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The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate
Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx
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The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate D. Peng ⇑, R. Jones Centre of Expertise Structural Mechanics, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Victoria 3800, Australia
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Fourier transform Interacting unequal crack Singular integral equation Stress intensity factor
a b s t r a c t This paper uses a Fourier transform technique to solve the problem of two interacting collinear unequal cracks in a finite width plate. This approach reduces the problem to the solution of two coupled integral equations each with a singular kernel which is the solved using Cauchy–Chebyshev polynomials. The solution is first validated by comparing the solutions for the case of two equal cracks in a finite width plate and for the case of two unequal cracks in an infinite width plate with published solutions. The problem of (interacting) cracks that grow from collocated corrosion pits that lie close to the edge of a structural member is then studied. Here it is shown that the closer that the two cracks are to a boundary accelerates crack growth towards the boundary and decelerates linking of the cracks. Ó 2014 Published by Elsevier Ltd.
1. Introduction The problem of crack interaction and its effect on structural integrity is a universal problem that has particular relevance to aging rail and aircraft structures. However, as revealed in [1], an analysis of a structure in which crack interaction is neglected can be very non-conservative as the cumulative effect of interacting cracks may significantly degrade structural integrity [2]. Due to the complexity of the problem, there is limited literature available on the interaction of multiple unequal cracks in a finite geometry. Exact solutions obtained by analytical methods are restricted to an infinite geometry [3–10]. In such cases the approaches used include: the boundary conditions perturbation technique, the complex potential method and the singular integral equation approach. For problems involving crack interaction in finite geometries analytical solutions are currently limited to symmetrical equal size cracks [11–17]. With the rapid advancement of computer technology numerical methods have been used to solve cracks interaction with finite boundary, such as finite element alternating technique algorithm [18,19], alternating indirect boundary element technique [20,21], the hybrid finite element method [22], and the reflecting shadow method [23]. The present paper derives an analytical solution to the problem of two interacting unequal collinear cracks in a finite width plate, see Fig. 1. A Fourier transform method, which uses dislocation density functions, is used to determine the associated singular integral equations which are then solved using Cauchy–Chebyshev
polynomials. This approach is first validated by comparing the solutions for the case of two equal cracks in a finite width plate and for the case of two unequal cracks in an infinite width plate with published solutions. The problem of two (interacting) cracks that grow from (nearby) collocated corrosion pits that lie close to the edge of a structural member is then studied. Here it is shown that the closer that the two cracks are to a boundary accelerates crack growth towards the boundary and decelerates linking of the cracks.
2. Basic equations of elasticity Consider an infinitely long homogeneous isotropic elastic plate with a width H that contains two collinear unequal cracks that lie perpendicular to the edges of the plate, see Fig. 1. The displacement components in the X, Y directions are defined as u and v respectively. As the displacement components are symmetric about the Y axis, the displacement components are not symmetric about the X axis. Therefore, a general Fourier transform with respect to y axis is introduced. The displacements may thus be expressed in terms of the following Fourier integrals [24]:
rffiffiffiffi Z 2
rffiffiffiffiffiffiffi Z 1 1 uðx; yÞ ¼ f 1 ða; yÞsinaxda þ g ðx; cÞeicy dc 2p 1 1 p 0 rffiffiffiffi Z rffiffiffiffiffiffiffi Z 1 2 1 1 v ðx; yÞ ¼ f 2 ða; yÞcosaxda þ g ðx; cÞeicy dc 2p 1 2 p 0 1
ð1Þ ð2Þ
⇑ Corresponding author. http://dx.doi.org/10.1016/j.tafmec.2014.10.007 0167-8442/Ó 2014 Published by Elsevier Ltd.
Please cite this article in press as: D. Peng, R. Jones, The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.007
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D. Peng, R. Jones / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx
Here:
rffiffiffiffi Z 2
rffiffiffiffi Z 2
1
The corresponding stress field are obtained by using the following relations [25].
1
uðx;yÞsinaxdx; f 2 ða;yÞ ¼ v ðx;yÞcosaxdx ð3Þ p p rffiffiffiffiffiffiffi 0Z rffiffiffiffiffiffiffi Z 0 1 1 1 1 uðx;yÞeicy dy g 2 ðx; cÞ ¼ v ðx; yÞeicy dy ð4Þ g 1 ðx; cÞ ¼ 2p 1 2p 1
f 1 ða;yÞ ¼
The unknown functions f1(a, y), f2(a, y), g1 (x, c) and g2(x, c) can be determined by boundary conditions. In the absence of body forces the equations of equilibrium [25] can be expressed as follow:
@ @u @ v ¼0 lr2 u þ ðk þ lÞ þ @x @x @y @ @u @ v ¼0 lr2 v þ ðk þ lÞ þ @y @x @y tE k¼ ð1 þ tÞð1 2tÞ
8 r ðx; yÞ ¼ ðk þ 2lÞ @u þ k @@yv > @x > < xx ryy ðx; yÞ ¼ ðk þ 2lÞ @@yv þ k @u @x > > : s ðx; yÞ ¼ @u þ @ v xy
(rffiffiffiffi Z 2
rxx ðx; yÞ ¼ l
ð7Þ
ð10Þ
g 2 ðx; cÞ ¼ ðA5 þ A6 xÞecx þ ðA7 þ A8 xÞecx x P 0
ð11Þ
c
ð9Þ
