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Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids
Accepted Manuscript Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids Shiwang Fan, Junhong Guo, Jing Yu PII: DOI: Reference:
S1000-9361(16)30160-1 http://dx.doi.org/10.1016/j.cja.2016.08.018 CJA 689
To appear in:
Chinese Journal of Aeronautics
Received Date: Revised Date: Accepted Date:
15 January 2016 3 August 2016 3 August 2016
Please cite this article as: S. Fan, J. Guo, J. Yu, Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids, Chinese Journal of Aeronautics (2016), doi: http://dx.doi.org/10.1016/j.cja.2016.08.018
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Chinese Journal of Aeronautics 28 (2015) xx-xx
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Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids Fan Shiwanga, Guo Junhonga,*, Yu Jinga,b a School of Science, Inner Mongolia University of Technology, Hohhot 010051, China College of General Education, Inner Mongolia Normal University, Hohhot 010022, China
b
Received 15 January 2016; revised 9 March 2016; accepted 3 May 2016
Abstract
By constructing a new numerical conformal mapping and using the Stroh-type formulism, an anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids is investigated. The explicit expressions of the complex potential function, field intensity factors, energy release rates and mechanical strain energy release rate near the crack tip are obtained under the assumptions that the surfaces of the cracks and hole are electrically permeable and electrically impermeable. Numerical examples were presented to show the influences of the geometrical parameters of defects and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under two electrical boundary conditions. Keywords: Piezoelectric material; Hole; Cracks; Conformal mapping; Stroh-type formulism 1. Introduction Due to the excellent coupling between mechanical and electrical fields, the piezoelectric devices have been used in aviation and aerospace industry, such as structural health monitoring, precision positioning and vibration control. The actuators and sensor in smart structures, which scale down favorably in terms of power output and efficiency, have yielded the novel compact piezoelectric hydraulic pumps in the aerospace industry 1 . However, the brittleness of piezoelectric materials inevitably leads to many kinds of defects (e.g., cracks and holes) during the processing, manufacturing and in-service periods. Therefore, it is of great significance to understand the fracture behavior of the complicated defects in piezoelectric materials, especially for cracks emanating from holes. In past decades, many crack problems was considered by researchers. For example, Gao and Yu 2 addressed the generalized two-dimensional plane problems of a semi-infinite crack in a piezoelectric medium subjected to a line force and a line charge based on the Stroh formalism, and obtained the explicit expressions of the field intensity factors and the Green’s functions. By comparing the electrically impermeable and permeable boundary assumptions, Wang and Mai 3 pointed out that the electrically impermeable boundary was a reasonable one in engineering applications. Li and Lee 4-5 analyzed the electroelastic behavior of a piezoelectric ceramic strip containing an anti-plane shear crack. By utilizing the integral transform, Li and his coauthors 6-8 considered the anti-plane interface cracks in two bonded dissimilar piezoelectric layers under the electrically permeable and impermeable assumptions. Guo et al. 9 studied the anti-plane problem of a semi-infinite crack in a piezoelectric strip by using the complex variable function method and the technique of the conformal mapping. For the hole problems, Gao and Fan 10 investigated the two-dimensional problems of an elliptical hole in a piezoelectric material based on the complex potential approach and obtained the explicit solutions in closed form under remotely uniform mechanical and electrical loads. Dai et al. 11 performed the stress concentration around an elliptic hole in transversely isotropic piezoelectric solids subjected *
Corresponding author..+86-471-6576143 E-mail address: [email protected]
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to the uniform mechanical and electrical loads at infinity. The results showed that the electromechanical coupling effect was helpful to reduce the stress concentration. For the cracks emanating from holes, Wang and Gao 12 analyzed the fracture problem of one and two cracks originating from a circular hole in an infinite piezoelectric solid by using the complex variable method. By constructing a new conformal mapping and using the Stroh-type formulism, Guo et al. 13 considered the anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material under the electrically impermeable boundary condition. Also, they analyzed the anti-plane fracture behavior of the two non-symmetrical collinear cracks emanating from an elliptical hole 14 and the multiple cracks emanating from a circular hole15 in a piezoelectric solid under different electrical boundary conditions. Based on the extended Stroh formalism and the boundary element method, Liang and Sun 16 successfully identified the hole/crack size, location, and orientation in finite circular piezoelectric plates by using the strain/electrical field data, stress/electrical displacement data, or displacement/electrical charge data under the static loadings. The key for using the complex variable method to consider the problems of complicated defects such as cracks emanating from holes is to find a suitable conformal mapping. As mentioned above, the previous literature is limited to the relatively simple hole shapes, such as circular or elliptical holes. Thus, the exact solutions of these defects can be derived by the conformal mapping. However, it is very difficult to find an exact conformal mapping for more complicated defects such as cracks emanating from square or triangle holes due to the intricacy of structure and difficulty of mathematics. Considering this, we can effectively solve the complicated defects by constructing a new numerical conformal mapping. Recently, Wang et al. 17 studied an anti-plane problem of piezoelectric solids containing a regular triangle hole with smooth vertices which emanates an edge crack by constructing a new numerical conformal mapping. Wang et al. 18 considered the anti-plane problems of two cracks emanating from a rhombus hole and a cracked half circular hole at the edge of a half plane in a piezoelectric solid by deriving an approximate mapping function and using the Laurent series. For the classical elasticity, Yan et al. 19 and Miao et al. 20 presented the interaction of two collinear cracks emanating from a square hole in a rectangular plate and an infinite plate by using a hybrid displacement discontinuity method and a generalized Bueckner’s principle. The results show that the geometric parameters of defects have a great influence on the failure of materials. Nevertheless, no investigation on four edge cracks emanating from a square hole has been reported up to now in any material including piezoelectric material. In fact, an increase of the number of cracks shows a different and interesting fracture behavior, which can provide an important reference in engineering practice. Therefore, this paper focuses on the anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids by constructing a new numerical conformal mapping, and the explicit solutions of the electroelastic fields are derived finally. 2. Basic equations For the transversely isotropic piezoelectric solids, the poling direction is along the positive x3 -direction and the isotropic plane is in the x1ox2 -plane. The anti-plane deformation is determined by the out-of-plane displacement and the in-plane electrical potential, which are the functions of x1 and x2 . The generalized Hooke’s law for a two-dimensional anti-plane problem of piezoelectric solids is σ 3i = c44γ 3i − e15 Ei (1) Di = e15γ 3i + ε11 Ei Under the assumption of small deformation, the gradient equation can be expressed as γ 3i = u3,i (2) Ei = −φ,i The elastic and electrostatic equilibrium equations in the absence of the body force are σ 3i ,i = 0 (3) Di ,i = 0 In Eqs. (1)-(3), i is from 1 to 2, the repeated indices denote summation, and a comma in the subscripts denotes a partial differentiation; c44 , e15 and ε11 are the elastic constants, piezoelectric constants and dielectric permittivity, respectively; σ 3i , γ 3i , u3 , Di , Ei and φ are the stress, strain, elastic displacement, electrical displacement, electrical field and electrical potential, respectively. Substituting Eqs. (1) and (2) into Eq. (3) leads to the following governing equation for the two-dimensional an-
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ti-plane problem of piezoelectric solids:
B0 ∇ 2 u = 0 2
2
2 1
2
2 2
where ∇ = ∂ / ∂ x + ∂ / ∂ x
(4) T
is two-dimensional Laplace operator, u = [u3 ,φ ]
is the generalized displace-
ment vector, and the material matrix B0 can be expressed as
e15 c B0 = 44 (5) ε11 e − 15 By introducing a generalized stress function vector φ , the general solution 21 to Eqs. (4) and (1) can be written in the Stroh formalism as u = Af ( z ) + Af ( z ) (6) φ = Bf ( z ) + Bf ( z ) where z = x1 + ix2 , f ( z ) is an analytic function to be determined by the boundary conditions, B = iB0 , and
A = I is a 2 × 2 unit matrix. It is found that once the analytic function vector f ( z ) is determined, the stress, strain, displacement, elastic displacement, electrical field and electrical potential for the anti-plane problem of piezoelectric solids with defects can be obtained from Eqs. (1), (2) and (6). 3. Description of problem and conformal mapping 3.1. Description of problem
Let us consider four edge cracks emanating from a square hole embedded in a transversely isotropic and infinite piezoelectric solid with 6mm crystals. The defect penetrates through the piezoelectric solid along the poling direction ( x3 -axis). a is the side length of square hole, L1 and L2 are the lengths of two horizontal cracks, and L3 is the length of vertical cracks (Fig. 1). When the piezoelectric solid containing four cracks emanating from a square hole is subjected to remote anti-plane mechanical and in-plane electrical loadings, i.e., load matrix Σ 2∞ = [σ 32∞ , D2∞ ]T , only two horizontal cracks belong to anti-plane deformation. But the existing of the vertical cracks has a great effect on the stress concentration of the horizontal cracks. Thus, the anti-plane deformation of the horizontal cracks is studied in this work.
