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Stress intensity factors for interacting cracks and complex crack configurations in linear elastic media
Engineering Failure Analysis 14 (2007) 179–195 www.elsevier.com/locate/engfailanal
Stress intensity factors for interacting cracks and complex crack configurations in linear elastic media Xiangqiao Yan
*
Research Laboratory on Composite Materials, Harbin Institute of Technology, Harbin 150001, PR China Received 26 August 2005; accepted 31 October 2005 Available online 3 February 2006
Abstract In this paper, an effective numerical method for analyzing interacting multiple cracks and complex crack configurations in infinite linear elastic media is presented. By extending Bueckner’s principle suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple-crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by taking into account the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author. Test examples are included to illustrate that the method is very simple and effective for analyzing interacting multiple cracks and complex crack configurations in an infinite linear elastic media under remote uniform stresses. Specifically, analysis of perpendicular cracks under general in-plane loading is performed using the numerical approach and many numerical results are given in the form of tables. Ó 2006 Published by Elsevier Ltd. Keywords: Crack; Boundary element; Stress intensity factors; Crack-tip element; Displacement discontinuity
1. Introduction There are several situations in fracture mechanics, which involve a complicated arrangement of cracks that is not amenable to a simple method of analysis. In some cases, the difficulty lies in having many cracks interacting with each other, e.g., when a single crack is embedded in a microcrack array. In other cases, as in crack branching phenomena, the complexity of the problem is due to the presence of irregular crack shapes consisting of several segments which form what is sometimes called a zig-zag or nonlinear crack. In relatively simple situations of multiple cracks, such as aligned cracks and crack branches, classical methods of analysis are applicable and they lead to elegant exact solutions, e.g., Erdogan [1] and Sih [2]. However, approximate methods are unavoidable in more complicated situations. Some existing studies have employed the representation of cracks by dislocations [3–5], which leads to integral equations which *
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1350-6307/$ - see front matter Ó 2006 Published by Elsevier Ltd. doi:10.1016/j.engfailanal.2005.10.008
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X. Yan / Engineering Failure Analysis 14 (2007) 179–195
Real case
Model 2a2
d2 [i] d1
2a1
2a2 d2 [ii]
2a1
Fig. 1. Crack configurations and respective geometric models [20].
can be solved in an approximate way. Other investigators of zig-zag crack configurations use methods based on polynomial approximations and truncation of a conformal mapping function [6,7]. A particularly simple treatment of crack interaction phenomena was introduced by Kachanov [8,9] who showed that many multiple crack problems can be solved with the help of a superposition procedure which leads to a system of linear algebraic equations for certain equilibrated crack-line tractions. Another variant of a crack interaction method was given by Horii and Nemat-Nasser [10]. Benveniste et al. [11] presented a general and simple method for computation of stress fields and the stress intensity factors in linear elastic media, which contain several cracks arranged in a complicated configuration. The method uses a superposition technique, which replaces a configuration of N cracks by means of N different problems, each involving an isolated crack located in an infinite medium and loaded by unknown traction. Such representations were used by Collins [12], Datsysin and Savruk [13], Cross [14], Chudnovsky and Kachanov [15], Chudnovsky et al. [3,4], Horii and Nemat-Nasser [10], and Chen [16]. In [11], a polynomial expansion for the unknown crack line tractions allows one to choose the number of suitable polynomials required for any desired accuracy. In this paper, a numerical method for analyzing interacting multiple cracks and complex crack configurations in infinite linear elastic media is presented. By extending Bueckner’s principle [17] suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple-crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by taking into account the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements [18] proposed recently by the author. Test examples are included to illustrate that the method is very simple and effective for analyzing interacting multiple cracks and complex crack configurations. Specifically, analysis of perpendicular cracks under general in-plane loading is performed using the numerical approach. These types of flaws may develop in the structural components under general mode I and II loading conditions. Evidence of this result is available from the test samples (Hastelloy-X tubing) under tension–torsion loading conducted by Jordan and Chen [19] in the Fatigue Testing Laboratory at the University of Connecicut. Fig. 1 shows some of the representative crack configurations due to such loading [20]. The crack geometries provide the original interest in predicting the mixedmode solutions to the multiple perpendicular crack problems. 2. Description of the present numerical approach The present numerical approach involves a generation of Bueckner’s principle and a displacement discontinuity method with crack-tip elements proposed recently by the author.
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
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2.1. A generalization of Bueckner’s principle Now consider a two-dimensional system containing a finite number of nonintersecting cracks. Specifically, consider an infinite elastic plate containing M arbitrarily oriented cracks under remote uniform stresses (i) 1 1 r1 and /(i) be the half-length of the ith crack yy ; rxx ; rxy . Let (x, y) be the global Cartesian coordinates. Let a ðiÞ ðiÞ and its orientation angle with respect to the x-axis, and ðxc ; y c Þ be the coordinates of the ith crack center. Denote the local Cartesian coordinate system associated with the ith crack by s(i) and t(i). The boundary conditions for the surfaces of the cracks are traction free, giving ðiÞ rðiÞ ss ¼ 0; rts ¼ 0;
i ¼ 1; 2; . . . ; M
ð1Þ
We shall refer to the above-described boundary value problem as the original problem. Bueckner [17] derived an important result, which is related to the principle of superposition. He demonstrated the equivalence of stress intensity factors resulting from external loading on a body and those resulting from internal tractions on the crack face. The stress intensity factor for a crack in a loaded body may be determined by considering the crack to be in an unloaded body with applied tractions on the crack surface only. These surface tractions are equal in magnitude but opposite in sign to those evaluated along the line of the crack site in the uncracked configuration. Here, we try to extend Bueckner’s principle [17] suited for a crack to a general system containing multiple interacting cracks. The original problem (see Fig. 2(a)) is divided into a homogeneous problem (the one without cracks) (Fig. 2 (b)) subjected to remote loads and a multiple-crack problem (see Fig. 2 (c)) in an unloaded body with applied tractions on the crack surfaces. The applied tractions on the ith crack surface are equal in magnitude but opposite in sign to those evaluated along the line of the ith crack site in the uncracked configuration, which are
∞
σ yy
∞
σ xy
σxx∞
=
a ∞
σ yy
∞
σ xy
− σ ss0(i) , −σ ts0 (i) σ xx∞
b
+
c K(a)=K(c)
Fig. 2. A generalization of Bueckner’s principle.
182
X. Yan / Engineering Failure Analysis 14 (2007) 179–195 ðiÞ
ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ 1 1 rss0 ¼ r1 xx sin / sin / 2rxy sin / cos / þ ryy cos / cos / ðiÞ
ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ 1 1 rts0 ¼ ðr1 yy rxx Þ sin / cos / þ rxy ðcos / cos / sin / sin / Þ
ð2Þ
which are called initial stresses here. Thus, the results in terms of stress intensity factors (SIFs) can be obtained by considering the latter problem (see Fig. 2(c)), which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author [18]. 2.2. Brief description of the displacement discontinuity method with crack-tip elements Here, the displacement discontinuity method with crack-tip elements proposed recently by the author [18] is described briefly. It consists of the (non-singular) constant displacement discontinuity element presented by Crouch and Starfield [21] and the crack-tip displacement discontinuity elements due to the author [22]. 2.2.1. Brief introduction of constant displacement discontinuity element [21] The displacement discontinuity Di is defined as the difference in displacement between the two sides of the segment [21]: Dx ¼ ux ðx; 0 Þ ux ðx; 0þ Þ Dy ¼ uy ðx; 0 Þ uy ðx; 0þ Þ
ð3Þ
The solution to the subject problem is given by Crouch and Starfield [21]. The displacements and stresses can be written as ux ¼ Dx ½2ð1 mÞF 3 ðx; yÞ yF 5 ðx; yÞ þ Dy ½ð1 2mÞF 2 ðx; yÞ yF 4 ðx; yÞ uy ¼ Dx ½ð1 2mÞF 2 ðx; yÞ yF 4 ðx; yÞ þ Dy ½2ð1 mÞF 3 ðx; yÞ yF 5 ðx; yÞ
ð4Þ
and rxx ¼ 2GDx ½2F 4 ðx; yÞ þ yF 6 ðx; yÞ þ 2GDy ½F 5 ðx; yÞ þ yF 7 ðx; yÞ ryy ¼ 2GDx ½yF 6 ðx; yÞ þ 2GDy ½F 5 ðx; yÞ yF 7 ðx; yÞ rxy ¼ 2GDx ½F 5 ðx; yÞ þ yF 7 ðx; yÞ þ 2GDy ½yF 6 ðx; yÞ
ð5Þ
G and m in these equations are shear modulus and the Poisson’s ratio, respectively. Functions F2 through F1 are described in Ref. [21]. Eqs. (4) and (5) are used by Crouch and Starfield [21] to set up a constant displacement discontinuity boundary element method. 2.2.2. Crack-tip displacement discontinuity elements By using Eqs. (4) and (5), recently, the author [22] presented the crack-tip displacement discontinuity elements, which can be classified as the left and the right crack-tip displacement discontinuity elements to deal with crack problems in general plane elasticity. The following gives basic formulas of the left crack-tip displacement discontinuity element. For the left crack-tip displacement discontinuity element, its-displacement discontinuity functions are chosen as 1 1 aþn 2 aþn 2 Dx ¼ H s ; Dy ¼ H n ð6Þ a a where Hs and Hn are the tangential and normal displacement discontinuity quantities at the center of the element, respectively. Here, it is noted that the element has the same unknowns as the two-dimensional constant displacement discontinuity element. But it can be seen that the displacement discontinuity functions defined in 6 can model the displacement fields around the crack tip. The stress field determined by the displacement discontinuity functions 6 possesses r1/2 singularity around the crack tip. Based on Eqs. (4) and (5), the displacements and stresses at a point (x, y) due to the left crack-tip displacement discontinuity element can be obtained from the differential viewpoint,
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
ux ¼ H s ½2ð1 mÞB3 ðx; yÞ yB5 ðx; yÞ þ H n ½ð1 2mÞB2 ðx; yÞ yB4 ðx; yÞ uy ¼ H s ½ð1 2mÞB2 ðx; yÞ yB4 ðx; yÞ þ H n ½2ð1 mÞB3 ðx; yÞ yB5 ðx; yÞ
183
ð7Þ
and rxx ¼ 2GH s ½2B4 ðx; yÞ þ yB6 ðx; yÞ þ 2GH n ½B5 ðx; yÞ þ yB7 ðx; yÞ ryy ¼ 2GH s ½yB6 ðx; yÞ þ 2GH n ½B5 ðx; yÞ yB7 ðx; yÞ
ð8Þ
rxy ¼ 2GH s ½B5 ðx; yÞ þ yB7 ðx; yÞ þ 2GH n ½yB6 ðx; yÞ where functions B2 through B7 are described in Ref. [22]. It can be seen by comparing Eqs. (7) and (8) with Eqs. (4) and (5) that the displacements and stresses due to the crack-tip displacement discontinuity possess the same forms as those due to a constant displacement discontinuity, with Fi(x, y) (i = 2, 3, . . ., 7) in Eqs. (4) and (5) being replaced by Bi(x, y) (i = 2, 3, , 7), Dx and Dy by Hs and Hn, respectively. This enables the boundary element implementation to be easy. For the right crack tip, formulas similar to Eqs. (6)–(8) can be obtained and are not given here. 2.2.3. Implementation of the displacement discontinuity method with crack-tip elements Crouch and Starfield [21] used Eqs. (4) and (5) to set up a constant displacement discontinuity boundary element method. Similarly, we can use Eqs. (7) and (8) to set up boundary element equations associated with the crack-tip elements. The constant displacement discontinuity elements together with the cracktip elements are combined easily to form a very effective numerical approach for calculating stress intensity factors in general plane crack problems. In the boundary element implementation the left or the right crack-tip displacement discontinuity element is placed locally at the corresponding left or right each crack tip on top of the ordinary non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called a displacement discontinuity method with crack-tip elements. 3. Computational formulas of stress intensity factors and some typical examples The objective of many analyses of linear elastic crack problems is to obtain the stress intensity factors (SIFs) KI and KII at the crack tips. Based on the displacement field around the crack tip, the following formulas exist: pffiffiffiffiffiffi G 2p limfDy ðrÞ=r0:5 g KI ¼ 4ð1 mÞ r!0 ð9Þ pffiffiffiffiffiffi G 2p 0:5 limfDx ðrÞ=r g K II ¼ 4ð1 mÞ r!0 where Dy(r) and Dx(r) are the normal and shear components of displacement discontinuity at a distance r from the crack tip(s). For the practical purpose, the limits in Eq. (9) can be approximated by simply evaluating the expression for a fixed value of r, small in relation to the size of the crack. By means of the crack-tip displacement discontinuity functions defined in Eq. (6), thus, the approximate formulas of the stress intensity factors KI and KII can be obtained by letting r in Eq. (9) be a, a half length of crack-tip element. pffiffiffiffiffiffi pffiffiffiffiffiffi 2pGH n 2pGH s pffiffiffi ; K II ¼ pffiffiffi KI ¼ ð10Þ 4ð1 mÞ a 4ð1 mÞ a To prove the efficiency of the suggested approach, some typical examples are given here. 3.1. Two collinear cracks with same length Here, we concern with the interaction of two collinear cracks with same length subjected to remote uniform stress r (see Fig. 3). The symmetry condition for this problem is available and the following cases are considered
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X. Yan / Engineering Failure Analysis 14 (2007) 179–195
σ
A
2a
2a
B d
σ Fig. 3. Schematic of two collinear cracks with same length subjected to remote uniform stress r.