unknown coefficients A1 to A6 can be determined from boundary conditions. The formulas (1) and (2) become the following Fourier integrals: 1
½ðA1 þ A2 yÞeay þ ½ðA3 þ A4 yÞeay sinaxda 0 rffiffiffiffiffiffi ffiZ 1 1 jcj j i A5 þ þ x A6 ejcjxicy dc 2p 1 c jcj
uðx; yÞ ¼
rffiffiffiffi Z 2
i þ y A2 eay p 0 a h j i o y A4 eay cosaxda þ A3 þ rffiffiffiffiffiffiffi Z a 1 1 þ ðA5 þ xA6 Þejcjxicy dc 2p 1 1
nh
A1 þ
(rffiffiffiffi Z 2
1
p
½ð2aA1 þ h1 j 2ayiA2 Þeay
0
þð2aA3 þ h1 j þ 2ayiA4 Þeay sinaxda ) rffiffiffiffiffiffiffi Z 1 1 1j jcjxicy þ2i cA5 þ cx A6 e dc ð18Þ 2p 1 2 Here j = 3 4m for plane strain and j = (3 m)/(1 + m) for generalised plane stress. The stress and mixed boundary conditions may be expressed as
8 > : y¼H
sxy ð0; yÞ ¼ 0 ryy ðx; 0Þ ¼ 0 sxy ðx; 0Þ ¼ 0 ryy ðx; HÞ ¼ 0 sxy ðx; HÞ ¼ 0
ð19Þ
and
p
v ðx; yÞ ¼
1
sxy ðx; yÞ ¼ l
ð12Þ
The Kolosov’s material constant j = 3 4t is for plane strain and j = (3 t)/(1 + t) is for plane stress. Here m is Poisson’s ratio. The
(rffiffiffiffi Z 2
ð16Þ
½ð2aA1 h1 þ j þ 2ayiA2 Þeay p 0 þð2aA3 þ h1 þ j 2ayiA4 Þeay cosaxda ) rffiffiffiffiffiffiffi Z 1 1 3j jcjxicy þ2i cA5 þ cx A6 e dc ð17Þ 2p 1 2
The condition that as x ? 1 the stresses should be finite requires that
A7 ¼ A8 ¼ 0
½ð2aA1 þ hj 3 þ 2ayiA2 Þeay
0
ryy ðx; yÞ ¼ l
ð8Þ
c
p
1
þð2aA3 þ h3 j þ 2ayiA4 Þeay cosaxda ) rffiffiffiffiffiffiffi Z 1 1 1þj þ2i cA5 þ þ cx A6 ejcjxicy dc 2p 1 2
ð6Þ
f 1 ða; yÞ ¼ ðA1 þ A2 yÞeay þ ðA3 þ A4 yÞeay h j i h j i f 2 ða; yÞ ¼ A1 þ þ y A2 eay þ A3 þ y A4 eay a a
j j þ x A6 ecx A7 þ x A 8 e cx x P 0 g 1 ðx; cÞ ¼ i A5 þ
@x
This yields:
ð5Þ
Here l, t and E are shear modulus, Poisson’s ratio and modulus of elasticity respectively. Substituting Eqs. (3) and (4) into Eqs. (5)– (7) yields:
rffiffiffiffi Z 2
@y
ð15Þ
ð13Þ
limx!0 rxx ðx; yÞ ¼ PðyÞ y 2 L uð0; yÞ ¼ 0
j
ð14Þ
y2L
where P(y) is the crack surface traction which can be an arbitrary function of y. Here, L and L denote the crack face and non-cracked boundary (along the centre line of the plate) respectively. For the specific problem addressed in this paper the mixed boundary conditions (20) can be expressed as:
limx!0 rxx ðx; yÞ ¼ pj ðyÞ aj 6 y 6 bj j ¼ 1; 2 uj ð0; yÞ ¼ 0
Y
ð20Þ
y < a1 ; b1 < y < a2 ; b2 < y 6 H ð21Þ
where pj(y), j = 1, 2, are the crack face tractions on crack 1 and 2 respectively. Using the boundary conditions (19) the unknown coefficients A1 A6 in Eqs. (13)–(18), can be expressed in the term of a single parameter A⁄, viz:
H
b2 a2 a1
b1
o Fig. 1. Two collinear asymmetric internal cracks in finite stripe.
X
Z 1 1 A1 ¼ pffiffiffiffiffiffiffi ðð1 2ajHÞeaH eaH ÞðeaH eicH Þða icÞ 4 2p 1 þðð2aH þ jÞeaH jeaH ÞðeaH eicH Þða þ icÞ þða icÞ
A c2 dc
Dða2 þ c2 Þ
2
ð22Þ
Please cite this article in press as: D. Peng, R. Jones, The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.007
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D. Peng, R. Jones / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx
1 A2 ¼ pffiffiffiffiffiffiffi 2 2p þðe
aH
Z
1
h
xðx; y; cÞ ¼
2aHeaH ðeaH eicH Þða icÞ
1
eaH ÞðeaH eicH Þða þ icÞ
A c2 dc
Dða þ c 2
2 Þ2
ð23Þ
A c2 dc
1 A4 ¼ pffiffiffiffiffiffiffi 2 2p
Z
Dða2 þ c2 Þ
1
þ2aHeaH ðeaH eicH Þða þ icÞ ð1 jÞi A5 ¼ 8 A6 ¼
jcji 4
aH
A2 ¼ ðB1 e
rffiffiffiffi
p 2
A
i
ð25Þ
2 Þ2
Dða2 þ c
ð26Þ
p
A
D ¼ ðe
A4 ¼ ðB1 þ B2 Þe
e
aH 2
Þ ð2aHÞ
aH
þ ðB3 e
aH
þ B4 e
þ DÞe
ð32Þ ay
ð33Þ ð34Þ
þ ðB3 þ B4 Þe
ay
ð35Þ ð36Þ
B2 ¼ ð2ay 3ÞðeaH eaH Þ þ 2aHeaH
ð37Þ
aH
e
aH
aH
Þ þ 2aHe
B4 ¼ 2aHð2ay þ 3ÞeaH þ eaH eaH
ð38Þ ð39Þ
As a result the mixed boundary conditions, i.e. Eq. (21), can be expressed as:
(
R1 limx!0 p1ffiffiffiffi jcjA xðx; y; cÞdc ¼ PðyÞ=l y 2 L 2p 1 R j 1 A eicy dc ¼ 0 y2L 1þ 1 16
ð40Þ
3. Singular integral equations and stress intensity factors
2
ð28Þ
Substituting Eqs. (22)–(27) into Eqs. (13) and (16) the rxx(x, y) stress and displacement u(0, y) now become
Z
DÞe
ay
B1 ¼ eaH eaH þ 2aHeaH ð2ay 3Þ
ð27Þ
Here we have defined the quantity D to be aH
þ B2 e ay
rffiffiffiffi 2
aH
B3 ¼ ð2ay þ 3Þðe
A c2 dc
ð31Þ
A3 ¼ ðB1 þ B2 Þeay þ ðB3 þ B4 Þeay
ðeaH eaH ÞðeaH eicH Þða icÞ
1
1
A1 ¼ ðB1 eaH þ B2 eaH þ DÞeay þ ðB3 eaH þ B4 eaH þ DÞeay
ð24Þ
2
Z
½aA1 þ icaA2 ðaA3 þ icA4 ÞeicH p 0 a2 cos axda 1 þ ð1 þ jcjxÞejcjxicy 2 2 2Dða2 þ c2 Þ jcj
And
Z 1 1 A3 ¼ pffiffiffiffiffiffiffi ððj 2aHÞeaH jeaH ÞðeaH eicH Þða icÞ 4 2p 1 þðð2aHj þ 1ÞeaH eaH ÞðeaH eicH Þða þ icÞ þða icÞ
rffiffiffiffi 2
1
1 rxx ðx; yÞ ¼ pffiffiffiffiffiffiffi l jcjA xðx; y; cÞdc 2p 1 Z 1 þ j 1 icy uð0; yÞ ¼ A e dc 16 1
ð29Þ
Define the dislocation density function u as:
@ 1þj i uð0; yÞ ¼ @y 16 Z 8 i /ðtÞeict dt A ¼ ð1 þ jÞcp
/ðyÞ ¼
ð30Þ
1
cA eicy dc
ð41Þ
1
t2L
ð42Þ
The first term in Eq. (40) can now be expressed as:
8 i x!0 ð1 þ jÞp
where
Z
lim
¼
PðyÞ
Z
Z 1 1 jcj /ðtÞ pffiffiffiffiffiffiffi A xðx; y; cÞeict dc dt 2p 1 c
y 2 L; t 2 L
l
ð43Þ
To simplify the analysis we use the residue theorem [26] to solve the infinite integral contained in Eq. (31). After somewhat routine manipulation the problem can be reduced to the following system of integral equations for the function /(t):
σ
Z
1 ð1 þ jÞp þ Kðt; yÞ /ðtÞdt ¼ PðyÞ y 2 L; t 2 L ty 4l
ð44Þ
where 2L
Kðt; yÞ ¼
2L
1 2
Z
1
½aA1 eat ð1 atÞA2 eat þ aðH tÞA3 eaðHtÞ
0
1 da y 2 L; t 2 L D p1 ðyÞ; y 2 a1 ; b1 PðyÞ ¼ p2 ðyÞ; y 2 a2 ; b2
þ h1 aðH tÞiA4 eaðHtÞ a
b
/ðtÞ ¼ H
H
/1 ðtÞ; t 2 a1 ; b1 /2 ðtÞ; t 2 a2 ; b2
;
ð45Þ ð46Þ
Here p1(y) and p2(y) are the tractions on the crack surfaces. The mixed boundary value problem may thus be reduced to the following singular integral equations: Z b2 1 ð1 þ jÞp þ K 11 ðt;yÞ /1 ðtÞdt þ K 12 ðt; yÞ/2 ðtÞdt ¼ p1 ðyÞ ty 4l a1 a2 Z b2 Z b1 1 ð1 þ jÞp K 21 ðt; yÞ/2 ðtÞdt þ þ K 22 ðt; yÞ /2 ðtÞdt ¼ p2 ðyÞ ty 4l a1 a2 Z
σ Fig. 2. A finite width plate with two equal collinear cracks.
