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Stress intensity factors of parallel cracks in a finite width sheet
&@eering Fracture M&an&s Vol. 35, No. 6, pp. 1073-1079,1990 Printed in Great Britain.
0013-7944/90 83.00+ 0.00 0 1990Pergamon Press pk.
STRESS INTENSITY FACTORS OF PARALLEL IN A FINITE WIDTH SHEET Z. D. JIANG, A. ZEGHLOUL,
CRACKS
G. BEZINE and J. PETIT
Laboratoire de Mecanique et de Physique des Materiaux, URA CNRS 863Ecole Nationale Sup&ieure de Mecanique et d’ACrotechnique, 86634 Poitiers Cedex, France Abstrac-Two and three parallel cracks in a finite sheet subjected to remote tensile loading have been studied. This paper presents empirical stress intensity factor formulae for these crack configurations. The stress intensity factors used to develop these formulae were obtained from finite element analysis. For central cracks and edge cracks, the formulae were within 1 and 3% of the finite element results, respectively.
1. INTRODUCTION SINCE fracture mechanics was introduced, i.e. ever since the stress intensity factor solution of a crack in an infinite body has been found, research on the problem of cracks may be broadly divided into two fields: one is to study a single crack in a finite body, the other is to study more than one crack, but in an infinite or in a semi-infinite body. A vast amount of results have been obtained in both fields. But for the analysis of multiple cracks in a finite body, little attention has been paid thus far. This paper discusses two and three parallel cracks in a sheet with finite width, this sheet is subjected to uniform tensile stress in the longitudinal direction remote from the cracks (see Fig. 1). Figure l(a) shows the cracks in the centre of the sheet and Fig. l(b) shows the cracks on an edge of the sheet. We compute the stress intensity factors for a wide range of configuration parameters using the finite element method. On the basis of these results, we present empirical stress intensity factor formulae for two and three parallel cracks.
2. FINITE ELEMENT
ANALYSIS
We use the finite element code “Abaqus”[l] and the 8 node quadrilateral isoparametric “CPESR” elements with a 2 x 2 Gauss integration rule. Figure 2 shows an example of finite element idealization. The dimension of the element that is closed to the tips of cracks is less than l/20 of the crack length. For all of the crack lengths 2a, H is always kept at H/a > 3; as the calculation shows, when H/a > 3, the variation of H has almost no influence on the results. We first calculate the J-integral. For a cracked body subjected to a remotely applied tensile load, the path independent integral defined by Rice[2] is:
(1) where r is a line contour taken from the lower crack surface in a counter clockwise direction to the upper crack surface, U is the displacement vector, ds is an increment arclength along the contour, w is the strain energy density, T is dependent of the outward normal nj along r, and is given by T = ap,. For plane strain conditions, the stress intensity factor can be expressed in terms of the J-integral as: K+--.-
EJ
l-9 where E, v, are the Young’s modulus and Poisson ratio, respectively. 1073
Z. D. JIANG, et al.
1074
4 4 + $ t
++i+++
w
b
0
0
Fig. 1. Parallel cracks in a finite sheet.
Fig. 2. Finite element modei.
We express the stress intensity factor K, as: JG = KP,,@iw, b/a)
(3)
where K(, is the stress intensity factor for an isolated crack in an infinite sheet and is given by: &=C+.
(4)
The correction factor I$ is a function of W (width of sheet), b (interval between cracks) and n (number of cracks). Tables 1 and 2 present the results for F2, and Tables 3-6 the results for F3.
