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Mixed mode transient analysis of functionally graded piezoelectric plane weakened by multiple cracks
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
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Mixed mode transient analysis of functionally graded piezoelectric plane weakened by multiple cracks S. Azizi, R. Bagheri
T
⁎
Department of Mechanical Engineering, Mechatronics Faculty, Karaj Branch, Islamic Azad University, Karaj, Alborz, Iran
ARTICLE INFO
ABSTRACT
Keywords: Transient loading Functionally graded piezoelectric materials Dislocation technique Mixed mode stress intensity factors
In this paper, a functionally graded piezoelectric (FGP) plane weakened by multiple parallel cracks is investigated under mixed mode mechanical and electrical impacts. The system contains Volterra type glide and climb edge dislocations. The material properties of the system are assumed to vary exponentially along the xaxis. At first, the stress field and electric displacement in FGP plane are derived by using a single dislocation. Then, a system of singular integral equations with a simple Cauchy kernel of the transient dynamic problem is obtained by determining a distributed dislocation density on the crack surface and using the Fourier and Laplace integral transforms, which are solved numerically using the Lobatto-Chebyshev integration formula. The dynamic field intensity factors are determined through numerical Laplace inversion and the distributed dislocation technique (DDT). Finally, various examples are provided to investigate the influence of the material properties, electromechanical coupling factor, loading conditions and cracks arrangement on the dynamic fracture behavior of the interacting cracks.
1. Introduction As the application of piezoelectric materials possessing passive, active and elastic properties have risen sharply, a great number of analytical, numerical and experimental studies on this type of materials are conducted in the field of engineering sciences [1,2]. The physical properties of functionally graded piezoelectric materials (FGPMs) are non-homogeneous and the electro- mechanical parameters may vary continuously in particular directions. The main advantage of FGPMs over conventional composites is the absence of any interface. It is generally well-known that FGPMs are very brittle and capable to fracture with low toughness. Thus, the investigation of the fracture analyses of this type of materials under mechanical and electrical loadings is a major task. Most of the studies, however, are related to static [3–10] or quasi-static conditions [11–17]; but it may be very important to study the response of cracks under dynamic loadings. Dynamic loadings are mainly categorized into two groups including harmonic loading [18–20] and impact loading. Impact loads applied on the cracked structures lead to catastrophic failure of the system. The problem of a finite crack in a strip of FGPM was studied by Ueda [21]. The electro-elastic properties of the FGPM are assumed to vary continuously along the thickness of the strip which is under in-plane mechanical and electrical impact loadings. Wang and Mai [22]
⁎
considered the mixed mode crack problem in an infinite functionally graded material. In this investigation, the influence of such factors including material inhomogeneity constants and crack spacing on the stress intensity factors, both at transient state and steady state are studied. The problem for a finite crack in an infinite piezoelectric solid under an impact mechanical loading, or an impact electrical loading or a combination of both is studied by Garcı́a-Sánchez et al. [23]. The effects of the electric field, material properties and electromechanical coupling coefficient were studied on stress and electric displacement intensity factors. Feng et al. [24] obtained the dynamic response of an interfacial crack between two dissimilar magneto-electro-elastic layers under magnetic, electrical and mechanical impact loadings. Their investigation showed that a negative magnetic loading (or electrical loading) is generally prone to inhibit the crack extension rather than a positive one for a magnetically (or electrically) impermeable interfacial crack. The in-plane crack problem of a magneto-electric material under in-plane mechanical, electric and magnetic impacts was investigated by Zhong et al. [25]. The Laplace and Fourier transforms are applied to reduce the associated initial and mixed-boundary value problem to dual integral equations, and then to singular integral equations with Cauchy kernel. The effects of applied magnetics and electric impacts on crack growth were investigated. Liu and Zhong [26] focused on the dynamic analysis of two collinear dielectric cracks in a piezoelectric material
Corresponding author. E-mail address: [email protected] (R. Bagheri).
https://doi.org/10.1016/j.tafmec.2019.02.014 Received 27 October 2018; Received in revised form 24 January 2019; Accepted 18 February 2019 Available online 19 February 2019 0167-8442/ © 2019 Elsevier Ltd. All rights reserved.
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
under the action of in-plane electro-mechanical impacts. They found that the positive electric field enhances the dynamic dielectric crack growth and a negative one impedes the dynamic dielectric crack growth in a piezoelectric solid. Rojas-Díaz et al. [27] analyzed the transient response of the cracked composite materials made of piezoelectric and piezomagnetic phases. They investigated the effect of the combined application of electric, magnetic and mechanical loads on the dynamic field intensity factors. The problem of two collinear mixed-mode limited-permeable cracks embedded in an infinite medium made of an FGPM with crack surfaces subjected to electro-mechanical loadings is investigated by Jamia et al. [28]. They studied the effects of the interaction of two cracks, material gradient parameter describing FGPMs and lattice parameter on the mechanical stress and electric displacement field near crack tips. In accordance with the above studies, there is no promising examination regarding the multiple cracks interaction with arbitrary arrangement in the piezoelectric materials. The distributed dislocation method has been introduced as a powerful tool to obtain the field intensity factors in the materials weakened by multiple cracks. The transient dynamic stress intensity factor for multiple curved cracks under impact loadings is studied by Ayatollahi and Monfared [29]. By means of the Cagniard-de Hoop method, the problem was solved and dynamic SIF was obtained. Analysis of multiple interacting cracks in exponentially graded layers under mode III transient deformation also in orthotropic layers under mixed mode transient loadings were investigated by Vafa et al. [30], Vafa and Fariborz [31], respectively. Bagheri [32] provided the calculation of the field intensity factors for multiple horizontal cracks in a piezoelectric half-plane under an anti-plane transient loading based on the distribution of screw dislocations. In this paper, we consider the dynamic problem of multiple parallel cracks in an FGPM under mixed-mode time-dependent loading. The problem is solved under assumption of impermeable boundary condition and the material properties of the system are assumed to change exponentially along the x-axis. The integral transform and distributed dislocation with time-dependent Burgers vector are used to reduce the problem to Cauchy-type singular integral equations, which is solved numerically by using the Lobatto–Chebyshev collocation method. A numerical Laplace transform inversion technique, described by Stehfest [33], is then used to obtain the dynamic field intensity factors at crack tips. The solutions are in agreement with those available in literature. The main objective of the paper is to study the effect of the time variation, the gradient of the material property and cracks interactions on the dynamic field intensity factors.
the FGPMs in two dimensional elasticity undergoing in-plane deformation are as follows:
y, t ) = c11 (x )
u x
yy (x , y , t ) = c13 (x )
u x
xx (x ,
xy (x , y, t ) = c44 (x )(
u y
Dx (x , y , t ) = e15 (x )(
u y u x
Dy (x , y , t ) = e31 (x )
+ c13 (x )
v y
+ e31 (x )
y
+ c33 (x )
v y
+ e33 (x )
y
+
v ) x
+
v ) x
+ e15 (x )
x
11 (x ) x
+ e33 (x )
v y
33 (x ) y
(1)
where xx , yy and xy are the in-plane stress tensors, Dx and Dy are the in-plane electric displacements. Also, c11 (x ) , c13 (x ) , c33 (x ) and c44 (x ) are the elastic modules, e31 (x ) , e33 (x ) and e15 (x ) are the piezoelectric parameters, and 11 (x ) and 33 (x ) are the dielectric permittivity of the FGPMs. To make the analysis tractable, the material properties of the FGPM plane are supposed to be one-dimensionally dependent as:
c11 (x ) = c110 e x , c13 (x ) = c130 e x , c33 (x ) = c330 e x , c44 (x ) = c440 e x , e31 (x ) = e310 e x , e33 (x ) = e330 e x , e15 (x ) = e150 e x , x x 11 (x ) = 110 e , 33 (x ) = 330 e
(2)
where cij0 are the elastic modules, eij0 are the piezoelectric parameters, and ij0 are the dielectric permittivity at x = 0 and is a positive or negative constant. By neglecting body forces and electric charge density, the equations of motion and Maxwell equation for the FGPMs are in the following forms:
c110
2u
x2
+ (c130 + c440)
2v x y
u x
v y
+ (c110 c440
2v x2
+ c130
+ (c130 + c440)
2u x y
2u
+ c440
+ e310 + c330
y2 y
+ e150
+ e150 2
110 x 2
2
330 y 2
+ e150
2v x2
+ e330
x y
2u 0 t2 ,
)= 2v y2
2
+ (e310 + e150)
2v y2
+ e150 (
+ e330
x2
+ (e310 + e150) u y
2
y2
+ c440 (
u y
+
2v 0 t2 ,
=
x
2
+
v ) x
v ) x
.
2u x y
110 x
= 0. (3)
In the above equations, it is assumed that the mass density of the medium varies exponentially along the x-axis with (x ) = 0 e x , where 0 is the density at x = 0 . Let an edge dislocation with time-dependent Burgers vector be located at the origin coordinates. The dislocation cut created along the positive direction of the x -axis, while glide, climb and electric dislocation of the edge dislocation are represented bx (t ) , by (t ) and b (t ) , respectively. Thus, the electromechanical conditions at the dislocation path can be described as:
2. Statement of the problem We consider a plane made of functionally graded piezoelectric materials (FGPMs) weakened by the electro-elastic Volterra types climb and glide edge dislocations shown in Fig. 1. The problem is formulated in the framework of linear piezoelectricity. The constitutive relations of
u (x , 0+, t ) v (x , 0+, t )
u (x , 0 , t ) = bx (t ) H (x ), v (x , 0 , t ) = by (t ) H (x ),
(x , 0+, t )
(x , 0 , t ) = b (t ) H (x ),
yy (x ,
0+, t ) =
yy (x ,
0 , t ),
xy (x ,
0+, t ) =
xy (x ,
0 , t ),
Dy (x , 0+, t ) = Dy (x , 0 , t ),
(4)
where H (·) is the Heaviside step function. It is of importance that the electric impermeability condition is adopted by introducing the jumps of electric potential across the dislocation path. It is convenient to solve the problem in the half-plane ( y 0 ), which necessitates decomposition of the problem into symmetric and anti-symmetric parts. The corresponding boundary conditions (4) for the symmetric problem is reduced to
v (x , 0, t ) =
Fig. 1. Functionally graded piezoelectric plane weakened by electro-elastic dislocations. 128
b y (t ) 2
H (x ),
(x , 0, t ) =
b (t ) 2
H (x ),
xy (x ,
0, t ) = 0, (5)
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
And for the anti-symmetric problem the conditions are
bx ( t ) H (x ), 2
u (x , 0, t ) =
yy (x ,
0, t ) = 0,
where
Dy (x , 0, t ) = 0,
2u
u x
+ (c110 0s
2u
+ e310
y
+ (e310 + e150 )
y2
1
2
x y 2
)
3 2u
+ (c130 + c440)
x2
+ c440 ( 0s
x y
v y
+ c130
2u
+ c440
2v
u y
v x
+
2v
+ c330
x y
) + e150
+ e150
y2
2
+ e330
x2
2
y2
x
= 0, 2
110
2
330
x2
u y
+ e150 (
2v
+ e150
y2
v x
+
+ e330
x2
)
110
2v
y2
+ (e310 + e150)
d2U dy2
+ [i (c130 + c440) + c130] 2) c
+ [(i d2V c330 2 dy
0s
110
2] U
dV dy
+ [i (c130 + c440) +
(7)
d2 330 dy 2
d2V dy 2
+ e330
+ [i (e310 + e150) + e310]
+
d2 e330 2 dy
2) e 150 V
dU dy
2] V
+(
i )
110
Aj ( , s ) e
=0
V ( , y, s ) =
j=1 3
( , y, s) =
(9)
j =1
4 j
r1
where mj =
+ r2
r4 3j 2 j
=
2 j
+ r3
+ r5
,
nj
j
+ mj (
1+ i
[
5
2)
j
+i (
2)
+ (i 5
+
7)] j
3
s 2/ c 2
1
=
2
=
3
=
4
+ 1
3 1
3 1
3
+
2
2
3
21/3p 3(q + r )1/3
,
j
{1, 2, 3}
= 0, +
(q + r )1/3 3 × 21/3
(1 + i 3 ) p 3 × 22/3 (q + r )1/3 (1
i 3 )p
3 × 22/3 (q + r )1/3
+
8
=
330
c440
=
9
=
,
,
H2 H7) + H1 H5 8
2 4 8+ 6
,
,
c130 + c440 , c440
110
c440
3
c110 , c440
=
4
=
c330 , c440
5
=
e310 , c440
6
=
e330 , c440
c440
, c=
0
1 2
(
1 2
(
1 2
(
3
Aj e
jy
j=1 3 j =1
) ei xd
mj Aj e
jy
nj Aj e
jy
3 j=1
) ei x d
) ei xd
(13)
e x c440 2
(
e x c440 2
(
e x c440 2
(
Dx (x , y , s ) =
e x c440 2
(
Dy (x , y , s ) =
e x c440 2
(
y, s ) = y, s) = y, s ) =
3
b1j Aj e
jy
) e i xd
b2j Aj e
jy
) ei x d
b3j Aj e
jy
) ei xd
b4j Aj e
jy
) ei x d
b5j Aj e
jy
) e i xd
j=1 3 j=1 3 j=1 3 j=1 3 j=1
(14)
b1j = i
3
+
1 mj j
+
5 nj j
b 2j = i
1
+
4 mj j
+
6 nj j
b3j =
j
b4j =
7 j
6 × 21/3 6 × 21/3
5
+i +
7 mj 6 mj j
7 nj
i
9 nj 8 nj j
j = 1, 2, 3
(15)
A1 ( , s ) =
(
A2 ( , s ) =
(
A3 ( , s ) =
(
( ) i/ ) [A11 bx (s ) 2 ( ) i/ ) [A21 bx (s ) 2 ( ) i/ ) [A31 bx (s ) 2
+ A12 by (s ) + A13 b (s )] + A22 b y (s ) + A23 b (s )] + A32 by (s ) + A33 b (s )]
(16)
(·) is the Dirac delta function and the functions where Aij , i, j {1, 2, 3} are given in Appendix A. Substituting unknown functions (16) into stress fields and electric displacements (14) yields
( 1 + i 3 )(q + r )1/3 (1 + i 3 )(q + r )1/3
+ i mj + i
The unknown functions Ai ( , s ), i = {1, 2, 3} may be obtained by utilizing boundary conditions (5) and (6), respectively to Eqs. (13) and (14) for the climb and glide dislocations. The expressions for these coefficients in the FGP plane containing both dislocations are
(10)
1
e150 , c440
b5j = i
and d ri, (i = 1, 2, ...,5) are presented in Appendix A. In Eq. (9), functions Aj , (j = 1, 2, 3) are arbitrary unknowns. The characteristic equation and its roots are found to be: 6
=
2
H1 4 )
where
jy.