b1
ð47Þ ð48Þ
The solution of the singular integral Eqs. (47) and (48) is subject to compatibility conditions
Please cite this article in press as: D. Peng, R. Jones, The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.007
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D. Peng, R. Jones / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx
Table 1 Non-dimensional stress intensity factor for symmetric collinear internal cracks in a sheet. a/H
b/H
0.1 0.2 0.4 0.5 0.1 0.5 0.5
0.5 0.6 0.8 0.9 0.9 0.95 0.98
p K(a)/r L
p K(b)/r L
Present
Collins and Cartwright [9]
Present
Collins and Cartwright [9]
1.145 1.136 1.14 1.16 1.69 1.22 1.27
1.176 1.109 1.097 1.127 1.678 1.194 1.268
1.109 1.13 1.15 1.23 1.65 1.51 1.91
1.115 1.094 1.122 1.221 1.694 1.445 1.875
σ
Z
bj
/j ðtÞdt ¼ 0 j ¼ 1; 2
ð49Þ
aj
A
After normalising transformations
2a2
2a1 B
C
D
nj ¼
b
2t ðbj þ aj Þ ; bj aj
σ
Z
1.90
a2/b = 0.2 a2/b = 0.2, [30] a2/b = 2
KI/K01
1.70 1.50
a2/b = 2, [30] a2/b = 5
1.30 1.10
0.6
0.8
1.0
a2/b = 5, [30]
a1/b
Non-Dimensional Stress Intensity Factors
3.40
a2/b = 0.2 a2/b = 0.2, [30] a2/b = 2
2.90
KI/K01
j ¼ 1; 2
the
ð50Þ
2.40
a2/b = 2, [30] a2/b = 5
1.90 1.40 0.4
ð51Þ
1
/j ðnj Þdnj ¼ 0 j ¼ 1; 2
ð52Þ
1
Let us express the unknown functions in the form: 1=2
F j ðtÞ
ð53Þ
Subsequently, the solution of the integral Eqs. (51) and (52) can be obtained by applying several different methods [27,28]. In this paper we will use the Lobatto–Chebyshev method [29]. This yield the following system of algebraic equations, viz: n X 2 Z p X
n1
1
1
s¼1 j¼1
F j ðnjs Þhs ¼
"
# m X dij m fm qk þ km e f ; n ; g ij q js ir njs gir q¼1 q
ð1 þ jÞp pi ðgir Þ i ¼ 1; 2 4l
n X F j ðns Þhs ¼ 0 j ¼ 1; 2
ð54Þ
where
( hs ¼
0.2
through
s¼1
Fig. 4. KI for tip A of two unequal length collinear cracks in a sheet subjected a uniform uniaxial tensile stress.
0.90 0.0
2y ðbj þ aj Þ ; bj aj
(aj, bj)
dij ð1 þ jÞp þ kij ðnj ; gi Þ /j ðnj Þdnj ¼ pi ðgi Þ i ¼ 1;2 nj gi 4l
/j ðtÞ ¼ ð1 t 2 Þ
Non-Dimensional Stress Intensity Factors
0.4
1
j¼1
Fig. 3. Two unequal collinear cracks in an infinite plate under a uniform uniaxial tensile stress.
0.2
intervals
Eqs. (47) and (48) can be written as 2 Z 1 X
0.90 0.0
gj ¼
the
0.6
0.8
1.0
a2/b = 5, [30]
a1/b Fig. 5. KI for tip B of two unequal length collinear cracks in a sheet subjected a uniform uniaxial tensile stress.
1 ; 2
s ¼ 1; n 1; s ¼ 2; 3; . . . ; n 1
njs ¼ cos
p
2n pr gir ¼ cos n
ð2s 1Þ s ¼ 2; . . . ; n 1; j ¼ 1; 2
ð55Þ
r ¼ 2; . . . ; n 1; i ¼ 1; 2
m Here km q are the weights and fq the associated locations, i.e. the roots of Laguerre polynomial, given in [26] for a particular value of m. By solving this set of 2n 2n linear algebraic equations, i.e. Eq. (54), we can determine the values of F1(±1) and F2(±1) and thereby determine the stress intensity factors at the tip of crack, viz:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi bj aj F j ð1Þ j ¼ 1; 2 2p rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4l bj aj Kðbj Þ ¼ F j ð1Þ j ¼ 1; 2 1þj 2p
Kðaj Þ ¼
4l 1þj
ð56Þ ð57Þ
Please cite this article in press as: D. Peng, R. Jones, The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.007
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D. Peng, R. Jones / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx
Non-Dimensional Stress Intensity Factors
Non-Dimensional Stress Intensity Factors
a2/b = 0.2
1.40
a2/b = 0.2, [30]
1.30
a2/b = 2
1.7
a2/b = 2, [30]
1.20
K I/K 01
KI/K02
1.50
a2/b = 5
1.10
a2/b = 5, [30]
1.5
0.90 0.0
0.2
0.4
0.6
0.8
0.9
1.0
a1/b
1.30
4.0
5.0
6.0
1.3
Tip C
1.2
Tip D
1.1
1.10
1
0.90 0.0
0.2
0.4
0.6
0.8
1.0
a1/b
1
2
σ 2a2
2a1 B
C
D
......
8
b
0.9
0.0
1.0
2.0
3.0
4.0
5.0
6.0
S/a 1
Fig. 7. KI for tip D of two unequal length collinear cracks in a sheet subjected a uniform uniaxial tensile stress.
