By virtue of the constitutive equations (7) and (8), it is not difficult to obtain the series expressions for the components of the antiplane stress and the inplane electric displacement. For example, we have
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τzy (x, y) = −2β Dy (x, y) = −2β
∞ n=1 ∞
n(c44 An + e15 Bn ) e−2nβy cos(2nβx) + c44 A0 + e15 B0 ,
(18)
n(e15 An − ε11 Bn ) e−2nβy cos(2nβx) + e15 A0 − ε11 B0 .
(19)
n=1
Hence, from the remote boundary conditions we get case 1: case 2: case 3:
ε τ + e15 D0 A0 = 11 0 , 2 c44 ε11 + e15
e τ − c44 D0 B0 = 15 0 ; 2 c44 ε11 + e15
(20)
A0 = γ0 , τ + e15 E0 , A0 = 0 c44
B0 = −E0 ;
(21)
B0 = −E0 ;
(22)
e γ − D0 B0 = 15 0 . ε11
(23)
A0 = γ0 ,
case 4:
Furthermore, in view of (15), (16), (18) and (19), using the boundary conditions (10) to (13) yields unknown coefficients Bn = 0 (n 1) and the system of simultaneous triple series equations for An , ∞ n=1 ∞
An cos(2nβx) = 0, nAn cos(2nβx) =
n=1
x a, b x h,
P , 2βc44
(24)
a < x < b,
(25)
where P = c44 A0 + e15 B0 , or case 1:
P = τ0 ;
(26)
case 2:
P = c44 γ0 − e15 E0 ;
(27)
case 3:
P = τ0 ; e2 P = c44 + 15 γ0 − ε11
(28)
case 4:
e15 D0 . ε11
(29)
To solve the resulting triple series equations for An , we denote g(x) =
∂w(x, 0) , ∂x
(30)
which allows us to express An in terms of g(x) An = −
1 hnβ
b g(u) sin(2nβu) du.
(31)
a
Upon substituting (31) into (25), since ∞
sin(2nβu) cos(2nβx) = −
n=1
1 sin(2βu) , 2 cos(2βu) − cos(2βx)
(32)
we obtain a singular integral equation for g(x) of the form 1 h
b a
sin(2βu)g(u) P , du = cos(2βu) − cos(2βx) c44
a < x < b.
(33)
We now introduce new variables: u1 = cos(2βu),
x1 = cos(2βx),
a1 = cos(2βa),
b1 = cos(2βb).
(34)
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Taking into account β = π/2h, then Eq. (33) can be rewritten as the following form 1 π
a1 b1
where
G(u1 ) P du1 = , u1 − x1 c44
b1 < x < a1 ,
(35)
G(u1 ) = G cos(2βu) ≡ g(u).
(36)
Eq. (35) is a standard singular integral equation with Cauchy kernel. By using the techniques described in Muskhelisvili (1953), the solution to Eq. (35) is found to be u1 − b1 a1 − u1 P C , (37) − +√ G(u1 ) = 2c44 a1 − u1 u1 − b1 (a1 − u1 )(u1 − b1 ) where C is an arbitrary constant, which can be determined for the problem under consideration by the requirement that b g(u) du = 0.
(38)
a
After some manipulations, we get C=
P 2 cos (βa) + cos2 (βb) − 2χ cos2 (βb) , c44
(39)
χ=
Π(c, k) , K(k)
(40)
with
and then P (u1 + 1) − (b1 + 1)χ , √ c44 (a1 − u1 )(u1 − b1 ) where K(k) and Π(c, k) are the complete elliptical integrals of the first and third kinds, respectively, i.e., G(u1 ) =
π/2
K(k) = 0
with
k=
π/2
1
Π(c, k) =
dθ,
1 − k 2 sin2 (θ)
tan2 (βb) − tan2 (βa) , tan(βb)
0
c=
1
dθ,
(41)
(42)
[1 + c sin2 (θ)] 1 − k 2 sin2 (θ)
tan2 (βa) − tan2 (βb) . sec2 (βb)
(43)
Now substituting (41) into (31), then into (18) and (19), we obtain the explicit expression for the antiplane shear stress and the electric displacement along the crack line in terms of the original variables as follows: τzy (x, 0) = Dy (x, 0) =
P [sec2 (βb) − χ sec2 (βx)]
cos(βb)
[tan2 (βa) − tan2 (βx)][tan2 (βb) − tan2 (βx)] cos(βa)
,
2 cos(βb) c44 ε11 + e15 B0 , − c44 c44 [tan2 (βa) − tan2 (βx)][tan2 (βb) − tan2 (βx)] cos(βa)
e15 P [sec2 (βb) − χ sec2 (βx)]
(44) (45)
for 0 < x < a, and τzy (x, 0) =
P [χ sec2 (βx) − sec2 (βb)]
cos(βb)
[tan2 (βx) − tan2 (βa)][tan2 (βx) − tan2 (βb)] cos(βa)
,
2 cos(βb) c44 ε11 + e15 B0 , − c44 c44 [tan2 (βx) − tan2 (βa)][tan2 (βx) − tan2 (βb)] cos(βa) for b < x < h, where P is given by (26)–(29), respectively, for four cases under consideration.