jy
nj Aj ( , s ) e
7
xy (x ,
jy
mj Aj ( , s ) e
=
yy (x ,
j=1 3
1
xx (x ,
1 , and is the Fourier variable, U , V and are Fourier where i = transforms of the displacement components u , v and electrical potential respectively. Because the stresses, displacement components and electrical potential must vanish as (x 2 + y 2 ) , therefore, the solution to the differential equation (8) is written as follows 3
2 4 8+ 6
c130 , c440
H1 H8 4 + H6 (2H1 6
H5 H8
H5
The stress fields and electric displacements in the Laplace domain are obtained from the Eqs. (1) and (13) as follows:
(8)
U ( , y, s ) =
=
v (x , y , s ) =
d dy
=0 2
H4 H6) + H2 H4 H8
H1 ( H62 + H5 H8)
u (x , y , s ) =
2) e 150
+ (i 0s
+ [i (e150 + e310) + e150] + (i
=
(x , y , s ) =
2) c 440
+ [(i
H62 + H3 (H5 H7
H1 6) + H3 ( H7 4 + H4 6 ) + 8 (H2 H4 2 4 8+ 6
(12)
=0
dU c440] dy
+ 27 3 ,
and the functions Hi , (i = 1, 2, ...,8) are given in Appendix A. By using the inverse Fourier transform, the displacement fields and electrical potential (9) are written as:
where the superscript ∗ denotes the Laplace transform, s is Laplace variable at the time transform domain. Mixed boundary value problem (7) is solved by employing the method of the Fourier transform to the space variable x and is reduced to
c440
1 2
4 H8 + 6 ( 2H6 + H2 H7
=
2u x y
= 0.
x
2,
9
4p 3 + q 2 ,
r=
= 0,
2v
c440
2v
+ (c130 + c440 )
x2
3 3 1
q=2
By using the Laplace transform the time variable in Eq. (3) is eliminated, and it becomes
c110
2 1
p=
(6)
(11) 129
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
xx (x , y , s ) =
e xc 440 [bx (s) 4
3 j y + b y (s ) j = 1 b1j Aj1 e
3 j y + b (s ) j = 1 b1j Aj2 e
3 jy j = 1 b1j Aj3 e ]
=0
+ yy (x , y, s ) =
e x c440 [bx (s ) 4
3 j y + b y (s ) j = 1 b2j Aj1 e
3 j y + b y (s ) j = 1 b3j Aj1 e
e x c440 D y (x , y, s ) = [bx (s ) 4
3 j y + b y (s ) j = 1 b5j Aj1 e
3 jy j = 1 b2j Aj3 e ]
+ =0
f yy (x , y , s, ) d
fxx (x , y, s, ) d e x c440 [bx (s ) xy (x , y, s ) = 4
3 j y + b (s ) j = 1 b2j Aj2 e
3 j y + b (s ) j = 1 b3j Aj2 e
3 jy j = 1 b3j Aj3 e ]
=0
e xc 440 + Dx (x , y , s ) = [bx (s ) 4
3 j y + b y (s ) j = 1 b4j Aj1 e
fxy (x , y, s, ) d
3 j y + b (s ) j = 1 b4j Aj 2 e
3 jy j = 1 b4j Aj3 e ]
+ =0
fdx (x , y , s, ) d
3 j y + b (s ) j = 1 b5j Aj2 e
3 jy j = 1 b5j Aj3 e ]
+ =0
(17)
fdy (x , y, s, ) d
where fij (x , y, s, ), i {x , y , d}, j {x , y} are defined in Appendix A. The singular nature of the kernels fij may be determined by studying the asymptotic behavior of the integrands of integrals in (17). By adding and subtracting the asymptotic expressions of the integrands using asymptotic expressions for large value , we find:
fij (x , y, s, ) = fij (x , y , s, ) + [fij (x , y, s, ) Singular Part
fij (x , y, s, )]
D y (x , y , s ) =
=
[
3
3 jy
b1j Aj1 e
+ by (s )
j= 1
3
b1j Aj2 e
jy
+ b (s )
j =1
b1j Aj3 e
jy
j= 1
] =0
e x c440 33 y 11 y 22 y + {bx (s )[ 2 1 1 + 22 2 + 23 3 ] x + ( 11 y ) 2 x + ( 22 y ) 2 x + ( 33 y )2 2 x x x by (s )[ 2 4 1 + 2 5 2 + 2 6 3 ] x + ( 11 y )2 x + ( 22 y )2 x + ( 33 y ) 2 x x x b (s )[ 2 7 1 + 2 8 2 + 2 9 3 ]} + [fxx (x , y, s , ) x + ( 11 y )2 x + ( 22 y )2 x + ( 33 y )2
3 3 3 e xc 440 bx (s ) b2j Aj1 e j y + b y (s ) b2j Aj2 e j y + b (s ) b2j Aj3 e j y 4
[
j =1
j=1
j= 1
[f yy (x , y, s , )
] =0
xi (p) = xic + pli, yi (p) = yic, 1
f yy (x , y , s , )] d
e x c440 [bx (s ) 4
3 j y + b y (s ) j = 1 b3j Aj1 e
3 j y + b (s ) j = 1 b3j Aj 2 e
3 jy j = 1 b3j Aj3 e ]
x x x e x c440 { bx (s )[ 2 1 7 2 + 2 2 8 2 + 2 3 9 2 ] 2 x + ( 11 y ) x + ( 22 y ) x + ( 33 y ) y y y y y b y (s )[ 24 7 11 2 + 2 5 8 22 2 + 26 9 33 2 ] b (s )[ 2 7 7 11 2 + 2 8 8 22 2 x + ( 11 y ) x + ( 22 y ) x + ( 33 y ) x + ( 11 y ) x + ( 22 y ) 33 y + 29 9 ]}+ x + ( 33 y )2
+
[fxy (x , y , s , ) (x , y , s ) =
e xc 440 bx (s ) 4
[
e xc 440 + { bx (s )[ 2 1 2 x + 4 10 11 y b y (s )[ 2 + x + ( 11 y ) 2 7 10 11 y b (s )[ 2 + x + ( 11 y ) 2 +
[fdx (x , y , s, )
3
b4j Aj1 e j y + b y (s )
j =1
j =1
10 x
( 11 y ) 2 5 11
x2 + (
8 11
x2 + (
b4j Aj 2 e j y + b (s )
3 j =1
x 2 + ( 22 y )2
5 14 x
x 2 + ( 22 y )2 8 14 x
x 2 + ( 22 y )2
+ +
+
x 2 + ( 33 y ) 2
6 15 x
x 2 + ( 33 y )2 9 15 x
x 2 + ( 33 y )2
=0
]
] ]} +
[fdy (x , y, s , )
(19)
p
1,
i
{1, 2, 3, ...,N }
(20)
=0
yy (x i (p), yi (p),
Dx
s) =
b4j Aj3 e j y
N k=1
ak
1 1
11 [k yyik (p , q, s ) bxk (q, s )
12 13 +k yyik (p, q, s ) byk (q, s ) + k yyik (p , q, s ) b k (q, s )] dq xy (x i (p), yi (p),
s) =
N k=1
ak
1 1
11 [k xyik (p , q, s ) bxk (q, s )
12 13 +k xyik (p, q, s ) b yk (q, s ) + k xyik (p , q, s ) b k (q, s )] dq
fxy (x , y , s , )] d 3
+
2
3 15 33 y
where (xic , yic ) and li are the coordinates of the center and half length of the i-th cracks, respectively. Employing the principle of superposition, the singular integral equations are
xy (x , y , s )
=
11 y )
2 14 22 y
]
In the framework of linear theory, the current problem can be treated as the superposition of two subproblem. Subproblem I considers the medium without any cracks under the action of mixed mode mechanical and electrical impacts at far field. While subproblem II concerns with the FGP plane with distributed electro-mechanical dislocations on the x-axis. The crack faces subjected to the distributed electromechanical dislocations cancel out the stress and electric displacement induced by subproblem I. Therefore, the above dislocation solutions are used to construct integral equations for the transient analysis of an FGPM plane weakened by N cracks. The geometry of cracks are presented in parametric form as
e xc 440 22 y 33 y 11 y + {bx (s )[ 2 1 4 + 22 5 + 23 6 ] 2 x + ( 11 y ) 2 x + ( 22 y )2 x + ( 33 y ) 2 5 5x 6 6x 4 4x b y (s )[ 2 + 2 + 2 ] x + ( 11 y )2 x + ( 22 y )2 x + ( 33 y )2 7 4x 8 5x 9 6x b (s )[ 2 + 2 + 2 ]} x + ( 11 y )2 x + ( 22 y )2 x + ( 33 y )2 +
11 y
j =1
3. Analysis of multiple parallel cracks
fxx (x , y, s, )] d
yy (x , y , s ) =
j =1
where fij (x , y , s, ), i {x , y , d}, j {x , y} , i, i {1, 2, ...,9} and {1, 2, ...,15} are given in Appendix B. All integrals in Eq. (19) i, i decay exponentially as | | ; therefore, integrals may be calculated numerically. It can be seen from Eq. (19) that the stress and electric displacement components exhibit the familiar Cauchy-type singularity at the locations of electro-mechanical dislocation.
