The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material

Mechanics Research Communications 65 (2015) 17–23

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The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material Yong-Jian Wang a,b , Cun-Fa Gao b,∗ , Haopeng Song b a b

College of Engineering, Nanjing Agricultural University, Nanjing 210031, China State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China

a r t i c l e

i n f o

Article history: Received 27 March 2014 Received in revised form 11 January 2015 Accepted 16 January 2015 Available online 24 January 2015 Keywords: Piezoelectric materials Arbitrary hole Edge cracks Complex variable method

a b s t r a c t In this paper, a general and simple way was found to solve the problem of an arbitrary hole with edge cracks in transversely isotropic piezoelectric materials based on the complex variable method and the method of numerical conformal mapping. Firstly, the approximate mapping function which maps the outside of the arbitrary hole and the cracks into the outside of a circular hole is derived after a series of conformal mapping process. Secondly, based on the assumption that the surface of the cracks and hole is electrically impermeable and traction-free, the approximate expressions for the complex potential, fields intensity factors and energy release rates are presented, respectively. Thirdly, under the in-plane electric loading together with the out-plane mechanical loading, the influences of the hole size, crack length and mechanical/electric loading on the fields intensity factors and energy release rates are analyzed. Finally, some particular holes with edge cracks are studied in numerical analysis. The result shows that, the mechanical loading always promotes crack growth, while the electric loading may retard crack growth. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Structures with holes in piezoelectric materials have been widely used in practical engineering. Under the complicated loading environment, this structure can produce the so-called “stress concentration” phenomenon, which leads to the damage of material. Therefore, it is important to study the cracked hole problem in piezoelectric materials. Many literature can be found for the fracture problems in piezoelectric materials, such as: McMeeking [1], Zhang and Gao [2], Schneider [3], Suo et al. [4], and Li et al. [5]. However, only a little literature focused on the cracked hole problems. Wang and Gao [6] modified the mapping function in Bowie [7] and then studied the mode III fracture problem of edge cracks originating from a circular hole in an infinite piezoelectric solid. Guo et al. [8,9] obtained the exact solutions for anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material under the electrically impermeable and permeable assumption. By using the complex variable method, Lu [10] obtained the analytical expression for the complex potential in a half plane. Hasebe et al. [24] studied the oblique crack at the edge of a half plane, and the analytical solution was presented. However, those literatures are only concerning about a certain shape hole, e.g.: circular hole,

∗ Corresponding author. Tel.: +86 25 84896237. E-mail address: [email protected] (C.-F. Gao). http://dx.doi.org/10.1016/j.mechrescom.2015.01.005 0093-6413/© 2015 Elsevier Ltd. All rights reserved.

elliptical hole and half plane. The solution of an arbitrary cracked hole remains unclear. Gao and Noda [11] studied the anti-plane problem of an infinite piezoelectric material contains an arbitrary hole by using the Faber series and the complex variable method. According to Gao and Noda’s method, the mapping function is the key to solve the cracked hole problem. In order to solve the arbitrary cracked hole problem, the method of numerical conformal mapping is used to obtain the approximate mapping function in this paper. Although, the method of numerical conformal mapping is not a new topic in mathematics, and can be found in many literatures [12–15], but the method used in this paper is more direct and simple. In this paper, some basic equations are given in Section 2, and detailed information of numerical conformal mapping function can be seen in Section 3. The complex potentials, fields intensity factors and energy release rate based on the impermeable and tractionfree boundary conditions are derived in Section 4. In Section 5, two particular holes are studied to investigate the influence of the shape on the fields intensity factors and the energy release rate. Finally, main conclusions are summarized in Section 6. 2. Basic equations In this paper, a transversely isotropic piezoelectric solid with the poling direction along the positive axis x3 and the isotropic plane in the x1 –x2 plane is considered. For the anti-plane problem, all the physical parameters depend on only two coordinate parameters

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Y.-J. Wang et al. / Mechanics Research Communications 65 (2015) 17–23

σ 32∞ S

x2 L2

D1∞

L1

o

x1

σ31∞

D2∞ Fig. 2. Cracks at the edge of a square hole in an infinite piezoelectric solid.

