- Email: [email protected]
Influence of a micro-crack on the finite macro-crack
Engineering Fracture Mechanics 177 (2017) 95–103
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Influence of a micro-crack on the finite macro-crack Li Xiaotao, Li Xu, Jiang Xiaoyu ⇑ School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China
a r t i c l e
i n f o
Article history: Received 21 October 2016 Received in revised form 25 March 2017 Accepted 26 March 2017 Available online 29 March 2017 Keywords: Macro-crack Micro-crack Distributed dislocation technique Stress intensity factor Crack propagation direction
a b s t r a c t The numerical solution of an infinite elastic plane containing a macro-crack and an arbitrary oriented micro-crack is presented based on the distributed dislocation technique and Gauss-Chebyshev quadrature method, which is verified by finite element method. The stress intensity factor (SIF), stresses field and strain energy density near the macrocrack tip are obtained. The effect of the micro-crack on SIF of the macro-crack is analyzed and the macro-crack propagation direction is predicted based on the minimum strain energy density criterion (SED). The results show that as the micro-crack length decreases or the distance between micro-crack and macro-crack increases, the effect of the microcrack on SIF of the macro-crack will be getting weaker. The micro-crack increases SIF of the macro-crack at some orientation, while it decreases SIF at other orientation. The micro-crack acts an attraction effect on the macro-crack propagation at some orientation, while it acts a repulsion effect at other orientation. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The modes of crack failure have three types that are opening mode, sliding mode and tearing mode. The mode I failure is the most dangerous, so the mode I mechanism of crack failure has been studied extensively. Generally, various microdefects, such as micro-cracks, dislocations and inclusions, exist in the materials, because it is difficult to avoid damage of materials in the process of manufacturing or application. When the micro-defects locate near the macro-crack tip, the direction and rate of the macro-crack propagation may be influenced. In the case, the mode I crack will not propagate along the direction of the crack face. So influences of the micro-crack on the macro-crack were researched in many literatures. Kachanov [1–3] proposed a simple solution to cracks interaction problem. The core idea of the method was that the actual tractions on individual cracks were replaced approximately by the average tractions. Based on the complex potentials and superposition principle, Gong [4–6] investigated the interaction problem between an arbitrary located and oriented micro-crack and a semi-infinite main crack; Hori and Nemat-Nasser [7] and Meguid et al. [8] studied the problem of interaction between micro-cracks and a finite macro-crack. Xiangqiao [9,10] presented a numerical approach for the interaction problem of macro-crack with micro-crack in an infinite elastic plane. The distributed dislocation technique was described in detail by the classic book [11,12], and it was applied to study various crack problems. The basic idea of the distributed dislocation technique was that cracks could be replaced equivalently by the continuous distributed dislocation. Based on the method, Erdogan et al. [13] presented the solution of a circular inhomogeneity and a crack in an infinite plane, and investigated the interaction problem between an arbitrarily orientated crack
⇑ Corresponding author. E-mail address: [email protected] (X. Jiang). http://dx.doi.org/10.1016/j.engfracmech.2017.03.037 0013-7944/Ó 2017 Elsevier Ltd. All rights reserved.
96
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
and an inhomogeneity; Jin and Keer [14] considered the multiple crack problem in an elastic half plane, and investigated the interaction among the edge cracks. Tao et al. [15] investigated the problem of an infinite plane containing a Griffith crack, a circular inhomogeneity and an edge dislocation under the uniaxial tensile load. Mousavi and Lazar [16] studied crack problems by applying nonsingular dislocations in nonlocal elasticity and obtained nonsingular stresses of modes I, II and III cracks. Sharma et al. [17] analyzed a list of inclined cracks in a two dimensional piezoelectric strip under various electrical boundary conditions. The criterion of mixed mode crack propagation direction has been focused on by many researchers. From the perspective of the experiment, Richard et al. [18,19] obtained an empirical formula by collecting a large amount of experimental data. From the perspective of the stress, Erdogan and Sih [20] proposed the maximum circumferential tensile stress criterion. And then they conducted on tensile fracture experiments using thin plexiglass plates, which verified their theory. The maximum circumferential tensile stress criterion depends on only the circumferential tensile stress, while it is independent of the material parameters. From the perspective of the energy, Palaniswamy and Knauss [21] proposed the maximum energy release rate criterion. Sih [22] proposed the SED criterion which considered the effect of the material parameters. Malíková et al. [23] shown that the SED criterion was accurate enough for most geometries and distances from the crack tip. So the SED criterion is applied to predict crack propagation direction in this paper. This paper is organized as follows. In the section two, based on the distributed dislocation technique, the solution of an infinite plane containing a macro-crack and an arbitrary oriented micro-crack is presented. In the section three, the finite element models are established to verify theoretical derivation in section two. In the section four, the influence of various parameters on SIF of the macro-crack and the macro-crack propagation direction are analyzed. In the section five, some conclusions are drawn. 2. Formulation 2.1. Dislocation influence function in an infinite elastic plane containing a macro-crack and an arbitrary oriented micro-crack There is an edge dislocation at the origin, with Burgers vector components bx and by . The stresses at a point ðx; yÞ induced by the dislocation can be obtained by Hill’s book [12].
2l fb G ðx; yÞ þ by Gyxx ðx; yÞg pðj þ 1Þ x xxx 2l r yy ðx; yÞ ¼ fb G ðx; yÞ þ by Gyyy ðx; yÞg pðj þ 1Þ x xyy 2l r xy ðx; yÞ ¼ fb G ðx; yÞ þ by Gyxy ðx; yÞg pðj þ 1Þ x xxy
r xx ðx; yÞ ¼
ð1Þ
where l is shear modulus. j is the Kolosov’s constant: j ¼ ð3 mÞ=ð1 þ mÞ for plane stress, j ¼ 3 4m for plane strain, m being the Poisson’s ratio. Gðx; yÞ is the dislocation influence function. The first subscript on the G denotes the Burgers vector, and the last two denote the associated traction. The expressions of Gðx; yÞ can be given by
y x y ð3x2 þ y2 Þ; Gyxx ðx; yÞ ¼ 4 ðx2 y2 Þ; Gxyy ðx; yÞ ¼ 4 ðx2 y2 Þ; r4 r r x x y Gyyy ðx; yÞ ¼ 4 ðx2 þ 3y2 Þ; Gxxy ðx; yÞ ¼ 4 ðx2 y2 Þ; Gyxy ðx; yÞ ¼ 4 ðx2 y2 Þ: r r r
Gxxx ðx; yÞ ¼
ð2Þ
where r2 ¼ x2 þ y2 . The problem of an infinite plane containing a macro-crack and an arbitrary oriented micro-crack will be analyzed in the following. The present problem can be divided into two sub-problems through the superposition principle. The first refers to the problem of an infinite medium in the absence of cracks under remote loadings. The second can be described as the problem of an infinite elastic plane containing a macro-crack and an arbitrary oriented micro-crack without external loads, and the two cracks are replaced by the continuous distributed dislocations. Firstly, the solution of the second sub-problem will be presented. An edge dislocation is located at ðn; 0Þ. The Burgers vector of the edge dislocation is b, having components bx and by , as shown in Fig. 1. In order to convenient statement, the macro-crack is called as crack 1 and the micro-crack is called as crack 2. The stresses at a point ðx1 þ n; y1 Þ induced by the edge dislocation at a point ðn; 0Þ can be given by
o 2l n 12 bx G12 xxx ðx1 ; y1 Þ þ by Gyxx ðx1 ; y1 Þ pðj þ 1Þ o 2l n 12 r yy ðx0 ; nÞ ¼ bx G12 xyy ðx1 ; y1 Þ þ by Gyyy ðx1 ; y1 Þ pðj þ 1Þ o 2l n 12 r xy ðx0 ; nÞ ¼ bx G12 xxy ðx1 ; y1 Þ þ by Gyxy ðx1 ; y1 Þ pðj þ 1Þ
r xx ðx0 ; nÞ ¼
ð3Þ
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
97
Fig. 1. The macro-crack and an arbitrary oriented micro-crack in an infinite plane.