2a=d ¼ 0:05; 0:1; 0:2; 0:3; 0:4; 0:5; 0:6; 0:7; 0:8; 0:9 Regarding the discretization of boundary elements, the number of elements discretized for each case is 30. The SIFs at crack tips A and B are normalized by pffiffiffiffiffiffi pffiffiffiffiffiffi F A ¼ K IA =ðr paÞ; F B ¼ K IB =ðr paÞ and are given in Table 1. For the comparison purpose, Table 1 gives also the exact solutions reported in Ref. [23]. It is found from Table 1 that the present numerical results are in very excellent agreement with the exact ones. 3.2. A finite main crack interaction with a collinear microcrack Shown in Fig. 4 is a finite main crack and its collinear microcrack under uniform far-field tension r normal to the crack faces, where the length of the main crack A 0 A is denoted by 2a, the length of the microcrack BB 0 by 2c and length h is utilized to specify the location of the center of the microcrack BB 0 . In this analysis, let the length of the main crack 2a be constant. Thus, ratios c/a and c/d can be used to indicate, respectively, the relative magnitude of the microcrack size and the relative distance away from the crack tip A to the center of he microcrack BB 0 . The following cases are considered: c=a ¼ 0:05 c=d ¼ 0:1; 0:333; 0:50; 0:667; 0:714; 0:769; 0:833; 0:909 Regarding the discretization of boundary elements, the number of elements discretized on a microcrack is kept constant 30 and boundary elements on the other boundaries are discretized according to the limitation condition that the boundary elements all have approximately the same length [22]. The present numerical results of the SIFs at crack tips A, B are normalized by
Table 1 pffiffiffiffiffiffi SIFs normalized by r pa for two collinear cracks with same length 2a/d
0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
FA
FB
Present
Ref. [23]
Present
Ref. [23]
0.9960 0.9969 1.0003 1.0058 1.0134 1.0234 1.0363 1.0531 1.0760 1.1118
1.00031 1.00120 1.00462 1.01017 1.01787 1.02795 1.04094 1.05786 1.08107 1.11741
0.9960 0.9970 1.0013 1.0094 1.0225 1.0429 1.0747 1.1263 1.2190 1.4334
1.00032 1.00132 1.00566 1.01383 1.02717 1.04796 1.08040 1.13326 1.22894 1.45387
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
185
σ 2c
A,
A 2a
B
B,
d
σ Fig. 4. A collinear microcrack in the vicinity of a finite main crack.
pffiffiffiffiffiffi F A ¼ K IA =ðr paÞ;
pffiffiffiffiffiffi F B ¼ K IB =ðr paÞ
and are given in Table 2. Obviously, the present numerical results can be used to reveal approximately the interaction of a semi-infinite main crack with a collinear microcrack. The analysis solution to the interaction of a semi-infinite main crack with a collinear microcrack was obtained in Refs. [24–26] and some digital results given by Gong and Horii [24] were also listed in Table 2. It is found from the Table 2 that the present numerical results are in very good agreement with analysis ones. 3.3. T-shaped crack Shown in Fig. 5 is a T-shaped crack under a uniform remote field stress. The following cases are considered here c=a ¼ 0:0219; 0:0492; 0:100; 0:200; 0:400; 0:600; 0:800; 1:00. The symmetry condition about y-axis for the cases shown in Fig. 5(a) and (b) is available. Regarding the discretization of boundary elements, the number of elements discretized on OA and OB segments is denoted by na and nb, respectively. Table 3 lists the variation of na and nb with c/a. The present numerical results of the SIFs at crack tips A and B are normalized by pffiffiffiffiffiffi F IA ¼ K IA =ðr paÞ pffiffiffiffiffiffi F IIA ¼ K IIA =ðr paÞ pffiffiffiffiffiffiffiffiffiffi F IB ¼ K IB =ðr pc=2Þ and are given in Tables 4 and 5. For the comparison purpose, Tables 4 and 5 list also those obtained by Kitagawa and Yuuki [7] by using a conformal mapping method (see also p. 273 in Ref. [23]). It is found from Tables 4 and 5 that the present numerical results are in very excellent agreement with those reported in Ref. [7].
Table 2 Normalized SIFs for the semi-infinite main crack interaction with a collinear microcrack c/d
FA Present
Ref. [24]
Present
Ref. [24]
0.1 0.333 0.500 0.667 0.714 0.769 0.833 0.909
1.0012 1.0297 1.0758 1.1636 1.2033 1.2640 1.3691 1.6092
– 1.030 1.076 1.167 1.209 1.274 1.387 1.652
0.3019 0.4917 0.6332 0.8131 0.8800 0.9732 1.1193 1.4155
– 0.457 0.611 0.805 0.877 0.979 1.138 1.469
FB
186
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
Fig. 5. Schematic of a T-shaped crack under a uniform remote field stress.
Table 3 Variation of the number of elements discretized on OA and OB segments with c/a c/a
.0219
.0492
.100
.200
.400
.600
.800
1.00
na nb
456 10
203 10
100 10
50 10
37 15
33 20
25 20
20 20
The symmetry condition about y-axis for the case shown in Fig. 5 (c) is not available. The discretization of boundary elements can be performed similarly. The present numerical results of the SIFs at crack tips A and B are normalized by
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
187
Table 4 Normalized SIFs for a T-shaped crack under uniform tension along x-axis c/a
0.0219 0.0492 0.100 0.200 0.400 0.600 0.800 1.00
FIB
FIA
FIIA
Present
Ref. [7]
Present
Ref. [7]
Present
Ref. [7]
1.5654 1.5636 1.5564 1.5297 1.4475 1.3502 1.2647 1.2001
1.584 1.583 1.576 1.549 1.458 1.358 1.272 1.208
0.0000 0.0000 0.0002 0.0018 0.0124 0.0324 0.0568 0.0809
0.000 0.000 0.000 0.002 0.013 0.033 0.057 0.081
0.0003 0.0015 0.0061 0.0234 0.0824 0.1537 0.2210 0.2787
0.000 0.002 0.006 0.024 0.083 0.154 0.221 0.279
Table 5 Normalized SIFs for a T-shaped crack under uniform tension along y-axis FIA
FIIA
c/a
FIB Present
Ref. [7]
Present
Ref. [7]
Present
Ref. [7]
0.0219 0.0492 0.100 0.200 0.400 0.600 0.800 1.00
1.5244 1.4719 1.3716 1.1705 0.7955 0.5032 0.3047 0.1780
1.542 1.489 1.384 1.178 0.795 0.501 0.303 0.176
0.9983 0.9986 0.9983 0.9985 1.0045 1.0131 1.0193 1.0228
1.000 1.000 1.000 1.002 1.008 1.017 1.024 1.029
0.0003 0.0014 0.0056 0.0197 0.0576 0.0899 0.1100 0.1209
0.000 0.001 0.006 0.020 0.058 0.090 0.111 0.122
pffiffiffiffiffiffi F IA ¼ K IA =ðs= paÞ pffiffiffiffiffiffi F IIA ¼ K IIA =ðs paÞ pffiffiffiffiffiffiffiffiffiffi F IB ¼ K IB =ðs pc=2Þ and are given in Table 6. For the comparison purpose, Table 6 list also those obtained by Kitagawa and Yuuki [7] by using a conformal mapping method (see also p. 273 in Ref. [23]). It is found from Table 6 that the present numerical results are in very excellent agreement with those reported in Ref. [7]. 3.4. A H-shaped crack Shown in Fig. 6 is a H-shaped crack under a uniform remote field stress. The symmetry condition about x-axis and y-axis for this problem is available. The following cases are considered here b=a ¼ 0:02; 0:05; 0:1; 0:2; 0:5; 0:7; 1:0; 1:5; 2:0; 2:5; 3:0; 3:5; 4:0; 4:5; 5:0; 5:5; 6:0; 7:0; 8:0; 9:0; 10:0 Table 6 Normalized SIFs for a T-shaped crack under pure shear c/a
FIIB Present
Ref. [7]
Present
Ref. [7]
Present
Ref. [7]
0.0219 0.0492 0.100 0.200 0.400 0.600 0.800 1.00
0.0429 0.0916 0.1842 0.3621 0.6855 0.9342 1.1039 1.2126
– 0.093 0.189 0.379 0.700 0.950 1.120 1.228
0.0000 0.0000 0.0000 0.0001 0.0017 0.0096 0.0289 0.0612
0.000 0.000 0.000 0.000 0.002 0.010 0.029 0.061
0.9977 0.9984 0.9981 0.9967 0.9918 0.9784 0.9549 0.9252
1.000 1.000 1.000 1.000 0.995 0.982 0.960 0.931
FIA
FIIA
188
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
Fig. 6. A H-shaped crack under a uniform remote field stress.