y, s )
e x c440 bx ( s ) 4
j =1
fdy (x , y , s , )] d
The stress fields and electric displacements components (17) by view of Eq. (18) and after very lengthy analysis, lead to xx (x ,
[
e xc 440 + {bx (s )[ 21 13 2 x +( x b y (s )[ 2 4 13 2 + x + ( 11 y ) x b (s )[ 2 7 13 2 + x + ( 11 y )
(18)
Non sin gular Part
3 3 3 e xc 440 bx (s ) b5j Aj1 e j y + b y (s ) b5j Aj 2 e j y + b (s ) b5j Aj3 e j y 4
Dy (x i (p), yi (p), s ) =
]
N k=1
ak
1 1
11 [kdyik (p , q, s ) bxk (q, s )
12 +kdyik (p , q , s ) byk (q , s )
=0
x x + 2 2 11 2 + 2 3 12 2 ] x + ( 22 y ) x + ( 33 y ) 22 y 33 y + 26 12 ] 2 x + ( 33 y ) 2 22 y ) 22 y 9 12 33 y + 2 ]} 2 x + ( 33 y ) 2 22 y )
13 + kdyik (p , q , s ) b k (q , s )] dq
p
i = 1, 2, ...,N ,
1
1 (21)
where bxk (q, s ) , byk (q , s ) and b k (q , s ) are the Laplace transforms of the dislocation density functions on the face of k-th crack, kiklm, l = 1, m = 1, 2, 3, i , k = x , y, d are coefficients of bx (s ) , by (s ) and b (s ) in Eq. (19). The kernels in Eq. (21) exhibit Cauchy type singularity p and may be expressed as for i = k as q
fdx (x , y , s , )] d
130
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
akl, 1i (q, s ) + p q
1l kryii (p , q , s ) =
k, l
q)m
akl, mi (q, s )(p m=0
{1, 2, 3}, r
(22)
{x , y , d }
The coefficients of singular terms akl, 1i (q, s ) are obtained by means of the Taylor series expansion of x i (q) and yi (q) in the vicinity of q and are
a11,
1i
= 0, a12,
a21,
1i
=
a31,
1i
c440 2 li
1i
c440 2 li
=
3 , j=1 j j+6
= 0, a32,
1i
c440 2 li
=
3 , j =1 j+3 j+3
a22,
1i
a13,
= 0, a23,
3 , j = 1 j + 3 j + 12
1i
a33,
=
1i
c440 2 li
Fig. 2. Geometry of one crack subjected to uniform normal and electric impact traction.
3 j=1 j +6 j +3
=0 1i
=
c440 2 li
3 , j = 1 j + 6 j + 12
c440 2 ai
kIR (s ) =
[gyi (1, s )
(23)
kIIR (s ) =
c440 g (1, 2 ai xi
The integral equation (21) must be solved under the following single-valued conditions out of each crack faces:
kDR (s ) =
c440 2 ai
1 1
bkj (q, s ) dq = 0, k
{x , y , d}, j
{1, 2, ...,N }
(24)
By observing that the fundamental solution of the stress fields and the electric displacements has a square root singularity at crack tips, the unknown dislocation densities on the surface of impermeable cracks, are taken as (Delale and Erdogan, [34])
gki (q , s ) e
bki (q, s ) =
1
q2
x
, k
{x , y , d},
1 < q < 1, i
1h kmyij (p , q, s )
gij (q, s ) 1
q2
n
m
r
2r
0
= lim r
yy (r ,
{1, 2, ...,N }
1
r=1
c440 2 ai
{x , y , d}, h
, s ), kII (s ) = lim r
0
[gyi ( 1, s )
3 j = 1 j + 3 j + 12 3 j =1 j+3 j+3
+ g i (1, s )
3 ], j = 1 j + 6 j + 12
+ g i ( 1, s )
3 ], j =1 j+6 j+3
3 , j=1 j j +6 3 j = 1 j + 3 j + 12
3 ], j = 1 j + 6 j + 12
+ g i ( 1, s )
{1, 2, ...,N }
kjki (t )
ln 2 t
M
Hm kjki ( m=1
ln 2 m ), j = I , II , D , k = R, L , t > 0 t
M
Hm = ( 1) 2 + m
min( M , m) 2
M
n 2 (2n)!
M n = [0.5(m + 1)] ( 2
n ) ! n ! (n
1) ! (m
n) !(2n
2r
xy (r ,
m)!
. (30)
In the above equation [·] signifies the integer part of the quantity.
(26)
{1, 2, 3}
(29)
where M is a chosen positive even number, and Hm is given by
1h er kmyij (pl , qr , s ) gij (qr , s ),
4. Results and discussion This section is divided into two main parts. The first part deals with the verification of the resulting analytical solutions while the second part tackles the examination of the effect of some important parameters such as loading conditions, the size of the cracks, geometrical parameters, material properties and interactions of cracks on the field intensity factors. In many engineering applications, the piezoelectric materials are loaded by combining electrical and mechanical loads. Therefore, the electromechanical coupling factor is defined by D = e150 D0 / 0 110 to combine the normal traction 0 and electrical displacement D0 . In the following examples, the dynamic field intensity factors is normalized by k 0 = 0 l for normal load, k 0 = 0 l for shear load and k 0D = e330 l / c330 for electric load, where l is the half length of the crack. Also, the time is normalized by t 0 = l/ c440 / 0 . In the numerical analysis, the piezoelectric material properties for PZT-6B are shown in Table 1. The preceding formulation lets the analysis to be performed for any number of parallel cracks subjected to mixed mode mechanical and electrical impacts. The number of points for the inversion of Laplace transform, using Stehfest’s method, is M = 8. In the first arrangement of
, s ), kD (s )
2r Dy (r , , s )
0
kDL (s ) =
1, s )
3 , j=1 j j+6
(28)
where the collocation points are chosen as (2l 1) (r 1) qr = cos[ n 1 ], r {1, 2, ...,n} , pl = cos[ 2(n 1) ], l {1, 2, ...,n 1} , er = 0.5 for r = 1, n and er = 1 for 1 < r < n . The modes I, II stress intensity factors and the electric displacement intensity factor are defined as
kI (s ) = lim
c440 g ( 2 ai xi
n
dq =
[gyi ( 1, s )
3 ], j=1 j+6 j+3
+ g i (1, s )
The numerical inversion of the Laplace transform is carried out based on the algorithm developed by Stehfest [33] as follows:
By replacing Eq.(25) into Eqs. (21) and (24), and using the Lobatto–Chebyshev integration formula, The discretization singular integral equations lead to
1
[gyi (1, s )
kIIL (s ) =
i
(25)
1
c440 2 ai
kIL (s ) =
s)
3 j=1 j +3 j+3
(27)
where r is the distance from the crack tip, = 0 exhibits the right side and = exhibits the left side crack tips, respectively. By replacing Eqs. (22) and (25) into (21) and resultant equations into (27) the field intensity at cracks tips for the i-th crack yield
Table 1 The relevant material properties. c110 × 1010
16.8
( ) N m2
c130 × 1010 6
( ) N m2
c330 × 1010
16.3
( ) N m2
c440 × 1010
2.71
( ) N m2
e310
( ) C m2
0.9
131
e330 7.1
( ) C m2
e150
4.6
( ) C m2
110
36
× 10
10
( ) C m2
330
34
× 10
10
( ) C Vm
( ) kg m3
7550
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
Fig. 3. The variation of mode-I DSIFs versus dimensionless time for an isotropic piezoelectric plane.
Fig. 4. The variation of the normalized EDIF with t / t 0 for different electromechanical coupling factor for an isotropic piezoelectric plane.
Fig. 5. The variation of the normalized EDIF with t / t 0 for different electromechanical coupling factor for an FGP plane.
the crack (Fig. 2), a straight crack with length 2l = 2(cm) which is located along the x-axis under uniform normal and electric impact traction applied on the crack face is displayed. The validity of the analysis is verified by comparing our results with those available in the literature. In the first example, an isotropic piezoelectric plane ( l = 0 ) containing a crack is considered. The effect of the constant normal traction without electrical loading on the crack face on the mode-I dynamic stress intensity factors (DSIFs) is studied. It
can be seen from Fig. 3 that our results are in excellent agreement with previous investigations (Ueda [21]). As the second verification of results, problems of an isotropic piezoelectric and an FGP plane weakened by a crack with various electromechanical coupling factors are shown in Figs. 4 and 5, respectively. The results are compared with those obtained by Garcı́a-Sánchez et al. [23]. As we may observe, our results are in excellent agreement with the previous investigation. As can be seen, the normalized electric 132
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
Fig. 6. The effect of l on the normalized electric displacement intensity factors for
D
= 1.
Fig. 7. The variation of mode-I DSIFs versus dimensionless time for an FGP plane without electric loading.
Fig. 8. The effect of
D
on the normalized mode-I DSIFs under uniform combining electrical and normal mechanical impact traction for l = 1.
displacement intensity factor (EDIF), kD /k 0D , is only weakly dependent on the dimensionless time, which is a consequence of the quasi-static assumption of the electrical fields. This phenomenon is observed with the Garcı́a-Sánchez et al. [23] and Bagheri [32]. Also, a comparison of Figs. 4 and 5 reveals that a higher value of kD /k 0D for an FGPM is obtained. In the remaining of this section, more examples are presented to demonstrate the applicability of the procedure. The variation of the normalized EDIFs against dimensionless time
(i.e., t / t 0 ) for three values l = 0.25, 0.5, 1.0 is depicted in Fig. 6. From Fig. 6, we observe that the normalized EDIFs, kD /k 0D , for the left tip of the crack decreases as the l increases, while for the right tip R, due to enhancement of the shear modulus of plane in the positive x-axis direction, an increase for kD /k 0D is observed. As can be seen, the normalized EDIFs are nearly independent of the normalized time for impermeable case. The results showed in Fig. 7, illustrate the influence of the 133
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
Fig. 9. Variation of mode-II DSIFs under uniform combining electrical and normal mechanical impact traction for l = 1.
length is taken to be 2l = 2(cm) . According to Fig. 7, the trend of variation of kI / k 0 is similar to kD /k 0D for various FGPM constant. The normalized mode I DSIFs rises rapidly with time, reaching a peak, then decreases in magnitude to reach static value. In other words, at t = the effect of the transient loading on the crack surface disappears, which causes the SIFs to become independent of time and converge into static value in an infinite FG plane. Figs. 8 and 9 display the effect of applied electric field of 0 < D < 2 on the non-dimensional mode I and II DSIFs for right tip of the crack with variation of t / t 0 . It is interesting to note that the normalized DSIF kIR/ k 0 and kIIR/ k 0 increase quickly at first with time, to reach their maximum value and then get approximately stable. Finally, the DSIF approaches the corresponding static value. In conclusion, in an FGP plane with a crack lying along the x-axis under uniform combining
Fig. 10. Geometry of one crack subjected to uniform shear impact traction
dimensionless non-homogeneity parameter l on the variation of the normalized mode-I DSIFs versus the t / t 0 . The electro-mechanical coupling factor of the FGP plane is supposed as D = 0 , while the crack
Fig. 11. Variation of mode-II DSIFs under impact shear traction for
D
= 0.
Fig. 12. Geometry of two aligned cracks subjected to uniform normal and electric impact traction. 134
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
respectively. The interaction between two cracks in the FGP plane with non-homogeneity parameter l = 1 and d/ l = 1.05 is studied in Figs. 13–15. It is worth mentioning that the lack of symmetry produces mode-II DSIFs even where cracks are subjected to normal traction. As it might be observed, the DSIFs for the crack tips L1 and R2 at large values of t / t 0 , are higher than L2 and R1, which are attributed to the stronger interaction. Also, compared with the single crack problem, higher values of the DSIF are observed due to the interaction effect between cracks. Also, the DSIF is the lowest at the crack tip, which is the farthest crack tip located in the less stiffer zone (see Fig. 16). In the next example, the medium contains two stacked identical cracks with lengths 2l shown in Fig. 17. The cracks lines are located on the vertical distances with d/ l = 0.1, 0.15. The variation of the normalized mode-I DSIFs versus dimensionless time t / t 0 , for D = 0.0 and l = 1.0 , are shown in Fig. 18. Under normal loading, the interaction effect is weaker and produces a shielding effect, thus DSIFs of two cracks are less than that of DSIFs of one crack with the same conditions. Two parallel shifted cracks under normal impact traction are shown in Fig. 19. The dimensionless modes I and II DSIFs versus dimensionless time are plotted in Figs. 20 and 21. In this case, non-homogeneity parameter and center distance are l = 1.0 and d/ l = 0.2 , respectively. We observed that, the magnitudes of DSIFs for the crack tip, located in a stiffer zone are higher than those other tips. It is interesting to note that, the shielding effect phenomenon is not seen in this case.