Fig. 1.
Four edge cracks emanating from a square hole in piezoelectric solids.
The mechanical/electrical boundary along the surface of the cracks and hole can be expressed as Bf ( z ) + Bf ( z ) = t s ds ∫s (7) T t s = (t3 , − Dn ) where S is the boundary along the surface of defect, t3 and Dn represent the anti-plane traction and electrical displacement of normal direction along the surface of the boundary, respectively.
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3.2. Conformal mapping
To solve the above boundary value problem (i.e., Eqs. (6) and (7)) efficiently, we construct a new numerical conformal mapping which transforms the boundary of complicated defect (Fig. 1) in the z-plane into a unit circle in the ζ -plane. Firstly, Savin 22 proposed a conformal mapping:
1 1 1 −11 z = ω (ζ ) = R ζ − ζ −3 + ζ −7 − ζ + ... (8) 6 56 176 which maps the exterior of the square hole in z -plane onto the exterior of a circular hole in ζ -plane. R is a constant related to the side length of square hole, i.e., R = 0.59136a . It is noted that the square hole is not an exact one, but it is a regular one with smooth vertices, and the radius of curvature r of its corners is r = 0.014a . The deviation becomes smaller if more items are taken in Eq. (8). For the current problem, the former four items have a high accuracy.
Fig. 2.
Schematic diagram of conformal mapping.
Inspired by the conformal mapping Eq. (8), we propose a new mapping (Fig. 2) which maps the exterior region of four cracks emanating from a square hole in z -plane onto the interior region of a unit circle in ζ -plane as follows:
1 1 1 z = ω (ζ ) = R µ (ζ ) − µ (ζ )−3 + µ (ζ ) −7 − µ (ζ )−11 (9) 6 56 176 in which the mapping function µ (ζ ) that maps outer region of four cracks emanating from a circular hole in z1 -plane onto the interior of a unit circle in ζ -plane was proposed firstly by Guo et al. 23 , i.e.,
µ (ζ ) =
2 1 2 2 2 2 2 2 2 2 c1 + c3 (1 + ζ ) + c2 + c3 (1 − ζ ) − 16c3 ζ 4ζ
2 + c12 + c32 (1 + ζ ) 2 + c22 + c32 (1 − ζ ) 2 − 16(c32 + 1)ζ 2
(10)
The real parameters ci (i = 1, 2,3) in Eq. (10) can be determined by
ck = [(1 + lk ) + (1 + lk ) −1 ] / 2 −1 c3 = [(1 + l3 ) − (1 + l3 ) ] / 2
k = 1, 2
(11)
where li are the lengths of cracks in z1 -plane (Fig. 2), which can be determined by the following relationship:
2 1 1 1 (1 + li ) −11 a + Li = R (1 + li ) − (1 + li ) −3 + (1 + li ) −7 − 2 6 56 176
(12)
where ζ = 1 in ζ -plane is the corresponding point of the right crack tip z1 = 2a / 2 + L1 = ω (1) in z -plane. As shown in Fig. 2, Eq. (9) maps the points A ~ H in z -plane onto the corresponding points A1 ~ H1 in
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z1 -plane, respectively, and then Eq. (10) maps the points A1 ~ H1 in z1 -plane onto the corresponding points A±′ ~ H ±′ in ζ -plane, respectively, in which the plus sign underneath indices of the alphabet denotes above or right plane, and the minus sign stands for below or left plane. 4. Explicit solution to problem 4.1. Complex potentials
For the current problem, the complex potential vector can be taken as 24 f ( z ) = c ∞ z + f0 ( z )
(13) ∞
∞
where c stands for a complex constant to be determined by the remote loading conditions, c z is the complex potential vector related to the external loads without defect, f 0 ( z ) is an unknown complex vector which stands for the potential vector disturbed by defect, and f 0 (∞ ) = 0 . Then, we solve the unknown complex potential vector
f 0 ( z ) under the boundary conditions. Differentiating Eq. (6) with respect to x1 and defining F ( z ) = df ( z ) dz , one has
u,1 = AF ( z ) + AF ( z ) (14) φ,1 = BF ( z ) + BF ( z ) Inserting Eq. (13) into Eq. (14) and then taking z → ∞ result in u ∞ = [γ ∞ , − E ∞ ]T = Ac ∞ + Ac ∞ 31 1 ,1 (15) ∞ φ,1 = [σ 32∞ , D2∞ ]T = Bc ∞ + Bc ∞ Thus, the constant vector c ∞ can be obtained from Eq. (15) as Ac ∞ = {[γ 31∞ , − E1∞ ]T + i[γ 32∞ , − E2∞ ]T } / 2 (16) ∞ ∞ ∞ T ∞ ∞ T Bc = {[σ 32 , D2 ] + i[σ 31 , D1 ] } / 2 If the electrically impermeable boundary condition is adopted and the surfaces of the crack and hole are free of mechanical loads, Eq. (7) can be written as Bf ( z ) + Bf ( z ) = 0 (17) Substituting Eq. (13) into Eq. (17), we have Bf 0 ( z ) + Bf 0 ( z ) = −( Bc∞ z + Bc ∞ z ) (18) Combined with conformal mapping z = ω (ζ ) , in the ζ -plane, Eq. (18) can be transformed into (19) Bf 0 (σ ) + Bf 0 (σ ) = − Bc ∞ω (σ ) + Bc ∞ω (σ ) where σ = eiθ is the point on the unit circle, and f 0 (σ ) = f 0 (ω (σ )) is defined. Inserting σ = σ −1 into Eq. (10), we have µ (σ ) = µ (σ ) . It can be found from Eq. (9) that ω (σ ) = ω (σ ) . Then, substituting the second formula of Eq. (15) into Eq. (19) results in (20) Bf 0 (σ ) + Bf 0 (σ ) = −φ,1∞ω (σ ) Multiplying
1 dσ , in which ζ < 1 at the two sides of Eq. (20), we have 2πi ∫τ σ − ζ
Bf 0 (σ ) Bf 0 (σ ) 1 1 1 ω (σ ) dσ + dσ = −φ,1∞ dσ (21) ∫ ∫ ∫ τ τ τ 2πi σ − ζ 2πi σ − ζ 2π i σ − ζ where f 0 (ζ ) is an analytic function inside the unit circle and ω (ζ ) is analytic inside the unit circle except for the point ζ = 0 . According to the Cauchy integral formula for an arbitrary point within ζ < 1 , Eq. (21) becomes ω (σ ) 1 dσ Bf 0 (ζ ) = −φ,1∞ ζ <1 (22) ∫ τ 2πi σ − ζ Using the residue theorem in complex variable function, we derive the right integral of Eq. (22) as
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ω (σ ) 1 R dσ = ω (ζ ) − ( c12 + c32 + c22 + c32 ) 2πi ∫τ σ − ζ 2ζ Substituting Eq. (23) into Eq. (22), we have R ( c12 + c32 + c22 + c32 ) Bf 0 (ζ ) = −φ,1∞ ω (ζ ) − 2ζ Differentiating Eq. (24) with respect to
ζ and defining
(23)
(24)
F0 (ζ ) = df 0 (ζ ) dζ , we find
R BF0 (ζ ) = −φ,1∞ ω ′(ζ ) + 2 c12 + c32 + c22 + c32 2ζ ′ in which R = 0.59136a and ω (ζ ) can be obtained from Eq. (9) as
)
(
(25)
1 1 1 µ (ζ ) −12 2 8 16 Also, µ ′(ζ ) in Eq. (26) can be determined from Eq. (10), that is,
ω ′(ζ ) = Rµ ′(ζ ) 1 + µ (ζ ) −4 − µ (ζ )−8 +
N (ζ 2 − 1)
µ ′(ζ ) =
4ζ
2
(
N − 16c32ζ 2 + N − 16(c32 + 1)ζ 2 2 3
2
2 3
)
2
( N − 16c ζ )[ N − 16(c + 1)ζ ]
(26)
(27)
where N = [ c12 + c32 (1 + ζ ) 2 + c22 + c32 (1 − ζ ) 2 ]2 . For the electrically permeable case, the square hole and the cracks are free of traction on their surfaces and are filled with air of a dielectric permittivity. Therefore, the boundary condition 17 on the surfaces of hole and cracks can be expressed as σ 32 = 0 (28) 0 D2 = D2 where D20 is unknown constant to be determined from the condition φ =0 Similar to the treatment of Guo et al.