where, as before, the superscripts 12 on the G denote the associated stresses at the position on crack 2 due to the distributed dislocation along crack 1. The expressions of the influence functions G12 ðx1 ; y1 Þ can be obtained by formula (2). x1 and y1 are defined in Fig. 1 and their expressions can be given by
x1 ¼ d cos h n þ x0 cos a
ð4Þ
y1 ¼ d sin h þ x0 sin a
The stresses in the local coordinate system o0 x0 y0 can be obtained by Mohr stress transformation.
o 2l n 12 0 0 bx G12 xy0 y0 ðx ; nÞ þ by Gyy0 y0 ðx ; nÞ pðj þ 1Þ o 2l n 0 0 r x0 y0 ðx ; nÞ ¼ b G12 ðx0 ; nÞ þ by G12 yx0 y0 ðx ; nÞ pðj þ 1Þ x xx0 y0
r y0 y0 ðx0 ; nÞ ¼
ð5Þ
where 12 12 12 0 2 G12 xy0 y0 ðx ; nÞ ¼ Gxxx ðx1 ; y1 Þsin a þ Gxyy ðx1 ; y1 Þcos a Gxxy ðx1 ; y1 Þ sin 2a 2
12 12 12 0 2 G12 yy0 y0 ðx ; nÞ ¼ Gyxx ðx1 ; y1 Þsin a þ Gyyy ðx1 ; y1 Þcos a Gyxy ðx1 ; y1 Þ sin 2a h i 12 12 12 0 G12 xx0 y0 ðx ; nÞ ¼ Gxyy ðx1 ; y1 Þ Gxxx ðx1 ; y1 Þ sin a cos a þ Gxxy ðx1 ; y1 Þ cos 2a h i 12 12 12 0 G12 yx0 y0 ðx ; nÞ ¼ Gyyy ðx1 ; y1 Þ Gyxx ðx1 ; y1 Þ sin a cos a þ Gyxy ðx1 ; y1 Þ cos 2a 2
ð6Þ
11 11 0 0 0 For formula (6), setting d ¼ 0 and h ¼ a ¼ 0, the dislocation influence functions G11 xy0 y0 ðx ; nÞ, Gyy0 y0 ðx ; nÞ, Gxx0 y0 ðx ; nÞ and 0 G11 yx0 y0 ðx ; nÞ
can be obtained. In the same way, the associated tractions at position on crack 1 due to the distributed dislocation along crack 2 can be obtained. 2.2. The singular integral equations about the dislocation density function By distributing continuous dislocations along crack 1, in the local coordinate system o0 x0 y0 , the traction components along crack 2 induced by the total dislocations along crack 1 can be given by
r y0 y0 ðx0 Þ ¼
2l pðj þ 1Þ
2l r x0 y0 ðx Þ ¼ pðj þ 1Þ 0
Z
Z
a1
h
a1 a1
h
a1
i 12 1 0 0 B1x ðnÞG12 xy0 y0 ðx ; nÞ þ By ðnÞGyy0 y0 ðx ; nÞ dn
i 12 1 0 0 B1x ðnÞG12 xx0 y0 ðx ; nÞ þ By ðnÞGyx0 y0 ðx ; nÞ dn
ð7Þ
1
where B ðnÞ is the dislocation density function along crack 1. In the same way, the tractions components on crack 1 due to the distributed dislocations along crack 2 can obtained. The first sub-problem refers to an infinite plane in the absence of cracks subjected to remote loadings. The stresses com~ yy and r ~ xy . It is ensured that the crack face is traction-free, so the integral ponents of the sub-problem are assumed to be r equations can be established based on the superposition principle.
r~ ij ðxÞ þ r ij ðxÞ ¼ 0 a1 < x < a1 ; ij ¼ yy or xy r~ ij ðx0 Þ þ r ij ðx0 Þ ¼ 0 a2 < x0 < a2 ; ij ¼ y0 y0 or x0 y0
ð8Þ
98
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
Further
jþ1~ 1 rij ðxÞ ¼ 2l p
Z
1
p
a2
h
a2
Z
p
a2
a2
h
a1
a1
h
i 11 1 B1x ðnÞG11 xij ðx; nÞ þ By ðnÞGyij ðx; nÞ dnþ
i 21 2 0 0 0 0 B2x0 ðn0 ÞG21 x0 ij ðx; n Þ þ By0 ðn ÞGy0 ij ðx; n Þ dn
jþ1~ 0 1 rij ðx Þ ¼ 2l p 1
Z
Z
a1
a1
h
0 B1x ðnÞG12 xij ðx ; nÞ
þ
a1 < x < a1 ; i
0 B1y ðnÞG12 yij ðx ; nÞ
i 22 2 0 0 0 0 0 0 B2x0 ðn0 ÞG22 x0 ij ðx ; n Þ þ By0 ðn ÞGy0 ij ðx ; n Þ dn
ij ¼ yy or xy ð9Þ
dnþ
a2 < x0 < a2 ;
ij ¼ y0 y0 or x0 y0
2.3. Numerical solution of singular integral equations The singular integral Eq. (9) can be solved by Gauss-Chebyshev quadrature [24,25]. Letting n ¼ a1 s, x ¼ a1 t, n0 ¼ a2 s and x ¼ a2 t, the integral over ½a1 ; a1 and ½a2 ; a2 may be normalized over ½1; 1. According to Hill’s book [11], the unknown function in present problem is ‘singular-at-both-ends’, so the form of dislocation density function may be given by 0
/ðsÞ BðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 s2 Þ
ð10Þ
where /ðsÞ is an unknown function. Eq. (9) can be replaced approximately by a series of algebraic equations by employing Gauss-Chebyshev quadrature method. It can be written as N n h i h io jþ1~ 1X 11 21 21 1 2 2 rij ðtk Þ ¼ a1 /1x ðsI ÞG11 xij ðt k ; sI Þ þ /y ðsI ÞGyij ðt k ; sI Þ þ a2 /x0 ðsI ÞGx0 ij ðt k ; sI Þ þ /y0 ðsI ÞGy0 ij ðt k ; sI Þ 2l N I¼1
ij ¼ yy or xy;
k ¼ 1...N 1
N n h i h io jþ1~ 1X 12 22 22 1 2 2 rij ðtk Þ ¼ a1 /1x ðsI ÞG12 xij ðt k ; sI Þ þ /y ðsI ÞGyij ðt k ; sI Þ þ a2 /x0 ðsI ÞGx0 ij ðt k ; sI Þ þ /y0 ðsI ÞGy0 ij ðt k ; sI Þ 2l N I¼1
ij ¼ y0 y0 or x0 y0 ;
ð11Þ
k ¼ 1...N 1
where
sI ¼ cos½pð2I 1Þ=2N I ¼ 1; 2; ; N t k ¼ cos½pk=N k ¼ 1; 2; ; N 1
ð12Þ
For Eq. (11), there are 4N unknowns /ðsI Þ while only 4(N-1) algebraic equations can be found. So four extra equations must be established to solve Eq. (11). Notice that, on the crack, there is on net dislocation. N N N N X X X X /1x ðsI Þ ¼ /1y ðsI Þ ¼ /2x0 ðsI Þ ¼ /2y0 ðsI Þ ¼ 0 I¼1
I¼1
I¼1
ð13Þ
I¼1
Then, SIFs at crack tip may be given by
pffiffiffiffiffiffi 2l K I ð1Þ ¼ pa / ð1Þ jþ1 y pffiffiffiffiffiffi 2l K II ð1Þ ¼ pa / ð1Þ jþ1 x
ð14Þ
where
N 1X 2I 1 2I 1 /ðsI Þ sin pð2N 1Þ sin N I¼1 4N 4N N 1X 2I 1 2I 1 /ð1Þ ¼ /ðsNþ1I Þsin pð2N 1Þ sin N I¼1 4N 4N
/ðþ1Þ ¼
ð15Þ
Next, the stress field will be calculated. An edge dislocation is located at position ðn; 0Þ. For an arbitrary point (x, y), the dislocation influence function can be written as
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
xn ððx nÞ2 y2 Þ r4 y xn G1xyy ðx; yÞ ¼ 4 ððx nÞ2 y2 Þ; G1yyy ðx; yÞ ¼ 4 ððx nÞ2 þ 3y2 Þ r r xn y 2 1 1 2 Gxxy ðx; yÞ ¼ 4 ððx nÞ y Þ; Gyxy ðx; yÞ ¼ 4 ððx nÞ2 y2 Þ r r
G1xxx ðx; yÞ ¼
y ð3ðx nÞ2 þ y2 Þ; r4
99
G1yxx ðx; yÞ ¼
ð16Þ
where r 2 ¼ ðx nÞ2 þ y2 . By the same method, G2 ðx; yÞ can be obtained. And then, the stress field may be given by
rxx ðx; yÞ ¼ r~ xx ðx; yÞ þ
N n h i h io 1 2l X a1 /1x ðsI ÞG1xxx ðx; yÞ þ /1y ðsI ÞG1yxx ðx; yÞ þ a2 /2x0 ðsI ÞG2x0 xx ðx; yÞ þ /2y0 ðsI ÞG2y0 xx ðx; yÞ N j þ 1 I¼1
rxy ðx; yÞ ¼ r~ xy ðx; yÞ þ
N n h i h io 1 2l X a1 /1x ðsI ÞG1xxy ðx; yÞ þ /1y ðsI ÞG1yxy ðx; yÞ þ a2 /2x0 ðsI ÞG2x0 xy ðx; yÞ þ /2y0 ðsI ÞG2y0 xy ðx; yÞ N j þ 1 I¼1
ryy ðx; yÞ ¼ r~ yy ðx; yÞ þ
N n h i h io 1 2l X a1 /1x ðsI ÞG1xyy ðx; yÞ þ /1y ðsI ÞG1yyy ðx; yÞ þ a2 /2x0 ðsI ÞG2x0 yy ðx; yÞ þ /2y0 ðsI ÞG2y0 yy ðx; yÞ N j þ 1 I¼1
I ¼ 1; 2 . . . N ð17Þ The polar coordinate system ðR; bÞ is established at the macro-crack tip, as shown in Fig. 1. The stress components in cylindrical coordinate may be given by
rrr ¼ rxx ðx; yÞ cos2 b þ ryy ðx; yÞ sin2 b þ rxy ðx; yÞ sin2 ð2bÞ rrh ¼ ½rxx ðx; yÞ ryy ðx; yÞ sin b cos b þ rxy ðx; yÞ cosð2bÞ rhh ¼ rxx ðx; yÞ sin2 b þ ryy ðx; yÞ cos2 b rxy ðx; yÞ sinð2bÞ
ð18Þ
x ¼ a1 þ R cos b y ¼ R sin b
ð19Þ
where
2.4. The SED criterion From the energy point of view, Sih [22] proposed the SED, and he thought that the crack propagated in the direction where the strain energy density reached its minimum. So the criterion can be given by
