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Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix
Theoretical and Applied Fracture Mechanics xxx (2015) xxx–xxx
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Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix Hui-Feng Yang a,b, Cun-Fa Gao a,⇑ a b
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China College of Ship and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, PR China
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Crack Smart materials Piezoelectric inclusions Electroelastic fields
a b s t r a c t In this paper, a general method is presented for evaluating the interaction between multiple piezoelectric inclusions and a nearby crack in a non-piezoelectric elastic matrix. The elastic matrix is subjected to a uniform far field in-plane tension and all inclusions are subjected to an out-of-plane uniform electric field. The crack in the elastic matrix is treated as a continuous distribution of edge dislocations, and then the solution of a unit edge dislocation interacting with multiple piezoelectric inclusions in an elastic medium is derived as the Green function. The problem is formulated into a set of singular integral equations which are solved by a numerical method, and the stress intensity factors (SIFs) at the crack are obtained in terms of the dislocation density functions evaluated from the singular integral equations. Numerical examples are given for a few typical arrays of piezoelectric inclusions with various material properties and geometric parameters. The results indicate that the applied uniform electric-field plays an important role in the interaction between multiple piezoelectric inclusions and the matrix crack., Moreover, it is found that the influences of ‘softer’ piezoelectric inclusions on the SIFs are quite different from that of ‘harder’ piezoelectric inclusions, and the SIFs at the crack tip are greatly affected by the geometry and array of piezoelectric inclusions. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Fiber-reinforced composites are widely used in engineering fields such as aerospace structures, submersible vehicles and offshore structures due to their relatively high strength and stiffness and low density. However fibers (inclusions) introduced as strengthening material phases destroy the homogeneity of the matrix and lead to local stress concentration around the inclusions. Meanwhile, unavoidable defects such as micro-cracks in the matrix may worsen the performance of the composites. Thus it is necessary to study the interaction between inclusions and matrix cracks for developing high performance composites. The inclusion-crackmatrix interaction problems have received a considerable attention recently. Earlier researchers, such as Tamate [1], Atkinson [2], Erdogan et al. [3], and Hsu and Shivakumar [4] analytically investigated the problem of a circular inclusion interacting with a crack in elastic matrix. Then Erdogan and Gupta [5], Bhargava and Bhargava [6], Isida and Noguchi [7], Wu and Chen [8], Kim ⇑ Corresponding author at: State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China. Tel.: +86 25 8489 6237. E-mail address: [email protected] (C.-F. Gao).
and Sudak [9], Liu and Ru [10] studied more complicated cases with multiple cracks, an elliptical inclusion or a three-phase circular inclusion. Recently, Li et al. [11] studied a screw dislocation interacting with a nanoscale circular inclusion and a model III crack by the complex potential function method. Numerical approaches such as finite element method (FEM) [12–14] and boundary element method (BEM) [15–17] were also developed to deal with more general situations with increasing technical complexity. In addition, the method of distributed dislocation is also an effective tool for a crack contained in a matrix [18–21], and alternatively a crack can also be simulated by body force method [22]. Recently there has been growing interest in ‘smart materials’ due to their intrinsic electromechanical coupling behavior. In general, piezoelectric fibers, based on ferroelectric crystals such as lead zirconate titanate (PZT) and barium titanate (BaTiO3), are widely employed as electromechanical sensors, transducers and actuators [23,24] in smart composites. Because of practical relevance of piezoelectric composites for engineering applications, electro-elastic analysis of such materials has become one of the most popular research topics. For example, Tan and Tong [25] proposed two micro-electromechanics models (a rectangle model and a rectangle-cylinder model) to predict elastic, piezoelectric and dielectric
http://dx.doi.org/10.1016/j.tafmec.2014.12.005 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
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constants of piezoelectric fiber-reinforced composite (PFRC) under single load or multiple loads. Yang and Gao [26] studied electroelastic fields in an infinite matrix with N coated piezoelectric inclusions based on the complex variable method. Dunn and Wienecke [27] studied electro-elastic field inside and around inclusions and inhomogeneities embedded in a transversely isotropic piezoelectric solids using Eshelby’s approach. It is noticed that most existing research works are restricted to the problems of piezoelectric inclusions in the absence of any cracks in the matrix. Practically, however, matrix cracking may also exist in piezoelectric composites during manufacturing process or under applied tensile mechanical stress or electrical loading. Little work has been done on piezoelectric inclusions interacting with a matrix crack, except Xiao and Bai [28] who provided a solution for a single piezoelectric inclusion interacting with a matrix crack. In many practical problems, multiple piezoelectric fibers are used as sensors or actuators in ‘smart materials’. Although the single inclusion model is adequate for sparsely arrayed fibers, it is certainly inadequate densely arrayed fibers. The interaction between multiple fibers and a matrix crack represents a significant research topic of practical relevance. The purpose of this work is to present a solution to the problem of closely arrayed multiple piezoelectric inclusions interacting with a near-by crack in a non-piezoeletric elastic matrix. In Section 2, a procedure is used to decompose the problem into two sub-problems. Section 3 solves the two sub-problems respectively and then provides a solution of the original problem. Some numerical results are given in Section 4 to show the influence of the applied electric field, array of inclusions, geometric parameters and material properties on the SIFs at the matrix crack. Main conclusions are summarized in Section 5. 2. Problem description
σ 22∞
x2 z10
E3∞
E3∞
E3∞
E
E3∞
c
o
E3∞
σ 11∞
x1
E3∞
zn0
z30
Fig. 2. Sub-problem I.
As the matrix is pure elastic material, there is no mechanicalelectric coupling inside the matrix. By employing the superposition principle of elasticity [29], the solution of the present problem can be obtained as the sum of two sub-problems, as shown in Fig. 1. The sub-problem I shown in Fig. 2 is the piezoelectric inclusions embedded in the elastic matrix without the matrix crack. For the sub-problem II shown in Fig. 3, the only external loads are the crack surface tractions which are equal in magnitude and opposite in sign to the stresses obtained in the sub-problem I along the crack faces. The superposition of sub-problem I and sub-problem II thus gives the solution of the original problem. 3. Solution procedure The sub-problem I has been solved in ref. [30]. In the matrix (an infinite plane with N circular holes), the complex potentials can be written in the form of power series N X 1 X
aðrÞ k
r¼1 k¼1
wIm ðzÞ
R z zr0
k ð1Þ
k N X 1 X R ðrÞ ¼ ðB2 þ iC 2 Þz þ bk ; z zr0 r¼1 k¼1 ðrÞ
ðrÞ
where ak and bk are unknown coefficients, zr0 is the centre of the r-th inclusion, R stands for a reference length which may be defined as R = min{R1, R2, R3, . . ., RN}, and Bi, Ci (i = 1, 2) are related to the applied uniform stresses at infinity: 1 1 1 1 B1 ¼ ðr1 11 þ r22 Þ=4; C 1 ¼ 0; B2 ¼ ðr22 r11 Þ=2; C 2 ¼ r12 :
ð2Þ
On the other hand, the complex potentials inside the inclusions lp can be expressed as
/If ðzÞ ¼ wIf ðzÞ ¼
1 X
a^ ðpÞ k
k¼0 1 X
z z k p0
R
k ^ðpÞ z zp0 b k R k¼0
; ð3Þ ðp ¼ 1; 2; . . . ; NÞ
σ 12∞
x2 σ 11∞
z10
z 20
x1
E3∞
σ 12∞
z 20
o
z 20 −c
z30
z10 ∞ 3
/Im ðzÞ ¼ ðB1 þ iC 1 Þz þ
In a rectangular coordinate system xj (j = 1, 2, 3), we consider an infinite isotropic elastic matrix containing N parallel cylindrical piezoelectric inclusions (fibers) aligned along the x3 direction. A through-x3 direction matrix crack of length 2c along the x1-axis locates along the x1-axis and is centered at the origin of the x1–x2 plane, as shown in Fig. 1. It is assumed that the elastic matrix is isotropic, while the piezoelectric inclusions are transversely isotropic and polarized along the symmetry axis x3. The matrix is subjected to far-field mechanical stresses and the inclusions are loaded by a uniform electric field E1 3 in the x3 direction. Additionally, all inclusions are assumed to be perfectly bounded to the matrix. The cross-section of the system is shown in Fig. 1, where the regions occupied by the matrix and the inclusions are denoted by subscripts ‘m’ and ‘f’, respectively, the shear modulus and the Poisson’s ratio of the elastic matrix and the inclusions are lm and lf, and mm and mf, respectively, and Rr (r = 1, 2, 3, . . ., N) represents the radius of the rth inclusion lr.
σ 22∞
x2
−c o
zn0
Fig. 1. The interaction between multiple circular piezoelectric inclusions and a crack in an infinite elastic matrix.
z30
c
x1
zn0 Fig. 3. Sub-problem II.