Table 7 Variation of the number of elements used on OB and BA segments with b/a b/a
0.02
0.5
0.1
0.2
0.5
0.7
1.0
1.5
2.0
2.5
3.0
na nb
500 10
200 10
100 10
50 10
40 20
28 20
20 20
20 30
20 40
20 50
20 60
b/a
3.5
4.0
4.5
5.0
5.5
6.0
7.0
8.0
9.0
10.0
na nb
20 70
20 80
20 90
20 100
20 110
20 120
20 140
20 160
20 180
20 200
Table 8 Normalized SIFs for a H-crack under uniform normal stress at infinity b/a
0.02
0.05
0.1
0.2
0.5
0.7
1.0
1.5
2.0
2.5
3.0
FIA FIIA
0.2595 0.3453
0.1884 0.3585
0.1152 0.3746
0.0264 0.3989
0.0835 0.4379
0.1034 0.4458
0.1109 0.4433
0.1132 0.4309
0.1146 0.4218
0.1155 0.4157
0.1162 0.4113
b/a
3.5
4.0
4.5
5.0
5.5
6.0
7.0
8.0
9.0
10.0
FIA FIIA
0.1167 0.4080
0.1170 0.4053
0.1173 0.4031
0.1176 0.4013
0.1178 0.3998
0.1179 0.3985
0.1182 0.3963
0.1184 0.3947
0.1185 0.3933
0.1186 0.3921
Table 9 Normalized SIFs for a H-crack under uniform normal stress at infinity [5] b/a
0.01
0.05
0.1
0.02
0.3
0.5
0.7
1.0
1.5
2.0
3.0
5.0
7.0
10.0
FIA FIIA
0.289 0.327
0.180 0.346
0.108 0.362
0.021 0.387
0.031 0.405
0.084 0.427
0.102 0.434
0.108 0.431
0.110 0.419
0.111 0.410
0.113 0.400
0.115 0.389
0.118 0.383
0.122 0.376
Regarding the discretization of boundary elements, the number of elements discretized on OB and BA segments is denoted by na and nb, respectively. Table 7 lists the variation of na and nb with b/a. The present numerical results of the SIFs at the crack tip A are normalized by pffiffiffiffiffiffi F IA ¼ K IA =ðr paÞ pffiffiffiffiffiffi F IIA ¼ K IIA =ðr paÞ
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189
and are given in Table 8. For the comparison purpose, Table 9 lists those obtained by Vitel [5] by using a dislocation distribution method. From Tables 8 and 9, it is found that the present numerical results are in very excellent agreement with those reported in Ref. [5]. 4. Interaction of perpendicular cracks in general in-plane loading In this section, the numerical approach described in this paper is used to study the interaction of perpendicular cracks in general in-plane loading. The first case is shown in Fig. 7. The following parameters are considered a ¼ 15 ; 30 ; 45 r ¼ 3:0; 2:5; 2:0; 1:5; 1:0; 0:543; 0:495; 0:354; 0:283; 0:075; 0:050; 0:025; 0:018; 0:014; 0:010 k¼ 2a Regarding the discretization of boundary elements, the total number of elementspdiscretized on cracks AB ffiffiffiffiffiffi and CD is 60 for all cases. The SIFs at crack tips A, B, C and D normalized by r pa are denoted by FIA, FIIA, FIB, FIIB, FIC, FIIC, FID and FIID, respectively, and are listed in Table 10. In order to illustrate the correctness of the present numerical results given in Tables 10, 11 lists the exact results [23] of the SIFs pffiffiffiffiffiffi normalized by r pa for a center-inclined crack in an infinite plate under uniform tension r where b is the angle between the crack plane and the load direction and a is half-crack length. Because when the parameter k is large the crack AB shown in Fig. 7 can be regarded as the inclined crack in an infinite plate under uniform tension r where the angle between the crack plane and the load direction is (90° a) and the crack CD can also be regarded as the inclined crack in an infinite plate under uniform tension r where the angle between the crack plane and the load direction is a, then FIA FIB, FIIA FIIB, FIC FID, and FIIC FIID hold and at this moment FIA, FIIA, FIB, FIIB, FIC, FIIC, FID and FIID should be in very agreement with those listed in Table 11. From Tables 10 and 11, it is found that the present numerical results indeed satisfy these conditions. In order to further illustrate the correctness of the present numerical results given pffiffiffiffiffiffi in Tables 10, 12 provides the comparison of the present numerical results of the SIFs normalized by r pa at the crack tip B for the case of a = 45° shown in Fig. 7 with analytical or numerical results available in literatures [20,14,27,28]. From Table 12, it is found that: (1) In the range of k from 0.543 to 3.0, the present numerical results are in very good agreement with analytical ones obtained by Gross [14]. While approximate analytical results by Hasan and Jordan [20] seem unreasonable because when the parameter k is large, for example, k = 3.0, they do not tend to the exact ones listed in Table 11; (2) In the range of k from 0.283 to 0.495, the present numerical results are in very good agreement with numerical ones obtained by Annigeri and Cleary [27,28] by using the Surface Integral and Finite Element Hybrid Method (SAFE). While approximate analytical results by Hasan and Jordan [20] have some
σ D α 2a C
r B
A 2a
d
d σ
Fig. 7. Two perpendicular cracks with equal length under general in-plane loading.
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X. Yan / Engineering Failure Analysis 14 (2007) 179–195
Table 10 pffiffiffiffiffiffi SIFs normalized by r pa for the crack problem shown in Fig. 7 a°
k
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
15
3.0 2.5 2.0 1.5 1.0 0.543 0.495 0.354 0.283 0.075 0.05 0.025 0.018 0.014 0.01
0.9280 0.9284 0.9290 0.9300 0.9321 0.9374 0.9384 0.9426 0.9459 0.9678 0.9744 0.9845 0.9887 0.9917 0.9953
0.2469 0.2465 0.2458 0.2446 0.2422 0.2369 0.2360 0.2323 0.2298 0.2180 0.2164 0.2158 0.2166 0.2177 0.2202
0.9292 0.9302 0.9320 0.9357 0.9450 0.9734 0.9798 1.0078 1.0313 1.2592 1.3612 1.5703 1.6799 1.7635 1.8684
0.2464 0.2457 0.2445 0.2424 0.2379 0.2280 0.2263 0.2200 0.2159 0.2010 0.1978 0.1819 0.1688 0.1617 0.1692
0.0607 0.0588 0.0560 0.0515 0.0440 0.0345 0.0336 0.0325 0.0336 0.0587 0.0641 0.0594 0.0564 0.0613 0.0871
0.2534 0.2549 0.2572 0.2610 0.2680 0.2840 0.2872 0.3016 0.3140 0.4626 0.5504 0.7926 0.9595 1.1061 1.3118
0.0609 0.0592 0.0566 0.0524 0.0453 0.0346 0.0332 0.0290 0.0271 0.0255 0.0269 0.0299 0.0315 0.0329 0.0350
0.2553 0.2578 0.2618 0.2690 0.2836 0.3153 0.3208 0.3419 0.3564 0.4342 0.4532 0.4806 0.4916 0.4992 0.5083
30
3.0 2.5 2.0 1.5 1.0 0.543 0.495 0.354 0.283 0.075 0.05 0.025 0.018 0.014 0.01
0.7460 0.7462 0.7466 0.7473 0.7486 0.7524 0.7533 0.7569 0.7599 0.7820 0.7890 0.7998 0.8044 0.8077 0.8119
0.4291 0.4288 0.4285 0.4282 0.4281 0.4292 0.4295 0.4311 0.4324 0.4441 0.4487 0.4572 0.4616 0.4652 0.4705
0.7477 0.7490 0.7511 0.7556 0.7668 0.8006 0.8080 0.8404 0.8671 1.1105 1.2130 1.4136 1.5166 1.5969 1.7037
0.4280 0.4272 0.4259 0.4236 0.4190 0.4105 0.4094 0.4063 0.4056 0.4376 0.4629 0.5271 0.5700 0.6122 0.6881
0.2463 0.2458 0.2454 0.2451 0.2466 0.2585 0.2618 0.2773 0.2910 0.4174 0.4641 0.5412 0.5793 0.6142 0.6753
0.4320 0.4325 0.4332 0.4343 0.4363 0.4429 0.4446 0.4534 0.4622 0.5886 0.6665 0.8796 1.0266 1.1575 1.3478
0.2454 0.2444 0.2429 0.2406 0.2366 0.2312 0.2306 0.2294 0.2294 0.2380 0.2419 0.2487 0.2518 0.2542 0.2574
0.4339 0.4354 0.4378 0.4423 0.4520 0.4744 0.4785 0.4943 0.5055 0.5689 0.5854 0.6101 0.6204 0.6277 0.6368
45
3.0 2.5 2.0 1.5 1.0 0.543 0.495 0.354 0.283 0.075 0.05 0.025 0.018 0.014 0.01
0.4955 0.4953 0.4949 0.4943 0.4934 0.4934 0.4942 0.4953 0.4970 0.5149 0.5202 0.5298 0.5347 0.5370 0.5410
0.4969 0.4974 0.4982 0.5000 0.5043 0.5154 0.5183 0.5263 0.5326 0.5726 0.5831 0.6006 0.6094 0.6143 0.6220
0.4972 0.4979 0.4992 0.5022 0.5106 0.5383 0.5452 0.5723 0.5954 0.8024 0.8834 1.0351 1.1138 1.1739 1.2641
0.4953 0.4950 0.4944 0.4932 0.4910 0.4886 0.4893 0.4911 0.4951 0.5795 0.6337 0.7811 0.8840 0.9747 1.1170
0.4972 0.4979 0.4992 0.5022 0.5106 0.5383 0.5452 0.5723 0.5954 0.8024 0.8845 1.0351 1.1138 1.1739 1.2641
0.4953 0.4950 0.4944 0.4932 0.4910 0.4886 0.4893 0.4911 0.4951 0.5795 0.6342 0.7811 0.8840 0.9747 1.1170
0.4955 0.4953 0.4949 0.4943 0.4934 0.4934 0.4942 0.4953 0.4970 0.5149 0.5203 0.5298 0.5347 0.5370 0.5410
0.4969 0.4974 0.4982 0.5000 0.5043 0.5154 0.5183 0.5263 0.5326 0.5726 0.5830 0.6006 0.6094 0.6143 0.6220
Table 11 pffiffiffiffiffiffi SIFs normalized by r pa for a center-inclined crack in an infinite plate [23] b° 15
30
45
60
75
FI
FII
FI
FII
FI
FII
FI
FII
FI
FII
0.0670
0.2500
0.2500
0.4330
0.5000
0.5000
0.7500
0.4330
0.9330
0.2500
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Table 12 pffiffiffiffiffiffi Comparison of SIFs normalized by r pa at the crack tip B for the case of a = 45° in Fig. 7 with analytical or numerical results available in literatures k
Present solution
Ref. [14]
Refs. [27,28]
Ref. [20]
FIB
FIIB
FIB
FIIB
FIB
FIIB
FIB
FIIB
3.0 2.5 2.0 1.5 1.0 0.543 0.495 0.354 0.283 0.200 0.150 0.100 0.075 0.050 0.025 0.018 0.014 0.010
0.4972 0.4979 0.4992 0.5022 0.5106 0.5383 0.5452 0.5723 0.5954 0.6382 0.6799 0.7486 0.8024 0.8834 1.0351 1.1138 1.1739 1.2641
0.4953 0.4950 0.4944 0.4932 0.4910 0.4886 0.4893 0.4911 0.4951 0.5060 0.5203 0.5505 0.5795 0.6337 0.7811 0.8840 0.9747 1.1170
0.5012 0.5019 0.5033 0.5066 0.5158 0.5466 – – – – – – – – – – – –
0.4992 0.4987 0.4979 0.4961 0.4916 0.4790 – – – – – – – – – – – –
– – – – – – 0.5363 0.5653 0.5843 – – – 0.2275 0.1969 0.1530 – – 0.1226
– – – – – – 0.4861 0.4889 0.4905 – – – 0.3995 0.3946 0.3864 – – 0.2767
0.4856 0.4871 0.4911 0.4995 0.5187 0.5637 0.5720 0.6035 0.6256 – – – 0.7536 0.7886 0.8735 – – 0.9054
0.5547 0.5547 0.5580 0.5616 0.5644 0.5620 0.5610 0.5550 0.5500 – – – 0.5063 0.4913 0.4663 – – 0.4368
difference from the numerical ones. By the way, it is pointed out that Gross’ solution has a significant limitation of calculating the stress intensity factors at k 6 0.4429 where the polynomial solution does not converge [20]; (3) It is interesting to note that at k < 0.283 the SAFE code generates a sharp decrease in the magnitude of the normalized mode I SIFs. Discussion [20] with the originator of the SAFE code revealed that it is difficult to obtain the collocation of the integration points of a crack placed at a very close distance (e.g. at k < 0.283) to another crack. Hence, it is reasonable to expect that at this small distance between two cracks, the SAFE code is unable to provide better results; (4) At k < 0.283, the present numerical results of the normalized mode I SIFs are in tendency in agreement with the approximate analytical ones obtained by Hasan and Jordan [20], but their differences increase with the decrease of the parameter k. However, the present numerical results of the normalized mode II SIFs are in tendency contrary to the approximate analytical ones obtained by Hasan and Jordan [20]. In order to completely examine the variation of normalized mode I and II SIFs at the crack tip B for the case of a = 45° shown in Fig. 7 with the parameter k, the normalized mode I and II SIFs corresponding to
