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Stress intensity factors for radial cracks in thick walled cylinders—III. Asymmetrical cracks
Engineering Frocrure Mechanics Vol. 49, No. 6, pp. 839-847, 1994 Copyright 0 1994 Elsevier Science Ltd 00117944@4)00134-0 Printed in Great Britain. All rights reserved . _ 0013-7944/94 -S7.00 + 0.00
Pergamon
STRESS INTENSITY THICK WALLED
FACTORS FOR RADIAL CRACKS CYLINDERS-III. ASYMMETRICAL CRACKS
IN
H. M. SHU, J. PETIT and G. BEZINE Laboratoire de Mecanique et Physique des Matdriaux URA CNRS 863-ENSMA, BP 109, 86960 Futuroscope, CEDEX, France Abstract-For many years, a great amount of work has been done on the problems of symmetrical multiple radial cracks in the thick walled cylinder. But, up to now, nothing has been done on asymmetrical multiple radial cracks in thick walled cylinders. This paper deals with two and three asymmetrical radial cracks, and the stress intensity factors (SIF) due to two kinds of internal pressures and full autofrettage process has been calculated using finite element method and the influence of crack configurations on SIF is discussed.
1. INTRODUCTION I of this study [I] deals with the multiple radial cracks in thick walled cylinders and the stress intensity factors (SIF) have been calculated using finite element method for a wide range of crack configuration parameters. However, in real situations, cracks are not always symmetrical. This third part studies two and three asymmetrical inner radial cracks in the thick walled cylinders undergoing two kinds of internal pressures and full autofrettage and the SIF for a wide range of parameter configuration has been calculated using finite element method. The influence of crack configurations on the SIF is discussed. PART
2. FINITE ELEMENT
ANALYSIS
Figure 1 shows the crack configurations, where r, and r2 are the inner and outer radius, respectively, with rl = 52.5 mm and r2 = 135 mm, which is the real dimension of a barrel; a is the length of cracks, w (= r2 - rl ) the width of wall thick, with 0.05 < a/w < 0.8. The cylinders undergo three kinds of loading, namely (1) the internal pressure pi on inner surface (no crack face pressure); (2) external tension p0 on outer surface and (3) full autofrettage which is simulated by a thermal loading [2-51. The finite element code ABAQUS was used with eight nodes quadrilateral and six nodes trilateral isoparametric elements. Only the upper half of the cylinder is considered. For the situation of three cracks in a cylinder undergoing internal pressure and external tension, when cracks are longer and the angle 0 between cracks 1 and 2 is smaller, the faces of crack 1 could present some overlapping. Therefore, to prevent this situation, GAP elements between the opposite nodes on both faces of crack 1 are added. We first calculate the J-integral around the cracks tip in a cylinder due to internal pressure, external tension and full autofrettage, respectively, and the SIF can be obtained by relation K: = EJ/(I - v2) (for plane strain) The SIF K, is expressed as K, = K~F(u/~,
8)
Fi (a, w, 0 ) for internal pressure where & = {zz
The correction
and
F = {;;,;,;;~;;~~~~;;a
factor F is a function of w, a and 8. 839
(1)
840
H. M. SHU et al.
Fig. 1.
3. NUMERICAL RESULTS AND DISCUSSION Table 1 presents the results calculated of correction factors F for a single crack under three kinds of loading; Tables 2-4 present the correction factors F for two asymmetrical cracks due to three kinds of loading, respectively; Tables 5-7 present the correction factors F for three cracks due to three kinds of loading, respectively.
Table 1. Correction factors F for a single crack due to three kinds of load 0.05 s F0 F.
1.3516 2.3766 -0.9381
0.10
0.20
1.2424 2.2786 -0.8219
0.30
1.1095 2.1679 -0.6464
1.0344 2.1320 -0.5163
a/w 0.40 0.9934 2.1413 -0.4111
0.50 0.9776 2.1995 -0.3210
0.60 0.9851 2.2905 -0.2399
0.70
0.80
1.0213 2.4586 -0.1620
1.1101 2.7619 - 0.0798
Table 2. Correction factors F, for two asymmetrical cracks due to internal pressure 8”
2.5 5 7.5 10 15 20 25 30 45 60 75 90
0.05
1.0968 1.1225 1.1941 1.2493 1.3044 1.3224 1.3275 1.3282 1.3529 1.3546 1.3595 1.3632
0.10
0.20
0.30
0.9337 0.9782 1.0158 1.0540 1.1097 1.1697 1.1788 1.1889 1.2303 1.2603 1.2830 1.2857
0.8162 0.8414 0.8629 0.8809 0.9040 0.9399 0.9626 0.983 1 1.0564 1.1365 1.2011 1.2219
0.7552 0.7722 0.7851 0.7951 0.8044 0.8237 0.8393 0.8616 0.9560 1.0759 1.1840 1.2263
a/w
0.40
0.50
0.60
0.70
0.80
0.7221 0.7343 0.7414 0.7459 0.7484 0.7626 0.7761 0.7985 0.9067 1.0569 1.2069 1.2726
0.7093 0.7176 0.7204 0.7214 0.7219 0.7385 0.7534 0.7787 0.8971 1.0711 1.2610 1.3515
0.7147 0.7200 0.7200 0.7200 0.7253 0.7507 0.7684 0.7975 0.9238 1.1173 1.3442 1.4606
0.7432 0.7469 0.7473 0.7513 0.7714 0.8102 0.8395 0.8597 0.9925 1.2026 1A626 1.6035
0.8163 0.8227 0.8344 0.8532 0.8958 0.9469 0.9673 0.9909 1.1313 1.3573 1.6442 1.8037
Table 3. Correction factors F, for two asymmetrical cracks due to external tension 8” 2.5 5 7.5 10 15 20 25 30 45 60 75 90
0.05
0.10
0.20
0.30
