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A finite crack in a semi-infinite strip of a graded piezoelectric material under electric loading
European Journal of Mechanics A/Solids 25 (2006) 250–259
A finite crack in a semi-infinite strip of a graded piezoelectric material under electric loading Sei Ueda ∗ Department of Mechanical Engineering, Osaka Institute of Technology, 5-16-1, Omiya, Asahi-ku, Osaka 535-8585, Japan Received 28 June 2005; accepted 12 September 2005 Available online 20 October 2005
Abstract In this paper, the mixed-mode crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electric loading. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity. 2005 Elsevier SAS. All rights reserved. Keywords: Functionally graded piezoelectric material; Parallel crack; Integral transform
1. Introduction In designing with piezoelectric materials, it is important to take into consideration imperfections, such as cracks, that are often pre-existing or are generated by external loads during the service life. Thus fracture of piezoelectric materials has received much attention. On the other hand, the concept of the well-known functionally graded materials (FGMs) can be extended to a piezoelectric material to improve its reliability (Wu et al., 1996). These new kinds of materials with continuously varying properties may be called functionally graded piezoelectric materials (FGPMs). Fracture of FGPMs has been studied under thermal load (Wang and Noda, 2001), anti-plane mechanical and in-plane electric loads (Li and Weng, 2002; Wang and Zhang, 2004), and in-plane mechanical and electric impact loads (Chen et al., 2003). Moreover, the present author studied mode III crack problem in an FGPM strip with elastic surface layers (Ueda, 2003), and thermally induced fracture of an FGPM strip (Ueda, 2004). One of the practical problems in this area is that of an internal crack in a FGPM subjected to in-plane loading in the nonhomogeneous direction. The problem in this case is always a mixed-mode problem. In this paper, we consider a simple case of these problems, namely that of an FGPM strip containing a crack parallel to the boundaries. It is assumed that the elastic stiffness, piezoelectric, and dielectric constants of the strip vary continuously in the thickness direction. The problem is solved under in-plane electric loading. Integral transform technique is used to
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E-mail address: [email protected] (S. Ueda). 0997-7538/$ – see front matter 2005 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2005.09.001
S. Ueda / European Journal of Mechanics A/Solids 25 (2006) 250–259
251
Fig. 1. Geometry of the crack problem in a functionally graded piezoelectric strip.
reduce the problem to the solutions of a system of singular integral equations (Erdogan and Wu, 1997). Numerical calculations are carried out (Erdogan et al., 1972), and the stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity. 2. Formulation of the problem A finite crack of length 2c is embedded in an infinite long functionally graded piezoelectric strip of thickness h. The in-plane electric loading is shown in Fig. 1. In the figure, D0 is a uniform electric displacement applied externally. A set of Cartesian coordinates is denoted by (x, y, z) with its origin at the mid-point of the crack face and x-direction along the crack line, where z is the poling axis. In the following, the subscripts x, y, z will be used to refer to the direction of coordinates. As shown in Fig. 1 with the shaded region showing an increasing intensity, the material property parameters are taken to vary continuously along the z-direction inside the strip. To achieve the objective of obtaining the stress and electric fields in the FGPM, the material properties, such as the elastic stiffness constants ckl (z), piezoelectric constants ekl (z) and dielectric constants εkk (z), are one-dimensionally dependent as following equations. ckl (z), ekl (z), εkk (z) = (ckl0 , ekl0 , εkk0 ) exp β(z + h2 )
(−h2 z h1 )
(1)
where β is a positive or negative constant, and the subscript 0 indicates the properties at the plane z = −h2 . Then the equations of plane electroelasticity for the nonhomogeneous medium may be expressed as ∂uxi (x, z) ∂uzi (x, z) ∂φi (x, z) + c13 (z) + e31 (z) , ∂x ∂z ∂z ∂uxi (x, z) ∂uzi (x, z) ∂φi (x, z) + c33 (z) + e33 (z) , σzzi (x, z) = c13 (z) ∂x ∂z ∂z ∂uxi (x, z) ∂uzi (x, z) ∂φi (x, z) σzxi = c44 (z) + + e15 (z) ∂z ∂x ∂x σxxi (x, z) = c11 (z)
∂uxi (x, z) ∂uzi (x, z) ∂φi (x, z) Dxi (x, z) = e15 (z) + − ε11 (z) , ∂z ∂x ∂x
∂uxi (x, z) ∂uzi (x, z) ∂φi (x, z) Dzi (x, z) = e31 (z) + e33 (z) − ε33 (z) ∂x ∂z ∂z
(i = 1, 2),
(i = 1, 2),
(2)
(3)
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∂ 2 uxi ∂ 2 uxi ∂ 2 uzi ∂ 2 φ1 c110 + c440 + (c130 + c440 ) + (e310 + e150 ) 2 2 ∂x∂z ∂x∂z ∂x ∂z
∂uxi ∂uzi ∂φi + + e150 = 0, + β c440 ∂z ∂x ∂x ∂ 2 uzi ∂ 2 uzi ∂ 2 φi ∂ 2 φi ∂ 2 uxi c440 + c + (c + c ) + e + e 330 130 440 150 330 ∂x∂z ∂x 2 ∂z2 ∂x 2 ∂z2