Table 13 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 45° in Fig. 7 k
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.283 0.200 0.150 0.100 0.075 0.050 0.025 0.018 0.014 0.010
0.4970 0.5007 0.5043 0.5106 0.5149 0.5202 0.5298 0.5347 0.5370 0.5410
0.5326 0.5429 0.5515 0.5643 0.5726 0.5831 0.6006 0.6094 0.6143 0.6220
0.5954 0.6382 0.6799 0.7486 0.8024 0.8834 1.0351 1.1138 1.1739 1.2641
0.4951 0.5060 0.5203 0.5505 0.5795 0.6337 0.7811 0.8840 0.9747 1.1170
0.5954 0.6382 0.6799 0.7486 0.8024 0.8845 1.0351 1.1138 1.1739 1.2641
0.4951 0.5060 0.5203 0.5505 0.5795 0.6342 0.7811 0.8840 0.9747 1.1170
0.4970 0.5007 0.5043 0.5106 0.5149 0.5203 0.5298 0.5347 0.5370 0.5410
0.5326 0.5429 0.5515 0.5643 0.5726 0.5830 0.6006 0.6094 0.6143 0.6220
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X. Yan / Engineering Failure Analysis 14 (2007) 179–195
σ D 2a2
α C A
d2 B
2a1
σ Fig. 8. Two perpendicular cracks under general in-plane loading. Table 14 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 0° shown in Fig. 8 a2/a1
d2/a2
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.1
0.1 0.5 1.0 5.0 10.0
1.0007 0.9996 0.9992 0.9989 0.9988
0.0019 0.0002 0.0004 0.0008 0.0004
1.3150 1.0679 1.0287 1.0010 0.9990
0.0696 0.0184 0.0077 0.0008 0.0003
0.1071 0.0108 0.0427 0.0604 0.0308
0.5249 0.3086 0.2392 0.1319 0.0900
0.0353 0.0546 0.0631 0.0547 0.0259
0.2634 0.2177 0.1910 0.1220 0.0836
0.5
0.1 0.5 1.0 5.0 10.0
1.0118 1.0083 1.0064 0.9968 0.9958
0.0231 0.0177 0.0121 0.0002 0.0000
1.3237 1.0638 1.0238 0.9965 0.9958
0.0446 0.0187 0.0106 0.0006 0.0000
0.1798 0.1575 0.1156 0.0085 0.0084
0.5900 0.3509 0.2660 0.0552 0.0120
0.1146 0.0835 0.0535 0.0107 0.0075
0.3001 0.2308 0.1830 0.0391 0.0094
1.0
0.1 0.5 1.0 5.0 10.0
1.0390 1.0218 1.0090 0.9924 0.9922
0.0516 0.0243 0.0089 0.0000 0.0000
1.3072 1.0474 1.0102 0.9923 0.9921
0.0531 0.0232 0.0105 0.0001 0.0000
0.2311 0.1328 0.0535 0.0112 0.0040
0.5986 0.3326 0.2166 0.0152 0.0024
0.0520 0.0183 0.0029 0.0090 0.0034
0.2419 0.1582 0.1039 0.0098 0.0019
Table 15 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 30° shown in Fig. 8 a2/a1
d2/a2
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.1
0.1 0.5 1.0 5.0 10.0
0.7549 0.7524 0.7514 0.7502 0.7502
0.4458 0.4396 0.4371 0.4334 0.4327
1.2017 0.8641 0.8027 0.7555 0.7517
0.6223 0.4838 0.4555 0.4349 0.4335
0.8354 0.4078 0.2813 0.0943 0.0717
0.6341 0.3771 0.3028 0.2093 0.1904
0.3236 0.2362 0.1877 0.0849 0.0714
0.3082 0.2746 0.2526 0.2043 0.1877
0.5
0.1 0.5 1.0 5.0 10.0
0.7678 0.7611 0.7602 0.7555 0.7503
0.4638 0.4394 0.4306 0.4304 0.4321
1.2857 0.8926 0.8194 0.7549 0.7495
0.5883 0.4794 0.4561 0.4356 0.4331
0.5428 0.2476 0.1803 0.1785 0.1805
0.7960 0.5322 0.4677 0.3497 0.3168
0.2053 0.1700 0.1622 0.1804 0.1801
0.4913 0.4500 0.4243 0.3385 0.3145
1.0
0.1 0.5 1.0 5.0 10.0
0.8084 0.7965 0.7906 0.7540 0.7472
0.4414 0.4154 0.4148 0.4321 0.4314
1.3740 0.9146 0.8257 0.7511 0.7465
0.5920 0.4892 0.4631 0.4351 0.4317
0.4176 0.2342 0.2280 0.2550 0.2510
0.9510 0.6655 0.5857 0.4466 0.4337
0.2476 0.2427 0.2490 0.2539 0.2507
0.6159 0.5544 0.5159 0.4415 0.4330
k = 0.200, 0.150, 0.100, 0.018 and 0.014 are added in Table 12 comparing with the data provided by Hasanp and ffiffiffiffiffiffi Jordan [20]. For the case of a = 45° shown in Fig. 7, the SIFs at crack tips A, B, C and D normalized by r pa corresponding to k = 0.283, 0.200, 0.150, 0.100, 0.075, 0.050, 0.025, 0.018, 0.014 and 0.010 are further given in
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
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Table 16 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 45° shown in Fig. 8 a2/a1
d2/a2
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.1
0.1 0.5 1.0 5.0 10.0
0.5047 0.5021 0.5010 0.4995 0.4995
0.5152 0.5082 0.5054 0.5007 0.4995
0.9125 0.6106 0.5523 0.5055 0.5015
0.7079 0.5566 0.5254 0.5015 0.4999
1.0218 0.5484 0.4097 0.1974 0.1604
0.5336 0.3198 0.2605 0.1930 0.1868
0.4604 0.3622 0.3078 0.1843 0.1583
0.2538 0.2347 0.2202 0.1908 0.1860
0.5
0.1 0.5 1.0 5.0 10.0
0.5132 0.5052 0.5041 0.5060 0.5013
0.5646 0.5279 0.5113 0.4963 0.4981
1.0145 0.6512 0.5777 0.5078 0.5007
0.6871 0.5516 0.5234 0.5018 0.4994
0.8958 0.5024 0.4019 0.3510 0.3533
0.6890 0.4804 0.4399 0.3832 0.3610
0.4379 0.3784 0.3563 0.3524 0.3532
0.4467 0.4302 0.4205 0.3761 0.3593
1.0
0.1 0.5 1.0 5.0 10.0
0.5369 0.5304 0.5327 0.5078 0.4993
0.5734 0.5111 0.4911 0.4975 0.4973
1.1318 0.6908 0.5961 0.5056 0.4986
0.6833 0.5555 0.5276 0.5014 0.4979
0.8287 0.5352 0.4935 0.4996 0.4977
0.8606 0.6419 0.5956 0.5095 0.4993
0.5203 0.4976 0.4962 0.4992 0.4975
0.6122 0.5787 0.5564 0.5056 0.4987
Table 17 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 60° shown in Fig. 8 a2/a1
d2/a2
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.1
0.1 0.5 1.0 5.0 10.0
0.2548 0.2525 0.2515 0.2499 0.2498
0.4491 0.4426 0.4400 0.4353 0.4337
0.5638 0.3414 0.2949 0.2557 0.2521
0.6253 0.4874 0.4584 0.4351 0.4334
1.0134 0.5888 0.4677 0.2821 0.2442
0.3597 0.2177 0.1803 0.1430 0.1453
0.5173 0.4290 0.3809 0.2696 0.2415
0.1643 0.1602 0.1541 0.1429 0.1458
0.5
0.1 0.5 1.0 5.0 10.0
0.2566 0.2479 0.2462 0.2554 0.2527
0.5258 0.4830 0.4611 0.4301 0.4315
0.6622 0.3867 0.3244 0.2607 0.2528
0.6251 0.4831 0.4539 0.4342 0.4328
1.1135 0.7037 0.5975 0.5255 0.5275
0.4696 0.3452 0.3296 0.3222 0.3108
0.6386 0.5716 0.5434 0.5263 0.5275
0.3163 0.3244 0.3283 0.3189 0.3098
1.0
0.1 0.5 1.0 5.0 10.0
0.2526 0.2514 0.2623 0.2606 0.2522
0.5751 0.4823 0.4415 0.4300 0.4309
0.7828 0.4366 0.3523 0.2605 0.2518
0.6132 0.4772 0.4516 0.4341 0.4316
1.1551 0.8135 0.7544 0.7461 0.7457
0.6130 0.4903 0.4760 0.4390 0.4325
0.7846 0.7516 0.7451 0.7461 0.7456
0.4725 0.4689 0.4626 0.4366 0.4321
Table 13. From Table 13, the author considers that the present numerical results are effective even though the parameter is very small. The second example taken here is shown in Fig. 8. The following cases are considered: a ¼ 0 ; 30 ; 45 ; 60 ; 90 a2 =a1 ¼ 0:1; 0:5; 1:0 d 2 =a2 ¼ 0:1; 0:5; 1:0; 5:0; 10:0 In this analysis, the length of crack AB can be taken constant, while the length of crack CD varies according to the relationship above. Regarding the discretization of boundary elements, the number of elements discretized on crack CD is kept constant 30, while the number of elements discretized on crack AB is chosen as 300, 60 and 30, respectively, corresponding to a2/a1 = 0.1,0.5, and 1.0. The present numerical results of the SIFs norpffiffiffiffiffiffi malized by r pa at crack tips A, B, C and D are listed in Tables 14–18, respectively, corresponding to a = 0°, 30°, 45°, 60° and 90°.
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X. Yan / Engineering Failure Analysis 14 (2007) 179–195
Table 18 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 90° shown in Fig. 8 a2/a1
d2/a2
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.1
0.1 0.5 1.0 5.0 10.0
0.0018 0.0011 0.0007 0.0005 0.0009
0.0085 0.0062 0.0053 0.0031 0.0016
0.0414 0.0239 0.0142 0.0026 0.0012
0.0757 0.0256 0.0135 0.0012 0.0000
0.4629 0.3512 0.3300 0.3150 0.3142
0.0223 0.0096 0.0053 0.0006 0.0001
0.3520 0.3310 0.3232 0.3148 0.3142
0.0238 0.0107 0.0057 0.0006 0.0001
0.5
0.1 0.5 1.0 5.0 10.0
0.0094 0.0169 0.0203 0.0021 0.0018
0.1010 0.0694 0.0489 0.0007 0.0012
0.0791 0.0533 0.0350 0.0093 0.0036
0.1183 0.0262 0.0062 0.0023 0.0006
0.9609 0.7544 0.7187 0.7025 0.7024
0.0611 0.0223 0.0098 0.0001 0.0000
0.7520 0.7196 0.7088 0.7025 0.7024
0.0492 0.0200 0.0086 0.0001 0.0000
1.0
0.1 0.5 1.0 5.0 10.0
0.0714 0.0670 0.0465 0.0069 0.0034
0.2158 0.1095 0.0444 0.0042 0.0010
0.1271 0.0929 0.0647 0.0123 0.0039
0.0957 0.0007 0.0125 0.0020 0.0004
1.2432 1.0256 0.9991 0.9935 0.9933
0.0756 0.0171 0.0026 0.0001 0.0000
1.0218 0.9995 0.9950 0.9934 0.9933
0.0442 0.0124 0.0026 0.0001 0.0000
5. Conclusions A numerical method for analyzing interacting multiple cracks and complex crack configurations in infinite linear elastic media is presented in this paper. Test examples show that the method is very simple and effective for analyzing interacting multiple cracks and complex crack configurations. Specifically, numerical analysis of perpendicular cracks under general in-plane loading is performed and many numerical results are given in the form of tables. Acknowledgement Special thanks are due to the Natural Science Foundation (10272037) of China for supporting the present work. References [1] Erdogan F. On the stress distribution in plates with collinear cuts under arbitrary loads. In: Proceedings, fourth US national congress of applied mechanics; 1962. p. 547–53. [2] Sih GC. Stress distribution near internal crack tips for longitudinal shear problem. ASME J Appl Mch 1965;32:51–8. [3] Chudnovsky A, Dolgoposky A, Kachanov M. Elastic interaction of a crack with a microcrack array – I. Formulation of the problem and general form of the solution. Int J Solids Struct 1987;23:1–10. [4] Chudnovsky A, Dolgoposky A, Kachanov M. Elastic interaction of a crack with a microcrack array – II. Elastic solution for two crack configurations (piecewise constant and linear approximations), Formulation of the problem and general form of the solution. Int J Solids Struct 1987;23:11–21. [5] Vitek V. Plane strain stress intensity factors for branched cracks. Int J Fract 1977;13:481–501. [6] Kitagawa H, Yuuki R. Analysis of the stress intensity factors for doubly symmetric bent cracks and forked cracks, trans. Japan Soc Mech Engrs 1978;44–386:3346–53. [7] Kitagawa H, Yuuki R. Stress intensity factors for branched cracks in a two dimensional stress state. Trans Japan Soc Mech Engrs 1975;41–436:1641–9. [8] Kachanov M. A simple techniques of stress analysis in elastic solids with many cracks. Int J Fract 1985;28:R11–9. [9] Kachanov M. Elastic Solids with many cracks: a simple method of analysis. Int J Solids Struct 1987;23:23–44. [10] Horii H, Nemat-Nasser S. Elastic fields of interacting inhomogeneities. Int J Solids Struct 1985;21:731–45. [11] Benveniste Y, Dvorak GJ, Zarzour J, Wung ECJ. On interacting cracks and complex crack configurations in linear elastic media. Eng Fract Mech 1989;25:1279–93. [12] Collins WD. Some axially symmetric stress distribution in elastic solids containing penny-shaped cracks. Proc R Soc Ser A 1962;266:359–86. [13] Datsyshin AP, Savruk MP. A system of arbitrary oriented cracks in elastic solids. J Appl Math Mech 1973;37:306–13.