23
(29)
0 2
, the D
can be obtained as e D20 = D2∞ − 15 σ 32∞ (30) c44 Therefore, we obtain the complex potential vector f ( z ) with defect by inserting Eqs. (16) and (24) into Eq. (13). Furthermore, all the electroelastic quantities of the anti-plane problem for four edge cracks emanating from a square hole in piezoelectric solids can be derived. Due to their long and complex expressions, the electroelastic quantities, such as stress and electrical displacement, are no longer given here. Next, we will give the explicit solution of the field intensity factors and the energy release rate at the crack tip.
4.2. Field intensity factors For the electrically impermeable boundary, the vector of field intensity factors is defined as T
κm = [ kσ , k D ] = lim 2π( z − z1 ) φ,1 z → z1
(31)
where z1 = 2a / 2 + L1 is the right crack tip in z -plane, and kσ and kD are the stress and electrical displacement intensity factors, respectively. From Eq. (27), we have µ ′(1) = 0 as ζ → 1 . Besides, we note that ω ′(1) = 0 . By using the L'Hospital rule, Eq. (31) becomes BF0 (ζ ) (32) κm = 2 π lim ζ →1 ω ′′(ζ ) Inserting Eq. (25) into Eq. (32) and noting Eq. (15), we obtain the expressions of the field intensity factors near the crack tip under the electrically impermeable boundary as follows:
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c12 + c32 + c22 + c32
T
κm = [ kσ , k D ] = π R If Eq. (33) is divided by
ω ′′(1)
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[σ 32∞ , D2∞ ]T
(33)
∞ , D2∞ ]T , the dimensionless field intensity factors can be written as πL ′[σ 32 R (34) K= c12 + c32 + c22 + c32 L′ω ′′(1)
(
)
in which L′ = ( 2a + L1 + L2 ) / 2 is the equivalent crack length. The second derivative of conformal mapping at the right crack tip is 1 1 1 ω ′′(1) = Rµ ′′(1) 1 + µ (1) −4 − µ (1) −8 + µ (1) −12 (35) 8 16 2 where µ (1) = c + c 2 − 1 1 1 (36) c12 + c32 ( c12 + c32 + c22 + c32 )(c1 + c12 − 1) µ ′′(1)= 2c1 c12 − 1 For the electrically permeable boundary, the field intensity factors can be defined similar to Eq. (36) as BF p (ζ ) (37) κp = [kσp , kDp ]T = 2 π lim 0 ζ →1 ω ′′(ζ ) in which kσp and kDp are the stress and electrical displacement intensity factors under the electrically permeable boundary at the crack tip, respectively. Repeating the process of Eqs. (13)-(25), we have R (38) BF0p (ζ ) = − Σ 2∞ ω ′(ζ ) + 2 c12 + c32 + c22 + c32 2ζ
)
(
∞ , D2∞ − D20 ]T . Inserting Eq. (38) into Eq. (37) and noting Eq. (34), we obtain the expressions of where Σ 2∞ = [σ 32 the field intensity factors near the crack tip under the electrically permeable boundary as ∞ , D2∞ − D20 ]T κp = [ kσp , kDp ]T = πL ′K [σ 32 (39)
Substituting Eq. (30) into Eq. (39), we can rewrite the expressions as T
e κp = [ k , k ] = πL′σ K 1, 15 (40) c44 Eq. (43) shows that the stress intensity factor only relates to the geometrical parameters of defects and the mechanical load, but the electrical displacement intensity factor is also dependent on the material constants. The electrical load has no effect on the electrical displacement intensity factors for the electrically permeable boundary, which is different from those for the electrically impermeable one. The similar conclusion can be found in previous work. p σ
p T D
∞ 32
12-13
4.3. Energy release rate and mechanical strain energy release rate For the electrically impermeable crack, the energy release rate is equivalent to the J -integral, i.e., 1 J = ( kσ ks − kD k E ) (41) 2 where ks and k E are the strain and electrical field intensity factors at the crack tip, respectively, which have the following relationship with the stress and electrical displacement intensity factors −e15 c T [ kσ , kD ]T = 44 [k s , k E ] e ε 15 11
(42)
Inserting Eqs. (33) and (42) into Eq. (41), and noting Eq. (34), we can obtain the energy release rate at the right crack tip as
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πL′K 2 [ε11 (σ 32∞ ) 2 + 2e15σ 32∞ D2∞ − c44 ( D2∞ ) 2 ] (43) 2 2(e15 + ε 11c44 ) In order to compare the results under different fracture criterions, we also present the mechanical strain energy release rates: k k πL′K 2 [ε11 (σ 32∞ ) 2 + e15σ 32∞ D2∞ ] G= σ s = (44) 2 2(e152 + ε11c44 ) However, for the electrically permeable case, the energy release rate is equivalent to the mechanical strain energy release rate, i.e., πL′K 2 ∞ 2 J p = Gp = (45) (σ 32 ) 2c44 It can be found from Eq. (33) that the stress and electrical displacement intensity factors decouple each other. However, the energy release rate and mechanical strain energy release rate can show the interplay between the mechanical and electrical fields. J=
5. Numerical examples
To discuss the effects of the geometrical parameters of complicated defect considered in this paper and the combined mechanical/electrical loads on the dimensionless energy release rate and the mechanical strain energy release rate under two electrically boundary conditions, we take PZT-7 as a model material with the following material constants c44 = 2.50 × 1010 N/m 2 , e15 = 13.50C/m 2 , ε11 = 171 × 10−10 C/Vm and J cr = 5.0N/m , where J cr stands for the critical energy release rate. 25
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Fig. 3.