@S @2S ¼ 0; 2 > 0; @b @b
where S ¼
1 2l
jþ1 8
ðrRR þ rbb Þ2 rRR rbb þ r2Rb
ð20Þ
3. Finite element method verification In the section, the solution to the problem derived in section two are verified by the finite element software ANSYS. Two cases are considered. The first case is that a micro-crack is parallel to the macro-crack in an infinite square plane under mode I load, that is a ¼ 0. The second is that a micro-crack is located at the radial direction of the macro-crack in an infinite square plane subjected to mode I load, that is h ¼ a. Taking the first case as an example, the finite element model is established, as shown in Fig. 2. The elasticity modulus is 210 GPa, and Poisson’s ratio is 0.3. The element type is plane183. Classical singular element is applied at the crack tip. There are 20 elements around the crack tip. The length of the macro-crack and the microcrack are also 10 mm. The length of the model is 200 mm, which is relatively large in comparison to length of cracks. So it is considered as an infinite plane. The model is subjected to tensile load in the y direction. Comparing with the theoretical solution, the calculation of finite element method is elastic. 4. Results and discussion In order to shown the numerical results clearly, the SIF of the macro-crack is normalized by
K0 ¼ r
pffiffiffiffiffiffiffiffi pa1
The mechanics behavior of the right macro-crack tip is only analyzed in the following.
ð21Þ
100
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
Fig. 2. The finite element model of the parallel micro-crack
4.1. Influence of the micro-crack orientation on SIF of the macro-crack As the section three introduced, the first case is a ¼ 0, that is parallel micro-crack. And the second is h ¼ a, that is radial micro-crack. The variation of normalized SIF K I =K 0 versus micro-crack orientation h under uniaxial tensile load is depicted for different values of distance d=a1 in Fig. 3. And comparison between theory and FEM results is shown. Solid lines denote theoretical results; dash lines denote FEM results. a2 =a1 is taken as 1. The results in Fig. 3 show that the parallel micro-crack increases SIF when h is less than about 37.5 degree, and decreases SIF when h is more than about 45 degree. While the radial micro-crack increases SIF when h is less than about 60 degree, and it has a little effect when h is more than about 60 degrees. The shielding effect of the parallel micro-crack is more apparent than that of the radial micro-crack. What’s more, Fig. 3 shows that the results of current theoretical method are very consistent with FEM results. The theoretical method is efficient to solve the crack problem. For different crack length, location, orientation and angle, the crack parameters only need to change, and the crack model need not to re-establish. So all results in the below section are obtained by the theoretical method.
4.2. Influence of the distance between micro-crack and macro-crack on SIF of the macro-crack The variation of normalized SIF K I =K 0 versus distance d=a1 under uniaxial tensile load is shown for different values of the micro-crack orientation h in Fig. 4. a2 =a1 is taken as 1. The results show that the effect of the micro-crack on SIF of the macrocrack will be getting weaker as the distance between the micro-crack and the macro-crack increases, which agrees with the results in Fig. 3.
(a) 1.3
(b)
K /K0
1.2
1.1
1.0
d/a1=2.5 d/a1=3 d/a1=4 d/a1=2.5 d/a1=3 d/a1=4
1.25 1.20
K /K0
d/a1=2.5 d/a1=3 d/a1=4 d/a1=2.5 d/a1=3 d/a1=4
1.15 1.10 1.05
0.9 1.00 0.8
0
15
30
45
θ (degree)
60
75
90
0
15
30
45
60
θ (degree)
Fig. 3. K I =K 0 versus h under uniaxial tensile load. (a) the parallel micro-crack; (b) the radial micro-crack.
75
90
101
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
(a)
(b) θ=15 degrees θ=45 degrees θ =75 degrees
1.4
K /K0
1.2
K /K0
θ=15 degrees θ=45 degrees θ=75 degrees
1.6
1.0
1.4
1.2
0.8
1.0
2.0
2.5
3.0
3.5
4.0
4.5
5.0
2.0
2.5
3.0
d/a1
3.5
4.0
d/a1
4.5
5.0
Fig. 4. K I =K 0 versus d=a1 under uniaxial tensile load. (a) the parallel micro-crack; (b) the radial micro-crack.
4.3. Influence of the micro-crack length on SIF of the macro-crack The variation of normalized SIF K I =K 0 versus micro-crack length a2 =a1 under uniaxial tensile load is shown for different values of the micro-crack orientation h in Fig. 5. d=a1 is taken as 2.5. On the whole, as the micro-crack length increases, the effect of micro-crack on SIF at the macro-crack tip will be getting stronger. There is almost no effect on SIF of the macro-crack when a2 =a1 is less than 0.1 for d=a1 ¼ 2:5. For the radial micro-crack, the effect of the micro-crack length on SIF of the macrocrack is little when h is more than 60 degrees, which is consistent with the results in Fig. 3(b). 4.4. Influence of the micro-crack on the macro-crack propagation direction As shown in Fig. 1, a polar coordinate system ðR; bÞ is established at the macro-crack tip. The strain energy density can be obtained by the Eqs. (17)(20). The variation of normalized the strain energy density 2Sl=r2 versus b under uniaxial tensile load is shown for different values of the micro-crack orientation h in Fig. 6. d=a1 is taken as 2.5 and R=a1 is taken as 0.1. And then the macro-crack propagation direction can be predicted by the SED criterion described in Section 2.4. The variation of the macro-crack propagation direction versus micro-crack orientation h under uniaxial tensile load is plotted for different values of the micro-crack length a2 =a1 in Fig. 7. The results show that the mode I macro-crack will not propagate along the direction of crack face when the micro-crack locates in the vicinity of the macro-crack tip. For the first case, the parallel micro-crack acts an attraction effect on the macro-crack propagation at about 15 < h < 75 degrees, while the parallel micro-crack acts a repulsion effect on the macro-crack propagation at about h < 10 and h > 75 degrees. For the second case, the radial micro-crack acts an attraction effect on the macro-crack propagation at about h > 15 degrees, while the radial micro-crack acts a repulsion effect on the macro-crack propagation at about h < 15 degrees. As the micro-crack length increases, the influence on macro-crack propagation direction will be getting stronger.