∂uxi ∂uzi ∂φi = 0, + β c130 + c330 + e330 ∂x ∂z ∂z ∂ 2 uzi ∂ 2 uzi ∂ 2 φi ∂ 2 φi ∂ 2 uxi e150 + e + (e + e ) − ε − ε 330 150 330 110 330 ∂x∂z ∂x 2 ∂z2 ∂x 2 ∂z2
∂uxi ∂uzi ∂φi =0 + β e310 + e330 − ε330 ∂x ∂z ∂z
(i = 1, 2),
(4)
where φi (x, z) is the electric potential, uxi (x, z), uzi (x, z) are the displacement components, and σxxi (x, z), σzzi (x, z), σzxi (x, z), Dxi (x, z), Dzi (x, z) (i = 1, 2) are the stress and electric displacement components. The subscript i = 1, 2 denote, respectively, the electroelastic fields in 0 z h1 and −h2 z 0. There are two idealized crack face electric boundary conditions that are impermeable and permeable conditions. Recently, Wang and Mai (2004) show that the electrically impermeable boundary is a reasonable one for engineering problems. Thus, through a proper superposition, the symmetry and boundary conditions can be written as |x| < c , σzz1 (x, 0) = σzz2 (x, 0) = 0 (5) c |x| < ∞ , uz1 (x, 0) = uz2 (x, 0) σzx1 (x, 0) = σzx2 (x, 0) = 0 |x| < c , (6) c |x| < ∞ , ux1 (x, 0) = ux2 (x, 0) Dz1 (x, 0) = Dz2 (x, 0) = −D0 |x| < c , (7) c |x| < ∞ , φ1 (x, 0) = φ2 (x, 0) σzz1 (x, 0) = σzz2 (x, 0) |x| < ∞ , (8) (9) σzx1 (x, 0) = σzx2 (x, 0) |x| < ∞ , (10) Dz1 (x, 0) = Dz2 (x, 0) |x| < ∞ , (11) σzz1 (x, h1 ) = 0 |x| < ∞ , (12) σzx1 (x, h1 ) = 0 |x| < ∞ , (13) Dz1 (x, h1 ) = 0 |x| < ∞ , (14) σzz2 (x, −h2 ) = 0 |x| < ∞ , (15) σzx2 (x, −h2 ) = 0 |x| < ∞ , (16) Dz2 (x, −h2 ) = 0 |x| < ∞ . 3. Method of solutions The general solution of the displacement components and the electric potential is obtained by solving the governing equations(4) using the Fourier integral transform techniques: 6 ∞ |s| i aij (s)Aij (s) exp sγij (s)z exp(−isx) ds uxi (x, z) = 2π s
(i = 1, 2),
(17)
j =1−∞
6 ∞ 1 uzi (x, z) = Aij (s) exp sγij (s)z exp(−isx) ds 2π j =1−∞
(i = 1, 2),
(18)
S. Ueda / European Journal of Mechanics A/Solids 25 (2006) 250–259 6 1 bij (s)Aij (s) exp sγij (s)z exp(−isx) ds 2π
253
∞
φi (x, z) = −
(i = 1, 2)
(19)
j =1 0
where Aij (s) (i = 1, 2, j = 1, 2, . . . , 6) are the unknowns to be solved. The known functions γij (s), aij (s), bij (s) (i = 1, 2, j = 1, 2, . . . , 6) are given in Appendix A. Substituting Eqs. (17)–(19) into Eqs. (2) and (3), one obtains the stress and electric displacement expressions. The problem may be reduced to a system of singular integral equations by defining the following new unknown functions Gl (x) (l = 1, 2, 3) (Erdogan and Wu, 1997): ∂ u (x, 0) − uz2 (x, 0) |x| < c , G1 (x) = ∂x z1 (20) 0 c |x| < ∞ , ∂ ux1 (x, 0) − ux2 (x, 0) |x| < c , (21) G2 (x) = ∂x 0 c |x| < ∞ , ∂ − φ1 (x, 0) − φ2 (x, 0) |x| < c , (22) G3 (x) = ∂x 0 c |x| < ∞ . Making use of the first boundary conditions (5)–(7) with Eqs. (8)–(16), we have the following system of singular integral equations for the determination of the unknown functions Gl (t) (l = 1, 2, 3): 1 π 1 π 1 π
c −c
∞ ∞ Z13 Z11 + M11 (t, x) G1 (t) − M12 (t, x)G2 (t) + + M13 (t, x) G3 (t) dt = 0 |x| < c , t −x t −x
c
M21 (t, x)G1 (t) +
−c c
∞ Z22 + M22 (t, x) G2 (t) + M23 (t, x)G3 (t) dt = 0 |x| < c , t −x
∞ ∞ Z31 Z33 + M31 (t, x) G1 (t) − M32 (t, x)G2 (t) + + M33 (t, x) G3 (t) dt t −x t −x −c = D0 exp(−βh2 ) |x| < c ,
(23)
(24)
(25)
∞ (k, l = 1, 2, 3) are given in Appendix B. The singular intewhere the kernel functions Mkl (t, x) and the constants Zkl gral equations (23)–(25) are to be solved with the following subsidiary conditions obtained from the second boundary conditions (5)–(7).
c Gl (t) dt = 0 (l = 1, 2, 3).
(26)
−c
4. Solution of the singular integral equation and intensity factors To solve the singular integral equations (23)–(25) and the additional equations (26), we introduce the following functions Φl (ξ ) (l = 1, 2, 3): Gl (t) =
1 Φl (ξ ) {(1 + ξ )(1 − ξ )}1/2
(l = 1, 2, 3)
(27)
where ξ = t/c and η = x/c (−c < t, x < c, −1 < ξ, η < 1). Using the Gauss–Jacobi integration formula (Erdogan et al., 1972), the singular integral equations (23)–(25) may be approximated by
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N ∞ Z11 1 + M11 (cξm , cηn ) Φ1 (ξm ) − M12 (cξm , cηn )Φ2 (ξm ) π ξ m − ηn m=1
+
∞ Z13 + M13 (ξm , ηn ) Φ3 (ξm ) Wm = 0 (n = 1, 2, . . . , N − 1), ξm − ηn
N ∞ Z22 1 M21 (cξm , cηn )Φ1 (ξm ) + + M22 (cξm , cηn ) Φ2 (ξm ) π ξ m − ηn m=1 + M23 (cξm , cηn )Φ3 (ξm ) Wm = 0 (n = 1, 2, . . . , N − 1),
(28)
(29)
N ∞ Z31 1 + M31 (cξm , cηn ) Φ1 (ξm ) − M32 (cξm , cηn )Φ2 (ξm ) π ξ m − ηn m=1
∞ Z33 + M33 (cξm , cηn ) Φ3 (ξm ) Wm = D0 exp(−βh2 ) (n = 1, 2, . . . , N − 1) ξm − ηn where Wm (m = 1, 2, . . . , N ) are the weighting constants and ξm , ηn are the roots of (−1/2,−1/2) (ξm ) = 0 (m = 1, 2, . . . , N ), PN (1/2,1/2) PN −1 (ηn ) = 0 (n = 1, 2, . . . , N − 1). +
(−1/2,−1/2)
(30)
(31)
(1/2,1/2)
In Eq. (31), PN (ξm ) and PN −1 (ηn ) denote the Jacobi polynomials. Eqs. (28)–(30) provides us with 3(N − 1) linear algebraic equations for 3N unknowns Φl (ξm ) (l = 1, 2, 3, m = 1, 2, . . . , N ). The additional equation is obtained by using the additional conditions (26). N
Φl (ξm )Wm = 0 (l = 1, 2, 3).