Table 1. Correction factors F2 for two central cracks a/w bla
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
OS
0.10 0.25 0.50 1.00 I‘S0 2.00 2.50 3.00
0.125 0.747 0.782 ::z
0.728 0.752 0.787 0.850 0.903 0.936 0.957 0.970
0.734 0.759 0.795 0.860 0.914 0.947 0.968 0.980
0.743 0.768 0.806 0.874 0.928 0.962 0.983 0.997
0.754 0.782 0.821 0.891 0.947 0.982 f
0.769 0.798 0.840 0.913 0.971 1.008 1.030 1.043
0.788 0.819 0.863 0.939 1.001 1.040 1.061 1.071
0.810 0.844 0.891 0.872 1.038 I.078 1.096 1.102
0.838 0.874 0.924 1.017 1.084 1.121 1.134 1.138
0.871 0.911 0.965 1.070 1.138 1.170 1.178 1.179
0.929 0.950 0.963
:K!
Table 2. Correction factors F, for two edge cracks a/w
bla
0.05
0.1
0.1s
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.10 0.25 0.50
0.824 0.847 0.881 0.934 0.989 1.033 1.064 1.08s
0.862 0.890 0.930 0.994 1.057 1.106 1.139 1.158
0.92 1 0.956 1.006 1.086 l.lSS 1.209 1.237 1.251
1.001 1.045 1.109 1.208 1.288 1.333 1.352 1.358
1.103 1.161 1.243 1.362 1.443 1.477 1.486 1.486
1.231 1.307 1.412 1.550 1.623 1.643 1.645 1.64s
1.392 1.492 1.625 1.775 1.833 1.840 1.841 1.842
1.594 1.726 1.892 2.045 2.084 2.085 2.085 2.086
1.852 2.027 2.229 2.390 2.390 2.390 2.390 2.392
2.186 2.419 2.657 2.801 2.778 2.778 2.779 2.779
E 2:Oo 2.50 3.00
SIF of parallel cracks
1075
Table 3. Correction factors Fq for the middle crack of three central cracks
bla
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.25 0.50 1.00 2.00
0.471 0.562 0.704 0.873
0.476 0.570 0.713 0.887
0.484 0.581 0.729 0.906
0.494 0.595 0.750 0.931
0.508 0.613 0.773 0.964
0.525 0.635 0.801 1.006
0.546 0.654 0.836 1.055
0.572 0.683 0.882 1.109
0.603 0.724 0.943 1.161
Table 4. Correction factors F, for the outer crack of three central cracks ah b/a
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.25 0.50 1.00 2.00
0.717 0.756 0.834 0.925
0.724 0.765 0.838 0.940
0.735 0.778 0.856 0.960
0.749 0.795 0.879 0.986
0.767 0.817 0.906 1.015
0.789 0.844 0.940 1.049
0.813 0.876 0.981 1.088
0.844 0.917 1.028 1.135
0.881 0.964 1.084 1.177
Table 5. Correction factors F%for the central crack of three edge cracks a/w
bla
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.25 0.50 1.00 2.00
0.559 0.625 0.743 1.028
0.606 0.719 0.898 1.159
0.671 0.815 1.041 1.311
0.760 0.947 1.230 1.473
0.878 1.121 1.451 1.650
1.034 1.349 1.711 1.854
1.244 1.641 2.010 2.102
1.527 2.013 2.359 2.413
1.914 2.483 2.776 2.778
Table 6. Correction factors 4 for the outer crack of three edge cracks
alw bla
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.25 0.50 1.00 2.00
0.849 0.901 0.933 1.105
0.921 0.988 1.084 1.211
1.017 1.103 1.214 1.337
1.142 1.247 1.374 1.482
1.299 1.425 1.561 1.651
1.495 1.642 1.785 1.853
1.741 1.908 2.054 2.101
2.052 2.239 2.380 2.412
2.448 2.657 2.783 2.778
3. FORMULAE
AND DISCUSSION
According to selected results here obtained by using the finite element method, we propose an approximate relation to calculate correction factors F, for two parallel cracks situated in the centre of a sheet of the form: F = A0 + A,(a/w) + M+9’~s (5)
2 $Tm2
in which
A 0= 0.709 + 0.1&b/a) - O.O4(b/a)2 + O.O02(b/a)3 - 0.032(b/t~)~ A ,= -0.044- OM(b/a)+ 0.27(b/t1)~ A2 = 0.118 + 0.85(b /a) - 0.47(b /a)’ + 0.056@ /u)~. Figure 3 compares the values given by relation (5) with all these results obtained by finite element method. The difference is less than 1%. Benthem and Koiter[3] have presented a relation for a central crack in a finite width sheet: K, = &G(u/w)
= K,
1 - 0.5(a/w) + 0.326(a/w)’ Jz&v
*
Figure 3 also shows the correction factors G. It can be observed that, with b/a> 3, the interrelation almost vanishes.