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Gross D. Stress intensity factors of system of cracks (in German). Ingenieur – Archiv 1982;51:301–10. Chudnovsky A, Kachanov M. Interaction of a crack with a field of microcracks. Int J Eng Sci 1983;21:1009–18. Chen YZ. General case of multiple crack problems in an infinite plate. Eng Fract Mech 1984;20:591–7. Buckner HF. The propagation of cracks and the energy of elastic deformation. Trans ASME, J Appl Mech 1958;80:1225–30. Yan X. Stress intensity factors for cracks emanating form a triangular or square hole in an infinite plate by boundary elements. Eng Failure Anal 2005;12(3):362–75. Jordan EH, Chen CT. A unique elevated temperature tension–torsion fatigue test rig. Exp Mech 1987;27:172–83. Hasan MS, Jordan E. An approximate analysis of perpendicular cracks under general in-plane loading. Eng Fract Mech 1995;51:167–91. Crouch SL, Starfield AM. Boundary element method in solid mechanics, with application in rock mechanics and geological mechanics. London/Bonton/Sydney: Geore Allon & Unwin; 1983. Yan X. A special crack tip displacement discontinuity element. Mech Res Commun 2004;31(6):651–9. Murakami Y. Stress intensity factors handbook. New York: Pergamon Press; 1987. 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. Rose LRF. Microcrack interaction with a main crack. Int J Fract 1986;31:233–42. Rubinstein A. Macrocrack interaction with semi-infinite microcrack array. Int J Fract 1985;27:113–9. Annigeri BS, Cleary MP. Int J Num Meth Eng 1984:869–85. Annigeri BS, Cleary MP. Quasi-static fracture propagation during the surface integral finite element hybrid method. Presented at the ASME pressure vessels and piping conference, San Antonio (TX); June 1984.
Stress intensity factors for interacting cracks and complex crack configurations in linear elastic media Xiangqiao Yan
*
Research Laboratory on Composite Materials, Harbin Institute of Technology, Harbin 150001, PR China Received 26 August 2005; accepted 31 October 2005 Available online 3 February 2006
Abstract In this paper, an effective numerical method for analyzing interacting multiple cracks and complex crack configurations in infinite linear elastic media is presented. By extending Bueckner’s principle suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple-crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by taking into account the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author. Test examples are included to illustrate that the method is very simple and effective for analyzing interacting multiple cracks and complex crack configurations in an infinite linear elastic media under remote uniform stresses. Specifically, analysis of perpendicular cracks under general in-plane loading is performed using the numerical approach and many numerical results are given in the form of tables. Ó 2006 Published by Elsevier Ltd. Keywords: Crack; Boundary element; Stress intensity factors; Crack-tip element; Displacement discontinuity
1. Introduction There are several situations in fracture mechanics, which involve a complicated arrangement of cracks that is not amenable to a simple method of analysis. In some cases, the difficulty lies in having many cracks interacting with each other, e.g., when a single crack is embedded in a microcrack array. In other cases, as in crack branching phenomena, the complexity of the problem is due to the presence of irregular crack shapes consisting of several segments which form what is sometimes called a zig-zag or nonlinear crack. In relatively simple situations of multiple cracks, such as aligned cracks and crack branches, classical methods of analysis are applicable and they lead to elegant exact solutions, e.g., Erdogan [1] and Sih [2]. However, approximate methods are unavoidable in more complicated situations. Some existing studies have employed the representation of cracks by dislocations [3–5], which leads to integral equations which *
Tel.: +87 451 8422 7537; fax: +87 451 8622 1048. E-mail address: [email protected].
1350-6307/$ - see front matter Ó 2006 Published by Elsevier Ltd. doi:10.1016/j.engfailanal.2005.10.008
180
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
Real case
Model 2a2
d2 [i] d1
2a1
2a2 d2 [ii]
2a1
Fig. 1. Crack configurations and respective geometric models [20].
can be solved in an approximate way. Other investigators of zig-zag crack configurations use methods based on polynomial approximations and truncation of a conformal mapping function [6,7]. A particularly simple treatment of crack interaction phenomena was introduced by Kachanov [8,9] who showed that many multiple crack problems can be solved with the help of a superposition procedure which leads to a system of linear algebraic equations for certain equilibrated crack-line tractions. Another variant of a crack interaction method was given by Horii and Nemat-Nasser [10]. Benveniste et al. [11] presented a general and simple method for computation of stress fields and the stress intensity factors in linear elastic media, which contain several cracks arranged in a complicated configuration. The method uses a superposition technique, which replaces a configuration of N cracks by means of N different problems, each involving an isolated crack located in an infinite medium and loaded by unknown traction. Such representations were used by Collins [12], Datsysin and Savruk [13], Cross [14], Chudnovsky and Kachanov [15], Chudnovsky et al. [3,4], Horii and Nemat-Nasser [10], and Chen [16]. In [11], a polynomial expansion for the unknown crack line tractions allows one to choose the number of suitable polynomials required for any desired accuracy. In this paper, a numerical method for analyzing interacting multiple cracks and complex crack configurations in infinite linear elastic media is presented. By extending Bueckner’s principle [17] suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple-crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by taking into account the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements [18] proposed recently by the author. Test examples are included to illustrate that the method is very simple and effective for analyzing interacting multiple cracks and complex crack configurations. Specifically, analysis of perpendicular cracks under general in-plane loading is performed using the numerical approach. These types of flaws may develop in the structural components under general mode I and II loading conditions. Evidence of this result is available from the test samples (Hastelloy-X tubing) under tension–torsion loading conducted by Jordan and Chen [19] in the Fatigue Testing Laboratory at the University of Connecicut. Fig. 1 shows some of the representative crack configurations due to such loading [20]. The crack geometries provide the original interest in predicting the mixedmode solutions to the multiple perpendicular crack problems. 2. Description of the present numerical approach The present numerical approach involves a generation of Bueckner’s principle and a displacement discontinuity method with crack-tip elements proposed recently by the author.
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
181
2.1. A generalization of Bueckner’s principle Now consider a two-dimensional system containing a finite number of nonintersecting cracks. Specifically, consider an infinite elastic plate containing M arbitrarily oriented cracks under remote uniform stresses (i) 1 1 r1 and /(i) be the half-length of the ith crack yy ; rxx ; rxy . Let (x, y) be the global Cartesian coordinates. Let a ðiÞ ðiÞ and its orientation angle with respect to the x-axis, and ðxc ; y c Þ be the coordinates of the ith crack center. Denote the local Cartesian coordinate system associated with the ith crack by s(i) and t(i). The boundary conditions for the surfaces of the cracks are traction free, giving ðiÞ rðiÞ ss ¼ 0; rts ¼ 0;
i ¼ 1; 2; . . . ; M
ð1Þ
We shall refer to the above-described boundary value problem as the original problem. Bueckner [17] derived an important result, which is related to the principle of superposition. He demonstrated the equivalence of stress intensity factors resulting from external loading on a body and those resulting from internal tractions on the crack face. The stress intensity factor for a crack in a loaded body may be determined by considering the crack to be in an unloaded body with applied tractions on the crack surface only. These surface tractions are equal in magnitude but opposite in sign to those evaluated along the line of the crack site in the uncracked configuration. Here, we try to extend Bueckner’s principle [17] suited for a crack to a general system containing multiple interacting cracks. The original problem (see Fig. 2(a)) is divided into a homogeneous problem (the one without cracks) (Fig. 2 (b)) subjected to remote loads and a multiple-crack problem (see Fig. 2 (c)) in an unloaded body with applied tractions on the crack surfaces. The applied tractions on the ith crack surface are equal in magnitude but opposite in sign to those evaluated along the line of the ith crack site in the uncracked configuration, which are
∞
σ yy
∞
σ xy
σxx∞
=
a ∞
σ yy
∞
σ xy
− σ ss0(i) , −σ ts0 (i) σ xx∞
b
+
c K(a)=K(c)
Fig. 2. A generalization of Bueckner’s principle.
182
X. Yan / Engineering Failure Analysis 14 (2007) 179–195 ðiÞ
ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ 1 1 rss0 ¼ r1 xx sin / sin / 2rxy sin / cos / þ ryy cos / cos / ðiÞ
ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ 1 1 rts0 ¼ ðr1 yy rxx Þ sin / cos / þ rxy ðcos / cos / sin / sin / Þ
ð2Þ
which are called initial stresses here. Thus, the results in terms of stress intensity factors (SIFs) can be obtained by considering the latter problem (see Fig. 2(c)), which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author [18]. 2.2. Brief description of the displacement discontinuity method with crack-tip elements Here, the displacement discontinuity method with crack-tip elements proposed recently by the author [18] is described briefly. It consists of the (non-singular) constant displacement discontinuity element presented by Crouch and Starfield [21] and the crack-tip displacement discontinuity elements due to the author [22]. 2.2.1. Brief introduction of constant displacement discontinuity element [21] The displacement discontinuity Di is defined as the difference in displacement between the two sides of the segment [21]: Dx ¼ ux ðx; 0 Þ ux ðx; 0þ Þ Dy ¼ uy ðx; 0 Þ uy ðx; 0þ Þ
ð3Þ
The solution to the subject problem is given by Crouch and Starfield [21]. The displacements and stresses can be written as ux ¼ Dx ½2ð1 mÞF 3 ðx; yÞ yF 5 ðx; yÞ þ Dy ½ð1 2mÞF 2 ðx; yÞ yF 4 ðx; yÞ uy ¼ Dx ½ð1 2mÞF 2 ðx; yÞ yF 4 ðx; yÞ þ Dy ½2ð1 mÞF 3 ðx; yÞ yF 5 ðx; yÞ
ð4Þ
and rxx ¼ 2GDx ½2F 4 ðx; yÞ þ yF 6 ðx; yÞ þ 2GDy ½F 5 ðx; yÞ þ yF 7 ðx; yÞ ryy ¼ 2GDx ½yF 6 ðx; yÞ þ 2GDy ½F 5 ðx; yÞ yF 7 ðx; yÞ rxy ¼ 2GDx ½F 5 ðx; yÞ þ yF 7 ðx; yÞ þ 2GDy ½yF 6 ðx; yÞ
ð5Þ
G and m in these equations are shear modulus and the Poisson’s ratio, respectively. Functions F2 through F1 are described in Ref. [21]. Eqs. (4) and (5) are used by Crouch and Starfield [21] to set up a constant displacement discontinuity boundary element method. 2.2.2. Crack-tip displacement discontinuity elements By using Eqs. (4) and (5), recently, the author [22] presented the crack-tip displacement discontinuity elements, which can be classified as the left and the right crack-tip displacement discontinuity elements to deal with crack problems in general plane elasticity. The following gives basic formulas of the left crack-tip displacement discontinuity element. For the left crack-tip displacement discontinuity element, its-displacement discontinuity functions are chosen as 1 1 aþn 2 aþn 2 Dx ¼ H s ; Dy ¼ H n ð6Þ a a where Hs and Hn are the tangential and normal displacement discontinuity quantities at the center of the element, respectively. Here, it is noted that the element has the same unknowns as the two-dimensional constant displacement discontinuity element. But it can be seen that the displacement discontinuity functions defined in 6 can model the displacement fields around the crack tip. The stress field determined by the displacement discontinuity functions 6 possesses r1/2 singularity around the crack tip. Based on Eqs. (4) and (5), the displacements and stresses at a point (x, y) due to the left crack-tip displacement discontinuity element can be obtained from the differential viewpoint,
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
ux ¼ H s ½2ð1 mÞB3 ðx; yÞ yB5 ðx; yÞ þ H n ½ð1 2mÞB2 ðx; yÞ yB4 ðx; yÞ uy ¼ H s ½ð1 2mÞB2 ðx; yÞ yB4 ðx; yÞ þ H n ½2ð1 mÞB3 ðx; yÞ yB5 ðx; yÞ
183
ð7Þ
and rxx ¼ 2GH s ½2B4 ðx; yÞ þ yB6 ðx; yÞ þ 2GH n ½B5 ðx; yÞ þ yB7 ðx; yÞ ryy ¼ 2GH s ½yB6 ðx; yÞ þ 2GH n ½B5 ðx; yÞ yB7 ðx; yÞ
ð8Þ
rxy ¼ 2GH s ½B5 ðx; yÞ þ yB7 ðx; yÞ þ 2GH n ½yB6 ðx; yÞ where functions B2 through B7 are described in Ref. [22]. It can be seen by comparing Eqs. (7) and (8) with Eqs. (4) and (5) that the displacements and stresses due to the crack-tip displacement discontinuity possess the same forms as those due to a constant displacement discontinuity, with Fi(x, y) (i = 2, 3, . . ., 7) in Eqs. (4) and (5) being replaced by Bi(x, y) (i = 2, 3, , 7), Dx and Dy by Hs and Hn, respectively. This enables the boundary element implementation to be easy. For the right crack tip, formulas similar to Eqs. (6)–(8) can be obtained and are not given here. 2.2.3. Implementation of the displacement discontinuity method with crack-tip elements Crouch and Starfield [21] used Eqs. (4) and (5) to set up a constant displacement discontinuity boundary element method. Similarly, we can use Eqs. (7) and (8) to set up boundary element equations associated with the crack-tip elements. The constant displacement discontinuity elements together with the cracktip elements are combined easily to form a very effective numerical approach for calculating stress intensity factors in general plane crack problems. In the boundary element implementation the left or the right crack-tip displacement discontinuity element is placed locally at the corresponding left or right each crack tip on top of the ordinary non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called a displacement discontinuity method with crack-tip elements. 3. Computational formulas of stress intensity factors and some typical examples The objective of many analyses of linear elastic crack problems is to obtain the stress intensity factors (SIFs) KI and KII at the crack tips. Based on the displacement field around the crack tip, the following formulas exist: pffiffiffiffiffiffi G 2p limfDy ðrÞ=r0:5 g KI ¼ 4ð1 mÞ r!0 ð9Þ pffiffiffiffiffiffi G 2p 0:5 limfDx ðrÞ=r g K II ¼ 4ð1 mÞ r!0 where Dy(r) and Dx(r) are the normal and shear components of displacement discontinuity at a distance r from the crack tip(s). For the practical purpose, the limits in Eq. (9) can be approximated by simply evaluating the expression for a fixed value of r, small in relation to the size of the crack. By means of the crack-tip displacement discontinuity functions defined in Eq. (6), thus, the approximate formulas of the stress intensity factors KI and KII can be obtained by letting r in Eq. (9) be a, a half length of crack-tip element. pffiffiffiffiffiffi pffiffiffiffiffiffi 2pGH n 2pGH s pffiffiffi ; K II ¼ pffiffiffi KI ¼ ð10Þ 4ð1 mÞ a 4ð1 mÞ a To prove the efficiency of the suggested approach, some typical examples are given here. 3.1. Two collinear cracks with same length Here, we concern with the interaction of two collinear cracks with same length subjected to remote uniform stress r (see Fig. 3). The symmetry condition for this problem is available and the following cases are considered
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X. Yan / Engineering Failure Analysis 14 (2007) 179–195
σ
A
2a
2a
B d
σ Fig. 3. Schematic of two collinear cracks with same length subjected to remote uniform stress r.