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Variation of dimensionless energy release rate J/Jcr with ratio of crack length to side length of square hole
Fig. 3 shows the effect of the geometrical parameters of complicated defect on the dimensionless energy release ∞ rates of four cracks emanating from a square hole under given mechanical loading σ 32 = 6MPa and electrical loading D2∞ = 2 × 10−3 C/m 2 for electrically impermeable and permeable boundary conditions. It is seen from Fig. 3(a) that the energy release rate always increases with increasing length of right crack for two electrical boundary conditions. However, for the same electrical boundary condition and same lengths of all cracks, the energy release rate also increases with increasing length of side of the square hole. Moreover, the energy release rate under the electrically permeable boundary condition is greater than that under the impermeable one. It can be observed from Fig. 3(b) that, when the lengths of vertical cracks and right horizontal crack are fixed, the energy release rate increases with increasing length of the left crack and size of square hole. It is interesting to note that the vertical cracks have a great effect on the stress concentration of the horizontal cracks (Fig. 3(c)). Three important features can be found: 1) When the length of the left crack is greater than that of the right one, i.e., L2> L1, the energy release rate decreases with increasing length of vertical cracks. It means that the increase of the length of vertical cracks can retard the crack growth. 2) When the length of the left crack is smaller than that of the right one, i.e.,
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L2< L1, the energy release rate increases with increasing length of vertical cracks. It shows that the increase of the length of vertical cracks may promote the crack growth. 3) When the length of the left crack is equal to that of the right one, i.e., L2= L1, an increase of the length of vertical cracks has no effect on the energy release rate, which contributes to the geometrical symmetry of the defect. Furthermore, the energy release rate approaches a constant when L2/a ≥10, which is the case of a triangular crack emanating from a semi-infinite piezoelectric solid.
Fig. 4.
Effect of applied electromechanical loads on dimensionless energy release rate
Fig. 4 shows the effects of the applied electromechanical loads on the dimensionless energy release rates for given geometrical parameters of defect. For the electrically impermeable boundary condition, when the positive electrical load (or no electrical load) is applied on the solid, the energy release rate increases with increasing mechanical load, which indicates that the mechanical loads always promote the crack growth (Fig. 4(a)). When the negative electrical load is applied on it, the energy release rate decreases slowly first and then increases quickly with increasing mechanical load. For the electrically permeable boundary condition, however, the energy release rate only depends on the magnitude of mechanical load. In other words, the energy release rate is almost independent of the electrical load, which is also clear in Fig. 4(b). Fig. 4(b) shows that the energy release rate decreases with the increasing absolute value of negative electrical field, and it increases first and then decreases with the positive electrical field. It indicates that the negative electrical load always retards the crack growth, but the positive electrical load may either promote or retard the crack growth for the electrically impermeable boundary condition, which is greatly dependent on the mechanical load, while the energy release rate is a constant which is independent of the electrical load under the electrically permeable boundary condition.
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Fig. 5.
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Effect of applied electromechanical loads on dimensionless mechanical strain energy release rate
Fig. 5 shows the effects of the applied electromechanical loads on the mechanical strain energy release rates under two electrical boundary conditions, where Gcr = 5.0N/m denotes the critical mechanical strain energy release rate. When the geometrical parameters of defect and the electrical load are fixed, the mechanical strain energy release rate increases with increasing mechanical load. Moreover, the mechanical strain energy release rate under the electrically impermeable boundary condition is greater than that under the electrically permeable one when σ 32∞ ≤ 2.5 MPa . However, the mechanical strain energy release rate of the electrically permeable cracks will be ∞ wholly larger than that of the electrically impermeable ones when σ 32 ≥ 5.5 MPa . Thus, we can control the crack growth by adjusting the applied mechanical loads. We can observe from Fig. 5(b) that the mechanical strain energy release rate increases linearly with increasing electrical load for the electrically impermeable boundary condition. The result shows that the positive electrical load always promotes the crack growth and the negative electrical load retards the crack growth, which is in agreement with the experimental result. 26 Comparing Fig. 4(b) with Fig. 5(b), we can see that the effect of the electrical load on the crack growth is different for the electrically impermeable boundary condition by using the different fracture criterions.
6. Conclusion
In this paper, by constructing a new numerical conformal mapping and using the Stroh-type formalism, the explicit expressions of the complex potential function, the field intensity factors and energy release rates and the mechanical strain energy release rate at the crack tip are derived for the anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids. Numerical examples were presented to show the effects of the geometrical parameters of complicated defect and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under the electrically permeable and impermeable boundary conditions. In summary, some useful conclusions are drawn as follows: (1) The increases of the length of horizontal cracks (left and right cracks) and size of the square hole always
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promote the crack growth under two electrical boundary conditions. (2) The appearance of vertical cracks greatly affects the stress concentration of the horizontal cracks. The increase of the length of vertical cracks can promote or retard the horizontal crack propagation, which is strongly dependent on the lengths of the two horizontal cracks. (3) The mechanical load always promotes the crack growth. The electrical load has no effect on the crack growth under the electrically permeable boundary condition, while the effect of the electrical load on the crack growth depends on the fracture criterion under the electrically impermeable boundary condition. Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 11262012 and 11502123) and the Inner Mongolia Natural Science Foundation (Nos. 2015JQ01 and 2015MS0129) of China. References
1. Valdovinos J. Pediatric mechanical circulatory support applications for frequency-leveraged piezoelectric hydraulic pumps [dissertation]. Los Angeles (LA): University of California; 2014. 2. Gao CF, Yu JH. Two-dimensional analysis of a semi-infinite crack in piezoelectric media. Mech Res Commun 1998; 25(6): 695-700. 3. Wang BL, Mai YW. Impermeable crack and permeable crack assumptions, which one is more realistic?. J Appl Mech 2004;71(4):575-8. 4. Li XF. Electroelastic analysis of an anti-plane shear crack in a piezoelectric ceramic strip. Int J Solids Struct 2002; 39(5): 1097-117. 5. Li XF, Lee KY. Electroelastic behavior of a rectangular piezoelectric ceramic with an antiplane shear crack at arbitrary position. Eur J Mech A/Solids 2004; 23(4): 645-58. 6. Li XF, Tang GJ. Antiplane interface crack between two bonded dissimilar piezoelectric layers. Eur J Mech A/Solids 2003; 22(2): 231-42. 7. Li XF, Tang GJ. Electroelastic analysis of an interface antiplane shear crack in a layered piezoelectric plate. Int J Eng Sci 2003; 41(12): 1405-22. 8. Li XF, Wang BL. Anti-plane shear crack normal to and terminating at the interface of two bonded piezoelectric ceramics. Int J Solids Struct 2007; 44(11-12): 3796-810. 9. Guo JH, Liu P, Lu ZX, Qin TY. Anti-plane analysis of a semi-infinite crack in a piezoelectric strip. Appl Math Mech 2011; 32(1): 75-82. 10. Gao CF, Fan WX. Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack. Int J Solids Struct 1999; 36(17): 2527-40. 11. Dai LC, Guo WL, Wang X. Stress concentration at an elliptic hole in transversely isotropic piezoelectric solids. Int J Solids Struct 2006; 43(6): 1818-31. 12. Wang YJ, Gao CF. The mode-III cracks originating from the edge of a circular hole in a piezoelectric solid. Int J Solids Struct 2008; 45(16): 4590-9. 13. Guo JH, Lu ZX, Han HT, Yang ZY. Exact solutions for anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material. Int J Solids Struct 2009; 46(21): 3799-809. 14. Guo JH, Lu ZX, Han HT, Yang ZY. The behavior of two non-symmetrical permeable cracks emanating from an elliptical hole in a piezoelectric solid. Eur J Mech A-Solid 2010; 29(4): 654-63. 15. Guo JH, Lu ZX, Feng X. The fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials. Acta Mech 2010; 215(1-4): 119-134. 16. Liang YC, Sun YP. Hole/crack identification in circular piezoelectric plates. Procedia Eng 2014; 79: 194-203. 17. Wang WH, Guo JH, Xing YM. Anti-plane analysis of an edge crack emanating from a regular triangle hole with smooth vertices in piezoelectric materials. Acta Mater Compos Sin 2015; 32(2): 601-7. 18. Wang YJ, Gao CF, Song HP. The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material. Mech Res Commun 2015; 65: 17-23. 19. Yan X, Liu B, Yu J. Cracks emanating from a square hole in rectangular plate in tension. Fatigue Fract Eng Mater Struct 2012; 35(3): 238-46. 20. Miao CQ, Wei YT, Yan XQ. Two collinear square-hole cracks in an infinite plate in tension. Theor Appl Fract Mech 2015; 75: 32-8.