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
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^ðpÞ are unknown coefficients. Once all unknown ^ kðpÞ and b where a k coefficients are determined [28], the stress fields of sub-problem I can be calculated through
x2 z10
I 11
r ¼ rI22 ¼ rI12 ¼
Re½2/0Im ðzÞ z/00I m ðzÞ Re½2/0Im ðzÞ þ z/00I m ðzÞ þ 00I 0I Im½z/m ðzÞ þ wm ðzÞ:
w0Im ðzÞ; w0Im ðzÞ;
ð4Þ
N X 1 X
vðrÞ k
r¼1 k¼1
lnðz zd Þ wdm ðzÞ ¼ c
czd z zd
z 20
o
In sub-problem II, the crack can be simulated by an array of distributed edge dislocations with unknown densities Bx and By (the subscripts x and y represent the x and y components of the edge dislocations) based on Bueckner’s theorem [31], therefore the solution of an edge dislocation interaction with N piezoelectric inclusion in an elastic medium should be obtained first as the Green function. As shown in Fig. 4, a single edge dislocation with Burger’s vector b = bx + iby locates at the point Zd. Based on Yang and Gao [30], in the matrix, the complex potentials can be given by
/dm ðzÞ ¼ c lnðz zd Þ þ
zd
R z zr0
k
N X 1 X ðrÞ þ kk r¼1 k¼1
R z zr0
Fig. 4. An edge dislocation interacting with N piezoelectric inclusions in elastic media.
ðpÞ
e0 ¼
ð5Þ ;
N X 1 X
R zp0 zr0
n¼1
r¼1 r–p
ðpÞ
k
zn0
z30
e1 ¼
;
x1
n
vðrÞ n ;
nþ1 N X 1 X R ðnÞ vðrÞ n ; zp0 zr0 r¼1 n¼1 r–p
ðpÞ
ek ¼
n N X 1 X R ð1Þk C nk vðrÞ n zp0 zr0 r¼1 n¼1
ðk P 2Þ;
r–p
ðrÞ
ðrÞ
where vk and kk are unknown coefficients, c = lm(by ibx)/ p(1 + jm), jm = 3–4mm in the case of plane strain (assumed henceforth in this paper) and jm = (3 – mm)/(1 + mm) in the case of plane stress. On the other hand, the complex potentials inside the inclusions lp can be expressed as
/df ðzÞ ¼ wdf ðzÞ ¼
1 X
z z k p0 ; R z z k
v^ kðpÞ
k¼0 1 X
^kðpÞ k
k¼0
p0
R
ðpÞ ek
ðpÞ
f0 ¼
f1 ¼
^ kðpÞ
ðpÞ where v and ^ kk are unknown coefficients. The above unknown coefficients should satisfy the continuous conditions of displacements and tractions between the inclusions and the matrix, which give that
k k kþ2 k R zp Rp Rp Rp ðpÞ ðpÞ ðpÞ eðpÞ fk ek þ ðk þ 1Þ þ ðk þ 2Þ þ e kþ1 kþ2 Rp R R R R kþ2 k Rp Rp ^ðpÞ ðpÞ ðpÞ kk ¼ g k ðk þ 2Þ v^ kþ2 R R k k k2 Rp zp R R ðpÞ eðpÞ eðpÞ ek ðk 1Þ ðk 2Þ ðk1Þ ðk2Þ Rp R R Rp k k R ðpÞ Rp Rp ðpÞ f k þ v^ ðpÞ v^ k ¼ g ðpÞ k k Rp R R k k kþ2 R zp Rp Rp ðpÞ eðpÞ eðpÞ jm ek ðk þ 1Þ kþ1 þ ðk þ 2Þ kþ2 Rp R R R " k kþ2 k # Rp ðpÞ lm Rp Rp ^ðpÞ ðpÞ kk ¼ hk fk þ ðk þ 2Þ v^ ðpÞ kþ2 þ R lf R R k k k2 R zp R R eðpÞ eðpÞ jm p eðpÞ k þ ðk 1Þ ðk1Þ ðk 2Þ ðk2Þ Rp R R Rp " # k k R ðpÞ lm R Rp ðpÞ f k jf p v^ ðpÞ v^ k ¼ hðpÞ k k Rp lf R R ð7Þ
ðpÞ
fk ¼
n¼1
n
kðrÞ n ;
nþ1 R ðrÞ ðnÞ kn ; zp0 zr0 n¼1
N X 1 X r¼1 r–p
ðpÞ
ðpÞ
f k ¼ kk
ð1Þk C nk
n¼1
R zp0 zr0
n
kðrÞ n
ðk P 2Þ;
ðk P 1; p ¼ 1; 2; 3 . . . ; NÞ:
3 2 Rp zp0 c Rp Rp czd Rp ðpÞ ; g 1 ¼ c þ c zp0 zd zp0 zd Rp zp0 zd Rp zp0 zd
cRp
ðpÞ
g1 ¼
zp0 zd
cRp ; zp0 zd
kþ2 kþ1 Rp ð1Þk zp0 c Rp ðpÞ ; k P 2; g k ¼ ð1Þk c zp0 zd zp0 zd Rp ðpÞ
gk ¼
ðpÞ
k ð1Þk c Rp ; k P 2: k zp0 zd
h1 ¼ c ðpÞ
h1 ¼
Rp zp0 zd
3
2 zp0 c Rp Rp czd Rp þ ; þc zp0 zd Rp zp0 zd Rp zp0 zd
cRp jm cRp ; zp0 zd zp0 zd
kþ2 kþ1 Rp ð1Þk zp0 c Rp ðpÞ hk ¼ ð1Þk c þ ; k P 2; zp0 zd zp0 zd Rp ðpÞ
hk ¼ where
R zp0 zr0
N X 1 X r¼1 r–p
ðp ¼ 1; 2; . . . ; NÞ
ðk P 1; p ¼ 1; 2; 3 . . . ; NÞ
N X 1 X r¼1 r–p
ðpÞ
ð6Þ
ðpÞ
¼ vk
k ð1Þk cjm Rp ; k P 2: k zp0 zd
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
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Eq. (7) and their conjugated equations constitute a set of 8 N M linear equations concerning 8 N M unknown coefficients vðrÞ ; kðrÞ ; v^ ðpÞ ; ^kðpÞ ; v ðrÞ ; kðrÞ ; v^ ðpÞ ; ^kðpÞ ðr ¼ 1; 2; . . . ; N; p ¼ 1; 2; . . . ; N; k ¼ k
k
k
k
k
k
k
k
1; 2; . . . ; MÞ. After they are determined, all the complex potentials for the matrix and inclusions are known, and then the stress fields produced by the dislocation located at zd can be expressed as
ð8Þ
Substituting bx = 1, by = 0, z = x, zd = x0 into Eq. (8), the shear stress bx rbx 12 ðx; 0; x0 ; 0Þ and the normal stress r22 ðx; 0; x0 ; 0Þ at the point
(x, 0) on the crack faces produced by a unit glide edge dislocation at (x0, 0) can be given as
rbx 22 ðx; 0; x0 ; 0Þ ¼ k11 ðx; x0 Þ; 2lm 1 rbx þ k ðx; x Þ; 12 ðx; 0; x0 ; 0Þ ¼ pð1 þ km Þ x x0 12 0
ð9Þ
rby 12 ðx; 0; x0 ; 0Þ and the normal stress
rby 22 ðx; 0; x0 ; 0Þ at the point (x, 0) on the crack faces produced by a unit climb edge dislocation at (x0, 0), as follows
2lm 1 þ k ðx; x Þ; pð1 þ km Þ x x0 21 0
ð10Þ
rby 12 ðx; 0; x0 ; 0Þ ¼ k22 ðx; x0 Þ: where k21(x, x0), k22(x, x0) represent the regular part of the fundamental solution of the unit climb edge dislocation interacting with the N circular inclusions. Then the stress field of sub-problem II can be obtained through an integral of the fundamental solutions along the crack line, which gives that
rII22 ðx; 0Þ ¼ r
II 12 ðx; 0Þ
¼
Z
c
Zcc
Bx ðx0 Þrbx 22 ðx; 0; x0 ; 0Þdx0 þ Bx ðx0 Þr
c
bx 12 ðx; 0; x0 ; 0Þdx0
Bx ðx0 Þk11 ðx;x0 Þdx0 þ
c
þ
Z
c
Zcc
By ðx0 Þrby 22 ðx; 0; x0 ; 0Þdx0
By ðx0 Þ
c
2lm 1 þ k21 ðx; x0 Þ dx0 pð1 þ km Þ x x0
Let t ¼ x0 =c; s ¼ x=c, one can transform Eq. (14) into two simple Cauchy-type singular integral equations as follows: Z 1 Z 1 2lm 1 cBx ðtÞk11 ðs;tÞdt þ By ðtÞ þ ck21 ðs;tÞ dt ¼ rI22 ðs;0Þ pð1 þ km Þ s t 1 1 Z 1 Z 1 2lm 1 Bx ðtÞ þ ck12 ðs;tÞ dt þ cBy ðtÞk22 ðs;tÞdt ¼ rI12 ðs;0Þ pð1 þ km Þ s t 1 1 ð15Þ On the other hand, the additional condition (12) can be rewritten as: 1
Bx ðtÞdt ¼ 0
1 1
Z
By ðx0 Þr
c
where c 6 x 6 c; Bx(x0) and By(x0) are the dislocation density components at the point (x0, 0).The single-value condition of displacement vector requires that the density functions of the system satisfy the following relation:
By ðtÞdt ¼ 0
As the crack is embedded in a homogeneous isotropic elastic matrix, both crack tips have the square root singularities. According to Erdogan [32], it is apparent that Eq. (15) is two singular integral equations with index +1, and the fundamental solution of the singular equations is:
F x ðtÞ Bx ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 t2 F y ðtÞ By ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 t2
ð17Þ
where Fx(t) and Fy(t) are bounded functions in the interval [1, 1]. According to the Cause-Chebyshev solution [32], Eqs. (15) and (16) are discreted, and one get a group of 2n linear algebra equations with the 2n unknown Fx(tk), Fy(tk) (k = 1, 2, 3, . . ., n), which give that n X cp
n
F x ðt k Þk11 ðsr ; t k Þ þ
n X p k¼1
n
F y ðt k Þ
2lm 1 þ ck11 ðsr ; tk Þ pð1 þ km Þ sr tk
I 22 ðsr Þ
¼ r n X p
n
F x ðt k Þ
X n 2lm 1 cp þ ck12 ðsr ; tk Þ þ F y ðtk Þk22 ðsr ; t k Þ pð1 þ km Þ sr tk n k¼1
¼ rI12 ðsr Þ n X
F x ðtk Þ ¼ 0
k¼1 n X F y ðtk Þ ¼ 0
c
Bx ðx0 Þdx0 ¼ 0
c c
ð16Þ
1
k¼1
Z
c
¼ rI12 ðx;0Þ
k¼1 by 12 ðx; 0; x0 ; 0Þdx0
ð11Þ
Z
Z
¼ rI22 ðx;0Þ Z c Z c 2lm 1 Bx ðx0 Þ þ k12 ðx; x0 Þ dx0 þ By ðx0 Þk22 ðx;x0 Þdx0 pð1 þ km Þ x x0 c c
Z
where k11 ðx; x0 Þ, k12 ðx; x0 Þ represent the regular part of the fundamental solution of the unit glide edge dislocation interacting with the N circular inclusions. Similarly, inserting bx = 0, by = 1, z = x, zd = x0 into Eq. (8), one
rby 22 ðx; 0; x0 ; 0Þ ¼
c
ð14Þ
0d rd11 ¼ Re½2/0dm ðzÞ z/00d m ðzÞ wm ðzÞ; 0d rd22 ¼ Re½2/0dm ðzÞ þ z/00d m ðzÞ þ wm ðzÞ; 0d rd12 ¼ Im½z/00d m ðzÞ þ wm ðzÞ:
may obtain the shear stress
Z
k¼1
ð12Þ
ð18Þ
By ðx0 Þdx0 ¼ 0
c
where
The traction free boundary condition on the crack faces requires that the normal and shear stress components along the crack faces are zero, i.e.