S
3.0
1.4
K I/K 01
KI/K02
1.50
2.0
Non-Dimensional Stress Intensity Factors
1.5
a2/b = 0.2 a2/b = 0.2, [30] a2/b = 2 a2/b = 2, [30] a2/b = 5 a2/b = 5, [30]
1.70
1.0
Fig. 9. Non-dimensional stress intensity factors for tip A and B for various S.
Non-Dimensional Stress Intensity Factors
1.90
0.0
S/a1
Fig. 6. KI for tip C of two unequal length collinear cracks in a sheet subjected a uniform uniaxial tensile stress.
A
Tip B
1.1
1.00
2.10
Tip A
1.3
σ Fig. 8. The boundary 1 close to tip A of the crack AB with distance S.
4. Validation and examples Two simple problems are considered to both verify and validate the accuracy of the present method. The first problem involves two equal collinear cracks in a finite medium under a remote uniform stress (r), see Fig. 2. The results obtained by the present method are given in Table 1 together with those values available from the literature [11]. It can be seen from Table 1, that there is good agreement between the present results and the published solutions. The second example is concerned with the problem of two unequal collinear cracks in an infinite body subject to a remote uniform stress (r), see Fig. 3. Plots of magnification factors KI/K01 and KI/K02 (Here, pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi K 01 ¼ r pa1 and K 02 ¼ r pa2 .) versus. a1/b are shown for tips A,
Fig. 10. Non-dimensional stress intensity factors for tip C and D for various S.
B, C and D for various values of a2/b, in Figs. 4–7 respectively. The maximum difference between the present solution and the analytical solution given in [30] was less than 2%. The problem of aging aircraft is not unique to civil aircraft. Australia, NATO, and Eastern European countries have been experiencing aging aircraft related problems in their military fleets, particularly amongst their transport aircraft. Widespread fatigue damage (WFD), sometimes termed multiple site damage (MSD), is perhaps the most troubling structure-related issue associated with aging aircraft. The onset of MSD, predominantly due to fatigue and or fatigue corrosion interaction, causes a dramatic reduction in residual strength of an aircraft. Determination of the residual strength of structure requires an accurate knowledge of the stress intensity factor distribution around the crack front. When interacting cracks prevail, the lack of interaction effect in among multiple cracks may lead to an inadequate evaluation. When WFD close to the boundary of structure, the couple effect on the boundary and the interaction between cracks should be consider. Let us next consider the case when the boundary 1 is a finite distance (S) from tip A of crack AB, as shown in Fig. 8. In this example we have taken boundary 2 to be far from crack CD, see Fig. 8. In this initial investigation we have set a1 = a2 and set the ratio of a1/b and a2/b are equal to 0.8. This ratio was chosen to ensure that the two cracks were interacting. This problem may be considered to approximate (interacting) cracking from two closely collocated corrosion pits that lie close to the edge of a structural member. The computed magnification factors KI/K01 at the tips A, B, C and D for various distances S are shown in Figs. 9 and 10. The results obtained show that either an enhancement or a shielding effect on the stress intensity factor depending on the positions of the boundary and which crack tip is considered. In general we see that the stress intensity factor at crack tip A
Please cite this article in press as: D. Peng, R. Jones, The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.007
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D. Peng, R. Jones / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx
increased as the boundary gets closer whilst the stress intensity factors at tips B, C and D reduce. This suggests that the closer that the two cracks are to a boundary accelerates crack growth towards the boundary and decelerates linking of the two cracks. It should be noted that the analysis in this example is primary. A lot of problems, such as effect of crack sizes and what happens when the larger crack is closer to edge, should be further investigated.
5. Conclusion This paper has presented a simple procedure for analysing the effect of crack interaction associated with two unequal unsymmetric collinear cracks in a finite width plate. It has been shown how the mixed boundary value problem can be reduced to a set singular integral equations and an effective and simple numerical method, based on the use of Chebyshev polynomials, has been used to determine the associated stress intensity factors. This solution process has been validated by comparison with published solutions. We also show that for two (interacting) equal cracks the closer that two cracks are to a boundary accelerates crack growth towards the boundary and decelerates linking of the two cracks. It should be noted that the analysis in this paper is limited to linear fracture mechanic analysis. A more detailed study will subsequently be performed to evaluate the effect for different size cracks. Acknowledgements This paper was developed within the CRC for Infrastructure and Engineering Asset Management (CIEAM), established and supported under the Australian Government’s Cooperative Research Centres Program. References [1] S. Pitt, R. Jones, Multiple-site and widespread fatigue damage in aging aircraft, Eng. Fail. Anal. 4 (1997) 237–257. [2] W.R. Hendricks, The Aloha-airlines accident a new era for ageing aircraft, in: S.N. Atluri, S.G. Sampath, P. Tong (Eds.), Structural Integrity of Ageing Airplanes, Springer, Berlin, 1991, pp. 153–165. [3] F. Erdogan, V. Biricikoglu, Two bonded half planes with a crack going through the interface, Int. J. Eng. Sci. 11 (1973) 745–766. [4] M. Isida, H. Noguchi, An interface crack and an arbitrary array of cracks in bonded semi-infinite bodies under in-plane loads, Trans. Jpn. Soc. Mech. Eng. A 49 (1983) 137–146.