2a=d ¼ 0:05; 0:1; 0:2; 0:3; 0:4; 0:5; 0:6; 0:7; 0:8; 0:9 Regarding the discretization of boundary elements, the number of elements discretized for each case is 30. The SIFs at crack tips A and B are normalized by pffiffiffiffiffiffi pffiffiffiffiffiffi F A ¼ K IA =ðr paÞ; F B ¼ K IB =ðr paÞ and are given in Table 1. For the comparison purpose, Table 1 gives also the exact solutions reported in Ref. [23]. It is found from Table 1 that the present numerical results are in very excellent agreement with the exact ones. 3.2. A finite main crack interaction with a collinear microcrack Shown in Fig. 4 is a finite main crack and its collinear microcrack under uniform far-field tension r normal to the crack faces, where the length of the main crack A 0 A is denoted by 2a, the length of the microcrack BB 0 by 2c and length h is utilized to specify the location of the center of the microcrack BB 0 . In this analysis, let the length of the main crack 2a be constant. Thus, ratios c/a and c/d can be used to indicate, respectively, the relative magnitude of the microcrack size and the relative distance away from the crack tip A to the center of he microcrack BB 0 . The following cases are considered: c=a ¼ 0:05 c=d ¼ 0:1; 0:333; 0:50; 0:667; 0:714; 0:769; 0:833; 0:909 Regarding the discretization of boundary elements, the number of elements discretized on a microcrack is kept constant 30 and boundary elements on the other boundaries are discretized according to the limitation condition that the boundary elements all have approximately the same length [22]. The present numerical results of the SIFs at crack tips A, B are normalized by
Table 1 pffiffiffiffiffiffi SIFs normalized by r pa for two collinear cracks with same length 2a/d
0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
FA
FB
Present
Ref. [23]
Present
Ref. [23]
0.9960 0.9969 1.0003 1.0058 1.0134 1.0234 1.0363 1.0531 1.0760 1.1118
1.00031 1.00120 1.00462 1.01017 1.01787 1.02795 1.04094 1.05786 1.08107 1.11741
0.9960 0.9970 1.0013 1.0094 1.0225 1.0429 1.0747 1.1263 1.2190 1.4334
1.00032 1.00132 1.00566 1.01383 1.02717 1.04796 1.08040 1.13326 1.22894 1.45387
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
185
σ 2c
A,
A 2a
B
B,
d
σ Fig. 4. A collinear microcrack in the vicinity of a finite main crack.
pffiffiffiffiffiffi F A ¼ K IA =ðr paÞ;
pffiffiffiffiffiffi F B ¼ K IB =ðr paÞ
and are given in Table 2. Obviously, the present numerical results can be used to reveal approximately the interaction of a semi-infinite main crack with a collinear microcrack. The analysis solution to the interaction of a semi-infinite main crack with a collinear microcrack was obtained in Refs. [24–26] and some digital results given by Gong and Horii [24] were also listed in Table 2. It is found from the Table 2 that the present numerical results are in very good agreement with analysis ones. 3.3. T-shaped crack Shown in Fig. 5 is a T-shaped crack under a uniform remote field stress. The following cases are considered here c=a ¼ 0:0219; 0:0492; 0:100; 0:200; 0:400; 0:600; 0:800; 1:00. The symmetry condition about y-axis for the cases shown in Fig. 5(a) and (b) is available. Regarding the discretization of boundary elements, the number of elements discretized on OA and OB segments is denoted by na and nb, respectively. Table 3 lists the variation of na and nb with c/a. The present numerical results of the SIFs at crack tips A and B are normalized by pffiffiffiffiffiffi F IA ¼ K IA =ðr paÞ pffiffiffiffiffiffi F IIA ¼ K IIA =ðr paÞ pffiffiffiffiffiffiffiffiffiffi F IB ¼ K IB =ðr pc=2Þ and are given in Tables 4 and 5. For the comparison purpose, Tables 4 and 5 list also those obtained by Kitagawa and Yuuki [7] by using a conformal mapping method (see also p. 273 in Ref. [23]). It is found from Tables 4 and 5 that the present numerical results are in very excellent agreement with those reported in Ref. [7].
Table 2 Normalized SIFs for the semi-infinite main crack interaction with a collinear microcrack c/d
FA Present
Ref. [24]
Present
Ref. [24]
0.1 0.333 0.500 0.667 0.714 0.769 0.833 0.909
1.0012 1.0297 1.0758 1.1636 1.2033 1.2640 1.3691 1.6092
– 1.030 1.076 1.167 1.209 1.274 1.387 1.652
0.3019 0.4917 0.6332 0.8131 0.8800 0.9732 1.1193 1.4155
– 0.457 0.611 0.805 0.877 0.979 1.138 1.469
FB
186
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
Fig. 5. Schematic of a T-shaped crack under a uniform remote field stress.
Table 3 Variation of the number of elements discretized on OA and OB segments with c/a c/a
.0219
.0492
.100
.200
.400
.600
.800
1.00
na nb
456 10
203 10
100 10
50 10
37 15
33 20
25 20
20 20
The symmetry condition about y-axis for the case shown in Fig. 5 (c) is not available. The discretization of boundary elements can be performed similarly. The present numerical results of the SIFs at crack tips A and B are normalized by
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
187
Table 4 Normalized SIFs for a T-shaped crack under uniform tension along x-axis c/a
0.0219 0.0492 0.100 0.200 0.400 0.600 0.800 1.00
FIB
FIA
FIIA
Present
Ref. [7]
Present
Ref. [7]
Present
Ref. [7]
1.5654 1.5636 1.5564 1.5297 1.4475 1.3502 1.2647 1.2001
1.584 1.583 1.576 1.549 1.458 1.358 1.272 1.208
0.0000 0.0000 0.0002 0.0018 0.0124 0.0324 0.0568 0.0809
0.000 0.000 0.000 0.002 0.013 0.033 0.057 0.081
0.0003 0.0015 0.0061 0.0234 0.0824 0.1537 0.2210 0.2787
0.000 0.002 0.006 0.024 0.083 0.154 0.221 0.279
Table 5 Normalized SIFs for a T-shaped crack under uniform tension along y-axis FIA
FIIA
c/a
FIB Present
Ref. [7]
Present
Ref. [7]
Present
Ref. [7]
0.0219 0.0492 0.100 0.200 0.400 0.600 0.800 1.00
1.5244 1.4719 1.3716 1.1705 0.7955 0.5032 0.3047 0.1780
1.542 1.489 1.384 1.178 0.795 0.501 0.303 0.176
0.9983 0.9986 0.9983 0.9985 1.0045 1.0131 1.0193 1.0228
1.000 1.000 1.000 1.002 1.008 1.017 1.024 1.029
0.0003 0.0014 0.0056 0.0197 0.0576 0.0899 0.1100 0.1209
0.000 0.001 0.006 0.020 0.058 0.090 0.111 0.122
pffiffiffiffiffiffi F IA ¼ K IA =ðs= paÞ pffiffiffiffiffiffi F IIA ¼ K IIA =ðs paÞ pffiffiffiffiffiffiffiffiffiffi F IB ¼ K IB =ðs pc=2Þ and are given in Table 6. For the comparison purpose, Table 6 list also those obtained by Kitagawa and Yuuki [7] by using a conformal mapping method (see also p. 273 in Ref. [23]). It is found from Table 6 that the present numerical results are in very excellent agreement with those reported in Ref. [7]. 3.4. A H-shaped crack Shown in Fig. 6 is a H-shaped crack under a uniform remote field stress. The symmetry condition about x-axis and y-axis for this problem is available. The following cases are considered here b=a ¼ 0:02; 0:05; 0:1; 0:2; 0:5; 0:7; 1:0; 1:5; 2:0; 2:5; 3:0; 3:5; 4:0; 4:5; 5:0; 5:5; 6:0; 7:0; 8:0; 9:0; 10:0 Table 6 Normalized SIFs for a T-shaped crack under pure shear c/a
FIIB Present
Ref. [7]
Present
Ref. [7]
Present
Ref. [7]
0.0219 0.0492 0.100 0.200 0.400 0.600 0.800 1.00
0.0429 0.0916 0.1842 0.3621 0.6855 0.9342 1.1039 1.2126
– 0.093 0.189 0.379 0.700 0.950 1.120 1.228
0.0000 0.0000 0.0000 0.0001 0.0017 0.0096 0.0289 0.0612
0.000 0.000 0.000 0.000 0.002 0.010 0.029 0.061
0.9977 0.9984 0.9981 0.9967 0.9918 0.9784 0.9549 0.9252
1.000 1.000 1.000 1.000 0.995 0.982 0.960 0.931
FIA
FIIA
188
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
Fig. 6. A H-shaped crack under a uniform remote field stress.