1076
2. D. JIANG, et al. 1.5
0 x
finite element eq.(5) [2] G
results
1.3 b/a = 3
,
I
I
I
0.2
0.1
I
I
0.3
0.4
a/w
0.5
Fig. 3. Correction factors Fz for 2 central cracks.
For two parallel cracks situated on a sheet edge: F
=
Al + A,@/wY+ 4(a/w)4
2 $=m?
in which A,, = 0.79 + O.O7(bja) + 0.04(b/a)2 - 0.01 l(b/a)3 A , = 1.74 + 284(b/a) - l~I4(b/a)~ + 0.206(b/a)3 A, = 6.02 + 2.19(b/a) - 3.26(6/a)‘+ 0.828(b/a)3. Equation (6) is accurate to 3%. Figure 4 compares the results of eq. (6) with the finite element results. For edge cracks, as compared with central cracks, the stress intensity factors increase rapidly when a/w increases, but more slowly when b/a increases. For three parallel cracks situated in the centre of a sheet: (7)
F2 3.
r
o
finite element
results
b/a = 3
=2 2.5
IA.5 =0.25 =O.l
2. 1.5 1.
I
0.1
I
0.2
1
0.3
I
0.4
Fig. 4. Correction factors F2 for 2 edge cracks.
I
0.5
alw
SIF of parallel cracks
1077
where for the middle crack: A0 = 0.391 + 0.33(b/a) + o.o2(b/a)* - O.O33(b/a)3 A , = -0.551 + 2.17(b/a) - 2.87(b/~)~ + 0.945(b/a)3 A2 = 0.851 - 2.80(b/u) + 4.05@/~.2)~- 1.356@/~)~ for the two outer cracks: A0 = 0.67 + O.l6(b/u) + O.O3(b/u)’ - 0.027@/~)~ A, = 0.13 - 0.87(6/u) + 0.43(b/~)~ + O.O10(b/~)~ A2 = -0.16 + 1.84(b/u) - 1.06(b/~)~ + O.l02(b/~)~ eq. (7) is accurate to 1%. Isida[4] has calculated the stress intensity factors for odd numbers of parallel cracks in an infinite sheet to b/u > 1.25. Our results to the case b/u = 2, u/w = 0.1 are in close agreement with those of [4]; in this case the influence of the edge is only of 1%. For three parallel cracks situated on the edge of a sheet: F = Ao + A,WTs
+
A,@/w)~
3
(8) Jm&7
where for the middle crack: A,, = 0.378 + 0.68(b/u) - 0.72(b/u)2 + 0.256(b/u)3 A, = 0.970 + l.l6(b/u)
+ 4.99(b/u)2 - 2.391(b/u)3
A, = 1.347 + 50.48(b/u) - 73.69(b/u)‘+
24.190(b/u)3
and for the two outer cracks: A, = 0.69 + 0.42(b/u) - 0.47(b/u)’ + 0.17(b/u)3 A, = 1.99 + 1.98(6/u) + 0.82(b/u)2 - 0.73(b/u)3 A2 = 6.83 + 6.02(b/u) - 16.20(b/u)2 + 6.18(b/u)3. Equation (8) is accurate to 3%. A comparison between the proposed eqs (7) and (8) and the finite element results, is shown in Figs 5-8.