(a)
θ=0 degrees θ=30 degrees θ=45 degrees θ=60 degrees θ=90 degrees
1.20
1.15
K /K0
1.1
K /K0
(b)
θ=0 degrees θ=30 degrees θ=45 degrees θ=60 degrees θ=90 degrees
1.2
1.0
1.10
1.05 0.9 1.00 0.1
0.2
0.3
0.4
0.5
0.6
a 2 /a 1
0.7
0.8
0.9
1.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a 2 /a 1
Fig. 5. K I =K 0 versus a2 =a1 under uniaxial tensile load. (a) the parallel micro-crack; (b) the radial micro-crack.
0.9
1.0
102
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
(a)
7.5
(b)
6.0
θ=0 degrees θ=30 degrees θ=60 degrees θ=90 degrees
6.0
4.5
2Sμ/σ2
2Sμ/σ2
7.5
θ=0 degrees θ=30 degrees θ=60 degrees θ=90 degrees
4.5
3.0 3.0 1.5 -90
-60
-30
0
30
60
1.5
90
-90
-60
-30
β (degree)
0
30
60
90
β (degree)
Fig. 6. 2Sl=r2 versus b for a2 =a1 ¼ 1 under uniaxial tensile load. (a) the parallel micro-crack; (b) the radial micro-crack.
a2/a1=0.8 a2/a1=0.5 a2/a1=0.2
8
4
0
-4 0
15
30
45
θ (degree)
60
75
90
(b) macro-crack propagation direction (degree)
macro-crack propagation direction (degree)
(a) 12
a2/a1=0.8 a2/a1=0.5 a2/a1=0.2
8
4
0
-4
0
15
30
45
60
75
90
θ (degree)
Fig. 7. The macro-crack propagation direction versus h for d=a1 ¼ 2 under uniaxial tensile load. (a) the parallel micro-crack; (b) the radial micro-crack.
5. Conclusions In the paper, the problem of an infinite plane containing a macro-crack and an arbitrary oriented micro-crack is investigated based on the distributed dislocation technique and Gauss-Chebyshev quadrature method. Influence of the micro-crack on the macro-crack is analyzed. Some conclusions can be summarized in the following. (1) The parallel micro-crack increases SIF of the macro-crack at about h < 37:5 degrees, and decreases SIF of the macrocrack at about h > 45 degrees, while the radial micro-crack increases SIF of the macro-crack at about h < 60 degrees, and it has a little effect on the macro-crack when h > 60 degrees. (2) The effect of the micro-crack on SIF of the macro-crack will be getting weaker as the distance between the micro-crack and the macro-crack increases, while the effect of the micro-crack on SIF of the macro-crack will be getting stronger as the micro-crack length increases. (3) The parallel micro-crack acts an attraction effect on the macro-crack propagation at about 15 < h < 75 degrees, while it acts a repulsion effect on the macro-crack propagation at about h < 10 and h > 75 degrees. The radial micro-crack acts an attraction effect on the macro-crack propagation at about h > 15 degrees, while it acts a repulsion effect on the macro-crack propagation at about h < 15 degrees.
Acknowledgments The work is supported by the National Natural Science Foundation of China (11472230), the National Natural Science Foundation of China Key Project (U1134202/E050303) and Sichuan Provincial Youth Science and Technology Innovation Team (2013TD0004).
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
103
References [1] Kachanov M. A simple technique of stress analysis in elastic solids with many cracks. Int J Fract 1985;28:R11–9. [2] Kachanov M. Elastic solids with many cracks: a simple method of analysis. Int J Solids Struct 1987;23:23–43. [3] Chudnovsky A, Dolgopolsky A, Kachanov M. Elastic interaction of a crack with a microcrack array—II. Elastic solution for two crack configurations (piecewise constant and linear approximations). Int J Solids Struct 1987;23:11–21. [4] Gong SX, Horii H. General solution to the problem of microcracks near the tip of a main crack. J Mech Phys Solids 1989;37:27–46. [5] Gong SX, Meguid SA. Microdefect interacting with a main crack: a general treatment. Int J Mech Sci 1992;34:933–45. [6] Gong SX. On the main crack-microcrack interaction under mode III loading. Eng Fract Mech 1995;51:753–62. [7] Hori M, Nemat-Nasser S. Interacting micro-cracks near the tip in the process zone of a macro-crack. J Mech Phys Solids 1987;35:601–29. [8] Meguid SA, Gaultier PE, Gong SX. A comparison between analytical and finite element analysis of main crack-microcrack interaction. Eng Fract Mech 1991;38:451–65. [9] Xiangqiao Y. An effective numerical approach for interaction of macrocrack with microcracks. Chin J Theor Appl Mech 2006;38:119–24. [10] Xiangqiao Y. Microcrack interacting with a finite main crack. J Harbin Inst Technol 2006;38:503–6. [11] Hills D, Kelly P, Dai D, Korsunsky A. Solution of crack problems: the distributed dislocation technique, 1996. Kluwer Academic Publishers; 1996. [12] Hills DA, Kelly P, Dai D, Korsunsky A. Solution of crack problems: the distributed dislocation technique. Springer Science & Business Media 2013. [13] Erdogan F, Gupta G, Ratwani M. Interaction between a circular inclusion and an arbitrarily oriented crack. J Appl Mech 1974;41:1007–13. [14] Jin X, Keer LM. Solution of multiple edge cracks in an elastic half plane. Int J Fract 2006;137:121–37. [15] Tao Y, Fang Q, Zeng X, Liu Y. Influence of dislocation on interaction between a crack and a circular inhomogeneity. Int J Mech Sci 2014;80:47–53. [16] Mousavi SM, Lazar M. Distributed dislocation technique for cracks based on non-singular dislocations in nonlocal elasticity of Helmholtz type. Eng Fract Mech 2015;136:79–95. [17] Sharma K, Bui TQ, Bhargava R, Yu T, Lei J, Hirose S. Numerical studies of an array of equidistant semi-permeable inclined cracks in 2-D piezoelectric strip using distributed dislocation method. Int J Solids Struct 2016;80:137–45. [18] Richard H, Fulland M, Sander M. Theoretical crack path prediction. Fatigue Fract Eng Mater Struct 2005;28:3–12. [19] Richard H, Schramm B, Schirmeisen N-H. Cracks on mixed mode loading–theories, experiments, simulations. Int J Fatigue 2014;62:93–103. [20] Erdogan F, Sih G. On the crack extension in plates under plane loading and transverse shear. J Fluids Eng 1963;85:519–25. [21] Palaniswamy K, Knauss W. Propagation of a crack under general, in-plane tension. Int J Fract 1972;8:114–7. [22] Sih GC. Strain-energy-density factor applied to mixed mode crack problems. Int J Fract 1974;10:305–21. [23] Malíková L, Vesely´ V, Seitl S. Crack propagation direction in a mixed mode geometry estimated via multi-parameter fracture criteria. Int J Fatigue 2016;89:99–107. [24] Erdogan F, Gupta GD, Cook T. Numerical solution of singular integral equations. Methods of analysis and solutions of crack problems. Springer; 1973. p. 368–425. [25] Kaya AC, Erdogan F. On the solution of integral equations with strongly singular kernels. Q Appl Math 1986.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Influence of a micro-crack on the finite macro-crack Li Xiaotao, Li Xu, Jiang Xiaoyu ⇑ School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China
a r t i c l e
i n f o
Article history: Received 21 October 2016 Received in revised form 25 March 2017 Accepted 26 March 2017 Available online 29 March 2017 Keywords: Macro-crack Micro-crack Distributed dislocation technique Stress intensity factor Crack propagation direction