Table 7 Variation of the number of elements used on OB and BA segments with b/a b/a
0.02
0.5
0.1
0.2
0.5
0.7
1.0
1.5
2.0
2.5
3.0
na nb
500 10
200 10
100 10
50 10
40 20
28 20
20 20
20 30
20 40
20 50
20 60
b/a
3.5
4.0
4.5
5.0
5.5
6.0
7.0
8.0
9.0
10.0
na nb
20 70
20 80
20 90
20 100
20 110
20 120
20 140
20 160
20 180
20 200
Table 8 Normalized SIFs for a H-crack under uniform normal stress at infinity b/a
0.02
0.05
0.1
0.2
0.5
0.7
1.0
1.5
2.0
2.5
3.0
FIA FIIA
0.2595 0.3453
0.1884 0.3585
0.1152 0.3746
0.0264 0.3989
0.0835 0.4379
0.1034 0.4458
0.1109 0.4433
0.1132 0.4309
0.1146 0.4218
0.1155 0.4157
0.1162 0.4113
b/a
3.5
4.0
4.5
5.0
5.5
6.0
7.0
8.0
9.0
10.0
FIA FIIA
0.1167 0.4080
0.1170 0.4053
0.1173 0.4031
0.1176 0.4013
0.1178 0.3998
0.1179 0.3985
0.1182 0.3963
0.1184 0.3947
0.1185 0.3933
0.1186 0.3921
Table 9 Normalized SIFs for a H-crack under uniform normal stress at infinity [5] b/a
0.01
0.05
0.1
0.02
0.3
0.5
0.7
1.0
1.5
2.0
3.0
5.0
7.0
10.0
FIA FIIA
0.289 0.327
0.180 0.346
0.108 0.362
0.021 0.387
0.031 0.405
0.084 0.427
0.102 0.434
0.108 0.431
0.110 0.419
0.111 0.410
0.113 0.400
0.115 0.389
0.118 0.383
0.122 0.376
Regarding the discretization of boundary elements, the number of elements discretized on OB and BA segments is denoted by na and nb, respectively. Table 7 lists the variation of na and nb with b/a. The present numerical results of the SIFs at the crack tip A are normalized by pffiffiffiffiffiffi F IA ¼ K IA =ðr paÞ pffiffiffiffiffiffi F IIA ¼ K IIA =ðr paÞ
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
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and are given in Table 8. For the comparison purpose, Table 9 lists those obtained by Vitel [5] by using a dislocation distribution method. From Tables 8 and 9, it is found that the present numerical results are in very excellent agreement with those reported in Ref. [5]. 4. Interaction of perpendicular cracks in general in-plane loading In this section, the numerical approach described in this paper is used to study the interaction of perpendicular cracks in general in-plane loading. The first case is shown in Fig. 7. The following parameters are considered a ¼ 15 ; 30 ; 45 r ¼ 3:0; 2:5; 2:0; 1:5; 1:0; 0:543; 0:495; 0:354; 0:283; 0:075; 0:050; 0:025; 0:018; 0:014; 0:010 k¼ 2a Regarding the discretization of boundary elements, the total number of elementspdiscretized on cracks AB ffiffiffiffiffiffi and CD is 60 for all cases. The SIFs at crack tips A, B, C and D normalized by r pa are denoted by FIA, FIIA, FIB, FIIB, FIC, FIIC, FID and FIID, respectively, and are listed in Table 10. In order to illustrate the correctness of the present numerical results given in Tables 10, 11 lists the exact results [23] of the SIFs pffiffiffiffiffiffi normalized by r pa for a center-inclined crack in an infinite plate under uniform tension r where b is the angle between the crack plane and the load direction and a is half-crack length. Because when the parameter k is large the crack AB shown in Fig. 7 can be regarded as the inclined crack in an infinite plate under uniform tension r where the angle between the crack plane and the load direction is (90° a) and the crack CD can also be regarded as the inclined crack in an infinite plate under uniform tension r where the angle between the crack plane and the load direction is a, then FIA FIB, FIIA FIIB, FIC FID, and FIIC FIID hold and at this moment FIA, FIIA, FIB, FIIB, FIC, FIIC, FID and FIID should be in very agreement with those listed in Table 11. From Tables 10 and 11, it is found that the present numerical results indeed satisfy these conditions. In order to further illustrate the correctness of the present numerical results given pffiffiffiffiffiffi in Tables 10, 12 provides the comparison of the present numerical results of the SIFs normalized by r pa at the crack tip B for the case of a = 45° shown in Fig. 7 with analytical or numerical results available in literatures [20,14,27,28]. From Table 12, it is found that: (1) In the range of k from 0.543 to 3.0, the present numerical results are in very good agreement with analytical ones obtained by Gross [14]. While approximate analytical results by Hasan and Jordan [20] seem unreasonable because when the parameter k is large, for example, k = 3.0, they do not tend to the exact ones listed in Table 11; (2) In the range of k from 0.283 to 0.495, the present numerical results are in very good agreement with numerical ones obtained by Annigeri and Cleary [27,28] by using the Surface Integral and Finite Element Hybrid Method (SAFE). While approximate analytical results by Hasan and Jordan [20] have some
σ D α 2a C
r B
A 2a
d
d σ
Fig. 7. Two perpendicular cracks with equal length under general in-plane loading.
190
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
Table 10 pffiffiffiffiffiffi SIFs normalized by r pa for the crack problem shown in Fig. 7 a°
k
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
15
3.0 2.5 2.0 1.5 1.0 0.543 0.495 0.354 0.283 0.075 0.05 0.025 0.018 0.014 0.01
0.9280 0.9284 0.9290 0.9300 0.9321 0.9374 0.9384 0.9426 0.9459 0.9678 0.9744 0.9845 0.9887 0.9917 0.9953
0.2469 0.2465 0.2458 0.2446 0.2422 0.2369 0.2360 0.2323 0.2298 0.2180 0.2164 0.2158 0.2166 0.2177 0.2202
0.9292 0.9302 0.9320 0.9357 0.9450 0.9734 0.9798 1.0078 1.0313 1.2592 1.3612 1.5703 1.6799 1.7635 1.8684
0.2464 0.2457 0.2445 0.2424 0.2379 0.2280 0.2263 0.2200 0.2159 0.2010 0.1978 0.1819 0.1688 0.1617 0.1692
0.0607 0.0588 0.0560 0.0515 0.0440 0.0345 0.0336 0.0325 0.0336 0.0587 0.0641 0.0594 0.0564 0.0613 0.0871
0.2534 0.2549 0.2572 0.2610 0.2680 0.2840 0.2872 0.3016 0.3140 0.4626 0.5504 0.7926 0.9595 1.1061 1.3118
0.0609 0.0592 0.0566 0.0524 0.0453 0.0346 0.0332 0.0290 0.0271 0.0255 0.0269 0.0299 0.0315 0.0329 0.0350
0.2553 0.2578 0.2618 0.2690 0.2836 0.3153 0.3208 0.3419 0.3564 0.4342 0.4532 0.4806 0.4916 0.4992 0.5083
30
3.0 2.5 2.0 1.5 1.0 0.543 0.495 0.354 0.283 0.075 0.05 0.025 0.018 0.014 0.01
0.7460 0.7462 0.7466 0.7473 0.7486 0.7524 0.7533 0.7569 0.7599 0.7820 0.7890 0.7998 0.8044 0.8077 0.8119
0.4291 0.4288 0.4285 0.4282 0.4281 0.4292 0.4295 0.4311 0.4324 0.4441 0.4487 0.4572 0.4616 0.4652 0.4705
0.7477 0.7490 0.7511 0.7556 0.7668 0.8006 0.8080 0.8404 0.8671 1.1105 1.2130 1.4136 1.5166 1.5969 1.7037
0.4280 0.4272 0.4259 0.4236 0.4190 0.4105 0.4094 0.4063 0.4056 0.4376 0.4629 0.5271 0.5700 0.6122 0.6881
0.2463 0.2458 0.2454 0.2451 0.2466 0.2585 0.2618 0.2773 0.2910 0.4174 0.4641 0.5412 0.5793 0.6142 0.6753
0.4320 0.4325 0.4332 0.4343 0.4363 0.4429 0.4446 0.4534 0.4622 0.5886 0.6665 0.8796 1.0266 1.1575 1.3478
0.2454 0.2444 0.2429 0.2406 0.2366 0.2312 0.2306 0.2294 0.2294 0.2380 0.2419 0.2487 0.2518 0.2542 0.2574
0.4339 0.4354 0.4378 0.4423 0.4520 0.4744 0.4785 0.4943 0.5055 0.5689 0.5854 0.6101 0.6204 0.6277 0.6368
45
3.0 2.5 2.0 1.5 1.0 0.543 0.495 0.354 0.283 0.075 0.05 0.025 0.018 0.014 0.01
0.4955 0.4953 0.4949 0.4943 0.4934 0.4934 0.4942 0.4953 0.4970 0.5149 0.5202 0.5298 0.5347 0.5370 0.5410
0.4969 0.4974 0.4982 0.5000 0.5043 0.5154 0.5183 0.5263 0.5326 0.5726 0.5831 0.6006 0.6094 0.6143 0.6220
0.4972 0.4979 0.4992 0.5022 0.5106 0.5383 0.5452 0.5723 0.5954 0.8024 0.8834 1.0351 1.1138 1.1739 1.2641
0.4953 0.4950 0.4944 0.4932 0.4910 0.4886 0.4893 0.4911 0.4951 0.5795 0.6337 0.7811 0.8840 0.9747 1.1170
0.4972 0.4979 0.4992 0.5022 0.5106 0.5383 0.5452 0.5723 0.5954 0.8024 0.8845 1.0351 1.1138 1.1739 1.2641
0.4953 0.4950 0.4944 0.4932 0.4910 0.4886 0.4893 0.4911 0.4951 0.5795 0.6342 0.7811 0.8840 0.9747 1.1170
0.4955 0.4953 0.4949 0.4943 0.4934 0.4934 0.4942 0.4953 0.4970 0.5149 0.5203 0.5298 0.5347 0.5370 0.5410
0.4969 0.4974 0.4982 0.5000 0.5043 0.5154 0.5183 0.5263 0.5326 0.5726 0.5830 0.6006 0.6094 0.6143 0.6220
Table 11 pffiffiffiffiffiffi SIFs normalized by r pa for a center-inclined crack in an infinite plate [23] b° 15
30
45
60
75
FI
FII
FI
FII
FI
FII
FI
FII
FI
FII
0.0670
0.2500
0.2500
0.4330
0.5000
0.5000
0.7500
0.4330
0.9330
0.2500
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Table 12 pffiffiffiffiffiffi Comparison of SIFs normalized by r pa at the crack tip B for the case of a = 45° in Fig. 7 with analytical or numerical results available in literatures k
Present solution
Ref. [14]
Refs. [27,28]
Ref. [20]
FIB
FIIB
FIB
FIIB
FIB
FIIB
FIB
FIIB
3.0 2.5 2.0 1.5 1.0 0.543 0.495 0.354 0.283 0.200 0.150 0.100 0.075 0.050 0.025 0.018 0.014 0.010
0.4972 0.4979 0.4992 0.5022 0.5106 0.5383 0.5452 0.5723 0.5954 0.6382 0.6799 0.7486 0.8024 0.8834 1.0351 1.1138 1.1739 1.2641
0.4953 0.4950 0.4944 0.4932 0.4910 0.4886 0.4893 0.4911 0.4951 0.5060 0.5203 0.5505 0.5795 0.6337 0.7811 0.8840 0.9747 1.1170
0.5012 0.5019 0.5033 0.5066 0.5158 0.5466 – – – – – – – – – – – –
0.4992 0.4987 0.4979 0.4961 0.4916 0.4790 – – – – – – – – – – – –
– – – – – – 0.5363 0.5653 0.5843 – – – 0.2275 0.1969 0.1530 – – 0.1226
– – – – – – 0.4861 0.4889 0.4905 – – – 0.3995 0.3946 0.3864 – – 0.2767
0.4856 0.4871 0.4911 0.4995 0.5187 0.5637 0.5720 0.6035 0.6256 – – – 0.7536 0.7886 0.8735 – – 0.9054
0.5547 0.5547 0.5580 0.5616 0.5644 0.5620 0.5610 0.5550 0.5500 – – – 0.5063 0.4913 0.4663 – – 0.4368