[5] G. Vialaton, G. Lhermet, G. Vessiere, M. Boivin, J. Bahuaud, Field of stresses in an infinite plate containing two collinear cuts loaded at an arbitrary location, Eng. Fract. Mech. 8 (1976) 526–538. [6] W. Gunther, Asymptotic elastic solution for two straight cracks of arbitrary length and location, Theoret. Appl. Fract. Mech. 3 (1985) 247–255. [7] D. Peng, Q. Zhang, Q. Yang, Integral transform solution of multi-material structure with finite crack perpendicular to the interface, ACTA Mech. Solida Sin. 10 (1997) 330–337. [8] E.E. Gdoutos, Growth predictions of two interacting cracks, Theoret. Appl. Fract. Mech. 1 (1984) 133–137. [9] R.A. Collins, D.J. Cartwright, An analytical solution for two equal-length collinear strip yield cracks, Eng. Fract. Mech. 68 (2001) 915–924. [10] W. Xu, X.R. Xu, H. Wang, Weight functions and strip yield solution for two equal-length collinear cracks in an infinite sheet, Eng. Fract. Mech. 78 (2011) 2356–2368. [11] F. Delale, F. Erdogan, The problem of internal and edge cracks in an orthotropic strip, J. Appl. Mech. 44 (1977) 237–242. [12] G.G. Adams, Crack interaction in an infinite elastic strip, Int. J. Eng. Sci. 18 (1980) 455–462. [13] S.D. Ranjit, M.S. Brij, Two coplanar Griffith cracks in an infinitely long elastic strip, J. Elast. 11 (1981) 229–238. [14] A.N. Das, M.L. Ghosh, Four co-planar Griffith cracks in an infinite elastic medium, Eng. Fract. Mech. 43 (1992) 941–955. [15] M. Lowengrub, K.N. Srivastava, On two coplanar Griffith cracks in an infinite elastic medium, Int. J. Eng. Sci. 6 (1968) 359–362. [16] M. Lowengrub, K.N. Srivastava, Two coplanar Griffith cracks in an infinitely long elastic strip, Int. J. Eng. Sci. 6 (1968) 425–434. [17] Z.G. Zhou, Y.Y. Bai, X.W. Zhang, Two collinear Griffith cracks subjected to uniform tension in infinitely long strip, Int. J. Solids Struct. 36 (1999) 5597– 5609. [18] D. Peng, R. Jones, S. Pitt, Implementation of a new algorithm for 3D fracture analysis, Int. J. Fract. 113 (2002) 57–75. [19] J.H. Park, S.N. Atluri, Analysis of a cracked thin isotropic plate subjected to bending moment by using FEAM, KSME Int. J. 13 (1999) 912–917. [20] D.S. Dawicke, J.C. Newman Jr., Analysis and prediction of multiple-site damage (MSD) fatigue crack growth, NASA Technical paper, vol. 3231, 1992. [21] H. Rajiyah, S.N. Atluri, Evaluation of K-factors and weight functions for 2-D mixed-mode multiple cracks by the boundary element alternating method, Eng. Fract. Mech. 32 (1989) 911–922. [22] P. Tong, R. Greif, L. Chen, Residual strength of aircraft panels with multiple site damage, Comput. Mech. 13 (1994) 285–294. [23] P.S. Theocaris, Interaction between collinear asymmetric cracks, J. Strain Anal. 7 (1972) 186–193. [24] I.N. Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951. [25] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, third ed., McGraw-Hill Book Company, New York, 1970. [26] A. Jeffrey, Complex Analysis and Applications, second ed., Chapman & Hall/CRC Taylor & Francis Group, Boca Raton, FL, USA, 2006. [27] F. Erdogan, G.D. Gupra, On the numerical solution of singular integral equations, Q. Appl. Math. 29 (1972) 525–534. [28] A.C. Kaya, F. Erdogan, On the solution of integral equations with strongly singular kernels, Q. Appl. Math. 45 (1987) 105–122. [29] P.S. Theocaris, N.I. Ioakimidis, Numerical integration methods for the solution of singular integral equations, Q. Appl. Math. 35 (1977) 173–183. [30] D.P. Rooke, D.J. Cartwright, Compendium of stress intensity factors, in: Great Britain Ministry of Defence, Procurement Executive, 1976, pp. 112–119.
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The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate D. Peng ⇑, R. Jones Centre of Expertise Structural Mechanics, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Victoria 3800, Australia
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Fourier transform Interacting unequal crack Singular integral equation Stress intensity factor
a b s t r a c t This paper uses a Fourier transform technique to solve the problem of two interacting collinear unequal cracks in a finite width plate. This approach reduces the problem to the solution of two coupled integral equations each with a singular kernel which is the solved using Cauchy–Chebyshev polynomials. The solution is first validated by comparing the solutions for the case of two equal cracks in a finite width plate and for the case of two unequal cracks in an infinite width plate with published solutions. The problem of (interacting) cracks that grow from collocated corrosion pits that lie close to the edge of a structural member is then studied. Here it is shown that the closer that the two cracks are to a boundary accelerates crack growth towards the boundary and decelerates linking of the cracks. Ó 2014 Published by Elsevier Ltd.
1. Introduction The problem of crack interaction and its effect on structural integrity is a universal problem that has particular relevance to aging rail and aircraft structures. However, as revealed in [1], an analysis of a structure in which crack interaction is neglected can be very non-conservative as the cumulative effect of interacting cracks may significantly degrade structural integrity [2]. Due to the complexity of the problem, there is limited literature available on the interaction of multiple unequal cracks in a finite geometry. Exact solutions obtained by analytical methods are restricted to an infinite geometry [3–10]. In such cases the approaches used include: the boundary conditions perturbation technique, the complex potential method and the singular integral equation approach. For problems involving crack interaction in finite geometries analytical solutions are currently limited to symmetrical equal size cracks [11–17]. With the rapid advancement of computer technology numerical methods have been used to solve cracks interaction with finite boundary, such as finite element alternating technique algorithm [18,19], alternating indirect boundary element technique [20,21], the hybrid finite element method [22], and the reflecting shadow method [23]. The present paper derives an analytical solution to the problem of two interacting unequal collinear cracks in a finite width plate, see Fig. 1. A Fourier transform method, which uses dislocation density functions, is used to determine the associated singular integral equations which are then solved using Cauchy–Chebyshev
polynomials. This approach is first validated by comparing the solutions for the case of two equal cracks in a finite width plate and for the case of two unequal cracks in an infinite width plate with published solutions. The problem of two (interacting) cracks that grow from (nearby) collocated corrosion pits that lie close to the edge of a structural member is then studied. Here it is shown that the closer that the two cracks are to a boundary accelerates crack growth towards the boundary and decelerates linking of the cracks.
2. Basic equations of elasticity Consider an infinitely long homogeneous isotropic elastic plate with a width H that contains two collinear unequal cracks that lie perpendicular to the edges of the plate, see Fig. 1. The displacement components in the X, Y directions are defined as u and v respectively. As the displacement components are symmetric about the Y axis, the displacement components are not symmetric about the X axis. Therefore, a general Fourier transform with respect to y axis is introduced. The displacements may thus be expressed in terms of the following Fourier integrals [24]:
rffiffiffiffi Z 2
rffiffiffiffiffiffiffi Z 1 1 uðx; yÞ ¼ f 1 ða; yÞsinaxda þ g ðx; cÞeicy dc 2p 1 1 p 0 rffiffiffiffi Z rffiffiffiffiffiffiffi Z 1 2 1 1 v ðx; yÞ ¼ f 2 ða; yÞcosaxda þ g ðx; cÞeicy dc 2p 1 2 p 0 1
ð1Þ ð2Þ
⇑ Corresponding author. http://dx.doi.org/10.1016/j.tafmec.2014.10.007 0167-8442/Ó 2014 Published by Elsevier Ltd.
Please cite this article in press as: D. Peng, R. Jones, The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.007
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D. Peng, R. Jones / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx
Here:
rffiffiffiffi Z 2
rffiffiffiffi Z 2
1
The corresponding stress field are obtained by using the following relations [25].