a b s t r a c t The numerical solution of an infinite elastic plane containing a macro-crack and an arbitrary oriented micro-crack is presented based on the distributed dislocation technique and Gauss-Chebyshev quadrature method, which is verified by finite element method. The stress intensity factor (SIF), stresses field and strain energy density near the macrocrack tip are obtained. The effect of the micro-crack on SIF of the macro-crack is analyzed and the macro-crack propagation direction is predicted based on the minimum strain energy density criterion (SED). The results show that as the micro-crack length decreases or the distance between micro-crack and macro-crack increases, the effect of the microcrack on SIF of the macro-crack will be getting weaker. The micro-crack increases SIF of the macro-crack at some orientation, while it decreases SIF at other orientation. The micro-crack acts an attraction effect on the macro-crack propagation at some orientation, while it acts a repulsion effect at other orientation. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The modes of crack failure have three types that are opening mode, sliding mode and tearing mode. The mode I failure is the most dangerous, so the mode I mechanism of crack failure has been studied extensively. Generally, various microdefects, such as micro-cracks, dislocations and inclusions, exist in the materials, because it is difficult to avoid damage of materials in the process of manufacturing or application. When the micro-defects locate near the macro-crack tip, the direction and rate of the macro-crack propagation may be influenced. In the case, the mode I crack will not propagate along the direction of the crack face. So influences of the micro-crack on the macro-crack were researched in many literatures. Kachanov [1–3] proposed a simple solution to cracks interaction problem. The core idea of the method was that the actual tractions on individual cracks were replaced approximately by the average tractions. Based on the complex potentials and superposition principle, Gong [4–6] investigated the interaction problem between an arbitrary located and oriented micro-crack and a semi-infinite main crack; Hori and Nemat-Nasser [7] and Meguid et al. [8] studied the problem of interaction between micro-cracks and a finite macro-crack. Xiangqiao [9,10] presented a numerical approach for the interaction problem of macro-crack with micro-crack in an infinite elastic plane. The distributed dislocation technique was described in detail by the classic book [11,12], and it was applied to study various crack problems. The basic idea of the distributed dislocation technique was that cracks could be replaced equivalently by the continuous distributed dislocation. Based on the method, Erdogan et al. [13] presented the solution of a circular inhomogeneity and a crack in an infinite plane, and investigated the interaction problem between an arbitrarily orientated crack
⇑ Corresponding author. E-mail address: [email protected] (X. Jiang). http://dx.doi.org/10.1016/j.engfracmech.2017.03.037 0013-7944/Ó 2017 Elsevier Ltd. All rights reserved.
96
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
and an inhomogeneity; Jin and Keer [14] considered the multiple crack problem in an elastic half plane, and investigated the interaction among the edge cracks. Tao et al. [15] investigated the problem of an infinite plane containing a Griffith crack, a circular inhomogeneity and an edge dislocation under the uniaxial tensile load. Mousavi and Lazar [16] studied crack problems by applying nonsingular dislocations in nonlocal elasticity and obtained nonsingular stresses of modes I, II and III cracks. Sharma et al. [17] analyzed a list of inclined cracks in a two dimensional piezoelectric strip under various electrical boundary conditions. The criterion of mixed mode crack propagation direction has been focused on by many researchers. From the perspective of the experiment, Richard et al. [18,19] obtained an empirical formula by collecting a large amount of experimental data. From the perspective of the stress, Erdogan and Sih [20] proposed the maximum circumferential tensile stress criterion. And then they conducted on tensile fracture experiments using thin plexiglass plates, which verified their theory. The maximum circumferential tensile stress criterion depends on only the circumferential tensile stress, while it is independent of the material parameters. From the perspective of the energy, Palaniswamy and Knauss [21] proposed the maximum energy release rate criterion. Sih [22] proposed the SED criterion which considered the effect of the material parameters. Malíková et al. [23] shown that the SED criterion was accurate enough for most geometries and distances from the crack tip. So the SED criterion is applied to predict crack propagation direction in this paper. This paper is organized as follows. In the section two, based on the distributed dislocation technique, the solution of an infinite plane containing a macro-crack and an arbitrary oriented micro-crack is presented. In the section three, the finite element models are established to verify theoretical derivation in section two. In the section four, the influence of various parameters on SIF of the macro-crack and the macro-crack propagation direction are analyzed. In the section five, some conclusions are drawn. 2. Formulation 2.1. Dislocation influence function in an infinite elastic plane containing a macro-crack and an arbitrary oriented micro-crack There is an edge dislocation at the origin, with Burgers vector components bx and by . The stresses at a point ðx; yÞ induced by the dislocation can be obtained by Hill’s book [12].
2l fb G ðx; yÞ þ by Gyxx ðx; yÞg pðj þ 1Þ x xxx 2l r yy ðx; yÞ ¼ fb G ðx; yÞ þ by Gyyy ðx; yÞg pðj þ 1Þ x xyy 2l r xy ðx; yÞ ¼ fb G ðx; yÞ þ by Gyxy ðx; yÞg pðj þ 1Þ x xxy
r xx ðx; yÞ ¼
ð1Þ
where l is shear modulus. j is the Kolosov’s constant: j ¼ ð3 mÞ=ð1 þ mÞ for plane stress, j ¼ 3 4m for plane strain, m being the Poisson’s ratio. Gðx; yÞ is the dislocation influence function. The first subscript on the G denotes the Burgers vector, and the last two denote the associated traction. The expressions of Gðx; yÞ can be given by
y x y ð3x2 þ y2 Þ; Gyxx ðx; yÞ ¼ 4 ðx2 y2 Þ; Gxyy ðx; yÞ ¼ 4 ðx2 y2 Þ; r4 r r x x y Gyyy ðx; yÞ ¼ 4 ðx2 þ 3y2 Þ; Gxxy ðx; yÞ ¼ 4 ðx2 y2 Þ; Gyxy ðx; yÞ ¼ 4 ðx2 y2 Þ: r r r
Gxxx ðx; yÞ ¼
ð2Þ
where r2 ¼ x2 þ y2 . The problem of an infinite plane containing a macro-crack and an arbitrary oriented micro-crack will be analyzed in the following. The present problem can be divided into two sub-problems through the superposition principle. The first refers to the problem of an infinite medium in the absence of cracks under remote loadings. The second can be described as the problem of an infinite elastic plane containing a macro-crack and an arbitrary oriented micro-crack without external loads, and the two cracks are replaced by the continuous distributed dislocations. Firstly, the solution of the second sub-problem will be presented. An edge dislocation is located at ðn; 0Þ. The Burgers vector of the edge dislocation is b, having components bx and by , as shown in Fig. 1. In order to convenient statement, the macro-crack is called as crack 1 and the micro-crack is called as crack 2. The stresses at a point ðx1 þ n; y1 Þ induced by the edge dislocation at a point ðn; 0Þ can be given by
o 2l n 12 bx G12 xxx ðx1 ; y1 Þ þ by Gyxx ðx1 ; y1 Þ pðj þ 1Þ o 2l n 12 r yy ðx0 ; nÞ ¼ bx G12 xyy ðx1 ; y1 Þ þ by Gyyy ðx1 ; y1 Þ pðj þ 1Þ o 2l n 12 r xy ðx0 ; nÞ ¼ bx G12 xxy ðx1 ; y1 Þ þ by Gyxy ðx1 ; y1 Þ pðj þ 1Þ
r xx ðx0 ; nÞ ¼
ð3Þ
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
97
Fig. 1. The macro-crack and an arbitrary oriented micro-crack in an infinite plane.