(32)
m=1
Φl (ξm ) (l = 1, 2, 3, m = 1, 2, . . . , N) are numerically evaluated from Eqs. (28)–(30) and (32). The values of Φl (1) (l = 1, 2, 3) which are necessary for obtaining the intensity factors are determined from quadratic extrapolations by employing the values of Φl (ξN −3 ), Φl (ξN −2 ) and Φl (ξN −1 ) (l = 1, 2, 3). Thus, the stress intensity factors KI , KII and the electric displacement intensity factor KD are defined and evaluated as: 1/2 ∞ ∞ Φ (1) , σzz1 (x, 0) = (πc)1/2 Z11 Φ1 (1) + Z13 KI = limx→c+ 2π(c − x) 3 1/2 ∞ 1/2 (33) σzx1 (x, 0) = (πc) Z22 Φ2 (1), KII = limx→c+ 2π(c − x) 1/2 ∞ Φ (1) + Z ∞ Φ (1) . KD = limx→c+ 2π(c − x) Dz1 (x, 0) = (πc)1/2 Z31 1 33 3 5. Numerical results and discussion To examine the effect of electroelastic interactions on the elastic and electric fields, the solutions of the singular integral equations have been computed numerically. For the numerical calculations, the material is considered to be cadmium selenide, with the following properties (Ashida and Tauchert, 1998): c110 = 7.41 × 1010 [Pa], c130 = 3.93 × 1010 [Pa], 10 c330 = 8.36 × 10 [Pa], 10 c440 = 1.32 × 10 [Pa], 2 (34) e310 = −0.16 [C/m ], e330 = 0.347 [C/m2 ], e150 = −0.138 [C/m2 ], −10 ε110 = 0.825 × 10 [C/Vm], ε330 = 0.903 × 10−10 [C/Vm].
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The electroelastic properties of the FGPM are obtained from Eq. (1) once the dimensions h and h2 = h − h1 are specified. If the material and geometry lack symmetry, the problem is one of mixed-mode, meaning that generally KI , KII and KD are non-zero. Moreover, it should be noted that in the example considered x = 0 is a plane of symmetry with respect to the external load as well as the material properties. Therefore, the stress and electric displacement intensity factors are calculated at x = c only and following symmetry conditions are valid: [KI ]x=−c = [KI ]x=c , [KII ]x=−c = −[KII ]x=c , (35) [KD ]x=−c = [KD ]x=c . The following normalizing stress is used to present the results of the stress intensity factors in dimensionless form: e330 D0 . (36) σ0 = − ε330 Figs. 2(a)–(c) show the effect of the normalized crack length c/ h on the normalized stress intensity factors (KI , KII )/σ0 (πc)1/2 and the normalized electric displacement intensity factor KD /D0 (πc)1/2 for the normalized material parameter βh = −1.0, 0.0, 1.0. The solid, dashed and dotted lines indicate the results for the normalized crack location h1 / h = 0.25, 0.5, 0.75, respectively. For the purpose of checking the accuracy of the results, numerical calculations are carried out with N = 40 and 50, and the absolute errors between them are smaller than 1.0 × 10−4 . No more than N = 40 terms in Eqs. (28)–(30) and (32) are then found to be necessary in obtaining the highly accurate values of intensity factors. The values of KI /σ0 (πc)1/2 and KD /D0 (πc)1/2 increase monotonically with increasing
(a)
(b)
(c) Fig. 2. (a) The effect of the crack length on the stress intensity factor KI . (b) The effect of the crack length on the stress intensity factor KII . (c) The effect of the crack length on the electric displacement intensity factor KD .
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S. Ueda / European Journal of Mechanics A/Solids 25 (2006) 250–259
(a)
(b)
(c) Fig. 3. (a) The effect of the crack location on the stress intensity factor KI . (b) The effect of the crack location on the stress intensity factor KII . (c) The effect of the crack location on the electric displacement intensity factor KD .
c/ h, but the absolute value of KII /σ0 (πc)1/2 increases at first, reaches maximum value and then decreases with increasing c/ h except for h1 / h = 0.25, βh = 1.0 and h1 / h = 0.75, βh = −1.0. For these cases, KII /σ0 (πc)1/2 increases slightly or decreases in 0.0 < c/ h < 0.1. The effect of βh on KI and KD is not so large. Because h1 / h=0.25 h1 / h=0.75 h1 / h=0.5 h1 / h=0.25 h1 / h=0.75 = [KI ]βh=−1.0 , [KII ]βh=0.0 = 0.0, [KII ]βh=1.0 = −[KII ]βh=−1.0 , of symmetry, it is found that [KI ]βh=1.0 h / h=0.25
h / h=0.75
1 1 = [KD ]βh=±1.0 , and so on. [KD ]βh=±1.0 Figs. 3(a)–(c) give the plots of (KI , KII )/σ0 (πc)1/2 and KD /D0 (πc)1/2 versus h1 / h for βh = −1.0, 0.0, 1.0 at c/ h = 0.5, respectively. If the crack approaches the free boundaries (h1 / h → 0.0 or h1 / h → 1.0), these intensity factors tend to increase, and the effect of βh on KD is small. KI for βh = 0.0 and KD are symmetric with respect to h1 / h = 0.5. The value of KI for β = 1.0 at h1 / h = 0.5 − ξ (0 ξ < 0.4) is same as one for βh = −1.0 at h1 / h = 0.5 + ξ . Figs. 4(a)–(c) indicate the effect of βh on KI , KII and KD for h1 / h = 0.25, 0.5, 0.75 with c/ h = 0.5. It is clear that KD is relatively insensitive to βh, whereas KI and KII are affected by βh. As βh increases, the values of KI for h1 / h = 0.25, 0.75 and KII decrease or increase, monotonically. The changes of KI for h1 / h = 0.5 and KD are symmetric with respect to βh = 0.0.
6. Conclusions The mixed-mode problem of a finite crack in a strip of a functionally graded piezoelectric material is studied. The following facts can be found from the numerical results. Firstly, the normalized intensity factors are under the
S. Ueda / European Journal of Mechanics A/Solids 25 (2006) 250–259
(a)
257
(b)
(c) Fig. 4. (a) The effect of the material gradient on the stress intensity factor KI . (b) The effect of the material gradient on the stress intensity factor KII . (c) The effect of the material gradient on the electric displacement intensity factor KD .