1
uðx;yÞsinaxdx; f 2 ða;yÞ ¼ v ðx;yÞcosaxdx ð3Þ p p rffiffiffiffiffiffiffi 0Z rffiffiffiffiffiffiffi Z 0 1 1 1 1 uðx;yÞeicy dy g 2 ðx; cÞ ¼ v ðx; yÞeicy dy ð4Þ g 1 ðx; cÞ ¼ 2p 1 2p 1
f 1 ða;yÞ ¼
The unknown functions f1(a, y), f2(a, y), g1 (x, c) and g2(x, c) can be determined by boundary conditions. In the absence of body forces the equations of equilibrium [25] can be expressed as follow:
@ @u @ v ¼0 lr2 u þ ðk þ lÞ þ @x @x @y @ @u @ v ¼0 lr2 v þ ðk þ lÞ þ @y @x @y tE k¼ ð1 þ tÞð1 2tÞ
8 r ðx; yÞ ¼ ðk þ 2lÞ @u þ k @@yv > @x > < xx ryy ðx; yÞ ¼ ðk þ 2lÞ @@yv þ k @u @x > > : s ðx; yÞ ¼ @u þ @ v xy
(rffiffiffiffi Z 2
rxx ðx; yÞ ¼ l
ð7Þ
ð10Þ
g 2 ðx; cÞ ¼ ðA5 þ A6 xÞecx þ ðA7 þ A8 xÞecx x P 0
ð11Þ
c
ð9Þ
unknown coefficients A1 to A6 can be determined from boundary conditions. The formulas (1) and (2) become the following Fourier integrals: 1
½ðA1 þ A2 yÞeay þ ½ðA3 þ A4 yÞeay sinaxda 0 rffiffiffiffiffiffi ffiZ 1 1 jcj j i A5 þ þ x A6 ejcjxicy dc 2p 1 c jcj
uðx; yÞ ¼
rffiffiffiffi Z 2
i þ y A2 eay p 0 a h j i o y A4 eay cosaxda þ A3 þ rffiffiffiffiffiffiffi Z a 1 1 þ ðA5 þ xA6 Þejcjxicy dc 2p 1 1
nh
A1 þ
(rffiffiffiffi Z 2
1
p
½ð2aA1 þ h1 j 2ayiA2 Þeay
0
þð2aA3 þ h1 j þ 2ayiA4 Þeay sinaxda ) rffiffiffiffiffiffiffi Z 1 1 1j jcjxicy þ2i cA5 þ cx A6 e dc ð18Þ 2p 1 2 Here j = 3 4m for plane strain and j = (3 m)/(1 + m) for generalised plane stress. The stress and mixed boundary conditions may be expressed as
8 >
sxy ð0; yÞ ¼ 0 ryy ðx; 0Þ ¼ 0 sxy ðx; 0Þ ¼ 0 ryy ðx; HÞ ¼ 0 sxy ðx; HÞ ¼ 0
ð19Þ
and
p
v ðx; yÞ ¼
1
sxy ðx; yÞ ¼ l
ð12Þ
The Kolosov’s material constant j = 3 4t is for plane strain and j = (3 t)/(1 + t) is for plane stress. Here m is Poisson’s ratio. The
(rffiffiffiffi Z 2
ð16Þ
½ð2aA1 h1 þ j þ 2ayiA2 Þeay p 0 þð2aA3 þ h1 þ j 2ayiA4 Þeay cosaxda ) rffiffiffiffiffiffiffi Z 1 1 3j jcjxicy þ2i cA5 þ cx A6 e dc ð17Þ 2p 1 2
The condition that as x ? 1 the stresses should be finite requires that
A7 ¼ A8 ¼ 0
½ð2aA1 þ hj 3 þ 2ayiA2 Þeay
0
ryy ðx; yÞ ¼ l
ð8Þ
c
p
1
þð2aA3 þ h3 j þ 2ayiA4 Þeay cosaxda ) rffiffiffiffiffiffiffi Z 1 1 1þj þ2i cA5 þ þ cx A6 ejcjxicy dc 2p 1 2
ð6Þ
f 1 ða; yÞ ¼ ðA1 þ A2 yÞeay þ ðA3 þ A4 yÞeay h j i h j i f 2 ða; yÞ ¼ A1 þ þ y A2 eay þ A3 þ y A4 eay a a
j j þ x A6 ecx A7 þ x A 8 e cx x P 0 g 1 ðx; cÞ ¼ i A5 þ
@x
This yields:
ð5Þ
Here l, t and E are shear modulus, Poisson’s ratio and modulus of elasticity respectively. Substituting Eqs. (3) and (4) into Eqs. (5)– (7) yields:
rffiffiffiffi Z 2
@y
ð15Þ
ð13Þ
limx!0 rxx ðx; yÞ ¼ PðyÞ y 2 L uð0; yÞ ¼ 0
j
ð14Þ
y2L
where P(y) is the crack surface traction which can be an arbitrary function of y. Here, L and L denote the crack face and non-cracked boundary (along the centre line of the plate) respectively. For the specific problem addressed in this paper the mixed boundary conditions (20) can be expressed as:
limx!0 rxx ðx; yÞ ¼ pj ðyÞ aj 6 y 6 bj j ¼ 1; 2 uj ð0; yÞ ¼ 0
Y
ð20Þ
y < a1 ; b1 < y < a2 ; b2 < y 6 H ð21Þ
where pj(y), j = 1, 2, are the crack face tractions on crack 1 and 2 respectively. Using the boundary conditions (19) the unknown coefficients A1 A6 in Eqs. (13)–(18), can be expressed in the term of a single parameter A⁄, viz:
H
b2 a2 a1
b1
o Fig. 1. Two collinear asymmetric internal cracks in finite stripe.
X
Z 1 1 A1 ¼ pffiffiffiffiffiffiffi ðð1 2ajHÞeaH eaH ÞðeaH eicH Þða icÞ 4 2p 1 þðð2aH þ jÞeaH jeaH ÞðeaH eicH Þða þ icÞ þða icÞ
A c2 dc
Dða2 þ c2 Þ
2
ð22Þ
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D. Peng, R. Jones / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx
1 A2 ¼ pffiffiffiffiffiffiffi 2 2p þðe
aH
Z
1
h
xðx; y; cÞ ¼
2aHeaH ðeaH eicH Þða icÞ
1
eaH ÞðeaH eicH Þða þ icÞ
A c2 dc
Dða þ c 2
2 Þ2
ð23Þ
A c2 dc
1 A4 ¼ pffiffiffiffiffiffiffi 2 2p
Z
Dða2 þ c2 Þ
1
þ2aHeaH ðeaH eicH Þða þ icÞ ð1 jÞi A5 ¼ 8 A6 ¼
jcji 4
aH
A2 ¼ ðB1 e
rffiffiffiffi
p 2
A
i
ð25Þ
2 Þ2
Dða2 þ c
ð26Þ
p
A
D ¼ ðe
A4 ¼ ðB1 þ B2 Þe
e
aH 2
Þ ð2aHÞ
aH
þ ðB3 e
aH
þ B4 e
þ DÞe
ð32Þ ay
ð33Þ ð34Þ
þ ðB3 þ B4 Þe
ay
ð35Þ ð36Þ
B2 ¼ ð2ay 3ÞðeaH eaH Þ þ 2aHeaH
ð37Þ
aH
e
aH
aH
Þ þ 2aHe
B4 ¼ 2aHð2ay þ 3ÞeaH þ eaH eaH
ð38Þ ð39Þ
As a result the mixed boundary conditions, i.e. Eq. (21), can be expressed as:
(
R1 limx!0 p1ffiffiffiffi jcjA xðx; y; cÞdc ¼ PðyÞ=l y 2 L 2p 1 R j 1 A eicy dc ¼ 0 y2L 1þ 1 16
ð40Þ