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21. Zhang TY, Gao CF. Fracture behaviors of piezoelectric materials. Theor Appl Fract Mech 2004; 41(1-3): 339-79. 22. Savin GN. Stress concentration around hole. 1958. 23.Guo JH, Lu ZX, Lv J. Field intensity factors of a mode-Ⅲ non-symmetrical cracks originating from circular hole in piezoelectric composite material. Acta Mater Compos Sin 2014; 31(1): 241-7. 24. Sosa H. Plane problems in piezoelectric media with defects. Int J Solids Struct 1991; 28(4): 491-505. 25. Pak YE. Crack extension force in a piezoelectric material. J Appl Mech-T ASME 1990; 57(3): 647-53. 26. Park SB, Sun CT. Fracture criteria for piezoelectric ceramics. J Am Ceram Soc 1995; 78(6): 1475-80.
FAN Shiwang is studying as a postgraduate in Inner Mongolia University of Technology. He received the B.S. degree from Inner Mongolia University of Science and Technology in 2013. His main research interest is fracture mechanics of new materials. E-mail: [email protected] GUO Junhong is an associate professor in Inner Mongolia University of Technology. He received the Ph.D. degree from Beihang University in 2011. His main research interest is fracture mechanics of new materials. E-mail: [email protected]; [email protected] YU Jing is a lecturer in Inner Mongolia Normal University. She received the Ph.D. degree from Inner Mongolia University of Technology in 2015. Her main research interest is fracture mechanics of new materials. E-mail: [email protected]
S1000-9361(16)30160-1 http://dx.doi.org/10.1016/j.cja.2016.08.018 CJA 689
To appear in:
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Received Date: Revised Date: Accepted Date:
15 January 2016 3 August 2016 3 August 2016
Please cite this article as: S. Fan, J. Guo, J. Yu, Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids, Chinese Journal of Aeronautics (2016), doi: http://dx.doi.org/10.1016/j.cja.2016.08.018
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Chinese Journal of Aeronautics 28 (2015) xx-xx
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Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids Fan Shiwanga, Guo Junhonga,*, Yu Jinga,b a School of Science, Inner Mongolia University of Technology, Hohhot 010051, China College of General Education, Inner Mongolia Normal University, Hohhot 010022, China
b
Received 15 January 2016; revised 9 March 2016; accepted 3 May 2016
Abstract
By constructing a new numerical conformal mapping and using the Stroh-type formulism, an anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids is investigated. The explicit expressions of the complex potential function, field intensity factors, energy release rates and mechanical strain energy release rate near the crack tip are obtained under the assumptions that the surfaces of the cracks and hole are electrically permeable and electrically impermeable. Numerical examples were presented to show the influences of the geometrical parameters of defects and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under two electrical boundary conditions. Keywords: Piezoelectric material; Hole; Cracks; Conformal mapping; Stroh-type formulism 1. Introduction Due to the excellent coupling between mechanical and electrical fields, the piezoelectric devices have been used in aviation and aerospace industry, such as structural health monitoring, precision positioning and vibration control. The actuators and sensor in smart structures, which scale down favorably in terms of power output and efficiency, have yielded the novel compact piezoelectric hydraulic pumps in the aerospace industry 1 . However, the brittleness of piezoelectric materials inevitably leads to many kinds of defects (e.g., cracks and holes) during the processing, manufacturing and in-service periods. Therefore, it is of great significance to understand the fracture behavior of the complicated defects in piezoelectric materials, especially for cracks emanating from holes. In past decades, many crack problems was considered by researchers. For example, Gao and Yu 2 addressed the generalized two-dimensional plane problems of a semi-infinite crack in a piezoelectric medium subjected to a line force and a line charge based on the Stroh formalism, and obtained the explicit expressions of the field intensity factors and the Green’s functions. By comparing the electrically impermeable and permeable boundary assumptions, Wang and Mai 3 pointed out that the electrically impermeable boundary was a reasonable one in engineering applications. Li and Lee 4-5 analyzed the electroelastic behavior of a piezoelectric ceramic strip containing an anti-plane shear crack. By utilizing the integral transform, Li and his coauthors 6-8 considered the anti-plane interface cracks in two bonded dissimilar piezoelectric layers under the electrically permeable and impermeable assumptions. Guo et al. 9 studied the anti-plane problem of a semi-infinite crack in a piezoelectric strip by using the complex variable function method and the technique of the conformal mapping. For the hole problems, Gao and Fan 10 investigated the two-dimensional problems of an elliptical hole in a piezoelectric material based on the complex potential approach and obtained the explicit solutions in closed form under remotely uniform mechanical and electrical loads. Dai et al. 11 performed the stress concentration around an elliptic hole in transversely isotropic piezoelectric solids subjected *
Corresponding author..+86-471-6576143 E-mail address: [email protected]
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to the uniform mechanical and electrical loads at infinity. The results showed that the electromechanical coupling effect was helpful to reduce the stress concentration. For the cracks emanating from holes, Wang and Gao 12 analyzed the fracture problem of one and two cracks originating from a circular hole in an infinite piezoelectric solid by using the complex variable method. By constructing a new conformal mapping and using the Stroh-type formulism, Guo et al. 13 considered the anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material under the electrically impermeable boundary condition. Also, they analyzed the anti-plane fracture behavior of the two non-symmetrical collinear cracks emanating from an elliptical hole 14 and the multiple cracks emanating from a circular hole15 in a piezoelectric solid under different electrical boundary conditions. Based on the extended Stroh formalism and the boundary element method, Liang and Sun 16 successfully identified the hole/crack size, location, and orientation in finite circular piezoelectric plates by using the strain/electrical field data, stress/electrical displacement data, or displacement/electrical charge data under the static loadings. The key for using the complex variable method to consider the problems of complicated defects such as cracks emanating from holes is to find a suitable conformal mapping. As mentioned above, the previous literature is limited to the relatively simple hole shapes, such as circular or elliptical holes. Thus, the exact solutions of these defects can be derived by the conformal mapping. However, it is very difficult to find an exact conformal mapping for more complicated defects such as cracks emanating from square or triangle holes due to the intricacy of structure and difficulty of mathematics. Considering this, we can effectively solve the complicated defects by constructing a new numerical conformal mapping. Recently, Wang et al. 17 studied an anti-plane problem of piezoelectric solids containing a regular triangle hole with smooth vertices which emanates an edge crack by constructing a new numerical conformal mapping. Wang et al. 18 considered the anti-plane problems of two cracks emanating from a rhombus hole and a cracked half circular hole at the edge of a half plane in a piezoelectric solid by deriving an approximate mapping function and using the Laurent series. For the classical elasticity, Yan et al. 19 and Miao et al. 20 presented the interaction of two collinear cracks emanating from a square hole in a rectangular plate and an infinite plate by using a hybrid displacement discontinuity method and a generalized Bueckner’s principle. The results show that the geometric parameters of defects have a great influence on the failure of materials. Nevertheless, no investigation on four edge cracks emanating from a square hole has been reported up to now in any material including piezoelectric material. In fact, an increase of the number of cracks shows a different and interesting fracture behavior, which can provide an important reference in engineering practice. Therefore, this paper focuses on the anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids by constructing a new numerical conformal mapping, and the explicit solutions of the electroelastic fields are derived finally. 