Dy (x, 0) =
e15 P [χ sec2 (βx) − sec2 (βb)]
(46) (47)
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As expected, the stress and electric displacement exhibit a usual square-root singularity near the crack tips. In addition, clearly, the electric field is always uniform in the entire piezoelectric strip, namely Ey = −B0 . On the other hand, the crack sliding displacement can be determined in closed-form as well. With the solution (41) in hand, integrating according to the original variables u(u1 = cos(2βu)) between the limits x and b (a x b) yields the crack sliding displacement as w(x, 0) =
cos(βb) P χF (ψ, k) − Π(ψ, c, k) , c44 β tan(βb) cos(βa)
where F (ψ, k) and Π(ψ, c, k) are the incomplete elliptical integrals of the first and the third kinds, respectively, with tan2 (βb) − tan2 (βx) −1 , ψ = sin tan2 (βb) − tan2 (βa)
(48)
(49)
k and c being given in (43). It is readily seen that the crack sliding displacement is independent of the applied electric loading, the permittivity and the piezoelectric constant for the cases 1 and 3, i.e., the applied stress is prescribed. Otherwise, the crack sliding displacement depends on the applied electric field and the piezoelectric constant, and not on the permittivity for the case 2, i.e., the remote strain and the remote electric field are prescribed. For the case 4 when the remote strain and the electric displacement are prescribed, the crack sliding displacement depends on the applied electric loading, and three material constants c44 , ε11 , and e15 .
4. Intensity factors and energy release rate Of particular interest is the crack-tip field from the point of view of fracture mechanics, and for this purpose it is desirable to determine the intensity factors of the electroelastic field, which characterizes the singular field near the crack tips. For the problem considered, the intensity factors can be evaluated straightforwardly from the above expressions for electroelastic field. According to the definition of the intensity factors at the inner and outer crack tips, i.e.,
2π(a − x)q(x, 0), Kout = lim 2π(x − b)q(x, 0), (50) Kinn = lim x→a −
x→b+
where q(x, 0) stands for the stress or the electric displacement, respectively, we arrive at τ = PY , τ = PY , Kinn Kout a b e15 e D D P Ya , Kout = 15 P Yb , Kinn = c44 c44
with
cos2 (πa/(2h)) − χ cos2 (πb/(2h)) πa
, 2h sin(πa/(2h)) cos2 (πa/(2h)) − cos2 (πb/(2h)) πb cos(πb/(2h))[χ − 1]
Yb = 2h cot , 2 2h cos (πa/(2h)) − cos2 (πb/(2h))
Ya =
(51) (52)
2h tan
(53)
(54)
where P is given by (26)–(29), χ = Π(c, k)/K(k), K(k) and Π(c, k) being defined by (42), respectively. On the other hand, from the viewpoint of Griffith’s energy balance used in classical linear elastic fracture mechanics, the energy release rate, defined as the change of the energy of a cracked medium for an infinitesimal crack extension, is also a significant fracture criterion for the study of failure analysis of piezoelectric materials (Suo et al., 1992; Dascalu and Maugin, 1994; Gao et al., 1997). Assume that under applied electromechanical loadings the crack tip advances along the crack plane from x = b to x = b + δ (δ b). The energy release rate at the outer crack tip x = b per unit length during this process is given by Pak (1990) 1 Gout = lim 2δ δ→0
δ
τzy (r, 0) · 2w(δ − r) + Dy (x, 0) · 2φ(δ − r) dr,
(55)
0
where r is the distance from the crack tip x = b. Due to the electrically permeable assumption, the contribution of the second part in the integrand to the energy release rate Gout vanishes. As a consequence, the energy release rate reduces to the mechanical strain energy release rate proposed by Park and Sun (1995), which is in accordance with their experimental data.