Fig. 1. Cracks at the edge of an arbitrary hole in an infinite piezoelectric solid.

(x1 , x2 ). The mechanical loads perpendicular to the plane, while the electric loads parallel to the plane. Under this condition, one has [2,6,16] 31 = c44 D1 = e15

∂u3 ∂ + e15 , ∂x1 ∂x1

32 = c44

∂u3 ∂ − ε11 , ∂x1 ∂x1

D2 = e15

∂u3 ∂ + e15 , ∂x2 ∂x2

∂u3 ∂ − ε11 . ∂x2 ∂x2

z = ω() = b1  + b0 + b−1  −1 + b−2  −2 + · · ·, (1)

where, the volume force and charge density are ignored;  ij , Di , ui and  stand for the stress, electric displacement, displacement and the potential, respectively; cij , eij and εij are the elastic stiffness tensor, the piezoelectric coupling tensors and the dielectric permittivities, respectively. The general solution of Eq. (1) can be expressed by the generalized stress function ␸ and generalized displacement u, which is [2] T

¯ u = (u3 , ) = Af(z) + Af(z),

(2)

¯ ␸ = Bf(z) + Bf(z),

(3)

(z = x1 + ix2 ),

where, f(z) is an unknown complex vector; A and B stand for the material constant matrices, defined as A = I,



B = iB0 = i



c44

e15

e15

−ε11

(4)

.

Standardized the matrixes A and B, and the new matrixes A and B have the following relation [17]:



BT

AT

¯T B

¯T A



¯ A A ¯ B B



 =

I

0

0

I

 .

(5)

Once the complex potential f(z) is obtained based on the given boundary conditions, all the fields can be determined from (31 , D1 )T = −␸,2 , E1 = −u3,1 ,

(32 , D2 )T = ␸,1 ,

E2 = −u3,2 .

we use a numerical method to obtain the approximate conformal mapping function here. The Laurent series form extended at the infinite point of the mapping function z = ω() can be expressed as

(6)

where, bm =



ω() d  m+1

(7)

(m < 2).

Eq. (7) maps the outside of the cracks and hole into the outside of a circular hole. Let  =  −1 , and a new mapping function which maps the outside of the cracks and hole into the inside of a circular hole is obtained. If the tips of the cracked hole lie on the x1 axis, the two tips correspond to ±1 in the -plane. For an arbitrary cracked hole, the coefficients of Eq. (7) are unknown and not easy to be derived. However, these coefficients can be determined much easier for a hole without cracks. Some particular mapping functions can be found in Savin’s research [19], which can be used as the baseline mapping functions. After a series of conformal mapping process, the mapping function which maps the outside of the arbitrary hole and cracks into the outside of a circular hole are obtained. This solution avoids the complicated process of numerical mapping method from the original problem. Taking a square hole with two cracks as an example, as shown in Fig. 2. The mapping function which maps the outside of the square into the outside of a circular hole has the form as [19]:



z = ω() = R  −



1 −3 1 −7 1 −11 + ··· ,  +  −  6 56 176

(8)

where, R is a constant connected with the square length a, and R = 0.5914a according to the correspondence of the points. Eq. (8) is an approximate mapping function, the square has a radius of curvature at the corner, which is r = 0.014a. The accuracy of the result can be controlled by the items of Eq. (8), and in this paper, the first four items are selected. Fig. 3 shows the brief process of the conformal mapping. Based on Eq. (8), the mapping function which maps the outside of the cracks and square hole into the outside of a circular hole can be expressed as



z = ω() = R ( −1 ) +

where the comma stands for the partial differential, and Ei stands for the electric field.

+

1 −3 ( −1 ) 6

y L2



1 1 −7 −11 , ( −1 ) + ( −1 ) 56 176

3. Mapping function An infinite piezoelectric solid contains an arbitrary cracked hole is shown in Fig. 1. According to the Riemann theorem [18], there must be an analytic function which can maps an arbitrary simply connected domain into a circular hole. However, one cannot obtain the specific conformal mapping function by using this theorem. Therefore,

1 2i

o z - plane

(9)

η

y1

x L1

l2

o z1 -plane

l1 x1

o

ζ - plane

Fig. 3. Brief process of the conformal mapping of a square hole and cracks.