rII22 ðx; 0Þ ¼ rI22 ðx; 0Þ; rII12 ðx; 0Þ ¼ rI12 ðx; 0Þ: Substituting Eqs. (9)–(11) into Eq. (13) results in
ð13Þ
ð2k 1Þp ; k ¼ 1; 2; . . . ; n 2n rp sr ¼ cos ; r ¼ 1; 2; . . . ; n 1: 2n
t k ¼ cos
Once the function Fx(t), Fy(t) are evaluated, following Erdogan [3], the stress intensity factors (SIFs) at the both crack tips can be expressed as:
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
H.-F. Yang, C.-F. Gao / Theoretical and Applied Fracture Mechanics xxx (2015) xxx–xxx
5
Table 1 Elastic coefficients C ij ð1010 N=m2 Þ; piezoelectric coefficients ekl ðC=m2 Þ; dielectric coefficients emn ð1010 C=VmÞ.
PZT5H
C11
C33
C44
C12
C13
e13
e33
e15
e11
e33
12.6
11.7
3.53
5.5
5.3
6.5
23.3
17.0
151
130
pffiffiffiffiffiffi 2lm pc F y ð1Þ; ð1 þ km Þ pffiffiffiffiffiffi 2l pc F y ð1Þ; K I ðp2 Þ ¼ m ð1 þ km Þ pffiffiffiffiffiffi 2lm pc F x ð1Þ; K II ðp1 Þ ¼ ð1 þ km Þ pffiffiffiffiffiffi 2l p c F x ð1Þ; K II ðp2 Þ ¼ m ð1 þ km Þ
K I ðp1 Þ ¼
ð19Þ
where p1 is the right crack tip and p2 is the left crack tip. pffiffiffiffiffiffi They can be normalized by K 0I ¼ r1 22 pc , then
2lm F y ð1Þ; r1 22 ð1 þ km Þ 2lm ¼ 1 F ð1Þ; r22 ð1 þ km Þ y K 0I K II ðp1 Þ 2lm ¼ 1 F ð1Þ; r22 ð1 þ km Þ x K 0I K II ðp2 Þ 2lm ¼ 1 F x ð1Þ: 0 r ð1 þ km Þ KI 22 K I ðp1 Þ K 0I K I ðp2 Þ
Fig. 5. The SIFs for a crack interacting with a single piezoelectric inclusion, where Xiao’s solution is found in the Ref. [28].
¼
ð20Þ
4. Results and discussion Choosing the PZT-5H as a model material of inclusions, the value of material constants for PZT-5H are listed in Table 1 [33]. We assume that Poisson’s ratio of the matrix is mm = 0.3. To validate the present methods, we calculate the variation of K I =K 0I at the right tip of the crack located near a single piezoelectric inclusion with the distance factor d/R, where R is the radius of the piezoelectric inclusion and d is the distance between the respective centers of the crack and the inclusion. The results shown in Fig. 5 can be compared with the solution provided by Xiao and Bai [28]. It is found that the present solution agrees well with Xiao and Bai’s solution. To investigate the influence of multiple inclusions, as shown in Fig. 6, we compared the SIFs at the crack tip of two identical piezoelectric inclusions contained in the matrix with that of the single piezoelectric inclusion. In this case, we change the position of the two piezoelectric inclusions synchronously. It is found that two ‘softer’ piezoelectric inclusions (having lower Young’s modulus than the matrix) elevate the SIFs at the crack tip while two ‘harder’ piezoelectric inclusions reduce the SIFs, and the influence will be getting weaker as the distance between the crack and the inclusions increases. To investigate the influence of the electric loading on the crack, we calculate the variation of K I =K 0I for a crack located at the midway of two identical piezoelectric inclusions under the normalized 1 electric loading e31 E1 3 =r22 . Due to the electro-mechanical coupling effect of the piezoelectric material, the influence of the electric loading is significant, especially for the ‘softer’ inclusions. It is found from Fig. 7 that the influence of the electric loading on the SIFs is linear, and the SIFs decreases as the electric loading increases. This indicates that only when the electric filed increases in the opposite direction, the SIFs increase. The SIFs tend to be infinitely large when E1 3 increase along the negative direction of x3. Hence, the crack is dangerous when the value of E1 3 is negative.
Fig. 6. The SIFs for a crack located at the center of two same piezoelectric inclusions, change the position of the two inclusions synchronously.
Fig. 7. Variation of the SIFs for a crack interacting with two same piezoelectric inclusions with the electric loading.
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
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H.-F. Yang, C.-F. Gao / Theoretical and Applied Fracture Mechanics xxx (2015) xxx–xxx
Fig. 8. The SIFs at the right tip of a crack interacting with two different piezoelectric inclusions.
Fig. 9. The SIFs at the both tip of the crack interacting with two same piezoelectric inclusions, fix the left inclusion and change the position of the right one.
Fig. 10. The SIFs at the right tip of the crack interacting with the regular array I of three piezoelectric inclusions.
Fig. 11. The model II SIFs at the right tip of the crack interacting with the regular triangular array II of three piezoelectric inclusions.
Fig. 12. The model I SIFs at the right tip of the crack interacting with the regular triangular array II of three piezoelectric inclusions.
Compared with a single piezoelectric inclusion, the influence of the electric loading on the SIFs is significant for two ‘softer’ piezoelectric inclusions while the influence can be neglected for two ‘harder’ piezoelectric inclusions. We fix the distance between the crack and the edges of the two inclusions, and then change the radius of the right piezoelectric inclusion to examine the influence of the geometry of inclusions. Fig. 8 shows the variation of K I =K 0I at the right tip of the crack. It is found that for the ‘softer’ inclusions, the SIFs increase as the radius of the right inclusion increases. However, for the ‘harder’ inclusions, the SIFs decrease as the radius of the right inclusion increases. It implies that a bigger piezoelectric inclusion can reduce the SIFs or increase it, depending on the relative stiffness of the inclusions and the matrix. We fix the left inclusion and change the position of the right inclusion. What shown in Fig. 9 is the variation of K I =K 0I at both tips of the crack. For the ‘softer’ inclusions, when the right inclusion is closer to the crack than the left (d2/R < 2), the SIFs at the right tip are larger than that at the left tip; when the right inclusion moves far away from the crack (d2/R > 2), the SIFs at the right tip are smaller than that at the left tip. This indicates that the nearby crack tip is more dangerous than the distant crack tip, and therefore crack
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H.-F. Yang, C.-F. Gao / Theoretical and Applied Fracture Mechanics xxx (2015) xxx–xxx
will propagate toward the nearby inclusion. On the contrary, the SIFs at the nearby crack tip are smaller than that at the distant crack tip for the ‘harder’ piezoelectric inclusions. This means that the ‘harder’ piezoelectric inclusions always resist crack propagation toward the nearby inclusion. Fig. 10 shows the variation of K I =K 0I at the right tip of the crack interacting with the regular triangular array of three identical piezoelectric inclusions with the distance factor d/R, where d is the distance between the center of the crack and the right inclusion. In this case, the regular triangular array of three piezoelectric inclusions is symmetrical about x1. It is found that the position of the crack and the applied electric loading have a significant influence on the SIFs at the crack tip for the ‘softer’ inclusions. It is worth notice that when the crack locates between the two left inclusions, the values of the SIFs are negative under the electric 1 loading e31 E1 3 =r22 ¼ 2; which indicates that the electric loading makes the crack faces closed. One can also find that the influence of the electric loading on the crack is very weak for the ‘harder’ inclusions. We consider the case when the regular triangular array of three piezoelectric inclusions is not symmetrical about x1, in which the model II SIFs will not be equal to zero. Fig. 11 demonstrates that not only the ‘softer’ inclusions but also the ‘harder’ inclusions make the model II SIFs change rapidly and substantially. However, when the crack locates within the inclusions array, only in the case of ‘softer’ inclusions the electric loading plays an important role in the model II SIFs. As shown in Fig. 12, the electric loading makes the model I SIFs decrease considerably in the case of ‘softer’ inclusions, but there is hardly any meaningful change of the model I SIFs in the case of ‘harder’ inclusions.