2. Basic equations For the transversely isotropic piezoelectric solids, the poling direction is along the positive x3 -direction and the isotropic plane is in the x1ox2 -plane. The anti-plane deformation is determined by the out-of-plane displacement and the in-plane electrical potential, which are the functions of x1 and x2 . The generalized Hooke’s law for a two-dimensional anti-plane problem of piezoelectric solids is σ 3i = c44γ 3i − e15 Ei (1) Di = e15γ 3i + ε11 Ei Under the assumption of small deformation, the gradient equation can be expressed as γ 3i = u3,i (2) Ei = −φ,i The elastic and electrostatic equilibrium equations in the absence of the body force are σ 3i ,i = 0 (3) Di ,i = 0 In Eqs. (1)-(3), i is from 1 to 2, the repeated indices denote summation, and a comma in the subscripts denotes a partial differentiation; c44 , e15 and ε11 are the elastic constants, piezoelectric constants and dielectric permittivity, respectively; σ 3i , γ 3i , u3 , Di , Ei and φ are the stress, strain, elastic displacement, electrical displacement, electrical field and electrical potential, respectively. Substituting Eqs. (1) and (2) into Eq. (3) leads to the following governing equation for the two-dimensional an-
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ti-plane problem of piezoelectric solids:
B0 ∇ 2 u = 0 2
2
2 1
2
2 2
where ∇ = ∂ / ∂ x + ∂ / ∂ x
(4) T
is two-dimensional Laplace operator, u = [u3 ,φ ]
is the generalized displace-
ment vector, and the material matrix B0 can be expressed as
e15 c B0 = 44 (5) ε11 e − 15 By introducing a generalized stress function vector φ , the general solution 21 to Eqs. (4) and (1) can be written in the Stroh formalism as u = Af ( z ) + Af ( z ) (6) φ = Bf ( z ) + Bf ( z ) where z = x1 + ix2 , f ( z ) is an analytic function to be determined by the boundary conditions, B = iB0 , and
A = I is a 2 × 2 unit matrix. It is found that once the analytic function vector f ( z ) is determined, the stress, strain, displacement, elastic displacement, electrical field and electrical potential for the anti-plane problem of piezoelectric solids with defects can be obtained from Eqs. (1), (2) and (6). 3. Description of problem and conformal mapping 3.1. Description of problem
Let us consider four edge cracks emanating from a square hole embedded in a transversely isotropic and infinite piezoelectric solid with 6mm crystals. The defect penetrates through the piezoelectric solid along the poling direction ( x3 -axis). a is the side length of square hole, L1 and L2 are the lengths of two horizontal cracks, and L3 is the length of vertical cracks (Fig. 1). When the piezoelectric solid containing four cracks emanating from a square hole is subjected to remote anti-plane mechanical and in-plane electrical loadings, i.e., load matrix Σ 2∞ = [σ 32∞ , D2∞ ]T , only two horizontal cracks belong to anti-plane deformation. But the existing of the vertical cracks has a great effect on the stress concentration of the horizontal cracks. Thus, the anti-plane deformation of the horizontal cracks is studied in this work.
Fig. 1.
Four edge cracks emanating from a square hole in piezoelectric solids.
The mechanical/electrical boundary along the surface of the cracks and hole can be expressed as Bf ( z ) + Bf ( z ) = t s ds ∫s (7) T t s = (t3 , − Dn ) where S is the boundary along the surface of defect, t3 and Dn represent the anti-plane traction and electrical displacement of normal direction along the surface of the boundary, respectively.
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3.2. Conformal mapping
To solve the above boundary value problem (i.e., Eqs. (6) and (7)) efficiently, we construct a new numerical conformal mapping which transforms the boundary of complicated defect (Fig. 1) in the z-plane into a unit circle in the ζ -plane. Firstly, Savin 22 proposed a conformal mapping:
1 1 1 −11 z = ω (ζ ) = R ζ − ζ −3 + ζ −7 − ζ + ... (8) 6 56 176 which maps the exterior of the square hole in z -plane onto the exterior of a circular hole in ζ -plane. R is a constant related to the side length of square hole, i.e., R = 0.59136a . It is noted that the square hole is not an exact one, but it is a regular one with smooth vertices, and the radius of curvature r of its corners is r = 0.014a . The deviation becomes smaller if more items are taken in Eq. (8). For the current problem, the former four items have a high accuracy.
Fig. 2.
Schematic diagram of conformal mapping.
Inspired by the conformal mapping Eq. (8), we propose a new mapping (Fig. 2) which maps the exterior region of four cracks emanating from a square hole in z -plane onto the interior region of a unit circle in ζ -plane as follows:
1 1 1 z = ω (ζ ) = R µ (ζ ) − µ (ζ )−3 + µ (ζ ) −7 − µ (ζ )−11 (9) 6 56 176 in which the mapping function µ (ζ ) that maps outer region of four cracks emanating from a circular hole in z1 -plane onto the interior of a unit circle in ζ -plane was proposed firstly by Guo et al. 23 , i.e.,
µ (ζ ) =
2 1 2 2 2 2 2 2 2 2 c1 + c3 (1 + ζ ) + c2 + c3 (1 − ζ ) − 16c3 ζ 4ζ
2 + c12 + c32 (1 + ζ ) 2 + c22 + c32 (1 − ζ ) 2 − 16(c32 + 1)ζ 2
(10)
The real parameters ci (i = 1, 2,3) in Eq. (10) can be determined by
ck = [(1 + lk ) + (1 + lk ) −1 ] / 2 −1 c3 = [(1 + l3 ) − (1 + l3 ) ] / 2
k = 1, 2
(11)
where li are the lengths of cracks in z1 -plane (Fig. 2), which can be determined by the following relationship:
2 1 1 1 (1 + li ) −11 a + Li = R (1 + li ) − (1 + li ) −3 + (1 + li ) −7 − 2 6 56 176
(12)
where ζ = 1 in ζ -plane is the corresponding point of the right crack tip z1 = 2a / 2 + L1 = ω (1) in z -plane. As shown in Fig. 2, Eq. (9) maps the points A ~ H in z -plane onto the corresponding points A1 ~ H1 in
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z1 -plane, respectively, and then Eq. (10) maps the points A1 ~ H1 in z1 -plane onto the corresponding points A±′ ~ H ±′ in ζ -plane, respectively, in which the plus sign underneath indices of the alphabet denotes above or right plane, and the minus sign stands for below or left plane. 4. Explicit solution to problem 4.1. Complex potentials
For the current problem, the complex potential vector can be taken as 24 f ( z ) = c ∞ z + f0 ( z )
(13) ∞
∞
where c stands for a complex constant to be determined by the remote loading conditions, c z is the complex potential vector related to the external loads without defect, f 0 ( z ) is an unknown complex vector which stands for the potential vector disturbed by defect, and f 0 (∞ ) = 0 . Then, we solve the unknown complex potential vector
f 0 ( z ) under the boundary conditions. Differentiating Eq. (6) with respect to x1 and defining F ( z ) = df ( z ) dz , one has