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Using the results obtained above, the energy release rate at the outer crack tip can be evaluated in closed-form as P 2 h cot(πb/(2h)) cos2 (πb/(2h)) [χ − 1]2 . c44 cos2 (πa/(2h)) − cos2 (πb/(2h)) The same route as the above can result in the energy release rate at the inner crack tip Gout =
(56)
P 2 2h[cos2 (πa/(2h)) − χ cos2 (πb/(2h))]2 . (57) c44 sin(πa/ h)[cos2 (πa/(2h)) − cos2 (πb/(2h))] It is readily seen that 1 τ 2 (58) K , G= 2c44 in agreement with that obtained by Zhang and Tong (1996), Shindo et al. (1997), Hou and Mei (1998). Obviously, from the expressions for P given by (26) to (29), it is concluded that for the case when the remote stress is prescribed (cases 1 and 3), the intensity factors of the stress and the electric displacement, and the energy release rate are independent of the remote electric loading, while for the case when the remote strain is prescribed (cases 2 and 4), the remote electric loading has a remarkable effect on the intensity factors of the stress and the electric displacement, and the energy release rate. Ginn =
5. Special cases 5.1. A piezoelectric strip with a central crack First, we consider an infinitely long piezoelectric strip of width 2h with a central crack of length 2b normal to the strip boundaries. In this case, the constant a in the above expressions should be chosen to be 0, which gives πb , (59) χ − 1 = tan2 2h and we find the intensity factors of the stress and the electric displacement at the crack tip x = b as πb τ , (60) K = P 2h tan 2h πb e P 2h tan , (61) K D = 15 c44 2h and the energy release rate at the crack tip x = b as πb P 2h tan . (62) G= c44 2h It is pointed out that the numerical intensity factors of the stress and the electric displacement at the crack tip x = b have been determined by Shindo et al. (1996) via the numerical solution of a Fredholm integral equation. Nevertheless, here the explicit expressions for these intensity factors and the energy release rate are obtained. 5.2. A piezoelectric strip with two collinear edge cracks Secondly, consider two collinear edge cracks perpendicular to the strip boundaries and distributed symmetrically in a piezoelectric strip. For this case, taking b as the width of the strip, i.e., b = h, the intensity factors of the stress and the electric displacement at the crack tip x = a are πa τ , (63) K = P 2h cot 2h e15 P πa D , (64) K = 2h cot c44 2h and the energy release rate at the crack tip x = a is πa P 2h G= cot . c44 2h
(65)
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5.3. An infinite piezoelectric medium with two collinear cracks Finally, we consider an infinite piezoelectric medium with two collinear internal cracks of equal length b − a, which can be taken as the limiting case of the present study as h → ∞. In fact, it is readily proven from (42) that c
S(c, k) − K(k) , (66) Π(c, k) = K(k) + c + k2 where S(c, k) =
π/2 1 − k 2 sin2 (θ) 0
1 + c sin2 (θ)
dθ.
(67)
Thus, a direct evaluation from (51) and (52) as h → ∞ leads to the intensity factors of the stress and the electric displacement at the crack tips x = a and x = b as follows: τ = P √πa Kinn
λ − (a/b)2
, a/b 1 − (a/b)2
λ − (a/b)2 D = e15 P √πa
, Kinn c44 a/b 1 − (a/b)2
√ τ = P πb 1 − λ Kout , 1 − (a/b)2 √ D = e15 P πb 1 − λ Kout , c44 1 − (a/b)2
(68) (69)
respectively, and the energy release rate at the crack tips x = a and x = b as Ginn = where λ=
P 2 π(b2 λ − a 2 )2 , 2c44 a(b2 − a 2 )
E( 1 − (a/b)2 )
, K( 1 − (a/b)2 )
Gout =
P 2 πb3 (1 − λ)2 , 2c44 b2 − a 2
(70)
(71)
K(·) and E(·) being the complete elliptical integrals of the first, and the second kinds, respectively. As a check, if imposing the requirement that the piezoelectric constant e15 = 0, the piezoelectric strip then reduces to a purely elastic strip. The well-known results for the latter (see, e.g., Tada et al. (1985)) can be recovered from the results given in this section by setting e15 = 0.
6. Conclusions An infinitely long piezoelectric strip of finite width with two collinear cracks of equal length is considered. The cracks are assumed to be perpendicular to the strip boundaries and distributed symmetrically in the strip. Explicit expressions for the electroelastic field are obtained for four cases of uniform electromechanical loadings at infinity. The results indicate that the electric field is always uniform in the entire strip, and the stress and the electric displacement in the vicinity of the crack tips are dominated by a term behaving like r −1/2 , r being the distance from the crack tips. The intensity factors of the stress and the electric displacement, and the energy release rate are determined in closed-form. In particular, for the case when the applied stress is prescribed, the singular electroelastic field is independent of the applied electric loading, while for the case when the applied strain is prescribed, the applied electric loading can give rise to the singular electroelastic field.
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