difference from the numerical ones. By the way, it is pointed out that Gross’ solution has a significant limitation of calculating the stress intensity factors at k 6 0.4429 where the polynomial solution does not converge [20]; (3) It is interesting to note that at k < 0.283 the SAFE code generates a sharp decrease in the magnitude of the normalized mode I SIFs. Discussion [20] with the originator of the SAFE code revealed that it is difficult to obtain the collocation of the integration points of a crack placed at a very close distance (e.g. at k < 0.283) to another crack. Hence, it is reasonable to expect that at this small distance between two cracks, the SAFE code is unable to provide better results; (4) At k < 0.283, the present numerical results of the normalized mode I SIFs are in tendency in agreement with the approximate analytical ones obtained by Hasan and Jordan [20], but their differences increase with the decrease of the parameter k. However, the present numerical results of the normalized mode II SIFs are in tendency contrary to the approximate analytical ones obtained by Hasan and Jordan [20]. In order to completely examine the variation of normalized mode I and II SIFs at the crack tip B for the case of a = 45° shown in Fig. 7 with the parameter k, the normalized mode I and II SIFs corresponding to
Table 13 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 45° in Fig. 7 k
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.283 0.200 0.150 0.100 0.075 0.050 0.025 0.018 0.014 0.010
0.4970 0.5007 0.5043 0.5106 0.5149 0.5202 0.5298 0.5347 0.5370 0.5410
0.5326 0.5429 0.5515 0.5643 0.5726 0.5831 0.6006 0.6094 0.6143 0.6220
0.5954 0.6382 0.6799 0.7486 0.8024 0.8834 1.0351 1.1138 1.1739 1.2641
0.4951 0.5060 0.5203 0.5505 0.5795 0.6337 0.7811 0.8840 0.9747 1.1170
0.5954 0.6382 0.6799 0.7486 0.8024 0.8845 1.0351 1.1138 1.1739 1.2641
0.4951 0.5060 0.5203 0.5505 0.5795 0.6342 0.7811 0.8840 0.9747 1.1170
0.4970 0.5007 0.5043 0.5106 0.5149 0.5203 0.5298 0.5347 0.5370 0.5410
0.5326 0.5429 0.5515 0.5643 0.5726 0.5830 0.6006 0.6094 0.6143 0.6220
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X. Yan / Engineering Failure Analysis 14 (2007) 179–195
σ D 2a2
α C A
d2 B
2a1
σ Fig. 8. Two perpendicular cracks under general in-plane loading. Table 14 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 0° shown in Fig. 8 a2/a1
d2/a2
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.1
0.1 0.5 1.0 5.0 10.0
1.0007 0.9996 0.9992 0.9989 0.9988
0.0019 0.0002 0.0004 0.0008 0.0004
1.3150 1.0679 1.0287 1.0010 0.9990
0.0696 0.0184 0.0077 0.0008 0.0003
0.1071 0.0108 0.0427 0.0604 0.0308
0.5249 0.3086 0.2392 0.1319 0.0900
0.0353 0.0546 0.0631 0.0547 0.0259
0.2634 0.2177 0.1910 0.1220 0.0836
0.5
0.1 0.5 1.0 5.0 10.0
1.0118 1.0083 1.0064 0.9968 0.9958
0.0231 0.0177 0.0121 0.0002 0.0000
1.3237 1.0638 1.0238 0.9965 0.9958
0.0446 0.0187 0.0106 0.0006 0.0000
0.1798 0.1575 0.1156 0.0085 0.0084
0.5900 0.3509 0.2660 0.0552 0.0120
0.1146 0.0835 0.0535 0.0107 0.0075
0.3001 0.2308 0.1830 0.0391 0.0094
1.0
0.1 0.5 1.0 5.0 10.0
1.0390 1.0218 1.0090 0.9924 0.9922
0.0516 0.0243 0.0089 0.0000 0.0000
1.3072 1.0474 1.0102 0.9923 0.9921
0.0531 0.0232 0.0105 0.0001 0.0000
0.2311 0.1328 0.0535 0.0112 0.0040
0.5986 0.3326 0.2166 0.0152 0.0024
0.0520 0.0183 0.0029 0.0090 0.0034
0.2419 0.1582 0.1039 0.0098 0.0019
Table 15 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 30° shown in Fig. 8 a2/a1
d2/a2
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.1
0.1 0.5 1.0 5.0 10.0
0.7549 0.7524 0.7514 0.7502 0.7502
0.4458 0.4396 0.4371 0.4334 0.4327
1.2017 0.8641 0.8027 0.7555 0.7517
0.6223 0.4838 0.4555 0.4349 0.4335
0.8354 0.4078 0.2813 0.0943 0.0717
0.6341 0.3771 0.3028 0.2093 0.1904
0.3236 0.2362 0.1877 0.0849 0.0714
0.3082 0.2746 0.2526 0.2043 0.1877
0.5
0.1 0.5 1.0 5.0 10.0
0.7678 0.7611 0.7602 0.7555 0.7503
0.4638 0.4394 0.4306 0.4304 0.4321
1.2857 0.8926 0.8194 0.7549 0.7495
0.5883 0.4794 0.4561 0.4356 0.4331
0.5428 0.2476 0.1803 0.1785 0.1805
0.7960 0.5322 0.4677 0.3497 0.3168
0.2053 0.1700 0.1622 0.1804 0.1801
0.4913 0.4500 0.4243 0.3385 0.3145
1.0
0.1 0.5 1.0 5.0 10.0
0.8084 0.7965 0.7906 0.7540 0.7472
0.4414 0.4154 0.4148 0.4321 0.4314
1.3740 0.9146 0.8257 0.7511 0.7465
0.5920 0.4892 0.4631 0.4351 0.4317
0.4176 0.2342 0.2280 0.2550 0.2510
0.9510 0.6655 0.5857 0.4466 0.4337
0.2476 0.2427 0.2490 0.2539 0.2507
0.6159 0.5544 0.5159 0.4415 0.4330
k = 0.200, 0.150, 0.100, 0.018 and 0.014 are added in Table 12 comparing with the data provided by Hasanp and ffiffiffiffiffiffi Jordan [20]. For the case of a = 45° shown in Fig. 7, the SIFs at crack tips A, B, C and D normalized by r pa corresponding to k = 0.283, 0.200, 0.150, 0.100, 0.075, 0.050, 0.025, 0.018, 0.014 and 0.010 are further given in
X. Yan / Engineering Failure Analysis 14 (2007) 179–195
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Table 16 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 45° shown in Fig. 8 a2/a1
d2/a2
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.1
0.1 0.5 1.0 5.0 10.0
0.5047 0.5021 0.5010 0.4995 0.4995
0.5152 0.5082 0.5054 0.5007 0.4995
0.9125 0.6106 0.5523 0.5055 0.5015
0.7079 0.5566 0.5254 0.5015 0.4999
1.0218 0.5484 0.4097 0.1974 0.1604
0.5336 0.3198 0.2605 0.1930 0.1868
0.4604 0.3622 0.3078 0.1843 0.1583
0.2538 0.2347 0.2202 0.1908 0.1860
0.5
0.1 0.5 1.0 5.0 10.0
0.5132 0.5052 0.5041 0.5060 0.5013
0.5646 0.5279 0.5113 0.4963 0.4981
1.0145 0.6512 0.5777 0.5078 0.5007
0.6871 0.5516 0.5234 0.5018 0.4994
0.8958 0.5024 0.4019 0.3510 0.3533
0.6890 0.4804 0.4399 0.3832 0.3610
0.4379 0.3784 0.3563 0.3524 0.3532
0.4467 0.4302 0.4205 0.3761 0.3593
1.0
0.1 0.5 1.0 5.0 10.0
0.5369 0.5304 0.5327 0.5078 0.4993
0.5734 0.5111 0.4911 0.4975 0.4973
1.1318 0.6908 0.5961 0.5056 0.4986
0.6833 0.5555 0.5276 0.5014 0.4979
0.8287 0.5352 0.4935 0.4996 0.4977
0.8606 0.6419 0.5956 0.5095 0.4993
0.5203 0.4976 0.4962 0.4992 0.4975
0.6122 0.5787 0.5564 0.5056 0.4987
Table 17 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 60° shown in Fig. 8 a2/a1
d2/a2
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.1
0.1 0.5 1.0 5.0 10.0
0.2548 0.2525 0.2515 0.2499 0.2498
0.4491 0.4426 0.4400 0.4353 0.4337
0.5638 0.3414 0.2949 0.2557 0.2521
0.6253 0.4874 0.4584 0.4351 0.4334
1.0134 0.5888 0.4677 0.2821 0.2442
0.3597 0.2177 0.1803 0.1430 0.1453
0.5173 0.4290 0.3809 0.2696 0.2415
0.1643 0.1602 0.1541 0.1429 0.1458
0.5
0.1 0.5 1.0 5.0 10.0
0.2566 0.2479 0.2462 0.2554 0.2527
0.5258 0.4830 0.4611 0.4301 0.4315
0.6622 0.3867 0.3244 0.2607 0.2528
0.6251 0.4831 0.4539 0.4342 0.4328
1.1135 0.7037 0.5975 0.5255 0.5275
0.4696 0.3452 0.3296 0.3222 0.3108
0.6386 0.5716 0.5434 0.5263 0.5275
0.3163 0.3244 0.3283 0.3189 0.3098
1.0
0.1 0.5 1.0 5.0 10.0
0.2526 0.2514 0.2623 0.2606 0.2522
0.5751 0.4823 0.4415 0.4300 0.4309
0.7828 0.4366 0.3523 0.2605 0.2518
0.6132 0.4772 0.4516 0.4341 0.4316
1.1551 0.8135 0.7544 0.7461 0.7457
0.6130 0.4903 0.4760 0.4390 0.4325
0.7846 0.7516 0.7451 0.7461 0.7456
0.4725 0.4689 0.4626 0.4366 0.4321
Table 13. From Table 13, the author considers that the present numerical results are effective even though the parameter is very small. The second example taken here is shown in Fig. 8. The following cases are considered: a ¼ 0 ; 30 ; 45 ; 60 ; 90 a2 =a1 ¼ 0:1; 0:5; 1:0 d 2 =a2 ¼ 0:1; 0:5; 1:0; 5:0; 10:0 In this analysis, the length of crack AB can be taken constant, while the length of crack CD varies according to the relationship above. Regarding the discretization of boundary elements, the number of elements discretized on crack CD is kept constant 30, while the number of elements discretized on crack AB is chosen as 300, 60 and 30, respectively, corresponding to a2/a1 = 0.1,0.5, and 1.0. The present numerical results of the SIFs norpffiffiffiffiffiffi malized by r pa at crack tips A, B, C and D are listed in Tables 14–18, respectively, corresponding to a = 0°, 30°, 45°, 60° and 90°.
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X. Yan / Engineering Failure Analysis 14 (2007) 179–195
Table 18 pffiffiffiffiffiffi SIFs normalized by r pa for the case of a = 90° shown in Fig. 8 a2/a1
d2/a2
FIA
FIIA
FIB
FIIB
FIC
FIIC
FID
FIID
0.1
0.1 0.5 1.0 5.0 10.0
0.0018 0.0011 0.0007 0.0005 0.0009
0.0085 0.0062 0.0053 0.0031 0.0016
0.0414 0.0239 0.0142 0.0026 0.0012
0.0757 0.0256 0.0135 0.0012 0.0000
0.4629 0.3512 0.3300 0.3150 0.3142
0.0223 0.0096 0.0053 0.0006 0.0001
0.3520 0.3310 0.3232 0.3148 0.3142
0.0238 0.0107 0.0057 0.0006 0.0001
0.5
0.1 0.5 1.0 5.0 10.0
0.0094 0.0169 0.0203 0.0021 0.0018
0.1010 0.0694 0.0489 0.0007 0.0012
0.0791 0.0533 0.0350 0.0093 0.0036
0.1183 0.0262 0.0062 0.0023 0.0006
0.9609 0.7544 0.7187 0.7025 0.7024
0.0611 0.0223 0.0098 0.0001 0.0000
0.7520 0.7196 0.7088 0.7025 0.7024
0.0492 0.0200 0.0086 0.0001 0.0000
1.0
0.1 0.5 1.0 5.0 10.0
0.0714 0.0670 0.0465 0.0069 0.0034
0.2158 0.1095 0.0444 0.0042 0.0010
0.1271 0.0929 0.0647 0.0123 0.0039
0.0957 0.0007 0.0125 0.0020 0.0004
1.2432 1.0256 0.9991 0.9935 0.9933
0.0756 0.0171 0.0026 0.0001 0.0000
1.0218 0.9995 0.9950 0.9934 0.9933
0.0442 0.0124 0.0026 0.0001 0.0000
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