1.8264 1.9744 2.0538 2.2030 2.3181 2.3356 2.3519 2.4373 2.3902 2.3915 2.3972 2.4165
1.6397 1.7653 1.8515 1.9327 2.0570 2.1340 2.1650 2.1885 2.2469 2.3069 2.3463 2.3608
1.5322 1.6184 1.6766 1.7220 1.7958 1.8516 1.9005 1.9370 2.0709 2.2225 2.3472 2.3905
1.5009 1.5737 1.6139 1.6459 1.6957 1.7280 1.7623 1.8009 1.9855 2.220 1 2.4352 2.5213
a/w 0.40 1.5133 1.5747 1.6031 1.6253 1.6606 1.6854 1.7159 1.7547 1.9170 2.2769 2.5858 2.7228
0.50
0.60
0.70
0.80
1.5570 1.6092 1.6313 1.6460 1.6750 1.7054 1.7386 1.7838 2.0282 2.3906 2.7933 2.9907
1.6315 1.6813 1.6991 1.7172 1.7452 1.7960 1.8359 1.8910 2.1518 2.5651 3.0600 3.3101
1.7856 1.8143 1.8283 1.8514 1.9158 1.9957 2.0353 2.0915 2.3729 2.8223 3.4150 3.7296
2.0456 2.0684 2.1079 2.1685 2.2798 2.3602 2.4140 2.4719 2.8867 3.253 1 3.9293 4.3075
841
SIFs for radial cracks in thick walled cylinders-III Table 4. Correction factors F, for two asymmetrical cracks due to full autofrettage 8”
0.10
0.05
2.5 5 1.5 10 15 20 25 30 45 60 15 90
0.20
-0.8740 -0.9146 - 0.9509 -0.9882 - 1.0516 - 1.0927 -1.1094 -1.1209 -1.1528 -1.1861 - 1.2088 - 1.2156
- 1.0439 -1.1141 -1.1863 - 1.2399 - 1.2796 - 1.3135 - 1.3216 - 1.3230 - 1.3406 - 1.3436 - 1.3500 - 1.3575
a/w
0.40
0.30
-0.6697 -0.6868 -0.7024 -0.7159 -0.7396 -0.7640 -0.7882 -0.8069 - I .8699 -0.9443 - 1.0038 - I .0256
-0.5219 -0.5338 -0.5383 -0.5420 -0.5467 -0.5550 -0.5719 -0.5903 -0.6681 -0.7696 -0.8618 -0.8994
-0.4148 -0.4134 -0.4107 -0.4079 -0.4026 -0.4038 -0.4176 -0.4348 -0.5179 -0.6346 -0.7514 -0.8040
0.50 -0.3184 -0.3110 -0.3032 -0.2962 -0.2863 -0.2861 -0.3003 -0.3177 -0.4016 -0.5245 -0.6587 -0.7231
0.60
-0.2311 -0.2186 -0.2072 -0.1984 -0.1887 -0.1905 -0.2071 -0.2255 -0.3077 -0.4302 -0.5735 -0.6475
0.70
0.80
-0.1465 -0.1298 -0.1173 -0.1096 -0.1048 -0.1111 -0.1311 -0.1505 -0.2296 -0.3459 -0.4881 -0.5657
-0.0572 -0.0402 - 0.0336 -0.0346 -0.0452 - 0.0595 -0.0804 -0.0992 -0.1731 -0.2695 -0.3953 -0.4661
Table 5. Correction factors Fi for three asymmetrical cracks due to internal pressure U/W
8” 5
F” F, F2
10
F, F2
15
F, F2
20
F, F2
30
F, F2
40
F, F2
50
F, F2
60
F,
15
F,
F*
F2
90
F, F*
105
F, F*
120
F, F2
135
F, F*
150
F, F*
165
F, F*
175
F,
0.05
0.10
0.20
0.36.’
0.40
0.50
0.60
0.70
0.80
0.7388 1.0103 0.9051 1.0973 1.0494 1.1738 1.1477 1.2294 1.2475 1.2851 1.2892 1.3076 1.3043 1.3189 1.3123 1.3338 1.3201 1.3458 1.3297 1.3531
0.5844 0.8901 0.6950 0.9442 0.7841 0.9849 0.8667 1.0180 0.9983 1.0821 1.0717 1.1198 1.1093 1.1661 1.1321 1.1970 1.1602 I .2344 1.1910 1.2578 1.2246 1.2649 1.2538
0.4707 0.7749 0.5475 0.8004 0.6086 0.8160 0.6570 0.8275 0.1275 0.8562 0.1165 0.9007 0.8125 0.9575 0.8445 1.0189 0.9016 1.1036 0.9773 1.1586 1.0642 1.1710 1.1438 1.1438 1.2115 1.1030 1.2564 1.0587
0.4269 0.7128 0.4934 0.7223 0.5419 0.7254 0.5753 0.7289 0.6091 0.7511 0.6200 0.7966 0.6288 0.8639 0.6481 0.9446 0.7142 1.0630 0.8299 1.1376 0.9714 1.1457 1.0996 1.0996
0.4133 0.6760 0.4760 0.6742 0.5150 0.6707 0.5348 0.6726 0.5385 0.7020 0.5198 0.7619 0.5028 0.8475 0.5057 0.9489 0.5765 1.0935 0.7335 1.1729 0.9289 1.1652 1.0990 1.0990 I .2265 I .0270 1.3043 0.9663 1.3331 0.9474 1.3063 0.9358
0.4198 0.6565 0.4809 0.6468 0.5092 0.6420 0.5140 0.6484 0.4861 0.6963 0.4388 0.7823 0.397 1 0.8967 0.3855 I .0242 0.4717 1.1887 0.6819 1.2556 0.9312 1.2208 1.1351 1.1351 1.2816
0.4455 0.6523 0.5040 0.6388 0.5169 0.6402 0.5038 0.6599 0.4382 0.1420 0.3854 0.8540 0.3885 0.9686 0.3981 1.1081 0.4324
0.4970 0.6675 0.5463 0.6595 0.5350 0.6796 0.4968 0.7262 0.4523 0.8281 0.4830 0.9294 0.5148 1.0727 0.5434 1.2040 0.5840
0.5986 0.7216 0.6170 0.1449 0.5624 0.8140 0.5498 0.8323 0.6171 0.9765 0.6846 0.1050 0.7429 I .2349 0.1992 1.3722 0.8685
1.3428 I .3556 1.3515 1.3515 1.3618 1.3479 1.3683 1.3405 1.3732 1.3117 1.3693 1.1379
1.2538 1.2826
1.2383 1.3023 1.2174 1.3105 1.1512 1.2980 1.0074
1.2689 0.9953 1.2438 0.9226
1.2009 1.0400 1.2649 0.9838 1.2837 0.9487
1.2554 0.9113
1.3029
1.4663
1.6662
0.6794
0.7452 I .5583
0.9359 1.7980 1.2987 I .6523
1.3834
1.0562
0.9811 1.3128 1.2095 1.2095 1.3685 1.1283
I.3695 0.9938 14090 0.9768 1.3889 0.9853
1.0661 1.5112 1.0385 1.5015 1.0579
1.4629
1.0904 1.4461 1.3311 1.3311 1.4948 1.2514 1.5917 1.1896 I.6442 1.1475 1.6460 1.1561
1.5337 I.5337 1.6914 1.4603 1.7847 1.4078
1.8352 I.3524
1.8436 1.3055
(1) For two cracks Figures 2 and 3 show that when the angle between two cracks is smaller, the SIF (negative SIF for autofrettage) for two cracks is smaller than that of a single crack. As the angle 8 increases, the SIF increases monotonically. When angle 8 reaches 45”, the SIF is near to that of a single crack, and when 8 = 90”, the SIF reaches the largest magnitude. That means that when angle between two cracks is smaller, the interaction of two cracks results in a relaxation of stress field at the crack tips; and when angle between two cracks is larger the interaction of two cracks results in enhancement of stress field at the crack tips. (2) For three cracks Figure 4 shows that for three cracks, when 8 is smaller, the SIF of crack 1 due to external tension is very little and is less than that of a single crack. With the increase of the angle between