Fig. 13. Geometry of two aligned cracks subjected to uniform shear impact traction.
electrical and normal mechanical impact traction, the influence of the electromechanical coupling factor may be significant on the mode I and II DSIF. As can be seen, a considerable increase in kIR/ k 0 and kIIR/ k 0 is observed as D increases. In the next example, we consider a crack with length 2l as shown in Fig. 10. The calculated DSIFs at the crack tips are normalized by k 0 = 0 l for uniform crack face shear loading. According to Fig. 11, the trend of variation of kII / k 0 is similar to kI / k 0 and kD /k 0D for various FGPM exponents. As examples of multiple cracks, we consider two aligned cracks under normal and shear impact traction as shown in Figs. 12 and 13,
Fig. 14. Variation of mode-I DSIFs under impact normal load for two aligned cracks.
Fig. 15. Variation of mode-II DSIFs under impact normal load for two aligned cracks.
135
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
Fig. 16. Variation of mode-II DSIFs under impact shear load for two aligned cracks.
Fig. 19. Geometry of two parallel shifted cracks subjected to uniform normal and electric impact traction.
Fig. 17. Geometry of two stacked identical cracks subjected to uniform normal impact traction.
singular integral equations in Laplace domain, which are solved by numerical Laplace technique developed by Stehfest [33] to obtain the dislocation density on the crack face. The results reported in Section 4 indicate that the non-homogeneity parameters significantly influence the fracture behavior of an FGPM plane. The mode I DSIFs can be reduced to two stacked cracks compared with the single crack under normal load. The time of occurrence of the maxima of DSIF depends upon shear wave velocity of material, and it decreases as the wave velocity increases.
5. Concluding remarks The work presented in this article is directed toward developing analytical methods based on the DDT for a mixed-mode fracture analysis of FGPMs plane weakened by multiple parallel cracks. The solution of edge dislocation is first obtained in plane. The material properties are assumed to change continuously along the x- axis, and the crack faces are subjected to uniform normal, shear and electric impact loads. The Fourier transforms is applied to reduce the problem to a system of
Fig.18. Interaction of mode I of two stacked cracks under impact normal load. 136
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
Fig. 20. Interaction of mode I for two parallel shifted cracks under impact normal load.
Fig. 21. Interaction of mode II for two parallel shifted cracks under impact normal load.
Appendix A The expressions in Eqs. (10) and (12) are as follows:
r1 =
8 2
r2 = (
2
r3 =
9(
r4 =
5 6(
H2 = i
+i )+ 2)[
2)
+
H3 = i (
5
H4 = i
+
2
H6 =
H8 =
7)
2
9(
2)
+i (
+i )+
6 7
1 9
5 7
+
8s
2/ c 2
+(
2
i
)(
3)
+
2 8)
+i (
2 7
+
2 9)]
s 2 /c 2
+
5
s 2/ c 2 2)
7 (i
H7 = i (
i) )( + (1 + i) )
1
+
H5 = i
(1
1 8
5 7(
3 (i
2
(
)( s 2/ c 2 + (i
i
r5 = (i H1 =
2 5
i )
5
+
i
7)
+
+
7
2)
The functions Aij , i, j
{1, 2, 3} in Eq. (16) are as follows:
137
2 7
+
3 8
+
9)
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
A11 =
i 2 (m2 p1 + n2 p2 ) i 3 (m3 p1 + n3 p2 ) + (m3 n2 m2 n3) p3 2 3 m1 n2) 1 2 + n3 3 (m1 1 m2 2) + m3 3 (n2 2 n1 1)]
p3 [(m2 n1
i m3 n2 + i m2 n3 + n3 2 n2 3 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 n1 2) i m3 n2 7 + i m2 n3 7 + m2 3 m3 2 m1 (n3 2 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 n1 2)
A12 =
m1 (n3 2
A13 =
i 1 (m1 p1 + n1 p2 ) i 3 (m3 p1 + n3 p2 ) + (m3 n1 m1 n3) p3 1 3 m1 n2) 1 2 + n3 3 (m1 1 m2 2) + m3 3 (n2 2 n1 1)]
A21 =
A31 =
p3 [(m2 n1
A22 = A23 =
i m3 n1 i m1 n3 n3 1 + n1 3 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 n1 2) i m3 n1 7 i m1 n3 7 + m3 1 m1 3 m1 (n3 2 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 n1 2)
i 1 (m1 p1 + n1 p2 ) i 2 (m2 p1 + n2 p2 ) + (m2 n1 m1 n2) p3 1 2 m1 n2) 1 2 + n3 3 (m1 1 m2 2) + m3 3 (n2 2 n1 1)]
p3 [(m2 n1
A32 =
m1 (n3 2
A33 =
i m2 n1 + i m1 n2 + n2 1 n1 2 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 i m2 n1 7 + i m1 n2 7 m2 1 + m1 2 m1 (n3 2 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 m1 (n3 2
n1 2) n1 2)
where
p1 =
1 6 p2
4 5
=
5 6
+
1 8 p3
2 6
=
+
4 8
The integrands are appeared in Eq. (17) are
fxx (x , y , s , ) =
ie x c440 1 b x (s ) 4
=
ie x c440 1 b x (s ) 4
= = =
[ [
ie x c440 1 b x (s ) 4
[
ie x c440 1 b x (s ) 4
[
ie x c440 1 b x (s ) 4
[
3
3
b1j Aj1 e
jy
3
+ b y (s )
b1j Aj2 e
j=1
jy
+ b (s )
j=1
3
3
b2j Aj1 e
jy
b2j Aj2 e
jy
+ b (s )
j=1
3 jy
b3j Aj2 e
jy
+ b (s )
j =1
3 jy
b4j Aj2 e
jy
+ b (s )
j=1
3 jy
)
jy
b5j Aj3 e
jy
3
+ b y (s )
b5j Aj2 e
j=1
] ei x fdy (x, y, s,
b4j Aj3 e j=1
3
b5j Aj1 e
)
jy
3
+ b y (s )
j=1
] ei x fdx (x , y, s,
b3j Aj3 e j =1
3
b4j Aj1 e
)
jy
3
+ b y (s )
j=1
] ei x fxy (x, y, s,
b2j Aj3 e j=1
3
b3j Aj1 e
)
jy
3
+ b y (s )
j=1
] ei x fyy (x, y, s,
b1j Aj3 e j=1
jy
+ b (s )
j=1
j=1
] ei x
Appendix B The singular part of the integral appeared in Eq. (19) are:
fxx (x , y , s, ) d =
e xc440 4
(b x (s )
3 e j =1 j j
jj | | y
3 e j=1 j+6 j
i sgn( ) b (s ) fyy (x , y , s , ) d =
e x c440 4
(bx (s )
3 e j=1 j j+3
jj | | y
i sgn( ) b (s ) fxy (x , y , s , ) d =
e x c440 4
fdx (x , y, s, ) d =
e x c440 4
3 e j=1 j j+9
( i sgn( ) bx (s )
e x c440 4
( bx ( s )
jj | | y
3 e j = 1 j j + 12
i sgn( ) b (s )
3 e j=1 j+3 j+3
jj | | y
3 e j =1 j+3 j+6
jj | | y
3 e j=1 j +3 j+9
jj | | y
jj | | y ) e i x d
b y (s ) jj | | y ) e i x d
3 e j = 1 j + 3 j + 12
+ i sgn( ) by (s )
3 e j = 1 j + 6 j + 12
jj | | y ) e i x d
where 1
=
2
=
3
=
4
=
(p1 m22 + p2 n22) p3 [
11 22 (m11 n22
p3 [
11 22 (m11 n22
p3 [
11 22 (m11 n22
(p1 m11 + p2 n11)
11 n22
(p1 m33 + p2 n33)
11
11
11 33 (m33 n11
+ (m22 n33
m11 n33) + 33
11 33 (m33 n11
(p1 m22 + p2 n22)
m22 n11) +
33
11 33 (m33 n11
+ (p1 m33 + p2 n33)
m22 n11) +
(p1 m11 + p2 n11)
m33 (
22
m22 n11) +
+ (m33 n11
m11 n33) + 22
+ (m11 n22
m11 n33) +
(m33 n22 m22 n33 n33 22 + n22 33) + m22 ( 33 n11 11 n33) + m11 (
22 n11)
22 n33
jj | | y +
jj | | y ) e i x d
b y (s )
jj | | y
3 e j=1 j+6 j+9
b (s ) fdy (x , y, s, ) d =
3 e j =1 j+6 j+3
3 e j=1 j +6 j +6
b (s )
jj | | y+
jj | | y ) e i x d
+ i sgn( ) by (s )
3 e j =1 j j+6
( i sgn( ) bx (s )
3 e j=1 j+3 j
+ i sgn( ) b y (s )
m33 n22 ) p3 22 33 (m22 n33
m11 n33) p3 22 33 (m22 n33
m22 n11) p3 22 33 (m22 n33
22 33
m33 n22)] 11 33
m33 n22)] 11 22
m33 n22)]
33 n22 )
138
jj | | y+
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
5
=
6
=
7
=
8
=
9
=
j
=
m33 (
11 n22
22 n11)
(m11 n33 m33 n11 n11 33 + n33 11) + m22 ( 33 n11 11 n33) + m11 (
22 n33
33 n22 )
m33 (
11 n22
(m22 n11 m11 n22 n22 11 + n11 22 ) 22 n11) + m22 ( 33 n11 11 n33) + m11 (
22 n33
33 n22 )
m33 (
11 n22
(m33 n22 7 m22 n33 7 22 n11) + m22 ( 33 n11
m22 33 + m33 11 n33) + m11 (
22 n33
33 n22 )
m33 (
11 n22
(m11 n33 7 m33 n11 7 22 n11) + m22 ( 33 n11
m33 11 + m11 33) 11 n33) + m11 ( 22 n33
33 n22 )
m33 (
11 n22
(m22 n11 7 m11 n22 7 22 n11) + m22 ( 33 n11
m11 22 + m22 11) 11 n33) + m11 ( 22 n33
33 n22)
1 mjj jj
3
j+3
=
1
j+6
=
jj
j+9
=
7 jj
j + 12
=
mjj =
njj =
11
=
22
=
33
=
11
=
5 njj jj ,
4 mjj jj
+ mjj + +
7 mjj
4 8 jj
(
5 6
+
(
+
2 jj 5
11
+
2 8)
2 mjj jj
+
3
7) jj
j = 1, 2, 3 2 7
3 jj
+ (
3 8 5 7
+
(1 i 3 ) p11 3 × 22/3 (q11 + r11)1/3
11
=
+
+(
p11 =
2 7 3
3 9 2 9) jj
, j = 1, 2, 3
6 (2 7
2 2(
5
+
7)
(1 + i 3 )(q11 + r11)1/3 6 × 21/3 +
3 6)
5
+
7
)2
2
2 7( 5
+
+ +
4 8 2 11
3
4( 5
7)
+
2
+
7)
+
+
4 3 9
8( 4 3
2 6
+
9
+2
6 3 7
2 2
+ 1)
+
8 3
22
2 6
3 9 2 6 22
q11 = 2
3 11
r11 =
3 2 4p11 + q11
9
+ + 2 2 9
4 8
33
+
, j = 1, 2, 3
4 8
=
2 7
(1 + i 3 ) p11 ( 1 + i 3 )(q11 + r11)1/3 + 2/3 1/3 3 × 2 (q11 + r11) 6 × 21/3
3
2 7
2 9 ) jj
+
21/3p11 (q + r11)1/3 + 11 1/3 3(q11 + r11) 3 × 21/3
3
9
5 7
+
11
4
+2
j = 1, 2, 3
8 njj jj ,
6 7
+
3
2 5
+(
+
j = 1, 2, 3
j = 1, 2, 3 9 njj ,
6 mjj jj
5
j = 1, 2, 3
6 njj jj , 7 njj ,
22 )
11 22
+ 27
33
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Mixed mode transient analysis of functionally graded piezoelectric plane weakened by multiple cracks S. Azizi, R. Bagheri
T
⁎
Department of Mechanical Engineering, Mechatronics Faculty, Karaj Branch, Islamic Azad University, Karaj, Alborz, Iran
ARTICLE INFO
ABSTRACT
Keywords: Transient loading Functionally graded piezoelectric materials Dislocation technique Mixed mode stress intensity factors
In this paper, a functionally graded piezoelectric (FGP) plane weakened by multiple parallel cracks is investigated under mixed mode mechanical and electrical impacts. The system contains Volterra type glide and climb edge dislocations. The material properties of the system are assumed to vary exponentially along the xaxis. At first, the stress field and electric displacement in FGP plane are derived by using a single dislocation. Then, a system of singular integral equations with a simple Cauchy kernel of the transient dynamic problem is obtained by determining a distributed dislocation density on the crack surface and using the Fourier and Laplace integral transforms, which are solved numerically using the Lobatto-Chebyshev integration formula. The dynamic field intensity factors are determined through numerical Laplace inversion and the distributed dislocation technique (DDT). Finally, various examples are provided to investigate the influence of the material properties, electromechanical coupling factor, loading conditions and cracks arrangement on the dynamic fracture behavior of the interacting cracks.