great influence of the geometrical parameters h1 / h and c/ h. Especially, the intensity factors of crack near the free surface are very large. Secondly, the effect of the material nonhomogeneity on the stress intensity factors KI of the center crack and the electric displacement intensity factor KD is not so large. Finally, the influence of the normalized material parameter βh on the stress intensity factors KI and KII is dependent on the crack location. For example, the increase of βh is beneficial for reducing [KI ]h1 / h>0.5 and [KII ]h1 / h<0.5 , while the reverse response is observed for [KI ]h1 / h<0.5 and [KII ]h1 / h>0.5 . Appendix A The functions γij = γij (s) (i = 1, 2, j = 1, 2, . . . , 6) are the roots of the following characteristic equations: f3 (s)g3 (s) + g2 (s)f4 (s) γij6 + f3 (s)g2 (s) + g2 (s)f3 (s) + f2 (s)g3 (s) + g1 (s)f4 (s) γij5 + f3 (s)g1 (s) + g2 (s)f2 (s) + f2 (s)g2 (s) + g1 (s)f3 (s) + f1 (s)g3 (s) + g0 (s)f4 (s) γij4 + f3 (s)g0 (s) + g2 (s)f1 (s) + f2 (s)g1 (s) + g1 (s)f2 (s) + f1 (s)g2 (s) + g0 (s)f3 (s) + f0 (s)g3 (s) γij3 + f2 (s)g0 (s) + g2 (s)f0 (s) + f1 (s)g1 (s) + g1 (s)f1 (s) + f0 (s)g2 (s) + g0 (s)f2 (s) γij2 + f1 (s)g0 (s) + g1 (s)f0 (s) + f0 (s)g1 (s) + g0 (s)f1 (s) γij + f0 (s)g0 (s) + g0 (s)f0 (s) = 0 (i = 1, 2, j = 1, 2, . . . , 6)
(A.1)
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where [γ1j ] < [γ1j +1 ], [γ2j ] > [γ2j +1 ] (j = 1, 2, . . . , 5) and 2 f1 (s) = (c130 ε330 + e310 e330 )(β/s) − e150 (e150 + e310 ) − ε110 (c130 + c440 ), f0 (s) = −(e150 e310 + c130 ε110 )β/|s|, 2 f4 (s) = c330 ε330 + e330 , 2 )β/|s|, f3 (s) = 2(c330 ε330 + e330 2 2 f2 (s) = (c330 ε330 + e330 ) β/|s| − c440 ε330 − 2e150 e330 − c330 ε110 , f1 (s) = −(c440 ε330 + 2e150 e330 + c330 ε110 )β/|s|, 2 f0 (s) = c440 ε110 + e150 , g2 (s) = (e150 + e310 ) ε330 (c130 + c440 ) + e330 (e150 + e310 ) − c440 (e150 ε330 − e330 ε110 ), g1 (s) = e150 ε330 (c130 + c440 ) + e330 (e150 + e310 ) + (e150 + e310 )(c130 ε330 + e310 e330 ) − c440 (e150 ε330 − e330 ε110 ) β/|s|, 2 g0 (s) = e150 (c130 ε330 + e310 e330 )(β/s) + c110 (e150 ε330 − e330 ε110 ), 2 g3 (s) = −(e150 + e310 )(c330 ε330 + e330 ), 2 )β/|s|, g2 (s) = −(e310 + 2e150 )(c330 ε330 + e330 2 2 g1 (s) = −e150 (c330 ε330 + e330 ) β/|s| + (e150 + e310 )(c440 ε330 + e150 e310 ) − (e150 ε330 − e330 ε110 )(c130 + c440 ), 2 g0 (s) = e330 (e150 + c440 ε110 )β/|s|. f3 (s) = ε330 (c130 + c440 ) + e330 (e150 + e310 ), f2 (s) = ε330 (2c130 + c440 ) + e330 (e150 + 2e310 ) β/|s|,
(A.2)
(A.3)
The functions aij (s) and bij (s) (i = 1, 2, j = 1, 2, . . . , 6) are given by aij (s) = bij (s) =
g3 (s)γij3 + g2 (s)γij2 + g1 (s)γij + g0 (s) g2 (s)γij2 + g1 (s)γij + g0 (s)
,
{(c130 + c440 )γij + c130 β/|s|}aij (s) + c330 γij2 + c330 (β/|s|)γij − c440 2 e330 γij + e330 (β/|s|)γij − e150
(i = 1, 2, j = 1, 2, . . . , 6).
(A.4)
Appendix B The kernel functions Mkl (t, x) (k, l = 1, 2, 3) are given by ∞ ∞ Z sin s(t − x) ds (l = 1, 3: k = 1, 3, l = 2 : k = 1), (s) − Z kl kl 0 Mkl (t, x) = ∞ ∞ Z cos s(t − x) ds (l = 2: k = 1, 3, l = 1, 3 : k = 1) (s) − Z kl kl
(B.1)
0
where Zkl (s) =
6
j =1
pk+3lj (s)dj,l (s),
∞ Zkl = lim Zkl (s) s→∞
(k, l = 1, 2, 3).
(B.2)
S. Ueda / European Journal of Mechanics A/Solids 25 (2006) 250–259
In Eq. (B.2), the functions pkij (s) (i = 1, 2, j, k = 1, 2, . . . , 6) are p1ij (s) = −i, p2ij (s) = aij (s), p3ij (s) = −ibij (s), p4ij (s) = c130 aij (s) + γij (s) c330 − e330 bij (s) , (i = 1, 2, j = 1, 2, . . . , 6). p5ij (s) = c440 γij (s)aij (s) − 1 + e150 bij (s), p6ij (s) = e310 aij (s) + γij (s) e330 + ε330 bij (s)
259
(B.3)
The functions dm,n (s) (m, n = 1, 2, . . . , 12) are the element of a square matrix D = ∆−1 of order 12. The elements δm,n (s) (m, n = 1, 2, . . . , 12) of the square matrix ∆ are given by δk,j (s) = pk1j (s), (k = 1, 2, . . . , 6), δk,j +6 (s) = −pk2j (s) (j = 1, 2, . . . , 6). (B.4) δk+3,j (s) = pk1j (s) exp sγ1j (s)h1 , (k = 4, 5, 6) δk+6,j +6 (s) = −pk2j (s) exp −sγ2j (s)h2 References Ashida, F., Tauchert, T.R., 1998. Transient response of a piezothermoelastic circular disk under axisymmetric heating. Acta Mech. 128, 1–14. Chen, J., Liu, Z.X., Zou, Z.Z., 2003. Electromechanical impact of a crack in a functionally graded piezoelectric medium. Theoret. Appl. Fracture Mech. 39, 47–60. Erdogan, F., Gupta, G.D., Cook, T.S., 1972. In: Sih, G.C. (Ed.), Methods of Analysis and Solution of Crack Problems. Noordhoff, Leyden. Erdogan, F., Wu, B.H., 1997. The surface crack problem for a plate with functionally graded properties. Trans. ASME J. Appl. Mech. 64, 449–456. Li, C., Weng, G.J., 2002. Antiplane crack problem in functionally graded piezoelectric materials. Trans. ASME J. Appl. Mech. 69, 481–488. Ueda, S., 2003. Crack in a functionally graded piezoelectric strip bonded to elastic surface layers under electromechanical loading. Theoret. Appl. Fracture Mech. 40, 225–236. Ueda, S., 2004. Thermally induced fracture of a functionally graded piezoelectric layer. J. Thermal Stresses 27, 291–309. Wang, B.L., Mai, Y.W., 2004. Impermeable crack and permeable crack assumptions, which one is more realistic?. Trans. ASME J. Appl. Mech. 71, 575–578. Wang, B.L., Noda, N., 2001. Thermally induced fracture of a smart functionally graded composite structure. Theoret. Appl. Fracture Mech. 35, 93–109. Wang, B.L., Zhang, X.H., 2004. A mode III crack in functionally graded piezoelectric material strip. Trans. ASME J. Appl. Mech. 71, 327–333. Wu, C.M., Kahn, M., Moy, W., 1996. Piezoelectric ceramics with functionally gradients: a new application in material design. J. Amer. Ceramics Soc. 79, 809–812.