H. M. SHU et a!.
842 Table 6. Correction
factors
F, for three asymmetrical
cracks
due to external
tension
a/n B’ 5 10 15 20 30
F,
0.05
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
F,
1.2754 1.7588 1.5989 1.9398 1.8595 2.0747 2.0299 2.1624 2.2068 2.2646 2.2727 2.3237 2.2991 2.3290 2.3129 2.3494 2.3265 2.3658 2.3430 2.3838 2.3661 2.3877 2.3840 2.3856 2.3993 2.3749 2.4111 2.3619 2.4176 2.3068 2.4124
1.0054 1.5697 I .2507 1.6977 1.4357 1.7928 1.5906 1.8637 1.8357 1.9873 1.9703 2.0738 2.0385 2.1501 2.0797 2.1965 2.1302 2.2624 2.1851 2.3053 2.2466 2.3175 2.3028 2.3037 2.3506 2.2706 2.3859 2.2339 2.4007 2.1106 2.3782 I.8144
0.8799 1.4596 1.0717 I.5407 1.2095
0.8558 1.4253 1.0331
0.8775 1.4239 1.0526 1.4545 1.1536 1.4655 1.1994 1.4699 1.2219 1.5410 1.1909 1.6654 1.1594 1.8390 1.1660 2.0438 1.3079 2.3384 1.6228 2.5017 2.0149 2.4906 2.3620 2.3637 2.6123 2.2156 2.7683 2.0905 2.8275 2.0428 2.7740 1.9575
0.9345 1.4478 1.1119 1.4644 1.1934 1.4699 1.2108 1.4840 1.1694 1.5937 1.0798 1.7751 0.9978 2.0113 0.9756 2.2785 1.1538 2.5865 1.5878 2.7696 2.1030 2.7038 2.5313 2.5327 2.8272 2.3617 3.0089 2.2331 3.0923 2.1876 3.0511 2.1412
1.0302 1.4996
1.1916 1.5994
I .4869
1.2077
1.3569 1.6057
6 F,
F2 F, F2
;I F; F2
40
F, F2
50
F,
60
F,
F2
F2
75
F, F?
90
F, F2
105
F, F2
120
F, F2
135
F, F2
150
F, F2
165
F,
175
F,
F2
F2
1.9904
Table 7. Correction
1.4765
1.5935 1.3041 1.6173 14456 1A829 I .5408 1.7713 1.6095 1.8799 1.6701 I .9957 1.7772 2.1567 1.9191 2.2588 2.0824 2.2839 2.2361 2.2369 2.3590 2.1559 2.4432 2.0724 2.4668 1.9455 2.4203 1.7578
factors
1.1504 1.5038 1.2202 1.5112 1.2985
1.5669 1.3241 1.6601 1.3425 1.7930 1.3805 1.9513 1.5091 2.1852
1.7337 2.3303 2.0090 2.3498 2.2628 2.2642 2.4552 2.1422 2.5795 2.0336 2.6170 1.9332 2.5615 1.8258
F, for three asymmetrical
cracks
1.5154 I .2558 I .5242 1.2356 1.5675 1.1135
1.7529 1.0295 1.9803 1.0435 2.2203 1.0651 2.5229 1.1398 2.9512 1.6408 3.1258 2.2786 2.9904 2.7701 2.7867 3.0980 2.6007 3.2979 2.4714 3.4018 2.3973 3.3811 2.3763
1.3513 1.6734 1.2754 I .7759 1.2016 2.0090 1.2800 2.2154 I .3604 2.4702 1.4226 2.8069 1.5120 3.4043 1.8501 3.6141 2.5994 3.3904 3.1321 3.1467 3.4780 2.9653 3.6887 2.8685 3.8037 2.7232 3.8072 2.6860
1.7953 1.5854 1.8913 1.4804 2.0500 1.4496 2.1850 1.628 1 2.4117 1.7924 2.6207 1.926 1 2.8975 2.0488 3.2737 2.2023 3.9640 2.3522 4.2793 3.1631 3.9713 3.6991 3.7217 4.0411 3.5378 4.2496 3.4263 4.3624 3.2856 4.3800 3.1287
due to fuii autofrettage
alw
F,
0.05
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
5
FL
10 15
F, F, F,
20
F,
30
F, F2 F,
-0.7262 - 1.0020 - 0.8938 - 1.0890 -1.0406 -1.1663 -1.1403 - 1.2185 - 1.2409 - 1.2744 - 1.2789 - 1.2971 - 1.2941 - 1.3130 - 1.3022 - 1.3236 -1.3100 - I .3356 -1.3159 - 1.3382 -1.3331 - 1.3456 - 1.3444 - 1.3444 - 1.3523 - 1.3378 - 1.3589 - 1.3249 - 1.3629 - 1.3022 - 1.3596 -1.1269
-0.5344 -0.8332 -0.6371 -0.8820 -0.7230 -0.9192 -0.8042 -0.9529 -0.9331 - I.0157 - I .OO52 -1.0616 - 1.0424 - 1.0982 - 1.0646 -1.1282 - 1.0923 - 1.i651 -1.1225 -1.1881 -1.1556 - 1.1950 - 1.1861 - 1.1861 -1.2125 -1.1686 - 1.2317 - 1.1488 - 1.2399 - 1.0843 - 1.2272 -0.9446
-0.3630 - 0.6360 - 0.4209 -0.6513 -0.4697 -0.6608 -0.5101 -0.6681 -0.5722 -0.6915 -0.6173 -0.7315 -0.6505 -0.7836 -0.6798 -0.8396 -0.7321 -0.8931 -0.8015 -0.9675 -0.8812 -0.9789 -0.9418 -0.9418 - 1.0161 -0.9169 - 1.0573 -0.8765 - 1.0685 -0.8188 - 1.0451 -0.7616
-0.2676 -0.4981 -0.3063 -0.4956 -0.3363 -0.4909 -0.3573 -0.4886 -0.3781 -0.5009 -0.3852 -0.5367 -0.3922 -0.5929 -0.4086 -0.6610 - 0.4648 -0.7612 -0.5630 -0.8245 -0.6833 -0.8311 -0.7942 -0.7942 -0.8779 -0.7404 - 0.9323 -0.6939 -0.9477 -0.6667 -0.9227 -0.6519
-0.2018 -0.3870 -0.2278 -0.3729 -0.243 1 -0.3611 -0.2484 -0.3560 -0.2389 - 0.3694 -0.2190 -0.4107 -0.2036 -0.4748 -0.2050 -0.5528 -0.2600 -0.6646 -0.3828 - 0.7263 -0.5355 -0.7199 -0.6708 -0.6708 - 0.7679 -0.6119 -0.8287 -0.5649 -0.8497 -0.5563 -0.8285 -0.5702
-0.1503 -0.2911 -0.1649 -0.2697 -0.1660 -0.2561 -0.1571 -0.2531 -0.1216 -0.2749 -0.0793 -0.3280 - 0.0450 - 0.4045 - 0.0346 -0.4928 - 0.0925 - 0.5938 -0.2453 - 0.6547 -0.4231 -0.6295 -0.5708 -0.5708 -0.6729 -0.5122 -0.7355 -0.4683 -0.7618 - 0.4660 -0.7474 -0.5002
-0.1052 -0.2041 -0.1065 -0.1797 -0.0926 -0.1694 - 0.0696 -0.1724 -0.0091 - 0.2098 -0.0566 -0.2802 -0.1096 -0.3727 -0.1260 - 0.4705 - 0.0430 -0.5768 -0.1412 -0.5948 -0.3355 -0.5488 -0.4853 - 0.4853 -0.5848 -0.4300 -0.6456 -0.3907 - 0.6748 -0.3870 -0.6687 -0.4327
-0.0593 -0.1211 -0.0431 -0.0999 -0.0121 -0.0988 - 0.0262 -0.1129 -0.1197 -0.1732 -0.2160 -0.2686 -0.2871 -0.3810 -0.2934 -0.4827 -0.1562 -0.5565 -0.0663 -0.5328 -0.2643 -0.4685 - 0.4050 -0.4050 -0.4962 -0.3563 -0.5516 -0.3227 -0.5806 -0.3146 -0.5819 -0.3595
-0.0008 - 0.0398 -0.0404 -0.0375 -0.0945 -0.0534 -0.1577 -0.0815 -0.3104 -0.1786 -0.4561 -0.3140 -0.5223 -0.4449 -0.4653 -0.5229 - 0.2280 -0.5213 -0.0197 -0.4515 -0.2014 -0.3785 -0.3216 -0.3216 -0.3980 -0.2819 -04447 -0.2559 -0.4701 -0.2448 -0.4758 -0.2736
0”
40
F;
F;
6
F2
50
F, F2
60
F, F2
15 90 105
F, F, F, FZ F, F2
120 135 150 165
F, F? F, F* F, F, F, F2
175
F, FZ
SIFs for radial cracks in thick walled cylinders-111
5.