where, as before, the superscripts 12 on the G denote the associated stresses at the position on crack 2 due to the distributed dislocation along crack 1. The expressions of the influence functions G12 ðx1 ; y1 Þ can be obtained by formula (2). x1 and y1 are defined in Fig. 1 and their expressions can be given by
x1 ¼ d cos h n þ x0 cos a
ð4Þ
y1 ¼ d sin h þ x0 sin a
The stresses in the local coordinate system o0 x0 y0 can be obtained by Mohr stress transformation.
o 2l n 12 0 0 bx G12 xy0 y0 ðx ; nÞ þ by Gyy0 y0 ðx ; nÞ pðj þ 1Þ o 2l n 0 0 r x0 y0 ðx ; nÞ ¼ b G12 ðx0 ; nÞ þ by G12 yx0 y0 ðx ; nÞ pðj þ 1Þ x xx0 y0
r y0 y0 ðx0 ; nÞ ¼
ð5Þ
where 12 12 12 0 2 G12 xy0 y0 ðx ; nÞ ¼ Gxxx ðx1 ; y1 Þsin a þ Gxyy ðx1 ; y1 Þcos a Gxxy ðx1 ; y1 Þ sin 2a 2
12 12 12 0 2 G12 yy0 y0 ðx ; nÞ ¼ Gyxx ðx1 ; y1 Þsin a þ Gyyy ðx1 ; y1 Þcos a Gyxy ðx1 ; y1 Þ sin 2a h i 12 12 12 0 G12 xx0 y0 ðx ; nÞ ¼ Gxyy ðx1 ; y1 Þ Gxxx ðx1 ; y1 Þ sin a cos a þ Gxxy ðx1 ; y1 Þ cos 2a h i 12 12 12 0 G12 yx0 y0 ðx ; nÞ ¼ Gyyy ðx1 ; y1 Þ Gyxx ðx1 ; y1 Þ sin a cos a þ Gyxy ðx1 ; y1 Þ cos 2a 2
ð6Þ
11 11 0 0 0 For formula (6), setting d ¼ 0 and h ¼ a ¼ 0, the dislocation influence functions G11 xy0 y0 ðx ; nÞ, Gyy0 y0 ðx ; nÞ, Gxx0 y0 ðx ; nÞ and 0 G11 yx0 y0 ðx ; nÞ
can be obtained. In the same way, the associated tractions at position on crack 1 due to the distributed dislocation along crack 2 can be obtained. 2.2. The singular integral equations about the dislocation density function By distributing continuous dislocations along crack 1, in the local coordinate system o0 x0 y0 , the traction components along crack 2 induced by the total dislocations along crack 1 can be given by
r y0 y0 ðx0 Þ ¼
2l pðj þ 1Þ
2l r x0 y0 ðx Þ ¼ pðj þ 1Þ 0
Z
Z
a1
h
a1 a1
h
a1
i 12 1 0 0 B1x ðnÞG12 xy0 y0 ðx ; nÞ þ By ðnÞGyy0 y0 ðx ; nÞ dn
i 12 1 0 0 B1x ðnÞG12 xx0 y0 ðx ; nÞ þ By ðnÞGyx0 y0 ðx ; nÞ dn
ð7Þ
1
where B ðnÞ is the dislocation density function along crack 1. In the same way, the tractions components on crack 1 due to the distributed dislocations along crack 2 can obtained. The first sub-problem refers to an infinite plane in the absence of cracks subjected to remote loadings. The stresses com~ yy and r ~ xy . It is ensured that the crack face is traction-free, so the integral ponents of the sub-problem are assumed to be r equations can be established based on the superposition principle.
r~ ij ðxÞ þ r ij ðxÞ ¼ 0 a1 < x < a1 ; ij ¼ yy or xy r~ ij ðx0 Þ þ r ij ðx0 Þ ¼ 0 a2 < x0 < a2 ; ij ¼ y0 y0 or x0 y0
ð8Þ
98
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
Further
jþ1~ 1 rij ðxÞ ¼ 2l p
Z
1
p
a2
h
a2
Z
p
a2
a2
h
a1
a1
h
i 11 1 B1x ðnÞG11 xij ðx; nÞ þ By ðnÞGyij ðx; nÞ dnþ
i 21 2 0 0 0 0 B2x0 ðn0 ÞG21 x0 ij ðx; n Þ þ By0 ðn ÞGy0 ij ðx; n Þ dn
jþ1~ 0 1 rij ðx Þ ¼ 2l p 1
Z
Z
a1
a1
h
0 B1x ðnÞG12 xij ðx ; nÞ
þ
a1 < x < a1 ; i
0 B1y ðnÞG12 yij ðx ; nÞ
i 22 2 0 0 0 0 0 0 B2x0 ðn0 ÞG22 x0 ij ðx ; n Þ þ By0 ðn ÞGy0 ij ðx ; n Þ dn
ij ¼ yy or xy ð9Þ
dnþ
a2 < x0 < a2 ;
ij ¼ y0 y0 or x0 y0
2.3. Numerical solution of singular integral equations The singular integral Eq. (9) can be solved by Gauss-Chebyshev quadrature [24,25]. Letting n ¼ a1 s, x ¼ a1 t, n0 ¼ a2 s and x ¼ a2 t, the integral over ½a1 ; a1 and ½a2 ; a2 may be normalized over ½1; 1. According to Hill’s book [11], the unknown function in present problem is ‘singular-at-both-ends’, so the form of dislocation density function may be given by 0
/ðsÞ BðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 s2 Þ
ð10Þ
where /ðsÞ is an unknown function. Eq. (9) can be replaced approximately by a series of algebraic equations by employing Gauss-Chebyshev quadrature method. It can be written as N n h i h io jþ1~ 1X 11 21 21 1 2 2 rij ðtk Þ ¼ a1 /1x ðsI ÞG11 xij ðt k ; sI Þ þ /y ðsI ÞGyij ðt k ; sI Þ þ a2 /x0 ðsI ÞGx0 ij ðt k ; sI Þ þ /y0 ðsI ÞGy0 ij ðt k ; sI Þ 2l N I¼1
ij ¼ yy or xy;
k ¼ 1...N 1
N n h i h io jþ1~ 1X 12 22 22 1 2 2 rij ðtk Þ ¼ a1 /1x ðsI ÞG12 xij ðt k ; sI Þ þ /y ðsI ÞGyij ðt k ; sI Þ þ a2 /x0 ðsI ÞGx0 ij ðt k ; sI Þ þ /y0 ðsI ÞGy0 ij ðt k ; sI Þ 2l N I¼1
ij ¼ y0 y0 or x0 y0 ;
ð11Þ
k ¼ 1...N 1
where
sI ¼ cos½pð2I 1Þ=2N I ¼ 1; 2; ; N t k ¼ cos½pk=N k ¼ 1; 2; ; N 1
ð12Þ
For Eq. (11), there are 4N unknowns /ðsI Þ while only 4(N-1) algebraic equations can be found. So four extra equations must be established to solve Eq. (11). Notice that, on the crack, there is on net dislocation. N N N N X X X X /1x ðsI Þ ¼ /1y ðsI Þ ¼ /2x0 ðsI Þ ¼ /2y0 ðsI Þ ¼ 0 I¼1
I¼1
I¼1
ð13Þ
I¼1
Then, SIFs at crack tip may be given by
pffiffiffiffiffiffi 2l K I ð1Þ ¼ pa / ð1Þ jþ1 y pffiffiffiffiffiffi 2l K II ð1Þ ¼ pa / ð1Þ jþ1 x
ð14Þ
where
N 1X 2I 1 2I 1 /ðsI Þ sin pð2N 1Þ sin N I¼1 4N 4N N 1X 2I 1 2I 1 /ð1Þ ¼ /ðsNþ1I Þsin pð2N 1Þ sin N I¼1 4N 4N
/ðþ1Þ ¼
ð15Þ
Next, the stress field will be calculated. An edge dislocation is located at position ðn; 0Þ. For an arbitrary point (x, y), the dislocation influence function can be written as
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
xn ððx nÞ2 y2 Þ r4 y xn G1xyy ðx; yÞ ¼ 4 ððx nÞ2 y2 Þ; G1yyy ðx; yÞ ¼ 4 ððx nÞ2 þ 3y2 Þ r r xn y 2 1 1 2 Gxxy ðx; yÞ ¼ 4 ððx nÞ y Þ; Gyxy ðx; yÞ ¼ 4 ððx nÞ2 y2 Þ r r
G1xxx ðx; yÞ ¼
y ð3ðx nÞ2 þ y2 Þ; r4
99
G1yxx ðx; yÞ ¼
ð16Þ
where r 2 ¼ ðx nÞ2 þ y2 . By the same method, G2 ðx; yÞ can be obtained. And then, the stress field may be given by
rxx ðx; yÞ ¼ r~ xx ðx; yÞ þ
N n h i h io 1 2l X a1 /1x ðsI ÞG1xxx ðx; yÞ þ /1y ðsI ÞG1yxx ðx; yÞ þ a2 /2x0 ðsI ÞG2x0 xx ðx; yÞ þ /2y0 ðsI ÞG2y0 xx ðx; yÞ N j þ 1 I¼1
rxy ðx; yÞ ¼ r~ xy ðx; yÞ þ
N n h i h io 1 2l X a1 /1x ðsI ÞG1xxy ðx; yÞ þ /1y ðsI ÞG1yxy ðx; yÞ þ a2 /2x0 ðsI ÞG2x0 xy ðx; yÞ þ /2y0 ðsI ÞG2y0 xy ðx; yÞ N j þ 1 I¼1
ryy ðx; yÞ ¼ r~ yy ðx; yÞ þ
N n h i h io 1 2l X a1 /1x ðsI ÞG1xyy ðx; yÞ þ /1y ðsI ÞG1yyy ðx; yÞ þ a2 /2x0 ðsI ÞG2x0 yy ðx; yÞ þ /2y0 ðsI ÞG2y0 yy ðx; yÞ N j þ 1 I¼1
I ¼ 1; 2 . . . N ð17Þ The polar coordinate system ðR; bÞ is established at the macro-crack tip, as shown in Fig. 1. The stress components in cylindrical coordinate may be given by
rrr ¼ rxx ðx; yÞ cos2 b þ ryy ðx; yÞ sin2 b þ rxy ðx; yÞ sin2 ð2bÞ rrh ¼ ½rxx ðx; yÞ ryy ðx; yÞ sin b cos b þ rxy ðx; yÞ cosð2bÞ rhh ¼ rxx ðx; yÞ sin2 b þ ryy ðx; yÞ cos2 b rxy ðx; yÞ sinð2bÞ
ð18Þ
x ¼ a1 þ R cos b y ¼ R sin b
ð19Þ
where
2.4. The SED criterion From the energy point of view, Sih [22] proposed the SED, and he thought that the crack propagated in the direction where the strain energy density reached its minimum. So the criterion can be given by