A finite crack in a semi-infinite strip of a graded piezoelectric material under electric loading Sei Ueda ∗ Department of Mechanical Engineering, Osaka Institute of Technology, 5-16-1, Omiya, Asahi-ku, Osaka 535-8585, Japan Received 28 June 2005; accepted 12 September 2005 Available online 20 October 2005
Abstract In this paper, the mixed-mode crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electric loading. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity. 2005 Elsevier SAS. All rights reserved. Keywords: Functionally graded piezoelectric material; Parallel crack; Integral transform
1. Introduction In designing with piezoelectric materials, it is important to take into consideration imperfections, such as cracks, that are often pre-existing or are generated by external loads during the service life. Thus fracture of piezoelectric materials has received much attention. On the other hand, the concept of the well-known functionally graded materials (FGMs) can be extended to a piezoelectric material to improve its reliability (Wu et al., 1996). These new kinds of materials with continuously varying properties may be called functionally graded piezoelectric materials (FGPMs). Fracture of FGPMs has been studied under thermal load (Wang and Noda, 2001), anti-plane mechanical and in-plane electric loads (Li and Weng, 2002; Wang and Zhang, 2004), and in-plane mechanical and electric impact loads (Chen et al., 2003). Moreover, the present author studied mode III crack problem in an FGPM strip with elastic surface layers (Ueda, 2003), and thermally induced fracture of an FGPM strip (Ueda, 2004). One of the practical problems in this area is that of an internal crack in a FGPM subjected to in-plane loading in the nonhomogeneous direction. The problem in this case is always a mixed-mode problem. In this paper, we consider a simple case of these problems, namely that of an FGPM strip containing a crack parallel to the boundaries. It is assumed that the elastic stiffness, piezoelectric, and dielectric constants of the strip vary continuously in the thickness direction. The problem is solved under in-plane electric loading. Integral transform technique is used to
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E-mail address: [email protected] (S. Ueda). 0997-7538/$ – see front matter 2005 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2005.09.001
S. Ueda / European Journal of Mechanics A/Solids 25 (2006) 250–259
251
Fig. 1. Geometry of the crack problem in a functionally graded piezoelectric strip.
reduce the problem to the solutions of a system of singular integral equations (Erdogan and Wu, 1997). Numerical calculations are carried out (Erdogan et al., 1972), and the stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity. 2. Formulation of the problem A finite crack of length 2c is embedded in an infinite long functionally graded piezoelectric strip of thickness h. The in-plane electric loading is shown in Fig. 1. In the figure, D0 is a uniform electric displacement applied externally. A set of Cartesian coordinates is denoted by (x, y, z) with its origin at the mid-point of the crack face and x-direction along the crack line, where z is the poling axis. In the following, the subscripts x, y, z will be used to refer to the direction of coordinates. As shown in Fig. 1 with the shaded region showing an increasing intensity, the material property parameters are taken to vary continuously along the z-direction inside the strip. To achieve the objective of obtaining the stress and electric fields in the FGPM, the material properties, such as the elastic stiffness constants ckl (z), piezoelectric constants ekl (z) and dielectric constants εkk (z), are one-dimensionally dependent as following equations. ckl (z), ekl (z), εkk (z) = (ckl0 , ekl0 , εkk0 ) exp β(z + h2 )
(−h2 z h1 )
(1)
where β is a positive or negative constant, and the subscript 0 indicates the properties at the plane z = −h2 . Then the equations of plane electroelasticity for the nonhomogeneous medium may be expressed as ∂uxi (x, z) ∂uzi (x, z) ∂φi (x, z) + c13 (z) + e31 (z) , ∂x ∂z ∂z ∂uxi (x, z) ∂uzi (x, z) ∂φi (x, z) + c33 (z) + e33 (z) , σzzi (x, z) = c13 (z) ∂x ∂z ∂z ∂uxi (x, z) ∂uzi (x, z) ∂φi (x, z) σzxi = c44 (z) + + e15 (z) ∂z ∂x ∂x σxxi (x, z) = c11 (z)
∂uxi (x, z) ∂uzi (x, z) ∂φi (x, z) Dxi (x, z) = e15 (z) + − ε11 (z) , ∂z ∂x ∂x
∂uxi (x, z) ∂uzi (x, z) ∂φi (x, z) Dzi (x, z) = e31 (z) + e33 (z) − ε33 (z) ∂x ∂z ∂z
(i = 1, 2),
(i = 1, 2),
(2)
(3)
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∂ 2 uxi ∂ 2 uxi ∂ 2 uzi ∂ 2 φ1 c110 + c440 + (c130 + c440 ) + (e310 + e150 ) 2 2 ∂x∂z ∂x∂z ∂x ∂z
∂uxi ∂uzi ∂φi + + e150 = 0, + β c440 ∂z ∂x ∂x ∂ 2 uzi ∂ 2 uzi ∂ 2 φi ∂ 2 φi ∂ 2 uxi c440 + c + (c + c ) + e + e 330 130 440 150 330 ∂x∂z ∂x 2 ∂z2 ∂x 2 ∂z2
∂uxi ∂uzi ∂φi = 0, + β c130 + c330 + e330 ∂x ∂z ∂z ∂ 2 uzi ∂ 2 uzi ∂ 2 φi ∂ 2 φi ∂ 2 uxi e150 + e + (e + e ) − ε − ε 330 150 330 110 330 ∂x∂z ∂x 2 ∂z2 ∂x 2 ∂z2