FO l-
.... ..... a single crack 4.
3.
2.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 2.
cracks, the SIF of crack 1 increases. When 6 is larger, the SIF of crack 1 is larger than that of a single crack, and when the angle 8 tends to 180”, the SIF of crack 1 tends to the largest magnitude which is nearly equal to that of two symmetrical cracks. In our calculation, by computing the displacement of crack faces, it has been found that, generally, when a/w > 0.5 and 20” < 0 < 75”, the two faces of crack 1 come into contact partially due to internal pressure and external tension, and thus the SIF of crack 1 becomes very small.
-1.4
-1.2 4. -0.8 -0.6 900 750
-0.4
600
-0.2
..-..---..
a single crack
0.1
0.2
450 dW
0.3
0.4 Fig. 3
0.5
0.6
0.7
0.8
H. M. SHU ef al.
844
5. . ......... a single crack
/
1750 16S” :3 1w
4.
IO.50
3,
2.
1.
1
0.1
0.2
,
0.3
0.4
0.5
0;6
t
0.7
0.8
Fig. 4.
Figures 5 and 7 show that when B c 90”, the SIF of crack 2 (negative SIF for autofr~ttage) increase as the angle increases; when 8 > 90”, the SIF decreases as the angle 8 increases; and when 0 = 90”, the SIF reaches the largest magnitude which is nearly equal to that of two symmetrical cracks. When 0 comes close to 180”, changing the angle has little influence on the SIF for crack 2. Figure 6 presents the SIF for crack 1 due to full autofrettage. It can be observed that at some values of crack configuration parameters, the SIF of crack 1 can become positive; that means that the crack 1 is partially open near to the crack tip. From Figs 2-7, we can also conclude that when angle 0 is small (< 30”) and cracks are short (a/w < 0.2), the increase of 8 leads to an obvious increase of SIF; but when cracks are longer, the increase of 0 leads to a little changing of SIF. The SIF for external tension (internal pressure on inner surface and on crack faces) is much larger than that for internal pressure (no crack face pressure). When a cylinder undergoes internal or external pressure, the SIF increases as the length of the cracks increases; but when a cylinder undergoes an autofrettage process, when the cracks are shorter, the negative SIF increases as the length of cracks increases, when the cracks are little longer, the negative SIF decreases obviously as the length of cracks increases.
4. CONCLuSIONS (1) The SIF for two and three asymmetrical radial inner cracks in thick walled cylinders undergoing two kinds of internal pressure and full autofrettage has been calculated respectively. (2) For two asymmetrical cracks, the SIF (negative SIF for an autofrettaged cylinder) increases as the angle B between two cracks increases; and when 0 = 90”, i.e. when the cracks become the symmetrical, the SIF reaches the largest magnitude. (3) For crack 1 of three asymmetrical cracks, the SIF (negative SIF for a autofrettaged cylinder) increases as the angle 8 between cracks 1 and 2 increases; when 0 tends to 180” the SIF of crack 1 tends to the largest magnitude. For crack 2, when 0 > 90”, the SIF decreases as the angle 0 increases; when 8 c 90”, the SIF increases as the angle increases; and when 8 is about 90”, the SIF reaches the largest magnitude. The largest SIF of cracks 1 and 2 are nearly equal to that of two symmetrical cracks. (4) For three cracks, the two faces of crack 1 will contact partially at some values of crack parameters due to internal pressure and external tension; and they will open partially near to the crack tip due to autofrettage.
845
SIFs for radial cracks in thick walled cylinders-III
5.
........*. a single crack 4.
3.
2.
1.
L
I
0.1
5.
1
I
0.2
0.3
0.4
0.5
0;6
0.7
a/w
0.8
F*
4.
900 105Q Izoo
3.
169 179
........*. a singie crack
1350
lsoo
2,
1.
I
t
I
0.1
0.2
0.3
0.4
1
0.5
due to external tension Fig. 5.
0;6
0.7
0.8
a/w
846
H.
M. SHU el al.
-1.2
-0.8
-0.4
.......... 8 single
c&
-1.2
-0.8 17.50 169 UP 1350 120° 209
-0.4
w* 756
Fig. 6.
SIFs for radial cracks in thick walled cylinders-41
-1.2 -i.
-0.8 4.6 -0.4 -0.2
-0.8
,
0.1
,
I
I
,
I
0.2 0.3 0.4 0.S 0.6 0.7
t
s/w
0.8
Fig. 7.
REVERENCES 111 II. M. Shu, J. Petit and G. Bezine, Stress intensity factors for radial cracks in thick walled cylinders-4 Symmetrical cracks. Engng Fracture hfech. 49, 61 l-623 (1994). [2] M. Per1 and R. Arone, Stress intensity factors for iarge arrays of radial cracks in thick-walled steel cylinders. Engng Fracture Meek ZS, 341-348 (1986). [3] M. Per1 and R. Arone, Stress intensity factors for a radial multicracked partially autofrettaged pressurized thick-walled cylinder. J. Press. Yess. Tecfmol. 110, 147-154 (1988). [4] A, P. Parker and J. R. Farrow, On the equivalence of axisymmetric bending, thermal and autof~tta~ residual stress fields. J. Strain Anal. 115, 51-62 (1980).