F3 1.2 finite element
results
=1 0.8
0.6
=0.25
I
a/w 0.1
0.2
0.3
0.4
0.5
Fig. 5. Correction factors I$ for the middle crack of 3 central cracks.
1078
2. D. JIANG, ef al. F3
bla = 2
1.2
=1 1.
=0.5 =0.25
0.6 0.6 0
fiiiy,;lement
results
.
-
0.4
i
I
I
0.1
0.2
I
0.3
I
0.4
I alw 0.5
Fig. 6. Correction factors Fj for the outer crack of 3 central cracks.
F3 finite element
I
results
I
I
I
0.1
0.2
0.3
I
0.4
I
a/w
0.5
Fig. 7. Correction factors 4 for the middle crack of 3 edge cracks.
F3 3. -
bla = ? 0
2.5 -
--
finite element
results
zA.5
eq.(81
=0.25
2. 1.5 -
0.5
atw 0.1
0.2
0.3
0.4
0.5
Fig. 8. Correction factors F, for the outer crack of 3 edge cracks.
SIF of parallel cracks
1079
4. CONCLUSION (1) Using the results obtained by finite element method, we present relations for stress intensity factors for two (b/a < 3) and three (b/a < 2) parallel cracks in a finite sheet. For central cracks, the relations are accurate to 1% (a/w < 0.5), and for edge cracks to 3% (a/w < 0.45). (2) When there is more than one crack in a sheet, since the flexibility of the sheet increases, the stress intensity factors decrease. Only when the interval between two cracks is more than three times the crack length, the problem can be treated as two single cracks far from each other. (3) The formulae presented herein should be useful for calculating fracture loads and determining the stress intensity factors in fatigue tests for these types of crack configurations. REFERENCES [I] Abaqus, Finite Element Program developed by Hibbitt, Karlsson and Sorensen Inc. (USA). [2] J. R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. appl. Med. (June 1968). [3] J. P. Benthem and W. T. Keiter, Mechanics of Fracture (Edited by G. C. Sih), Vol. 1, chapter 3 (1973). [4] M. Isida, Bull. JSME, 13, 635 (1970).
0013-7944/90 83.00+ 0.00 0 1990Pergamon Press pk.
STRESS INTENSITY FACTORS OF PARALLEL IN A FINITE WIDTH SHEET Z. D. JIANG, A. ZEGHLOUL,
CRACKS
G. BEZINE and J. PETIT
Laboratoire de Mecanique et de Physique des Materiaux, URA CNRS 863Ecole Nationale Sup&ieure de Mecanique et d’ACrotechnique, 86634 Poitiers Cedex, France Abstrac-Two and three parallel cracks in a finite sheet subjected to remote tensile loading have been studied. This paper presents empirical stress intensity factor formulae for these crack configurations. The stress intensity factors used to develop these formulae were obtained from finite element analysis. For central cracks and edge cracks, the formulae were within 1 and 3% of the finite element results, respectively.
1. INTRODUCTION SINCE fracture mechanics was introduced, i.e. ever since the stress intensity factor solution of a crack in an infinite body has been found, research on the problem of cracks may be broadly divided into two fields: one is to study a single crack in a finite body, the other is to study more than one crack, but in an infinite or in a semi-infinite body. A vast amount of results have been obtained in both fields. But for the analysis of multiple cracks in a finite body, little attention has been paid thus far. This paper discusses two and three parallel cracks in a sheet with finite width, this sheet is subjected to uniform tensile stress in the longitudinal direction remote from the cracks (see Fig. 1). Figure l(a) shows the cracks in the centre of the sheet and Fig. l(b) shows the cracks on an edge of the sheet. We compute the stress intensity factors for a wide range of configuration parameters using the finite element method. On the basis of these results, we present empirical stress intensity factor formulae for two and three parallel cracks.