5. Conclusion The interaction of multiple piezoelectric inclusions embedded in an infinite non-piezoelectric elastic matrix with a matrix crack is investigated. Based on the superposition principle of elasticity, the solution of the original problem can be obtained as the sum of two sub-problems. The solution of sub-problem I can be obtained by a previous work. In sub-problem II, the crack is simulated by a continuous distribution of edge dislocations with unknown density functions. Thus, the solution of an edge dislocation interaction with multiple piezoelectric inclusions embedded in an elastic medium is derived as the Green functions, and then the stress field in the sub-problem II can be obtain through an integral of the Green functions along the crack line. Moreover the superposition of the stress fields of sub-problem I and sub-problem II is requested to satisfy the traction free boundary condition along the crack surface. Thus a set of singular integral equations are formulated which can be solved by a numerical method, and the stress intensity factors at the crack are derived in terms of the dislocation density functions evaluated from the integral equations. Finally, numerical examples are given for a few typical arrays of piezoelectric inclusions with various materials and geometric parameters. The main results can be summarized as follows: (1) The applied electric-field loading plays an important role in the interaction between multiple piezoelectric inclusions and the matrix crack. For example, the electric-field loading applied along the positive x3 direction may cause a crack closed while an electric-field loading applied in the opposite direction can cause crack growth. (2) The influence of ‘softer’ piezoelectric inclusions on the crack is quite different from that of ‘harder’ piezoelectric inclusions. In general, the ‘softer’ inclusions are more sensitive to the changes of the geometric parameters and the electric
7
loading. For example, the electric loading changes the SIFs greatly in the case of ‘softer’ piezoelectric inclusions, but it causes hardly any meaningful change of the SIFs in the case of, ‘harder’ inclusions. (3) The geometric and elastic parameters have a remarkable influence on the SIFs. From the above-mentioned cases, one can find that the relative stiffness of the inclusions and the matrix could have a decisive effect on matrix cracking in piezoelectric fiber-reinforced composites. (4) The SIFs at a matrix crack is greatly affected by the number and array of nearby piezoelectric inclusions. When the problem of multiple piezoelectric inclusions interacting with a matrix crack is concerned, the present method can be used to study whether the matrix crack will grow or not.
Acknowledgements The authors thank the support from the National Natural Science Foundation of China (11232007, 11472130) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
References [1] O. Tamate, The effect of a circular inclusion on the stresses around a line crack in a sheet under tension, Int. J. Fract. Mech. 4 (1968) 257–265. [2] C. Atkinson, The interaction between a crack and an inclusion, Int. J. Eng. Sci. 10 (1972) 127–136. [3] F. Erdogan, G.D. Gupta, M. Ratwani, Interaction between a circular inclusion and an arbitrarily oriented crack, ASME J. Appl. Mech. 41 (1974) 1007–1013. [4] Y.C. Hsu, V. Shivakumar, Interaction between an elastic circular inclusion and two symmetrically placed collinear cracks, Int. J. Fract. Mech. 12 (1976) 619– 630. [5] F. Erdogan, G.D. Gupta, The inclusion problem with a crack crossing the boundary, Int. J. Fract. 11 (1975) 13–27. [6] R.D. Bhargava, R.R. Bhargava, Elastic circular inclusion in an infinite plane containing two cracks, Int. J. Eng. Sci. 11 (1973) 437–449. [7] M. Isida, H. Noguchi, Plane problem of an arbitrary array of cracks in an infinite plate containing a circular inclusion, J. Jpn. Soc. Mech. Eng. 49 (1983) 147–155. [8] C.H. Wu, C.H. Chen, A crack in a confocal elliptic inhomogeneity embedded in an infinite medium, J. Appl. Mech. 57 (1990) 91–96. [9] K. Kim, L.J. Sudak, Interaction between a radial matrix crack and a three-phase circular inclusion with imperfect interface in plane elasticity, Int. J. Fract. 131 (2005) 155–172. [10] Y. Liu, C.Q. Ru, P. Schiavone, A. Mioduchowski, New phenomena concerning the effect of imperfect bonding on radial matrix cracking in fiber composites, Int. J. Eng. Sci. 39 (2001) 2033–2050. [11] J. Li, Q.H. Fang, Y.W. Liu, Crack interaction with a second phase nanoscale circular inclusion in an elastic matrix, Int. J. Eng. Sci. 72 (2013) 89–97. [12] R. Li, A. Chudnovsky, Variation of the energy release rate as a crack approaches and passes through an elastic inclusion, Int. J. Fract. 59 (1993) R69–R74. [13] P. Lipetzky, S. Schmauder, Crack-particle interaction in two-phase composites. Part I: Particle shape effects, Int. J. Fract. 65 (1994) 345–358. [14] P. Lipetzky, Z. Knesl, Crack-particle interaction in two-phase composites. Part II: Crack deflection, Int. J. Fract. 73 (1995) 81–92. [15] M.B. Bush, The interaction between a crack and a particle cluster, Int. J. Fract. 88 (1997) 215–232. [16] C. Wang, W. Libardi, J.B. Baldo, Analysis of crack extension paths and toughening in a two phase brittle particulate composite by the boundary element method, Int. J. Fract. 94 (1998) 177–188. [17] M.G. Knight, L.C. Wrobel, J.L. Henshall, L.A. Lacerda, A study of the interaction between a propagating crack and an uncoated/coated elastic inclusion using BE technique, Int. J. Fract. 114 (2002) 47–61. [18] D.A. Hills, M. Comninou, A normally loaded half plane with an edge crack, Int. J. Solids Struct. 21 (1985) 399–410. [19] J. Helsing, Stress intensity factors for a crack in front of an inclusion, Eng. Fract. Mech. 64 (1999) 245–253. [20] Z.M. Xiao, B.J. Chen, Stress intensity factor for a Griffith crack interacting with a coated inclusion, Int. J. Fract. 108 (2001) 193–205. [21] Y.S. Tao, Q.H. Fang, X. Zeng, Y.W. Liu, Influence of dislocation on interaction between a crack and a circular inhomogeneity, Int. J. Mech. Sci. 80 (2014) 47– 53. [22] D.H. Chen, The effect of an elliptical inclusion on a crack, Int. J. Fract. 85 (1997) 351–364. [23] J. Herbert, Ferroelectric Transducers and Sensors, Gordon and Breach Science Publishers, New York, 1982.
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
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H.-F. Yang, C.-F. Gao / Theoretical and Applied Fracture Mechanics xxx (2015) xxx–xxx
[24] B.Q. Dong, Z.J. Li, Cement-based piezoelectric ceramic smart composites, Compos. Sci. Technol. 65 (2005) 1363–1371. [25] P. Tan, L.Y. Tong, Micro-electromechanics models for piezoelectric- fiberreinforced composite materials, Compos. Sci. Technol. 61 (2001) 759–769. [26] B.H. Yang, C.F. Gao, Anti-plane electro-elastic fields in an infinite matrix with N coated-piezoelectric inclusions, Compos. Sci. Technol. 69 (2009) 2668–2674. [27] M.L. Dunn, H.A. Wienecke, Inclusion and inhomogeneities in transversely isotropic piezoelectric solids, Int. J. Solids Struct. 34 (1997) 3571–3582. [28] Z.M. Xiao, J. Bai, On piezoelectric inhomogeneity related problems-Part II: A circular piezoelectric inhomogeneity interacting with a nearby crack, Int. J. Eng. Sci. 37 (1999) 961–976.
[29] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1934. [30] B.H. Yang, C.F. Gao, Plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix, Int. J. Eng. Sci. 48 (2010) 518–528. [31] D.A. Hills, A.P. Kelly, D.A. Dai, A.M. Korsuncky, Solution of Crack Problems, the Distributed Dislocation Technique, Kluwer, Dordrecht, 1996. [32] F. Erdogan, Mixed boundary-value problems in mechanics, Mech. Today 4 (1990) 1–85. [33] Y.E. Pak, Linear electro-elastic fracture mechanics of piezoelectric materials, Int. J. Fract. 54 (1992) 79–100.