[.5] Pu. S. I, and Chen, P. C. ‘I. Stress intensity factors for radial cracks in pie-stressed, stein-hardening materials. J. Press. I/ess. Technol. IOS, 117-123 (1983). (Receiveci I? October 1993)
thick-walfed cylinder of
Pergamon
STRESS INTENSITY THICK WALLED
FACTORS FOR RADIAL CRACKS CYLINDERS-III. ASYMMETRICAL CRACKS
IN
H. M. SHU, J. PETIT and G. BEZINE Laboratoire de Mecanique et Physique des Matdriaux URA CNRS 863-ENSMA, BP 109, 86960 Futuroscope, CEDEX, France Abstract-For many years, a great amount of work has been done on the problems of symmetrical multiple radial cracks in the thick walled cylinder. But, up to now, nothing has been done on asymmetrical multiple radial cracks in thick walled cylinders. This paper deals with two and three asymmetrical radial cracks, and the stress intensity factors (SIF) due to two kinds of internal pressures and full autofrettage process has been calculated using finite element method and the influence of crack configurations on SIF is discussed.
1. INTRODUCTION I of this study [I] deals with the multiple radial cracks in thick walled cylinders and the stress intensity factors (SIF) have been calculated using finite element method for a wide range of crack configuration parameters. However, in real situations, cracks are not always symmetrical. This third part studies two and three asymmetrical inner radial cracks in the thick walled cylinders undergoing two kinds of internal pressures and full autofrettage and the SIF for a wide range of parameter configuration has been calculated using finite element method. The influence of crack configurations on the SIF is discussed. PART
2. FINITE ELEMENT
ANALYSIS
Figure 1 shows the crack configurations, where r, and r2 are the inner and outer radius, respectively, with rl = 52.5 mm and r2 = 135 mm, which is the real dimension of a barrel; a is the length of cracks, w (= r2 - rl ) the width of wall thick, with 0.05 < a/w < 0.8. The cylinders undergo three kinds of loading, namely (1) the internal pressure pi on inner surface (no crack face pressure); (2) external tension p0 on outer surface and (3) full autofrettage which is simulated by a thermal loading [2-51. The finite element code ABAQUS was used with eight nodes quadrilateral and six nodes trilateral isoparametric elements. Only the upper half of the cylinder is considered. For the situation of three cracks in a cylinder undergoing internal pressure and external tension, when cracks are longer and the angle 0 between cracks 1 and 2 is smaller, the faces of crack 1 could present some overlapping. Therefore, to prevent this situation, GAP elements between the opposite nodes on both faces of crack 1 are added. We first calculate the J-integral around the cracks tip in a cylinder due to internal pressure, external tension and full autofrettage, respectively, and the SIF can be obtained by relation K: = EJ/(I - v2) (for plane strain) The SIF K, is expressed as K, = K~F(u/~,
8)
Fi (a, w, 0 ) for internal pressure where & = {zz
The correction
and
F = {;;,;,;;~;;~~~~;;a
factor F is a function of w, a and 8. 839
(1)
840
H. M. SHU et al.
Fig. 1.
3. NUMERICAL RESULTS AND DISCUSSION Table 1 presents the results calculated of correction factors F for a single crack under three kinds of loading; Tables 2-4 present the correction factors F for two asymmetrical cracks due to three kinds of loading, respectively; Tables 5-7 present the correction factors F for three cracks due to three kinds of loading, respectively.
Table 1. Correction factors F for a single crack due to three kinds of load 0.05 s F0 F.
1.3516 2.3766 -0.9381
0.10
0.20
1.2424 2.2786 -0.8219
0.30
1.1095 2.1679 -0.6464
1.0344 2.1320 -0.5163
a/w 0.40 0.9934 2.1413 -0.4111
0.50 0.9776 2.1995 -0.3210
0.60 0.9851 2.2905 -0.2399
0.70
0.80
1.0213 2.4586 -0.1620
1.1101 2.7619 - 0.0798
Table 2. Correction factors F, for two asymmetrical cracks due to internal pressure 8”
2.5 5 7.5 10 15 20 25 30 45 60 75 90
0.05
1.0968 1.1225 1.1941 1.2493 1.3044 1.3224 1.3275 1.3282 1.3529 1.3546 1.3595 1.3632
0.10
0.20
0.30
0.9337 0.9782 1.0158 1.0540 1.1097 1.1697 1.1788 1.1889 1.2303 1.2603 1.2830 1.2857
0.8162 0.8414 0.8629 0.8809 0.9040 0.9399 0.9626 0.983 1 1.0564 1.1365 1.2011 1.2219
0.7552 0.7722 0.7851 0.7951 0.8044 0.8237 0.8393 0.8616 0.9560 1.0759 1.1840 1.2263
a/w
0.40
0.50
0.60
0.70
0.80
0.7221 0.7343 0.7414 0.7459 0.7484 0.7626 0.7761 0.7985 0.9067 1.0569 1.2069 1.2726
0.7093 0.7176 0.7204 0.7214 0.7219 0.7385 0.7534 0.7787 0.8971 1.0711 1.2610 1.3515
0.7147 0.7200 0.7200 0.7200 0.7253 0.7507 0.7684 0.7975 0.9238 1.1173 1.3442 1.4606
0.7432 0.7469 0.7473 0.7513 0.7714 0.8102 0.8395 0.8597 0.9925 1.2026 1A626 1.6035
0.8163 0.8227 0.8344 0.8532 0.8958 0.9469 0.9673 0.9909 1.1313 1.3573 1.6442 1.8037
Table 3. Correction factors F, for two asymmetrical cracks due to external tension 8” 2.5 5 7.5 10 15 20 25 30 45 60 75 90
0.05
0.10
0.20
0.30
1.8264 1.9744 2.0538 2.2030 2.3181 2.3356 2.3519 2.4373 2.3902 2.3915 2.3972 2.4165