∂uxi ∂uzi ∂φi =0 + β e310 + e330 − ε330 ∂x ∂z ∂z
(i = 1, 2),
(4)
where φi (x, z) is the electric potential, uxi (x, z), uzi (x, z) are the displacement components, and σxxi (x, z), σzzi (x, z), σzxi (x, z), Dxi (x, z), Dzi (x, z) (i = 1, 2) are the stress and electric displacement components. The subscript i = 1, 2 denote, respectively, the electroelastic fields in 0 z h1 and −h2 z 0. There are two idealized crack face electric boundary conditions that are impermeable and permeable conditions. Recently, Wang and Mai (2004) show that the electrically impermeable boundary is a reasonable one for engineering problems. Thus, through a proper superposition, the symmetry and boundary conditions can be written as |x| < c , σzz1 (x, 0) = σzz2 (x, 0) = 0 (5) c |x| < ∞ , uz1 (x, 0) = uz2 (x, 0) σzx1 (x, 0) = σzx2 (x, 0) = 0 |x| < c , (6) c |x| < ∞ , ux1 (x, 0) = ux2 (x, 0) Dz1 (x, 0) = Dz2 (x, 0) = −D0 |x| < c , (7) c |x| < ∞ , φ1 (x, 0) = φ2 (x, 0) σzz1 (x, 0) = σzz2 (x, 0) |x| < ∞ , (8) (9) σzx1 (x, 0) = σzx2 (x, 0) |x| < ∞ , (10) Dz1 (x, 0) = Dz2 (x, 0) |x| < ∞ , (11) σzz1 (x, h1 ) = 0 |x| < ∞ , (12) σzx1 (x, h1 ) = 0 |x| < ∞ , (13) Dz1 (x, h1 ) = 0 |x| < ∞ , (14) σzz2 (x, −h2 ) = 0 |x| < ∞ , (15) σzx2 (x, −h2 ) = 0 |x| < ∞ , (16) Dz2 (x, −h2 ) = 0 |x| < ∞ . 3. Method of solutions The general solution of the displacement components and the electric potential is obtained by solving the governing equations(4) using the Fourier integral transform techniques: 6 ∞ |s| i aij (s)Aij (s) exp sγij (s)z exp(−isx) ds uxi (x, z) = 2π s
(i = 1, 2),
(17)
j =1−∞
6 ∞ 1 uzi (x, z) = Aij (s) exp sγij (s)z exp(−isx) ds 2π j =1−∞
(i = 1, 2),
(18)
S. Ueda / European Journal of Mechanics A/Solids 25 (2006) 250–259 6 1 bij (s)Aij (s) exp sγij (s)z exp(−isx) ds 2π
253
∞
φi (x, z) = −
(i = 1, 2)
(19)
j =1 0
where Aij (s) (i = 1, 2, j = 1, 2, . . . , 6) are the unknowns to be solved. The known functions γij (s), aij (s), bij (s) (i = 1, 2, j = 1, 2, . . . , 6) are given in Appendix A. Substituting Eqs. (17)–(19) into Eqs. (2) and (3), one obtains the stress and electric displacement expressions. The problem may be reduced to a system of singular integral equations by defining the following new unknown functions Gl (x) (l = 1, 2, 3) (Erdogan and Wu, 1997): ∂ u (x, 0) − uz2 (x, 0) |x| < c , G1 (x) = ∂x z1 (20) 0 c |x| < ∞ , ∂ ux1 (x, 0) − ux2 (x, 0) |x| < c , (21) G2 (x) = ∂x 0 c |x| < ∞ , ∂ − φ1 (x, 0) − φ2 (x, 0) |x| < c , (22) G3 (x) = ∂x 0 c |x| < ∞ . Making use of the first boundary conditions (5)–(7) with Eqs. (8)–(16), we have the following system of singular integral equations for the determination of the unknown functions Gl (t) (l = 1, 2, 3): 1 π 1 π 1 π
c −c
∞ ∞ Z13 Z11 + M11 (t, x) G1 (t) − M12 (t, x)G2 (t) + + M13 (t, x) G3 (t) dt = 0 |x| < c , t −x t −x
c
M21 (t, x)G1 (t) +
−c c
∞ Z22 + M22 (t, x) G2 (t) + M23 (t, x)G3 (t) dt = 0 |x| < c , t −x
∞ ∞ Z31 Z33 + M31 (t, x) G1 (t) − M32 (t, x)G2 (t) + + M33 (t, x) G3 (t) dt t −x t −x −c = D0 exp(−βh2 ) |x| < c ,
(23)
(24)
(25)
∞ (k, l = 1, 2, 3) are given in Appendix B. The singular intewhere the kernel functions Mkl (t, x) and the constants Zkl gral equations (23)–(25) are to be solved with the following subsidiary conditions obtained from the second boundary conditions (5)–(7).
c Gl (t) dt = 0 (l = 1, 2, 3).
(26)
−c
4. Solution of the singular integral equation and intensity factors To solve the singular integral equations (23)–(25) and the additional equations (26), we introduce the following functions Φl (ξ ) (l = 1, 2, 3): Gl (t) =
1 Φl (ξ ) {(1 + ξ )(1 − ξ )}1/2
(l = 1, 2, 3)
(27)
where ξ = t/c and η = x/c (−c < t, x < c, −1 < ξ, η < 1). Using the Gauss–Jacobi integration formula (Erdogan et al., 1972), the singular integral equations (23)–(25) may be approximated by
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N ∞ Z11 1 + M11 (cξm , cηn ) Φ1 (ξm ) − M12 (cξm , cηn )Φ2 (ξm ) π ξ m − ηn m=1
+
∞ Z13 + M13 (ξm , ηn ) Φ3 (ξm ) Wm = 0 (n = 1, 2, . . . , N − 1), ξm − ηn
N ∞ Z22 1 M21 (cξm , cηn )Φ1 (ξm ) + + M22 (cξm , cηn ) Φ2 (ξm ) π ξ m − ηn m=1 + M23 (cξm , cηn )Φ3 (ξm ) Wm = 0 (n = 1, 2, . . . , N − 1),
(28)
(29)
N ∞ Z31 1 + M31 (cξm , cηn ) Φ1 (ξm ) − M32 (cξm , cηn )Φ2 (ξm ) π ξ m − ηn m=1
∞ Z33 + M33 (cξm , cηn ) Φ3 (ξm ) Wm = D0 exp(−βh2 ) (n = 1, 2, . . . , N − 1) ξm − ηn where Wm (m = 1, 2, . . . , N ) are the weighting constants and ξm , ηn are the roots of (−1/2,−1/2) (ξm ) = 0 (m = 1, 2, . . . , N ), PN (1/2,1/2) PN −1 (ηn ) = 0 (n = 1, 2, . . . , N − 1). +
(−1/2,−1/2)
(30)
(31)
(1/2,1/2)
In Eq. (31), PN (ξm ) and PN −1 (ηn ) denote the Jacobi polynomials. Eqs. (28)–(30) provides us with 3(N − 1) linear algebraic equations for 3N unknowns Φl (ξm ) (l = 1, 2, 3, m = 1, 2, . . . , N ). The additional equation is obtained by using the additional conditions (26). N
Φl (ξm )Wm = 0 (l = 1, 2, 3).