1. Introduction As the application of piezoelectric materials possessing passive, active and elastic properties have risen sharply, a great number of analytical, numerical and experimental studies on this type of materials are conducted in the field of engineering sciences [1,2]. The physical properties of functionally graded piezoelectric materials (FGPMs) are non-homogeneous and the electro- mechanical parameters may vary continuously in particular directions. The main advantage of FGPMs over conventional composites is the absence of any interface. It is generally well-known that FGPMs are very brittle and capable to fracture with low toughness. Thus, the investigation of the fracture analyses of this type of materials under mechanical and electrical loadings is a major task. Most of the studies, however, are related to static [3–10] or quasi-static conditions [11–17]; but it may be very important to study the response of cracks under dynamic loadings. Dynamic loadings are mainly categorized into two groups including harmonic loading [18–20] and impact loading. Impact loads applied on the cracked structures lead to catastrophic failure of the system. The problem of a finite crack in a strip of FGPM was studied by Ueda [21]. The electro-elastic properties of the FGPM are assumed to vary continuously along the thickness of the strip which is under in-plane mechanical and electrical impact loadings. Wang and Mai [22]
⁎
considered the mixed mode crack problem in an infinite functionally graded material. In this investigation, the influence of such factors including material inhomogeneity constants and crack spacing on the stress intensity factors, both at transient state and steady state are studied. The problem for a finite crack in an infinite piezoelectric solid under an impact mechanical loading, or an impact electrical loading or a combination of both is studied by Garcı́a-Sánchez et al. [23]. The effects of the electric field, material properties and electromechanical coupling coefficient were studied on stress and electric displacement intensity factors. Feng et al. [24] obtained the dynamic response of an interfacial crack between two dissimilar magneto-electro-elastic layers under magnetic, electrical and mechanical impact loadings. Their investigation showed that a negative magnetic loading (or electrical loading) is generally prone to inhibit the crack extension rather than a positive one for a magnetically (or electrically) impermeable interfacial crack. The in-plane crack problem of a magneto-electric material under in-plane mechanical, electric and magnetic impacts was investigated by Zhong et al. [25]. The Laplace and Fourier transforms are applied to reduce the associated initial and mixed-boundary value problem to dual integral equations, and then to singular integral equations with Cauchy kernel. The effects of applied magnetics and electric impacts on crack growth were investigated. Liu and Zhong [26] focused on the dynamic analysis of two collinear dielectric cracks in a piezoelectric material
Corresponding author. E-mail address: [email protected] (R. Bagheri).
https://doi.org/10.1016/j.tafmec.2019.02.014 Received 27 October 2018; Received in revised form 24 January 2019; Accepted 18 February 2019 Available online 19 February 2019 0167-8442/ © 2019 Elsevier Ltd. All rights reserved.
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
under the action of in-plane electro-mechanical impacts. They found that the positive electric field enhances the dynamic dielectric crack growth and a negative one impedes the dynamic dielectric crack growth in a piezoelectric solid. Rojas-Díaz et al. [27] analyzed the transient response of the cracked composite materials made of piezoelectric and piezomagnetic phases. They investigated the effect of the combined application of electric, magnetic and mechanical loads on the dynamic field intensity factors. The problem of two collinear mixed-mode limited-permeable cracks embedded in an infinite medium made of an FGPM with crack surfaces subjected to electro-mechanical loadings is investigated by Jamia et al. [28]. They studied the effects of the interaction of two cracks, material gradient parameter describing FGPMs and lattice parameter on the mechanical stress and electric displacement field near crack tips. In accordance with the above studies, there is no promising examination regarding the multiple cracks interaction with arbitrary arrangement in the piezoelectric materials. The distributed dislocation method has been introduced as a powerful tool to obtain the field intensity factors in the materials weakened by multiple cracks. The transient dynamic stress intensity factor for multiple curved cracks under impact loadings is studied by Ayatollahi and Monfared [29]. By means of the Cagniard-de Hoop method, the problem was solved and dynamic SIF was obtained. Analysis of multiple interacting cracks in exponentially graded layers under mode III transient deformation also in orthotropic layers under mixed mode transient loadings were investigated by Vafa et al. [30], Vafa and Fariborz [31], respectively. Bagheri [32] provided the calculation of the field intensity factors for multiple horizontal cracks in a piezoelectric half-plane under an anti-plane transient loading based on the distribution of screw dislocations. In this paper, we consider the dynamic problem of multiple parallel cracks in an FGPM under mixed-mode time-dependent loading. The problem is solved under assumption of impermeable boundary condition and the material properties of the system are assumed to change exponentially along the x-axis. The integral transform and distributed dislocation with time-dependent Burgers vector are used to reduce the problem to Cauchy-type singular integral equations, which is solved numerically by using the Lobatto–Chebyshev collocation method. A numerical Laplace transform inversion technique, described by Stehfest [33], is then used to obtain the dynamic field intensity factors at crack tips. The solutions are in agreement with those available in literature. The main objective of the paper is to study the effect of the time variation, the gradient of the material property and cracks interactions on the dynamic field intensity factors.
the FGPMs in two dimensional elasticity undergoing in-plane deformation are as follows:
y, t ) = c11 (x )
u x
yy (x , y , t ) = c13 (x )
u x
xx (x ,
xy (x , y, t ) = c44 (x )(
u y
Dx (x , y , t ) = e15 (x )(
u y u x
Dy (x , y , t ) = e31 (x )
+ c13 (x )
v y
+ e31 (x )
y
+ c33 (x )
v y
+ e33 (x )
y
+
v ) x
+
v ) x
+ e15 (x )
x
11 (x ) x
+ e33 (x )
v y
33 (x ) y
(1)
where xx , yy and xy are the in-plane stress tensors, Dx and Dy are the in-plane electric displacements. Also, c11 (x ) , c13 (x ) , c33 (x ) and c44 (x ) are the elastic modules, e31 (x ) , e33 (x ) and e15 (x ) are the piezoelectric parameters, and 11 (x ) and 33 (x ) are the dielectric permittivity of the FGPMs. To make the analysis tractable, the material properties of the FGPM plane are supposed to be one-dimensionally dependent as:
c11 (x ) = c110 e x , c13 (x ) = c130 e x , c33 (x ) = c330 e x , c44 (x ) = c440 e x , e31 (x ) = e310 e x , e33 (x ) = e330 e x , e15 (x ) = e150 e x , x x 11 (x ) = 110 e , 33 (x ) = 330 e
(2)
where cij0 are the elastic modules, eij0 are the piezoelectric parameters, and ij0 are the dielectric permittivity at x = 0 and is a positive or negative constant. By neglecting body forces and electric charge density, the equations of motion and Maxwell equation for the FGPMs are in the following forms:
c110
2u
x2
+ (c130 + c440)
2v x y
u x
v y
+ (c110 c440
2v x2
+ c130
+ (c130 + c440)
2u x y
2u
+ c440
+ e310 + c330
y2 y
+ e150
+ e150 2
110 x 2
2
330 y 2
+ e150
2v x2
+ e330
x y
2u 0 t2 ,
)= 2v y2
2
+ (e310 + e150)
2v y2
+ e150 (
+ e330
x2
+ (e310 + e150) u y
2
y2
+ c440 (
u y
+
2v 0 t2 ,
=
x
2
+
v ) x
v ) x
.
2u x y
110 x
= 0. (3)
In the above equations, it is assumed that the mass density of the medium varies exponentially along the x-axis with (x ) = 0 e x , where 0 is the density at x = 0 . Let an edge dislocation with time-dependent Burgers vector be located at the origin coordinates. The dislocation cut created along the positive direction of the x -axis, while glide, climb and electric dislocation of the edge dislocation are represented bx (t ) , by (t ) and b (t ) , respectively. Thus, the electromechanical conditions at the dislocation path can be described as:
2. Statement of the problem We consider a plane made of functionally graded piezoelectric materials (FGPMs) weakened by the electro-elastic Volterra types climb and glide edge dislocations shown in Fig. 1. The problem is formulated in the framework of linear piezoelectricity. The constitutive relations of
u (x , 0+, t ) v (x , 0+, t )
u (x , 0 , t ) = bx (t ) H (x ), v (x , 0 , t ) = by (t ) H (x ),
(x , 0+, t )
(x , 0 , t ) = b (t ) H (x ),
yy (x ,
0+, t ) =
yy (x ,
0 , t ),
xy (x ,
0+, t ) =
xy (x ,
0 , t ),
Dy (x , 0+, t ) = Dy (x , 0 , t ),
(4)
where H (·) is the Heaviside step function. It is of importance that the electric impermeability condition is adopted by introducing the jumps of electric potential across the dislocation path. It is convenient to solve the problem in the half-plane ( y 0 ), which necessitates decomposition of the problem into symmetric and anti-symmetric parts. The corresponding boundary conditions (4) for the symmetric problem is reduced to
v (x , 0, t ) =
Fig. 1. Functionally graded piezoelectric plane weakened by electro-elastic dislocations. 128
b y (t ) 2
H (x ),
(x , 0, t ) =
b (t ) 2
H (x ),
xy (x ,
0, t ) = 0, (5)
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
And for the anti-symmetric problem the conditions are
bx ( t ) H (x ), 2
u (x , 0, t ) =
yy (x ,
0, t ) = 0,
where
Dy (x , 0, t ) = 0,
2u
u x
+ (c110 0s
2u
+ e310
y
+ (e310 + e150 )
y2
1
2
x y 2
)
3 2u
+ (c130 + c440)
x2
+ c440 ( 0s
x y
v y
+ c130
2u
+ c440
2v
u y
v x
+
2v
+ c330
x y
) + e150
+ e150
y2
2
+ e330
x2
2
y2
x
= 0, 2
110
2
330
x2
u y
+ e150 (
2v
+ e150
y2
v x
+
+ e330
x2
)
110
2v
y2
+ (e310 + e150)
d2U dy2
+ [i (c130 + c440) + c130] 2) c
+ [(i d2V c330 2 dy
0s
110
2] U
dV dy
+ [i (c130 + c440) +
(7)
d2 330 dy 2
d2V dy 2
+ e330
+ [i (e310 + e150) + e310]
+
d2 e330 2 dy
2) e 150 V
dU dy
2] V
+(
i )
110
Aj ( , s ) e
=0
V ( , y, s ) =
j=1 3
( , y, s) =
(9)
j =1
4 j
r1
where mj =
+ r2
r4 3j 2 j
=
2 j
+ r3
+ r5
,
nj
j
+ mj (
1+ i
[
5
2)
j
+i (
2)
+ (i 5
+
7)] j
3
s 2/ c 2
1
=
2
=
3
=
4
+ 1
3 1
3 1
3
+
2
2
3
21/3p 3(q + r )1/3
,
j
{1, 2, 3}
= 0, +
(q + r )1/3 3 × 21/3
(1 + i 3 ) p 3 × 22/3 (q + r )1/3 (1
i 3 )p
3 × 22/3 (q + r )1/3
+
8
=
330
c440
=
9
=
,
,
H2 H7) + H1 H5 8
2 4 8+ 6
,
,
c130 + c440 , c440
110
c440
3
c110 , c440
=
4
=
c330 , c440
5
=
e310 , c440
6
=
e330 , c440
c440
, c=
0
1 2
(
1 2
(
1 2
(
3
Aj e
jy
j=1 3 j =1
) ei xd
mj Aj e
jy
nj Aj e
jy
3 j=1
) ei x d
) ei xd
(13)
e x c440 2
(
e x c440 2
(
e x c440 2
(
Dx (x , y , s ) =
e x c440 2
(
Dy (x , y , s ) =
e x c440 2
(
y, s ) = y, s) = y, s ) =
3
b1j Aj e
jy
) e i xd
b2j Aj e
jy
) ei x d
b3j Aj e
jy
) ei xd
b4j Aj e
jy
) ei x d
b5j Aj e
jy
) e i xd
j=1 3 j=1 3 j=1 3 j=1 3 j=1
(14)
b1j = i
3
+
1 mj j
+
5 nj j
b 2j = i
1
+
4 mj j
+
6 nj j
b3j =
j
b4j =
7 j
6 × 21/3 6 × 21/3
5
+i +
7 mj 6 mj j
7 nj
i
9 nj 8 nj j
j = 1, 2, 3
(15)
A1 ( , s ) =
(
A2 ( , s ) =
(
A3 ( , s ) =
(
( ) i/ ) [A11 bx (s ) 2 ( ) i/ ) [A21 bx (s ) 2 ( ) i/ ) [A31 bx (s ) 2
+ A12 by (s ) + A13 b (s )] + A22 b y (s ) + A23 b (s )] + A32 by (s ) + A33 b (s )]
(16)
(·) is the Dirac delta function and the functions where Aij , i, j {1, 2, 3} are given in Appendix A. Substituting unknown functions (16) into stress fields and electric displacements (14) yields
( 1 + i 3 )(q + r )1/3 (1 + i 3 )(q + r )1/3
+ i mj + i
The unknown functions Ai ( , s ), i = {1, 2, 3} may be obtained by utilizing boundary conditions (5) and (6), respectively to Eqs. (13) and (14) for the climb and glide dislocations. The expressions for these coefficients in the FGP plane containing both dislocations are
(10)
1
e150 , c440
b5j = i
and d ri, (i = 1, 2, ...,5) are presented in Appendix A. In Eq. (9), functions Aj , (j = 1, 2, 3) are arbitrary unknowns. The characteristic equation and its roots are found to be: 6
=
2
H1 4 )
where
jy.