1.6397 1.7653 1.8515 1.9327 2.0570 2.1340 2.1650 2.1885 2.2469 2.3069 2.3463 2.3608
1.5322 1.6184 1.6766 1.7220 1.7958 1.8516 1.9005 1.9370 2.0709 2.2225 2.3472 2.3905
1.5009 1.5737 1.6139 1.6459 1.6957 1.7280 1.7623 1.8009 1.9855 2.220 1 2.4352 2.5213
a/w 0.40 1.5133 1.5747 1.6031 1.6253 1.6606 1.6854 1.7159 1.7547 1.9170 2.2769 2.5858 2.7228
0.50
0.60
0.70
0.80
1.5570 1.6092 1.6313 1.6460 1.6750 1.7054 1.7386 1.7838 2.0282 2.3906 2.7933 2.9907
1.6315 1.6813 1.6991 1.7172 1.7452 1.7960 1.8359 1.8910 2.1518 2.5651 3.0600 3.3101
1.7856 1.8143 1.8283 1.8514 1.9158 1.9957 2.0353 2.0915 2.3729 2.8223 3.4150 3.7296
2.0456 2.0684 2.1079 2.1685 2.2798 2.3602 2.4140 2.4719 2.8867 3.253 1 3.9293 4.3075
841
SIFs for radial cracks in thick walled cylinders-III Table 4. Correction factors F, for two asymmetrical cracks due to full autofrettage 8”
0.10
0.05
2.5 5 1.5 10 15 20 25 30 45 60 15 90
0.20
-0.8740 -0.9146 - 0.9509 -0.9882 - 1.0516 - 1.0927 -1.1094 -1.1209 -1.1528 -1.1861 - 1.2088 - 1.2156
- 1.0439 -1.1141 -1.1863 - 1.2399 - 1.2796 - 1.3135 - 1.3216 - 1.3230 - 1.3406 - 1.3436 - 1.3500 - 1.3575
a/w
0.40
0.30
-0.6697 -0.6868 -0.7024 -0.7159 -0.7396 -0.7640 -0.7882 -0.8069 - I .8699 -0.9443 - 1.0038 - I .0256
-0.5219 -0.5338 -0.5383 -0.5420 -0.5467 -0.5550 -0.5719 -0.5903 -0.6681 -0.7696 -0.8618 -0.8994
-0.4148 -0.4134 -0.4107 -0.4079 -0.4026 -0.4038 -0.4176 -0.4348 -0.5179 -0.6346 -0.7514 -0.8040
0.50 -0.3184 -0.3110 -0.3032 -0.2962 -0.2863 -0.2861 -0.3003 -0.3177 -0.4016 -0.5245 -0.6587 -0.7231
0.60
-0.2311 -0.2186 -0.2072 -0.1984 -0.1887 -0.1905 -0.2071 -0.2255 -0.3077 -0.4302 -0.5735 -0.6475
0.70
0.80
-0.1465 -0.1298 -0.1173 -0.1096 -0.1048 -0.1111 -0.1311 -0.1505 -0.2296 -0.3459 -0.4881 -0.5657
-0.0572 -0.0402 - 0.0336 -0.0346 -0.0452 - 0.0595 -0.0804 -0.0992 -0.1731 -0.2695 -0.3953 -0.4661
Table 5. Correction factors Fi for three asymmetrical cracks due to internal pressure U/W
8” 5
F” F, F2
10
F, F2
15
F, F2
20
F, F2
30
F, F2
40
F, F2
50
F, F2
60
F,
15
F,
F*
F2
90
F, F*
105
F, F*
120
F, F2
135
F, F*
150
F, F*
165
F, F*
175
F,
0.05
0.10
0.20
0.36.’
0.40
0.50
0.60
0.70
0.80
0.7388 1.0103 0.9051 1.0973 1.0494 1.1738 1.1477 1.2294 1.2475 1.2851 1.2892 1.3076 1.3043 1.3189 1.3123 1.3338 1.3201 1.3458 1.3297 1.3531
0.5844 0.8901 0.6950 0.9442 0.7841 0.9849 0.8667 1.0180 0.9983 1.0821 1.0717 1.1198 1.1093 1.1661 1.1321 1.1970 1.1602 I .2344 1.1910 1.2578 1.2246 1.2649 1.2538
0.4707 0.7749 0.5475 0.8004 0.6086 0.8160 0.6570 0.8275 0.1275 0.8562 0.1165 0.9007 0.8125 0.9575 0.8445 1.0189 0.9016 1.1036 0.9773 1.1586 1.0642 1.1710 1.1438 1.1438 1.2115 1.1030 1.2564 1.0587
0.4269 0.7128 0.4934 0.7223 0.5419 0.7254 0.5753 0.7289 0.6091 0.7511 0.6200 0.7966 0.6288 0.8639 0.6481 0.9446 0.7142 1.0630 0.8299 1.1376 0.9714 1.1457 1.0996 1.0996
0.4133 0.6760 0.4760 0.6742 0.5150 0.6707 0.5348 0.6726 0.5385 0.7020 0.5198 0.7619 0.5028 0.8475 0.5057 0.9489 0.5765 1.0935 0.7335 1.1729 0.9289 1.1652 1.0990 1.0990 I .2265 I .0270 1.3043 0.9663 1.3331 0.9474 1.3063 0.9358
0.4198 0.6565 0.4809 0.6468 0.5092 0.6420 0.5140 0.6484 0.4861 0.6963 0.4388 0.7823 0.397 1 0.8967 0.3855 I .0242 0.4717 1.1887 0.6819 1.2556 0.9312 1.2208 1.1351 1.1351 1.2816
0.4455 0.6523 0.5040 0.6388 0.5169 0.6402 0.5038 0.6599 0.4382 0.1420 0.3854 0.8540 0.3885 0.9686 0.3981 1.1081 0.4324
0.4970 0.6675 0.5463 0.6595 0.5350 0.6796 0.4968 0.7262 0.4523 0.8281 0.4830 0.9294 0.5148 1.0727 0.5434 1.2040 0.5840
0.5986 0.7216 0.6170 0.1449 0.5624 0.8140 0.5498 0.8323 0.6171 0.9765 0.6846 0.1050 0.7429 I .2349 0.1992 1.3722 0.8685
1.3428 I .3556 1.3515 1.3515 1.3618 1.3479 1.3683 1.3405 1.3732 1.3117 1.3693 1.1379
1.2538 1.2826
1.2383 1.3023 1.2174 1.3105 1.1512 1.2980 1.0074
1.2689 0.9953 1.2438 0.9226
1.2009 1.0400 1.2649 0.9838 1.2837 0.9487
1.2554 0.9113
1.3029
1.4663
1.6662
0.6794
0.7452 I .5583
0.9359 1.7980 1.2987 I .6523
1.3834
1.0562
0.9811 1.3128 1.2095 1.2095 1.3685 1.1283
I.3695 0.9938 14090 0.9768 1.3889 0.9853
1.0661 1.5112 1.0385 1.5015 1.0579
1.4629
1.0904 1.4461 1.3311 1.3311 1.4948 1.2514 1.5917 1.1896 I.6442 1.1475 1.6460 1.1561
1.5337 I.5337 1.6914 1.4603 1.7847 1.4078
1.8352 I.3524
1.8436 1.3055
(1) For two cracks Figures 2 and 3 show that when the angle between two cracks is smaller, the SIF (negative SIF for autofrettage) for two cracks is smaller than that of a single crack. As the angle 8 increases, the SIF increases monotonically. When angle 8 reaches 45”, the SIF is near to that of a single crack, and when 8 = 90”, the SIF reaches the largest magnitude. That means that when angle between two cracks is smaller, the interaction of two cracks results in a relaxation of stress field at the crack tips; and when angle between two cracks is larger the interaction of two cracks results in enhancement of stress field at the crack tips. (2) For three cracks Figure 4 shows that for three cracks, when 8 is smaller, the SIF of crack 1 due to external tension is very little and is less than that of a single crack. With the increase of the angle between