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Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix Hui-Feng Yang a,b, Cun-Fa Gao a,⇑ a b
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China College of Ship and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, PR China
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Crack Smart materials Piezoelectric inclusions Electroelastic fields
a b s t r a c t In this paper, a general method is presented for evaluating the interaction between multiple piezoelectric inclusions and a nearby crack in a non-piezoelectric elastic matrix. The elastic matrix is subjected to a uniform far field in-plane tension and all inclusions are subjected to an out-of-plane uniform electric field. The crack in the elastic matrix is treated as a continuous distribution of edge dislocations, and then the solution of a unit edge dislocation interacting with multiple piezoelectric inclusions in an elastic medium is derived as the Green function. The problem is formulated into a set of singular integral equations which are solved by a numerical method, and the stress intensity factors (SIFs) at the crack are obtained in terms of the dislocation density functions evaluated from the singular integral equations. Numerical examples are given for a few typical arrays of piezoelectric inclusions with various material properties and geometric parameters. The results indicate that the applied uniform electric-field plays an important role in the interaction between multiple piezoelectric inclusions and the matrix crack., Moreover, it is found that the influences of ‘softer’ piezoelectric inclusions on the SIFs are quite different from that of ‘harder’ piezoelectric inclusions, and the SIFs at the crack tip are greatly affected by the geometry and array of piezoelectric inclusions. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Fiber-reinforced composites are widely used in engineering fields such as aerospace structures, submersible vehicles and offshore structures due to their relatively high strength and stiffness and low density. However fibers (inclusions) introduced as strengthening material phases destroy the homogeneity of the matrix and lead to local stress concentration around the inclusions. Meanwhile, unavoidable defects such as micro-cracks in the matrix may worsen the performance of the composites. Thus it is necessary to study the interaction between inclusions and matrix cracks for developing high performance composites. The inclusion-crackmatrix interaction problems have received a considerable attention recently. Earlier researchers, such as Tamate [1], Atkinson [2], Erdogan et al. [3], and Hsu and Shivakumar [4] analytically investigated the problem of a circular inclusion interacting with a crack in elastic matrix. Then Erdogan and Gupta [5], Bhargava and Bhargava [6], Isida and Noguchi [7], Wu and Chen [8], Kim ⇑ Corresponding author at: State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China. Tel.: +86 25 8489 6237. E-mail address: [email protected] (C.-F. Gao).
and Sudak [9], Liu and Ru [10] studied more complicated cases with multiple cracks, an elliptical inclusion or a three-phase circular inclusion. Recently, Li et al. [11] studied a screw dislocation interacting with a nanoscale circular inclusion and a model III crack by the complex potential function method. Numerical approaches such as finite element method (FEM) [12–14] and boundary element method (BEM) [15–17] were also developed to deal with more general situations with increasing technical complexity. In addition, the method of distributed dislocation is also an effective tool for a crack contained in a matrix [18–21], and alternatively a crack can also be simulated by body force method [22]. Recently there has been growing interest in ‘smart materials’ due to their intrinsic electromechanical coupling behavior. In general, piezoelectric fibers, based on ferroelectric crystals such as lead zirconate titanate (PZT) and barium titanate (BaTiO3), are widely employed as electromechanical sensors, transducers and actuators [23,24] in smart composites. Because of practical relevance of piezoelectric composites for engineering applications, electro-elastic analysis of such materials has become one of the most popular research topics. For example, Tan and Tong [25] proposed two micro-electromechanics models (a rectangle model and a rectangle-cylinder model) to predict elastic, piezoelectric and dielectric
http://dx.doi.org/10.1016/j.tafmec.2014.12.005 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
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constants of piezoelectric fiber-reinforced composite (PFRC) under single load or multiple loads. Yang and Gao [26] studied electroelastic fields in an infinite matrix with N coated piezoelectric inclusions based on the complex variable method. Dunn and Wienecke [27] studied electro-elastic field inside and around inclusions and inhomogeneities embedded in a transversely isotropic piezoelectric solids using Eshelby’s approach. It is noticed that most existing research works are restricted to the problems of piezoelectric inclusions in the absence of any cracks in the matrix. Practically, however, matrix cracking may also exist in piezoelectric composites during manufacturing process or under applied tensile mechanical stress or electrical loading. Little work has been done on piezoelectric inclusions interacting with a matrix crack, except Xiao and Bai [28] who provided a solution for a single piezoelectric inclusion interacting with a matrix crack. In many practical problems, multiple piezoelectric fibers are used as sensors or actuators in ‘smart materials’. Although the single inclusion model is adequate for sparsely arrayed fibers, it is certainly inadequate densely arrayed fibers. The interaction between multiple fibers and a matrix crack represents a significant research topic of practical relevance. The purpose of this work is to present a solution to the problem of closely arrayed multiple piezoelectric inclusions interacting with a near-by crack in a non-piezoeletric elastic matrix. In Section 2, a procedure is used to decompose the problem into two sub-problems. Section 3 solves the two sub-problems respectively and then provides a solution of the original problem. Some numerical results are given in Section 4 to show the influence of the applied electric field, array of inclusions, geometric parameters and material properties on the SIFs at the matrix crack. Main conclusions are summarized in Section 5. 2. Problem description
σ 22∞
x2 z10
E3∞
E3∞
E3∞
E
E3∞
c
o
E3∞
σ 11∞
x1
E3∞
zn0
z30
Fig. 2. Sub-problem I.
As the matrix is pure elastic material, there is no mechanicalelectric coupling inside the matrix. By employing the superposition principle of elasticity [29], the solution of the present problem can be obtained as the sum of two sub-problems, as shown in Fig. 1. The sub-problem I shown in Fig. 2 is the piezoelectric inclusions embedded in the elastic matrix without the matrix crack. For the sub-problem II shown in Fig. 3, the only external loads are the crack surface tractions which are equal in magnitude and opposite in sign to the stresses obtained in the sub-problem I along the crack faces. The superposition of sub-problem I and sub-problem II thus gives the solution of the original problem. 3. Solution procedure The sub-problem I has been solved in ref. [30]. In the matrix (an infinite plane with N circular holes), the complex potentials can be written in the form of power series N X 1 X
aðrÞ k
r¼1 k¼1
wIm ðzÞ
R z zr0
k ð1Þ
k N X 1 X R ðrÞ ¼ ðB2 þ iC 2 Þz þ bk ; z zr0 r¼1 k¼1 ðrÞ
ðrÞ
where ak and bk are unknown coefficients, zr0 is the centre of the r-th inclusion, R stands for a reference length which may be defined as R = min{R1, R2, R3, . . ., RN}, and Bi, Ci (i = 1, 2) are related to the applied uniform stresses at infinity: 1 1 1 1 B1 ¼ ðr1 11 þ r22 Þ=4; C 1 ¼ 0; B2 ¼ ðr22 r11 Þ=2; C 2 ¼ r12 :
ð2Þ
On the other hand, the complex potentials inside the inclusions lp can be expressed as
/If ðzÞ ¼ wIf ðzÞ ¼
1 X
a^ ðpÞ k
k¼0 1 X
z z k p0
R
k ^ðpÞ z zp0 b k R k¼0
; ð3Þ ðp ¼ 1; 2; . . . ; NÞ
σ 12∞
x2 σ 11∞
z10
z 20
x1
E3∞
σ 12∞
z 20
o
z 20 −c
z30
z10 ∞ 3
/Im ðzÞ ¼ ðB1 þ iC 1 Þz þ
In a rectangular coordinate system xj (j = 1, 2, 3), we consider an infinite isotropic elastic matrix containing N parallel cylindrical piezoelectric inclusions (fibers) aligned along the x3 direction. A through-x3 direction matrix crack of length 2c along the x1-axis locates along the x1-axis and is centered at the origin of the x1–x2 plane, as shown in Fig. 1. It is assumed that the elastic matrix is isotropic, while the piezoelectric inclusions are transversely isotropic and polarized along the symmetry axis x3. The matrix is subjected to far-field mechanical stresses and the inclusions are loaded by a uniform electric field E1 3 in the x3 direction. Additionally, all inclusions are assumed to be perfectly bounded to the matrix. The cross-section of the system is shown in Fig. 1, where the regions occupied by the matrix and the inclusions are denoted by subscripts ‘m’ and ‘f’, respectively, the shear modulus and the Poisson’s ratio of the elastic matrix and the inclusions are lm and lf, and mm and mf, respectively, and Rr (r = 1, 2, 3, . . ., N) represents the radius of the rth inclusion lr.
σ 22∞
x2
−c o
zn0
Fig. 1. The interaction between multiple circular piezoelectric inclusions and a crack in an infinite elastic matrix.
z30
c
x1
zn0 Fig. 3. Sub-problem II.