3. Singular integral equations and stress intensity factors
2
ð28Þ
Substituting Eqs. (22)–(27) into Eqs. (13) and (16) the rxx(x, y) stress and displacement u(0, y) now become
Z
DÞe
ay
B1 ¼ eaH eaH þ 2aHeaH ð2ay 3Þ
ð27Þ
Here we have defined the quantity D to be aH
þ B2 e ay
rffiffiffiffi 2
aH
B3 ¼ ð2ay þ 3Þðe
A c2 dc
ð31Þ
A3 ¼ ðB1 þ B2 Þeay þ ðB3 þ B4 Þeay
ðeaH eaH ÞðeaH eicH Þða icÞ
1
1
A1 ¼ ðB1 eaH þ B2 eaH þ DÞeay þ ðB3 eaH þ B4 eaH þ DÞeay
ð24Þ
2
Z
½aA1 þ icaA2 ðaA3 þ icA4 ÞeicH p 0 a2 cos axda 1 þ ð1 þ jcjxÞejcjxicy 2 2 2Dða2 þ c2 Þ jcj
And
Z 1 1 A3 ¼ pffiffiffiffiffiffiffi ððj 2aHÞeaH jeaH ÞðeaH eicH Þða icÞ 4 2p 1 þðð2aHj þ 1ÞeaH eaH ÞðeaH eicH Þða þ icÞ þða icÞ
rffiffiffiffi 2
1
1 rxx ðx; yÞ ¼ pffiffiffiffiffiffiffi l jcjA xðx; y; cÞdc 2p 1 Z 1 þ j 1 icy uð0; yÞ ¼ A e dc 16 1
ð29Þ
Define the dislocation density function u as:
@ 1þj i uð0; yÞ ¼ @y 16 Z 8 i /ðtÞeict dt A ¼ ð1 þ jÞcp
/ðyÞ ¼
ð30Þ
1
cA eicy dc
ð41Þ
1
t2L
ð42Þ
The first term in Eq. (40) can now be expressed as:
8 i x!0 ð1 þ jÞp
where
Z
lim
¼
PðyÞ
Z
Z 1 1 jcj /ðtÞ pffiffiffiffiffiffiffi A xðx; y; cÞeict dc dt 2p 1 c
y 2 L; t 2 L
l
ð43Þ
To simplify the analysis we use the residue theorem [26] to solve the infinite integral contained in Eq. (31). After somewhat routine manipulation the problem can be reduced to the following system of integral equations for the function /(t):
σ
Z
1 ð1 þ jÞp þ Kðt; yÞ /ðtÞdt ¼ PðyÞ y 2 L; t 2 L ty 4l
ð44Þ
where 2L
Kðt; yÞ ¼
2L
1 2
Z
1
½aA1 eat ð1 atÞA2 eat þ aðH tÞA3 eaðHtÞ
0
1 da y 2 L; t 2 L D p1 ðyÞ; y 2 a1 ; b1 PðyÞ ¼ p2 ðyÞ; y 2 a2 ; b2
þ h1 aðH tÞiA4 eaðHtÞ a
b
/ðtÞ ¼ H
H
/1 ðtÞ; t 2 a1 ; b1 /2 ðtÞ; t 2 a2 ; b2
;
ð45Þ ð46Þ
Here p1(y) and p2(y) are the tractions on the crack surfaces. The mixed boundary value problem may thus be reduced to the following singular integral equations: Z b2 1 ð1 þ jÞp þ K 11 ðt;yÞ /1 ðtÞdt þ K 12 ðt; yÞ/2 ðtÞdt ¼ p1 ðyÞ ty 4l a1 a2 Z b2 Z b1 1 ð1 þ jÞp K 21 ðt; yÞ/2 ðtÞdt þ þ K 22 ðt; yÞ /2 ðtÞdt ¼ p2 ðyÞ ty 4l a1 a2 Z
σ Fig. 2. A finite width plate with two equal collinear cracks.
b1
ð47Þ ð48Þ
The solution of the singular integral Eqs. (47) and (48) is subject to compatibility conditions
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D. Peng, R. Jones / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx
Table 1 Non-dimensional stress intensity factor for symmetric collinear internal cracks in a sheet. a/H
b/H
0.1 0.2 0.4 0.5 0.1 0.5 0.5
0.5 0.6 0.8 0.9 0.9 0.95 0.98
p K(a)/r L
p K(b)/r L
Present
Collins and Cartwright [9]
Present
Collins and Cartwright [9]
1.145 1.136 1.14 1.16 1.69 1.22 1.27
1.176 1.109 1.097 1.127 1.678 1.194 1.268
1.109 1.13 1.15 1.23 1.65 1.51 1.91
1.115 1.094 1.122 1.221 1.694 1.445 1.875
σ
Z
bj
/j ðtÞdt ¼ 0 j ¼ 1; 2
ð49Þ
aj
A
After normalising transformations
2a2
2a1 B
C
D
nj ¼
b
2t ðbj þ aj Þ ; bj aj
σ
Z
1.90
a2/b = 0.2 a2/b = 0.2, [30] a2/b = 2
KI/K01
1.70 1.50
a2/b = 2, [30] a2/b = 5
1.30 1.10
0.6
0.8
1.0
a2/b = 5, [30]
a1/b
Non-Dimensional Stress Intensity Factors
3.40
a2/b = 0.2 a2/b = 0.2, [30] a2/b = 2
2.90
KI/K01
j ¼ 1; 2
the
ð50Þ
2.40
a2/b = 2, [30] a2/b = 5
1.90 1.40 0.4
ð51Þ
1
/j ðnj Þdnj ¼ 0 j ¼ 1; 2
ð52Þ
1
Let us express the unknown functions in the form: 1=2
F j ðtÞ
ð53Þ
Subsequently, the solution of the integral Eqs. (51) and (52) can be obtained by applying several different methods [27,28]. In this paper we will use the Lobatto–Chebyshev method [29]. This yield the following system of algebraic equations, viz: n X 2 Z p X
n1
1
1
s¼1 j¼1
F j ðnjs Þhs ¼
"
# m X dij m fm qk þ km e f ; n ; g ij q js ir njs gir q¼1 q
ð1 þ jÞp pi ðgir Þ i ¼ 1; 2 4l
n X F j ðns Þhs ¼ 0 j ¼ 1; 2
ð54Þ
where
( hs ¼
0.2
through
s¼1
Fig. 4. KI for tip A of two unequal length collinear cracks in a sheet subjected a uniform uniaxial tensile stress.
0.90 0.0
2y ðbj þ aj Þ ; bj aj
(aj, bj)
dij ð1 þ jÞp þ kij ðnj ; gi Þ /j ðnj Þdnj ¼ pi ðgi Þ i ¼ 1;2 nj gi 4l
/j ðtÞ ¼ ð1 t 2 Þ
Non-Dimensional Stress Intensity Factors
0.4
1
j¼1
Fig. 3. Two unequal collinear cracks in an infinite plate under a uniform uniaxial tensile stress.
0.2
intervals
Eqs. (47) and (48) can be written as 2 Z 1 X
0.90 0.0
gj ¼
the
0.6
0.8
1.0
a2/b = 5, [30]
a1/b Fig. 5. KI for tip B of two unequal length collinear cracks in a sheet subjected a uniform uniaxial tensile stress.