2. FINITE ELEMENT
ANALYSIS
We use the finite element code “Abaqus”[l] and the 8 node quadrilateral isoparametric “CPESR” elements with a 2 x 2 Gauss integration rule. Figure 2 shows an example of finite element idealization. The dimension of the element that is closed to the tips of cracks is less than l/20 of the crack length. For all of the crack lengths 2a, H is always kept at H/a > 3; as the calculation shows, when H/a > 3, the variation of H has almost no influence on the results. We first calculate the J-integral. For a cracked body subjected to a remotely applied tensile load, the path independent integral defined by Rice[2] is:
(1) where r is a line contour taken from the lower crack surface in a counter clockwise direction to the upper crack surface, U is the displacement vector, ds is an increment arclength along the contour, w is the strain energy density, T is dependent of the outward normal nj along r, and is given by T = ap,. For plane strain conditions, the stress intensity factor can be expressed in terms of the J-integral as: K+--.-
EJ
l-9 where E, v, are the Young’s modulus and Poisson ratio, respectively. 1073
Z. D. JIANG, et al.
1074
4 4 + $ t
++i+++
w
b
0
0
Fig. 1. Parallel cracks in a finite sheet.
Fig. 2. Finite element modei.
We express the stress intensity factor K, as: JG = KP,,@iw, b/a)
(3)
where K(, is the stress intensity factor for an isolated crack in an infinite sheet and is given by: &=C+.
(4)
The correction factor I$ is a function of W (width of sheet), b (interval between cracks) and n (number of cracks). Tables 1 and 2 present the results for F2, and Tables 3-6 the results for F3.
Table 1. Correction factors F2 for two central cracks a/w bla
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
OS
0.10 0.25 0.50 1.00 I‘S0 2.00 2.50 3.00
0.125 0.747 0.782 ::z
0.728 0.752 0.787 0.850 0.903 0.936 0.957 0.970
0.734 0.759 0.795 0.860 0.914 0.947 0.968 0.980
0.743 0.768 0.806 0.874 0.928 0.962 0.983 0.997
0.754 0.782 0.821 0.891 0.947 0.982 f
0.769 0.798 0.840 0.913 0.971 1.008 1.030 1.043
0.788 0.819 0.863 0.939 1.001 1.040 1.061 1.071
0.810 0.844 0.891 0.872 1.038 I.078 1.096 1.102
0.838 0.874 0.924 1.017 1.084 1.121 1.134 1.138
0.871 0.911 0.965 1.070 1.138 1.170 1.178 1.179
0.929 0.950 0.963
:K!
Table 2. Correction factors F, for two edge cracks a/w
bla
0.05
0.1
0.1s
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.10 0.25 0.50
0.824 0.847 0.881 0.934 0.989 1.033 1.064 1.08s
0.862 0.890 0.930 0.994 1.057 1.106 1.139 1.158
0.92 1 0.956 1.006 1.086 l.lSS 1.209 1.237 1.251
1.001 1.045 1.109 1.208 1.288 1.333 1.352 1.358
1.103 1.161 1.243 1.362 1.443 1.477 1.486 1.486
1.231 1.307 1.412 1.550 1.623 1.643 1.645 1.64s
1.392 1.492 1.625 1.775 1.833 1.840 1.841 1.842
1.594 1.726 1.892 2.045 2.084 2.085 2.085 2.086
1.852 2.027 2.229 2.390 2.390 2.390 2.390 2.392
2.186 2.419 2.657 2.801 2.778 2.778 2.779 2.779
E 2:Oo 2.50 3.00
SIF of parallel cracks
1075
Table 3. Correction factors Fq for the middle crack of three central cracks
bla
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.25 0.50 1.00 2.00