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^ðpÞ are unknown coefficients. Once all unknown ^ kðpÞ and b where a k coefficients are determined [28], the stress fields of sub-problem I can be calculated through
x2 z10
I 11
r ¼ rI22 ¼ rI12 ¼
Re½2/0Im ðzÞ z/00I m ðzÞ Re½2/0Im ðzÞ þ z/00I m ðzÞ þ 00I 0I Im½z/m ðzÞ þ wm ðzÞ:
w0Im ðzÞ; w0Im ðzÞ;
ð4Þ
N X 1 X
vðrÞ k
r¼1 k¼1
lnðz zd Þ wdm ðzÞ ¼ c
czd z zd
z 20
o
In sub-problem II, the crack can be simulated by an array of distributed edge dislocations with unknown densities Bx and By (the subscripts x and y represent the x and y components of the edge dislocations) based on Bueckner’s theorem [31], therefore the solution of an edge dislocation interaction with N piezoelectric inclusion in an elastic medium should be obtained first as the Green function. As shown in Fig. 4, a single edge dislocation with Burger’s vector b = bx + iby locates at the point Zd. Based on Yang and Gao [30], in the matrix, the complex potentials can be given by
/dm ðzÞ ¼ c lnðz zd Þ þ
zd
R z zr0
k
N X 1 X ðrÞ þ kk r¼1 k¼1
R z zr0
Fig. 4. An edge dislocation interacting with N piezoelectric inclusions in elastic media.
ðpÞ
e0 ¼
ð5Þ ;
N X 1 X
R zp0 zr0
n¼1
r¼1 r–p
ðpÞ
k
zn0
z30
e1 ¼
;
x1
n
vðrÞ n ;
nþ1 N X 1 X R ðnÞ vðrÞ n ; zp0 zr0 r¼1 n¼1 r–p
ðpÞ
ek ¼
n N X 1 X R ð1Þk C nk vðrÞ n zp0 zr0 r¼1 n¼1
ðk P 2Þ;
r–p
ðrÞ
ðrÞ
where vk and kk are unknown coefficients, c = lm(by ibx)/ p(1 + jm), jm = 3–4mm in the case of plane strain (assumed henceforth in this paper) and jm = (3 – mm)/(1 + mm) in the case of plane stress. On the other hand, the complex potentials inside the inclusions lp can be expressed as
/df ðzÞ ¼ wdf ðzÞ ¼
1 X
z z k p0 ; R z z k
v^ kðpÞ
k¼0 1 X
^kðpÞ k
k¼0
p0
R
ðpÞ ek
ðpÞ
f0 ¼
f1 ¼
^ kðpÞ
ðpÞ where v and ^ kk are unknown coefficients. The above unknown coefficients should satisfy the continuous conditions of displacements and tractions between the inclusions and the matrix, which give that
k k kþ2 k R zp Rp Rp Rp ðpÞ ðpÞ ðpÞ eðpÞ fk ek þ ðk þ 1Þ þ ðk þ 2Þ þ e kþ1 kþ2 Rp R R R R kþ2 k Rp Rp ^ðpÞ ðpÞ ðpÞ kk ¼ g k ðk þ 2Þ v^ kþ2 R R k k k2 Rp zp R R ðpÞ eðpÞ eðpÞ ek ðk 1Þ ðk 2Þ ðk1Þ ðk2Þ Rp R R Rp k k R ðpÞ Rp Rp ðpÞ f k þ v^ ðpÞ v^ k ¼ g ðpÞ k k Rp R R k k kþ2 R zp Rp Rp ðpÞ eðpÞ eðpÞ jm ek ðk þ 1Þ kþ1 þ ðk þ 2Þ kþ2 Rp R R R " k kþ2 k # Rp ðpÞ lm Rp Rp ^ðpÞ ðpÞ kk ¼ hk fk þ ðk þ 2Þ v^ ðpÞ kþ2 þ R lf R R k k k2 R zp R R eðpÞ eðpÞ jm p eðpÞ k þ ðk 1Þ ðk1Þ ðk 2Þ ðk2Þ Rp R R Rp " # k k R ðpÞ lm R Rp ðpÞ f k jf p v^ ðpÞ v^ k ¼ hðpÞ k k Rp lf R R ð7Þ
ðpÞ
fk ¼
n¼1
n
kðrÞ n ;
nþ1 R ðrÞ ðnÞ kn ; zp0 zr0 n¼1
N X 1 X r¼1 r–p
ðpÞ
ðpÞ
f k ¼ kk
ð1Þk C nk
n¼1
R zp0 zr0
n
kðrÞ n
ðk P 2Þ;
ðk P 1; p ¼ 1; 2; 3 . . . ; NÞ:
3 2 Rp zp0 c Rp Rp czd Rp ðpÞ ; g 1 ¼ c þ c zp0 zd zp0 zd Rp zp0 zd Rp zp0 zd
cRp
ðpÞ
g1 ¼
zp0 zd
cRp ; zp0 zd
kþ2 kþ1 Rp ð1Þk zp0 c Rp ðpÞ ; k P 2; g k ¼ ð1Þk c zp0 zd zp0 zd Rp ðpÞ
gk ¼
ðpÞ
k ð1Þk c Rp ; k P 2: k zp0 zd
h1 ¼ c ðpÞ
h1 ¼
Rp zp0 zd
3
2 zp0 c Rp Rp czd Rp þ ; þc zp0 zd Rp zp0 zd Rp zp0 zd
cRp jm cRp ; zp0 zd zp0 zd
kþ2 kþ1 Rp ð1Þk zp0 c Rp ðpÞ hk ¼ ð1Þk c þ ; k P 2; zp0 zd zp0 zd Rp ðpÞ
hk ¼ where
R zp0 zr0
N X 1 X r¼1 r–p
ðp ¼ 1; 2; . . . ; NÞ
ðk P 1; p ¼ 1; 2; 3 . . . ; NÞ
N X 1 X r¼1 r–p
ðpÞ
ð6Þ
ðpÞ
¼ vk
k ð1Þk cjm Rp ; k P 2: k zp0 zd
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
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Eq. (7) and their conjugated equations constitute a set of 8 N M linear equations concerning 8 N M unknown coefficients vðrÞ ; kðrÞ ; v^ ðpÞ ; ^kðpÞ ; v ðrÞ ; kðrÞ ; v^ ðpÞ ; ^kðpÞ ðr ¼ 1; 2; . . . ; N; p ¼ 1; 2; . . . ; N; k ¼ k
k
k
k
k
k
k
k
1; 2; . . . ; MÞ. After they are determined, all the complex potentials for the matrix and inclusions are known, and then the stress fields produced by the dislocation located at zd can be expressed as
ð8Þ
Substituting bx = 1, by = 0, z = x, zd = x0 into Eq. (8), the shear stress bx rbx 12 ðx; 0; x0 ; 0Þ and the normal stress r22 ðx; 0; x0 ; 0Þ at the point
(x, 0) on the crack faces produced by a unit glide edge dislocation at (x0, 0) can be given as
rbx 22 ðx; 0; x0 ; 0Þ ¼ k11 ðx; x0 Þ; 2lm 1 rbx þ k ðx; x Þ; 12 ðx; 0; x0 ; 0Þ ¼ pð1 þ km Þ x x0 12 0
ð9Þ
rby 12 ðx; 0; x0 ; 0Þ and the normal stress
rby 22 ðx; 0; x0 ; 0Þ at the point (x, 0) on the crack faces produced by a unit climb edge dislocation at (x0, 0), as follows
2lm 1 þ k ðx; x Þ; pð1 þ km Þ x x0 21 0
ð10Þ
rby 12 ðx; 0; x0 ; 0Þ ¼ k22 ðx; x0 Þ: where k21(x, x0), k22(x, x0) represent the regular part of the fundamental solution of the unit climb edge dislocation interacting with the N circular inclusions. Then the stress field of sub-problem II can be obtained through an integral of the fundamental solutions along the crack line, which gives that
rII22 ðx; 0Þ ¼ r
II 12 ðx; 0Þ
¼
Z
c
Zcc
Bx ðx0 Þrbx 22 ðx; 0; x0 ; 0Þdx0 þ Bx ðx0 Þr
c
bx 12 ðx; 0; x0 ; 0Þdx0
Bx ðx0 Þk11 ðx;x0 Þdx0 þ
c
þ
Z
c
Zcc
By ðx0 Þrby 22 ðx; 0; x0 ; 0Þdx0
By ðx0 Þ
c
2lm 1 þ k21 ðx; x0 Þ dx0 pð1 þ km Þ x x0
Let t ¼ x0 =c; s ¼ x=c, one can transform Eq. (14) into two simple Cauchy-type singular integral equations as follows: Z 1 Z 1 2lm 1 cBx ðtÞk11 ðs;tÞdt þ By ðtÞ þ ck21 ðs;tÞ dt ¼ rI22 ðs;0Þ pð1 þ km Þ s t 1 1 Z 1 Z 1 2lm 1 Bx ðtÞ þ ck12 ðs;tÞ dt þ cBy ðtÞk22 ðs;tÞdt ¼ rI12 ðs;0Þ pð1 þ km Þ s t 1 1 ð15Þ On the other hand, the additional condition (12) can be rewritten as: 1
Bx ðtÞdt ¼ 0
1 1
Z
By ðx0 Þr
c
where c 6 x 6 c; Bx(x0) and By(x0) are the dislocation density components at the point (x0, 0).The single-value condition of displacement vector requires that the density functions of the system satisfy the following relation:
By ðtÞdt ¼ 0
As the crack is embedded in a homogeneous isotropic elastic matrix, both crack tips have the square root singularities. According to Erdogan [32], it is apparent that Eq. (15) is two singular integral equations with index +1, and the fundamental solution of the singular equations is:
F x ðtÞ Bx ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 t2 F y ðtÞ By ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 t2
ð17Þ
where Fx(t) and Fy(t) are bounded functions in the interval [1, 1]. According to the Cause-Chebyshev solution [32], Eqs. (15) and (16) are discreted, and one get a group of 2n linear algebra equations with the 2n unknown Fx(tk), Fy(tk) (k = 1, 2, 3, . . ., n), which give that n X cp
n
F x ðt k Þk11 ðsr ; t k Þ þ
n X p k¼1
n
F y ðt k Þ
2lm 1 þ ck11 ðsr ; tk Þ pð1 þ km Þ sr tk
I 22 ðsr Þ
¼ r n X p
n
F x ðt k Þ
X n 2lm 1 cp þ ck12 ðsr ; tk Þ þ F y ðtk Þk22 ðsr ; t k Þ pð1 þ km Þ sr tk n k¼1
¼ rI12 ðsr Þ n X
F x ðtk Þ ¼ 0
k¼1 n X F y ðtk Þ ¼ 0
c
Bx ðx0 Þdx0 ¼ 0
c c
ð16Þ
1
k¼1
Z
c
¼ rI12 ðx;0Þ
k¼1 by 12 ðx; 0; x0 ; 0Þdx0
ð11Þ
Z
Z
¼ rI22 ðx;0Þ Z c Z c 2lm 1 Bx ðx0 Þ þ k12 ðx; x0 Þ dx0 þ By ðx0 Þk22 ðx;x0 Þdx0 pð1 þ km Þ x x0 c c
Z
where k11 ðx; x0 Þ, k12 ðx; x0 Þ represent the regular part of the fundamental solution of the unit glide edge dislocation interacting with the N circular inclusions. Similarly, inserting bx = 0, by = 1, z = x, zd = x0 into Eq. (8), one
rby 22 ðx; 0; x0 ; 0Þ ¼
c
ð14Þ
0d rd11 ¼ Re½2/0dm ðzÞ z/00d m ðzÞ wm ðzÞ; 0d rd22 ¼ Re½2/0dm ðzÞ þ z/00d m ðzÞ þ wm ðzÞ; 0d rd12 ¼ Im½z/00d m ðzÞ þ wm ðzÞ:
may obtain the shear stress
Z
k¼1
ð12Þ
ð18Þ
By ðx0 Þdx0 ¼ 0
c
where
The traction free boundary condition on the crack faces requires that the normal and shear stress components along the crack faces are zero, i.e.