ξ

Y.-J. Wang et al. / Mechanics Research Communications 65 (2015) 17–23

where, () =

ε1 (1 + )2 + ε2 (1 − )2 +

1 4



 2

(ε21 − 1)(1 + )4 + 2(ε1 ε2 + 1)(1 −  2 ) + (ε22 − 1)(1 − )4

It is noted that there is a radical sign in the variable (), an appropriate root must be selected in order to get the right result. The real parameter ε can be derived by εi =

1 + li + (1 + li ) 2

where li respondence

−1

be

determined by the points,

the √

cor-

2 a + L1 = 2 √ −3 −7 −11 1 1 1 R (1 + l1 ) + 6 (1 + l1 ) + 56 (1 + l1 ) + 176 (1 + l1 ) , − 22 a − −3 −7 −11 1 1 L2 = R (−1 − l2 ) + 16 (−1 − l2 ) + 56 (−1 − l2 ) + 176 (−1 − l2 ) .

of







Taking Laurent series of Eq. (9), the first three coefficients are listed here: b−1 = R

ε + ε 1 2 2

−

2 ε1 + ε2



,

b1 = R

ε1 + ε2 , 2

b0 = R(ε1 − ε2 ).

Besides, the mapping function which maps the equilateral triangle hole into the outside of a circular hole was given by Savin [19],



z=R +



1 −2 2 −5 .  +  3 45

(10)

From Eq. (10), the mapping function which maps the outside of an equilateral triangle hole and a crack into the outside of a circular hole can also be derived. Due to the space limitation, the analysis of the equilateral triangle hole and a crack is not presented. 4. Solution 4.1. Complex potentials In this case, the potential vector can be expressed by the potential vector without defect and the disturbance by the defect in the infinite plane, it can be written as f(z) = c∞ z + f0 (z),

where, c∞ z stands for the potential vector without defect, and c∞ is a complex constant to be found from remote loading conditions. f0 (z) is an unknown complex vector which stands for the potential vector with defect, and moreover f0 (∞) = 0. Firstly, from Eqs. (2) and (3), one has ¯ u,1 = AF(z) + AF(z),

(12)

¯ ␸,1 = BF(z) + BF(z),

(13)

where F(z) = df(z)/dz.Inserting Eq. (11) into Eqs. (12) and (13), and then letting z → ∞, leads to ¯ ∞ = u∞ , Ac∞ + Ac ,1

(14)

Bc∞ + Bc∞ = ␸∞ ,1 ,

(15)

where ∞ , −E ∞ )T , u∞ = ( 31 ,1 1

(16)

∞ , D ∞ )T . = (32 ␸∞ ,1 2

From Eqs. (14) and (15), c∞

1 ∞ = (B−1 ␸∞ ,1 + u,1 ). 2

boundary along the surface of the cracks and the hole can be expressed as (18)

in which, the assumption that the boundary of the cracks and hole is electrically impermeable and traction-free has been taken. Substituting Eq. (11) into Eq. (18) results in ¯ 0 (z) = −(Bc∞ z + Bc∞ z). Bf0 (z) + Bf

¯ 0 () = −[Bc∞ ω() + Bc ¯ ∞ ω()], Bf0 () + Bf

Bf0 () = Bc∞ (−ω() + b1  + b0 ) − Bc∞ b1  −1 .



31



=

D1



32 D2





 =

∞ 31

D1∞ ∞ 32



 BF ()  0

− 2Im

 + 2Re

D2∞

ω ()

 BF ()  0 ω ()

,

(22)

.