u,1 = AF ( z ) + AF ( z ) (14) φ,1 = BF ( z ) + BF ( z ) Inserting Eq. (13) into Eq. (14) and then taking z → ∞ result in u ∞ = [γ ∞ , − E ∞ ]T = Ac ∞ + Ac ∞ 31 1 ,1 (15) ∞ φ,1 = [σ 32∞ , D2∞ ]T = Bc ∞ + Bc ∞ Thus, the constant vector c ∞ can be obtained from Eq. (15) as Ac ∞ = {[γ 31∞ , − E1∞ ]T + i[γ 32∞ , − E2∞ ]T } / 2 (16) ∞ ∞ ∞ T ∞ ∞ T Bc = {[σ 32 , D2 ] + i[σ 31 , D1 ] } / 2 If the electrically impermeable boundary condition is adopted and the surfaces of the crack and hole are free of mechanical loads, Eq. (7) can be written as Bf ( z ) + Bf ( z ) = 0 (17) Substituting Eq. (13) into Eq. (17), we have Bf 0 ( z ) + Bf 0 ( z ) = −( Bc∞ z + Bc ∞ z ) (18) Combined with conformal mapping z = ω (ζ ) , in the ζ -plane, Eq. (18) can be transformed into (19) Bf 0 (σ ) + Bf 0 (σ ) = − Bc ∞ω (σ ) + Bc ∞ω (σ ) where σ = eiθ is the point on the unit circle, and f 0 (σ ) = f 0 (ω (σ )) is defined. Inserting σ = σ −1 into Eq. (10), we have µ (σ ) = µ (σ ) . It can be found from Eq. (9) that ω (σ ) = ω (σ ) . Then, substituting the second formula of Eq. (15) into Eq. (19) results in (20) Bf 0 (σ ) + Bf 0 (σ ) = −φ,1∞ω (σ ) Multiplying
1 dσ , in which ζ < 1 at the two sides of Eq. (20), we have 2πi ∫τ σ − ζ
Bf 0 (σ ) Bf 0 (σ ) 1 1 1 ω (σ ) dσ + dσ = −φ,1∞ dσ (21) ∫ ∫ ∫ τ τ τ 2πi σ − ζ 2πi σ − ζ 2π i σ − ζ where f 0 (ζ ) is an analytic function inside the unit circle and ω (ζ ) is analytic inside the unit circle except for the point ζ = 0 . According to the Cauchy integral formula for an arbitrary point within ζ < 1 , Eq. (21) becomes ω (σ ) 1 dσ Bf 0 (ζ ) = −φ,1∞ ζ <1 (22) ∫ τ 2πi σ − ζ Using the residue theorem in complex variable function, we derive the right integral of Eq. (22) as
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ω (σ ) 1 R dσ = ω (ζ ) − ( c12 + c32 + c22 + c32 ) 2πi ∫τ σ − ζ 2ζ Substituting Eq. (23) into Eq. (22), we have R ( c12 + c32 + c22 + c32 ) Bf 0 (ζ ) = −φ,1∞ ω (ζ ) − 2ζ Differentiating Eq. (24) with respect to
ζ and defining
(23)
(24)
F0 (ζ ) = df 0 (ζ ) dζ , we find
R BF0 (ζ ) = −φ,1∞ ω ′(ζ ) + 2 c12 + c32 + c22 + c32 2ζ ′ in which R = 0.59136a and ω (ζ ) can be obtained from Eq. (9) as
)
(
(25)
1 1 1 µ (ζ ) −12 2 8 16 Also, µ ′(ζ ) in Eq. (26) can be determined from Eq. (10), that is,
ω ′(ζ ) = Rµ ′(ζ ) 1 + µ (ζ ) −4 − µ (ζ )−8 +
N (ζ 2 − 1)
µ ′(ζ ) =
4ζ
2
(
N − 16c32ζ 2 + N − 16(c32 + 1)ζ 2 2 3
2
2 3
)
2
( N − 16c ζ )[ N − 16(c + 1)ζ ]
(26)
(27)
where N = [ c12 + c32 (1 + ζ ) 2 + c22 + c32 (1 − ζ ) 2 ]2 . For the electrically permeable case, the square hole and the cracks are free of traction on their surfaces and are filled with air of a dielectric permittivity. Therefore, the boundary condition 17 on the surfaces of hole and cracks can be expressed as σ 32 = 0 (28) 0 D2 = D2 where D20 is unknown constant to be determined from the condition φ =0 Similar to the treatment of Guo et al.
23
(29)
0 2
, the D
can be obtained as e D20 = D2∞ − 15 σ 32∞ (30) c44 Therefore, we obtain the complex potential vector f ( z ) with defect by inserting Eqs. (16) and (24) into Eq. (13). Furthermore, all the electroelastic quantities of the anti-plane problem for four edge cracks emanating from a square hole in piezoelectric solids can be derived. Due to their long and complex expressions, the electroelastic quantities, such as stress and electrical displacement, are no longer given here. Next, we will give the explicit solution of the field intensity factors and the energy release rate at the crack tip.
4.2. Field intensity factors For the electrically impermeable boundary, the vector of field intensity factors is defined as T
κm = [ kσ , k D ] = lim 2π( z − z1 ) φ,1 z → z1
(31)
where z1 = 2a / 2 + L1 is the right crack tip in z -plane, and kσ and kD are the stress and electrical displacement intensity factors, respectively. From Eq. (27), we have µ ′(1) = 0 as ζ → 1 . Besides, we note that ω ′(1) = 0 . By using the L'Hospital rule, Eq. (31) becomes BF0 (ζ ) (32) κm = 2 π lim ζ →1 ω ′′(ζ ) Inserting Eq. (25) into Eq. (32) and noting Eq. (15), we obtain the expressions of the field intensity factors near the crack tip under the electrically impermeable boundary as follows:
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c12 + c32 + c22 + c32
T
κm = [ kσ , k D ] = π R If Eq. (33) is divided by
ω ′′(1)
·7·
[σ 32∞ , D2∞ ]T
(33)
∞ , D2∞ ]T , the dimensionless field intensity factors can be written as πL ′[σ 32 R (34) K= c12 + c32 + c22 + c32 L′ω ′′(1)
(
)
in which L′ = ( 2a + L1 + L2 ) / 2 is the equivalent crack length. The second derivative of conformal mapping at the right crack tip is 1 1 1 ω ′′(1) = Rµ ′′(1) 1 + µ (1) −4 − µ (1) −8 + µ (1) −12 (35) 8 16 2 where µ (1) = c + c 2 − 1 1 1 (36) c12 + c32 ( c12 + c32 + c22 + c32 )(c1 + c12 − 1) µ ′′(1)= 2c1 c12 − 1 For the electrically permeable boundary, the field intensity factors can be defined similar to Eq. (36) as BF p (ζ ) (37) κp = [kσp , kDp ]T = 2 π lim 0 ζ →1 ω ′′(ζ ) in which kσp and kDp are the stress and electrical displacement intensity factors under the electrically permeable boundary at the crack tip, respectively. Repeating the process of Eqs. (13)-(25), we have R (38) BF0p (ζ ) = − Σ 2∞ ω ′(ζ ) + 2 c12 + c32 + c22 + c32 2ζ
)
(
∞ , D2∞ − D20 ]T . Inserting Eq. (38) into Eq. (37) and noting Eq. (34), we obtain the expressions of where Σ 2∞ = [σ 32 the field intensity factors near the crack tip under the electrically permeable boundary as ∞ , D2∞ − D20 ]T κp = [ kσp , kDp ]T = πL ′K [σ 32 (39)
Substituting Eq. (30) into Eq. (39), we can rewrite the expressions as T
e κp = [ k , k ] = πL′σ K 1, 15 (40) c44 Eq. (43) shows that the stress intensity factor only relates to the geometrical parameters of defects and the mechanical load, but the electrical displacement intensity factor is also dependent on the material constants. The electrical load has no effect on the electrical displacement intensity factors for the electrically permeable boundary, which is different from those for the electrically impermeable one. The similar conclusion can be found in previous work. p σ
p T D
∞ 32
12-13
4.3. Energy release rate and mechanical strain energy release rate For the electrically impermeable crack, the energy release rate is equivalent to the J -integral, i.e., 1 J = ( kσ ks − kD k E ) (41) 2 where ks and k E are the strain and electrical field intensity factors at the crack tip, respectively, which have the following relationship with the stress and electrical displacement intensity factors −e15 c T [ kσ , kD ]T = 44 [k s , k E ] e ε 15 11
(42)
Inserting Eqs. (33) and (42) into Eq. (41), and noting Eq. (34), we can obtain the energy release rate at the right crack tip as
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πL′K 2 [ε11 (σ 32∞ ) 2 + 2e15σ 32∞ D2∞ − c44 ( D2∞ ) 2 ] (43) 2 2(e15 + ε 11c44 ) In order to compare the results under different fracture criterions, we also present the mechanical strain energy release rates: k k πL′K 2 [ε11 (σ 32∞ ) 2 + e15σ 32∞ D2∞ ] G= σ s = (44) 2 2(e152 + ε11c44 ) However, for the electrically permeable case, the energy release rate is equivalent to the mechanical strain energy release rate, i.e., πL′K 2 ∞ 2 J p = Gp = (45) (σ 32 ) 2c44 It can be found from Eq. (33) that the stress and electrical displacement intensity factors decouple each other. However, the energy release rate and mechanical strain energy release rate can show the interplay between the mechanical and electrical fields. J=
5. Numerical examples
To discuss the effects of the geometrical parameters of complicated defect considered in this paper and the combined mechanical/electrical loads on the dimensionless energy release rate and the mechanical strain energy release rate under two electrically boundary conditions, we take PZT-7 as a model material with the following material constants c44 = 2.50 × 1010 N/m 2 , e15 = 13.50C/m 2 , ε11 = 171 × 10−10 C/Vm and J cr = 5.0N/m , where J cr stands for the critical energy release rate. 25
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Fig. 3.