rII22 ðx; 0Þ ¼ rI22 ðx; 0Þ; rII12 ðx; 0Þ ¼ rI12 ðx; 0Þ: Substituting Eqs. (9)–(11) into Eq. (13) results in
ð13Þ
ð2k 1Þp ; k ¼ 1; 2; . . . ; n 2n rp sr ¼ cos ; r ¼ 1; 2; . . . ; n 1: 2n
t k ¼ cos
Once the function Fx(t), Fy(t) are evaluated, following Erdogan [3], the stress intensity factors (SIFs) at the both crack tips can be expressed as:
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
H.-F. Yang, C.-F. Gao / Theoretical and Applied Fracture Mechanics xxx (2015) xxx–xxx
5
Table 1 Elastic coefficients C ij ð1010 N=m2 Þ; piezoelectric coefficients ekl ðC=m2 Þ; dielectric coefficients emn ð1010 C=VmÞ.
PZT5H
C11
C33
C44
C12
C13
e13
e33
e15
e11
e33
12.6
11.7
3.53
5.5
5.3
6.5
23.3
17.0
151
130
pffiffiffiffiffiffi 2lm pc F y ð1Þ; ð1 þ km Þ pffiffiffiffiffiffi 2l pc F y ð1Þ; K I ðp2 Þ ¼ m ð1 þ km Þ pffiffiffiffiffiffi 2lm pc F x ð1Þ; K II ðp1 Þ ¼ ð1 þ km Þ pffiffiffiffiffiffi 2l p c F x ð1Þ; K II ðp2 Þ ¼ m ð1 þ km Þ
K I ðp1 Þ ¼
ð19Þ
where p1 is the right crack tip and p2 is the left crack tip. pffiffiffiffiffiffi They can be normalized by K 0I ¼ r1 22 pc , then
2lm F y ð1Þ; r1 22 ð1 þ km Þ 2lm ¼ 1 F ð1Þ; r22 ð1 þ km Þ y K 0I K II ðp1 Þ 2lm ¼ 1 F ð1Þ; r22 ð1 þ km Þ x K 0I K II ðp2 Þ 2lm ¼ 1 F x ð1Þ: 0 r ð1 þ km Þ KI 22 K I ðp1 Þ K 0I K I ðp2 Þ
Fig. 5. The SIFs for a crack interacting with a single piezoelectric inclusion, where Xiao’s solution is found in the Ref. [28].
¼
ð20Þ
4. Results and discussion Choosing the PZT-5H as a model material of inclusions, the value of material constants for PZT-5H are listed in Table 1 [33]. We assume that Poisson’s ratio of the matrix is mm = 0.3. To validate the present methods, we calculate the variation of K I =K 0I at the right tip of the crack located near a single piezoelectric inclusion with the distance factor d/R, where R is the radius of the piezoelectric inclusion and d is the distance between the respective centers of the crack and the inclusion. The results shown in Fig. 5 can be compared with the solution provided by Xiao and Bai [28]. It is found that the present solution agrees well with Xiao and Bai’s solution. To investigate the influence of multiple inclusions, as shown in Fig. 6, we compared the SIFs at the crack tip of two identical piezoelectric inclusions contained in the matrix with that of the single piezoelectric inclusion. In this case, we change the position of the two piezoelectric inclusions synchronously. It is found that two ‘softer’ piezoelectric inclusions (having lower Young’s modulus than the matrix) elevate the SIFs at the crack tip while two ‘harder’ piezoelectric inclusions reduce the SIFs, and the influence will be getting weaker as the distance between the crack and the inclusions increases. To investigate the influence of the electric loading on the crack, we calculate the variation of K I =K 0I for a crack located at the midway of two identical piezoelectric inclusions under the normalized 1 electric loading e31 E1 3 =r22 . Due to the electro-mechanical coupling effect of the piezoelectric material, the influence of the electric loading is significant, especially for the ‘softer’ inclusions. It is found from Fig. 7 that the influence of the electric loading on the SIFs is linear, and the SIFs decreases as the electric loading increases. This indicates that only when the electric filed increases in the opposite direction, the SIFs increase. The SIFs tend to be infinitely large when E1 3 increase along the negative direction of x3. Hence, the crack is dangerous when the value of E1 3 is negative.
Fig. 6. The SIFs for a crack located at the center of two same piezoelectric inclusions, change the position of the two inclusions synchronously.
Fig. 7. Variation of the SIFs for a crack interacting with two same piezoelectric inclusions with the electric loading.
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
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Fig. 8. The SIFs at the right tip of a crack interacting with two different piezoelectric inclusions.
Fig. 9. The SIFs at the both tip of the crack interacting with two same piezoelectric inclusions, fix the left inclusion and change the position of the right one.
Fig. 10. The SIFs at the right tip of the crack interacting with the regular array I of three piezoelectric inclusions.
Fig. 11. The model II SIFs at the right tip of the crack interacting with the regular triangular array II of three piezoelectric inclusions.
Fig. 12. The model I SIFs at the right tip of the crack interacting with the regular triangular array II of three piezoelectric inclusions.
Compared with a single piezoelectric inclusion, the influence of the electric loading on the SIFs is significant for two ‘softer’ piezoelectric inclusions while the influence can be neglected for two ‘harder’ piezoelectric inclusions. We fix the distance between the crack and the edges of the two inclusions, and then change the radius of the right piezoelectric inclusion to examine the influence of the geometry of inclusions. Fig. 8 shows the variation of K I =K 0I at the right tip of the crack. It is found that for the ‘softer’ inclusions, the SIFs increase as the radius of the right inclusion increases. However, for the ‘harder’ inclusions, the SIFs decrease as the radius of the right inclusion increases. It implies that a bigger piezoelectric inclusion can reduce the SIFs or increase it, depending on the relative stiffness of the inclusions and the matrix. We fix the left inclusion and change the position of the right inclusion. What shown in Fig. 9 is the variation of K I =K 0I at both tips of the crack. For the ‘softer’ inclusions, when the right inclusion is closer to the crack than the left (d2/R < 2), the SIFs at the right tip are larger than that at the left tip; when the right inclusion moves far away from the crack (d2/R > 2), the SIFs at the right tip are smaller than that at the left tip. This indicates that the nearby crack tip is more dangerous than the distant crack tip, and therefore crack
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will propagate toward the nearby inclusion. On the contrary, the SIFs at the nearby crack tip are smaller than that at the distant crack tip for the ‘harder’ piezoelectric inclusions. This means that the ‘harder’ piezoelectric inclusions always resist crack propagation toward the nearby inclusion. Fig. 10 shows the variation of K I =K 0I at the right tip of the crack interacting with the regular triangular array of three identical piezoelectric inclusions with the distance factor d/R, where d is the distance between the center of the crack and the right inclusion. In this case, the regular triangular array of three piezoelectric inclusions is symmetrical about x1. It is found that the position of the crack and the applied electric loading have a significant influence on the SIFs at the crack tip for the ‘softer’ inclusions. It is worth notice that when the crack locates between the two left inclusions, the values of the SIFs are negative under the electric 1 loading e31 E1 3 =r22 ¼ 2; which indicates that the electric loading makes the crack faces closed. One can also find that the influence of the electric loading on the crack is very weak for the ‘harder’ inclusions. We consider the case when the regular triangular array of three piezoelectric inclusions is not symmetrical about x1, in which the model II SIFs will not be equal to zero. Fig. 11 demonstrates that not only the ‘softer’ inclusions but also the ‘harder’ inclusions make the model II SIFs change rapidly and substantially. However, when the crack locates within the inclusions array, only in the case of ‘softer’ inclusions the electric loading plays an important role in the model II SIFs. As shown in Fig. 12, the electric loading makes the model I SIFs decrease considerably in the case of ‘softer’ inclusions, but there is hardly any meaningful change of the model I SIFs in the case of ‘harder’ inclusions.