(32)
m=1
Φl (ξm ) (l = 1, 2, 3, m = 1, 2, . . . , N) are numerically evaluated from Eqs. (28)–(30) and (32). The values of Φl (1) (l = 1, 2, 3) which are necessary for obtaining the intensity factors are determined from quadratic extrapolations by employing the values of Φl (ξN −3 ), Φl (ξN −2 ) and Φl (ξN −1 ) (l = 1, 2, 3). Thus, the stress intensity factors KI , KII and the electric displacement intensity factor KD are defined and evaluated as: 1/2 ∞ ∞ Φ (1) , σzz1 (x, 0) = (πc)1/2 Z11 Φ1 (1) + Z13 KI = limx→c+ 2π(c − x) 3 1/2 ∞ 1/2 (33) σzx1 (x, 0) = (πc) Z22 Φ2 (1), KII = limx→c+ 2π(c − x) 1/2 ∞ Φ (1) + Z ∞ Φ (1) . KD = limx→c+ 2π(c − x) Dz1 (x, 0) = (πc)1/2 Z31 1 33 3 5. Numerical results and discussion To examine the effect of electroelastic interactions on the elastic and electric fields, the solutions of the singular integral equations have been computed numerically. For the numerical calculations, the material is considered to be cadmium selenide, with the following properties (Ashida and Tauchert, 1998): c110 = 7.41 × 1010 [Pa], c130 = 3.93 × 1010 [Pa], 10 c330 = 8.36 × 10 [Pa], 10 c440 = 1.32 × 10 [Pa], 2 (34) e310 = −0.16 [C/m ], e330 = 0.347 [C/m2 ], e150 = −0.138 [C/m2 ], −10 ε110 = 0.825 × 10 [C/Vm], ε330 = 0.903 × 10−10 [C/Vm].
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The electroelastic properties of the FGPM are obtained from Eq. (1) once the dimensions h and h2 = h − h1 are specified. If the material and geometry lack symmetry, the problem is one of mixed-mode, meaning that generally KI , KII and KD are non-zero. Moreover, it should be noted that in the example considered x = 0 is a plane of symmetry with respect to the external load as well as the material properties. Therefore, the stress and electric displacement intensity factors are calculated at x = c only and following symmetry conditions are valid: [KI ]x=−c = [KI ]x=c , [KII ]x=−c = −[KII ]x=c , (35) [KD ]x=−c = [KD ]x=c . The following normalizing stress is used to present the results of the stress intensity factors in dimensionless form: e330 D0 . (36) σ0 = − ε330 Figs. 2(a)–(c) show the effect of the normalized crack length c/ h on the normalized stress intensity factors (KI , KII )/σ0 (πc)1/2 and the normalized electric displacement intensity factor KD /D0 (πc)1/2 for the normalized material parameter βh = −1.0, 0.0, 1.0. The solid, dashed and dotted lines indicate the results for the normalized crack location h1 / h = 0.25, 0.5, 0.75, respectively. For the purpose of checking the accuracy of the results, numerical calculations are carried out with N = 40 and 50, and the absolute errors between them are smaller than 1.0 × 10−4 . No more than N = 40 terms in Eqs. (28)–(30) and (32) are then found to be necessary in obtaining the highly accurate values of intensity factors. The values of KI /σ0 (πc)1/2 and KD /D0 (πc)1/2 increase monotonically with increasing
(a)
(b)
(c) Fig. 2. (a) The effect of the crack length on the stress intensity factor KI . (b) The effect of the crack length on the stress intensity factor KII . (c) The effect of the crack length on the electric displacement intensity factor KD .
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(a)
(b)
(c) Fig. 3. (a) The effect of the crack location on the stress intensity factor KI . (b) The effect of the crack location on the stress intensity factor KII . (c) The effect of the crack location on the electric displacement intensity factor KD .
c/ h, but the absolute value of KII /σ0 (πc)1/2 increases at first, reaches maximum value and then decreases with increasing c/ h except for h1 / h = 0.25, βh = 1.0 and h1 / h = 0.75, βh = −1.0. For these cases, KII /σ0 (πc)1/2 increases slightly or decreases in 0.0 < c/ h < 0.1. The effect of βh on KI and KD is not so large. Because h1 / h=0.25 h1 / h=0.75 h1 / h=0.5 h1 / h=0.25 h1 / h=0.75 = [KI ]βh=−1.0 , [KII ]βh=0.0 = 0.0, [KII ]βh=1.0 = −[KII ]βh=−1.0 , of symmetry, it is found that [KI ]βh=1.0 h / h=0.25
h / h=0.75
1 1 = [KD ]βh=±1.0 , and so on. [KD ]βh=±1.0 Figs. 3(a)–(c) give the plots of (KI , KII )/σ0 (πc)1/2 and KD /D0 (πc)1/2 versus h1 / h for βh = −1.0, 0.0, 1.0 at c/ h = 0.5, respectively. If the crack approaches the free boundaries (h1 / h → 0.0 or h1 / h → 1.0), these intensity factors tend to increase, and the effect of βh on KD is small. KI for βh = 0.0 and KD are symmetric with respect to h1 / h = 0.5. The value of KI for β = 1.0 at h1 / h = 0.5 − ξ (0 ξ < 0.4) is same as one for βh = −1.0 at h1 / h = 0.5 + ξ . Figs. 4(a)–(c) indicate the effect of βh on KI , KII and KD for h1 / h = 0.25, 0.5, 0.75 with c/ h = 0.5. It is clear that KD is relatively insensitive to βh, whereas KI and KII are affected by βh. As βh increases, the values of KI for h1 / h = 0.25, 0.75 and KII decrease or increase, monotonically. The changes of KI for h1 / h = 0.5 and KD are symmetric with respect to βh = 0.0.
6. Conclusions The mixed-mode problem of a finite crack in a strip of a functionally graded piezoelectric material is studied. The following facts can be found from the numerical results. Firstly, the normalized intensity factors are under the
S. Ueda / European Journal of Mechanics A/Solids 25 (2006) 250–259
(a)
257
(b)
(c) Fig. 4. (a) The effect of the material gradient on the stress intensity factor KI . (b) The effect of the material gradient on the stress intensity factor KII . (c) The effect of the material gradient on the electric displacement intensity factor KD .