(23)

Obviously, along the axis x1 , if  → ∞, then BF0 () → 0, Eqs. (22) and (23) tend to the external loads. And at the crack tip  = ±1, ω (±1) = 0 due to the requirement of field singularity at the crack tips, then Eqs. (22) and (23) tend to infinity. 4.2. Fields intensity factors The vector of fields intensity factors can be defined as T

k = (k , kD ) = lim



z→z1

2(z − z1 )␸,1 ,

(24)

where z1 = R + L. Inserting Eq. (13) into Eq. (24), results in



z→z1

It can be seen that c∞ is depended on the external loads and the generalized strain at infinity, where the electricity and the mechanics are auto decoupled. Secondly, the boundary conditions are used to solve the unknown complex potential function f0 (z). The mechanical-electric

(21)

Eq. (21) is the expression of the complex potential. It can be seen, by using the Laurent series, the complex potential only has the connection with the first two coefficients of the Laurent series. There is no need to solve all the coefficients. Finally, once the complex potential was obtained, all the fields can be determined by the following equations:

T

(17)

(20)

where  is the point on the unit circle, and f0 () = f0 (ω()) is defined. According to Eq. (11), f0 () is the boundary value of a holomorphic function outside the unit circle, ω() is the boundary value of a holomorphic function outside the unit circle except the infinity point. Inserting Eq. (7) into Eq. (20), and taking Cauchy integrals [20] at two sides of Eq. (20) gives:

k = (k , kD ) = 2 lim can be determined by

(19)

Along the surface of the cracks and hole in the -plane, Eq. (19) becomes

(11)

c∞

.

¯ Bf(z) + Bf(z) = 0,

(i = 1, 2).

can

19

2(z − z1 )BF0 (z).

(25)

The condition that BF0 (z) is real on the axis x1 has been considered in Eq. (25). In the -plane, the fields intensity factors can be written as √ Bc∞ + Bc∞ , k = 2 b1  ω (±1) where the L’Hospital rule has been used.

(26)

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Y.-J. Wang et al. / Mechanics Research Communications 65 (2015) 17–23

Taking the circular hole with edge cracks as an example. In Eq. (7), b1 = R(2/(1 − ε))1/n where n stands for the numbers of the cracks, and ε is a parameter depended on the shape. Then Eq. (26) becomes

1.1

√  k = 2R 

0.9

1/n





1 ω (1)

∞ 32

D2∞



,

(27)

0.8

g

2 1−ε

1

which has the same form with the Wang and Gaos’ work [6]. The dimensionless fields intensity factors can be derived from ∞ ∞ T √ Eq. (26) (divided by ae 32 ) as D2 g=



2b1 ae ω (±1)

,

0.7

0.6

0.5 a=1 a=0.1 a=0.01

0.4

(28)

0

where ae is the equivalent crack length, and g is the equivalent fields intensity factors.

0.2

0.4

0.6

0.8

1 L/a

1.2

1.4

1.6

1.8

2

Fig. 4. Distribution of the equivalent fields intensity factors when the size of the cracks and square change.

4.3. Energy release rate The energy release rate is equal to the J-integral, and it can be expressed as [16], k kS − kD kE , J= 2

(29)

where k , kS , kD and kE are the singular factors of stress, strain, electric displacement and electric field at the crack tip, respectively. Using Eq. (1), the relationship of singular factors can be expressed as k = c44 kS − e15 kE ,

(30)

kD = e15 kS + ε11 kE .

Thus, once k and kD are obtained from Eq. (26), kS and kE can be derived from Eq. (30), e15 kD + ε11 k e15 2 + ε11 c44

kE =

c44 kD − e15 k e15 2 + ε11 c44

(31)

Substituting Eqs. (26) and (31) into Eq. (29) leads to J=

ae ∞2 ∞ + 2e15 D2∞ 32 − c44 D2∞2 ]g 2 [ε11 32 2(e15 2 + ε11 c44 )

(32)

5. Numerical examples Take PZT-5H as the model material with the following material constants: c44 = 3.53 × 1010

N , m2

e15 = 17.0

ε11 = 151 × 10−10

C , Vm

Jcr = 5.0

C , m2

N m

(33)

where Jcr stands for the critical energy release rate. N is the force in Newtons, C is the charge in coulombs, V is the electric potential in volts, and m is the length in meters. Some different shapes with edge cracks are given to investigate the influence of the shape on the fields intensity factors and the energy release rate. 5.1. Problem of a square hole with edge cracks