jy
nj Aj ( , s ) e
7
xy (x ,
jy
mj Aj ( , s ) e
=
yy (x ,
j=1 3
1
xx (x ,
1 , and is the Fourier variable, U , V and are Fourier where i = transforms of the displacement components u , v and electrical potential respectively. Because the stresses, displacement components and electrical potential must vanish as (x 2 + y 2 ) , therefore, the solution to the differential equation (8) is written as follows 3
2 4 8+ 6
c130 , c440
H1 H8 4 + H6 (2H1 6
H5 H8
H5
The stress fields and electric displacements in the Laplace domain are obtained from the Eqs. (1) and (13) as follows:
(8)
U ( , y, s ) =
=
v (x , y , s ) =
d dy
=0 2
H4 H6) + H2 H4 H8
H1 ( H62 + H5 H8)
u (x , y , s ) =
2) e 150
+ (i 0s
+ [i (e150 + e310) + e150] + (i
=
(x , y , s ) =
2) c 440
+ [(i
H62 + H3 (H5 H7
H1 6) + H3 ( H7 4 + H4 6 ) + 8 (H2 H4 2 4 8+ 6
(12)
=0
dU c440] dy
+ 27 3 ,
and the functions Hi , (i = 1, 2, ...,8) are given in Appendix A. By using the inverse Fourier transform, the displacement fields and electrical potential (9) are written as:
where the superscript ∗ denotes the Laplace transform, s is Laplace variable at the time transform domain. Mixed boundary value problem (7) is solved by employing the method of the Fourier transform to the space variable x and is reduced to
c440
1 2
4 H8 + 6 ( 2H6 + H2 H7
=
2u x y
= 0.
x
2,
9
4p 3 + q 2 ,
r=
= 0,
2v
c440
2v
+ (c130 + c440 )
x2
3 3 1
q=2
By using the Laplace transform the time variable in Eq. (3) is eliminated, and it becomes
c110
2 1
p=
(6)
(11) 129
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
xx (x , y , s ) =
e xc 440 [bx (s) 4
3 j y + b y (s ) j = 1 b1j Aj1 e
3 j y + b (s ) j = 1 b1j Aj2 e
3 jy j = 1 b1j Aj3 e ]
=0
+ yy (x , y, s ) =
e x c440 [bx (s ) 4
3 j y + b y (s ) j = 1 b2j Aj1 e
3 j y + b y (s ) j = 1 b3j Aj1 e
e x c440 D y (x , y, s ) = [bx (s ) 4
3 j y + b y (s ) j = 1 b5j Aj1 e
3 jy j = 1 b2j Aj3 e ]
+ =0
f yy (x , y , s, ) d
fxx (x , y, s, ) d e x c440 [bx (s ) xy (x , y, s ) = 4
3 j y + b (s ) j = 1 b2j Aj2 e
3 j y + b (s ) j = 1 b3j Aj2 e
3 jy j = 1 b3j Aj3 e ]
=0
e xc 440 + Dx (x , y , s ) = [bx (s ) 4
3 j y + b y (s ) j = 1 b4j Aj1 e
fxy (x , y, s, ) d
3 j y + b (s ) j = 1 b4j Aj 2 e
3 jy j = 1 b4j Aj3 e ]
+ =0
fdx (x , y , s, ) d
3 j y + b (s ) j = 1 b5j Aj2 e
3 jy j = 1 b5j Aj3 e ]
+ =0
(17)
fdy (x , y, s, ) d
where fij (x , y, s, ), i {x , y , d}, j {x , y} are defined in Appendix A. The singular nature of the kernels fij may be determined by studying the asymptotic behavior of the integrands of integrals in (17). By adding and subtracting the asymptotic expressions of the integrands using asymptotic expressions for large value , we find:
fij (x , y, s, ) = fij (x , y , s, ) + [fij (x , y, s, ) Singular Part
fij (x , y, s, )]
D y (x , y , s ) =
=
[
3
3 jy
b1j Aj1 e
+ by (s )
j= 1
3
b1j Aj2 e
jy
+ b (s )
j =1
b1j Aj3 e
jy
j= 1
] =0
e x c440 33 y 11 y 22 y + {bx (s )[ 2 1 1 + 22 2 + 23 3 ] x + ( 11 y ) 2 x + ( 22 y ) 2 x + ( 33 y )2 2 x x x by (s )[ 2 4 1 + 2 5 2 + 2 6 3 ] x + ( 11 y )2 x + ( 22 y )2 x + ( 33 y ) 2 x x x b (s )[ 2 7 1 + 2 8 2 + 2 9 3 ]} + [fxx (x , y, s , ) x + ( 11 y )2 x + ( 22 y )2 x + ( 33 y )2
3 3 3 e xc 440 bx (s ) b2j Aj1 e j y + b y (s ) b2j Aj2 e j y + b (s ) b2j Aj3 e j y 4
[
j =1
j=1
j= 1
[f yy (x , y, s , )
] =0
xi (p) = xic + pli, yi (p) = yic, 1
f yy (x , y , s , )] d
e x c440 [bx (s ) 4
3 j y + b y (s ) j = 1 b3j Aj1 e
3 j y + b (s ) j = 1 b3j Aj 2 e
3 jy j = 1 b3j Aj3 e ]
x x x e x c440 { bx (s )[ 2 1 7 2 + 2 2 8 2 + 2 3 9 2 ] 2 x + ( 11 y ) x + ( 22 y ) x + ( 33 y ) y y y y y b y (s )[ 24 7 11 2 + 2 5 8 22 2 + 26 9 33 2 ] b (s )[ 2 7 7 11 2 + 2 8 8 22 2 x + ( 11 y ) x + ( 22 y ) x + ( 33 y ) x + ( 11 y ) x + ( 22 y ) 33 y + 29 9 ]}+ x + ( 33 y )2
+
[fxy (x , y , s , ) (x , y , s ) =
e xc 440 bx (s ) 4
[
e xc 440 + { bx (s )[ 2 1 2 x + 4 10 11 y b y (s )[ 2 + x + ( 11 y ) 2 7 10 11 y b (s )[ 2 + x + ( 11 y ) 2 +
[fdx (x , y , s, )
3
b4j Aj1 e j y + b y (s )
j =1
j =1
10 x
( 11 y ) 2 5 11
x2 + (
8 11
x2 + (
b4j Aj 2 e j y + b (s )
3 j =1
x 2 + ( 22 y )2
5 14 x
x 2 + ( 22 y )2 8 14 x
x 2 + ( 22 y )2
+ +
+
x 2 + ( 33 y ) 2
6 15 x
x 2 + ( 33 y )2 9 15 x
x 2 + ( 33 y )2
=0
]
] ]} +
[fdy (x , y, s , )
(19)
p
1,
i
{1, 2, 3, ...,N }
(20)
=0
yy (x i (p), yi (p),
Dx
s) =
b4j Aj3 e j y
N k=1
ak
1 1
11 [k yyik (p , q, s ) bxk (q, s )
12 13 +k yyik (p, q, s ) byk (q, s ) + k yyik (p , q, s ) b k (q, s )] dq xy (x i (p), yi (p),
s) =
N k=1
ak
1 1
11 [k xyik (p , q, s ) bxk (q, s )
12 13 +k xyik (p, q, s ) b yk (q, s ) + k xyik (p , q, s ) b k (q, s )] dq
fxy (x , y , s , )] d 3
+
2
3 15 33 y
where (xic , yic ) and li are the coordinates of the center and half length of the i-th cracks, respectively. Employing the principle of superposition, the singular integral equations are
xy (x , y , s )
=
11 y )
2 14 22 y
]
In the framework of linear theory, the current problem can be treated as the superposition of two subproblem. Subproblem I considers the medium without any cracks under the action of mixed mode mechanical and electrical impacts at far field. While subproblem II concerns with the FGP plane with distributed electro-mechanical dislocations on the x-axis. The crack faces subjected to the distributed electromechanical dislocations cancel out the stress and electric displacement induced by subproblem I. Therefore, the above dislocation solutions are used to construct integral equations for the transient analysis of an FGPM plane weakened by N cracks. The geometry of cracks are presented in parametric form as
e xc 440 22 y 33 y 11 y + {bx (s )[ 2 1 4 + 22 5 + 23 6 ] 2 x + ( 11 y ) 2 x + ( 22 y )2 x + ( 33 y ) 2 5 5x 6 6x 4 4x b y (s )[ 2 + 2 + 2 ] x + ( 11 y )2 x + ( 22 y )2 x + ( 33 y )2 7 4x 8 5x 9 6x b (s )[ 2 + 2 + 2 ]} x + ( 11 y )2 x + ( 22 y )2 x + ( 33 y )2 +
11 y
j =1
3. Analysis of multiple parallel cracks
fxx (x , y, s, )] d
yy (x , y , s ) =
j =1
where fij (x , y , s, ), i {x , y , d}, j {x , y} , i, i {1, 2, ...,9} and {1, 2, ...,15} are given in Appendix B. All integrals in Eq. (19) i, i decay exponentially as | | ; therefore, integrals may be calculated numerically. It can be seen from Eq. (19) that the stress and electric displacement components exhibit the familiar Cauchy-type singularity at the locations of electro-mechanical dislocation.