great influence of the geometrical parameters h1 / h and c/ h. Especially, the intensity factors of crack near the free surface are very large. Secondly, the effect of the material nonhomogeneity on the stress intensity factors KI of the center crack and the electric displacement intensity factor KD is not so large. Finally, the influence of the normalized material parameter βh on the stress intensity factors KI and KII is dependent on the crack location. For example, the increase of βh is beneficial for reducing [KI ]h1 / h>0.5 and [KII ]h1 / h<0.5 , while the reverse response is observed for [KI ]h1 / h<0.5 and [KII ]h1 / h>0.5 . Appendix A The functions γij = γij (s) (i = 1, 2, j = 1, 2, . . . , 6) are the roots of the following characteristic equations: f3 (s)g3 (s) + g2 (s)f4 (s) γij6 + f3 (s)g2 (s) + g2 (s)f3 (s) + f2 (s)g3 (s) + g1 (s)f4 (s) γij5 + f3 (s)g1 (s) + g2 (s)f2 (s) + f2 (s)g2 (s) + g1 (s)f3 (s) + f1 (s)g3 (s) + g0 (s)f4 (s) γij4 + f3 (s)g0 (s) + g2 (s)f1 (s) + f2 (s)g1 (s) + g1 (s)f2 (s) + f1 (s)g2 (s) + g0 (s)f3 (s) + f0 (s)g3 (s) γij3 + f2 (s)g0 (s) + g2 (s)f0 (s) + f1 (s)g1 (s) + g1 (s)f1 (s) + f0 (s)g2 (s) + g0 (s)f2 (s) γij2 + f1 (s)g0 (s) + g1 (s)f0 (s) + f0 (s)g1 (s) + g0 (s)f1 (s) γij + f0 (s)g0 (s) + g0 (s)f0 (s) = 0 (i = 1, 2, j = 1, 2, . . . , 6)
(A.1)
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where [γ1j ] < [γ1j +1 ], [γ2j ] > [γ2j +1 ] (j = 1, 2, . . . , 5) and 2 f1 (s) = (c130 ε330 + e310 e330 )(β/s) − e150 (e150 + e310 ) − ε110 (c130 + c440 ), f0 (s) = −(e150 e310 + c130 ε110 )β/|s|, 2 f4 (s) = c330 ε330 + e330 , 2 )β/|s|, f3 (s) = 2(c330 ε330 + e330 2 2 f2 (s) = (c330 ε330 + e330 ) β/|s| − c440 ε330 − 2e150 e330 − c330 ε110 , f1 (s) = −(c440 ε330 + 2e150 e330 + c330 ε110 )β/|s|, 2 f0 (s) = c440 ε110 + e150 , g2 (s) = (e150 + e310 ) ε330 (c130 + c440 ) + e330 (e150 + e310 ) − c440 (e150 ε330 − e330 ε110 ), g1 (s) = e150 ε330 (c130 + c440 ) + e330 (e150 + e310 ) + (e150 + e310 )(c130 ε330 + e310 e330 ) − c440 (e150 ε330 − e330 ε110 ) β/|s|, 2 g0 (s) = e150 (c130 ε330 + e310 e330 )(β/s) + c110 (e150 ε330 − e330 ε110 ), 2 g3 (s) = −(e150 + e310 )(c330 ε330 + e330 ), 2 )β/|s|, g2 (s) = −(e310 + 2e150 )(c330 ε330 + e330 2 2 g1 (s) = −e150 (c330 ε330 + e330 ) β/|s| + (e150 + e310 )(c440 ε330 + e150 e310 ) − (e150 ε330 − e330 ε110 )(c130 + c440 ), 2 g0 (s) = e330 (e150 + c440 ε110 )β/|s|. f3 (s) = ε330 (c130 + c440 ) + e330 (e150 + e310 ), f2 (s) = ε330 (2c130 + c440 ) + e330 (e150 + 2e310 ) β/|s|,
(A.2)
(A.3)
The functions aij (s) and bij (s) (i = 1, 2, j = 1, 2, . . . , 6) are given by aij (s) = bij (s) =
g3 (s)γij3 + g2 (s)γij2 + g1 (s)γij + g0 (s) g2 (s)γij2 + g1 (s)γij + g0 (s)
,
{(c130 + c440 )γij + c130 β/|s|}aij (s) + c330 γij2 + c330 (β/|s|)γij − c440 2 e330 γij + e330 (β/|s|)γij − e150
(i = 1, 2, j = 1, 2, . . . , 6).
(A.4)
Appendix B The kernel functions Mkl (t, x) (k, l = 1, 2, 3) are given by ∞ ∞ Z sin s(t − x) ds (l = 1, 3: k = 1, 3, l = 2 : k = 1), (s) − Z kl kl 0 Mkl (t, x) = ∞ ∞ Z cos s(t − x) ds (l = 2: k = 1, 3, l = 1, 3 : k = 1) (s) − Z kl kl
(B.1)
0
where Zkl (s) =
6
j =1
pk+3lj (s)dj,l (s),
∞ Zkl = lim Zkl (s) s→∞
(k, l = 1, 2, 3).
(B.2)
S. Ueda / European Journal of Mechanics A/Solids 25 (2006) 250–259
In Eq. (B.2), the functions pkij (s) (i = 1, 2, j, k = 1, 2, . . . , 6) are p1ij (s) = −i, p2ij (s) = aij (s), p3ij (s) = −ibij (s), p4ij (s) = c130 aij (s) + γij (s) c330 − e330 bij (s) , (i = 1, 2, j = 1, 2, . . . , 6). p5ij (s) = c440 γij (s)aij (s) − 1 + e150 bij (s), p6ij (s) = e310 aij (s) + γij (s) e330 + ε330 bij (s)
259
(B.3)
The functions dm,n (s) (m, n = 1, 2, . . . , 12) are the element of a square matrix D = ∆−1 of order 12. The elements δm,n (s) (m, n = 1, 2, . . . , 12) of the square matrix ∆ are given by δk,j (s) = pk1j (s), (k = 1, 2, . . . , 6), δk,j +6 (s) = −pk2j (s) (j = 1, 2, . . . , 6). (B.4) δk+3,j (s) = pk1j (s) exp sγ1j (s)h1 , (k = 4, 5, 6) δk+6,j +6 (s) = −pk2j (s) exp −sγ2j (s)h2 References Ashida, F., Tauchert, T.R., 1998. Transient response of a piezothermoelastic circular disk under axisymmetric heating. Acta Mech. 128, 1–14. Chen, J., Liu, Z.X., Zou, Z.Z., 2003. Electromechanical impact of a crack in a functionally graded piezoelectric medium. Theoret. Appl. Fracture Mech. 39, 47–60. Erdogan, F., Gupta, G.D., Cook, T.S., 1972. In: Sih, G.C. (Ed.), Methods of Analysis and Solution of Crack Problems. Noordhoff, Leyden. Erdogan, F., Wu, B.H., 1997. The surface crack problem for a plate with functionally graded properties. Trans. ASME J. Appl. Mech. 64, 449–456. Li, C., Weng, G.J., 2002. Antiplane crack problem in functionally graded piezoelectric materials. Trans. ASME J. Appl. Mech. 69, 481–488. Ueda, S., 2003. Crack in a functionally graded piezoelectric strip bonded to elastic surface layers under electromechanical loading. Theoret. Appl. Fracture Mech. 40, 225–236. Ueda, S., 2004. Thermally induced fracture of a functionally graded piezoelectric layer. J. Thermal Stresses 27, 291–309. Wang, B.L., Mai, Y.W., 2004. Impermeable crack and permeable crack assumptions, which one is more realistic?. Trans. ASME J. Appl. Mech. 71, 575–578. Wang, B.L., Noda, N., 2001. Thermally induced fracture of a smart functionally graded composite structure. Theoret. Appl. Fracture Mech. 35, 93–109. Wang, B.L., Zhang, X.H., 2004. A mode III crack in functionally graded piezoelectric material strip. Trans. ASME J. Appl. Mech. 71, 327–333. Wu, C.M., Kahn, M., Moy, W., 1996. Piezoelectric ceramics with functionally gradients: a new application in material design. J. Amer. Ceramics Soc. 79, 809–812.