1.1 L1/a=0.5

1.08

L1/a=1 L1/a=2

1.06 1.04 1.02 g

kS =

Fig. 5 shows the distribution of the equivalent fields intensity factors when the length of the cracks changes, and a = 0.01 m is used in this figure. For a pure crack (the hole size is small enough), no matter how long the crack is, the equivalent fields intensity factors should be a constant 1. However, in Fig. 5, the equivalent fields intensity factors at the right crack tip decreases slowly and tends to a fixed value (very close to the constant 1), as the length of the left crack (L2 ) increases. Which means the hole effect cannot be ignored, especially for the crack tip near the hole. However, the equivalent fields intensity factors is very close to the constant 1, if the crack length is long enough. In addition, the equivalent fields intensity factors at the right crack tip increases, and will trend to 1 (as shown in Fig. 4), as the length of the right crack (L1 ) increases. Besides, due to the numerical conformal mapping function, the curves in Figs. 4 and 5 are not smooth. Let a = 0.01 m, L1 = L2 = a, the distribution of the energy release rate with the external loads are shown in Fig. 6. It can be seen, the energy release rate has a parabolic relationship with the electric loads when the geometry is fixed. And the energy release rate is negative when the mechanical load is zero. That is, pure electric field retards crack growth. Let L1 = L2 = L, the distributions of energy release rate with the crack length under a given electric or mechanical load are shown in Figs. 7 and 8. It can be found that, the energy release rate always increases as the crack growth, and the mechanical load always

1 0.98 0.96

Let a = 0.01 m, 0.1 m, 1 m, L1 = L2 = L, the distribution of the equivalent fields intensity factors with the length of cracks is shown in Fig. 4. It can be seen, as the increases of crack length, the equivalent fields intensity factors tends to a constant 1. While the size of the square has only a little effect on the equivalent fields intensity factors under the same value of L/˛.

0.94 0.92 0

0.2

0.4

0.6

0.8

1 L2/a

1.2

1.4

1.6

1.8

2

Fig. 5. Distribution of the equivalent fields intensity factors when the length of the cracks change.

Y.-J. Wang et al. / Mechanics Research Communications 65 (2015) 17–23

21

σ 32∞ S

x2 L

R

o

x1

D2∞

Fig. 9. A cracked circular hole at the edge of a half plane.

∞ Fig. 6. Influence of the applied electric load D2∞ and mechanical load 32 on the energy release rate.

circular hole in a full plane into the outside of a circular hole was given by Wang and Gao [6]. Based on this function, a new mapping function which maps the outside of the cracks and circular hole in the right plane into the outside of a circular hole can be expressed as following: R z = ω(z1 ) = √ 1−ε [z12 + z1−2 + 1 + ε + (1 + z1−2 )(z14 + 2εz12 + 1) where +1 z1 = f () = − + −1



Fig. 7. Energy release rate at a given electric load D2∞ under combined mechanical ∞ . load 32

ε=2

2

(1 + ) − 1



2

+1 −1

− 1,

2

(1 + ) + 1

2

1/2 1/2

]

,

(34)

+ 1,

 = L/R.

where, u1 = 0 at the edge of the half plane is assumed, and the brief mapping process can be seen in Fig. 10. Taking Laurent series of Eq. (34), one has ω() =

+∞ 

cm  m ,

(35)

m=−∞

and the negative power items  can be expressed as b−1 (f() − a0 ), √ where a0 = −1 + 2, b−1 = 2/1 − εR. In this case, the complex potentials and the equivalent fields intensity factors can be expressed as: −1 Bf0 () = ␸∞ − a0 −1 ), ,1 b−1 (f ()



k3 = R

∞ combined electric Fig. 8. Energy release rate at a given under mechanical load 32 load D2∞ .