1 ; 2
s ¼ 1; n 1; s ¼ 2; 3; . . . ; n 1
njs ¼ cos
p
2n pr gir ¼ cos n
ð2s 1Þ s ¼ 2; . . . ; n 1; j ¼ 1; 2
ð55Þ
r ¼ 2; . . . ; n 1; i ¼ 1; 2
m Here km q are the weights and fq the associated locations, i.e. the roots of Laguerre polynomial, given in [26] for a particular value of m. By solving this set of 2n 2n linear algebraic equations, i.e. Eq. (54), we can determine the values of F1(±1) and F2(±1) and thereby determine the stress intensity factors at the tip of crack, viz:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi bj aj F j ð1Þ j ¼ 1; 2 2p rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4l bj aj Kðbj Þ ¼ F j ð1Þ j ¼ 1; 2 1þj 2p
Kðaj Þ ¼
4l 1þj
ð56Þ ð57Þ
Please cite this article in press as: D. Peng, R. Jones, The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.007
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Non-Dimensional Stress Intensity Factors
Non-Dimensional Stress Intensity Factors
a2/b = 0.2
1.40
a2/b = 0.2, [30]
1.30
a2/b = 2
1.7
a2/b = 2, [30]
1.20
K I/K 01
KI/K02
1.50
a2/b = 5
1.10
a2/b = 5, [30]
1.5
0.90 0.0
0.2
0.4
0.6
0.8
0.9
1.0
a1/b
1.30
4.0
5.0
6.0
1.3
Tip C
1.2
Tip D
1.1
1.10
1
0.90 0.0
0.2
0.4
0.6
0.8
1.0
a1/b
1
2
σ 2a2
2a1 B
C
D
......
8
b
0.9
0.0
1.0
2.0
3.0
4.0
5.0
6.0
S/a 1
Fig. 7. KI for tip D of two unequal length collinear cracks in a sheet subjected a uniform uniaxial tensile stress.
S
3.0
1.4
K I/K 01
KI/K02
1.50
2.0
Non-Dimensional Stress Intensity Factors
1.5
a2/b = 0.2 a2/b = 0.2, [30] a2/b = 2 a2/b = 2, [30] a2/b = 5 a2/b = 5, [30]
1.70
1.0
Fig. 9. Non-dimensional stress intensity factors for tip A and B for various S.
Non-Dimensional Stress Intensity Factors
1.90
0.0
S/a1
Fig. 6. KI for tip C of two unequal length collinear cracks in a sheet subjected a uniform uniaxial tensile stress.
A
Tip B
1.1
1.00
2.10
Tip A
1.3
σ Fig. 8. The boundary 1 close to tip A of the crack AB with distance S.
4. Validation and examples Two simple problems are considered to both verify and validate the accuracy of the present method. The first problem involves two equal collinear cracks in a finite medium under a remote uniform stress (r), see Fig. 2. The results obtained by the present method are given in Table 1 together with those values available from the literature [11]. It can be seen from Table 1, that there is good agreement between the present results and the published solutions. The second example is concerned with the problem of two unequal collinear cracks in an infinite body subject to a remote uniform stress (r), see Fig. 3. Plots of magnification factors KI/K01 and KI/K02 (Here, pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi K 01 ¼ r pa1 and K 02 ¼ r pa2 .) versus. a1/b are shown for tips A,
Fig. 10. Non-dimensional stress intensity factors for tip C and D for various S.
B, C and D for various values of a2/b, in Figs. 4–7 respectively. The maximum difference between the present solution and the analytical solution given in [30] was less than 2%. The problem of aging aircraft is not unique to civil aircraft. Australia, NATO, and Eastern European countries have been experiencing aging aircraft related problems in their military fleets, particularly amongst their transport aircraft. Widespread fatigue damage (WFD), sometimes termed multiple site damage (MSD), is perhaps the most troubling structure-related issue associated with aging aircraft. The onset of MSD, predominantly due to fatigue and or fatigue corrosion interaction, causes a dramatic reduction in residual strength of an aircraft. Determination of the residual strength of structure requires an accurate knowledge of the stress intensity factor distribution around the crack front. When interacting cracks prevail, the lack of interaction effect in among multiple cracks may lead to an inadequate evaluation. When WFD close to the boundary of structure, the couple effect on the boundary and the interaction between cracks should be consider. Let us next consider the case when the boundary 1 is a finite distance (S) from tip A of crack AB, as shown in Fig. 8. In this example we have taken boundary 2 to be far from crack CD, see Fig. 8. In this initial investigation we have set a1 = a2 and set the ratio of a1/b and a2/b are equal to 0.8. This ratio was chosen to ensure that the two cracks were interacting. This problem may be considered to approximate (interacting) cracking from two closely collocated corrosion pits that lie close to the edge of a structural member. The computed magnification factors KI/K01 at the tips A, B, C and D for various distances S are shown in Figs. 9 and 10. The results obtained show that either an enhancement or a shielding effect on the stress intensity factor depending on the positions of the boundary and which crack tip is considered. In general we see that the stress intensity factor at crack tip A
Please cite this article in press as: D. Peng, R. Jones, The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.007
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D. Peng, R. Jones / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx
increased as the boundary gets closer whilst the stress intensity factors at tips B, C and D reduce. This suggests that the closer that the two cracks are to a boundary accelerates crack growth towards the boundary and decelerates linking of the two cracks. It should be noted that the analysis in this example is primary. A lot of problems, such as effect of crack sizes and what happens when the larger crack is closer to edge, should be further investigated.
5. Conclusion This paper has presented a simple procedure for analysing the effect of crack interaction associated with two unequal unsymmetric collinear cracks in a finite width plate. It has been shown how the mixed boundary value problem can be reduced to a set singular integral equations and an effective and simple numerical method, based on the use of Chebyshev polynomials, has been used to determine the associated stress intensity factors. This solution process has been validated by comparison with published solutions. We also show that for two (interacting) equal cracks the closer that two cracks are to a boundary accelerates crack growth towards the boundary and decelerates linking of the two cracks. It should be noted that the analysis in this paper is limited to linear fracture mechanic analysis. A more detailed study will subsequently be performed to evaluate the effect for different size cracks. Acknowledgements This paper was developed within the CRC for Infrastructure and Engineering Asset Management (CIEAM), established and supported under the Australian Government’s Cooperative Research Centres Program. References [1] S. Pitt, R. Jones, Multiple-site and widespread fatigue damage in aging aircraft, Eng. Fail. Anal. 4 (1997) 237–257. [2] W.R. Hendricks, The Aloha-airlines accident a new era for ageing aircraft, in: S.N. Atluri, S.G. Sampath, P. Tong (Eds.), Structural Integrity of Ageing Airplanes, Springer, Berlin, 1991, pp. 153–165. [3] F. Erdogan, V. Biricikoglu, Two bonded half planes with a crack going through the interface, Int. J. Eng. Sci. 11 (1973) 745–766. [4] M. Isida, H. Noguchi, An interface crack and an arbitrary array of cracks in bonded semi-infinite bodies under in-plane loads, Trans. Jpn. Soc. Mech. Eng. A 49 (1983) 137–146.
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Please cite this article in press as: D. Peng, R. Jones, The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.007