y, s )
e x c440 bx ( s ) 4
j =1
fdy (x , y , s , )] d
The stress fields and electric displacements components (17) by view of Eq. (18) and after very lengthy analysis, lead to xx (x ,
[
e xc 440 + {bx (s )[ 21 13 2 x +( x b y (s )[ 2 4 13 2 + x + ( 11 y ) x b (s )[ 2 7 13 2 + x + ( 11 y )
(18)
Non sin gular Part
3 3 3 e xc 440 bx (s ) b5j Aj1 e j y + b y (s ) b5j Aj 2 e j y + b (s ) b5j Aj3 e j y 4
Dy (x i (p), yi (p), s ) =
]
N k=1
ak
1 1
11 [kdyik (p , q, s ) bxk (q, s )
12 +kdyik (p , q , s ) byk (q , s )
=0
x x + 2 2 11 2 + 2 3 12 2 ] x + ( 22 y ) x + ( 33 y ) 22 y 33 y + 26 12 ] 2 x + ( 33 y ) 2 22 y ) 22 y 9 12 33 y + 2 ]} 2 x + ( 33 y ) 2 22 y )
13 + kdyik (p , q , s ) b k (q , s )] dq
p
i = 1, 2, ...,N ,
1
1 (21)
where bxk (q, s ) , byk (q , s ) and b k (q , s ) are the Laplace transforms of the dislocation density functions on the face of k-th crack, kiklm, l = 1, m = 1, 2, 3, i , k = x , y, d are coefficients of bx (s ) , by (s ) and b (s ) in Eq. (19). The kernels in Eq. (21) exhibit Cauchy type singularity p and may be expressed as for i = k as q
fdx (x , y , s , )] d
130
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
akl, 1i (q, s ) + p q
1l kryii (p , q , s ) =
k, l
q)m
akl, mi (q, s )(p m=0
{1, 2, 3}, r
(22)
{x , y , d }
The coefficients of singular terms akl, 1i (q, s ) are obtained by means of the Taylor series expansion of x i (q) and yi (q) in the vicinity of q and are
a11,
1i
= 0, a12,
a21,
1i
=
a31,
1i
c440 2 li
1i
c440 2 li
=
3 , j=1 j j+6
= 0, a32,
1i
c440 2 li
=
3 , j =1 j+3 j+3
a22,
1i
a13,
= 0, a23,
3 , j = 1 j + 3 j + 12
1i
a33,
=
1i
c440 2 li
Fig. 2. Geometry of one crack subjected to uniform normal and electric impact traction.
3 j=1 j +6 j +3
=0 1i
=
c440 2 li
3 , j = 1 j + 6 j + 12
c440 2 ai
kIR (s ) =
[gyi (1, s )
(23)
kIIR (s ) =
c440 g (1, 2 ai xi
The integral equation (21) must be solved under the following single-valued conditions out of each crack faces:
kDR (s ) =
c440 2 ai
1 1
bkj (q, s ) dq = 0, k
{x , y , d}, j
{1, 2, ...,N }
(24)
By observing that the fundamental solution of the stress fields and the electric displacements has a square root singularity at crack tips, the unknown dislocation densities on the surface of impermeable cracks, are taken as (Delale and Erdogan, [34])
gki (q , s ) e
bki (q, s ) =
1
q2
x
, k
{x , y , d},
1 < q < 1, i
1h kmyij (p , q, s )
gij (q, s ) 1
q2
n
m
r
2r
0
= lim r
yy (r ,
{1, 2, ...,N }
1
r=1
c440 2 ai
{x , y , d}, h
, s ), kII (s ) = lim r
0
[gyi ( 1, s )
3 j = 1 j + 3 j + 12 3 j =1 j+3 j+3
+ g i (1, s )
3 ], j = 1 j + 6 j + 12
+ g i ( 1, s )
3 ], j =1 j+6 j+3
3 , j=1 j j +6 3 j = 1 j + 3 j + 12
3 ], j = 1 j + 6 j + 12
+ g i ( 1, s )
{1, 2, ...,N }
kjki (t )
ln 2 t
M
Hm kjki ( m=1
ln 2 m ), j = I , II , D , k = R, L , t > 0 t
M
Hm = ( 1) 2 + m
min( M , m) 2
M
n 2 (2n)!
M n = [0.5(m + 1)] ( 2
n ) ! n ! (n
1) ! (m
n) !(2n
2r
xy (r ,
m)!
. (30)
In the above equation [·] signifies the integer part of the quantity.
(26)
{1, 2, 3}
(29)
where M is a chosen positive even number, and Hm is given by
1h er kmyij (pl , qr , s ) gij (qr , s ),
4. Results and discussion This section is divided into two main parts. The first part deals with the verification of the resulting analytical solutions while the second part tackles the examination of the effect of some important parameters such as loading conditions, the size of the cracks, geometrical parameters, material properties and interactions of cracks on the field intensity factors. In many engineering applications, the piezoelectric materials are loaded by combining electrical and mechanical loads. Therefore, the electromechanical coupling factor is defined by D = e150 D0 / 0 110 to combine the normal traction 0 and electrical displacement D0 . In the following examples, the dynamic field intensity factors is normalized by k 0 = 0 l for normal load, k 0 = 0 l for shear load and k 0D = e330 l / c330 for electric load, where l is the half length of the crack. Also, the time is normalized by t 0 = l/ c440 / 0 . In the numerical analysis, the piezoelectric material properties for PZT-6B are shown in Table 1. The preceding formulation lets the analysis to be performed for any number of parallel cracks subjected to mixed mode mechanical and electrical impacts. The number of points for the inversion of Laplace transform, using Stehfest’s method, is M = 8. In the first arrangement of
, s ), kD (s )
2r Dy (r , , s )
0
kDL (s ) =
1, s )
3 , j=1 j j+6
(28)
where the collocation points are chosen as (2l 1) (r 1) qr = cos[ n 1 ], r {1, 2, ...,n} , pl = cos[ 2(n 1) ], l {1, 2, ...,n 1} , er = 0.5 for r = 1, n and er = 1 for 1 < r < n . The modes I, II stress intensity factors and the electric displacement intensity factor are defined as
kI (s ) = lim
c440 g ( 2 ai xi
n
dq =
[gyi ( 1, s )
3 ], j=1 j+6 j+3
+ g i (1, s )
The numerical inversion of the Laplace transform is carried out based on the algorithm developed by Stehfest [33] as follows:
By replacing Eq.(25) into Eqs. (21) and (24), and using the Lobatto–Chebyshev integration formula, The discretization singular integral equations lead to
1
[gyi (1, s )
kIIL (s ) =
i
(25)
1
c440 2 ai
kIL (s ) =
s)
3 j=1 j +3 j+3
(27)
where r is the distance from the crack tip, = 0 exhibits the right side and = exhibits the left side crack tips, respectively. By replacing Eqs. (22) and (25) into (21) and resultant equations into (27) the field intensity at cracks tips for the i-th crack yield
Table 1 The relevant material properties. c110 × 1010
16.8
( ) N m2
c130 × 1010 6
( ) N m2
c330 × 1010
16.3
( ) N m2
c440 × 1010
2.71
( ) N m2
e310
( ) C m2
0.9
131
e330 7.1
( ) C m2
e150
4.6
( ) C m2
110
36
× 10
10
( ) C m2
330
34
× 10
10
( ) C Vm
( ) kg m3
7550
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
Fig. 3. The variation of mode-I DSIFs versus dimensionless time for an isotropic piezoelectric plane.
Fig. 4. The variation of the normalized EDIF with t / t 0 for different electromechanical coupling factor for an isotropic piezoelectric plane.
Fig. 5. The variation of the normalized EDIF with t / t 0 for different electromechanical coupling factor for an FGP plane.
the crack (Fig. 2), a straight crack with length 2l = 2(cm) which is located along the x-axis under uniform normal and electric impact traction applied on the crack face is displayed. The validity of the analysis is verified by comparing our results with those available in the literature. In the first example, an isotropic piezoelectric plane ( l = 0 ) containing a crack is considered. The effect of the constant normal traction without electrical loading on the crack face on the mode-I dynamic stress intensity factors (DSIFs) is studied. It
can be seen from Fig. 3 that our results are in excellent agreement with previous investigations (Ueda [21]). As the second verification of results, problems of an isotropic piezoelectric and an FGP plane weakened by a crack with various electromechanical coupling factors are shown in Figs. 4 and 5, respectively. The results are compared with those obtained by Garcı́a-Sánchez et al. [23]. As we may observe, our results are in excellent agreement with the previous investigation. As can be seen, the normalized electric 132
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
Fig. 6. The effect of l on the normalized electric displacement intensity factors for
D
= 1.
Fig. 7. The variation of mode-I DSIFs versus dimensionless time for an FGP plane without electric loading.
Fig. 8. The effect of
D
on the normalized mode-I DSIFs under uniform combining electrical and normal mechanical impact traction for l = 1.
displacement intensity factor (EDIF), kD /k 0D , is only weakly dependent on the dimensionless time, which is a consequence of the quasi-static assumption of the electrical fields. This phenomenon is observed with the Garcı́a-Sánchez et al. [23] and Bagheri [32]. Also, a comparison of Figs. 4 and 5 reveals that a higher value of kD /k 0D for an FGPM is obtained. In the remaining of this section, more examples are presented to demonstrate the applicability of the procedure. The variation of the normalized EDIFs against dimensionless time
(i.e., t / t 0 ) for three values l = 0.25, 0.5, 1.0 is depicted in Fig. 6. From Fig. 6, we observe that the normalized EDIFs, kD /k 0D , for the left tip of the crack decreases as the l increases, while for the right tip R, due to enhancement of the shear modulus of plane in the positive x-axis direction, an increase for kD /k 0D is observed. As can be seen, the normalized EDIFs are nearly independent of the normalized time for impermeable case. The results showed in Fig. 7, illustrate the influence of the 133
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
Fig. 9. Variation of mode-II DSIFs under uniform combining electrical and normal mechanical impact traction for l = 1.
length is taken to be 2l = 2(cm) . According to Fig. 7, the trend of variation of kI / k 0 is similar to kD /k 0D for various FGPM constant. The normalized mode I DSIFs rises rapidly with time, reaching a peak, then decreases in magnitude to reach static value. In other words, at t = the effect of the transient loading on the crack surface disappears, which causes the SIFs to become independent of time and converge into static value in an infinite FG plane. Figs. 8 and 9 display the effect of applied electric field of 0 < D < 2 on the non-dimensional mode I and II DSIFs for right tip of the crack with variation of t / t 0 . It is interesting to note that the normalized DSIF kIR/ k 0 and kIIR/ k 0 increase quickly at first with time, to reach their maximum value and then get approximately stable. Finally, the DSIF approaches the corresponding static value. In conclusion, in an FGP plane with a crack lying along the x-axis under uniform combining
Fig. 10. Geometry of one crack subjected to uniform shear impact traction
dimensionless non-homogeneity parameter l on the variation of the normalized mode-I DSIFs versus the t / t 0 . The electro-mechanical coupling factor of the FGP plane is supposed as D = 0 , while the crack
Fig. 11. Variation of mode-II DSIFs under impact shear traction for
D
= 0.
Fig. 12. Geometry of two aligned cracks subjected to uniform normal and electric impact traction. 134
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
respectively. The interaction between two cracks in the FGP plane with non-homogeneity parameter l = 1 and d/ l = 1.05 is studied in Figs. 13–15. It is worth mentioning that the lack of symmetry produces mode-II DSIFs even where cracks are subjected to normal traction. As it might be observed, the DSIFs for the crack tips L1 and R2 at large values of t / t 0 , are higher than L2 and R1, which are attributed to the stronger interaction. Also, compared with the single crack problem, higher values of the DSIF are observed due to the interaction effect between cracks. Also, the DSIF is the lowest at the crack tip, which is the farthest crack tip located in the less stiffer zone (see Fig. 16). In the next example, the medium contains two stacked identical cracks with lengths 2l shown in Fig. 17. The cracks lines are located on the vertical distances with d/ l = 0.1, 0.15. The variation of the normalized mode-I DSIFs versus dimensionless time t / t 0 , for D = 0.0 and l = 1.0 , are shown in Fig. 18. Under normal loading, the interaction effect is weaker and produces a shielding effect, thus DSIFs of two cracks are less than that of DSIFs of one crack with the same conditions. Two parallel shifted cracks under normal impact traction are shown in Fig. 19. The dimensionless modes I and II DSIFs versus dimensionless time are plotted in Figs. 20 and 21. In this case, non-homogeneity parameter and center distance are l = 1.0 and d/ l = 0.2 , respectively. We observed that, the magnitudes of DSIFs for the crack tip, located in a stiffer zone are higher than those other tips. It is interesting to note that, the shielding effect phenomenon is not seen in this case.