H. M. SHU et a!.
842 Table 6. Correction
factors
F, for three asymmetrical
cracks
due to external
tension
a/n B’ 5 10 15 20 30
F,
0.05
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
F,
1.2754 1.7588 1.5989 1.9398 1.8595 2.0747 2.0299 2.1624 2.2068 2.2646 2.2727 2.3237 2.2991 2.3290 2.3129 2.3494 2.3265 2.3658 2.3430 2.3838 2.3661 2.3877 2.3840 2.3856 2.3993 2.3749 2.4111 2.3619 2.4176 2.3068 2.4124
1.0054 1.5697 I .2507 1.6977 1.4357 1.7928 1.5906 1.8637 1.8357 1.9873 1.9703 2.0738 2.0385 2.1501 2.0797 2.1965 2.1302 2.2624 2.1851 2.3053 2.2466 2.3175 2.3028 2.3037 2.3506 2.2706 2.3859 2.2339 2.4007 2.1106 2.3782 I.8144
0.8799 1.4596 1.0717 I.5407 1.2095
0.8558 1.4253 1.0331
0.8775 1.4239 1.0526 1.4545 1.1536 1.4655 1.1994 1.4699 1.2219 1.5410 1.1909 1.6654 1.1594 1.8390 1.1660 2.0438 1.3079 2.3384 1.6228 2.5017 2.0149 2.4906 2.3620 2.3637 2.6123 2.2156 2.7683 2.0905 2.8275 2.0428 2.7740 1.9575
0.9345 1.4478 1.1119 1.4644 1.1934 1.4699 1.2108 1.4840 1.1694 1.5937 1.0798 1.7751 0.9978 2.0113 0.9756 2.2785 1.1538 2.5865 1.5878 2.7696 2.1030 2.7038 2.5313 2.5327 2.8272 2.3617 3.0089 2.2331 3.0923 2.1876 3.0511 2.1412
1.0302 1.4996
1.1916 1.5994
I .4869
1.2077
1.3569 1.6057
6 F,
F2 F, F2
;I F; F2
40
F, F2
50
F,
60
F,
F2
F2
75
F, F?
90
F, F2
105
F, F2
120
F, F2
135
F, F2
150
F, F2
165
F,
175
F,
F2
F2
1.9904
Table 7. Correction
1.4765
1.5935 1.3041 1.6173 14456 1A829 I .5408 1.7713 1.6095 1.8799 1.6701 I .9957 1.7772 2.1567 1.9191 2.2588 2.0824 2.2839 2.2361 2.2369 2.3590 2.1559 2.4432 2.0724 2.4668 1.9455 2.4203 1.7578
factors
1.1504 1.5038 1.2202 1.5112 1.2985
1.5669 1.3241 1.6601 1.3425 1.7930 1.3805 1.9513 1.5091 2.1852
1.7337 2.3303 2.0090 2.3498 2.2628 2.2642 2.4552 2.1422 2.5795 2.0336 2.6170 1.9332 2.5615 1.8258
F, for three asymmetrical
cracks
1.5154 I .2558 I .5242 1.2356 1.5675 1.1135
1.7529 1.0295 1.9803 1.0435 2.2203 1.0651 2.5229 1.1398 2.9512 1.6408 3.1258 2.2786 2.9904 2.7701 2.7867 3.0980 2.6007 3.2979 2.4714 3.4018 2.3973 3.3811 2.3763
1.3513 1.6734 1.2754 I .7759 1.2016 2.0090 1.2800 2.2154 I .3604 2.4702 1.4226 2.8069 1.5120 3.4043 1.8501 3.6141 2.5994 3.3904 3.1321 3.1467 3.4780 2.9653 3.6887 2.8685 3.8037 2.7232 3.8072 2.6860
1.7953 1.5854 1.8913 1.4804 2.0500 1.4496 2.1850 1.628 1 2.4117 1.7924 2.6207 1.926 1 2.8975 2.0488 3.2737 2.2023 3.9640 2.3522 4.2793 3.1631 3.9713 3.6991 3.7217 4.0411 3.5378 4.2496 3.4263 4.3624 3.2856 4.3800 3.1287
due to fuii autofrettage
alw
F,
0.05
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
5
FL
10 15
F, F, F,
20
F,
30
F, F2 F,
-0.7262 - 1.0020 - 0.8938 - 1.0890 -1.0406 -1.1663 -1.1403 - 1.2185 - 1.2409 - 1.2744 - 1.2789 - 1.2971 - 1.2941 - 1.3130 - 1.3022 - 1.3236 -1.3100 - I .3356 -1.3159 - 1.3382 -1.3331 - 1.3456 - 1.3444 - 1.3444 - 1.3523 - 1.3378 - 1.3589 - 1.3249 - 1.3629 - 1.3022 - 1.3596 -1.1269
-0.5344 -0.8332 -0.6371 -0.8820 -0.7230 -0.9192 -0.8042 -0.9529 -0.9331 - I.0157 - I .OO52 -1.0616 - 1.0424 - 1.0982 - 1.0646 -1.1282 - 1.0923 - 1.i651 -1.1225 -1.1881 -1.1556 - 1.1950 - 1.1861 - 1.1861 -1.2125 -1.1686 - 1.2317 - 1.1488 - 1.2399 - 1.0843 - 1.2272 -0.9446
-0.3630 - 0.6360 - 0.4209 -0.6513 -0.4697 -0.6608 -0.5101 -0.6681 -0.5722 -0.6915 -0.6173 -0.7315 -0.6505 -0.7836 -0.6798 -0.8396 -0.7321 -0.8931 -0.8015 -0.9675 -0.8812 -0.9789 -0.9418 -0.9418 - 1.0161 -0.9169 - 1.0573 -0.8765 - 1.0685 -0.8188 - 1.0451 -0.7616
-0.2676 -0.4981 -0.3063 -0.4956 -0.3363 -0.4909 -0.3573 -0.4886 -0.3781 -0.5009 -0.3852 -0.5367 -0.3922 -0.5929 -0.4086 -0.6610 - 0.4648 -0.7612 -0.5630 -0.8245 -0.6833 -0.8311 -0.7942 -0.7942 -0.8779 -0.7404 - 0.9323 -0.6939 -0.9477 -0.6667 -0.9227 -0.6519
-0.2018 -0.3870 -0.2278 -0.3729 -0.243 1 -0.3611 -0.2484 -0.3560 -0.2389 - 0.3694 -0.2190 -0.4107 -0.2036 -0.4748 -0.2050 -0.5528 -0.2600 -0.6646 -0.3828 - 0.7263 -0.5355 -0.7199 -0.6708 -0.6708 - 0.7679 -0.6119 -0.8287 -0.5649 -0.8497 -0.5563 -0.8285 -0.5702
-0.1503 -0.2911 -0.1649 -0.2697 -0.1660 -0.2561 -0.1571 -0.2531 -0.1216 -0.2749 -0.0793 -0.3280 - 0.0450 - 0.4045 - 0.0346 -0.4928 - 0.0925 - 0.5938 -0.2453 - 0.6547 -0.4231 -0.6295 -0.5708 -0.5708 -0.6729 -0.5122 -0.7355 -0.4683 -0.7618 - 0.4660 -0.7474 -0.5002
-0.1052 -0.2041 -0.1065 -0.1797 -0.0926 -0.1694 - 0.0696 -0.1724 -0.0091 - 0.2098 -0.0566 -0.2802 -0.1096 -0.3727 -0.1260 - 0.4705 - 0.0430 -0.5768 -0.1412 -0.5948 -0.3355 -0.5488 -0.4853 - 0.4853 -0.5848 -0.4300 -0.6456 -0.3907 - 0.6748 -0.3870 -0.6687 -0.4327
-0.0593 -0.1211 -0.0431 -0.0999 -0.0121 -0.0988 - 0.0262 -0.1129 -0.1197 -0.1732 -0.2160 -0.2686 -0.2871 -0.3810 -0.2934 -0.4827 -0.1562 -0.5565 -0.0663 -0.5328 -0.2643 -0.4685 - 0.4050 -0.4050 -0.4962 -0.3563 -0.5516 -0.3227 -0.5806 -0.3146 -0.5819 -0.3595
-0.0008 - 0.0398 -0.0404 -0.0375 -0.0945 -0.0534 -0.1577 -0.0815 -0.3104 -0.1786 -0.4561 -0.3140 -0.5223 -0.4449 -0.4653 -0.5229 - 0.2280 -0.5213 -0.0197 -0.4515 -0.2014 -0.3785 -0.3216 -0.3216 -0.3980 -0.2819 -04447 -0.2559 -0.4701 -0.2448 -0.4758 -0.2736
0”
40
F;
F;
6
F2
50
F, F2
60
F, F2
15 90 105
F, F, F, FZ F, F2
120 135 150 165
F, F? F, F* F, F, F, F2
175
F, FZ
SIFs for radial cracks in thick walled cylinders-111
5.