@S @2S ¼ 0; 2 > 0; @b @b
where S ¼
1 2l
jþ1 8
ðrRR þ rbb Þ2 rRR rbb þ r2Rb
ð20Þ
3. Finite element method verification In the section, the solution to the problem derived in section two are verified by the finite element software ANSYS. Two cases are considered. The first case is that a micro-crack is parallel to the macro-crack in an infinite square plane under mode I load, that is a ¼ 0. The second is that a micro-crack is located at the radial direction of the macro-crack in an infinite square plane subjected to mode I load, that is h ¼ a. Taking the first case as an example, the finite element model is established, as shown in Fig. 2. The elasticity modulus is 210 GPa, and Poisson’s ratio is 0.3. The element type is plane183. Classical singular element is applied at the crack tip. There are 20 elements around the crack tip. The length of the macro-crack and the microcrack are also 10 mm. The length of the model is 200 mm, which is relatively large in comparison to length of cracks. So it is considered as an infinite plane. The model is subjected to tensile load in the y direction. Comparing with the theoretical solution, the calculation of finite element method is elastic. 4. Results and discussion In order to shown the numerical results clearly, the SIF of the macro-crack is normalized by
K0 ¼ r
pffiffiffiffiffiffiffiffi pa1
The mechanics behavior of the right macro-crack tip is only analyzed in the following.
ð21Þ
100
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
Fig. 2. The finite element model of the parallel micro-crack
4.1. Influence of the micro-crack orientation on SIF of the macro-crack As the section three introduced, the first case is a ¼ 0, that is parallel micro-crack. And the second is h ¼ a, that is radial micro-crack. The variation of normalized SIF K I =K 0 versus micro-crack orientation h under uniaxial tensile load is depicted for different values of distance d=a1 in Fig. 3. And comparison between theory and FEM results is shown. Solid lines denote theoretical results; dash lines denote FEM results. a2 =a1 is taken as 1. The results in Fig. 3 show that the parallel micro-crack increases SIF when h is less than about 37.5 degree, and decreases SIF when h is more than about 45 degree. While the radial micro-crack increases SIF when h is less than about 60 degree, and it has a little effect when h is more than about 60 degrees. The shielding effect of the parallel micro-crack is more apparent than that of the radial micro-crack. What’s more, Fig. 3 shows that the results of current theoretical method are very consistent with FEM results. The theoretical method is efficient to solve the crack problem. For different crack length, location, orientation and angle, the crack parameters only need to change, and the crack model need not to re-establish. So all results in the below section are obtained by the theoretical method.
4.2. Influence of the distance between micro-crack and macro-crack on SIF of the macro-crack The variation of normalized SIF K I =K 0 versus distance d=a1 under uniaxial tensile load is shown for different values of the micro-crack orientation h in Fig. 4. a2 =a1 is taken as 1. The results show that the effect of the micro-crack on SIF of the macrocrack will be getting weaker as the distance between the micro-crack and the macro-crack increases, which agrees with the results in Fig. 3.
(a) 1.3
(b)
K /K0
1.2
1.1
1.0
d/a1=2.5 d/a1=3 d/a1=4 d/a1=2.5 d/a1=3 d/a1=4
1.25 1.20
K /K0
d/a1=2.5 d/a1=3 d/a1=4 d/a1=2.5 d/a1=3 d/a1=4
1.15 1.10 1.05
0.9 1.00 0.8
0
15
30
45
θ (degree)
60
75
90
0
15
30
45
60
θ (degree)
Fig. 3. K I =K 0 versus h under uniaxial tensile load. (a) the parallel micro-crack; (b) the radial micro-crack.
75
90
101
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
(a)
(b) θ=15 degrees θ=45 degrees θ =75 degrees
1.4
K /K0
1.2
K /K0
θ=15 degrees θ=45 degrees θ=75 degrees
1.6
1.0
1.4
1.2
0.8
1.0
2.0
2.5
3.0
3.5
4.0
4.5
5.0
2.0
2.5
3.0
d/a1
3.5
4.0
d/a1
4.5
5.0
Fig. 4. K I =K 0 versus d=a1 under uniaxial tensile load. (a) the parallel micro-crack; (b) the radial micro-crack.
4.3. Influence of the micro-crack length on SIF of the macro-crack The variation of normalized SIF K I =K 0 versus micro-crack length a2 =a1 under uniaxial tensile load is shown for different values of the micro-crack orientation h in Fig. 5. d=a1 is taken as 2.5. On the whole, as the micro-crack length increases, the effect of micro-crack on SIF at the macro-crack tip will be getting stronger. There is almost no effect on SIF of the macro-crack when a2 =a1 is less than 0.1 for d=a1 ¼ 2:5. For the radial micro-crack, the effect of the micro-crack length on SIF of the macrocrack is little when h is more than 60 degrees, which is consistent with the results in Fig. 3(b). 4.4. Influence of the micro-crack on the macro-crack propagation direction As shown in Fig. 1, a polar coordinate system ðR; bÞ is established at the macro-crack tip. The strain energy density can be obtained by the Eqs. (17)(20). The variation of normalized the strain energy density 2Sl=r2 versus b under uniaxial tensile load is shown for different values of the micro-crack orientation h in Fig. 6. d=a1 is taken as 2.5 and R=a1 is taken as 0.1. And then the macro-crack propagation direction can be predicted by the SED criterion described in Section 2.4. The variation of the macro-crack propagation direction versus micro-crack orientation h under uniaxial tensile load is plotted for different values of the micro-crack length a2 =a1 in Fig. 7. The results show that the mode I macro-crack will not propagate along the direction of crack face when the micro-crack locates in the vicinity of the macro-crack tip. For the first case, the parallel micro-crack acts an attraction effect on the macro-crack propagation at about 15 < h < 75 degrees, while the parallel micro-crack acts a repulsion effect on the macro-crack propagation at about h < 10 and h > 75 degrees. For the second case, the radial micro-crack acts an attraction effect on the macro-crack propagation at about h > 15 degrees, while the radial micro-crack acts a repulsion effect on the macro-crack propagation at about h < 15 degrees. As the micro-crack length increases, the influence on macro-crack propagation direction will be getting stronger.