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Variation of dimensionless energy release rate J/Jcr with ratio of crack length to side length of square hole
Fig. 3 shows the effect of the geometrical parameters of complicated defect on the dimensionless energy release ∞ rates of four cracks emanating from a square hole under given mechanical loading σ 32 = 6MPa and electrical loading D2∞ = 2 × 10−3 C/m 2 for electrically impermeable and permeable boundary conditions. It is seen from Fig. 3(a) that the energy release rate always increases with increasing length of right crack for two electrical boundary conditions. However, for the same electrical boundary condition and same lengths of all cracks, the energy release rate also increases with increasing length of side of the square hole. Moreover, the energy release rate under the electrically permeable boundary condition is greater than that under the impermeable one. It can be observed from Fig. 3(b) that, when the lengths of vertical cracks and right horizontal crack are fixed, the energy release rate increases with increasing length of the left crack and size of square hole. It is interesting to note that the vertical cracks have a great effect on the stress concentration of the horizontal cracks (Fig. 3(c)). Three important features can be found: 1) When the length of the left crack is greater than that of the right one, i.e., L2> L1, the energy release rate decreases with increasing length of vertical cracks. It means that the increase of the length of vertical cracks can retard the crack growth. 2) When the length of the left crack is smaller than that of the right one, i.e.,
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L2< L1, the energy release rate increases with increasing length of vertical cracks. It shows that the increase of the length of vertical cracks may promote the crack growth. 3) When the length of the left crack is equal to that of the right one, i.e., L2= L1, an increase of the length of vertical cracks has no effect on the energy release rate, which contributes to the geometrical symmetry of the defect. Furthermore, the energy release rate approaches a constant when L2/a ≥10, which is the case of a triangular crack emanating from a semi-infinite piezoelectric solid.
Fig. 4.
Effect of applied electromechanical loads on dimensionless energy release rate
Fig. 4 shows the effects of the applied electromechanical loads on the dimensionless energy release rates for given geometrical parameters of defect. For the electrically impermeable boundary condition, when the positive electrical load (or no electrical load) is applied on the solid, the energy release rate increases with increasing mechanical load, which indicates that the mechanical loads always promote the crack growth (Fig. 4(a)). When the negative electrical load is applied on it, the energy release rate decreases slowly first and then increases quickly with increasing mechanical load. For the electrically permeable boundary condition, however, the energy release rate only depends on the magnitude of mechanical load. In other words, the energy release rate is almost independent of the electrical load, which is also clear in Fig. 4(b). Fig. 4(b) shows that the energy release rate decreases with the increasing absolute value of negative electrical field, and it increases first and then decreases with the positive electrical field. It indicates that the negative electrical load always retards the crack growth, but the positive electrical load may either promote or retard the crack growth for the electrically impermeable boundary condition, which is greatly dependent on the mechanical load, while the energy release rate is a constant which is independent of the electrical load under the electrically permeable boundary condition.
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Fig. 5.
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Effect of applied electromechanical loads on dimensionless mechanical strain energy release rate
Fig. 5 shows the effects of the applied electromechanical loads on the mechanical strain energy release rates under two electrical boundary conditions, where Gcr = 5.0N/m denotes the critical mechanical strain energy release rate. When the geometrical parameters of defect and the electrical load are fixed, the mechanical strain energy release rate increases with increasing mechanical load. Moreover, the mechanical strain energy release rate under the electrically impermeable boundary condition is greater than that under the electrically permeable one when σ 32∞ ≤ 2.5 MPa . However, the mechanical strain energy release rate of the electrically permeable cracks will be ∞ wholly larger than that of the electrically impermeable ones when σ 32 ≥ 5.5 MPa . Thus, we can control the crack growth by adjusting the applied mechanical loads. We can observe from Fig. 5(b) that the mechanical strain energy release rate increases linearly with increasing electrical load for the electrically impermeable boundary condition. The result shows that the positive electrical load always promotes the crack growth and the negative electrical load retards the crack growth, which is in agreement with the experimental result. 26 Comparing Fig. 4(b) with Fig. 5(b), we can see that the effect of the electrical load on the crack growth is different for the electrically impermeable boundary condition by using the different fracture criterions.
6. Conclusion
In this paper, by constructing a new numerical conformal mapping and using the Stroh-type formalism, the explicit expressions of the complex potential function, the field intensity factors and energy release rates and the mechanical strain energy release rate at the crack tip are derived for the anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids. Numerical examples were presented to show the effects of the geometrical parameters of complicated defect and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under the electrically permeable and impermeable boundary conditions. In summary, some useful conclusions are drawn as follows: (1) The increases of the length of horizontal cracks (left and right cracks) and size of the square hole always
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promote the crack growth under two electrical boundary conditions. (2) The appearance of vertical cracks greatly affects the stress concentration of the horizontal cracks. The increase of the length of vertical cracks can promote or retard the horizontal crack propagation, which is strongly dependent on the lengths of the two horizontal cracks. (3) The mechanical load always promotes the crack growth. The electrical load has no effect on the crack growth under the electrically permeable boundary condition, while the effect of the electrical load on the crack growth depends on the fracture criterion under the electrically impermeable boundary condition. Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 11262012 and 11502123) and the Inner Mongolia Natural Science Foundation (Nos. 2015JQ01 and 2015MS0129) of China. References
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FAN Shiwang is studying as a postgraduate in Inner Mongolia University of Technology. He received the B.S. degree from Inner Mongolia University of Science and Technology in 2013. His main research interest is fracture mechanics of new materials. E-mail: [email protected] GUO Junhong is an associate professor in Inner Mongolia University of Technology. He received the Ph.D. degree from Beihang University in 2011. His main research interest is fracture mechanics of new materials. E-mail: [email protected]; [email protected] YU Jing is a lecturer in Inner Mongolia Normal University. She received the Ph.D. degree from Inner Mongolia University of Technology in 2015. Her main research interest is fracture mechanics of new materials. E-mail: [email protected]