2 1−ε

√





∞ 32

ω (−1)

D2∞



(36)



(37)

,

η

promotes crack growth. Although, a small gap exists in Figs. 7 and 8 because of the approximate conformal mapping function, it can be seen from the trend of the curves, that the energy release rate is zero if the crack does not exist. 5.2. Problem of a cracked circular hole at the edge of a half plane Fig. 9 shows a cracked circular hole at the edge of a half plane. The mapping function which maps the outside of the cracks and

ξ

ζ Fig. 10. Brief process of the conformal mapping of a circular hole with a crack in a half plane.

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Y.-J. Wang et al. / Mechanics Research Communications 65 (2015) 17–23

potentials are obtained under the assumption that the boundary is electrically impermeable and traction-free by Cauchy integral, together with the fields intensity factors and energy release rate. In the section of numerical examples, the cracked square, and the cracked circular hole at the edge of a half plane are studied. Some useful conclusions can be summarized as follows: (1) under the assumption that the boundary is electrically impermeable and traction-free, a singularity of r−1/2 is obtained; (2) the solutions for the half plan are the same as the solutions for the full plane, under the same conditions; (3) the dimensionless fields intensity factors tend to a constant 1 as the crack length growth; (4) energy release rate increases as the crack length growth; (5) mechanical loads always promote crack growth, while the applied electric loading can either promotes or retard crack growth, depending on the magnitude and the direction of electric loading.

1 0.9 0.8

stress intensity factor

0.7 0.6 0.5 0.4 0.3 0.2 present work Guo and Lus' work

0.1 0

0

0.5

1

1.5

2

2.5 L/R

3

3.5

4

4.5

5

Fig. 11. Comparison of the stress intensity factor between Eqs. (39) and (40).

Acknowledgements where,



ω (−1) = 

1   R = ω (1) z1 4 4



4+

√ 2(3 + ε)(1 + ε)−1/2

(1 − ε)(3 + ε + 2(2 + 2ε)1/2 )

.

(38)

It can be seen from Eq. (38), a constant 1/4 can be found compared with Wang and Gaos’ [6] work. And inserting Eq. (38) into Eq. (37), the result has the same form with Eq. (37) in Wang and Gaos’ [6] work. The equivalent fields intensity factors can be written



g=R

2 1−ε



1 ω (−1)

1 √ , ae

(39)

where ae = R + L/2. When R → 0, and ε = 1, Eq. (39) becomes g = 1. This is the classical solution of the fracture problem. From Eqs. (32) and (39), a conclusion can be found that the fields intensity factors and the energy release rate in the half plane are identical to the full plane. Some figures about the distribution of the equivalent fields intensity factors and the energy release rate can be found in Wang and Gaos’ [6] work. Guo and Lu [21] derived the result of the multiple cracks originating from a circular hole, and found that their result is equal to Eq. (40) in Wang and Gaos’ [6] work. In addition, the stress intensity factor agrees well with that of isotropic materials [22]. Besides, Guo and Lu [23] give the dimensionless fields intensity factors of a half plane with an edge crack originating from a half circular hole in a 1D hexagonal quasicrystals, i.e.:



K=

1−

 R 4 L

(40)

In Eq. (40), the dimensionless fields intensity factors K is unacceptable if L ≤ R. The comparison of Eqs. (39) and (40) can be seen in Fig. 11, in which only the mechanical loading is considered. It can be seen that when L > 2R, the stress intensity factors are almost the same. 6. Conclusion To the arbitrary cracked hole problem in the transversely isotropic piezoelectric materials under the out-plane shear loads and in-plan electrical loads, the complex variable method and the method of numerical conformal mapping are used in this paper. The mapping function which maps the outside of the cracks and arbitrary hole into the outside of a circular hole is given in the isotropic plane. Based on this mapping function, the complex

We thank the support from the National Natural Science Foundation of China (nos. 11232007 and 11202099), and Wang thanks the support from the Science and Technology Innovation Fund for the Youth of Nanjing Agricultural University (no. KJ2013040) and the Scientific Research Foundation, Nanjing Agricultural University (no. RCQD13-06).

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