FO l-
.... ..... a single crack 4.
3.
2.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 2.
cracks, the SIF of crack 1 increases. When 6 is larger, the SIF of crack 1 is larger than that of a single crack, and when the angle 8 tends to 180”, the SIF of crack 1 tends to the largest magnitude which is nearly equal to that of two symmetrical cracks. In our calculation, by computing the displacement of crack faces, it has been found that, generally, when a/w > 0.5 and 20” < 0 < 75”, the two faces of crack 1 come into contact partially due to internal pressure and external tension, and thus the SIF of crack 1 becomes very small.
-1.4
-1.2 4. -0.8 -0.6 900 750
-0.4
600
-0.2
..-..---..
a single crack
0.1
0.2
450 dW
0.3
0.4 Fig. 3
0.5
0.6
0.7
0.8
H. M. SHU ef al.
844
5. . ......... a single crack
/
1750 16S” :3 1w
4.
IO.50
3,
2.
1.
1
0.1
0.2
,
0.3
0.4
0.5
0;6
t
0.7
0.8
Fig. 4.
Figures 5 and 7 show that when B c 90”, the SIF of crack 2 (negative SIF for autofr~ttage) increase as the angle increases; when 8 > 90”, the SIF decreases as the angle 8 increases; and when 0 = 90”, the SIF reaches the largest magnitude which is nearly equal to that of two symmetrical cracks. When 0 comes close to 180”, changing the angle has little influence on the SIF for crack 2. Figure 6 presents the SIF for crack 1 due to full autofrettage. It can be observed that at some values of crack configuration parameters, the SIF of crack 1 can become positive; that means that the crack 1 is partially open near to the crack tip. From Figs 2-7, we can also conclude that when angle 0 is small (< 30”) and cracks are short (a/w < 0.2), the increase of 8 leads to an obvious increase of SIF; but when cracks are longer, the increase of 0 leads to a little changing of SIF. The SIF for external tension (internal pressure on inner surface and on crack faces) is much larger than that for internal pressure (no crack face pressure). When a cylinder undergoes internal or external pressure, the SIF increases as the length of the cracks increases; but when a cylinder undergoes an autofrettage process, when the cracks are shorter, the negative SIF increases as the length of cracks increases, when the cracks are little longer, the negative SIF decreases obviously as the length of cracks increases.
4. CONCLuSIONS (1) The SIF for two and three asymmetrical radial inner cracks in thick walled cylinders undergoing two kinds of internal pressure and full autofrettage has been calculated respectively. (2) For two asymmetrical cracks, the SIF (negative SIF for an autofrettaged cylinder) increases as the angle B between two cracks increases; and when 0 = 90”, i.e. when the cracks become the symmetrical, the SIF reaches the largest magnitude. (3) For crack 1 of three asymmetrical cracks, the SIF (negative SIF for a autofrettaged cylinder) increases as the angle 8 between cracks 1 and 2 increases; when 0 tends to 180” the SIF of crack 1 tends to the largest magnitude. For crack 2, when 0 > 90”, the SIF decreases as the angle 0 increases; when 8 c 90”, the SIF increases as the angle increases; and when 8 is about 90”, the SIF reaches the largest magnitude. The largest SIF of cracks 1 and 2 are nearly equal to that of two symmetrical cracks. (4) For three cracks, the two faces of crack 1 will contact partially at some values of crack parameters due to internal pressure and external tension; and they will open partially near to the crack tip due to autofrettage.
845
SIFs for radial cracks in thick walled cylinders-III
5.
........*. a single crack 4.
3.
2.
1.
L
I
0.1
5.
1
I
0.2
0.3
0.4
0.5
0;6
0.7
a/w
0.8
F*
4.
900 105Q Izoo
3.
169 179
........*. a singie crack
1350
lsoo
2,
1.
I
t
I
0.1
0.2
0.3
0.4
1
0.5
due to external tension Fig. 5.
0;6
0.7
0.8
a/w
846
H.
M. SHU el al.
-1.2
-0.8
-0.4
.......... 8 single
c&
-1.2
-0.8 17.50 169 UP 1350 120° 209
-0.4
w* 756
Fig. 6.
SIFs for radial cracks in thick walled cylinders-41
-1.2 -i.
-0.8 4.6 -0.4 -0.2
-0.8
,
0.1
,
I
I
,
I
0.2 0.3 0.4 0.S 0.6 0.7
t
s/w
0.8
Fig. 7.
REVERENCES 111 II. M. Shu, J. Petit and G. Bezine, Stress intensity factors for radial cracks in thick walled cylinders-4 Symmetrical cracks. Engng Fracture hfech. 49, 61 l-623 (1994). [2] M. Per1 and R. Arone, Stress intensity factors for iarge arrays of radial cracks in thick-walled steel cylinders. Engng Fracture Meek ZS, 341-348 (1986). [3] M. Per1 and R. Arone, Stress intensity factors for a radial multicracked partially autofrettaged pressurized thick-walled cylinder. J. Press. Yess. Tecfmol. 110, 147-154 (1988). [4] A, P. Parker and J. R. Farrow, On the equivalence of axisymmetric bending, thermal and autof~tta~ residual stress fields. J. Strain Anal. 115, 51-62 (1980).
[.5] Pu. S. I, and Chen, P. C. ‘I. Stress intensity factors for radial cracks in pie-stressed, stein-hardening materials. J. Press. I/ess. Technol. IOS, 117-123 (1983). (Receiveci I? October 1993)
thick-walfed cylinder of