Fig. 13. Geometry of two aligned cracks subjected to uniform shear impact traction.
electrical and normal mechanical impact traction, the influence of the electromechanical coupling factor may be significant on the mode I and II DSIF. As can be seen, a considerable increase in kIR/ k 0 and kIIR/ k 0 is observed as D increases. In the next example, we consider a crack with length 2l as shown in Fig. 10. The calculated DSIFs at the crack tips are normalized by k 0 = 0 l for uniform crack face shear loading. According to Fig. 11, the trend of variation of kII / k 0 is similar to kI / k 0 and kD /k 0D for various FGPM exponents. As examples of multiple cracks, we consider two aligned cracks under normal and shear impact traction as shown in Figs. 12 and 13,
Fig. 14. Variation of mode-I DSIFs under impact normal load for two aligned cracks.
Fig. 15. Variation of mode-II DSIFs under impact normal load for two aligned cracks.
135
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
Fig. 16. Variation of mode-II DSIFs under impact shear load for two aligned cracks.
Fig. 19. Geometry of two parallel shifted cracks subjected to uniform normal and electric impact traction.
Fig. 17. Geometry of two stacked identical cracks subjected to uniform normal impact traction.
singular integral equations in Laplace domain, which are solved by numerical Laplace technique developed by Stehfest [33] to obtain the dislocation density on the crack face. The results reported in Section 4 indicate that the non-homogeneity parameters significantly influence the fracture behavior of an FGPM plane. The mode I DSIFs can be reduced to two stacked cracks compared with the single crack under normal load. The time of occurrence of the maxima of DSIF depends upon shear wave velocity of material, and it decreases as the wave velocity increases.
5. Concluding remarks The work presented in this article is directed toward developing analytical methods based on the DDT for a mixed-mode fracture analysis of FGPMs plane weakened by multiple parallel cracks. The solution of edge dislocation is first obtained in plane. The material properties are assumed to change continuously along the x- axis, and the crack faces are subjected to uniform normal, shear and electric impact loads. The Fourier transforms is applied to reduce the problem to a system of
Fig.18. Interaction of mode I of two stacked cracks under impact normal load. 136
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
Fig. 20. Interaction of mode I for two parallel shifted cracks under impact normal load.
Fig. 21. Interaction of mode II for two parallel shifted cracks under impact normal load.
Appendix A The expressions in Eqs. (10) and (12) are as follows:
r1 =
8 2
r2 = (
2
r3 =
9(
r4 =
5 6(
H2 = i
+i )+ 2)[
2)
+
H3 = i (
5
H4 = i
+
2
H6 =
H8 =
7)
2
9(
2)
+i (
+i )+
6 7
1 9
5 7
+
8s
2/ c 2
+(
2
i
)(
3)
+
2 8)
+i (
2 7
+
2 9)]
s 2 /c 2
+
5
s 2/ c 2 2)
7 (i
H7 = i (
i) )( + (1 + i) )
1
+
H5 = i
(1
1 8
5 7(
3 (i
2
(
)( s 2/ c 2 + (i
i
r5 = (i H1 =
2 5
i )
5
+
i
7)
+
+
7
2)
The functions Aij , i, j
{1, 2, 3} in Eq. (16) are as follows:
137
2 7
+
3 8
+
9)
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
A11 =
i 2 (m2 p1 + n2 p2 ) i 3 (m3 p1 + n3 p2 ) + (m3 n2 m2 n3) p3 2 3 m1 n2) 1 2 + n3 3 (m1 1 m2 2) + m3 3 (n2 2 n1 1)]
p3 [(m2 n1
i m3 n2 + i m2 n3 + n3 2 n2 3 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 n1 2) i m3 n2 7 + i m2 n3 7 + m2 3 m3 2 m1 (n3 2 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 n1 2)
A12 =
m1 (n3 2
A13 =
i 1 (m1 p1 + n1 p2 ) i 3 (m3 p1 + n3 p2 ) + (m3 n1 m1 n3) p3 1 3 m1 n2) 1 2 + n3 3 (m1 1 m2 2) + m3 3 (n2 2 n1 1)]
A21 =
A31 =
p3 [(m2 n1
A22 = A23 =
i m3 n1 i m1 n3 n3 1 + n1 3 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 n1 2) i m3 n1 7 i m1 n3 7 + m3 1 m1 3 m1 (n3 2 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 n1 2)
i 1 (m1 p1 + n1 p2 ) i 2 (m2 p1 + n2 p2 ) + (m2 n1 m1 n2) p3 1 2 m1 n2) 1 2 + n3 3 (m1 1 m2 2) + m3 3 (n2 2 n1 1)]
p3 [(m2 n1
A32 =
m1 (n3 2
A33 =
i m2 n1 + i m1 n2 + n2 1 n1 2 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 i m2 n1 7 + i m1 n2 7 m2 1 + m1 2 m1 (n3 2 n2 3) + m2 (n1 3 n3 1) + m3 (n2 1 m1 (n3 2
n1 2) n1 2)
where
p1 =
1 6 p2
4 5
=
5 6
+
1 8 p3
2 6
=
+
4 8
The integrands are appeared in Eq. (17) are
fxx (x , y , s , ) =
ie x c440 1 b x (s ) 4
=
ie x c440 1 b x (s ) 4
= = =
[ [
ie x c440 1 b x (s ) 4
[
ie x c440 1 b x (s ) 4
[
ie x c440 1 b x (s ) 4
[
3
3
b1j Aj1 e
jy
3
+ b y (s )
b1j Aj2 e
j=1
jy
+ b (s )
j=1
3
3
b2j Aj1 e
jy
b2j Aj2 e
jy
+ b (s )
j=1
3 jy
b3j Aj2 e
jy
+ b (s )
j =1
3 jy
b4j Aj2 e
jy
+ b (s )
j=1
3 jy
)
jy
b5j Aj3 e
jy
3
+ b y (s )
b5j Aj2 e
j=1
] ei x fdy (x, y, s,
b4j Aj3 e j=1
3
b5j Aj1 e
)
jy
3
+ b y (s )
j=1
] ei x fdx (x , y, s,
b3j Aj3 e j =1
3
b4j Aj1 e
)
jy
3
+ b y (s )
j=1
] ei x fxy (x, y, s,
b2j Aj3 e j=1
3
b3j Aj1 e
)
jy
3
+ b y (s )
j=1
] ei x fyy (x, y, s,
b1j Aj3 e j=1
jy
+ b (s )
j=1
j=1
] ei x
Appendix B The singular part of the integral appeared in Eq. (19) are:
fxx (x , y , s, ) d =
e xc440 4
(b x (s )
3 e j =1 j j
jj | | y
3 e j=1 j+6 j
i sgn( ) b (s ) fyy (x , y , s , ) d =
e x c440 4
(bx (s )
3 e j=1 j j+3
jj | | y
i sgn( ) b (s ) fxy (x , y , s , ) d =
e x c440 4
fdx (x , y, s, ) d =
e x c440 4
3 e j=1 j j+9
( i sgn( ) bx (s )
e x c440 4
( bx ( s )
jj | | y
3 e j = 1 j j + 12
i sgn( ) b (s )
3 e j=1 j+3 j+3
jj | | y
3 e j =1 j+3 j+6
jj | | y
3 e j=1 j +3 j+9
jj | | y
jj | | y ) e i x d
b y (s ) jj | | y ) e i x d
3 e j = 1 j + 3 j + 12
+ i sgn( ) by (s )
3 e j = 1 j + 6 j + 12
jj | | y ) e i x d
where 1
=
2
=
3
=
4
=
(p1 m22 + p2 n22) p3 [
11 22 (m11 n22
p3 [
11 22 (m11 n22
p3 [
11 22 (m11 n22
(p1 m11 + p2 n11)
11 n22
(p1 m33 + p2 n33)
11
11
11 33 (m33 n11
+ (m22 n33
m11 n33) + 33
11 33 (m33 n11
(p1 m22 + p2 n22)
m22 n11) +
33
11 33 (m33 n11
+ (p1 m33 + p2 n33)
m22 n11) +
(p1 m11 + p2 n11)
m33 (
22
m22 n11) +
+ (m33 n11
m11 n33) + 22
+ (m11 n22
m11 n33) +
(m33 n22 m22 n33 n33 22 + n22 33) + m22 ( 33 n11 11 n33) + m11 (
22 n11)
22 n33
jj | | y +
jj | | y ) e i x d
b y (s )
jj | | y
3 e j=1 j+6 j+9
b (s ) fdy (x , y, s, ) d =
3 e j =1 j+6 j+3
3 e j=1 j +6 j +6
b (s )
jj | | y+
jj | | y ) e i x d
+ i sgn( ) by (s )
3 e j =1 j j+6
( i sgn( ) bx (s )
3 e j=1 j+3 j
+ i sgn( ) b y (s )
m33 n22 ) p3 22 33 (m22 n33
m11 n33) p3 22 33 (m22 n33
m22 n11) p3 22 33 (m22 n33
22 33
m33 n22)] 11 33
m33 n22)] 11 22
m33 n22)]
33 n22 )
138
jj | | y+
Theoretical and Applied Fracture Mechanics 101 (2019) 127–140
S. Azizi and R. Bagheri
5
=
6
=
7
=
8
=
9
=
j
=
m33 (
11 n22
22 n11)
(m11 n33 m33 n11 n11 33 + n33 11) + m22 ( 33 n11 11 n33) + m11 (
22 n33
33 n22 )
m33 (
11 n22
(m22 n11 m11 n22 n22 11 + n11 22 ) 22 n11) + m22 ( 33 n11 11 n33) + m11 (
22 n33
33 n22 )
m33 (
11 n22
(m33 n22 7 m22 n33 7 22 n11) + m22 ( 33 n11
m22 33 + m33 11 n33) + m11 (
22 n33
33 n22 )
m33 (
11 n22
(m11 n33 7 m33 n11 7 22 n11) + m22 ( 33 n11
m33 11 + m11 33) 11 n33) + m11 ( 22 n33
33 n22 )
m33 (
11 n22
(m22 n11 7 m11 n22 7 22 n11) + m22 ( 33 n11
m11 22 + m22 11) 11 n33) + m11 ( 22 n33
33 n22)
1 mjj jj
3
j+3
=
1
j+6
=
jj
j+9
=
7 jj
j + 12
=
mjj =
njj =
11
=
22
=
33
=
11
=
5 njj jj ,
4 mjj jj
+ mjj + +
7 mjj
4 8 jj
(
5 6
+
(
+
2 jj 5
11
+
2 8)
2 mjj jj
+
3
7) jj
j = 1, 2, 3 2 7
3 jj
+ (
3 8 5 7
+
(1 i 3 ) p11 3 × 22/3 (q11 + r11)1/3
11
=
+
+(
p11 =
2 7 3
3 9 2 9) jj
, j = 1, 2, 3
6 (2 7
2 2(
5
+
7)
(1 + i 3 )(q11 + r11)1/3 6 × 21/3 +
3 6)
5
+
7
)2
2
2 7( 5
+
+ +
4 8 2 11
3
4( 5
7)
+
2
+
7)
+
+
4 3 9
8( 4 3
2 6
+
9
+2
6 3 7
2 2
+ 1)
+
8 3
22
2 6
3 9 2 6 22
q11 = 2
3 11
r11 =
3 2 4p11 + q11
9
+ + 2 2 9
4 8
33
+
, j = 1, 2, 3
4 8
=
2 7
(1 + i 3 ) p11 ( 1 + i 3 )(q11 + r11)1/3 + 2/3 1/3 3 × 2 (q11 + r11) 6 × 21/3
3
2 7
2 9 ) jj
+
21/3p11 (q + r11)1/3 + 11 1/3 3(q11 + r11) 3 × 21/3
3
9
5 7
+
11
4
+2
j = 1, 2, 3
8 njj jj ,
6 7
+
3
2 5
+(
+
j = 1, 2, 3
j = 1, 2, 3 9 njj ,
6 mjj jj
5
j = 1, 2, 3
6 njj jj , 7 njj ,
22 )
11 22
+ 27
33
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