0.471 0.562 0.704 0.873
0.476 0.570 0.713 0.887
0.484 0.581 0.729 0.906
0.494 0.595 0.750 0.931
0.508 0.613 0.773 0.964
0.525 0.635 0.801 1.006
0.546 0.654 0.836 1.055
0.572 0.683 0.882 1.109
0.603 0.724 0.943 1.161
Table 4. Correction factors F, for the outer crack of three central cracks ah b/a
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.25 0.50 1.00 2.00
0.717 0.756 0.834 0.925
0.724 0.765 0.838 0.940
0.735 0.778 0.856 0.960
0.749 0.795 0.879 0.986
0.767 0.817 0.906 1.015
0.789 0.844 0.940 1.049
0.813 0.876 0.981 1.088
0.844 0.917 1.028 1.135
0.881 0.964 1.084 1.177
Table 5. Correction factors F%for the central crack of three edge cracks a/w
bla
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.25 0.50 1.00 2.00
0.559 0.625 0.743 1.028
0.606 0.719 0.898 1.159
0.671 0.815 1.041 1.311
0.760 0.947 1.230 1.473
0.878 1.121 1.451 1.650
1.034 1.349 1.711 1.854
1.244 1.641 2.010 2.102
1.527 2.013 2.359 2.413
1.914 2.483 2.776 2.778
Table 6. Correction factors 4 for the outer crack of three edge cracks
alw bla
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.25 0.50 1.00 2.00
0.849 0.901 0.933 1.105
0.921 0.988 1.084 1.211
1.017 1.103 1.214 1.337
1.142 1.247 1.374 1.482
1.299 1.425 1.561 1.651
1.495 1.642 1.785 1.853
1.741 1.908 2.054 2.101
2.052 2.239 2.380 2.412
2.448 2.657 2.783 2.778
3. FORMULAE
AND DISCUSSION
According to selected results here obtained by using the finite element method, we propose an approximate relation to calculate correction factors F, for two parallel cracks situated in the centre of a sheet of the form: F = A0 + A,(a/w) + M+9’~s (5)
2 $Tm2
in which
A 0= 0.709 + 0.1&b/a) - O.O4(b/a)2 + O.O02(b/a)3 - 0.032(b/t~)~ A ,= -0.044- OM(b/a)+ 0.27(b/t1)~ A2 = 0.118 + 0.85(b /a) - 0.47(b /a)’ + 0.056@ /u)~. Figure 3 compares the values given by relation (5) with all these results obtained by finite element method. The difference is less than 1%. Benthem and Koiter[3] have presented a relation for a central crack in a finite width sheet: K, = &G(u/w)
= K,
1 - 0.5(a/w) + 0.326(a/w)’ Jz&v
*
Figure 3 also shows the correction factors G. It can be observed that, with b/a> 3, the interrelation almost vanishes.
1076
2. D. JIANG, et al. 1.5
0 x
finite element eq.(5) [2] G
results
1.3 b/a = 3
,
I
I
I
0.2
0.1
I
I
0.3
0.4
a/w
0.5
Fig. 3. Correction factors Fz for 2 central cracks.
For two parallel cracks situated on a sheet edge: F
=
Al + A,@/wY+ 4(a/w)4
2 $=m?
in which A,, = 0.79 + O.O7(bja) + 0.04(b/a)2 - 0.01 l(b/a)3 A , = 1.74 + 284(b/a) - l~I4(b/a)~ + 0.206(b/a)3 A, = 6.02 + 2.19(b/a) - 3.26(6/a)‘+ 0.828(b/a)3. Equation (6) is accurate to 3%. Figure 4 compares the results of eq. (6) with the finite element results. For edge cracks, as compared with central cracks, the stress intensity factors increase rapidly when a/w increases, but more slowly when b/a increases. For three parallel cracks situated in the centre of a sheet: (7)
F2 3.
r
o
finite element
results
b/a = 3
=2 2.5
IA.5 =0.25 =O.l
2. 1.5 1.
I
0.1
I
0.2
1
0.3
I
0.4
Fig. 4. Correction factors F2 for 2 edge cracks.