(a)
θ=0 degrees θ=30 degrees θ=45 degrees θ=60 degrees θ=90 degrees
1.20
1.15
K /K0
1.1
K /K0
(b)
θ=0 degrees θ=30 degrees θ=45 degrees θ=60 degrees θ=90 degrees
1.2
1.0
1.10
1.05 0.9 1.00 0.1
0.2
0.3
0.4
0.5
0.6
a 2 /a 1
0.7
0.8
0.9
1.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a 2 /a 1
Fig. 5. K I =K 0 versus a2 =a1 under uniaxial tensile load. (a) the parallel micro-crack; (b) the radial micro-crack.
0.9
1.0
102
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
(a)
7.5
(b)
6.0
θ=0 degrees θ=30 degrees θ=60 degrees θ=90 degrees
6.0
4.5
2Sμ/σ2
2Sμ/σ2
7.5
θ=0 degrees θ=30 degrees θ=60 degrees θ=90 degrees
4.5
3.0 3.0 1.5 -90
-60
-30
0
30
60
1.5
90
-90
-60
-30
β (degree)
0
30
60
90
β (degree)
Fig. 6. 2Sl=r2 versus b for a2 =a1 ¼ 1 under uniaxial tensile load. (a) the parallel micro-crack; (b) the radial micro-crack.
a2/a1=0.8 a2/a1=0.5 a2/a1=0.2
8
4
0
-4 0
15
30
45
θ (degree)
60
75
90
(b) macro-crack propagation direction (degree)
macro-crack propagation direction (degree)
(a) 12
a2/a1=0.8 a2/a1=0.5 a2/a1=0.2
8
4
0
-4
0
15
30
45
60
75
90
θ (degree)
Fig. 7. The macro-crack propagation direction versus h for d=a1 ¼ 2 under uniaxial tensile load. (a) the parallel micro-crack; (b) the radial micro-crack.
5. Conclusions In the paper, the problem of an infinite plane containing a macro-crack and an arbitrary oriented micro-crack is investigated based on the distributed dislocation technique and Gauss-Chebyshev quadrature method. Influence of the micro-crack on the macro-crack is analyzed. Some conclusions can be summarized in the following. (1) The parallel micro-crack increases SIF of the macro-crack at about h < 37:5 degrees, and decreases SIF of the macrocrack at about h > 45 degrees, while the radial micro-crack increases SIF of the macro-crack at about h < 60 degrees, and it has a little effect on the macro-crack when h > 60 degrees. (2) The effect of the micro-crack on SIF of the macro-crack will be getting weaker as the distance between the micro-crack and the macro-crack increases, while the effect of the micro-crack on SIF of the macro-crack will be getting stronger as the micro-crack length increases. (3) The parallel micro-crack acts an attraction effect on the macro-crack propagation at about 15 < h < 75 degrees, while it acts a repulsion effect on the macro-crack propagation at about h < 10 and h > 75 degrees. The radial micro-crack acts an attraction effect on the macro-crack propagation at about h > 15 degrees, while it acts a repulsion effect on the macro-crack propagation at about h < 15 degrees.
Acknowledgments The work is supported by the National Natural Science Foundation of China (11472230), the National Natural Science Foundation of China Key Project (U1134202/E050303) and Sichuan Provincial Youth Science and Technology Innovation Team (2013TD0004).
X. Li et al. / Engineering Fracture Mechanics 177 (2017) 95–103
103
References [1] Kachanov M. A simple technique of stress analysis in elastic solids with many cracks. Int J Fract 1985;28:R11–9. [2] Kachanov M. Elastic solids with many cracks: a simple method of analysis. Int J Solids Struct 1987;23:23–43. [3] Chudnovsky A, Dolgopolsky A, Kachanov M. Elastic interaction of a crack with a microcrack array—II. Elastic solution for two crack configurations (piecewise constant and linear approximations). Int J Solids Struct 1987;23:11–21. [4] Gong SX, Horii H. General solution to the problem of microcracks near the tip of a main crack. J Mech Phys Solids 1989;37:27–46. [5] Gong SX, Meguid SA. Microdefect interacting with a main crack: a general treatment. Int J Mech Sci 1992;34:933–45. [6] Gong SX. On the main crack-microcrack interaction under mode III loading. Eng Fract Mech 1995;51:753–62. [7] Hori M, Nemat-Nasser S. Interacting micro-cracks near the tip in the process zone of a macro-crack. J Mech Phys Solids 1987;35:601–29. [8] Meguid SA, Gaultier PE, Gong SX. A comparison between analytical and finite element analysis of main crack-microcrack interaction. Eng Fract Mech 1991;38:451–65. [9] Xiangqiao Y. An effective numerical approach for interaction of macrocrack with microcracks. Chin J Theor Appl Mech 2006;38:119–24. [10] Xiangqiao Y. Microcrack interacting with a finite main crack. J Harbin Inst Technol 2006;38:503–6. [11] Hills D, Kelly P, Dai D, Korsunsky A. Solution of crack problems: the distributed dislocation technique, 1996. Kluwer Academic Publishers; 1996. [12] Hills DA, Kelly P, Dai D, Korsunsky A. Solution of crack problems: the distributed dislocation technique. Springer Science & Business Media 2013. [13] Erdogan F, Gupta G, Ratwani M. Interaction between a circular inclusion and an arbitrarily oriented crack. J Appl Mech 1974;41:1007–13. [14] Jin X, Keer LM. Solution of multiple edge cracks in an elastic half plane. Int J Fract 2006;137:121–37. [15] Tao Y, Fang Q, Zeng X, Liu Y. Influence of dislocation on interaction between a crack and a circular inhomogeneity. Int J Mech Sci 2014;80:47–53. [16] Mousavi SM, Lazar M. Distributed dislocation technique for cracks based on non-singular dislocations in nonlocal elasticity of Helmholtz type. Eng Fract Mech 2015;136:79–95. [17] Sharma K, Bui TQ, Bhargava R, Yu T, Lei J, Hirose S. Numerical studies of an array of equidistant semi-permeable inclined cracks in 2-D piezoelectric strip using distributed dislocation method. Int J Solids Struct 2016;80:137–45. [18] Richard H, Fulland M, Sander M. Theoretical crack path prediction. Fatigue Fract Eng Mater Struct 2005;28:3–12. [19] Richard H, Schramm B, Schirmeisen N-H. Cracks on mixed mode loading–theories, experiments, simulations. Int J Fatigue 2014;62:93–103. [20] Erdogan F, Sih G. On the crack extension in plates under plane loading and transverse shear. J Fluids Eng 1963;85:519–25. [21] Palaniswamy K, Knauss W. Propagation of a crack under general, in-plane tension. Int J Fract 1972;8:114–7. [22] Sih GC. Strain-energy-density factor applied to mixed mode crack problems. Int J Fract 1974;10:305–21. [23] Malíková L, Vesely´ V, Seitl S. Crack propagation direction in a mixed mode geometry estimated via multi-parameter fracture criteria. Int J Fatigue 2016;89:99–107. [24] Erdogan F, Gupta GD, Cook T. Numerical solution of singular integral equations. Methods of analysis and solutions of crack problems. Springer; 1973. p. 368–425. [25] Kaya AC, Erdogan F. On the solution of integral equations with strongly singular kernels. Q Appl Math 1986.