5. Conclusion The interaction of multiple piezoelectric inclusions embedded in an infinite non-piezoelectric elastic matrix with a matrix crack is investigated. Based on the superposition principle of elasticity, the solution of the original problem can be obtained as the sum of two sub-problems. The solution of sub-problem I can be obtained by a previous work. In sub-problem II, the crack is simulated by a continuous distribution of edge dislocations with unknown density functions. Thus, the solution of an edge dislocation interaction with multiple piezoelectric inclusions embedded in an elastic medium is derived as the Green functions, and then the stress field in the sub-problem II can be obtain through an integral of the Green functions along the crack line. Moreover the superposition of the stress fields of sub-problem I and sub-problem II is requested to satisfy the traction free boundary condition along the crack surface. Thus a set of singular integral equations are formulated which can be solved by a numerical method, and the stress intensity factors at the crack are derived in terms of the dislocation density functions evaluated from the integral equations. Finally, numerical examples are given for a few typical arrays of piezoelectric inclusions with various materials and geometric parameters. The main results can be summarized as follows: (1) The applied electric-field loading plays an important role in the interaction between multiple piezoelectric inclusions and the matrix crack. For example, the electric-field loading applied along the positive x3 direction may cause a crack closed while an electric-field loading applied in the opposite direction can cause crack growth. (2) The influence of ‘softer’ piezoelectric inclusions on the crack is quite different from that of ‘harder’ piezoelectric inclusions. In general, the ‘softer’ inclusions are more sensitive to the changes of the geometric parameters and the electric
7
loading. For example, the electric loading changes the SIFs greatly in the case of ‘softer’ piezoelectric inclusions, but it causes hardly any meaningful change of the SIFs in the case of, ‘harder’ inclusions. (3) The geometric and elastic parameters have a remarkable influence on the SIFs. From the above-mentioned cases, one can find that the relative stiffness of the inclusions and the matrix could have a decisive effect on matrix cracking in piezoelectric fiber-reinforced composites. (4) The SIFs at a matrix crack is greatly affected by the number and array of nearby piezoelectric inclusions. When the problem of multiple piezoelectric inclusions interacting with a matrix crack is concerned, the present method can be used to study whether the matrix crack will grow or not.
Acknowledgements The authors thank the support from the National Natural Science Foundation of China (11232007, 11472130) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
References [1] O. Tamate, The effect of a circular inclusion on the stresses around a line crack in a sheet under tension, Int. J. Fract. Mech. 4 (1968) 257–265. [2] C. Atkinson, The interaction between a crack and an inclusion, Int. J. Eng. Sci. 10 (1972) 127–136. [3] F. Erdogan, G.D. Gupta, M. Ratwani, Interaction between a circular inclusion and an arbitrarily oriented crack, ASME J. Appl. Mech. 41 (1974) 1007–1013. [4] Y.C. Hsu, V. Shivakumar, Interaction between an elastic circular inclusion and two symmetrically placed collinear cracks, Int. J. Fract. Mech. 12 (1976) 619– 630. [5] F. Erdogan, G.D. Gupta, The inclusion problem with a crack crossing the boundary, Int. J. Fract. 11 (1975) 13–27. [6] R.D. Bhargava, R.R. Bhargava, Elastic circular inclusion in an infinite plane containing two cracks, Int. J. Eng. Sci. 11 (1973) 437–449. [7] M. Isida, H. Noguchi, Plane problem of an arbitrary array of cracks in an infinite plate containing a circular inclusion, J. Jpn. Soc. Mech. Eng. 49 (1983) 147–155. [8] C.H. Wu, C.H. Chen, A crack in a confocal elliptic inhomogeneity embedded in an infinite medium, J. Appl. Mech. 57 (1990) 91–96. [9] K. Kim, L.J. Sudak, Interaction between a radial matrix crack and a three-phase circular inclusion with imperfect interface in plane elasticity, Int. J. Fract. 131 (2005) 155–172. [10] Y. Liu, C.Q. Ru, P. Schiavone, A. Mioduchowski, New phenomena concerning the effect of imperfect bonding on radial matrix cracking in fiber composites, Int. J. Eng. Sci. 39 (2001) 2033–2050. [11] J. Li, Q.H. Fang, Y.W. Liu, Crack interaction with a second phase nanoscale circular inclusion in an elastic matrix, Int. J. Eng. Sci. 72 (2013) 89–97. [12] R. Li, A. Chudnovsky, Variation of the energy release rate as a crack approaches and passes through an elastic inclusion, Int. J. Fract. 59 (1993) R69–R74. [13] P. Lipetzky, S. Schmauder, Crack-particle interaction in two-phase composites. Part I: Particle shape effects, Int. J. Fract. 65 (1994) 345–358. [14] P. Lipetzky, Z. Knesl, Crack-particle interaction in two-phase composites. Part II: Crack deflection, Int. J. Fract. 73 (1995) 81–92. [15] M.B. Bush, The interaction between a crack and a particle cluster, Int. J. Fract. 88 (1997) 215–232. [16] C. Wang, W. Libardi, J.B. Baldo, Analysis of crack extension paths and toughening in a two phase brittle particulate composite by the boundary element method, Int. J. Fract. 94 (1998) 177–188. [17] M.G. Knight, L.C. Wrobel, J.L. Henshall, L.A. Lacerda, A study of the interaction between a propagating crack and an uncoated/coated elastic inclusion using BE technique, Int. J. Fract. 114 (2002) 47–61. [18] D.A. Hills, M. Comninou, A normally loaded half plane with an edge crack, Int. J. Solids Struct. 21 (1985) 399–410. [19] J. Helsing, Stress intensity factors for a crack in front of an inclusion, Eng. Fract. Mech. 64 (1999) 245–253. [20] Z.M. Xiao, B.J. Chen, Stress intensity factor for a Griffith crack interacting with a coated inclusion, Int. J. Fract. 108 (2001) 193–205. [21] Y.S. Tao, Q.H. Fang, X. Zeng, Y.W. Liu, Influence of dislocation on interaction between a crack and a circular inhomogeneity, Int. J. Mech. Sci. 80 (2014) 47– 53. [22] D.H. Chen, The effect of an elliptical inclusion on a crack, Int. J. Fract. 85 (1997) 351–364. [23] J. Herbert, Ferroelectric Transducers and Sensors, Gordon and Breach Science Publishers, New York, 1982.
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005
8
H.-F. Yang, C.-F. Gao / Theoretical and Applied Fracture Mechanics xxx (2015) xxx–xxx
[24] B.Q. Dong, Z.J. Li, Cement-based piezoelectric ceramic smart composites, Compos. Sci. Technol. 65 (2005) 1363–1371. [25] P. Tan, L.Y. Tong, Micro-electromechanics models for piezoelectric- fiberreinforced composite materials, Compos. Sci. Technol. 61 (2001) 759–769. [26] B.H. Yang, C.F. Gao, Anti-plane electro-elastic fields in an infinite matrix with N coated-piezoelectric inclusions, Compos. Sci. Technol. 69 (2009) 2668–2674. [27] M.L. Dunn, H.A. Wienecke, Inclusion and inhomogeneities in transversely isotropic piezoelectric solids, Int. J. Solids Struct. 34 (1997) 3571–3582. [28] Z.M. Xiao, J. Bai, On piezoelectric inhomogeneity related problems-Part II: A circular piezoelectric inhomogeneity interacting with a nearby crack, Int. J. Eng. Sci. 37 (1999) 961–976.
[29] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1934. [30] B.H. Yang, C.F. Gao, Plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix, Int. J. Eng. Sci. 48 (2010) 518–528. [31] D.A. Hills, A.P. Kelly, D.A. Dai, A.M. Korsuncky, Solution of Crack Problems, the Distributed Dislocation Technique, Kluwer, Dordrecht, 1996. [32] F. Erdogan, Mixed boundary-value problems in mechanics, Mech. Today 4 (1990) 1–85. [33] Y.E. Pak, Linear electro-elastic fracture mechanics of piezoelectric materials, Int. J. Fract. 54 (1992) 79–100.
Please cite this article in press as: H.-F. Yang, C.-F. Gao, Interaction between multiple piezoelectric inclusions and a crack in a non-piezoelectric elastic matrix, Theor. Appl. Fract. Mech. (2015), http://dx.doi.org/10.1016/j.tafmec.2014.12.005