I
0.5
alw
SIF of parallel cracks
1077
where for the middle crack: A0 = 0.391 + 0.33(b/a) + o.o2(b/a)* - O.O33(b/a)3 A , = -0.551 + 2.17(b/a) - 2.87(b/~)~ + 0.945(b/a)3 A2 = 0.851 - 2.80(b/u) + 4.05@/~.2)~- 1.356@/~)~ for the two outer cracks: A0 = 0.67 + O.l6(b/u) + O.O3(b/u)’ - 0.027@/~)~ A, = 0.13 - 0.87(6/u) + 0.43(b/~)~ + O.O10(b/~)~ A2 = -0.16 + 1.84(b/u) - 1.06(b/~)~ + O.l02(b/~)~ eq. (7) is accurate to 1%. Isida[4] has calculated the stress intensity factors for odd numbers of parallel cracks in an infinite sheet to b/u > 1.25. Our results to the case b/u = 2, u/w = 0.1 are in close agreement with those of [4]; in this case the influence of the edge is only of 1%. For three parallel cracks situated on the edge of a sheet: F = Ao + A,WTs
+
A,@/w)~
3
(8) Jm&7
where for the middle crack: A,, = 0.378 + 0.68(b/u) - 0.72(b/u)2 + 0.256(b/u)3 A, = 0.970 + l.l6(b/u)
+ 4.99(b/u)2 - 2.391(b/u)3
A, = 1.347 + 50.48(b/u) - 73.69(b/u)‘+
24.190(b/u)3
and for the two outer cracks: A, = 0.69 + 0.42(b/u) - 0.47(b/u)’ + 0.17(b/u)3 A, = 1.99 + 1.98(6/u) + 0.82(b/u)2 - 0.73(b/u)3 A2 = 6.83 + 6.02(b/u) - 16.20(b/u)2 + 6.18(b/u)3. Equation (8) is accurate to 3%. A comparison between the proposed eqs (7) and (8) and the finite element results, is shown in Figs 5-8.
F3 1.2 finite element
results
=1 0.8
0.6
=0.25
I
a/w 0.1
0.2
0.3
0.4
0.5
Fig. 5. Correction factors I$ for the middle crack of 3 central cracks.
1078
2. D. JIANG, ef al. F3
bla = 2
1.2
=1 1.
=0.5 =0.25
0.6 0.6 0
fiiiy,;lement
results
.
-
0.4
i
I
I
0.1
0.2
I
0.3
I
0.4
I alw 0.5
Fig. 6. Correction factors Fj for the outer crack of 3 central cracks.
F3 finite element
I
results
I
I
I
0.1
0.2
0.3
I
0.4
I
a/w
0.5
Fig. 7. Correction factors 4 for the middle crack of 3 edge cracks.
F3 3. -
bla = ? 0
2.5 -
--
finite element
results
zA.5
eq.(81
=0.25
2. 1.5 -
0.5
atw 0.1
0.2
0.3
0.4
0.5
Fig. 8. Correction factors F, for the outer crack of 3 edge cracks.
SIF of parallel cracks
1079
4. CONCLUSION (1) Using the results obtained by finite element method, we present relations for stress intensity factors for two (b/a < 3) and three (b/a < 2) parallel cracks in a finite sheet. For central cracks, the relations are accurate to 1% (a/w < 0.5), and for edge cracks to 3% (a/w < 0.45). (2) When there is more than one crack in a sheet, since the flexibility of the sheet increases, the stress intensity factors decrease. Only when the interval between two cracks is more than three times the crack length, the problem can be treated as two single cracks far from each other. (3) The formulae presented herein should be useful for calculating fracture loads and determining the stress intensity factors in fatigue tests for these types of crack configurations. REFERENCES [I] Abaqus, Finite Element Program developed by Hibbitt, Karlsson and Sorensen Inc. (USA). [2] J. R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. appl. Med. (June 1968). [3] J. P. Benthem and W. T. Keiter, Mechanics of Fracture (Edited by G. C. Sih), Vol. 1, chapter 3 (1973). [4] M. Isida, Bull. JSME, 13, 635 (1970).