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Effect of boundaries parallel to crack on the SIFs of a semi-circular surface crack – A numerical study
Available online at www.sciencedirect.com
Engineering Failure Analysis 16 (2009) 538–544 www.elsevier.com/locate/engfailanal
Short communication
Effect of boundaries parallel to crack on the SIFs of a semi-circular surface crack – A numerical study Yonggang Zhao, Xiangqiao Yan * Research Laboratory on Composite Materials, Harbin Institute of Technology, Harbin 150001, PR China Received 6 January 2008; accepted 24 January 2008 Available online 6 February 2008
Abstract This note is concerned with a surface semi-circular crack by using finite element method. Attention is specifically paid to the effect of the boundaries parallel to the crack on the stress intensity factors (SIFs). The numerical results illustrate that when the boundaries parallel to the crack are closer to the crack surface the effect of the boundaries parallel to the crack on the SIFs must be considered. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Stress intensity factor; Surface crack; Finite element
1. Introduction An elastic surface crack problem is very important subject in fracture mechanics. Many results have been reported in the literature [1–12]. The numerical results obtained by Newman and his coworkers [1,2] by using finite element method, especially, have been widely used in practical applications. It is found that from the previous investigation results [1–12] that the effect of the boundaries parallel to the crack on the SIFs is not considered. From the investigations of two-dimensional crack problems reported in Refs [13,14], it is found that the effect of the boundaries parallel to the crack on the SIFs is not ignored when H/a (see Fig. 1) is less than a certain value. For a center crack problem (see Figs. 1 and 2) in rectangular plate in tension, for example, it is found from the numerical results of Ref. [13] that (1) even that at a/W = 0.2, the SIFs when H/W = 0.4 is 22.4% more than that when H/W = 2; (2) at a/W = 0.5, the SIFs when H/W = 0.4 is 89.3% more than that when H/W = 2. Based on the above consideration, in this note, we specially study the effect of the boundaries parallel to the crack on the SIFs for an elastic surface crack problem. Here, stress analysis is performed by using finite element analysis software Ansys. The SIFs are computed using the displacement correlation technique [15,16]. *
Corresponding author. E-mail address: [email protected] (X. Yan).
1350-6307/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2008.01.005
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
539
σ
H
2a
H 2W σ Fig. 1. Schematic of a center crack in rectangular plate in tension.
4.0
H/W=0.4
3.5
Normalized SIFs
3.0
2.5
0.7 2.0
1.0 1.5
2.0 1.0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
a/W Fig. 2. Normalized SIFs of center crack in rectangular plate in tension.
2. Numerical results and discussion For the purpose of illustrating the accuracy of the numerical results obtained in this section, first a surface semi-circular crack problem (see Fig. 3) in a finite elastic body is considered: W =a ¼ 5:0;
T =a ¼ 5:0;
H =a ¼ 5:0
The calculated SIFs is normalized by rffiffiffiffiffiffi pa Þ ðQ ¼ p2 =4Þ g ¼ K 1 =ðr Q and is given in Table 1. For the comparison purpose, Table 1 also lists those reported in Ref. [1], from which it is found that the agreement is very good. In order to study the effect of the boundaries parallel to the crack on the SIFs, then the following two cases are considered:
540
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
Fig. 3. The schematic of a surface semi-circular crack in a finite plate.
Table 1 Normalized SIFs of a semi-circular surface crack in the case of W/a = 5.0, T/a = 5.0, H/a = 5.0 2h/p
0
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1
Present Raju-Newman [1] Relative error (%)
1.1733 1.174 0.6
1.1226 1.145 1.96
1.0813 1.105 2.14
1.0702 1.082 1.09
1.0596 1.067 0.69
1.052 1.058 0.56
1.047 1.053 0.57
1.0444 1.05 0.53
1.0435 1.049 0.52
Case 1: T/a = 5.0, W/a = 5.0, 4.0, 3.0, 2.0, 1.6, 1.4, 1.2, H/a = 5.0, 4.0, 3.0, 2.0, 1.6, 1.4, 1.2. Case 2: W/a = 5.0, T/a = 5.0, 4.0, 3.0, 2.0, 1.6, 1.4, 1.2, H/a = 5.0, 4.0, 3.0, 2.0, 1.6, 1.4, 1.2. For case 1, the maximum normalized SIFs occur at the intersection point where the crack front curve intersects the free surface (i.e., h = 0), while the minimum normalized SIFs occur at the maximum depth (i.e.,
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
541
h = p/2). The numerical results are listed in Tables 2 and 3, respectively. For conveniently observing the effect of the boundaries parallel to the crack the SIFs, some of the numerical results given in Tables 2 and 3 are pictured in Figs. 4 and 5, respectively. From Tables 2 and 3 (also see Figs. 4 and 5), it can be seen the effect of the boundaries parallel to the crack on the SIFs has the following characterizations:
Table 2 SIFs for the concerned case at near the free surface (2h/p = 0) H/a
5.0 4.0 3.0 2.0 1.6 1.4 1.2
W/a 5.0
4.0
3.0
2.0
1.6
1.4
1.2
1.1733 1.1807 1.2034 1.2905 1.3725 1.4572 1.5731
1.1869 1.1930 1.2075 1.2921 1.3886 1.4753 1.6043
1.1921 1.1966 1.2204 1.3085 1.3966 1.4888 1.6145
1.2397 1.2407 1.2555 1.3452 1.4438 1.5248 1.6362
1.3164 1.3192 1.3246 1.4058 1.5073 1.5974 1.7336
1.3911 1.3921 1.4090 1.4916 1.6004 1.6853 1.8237
1.5592 1.5624 1.5848 1.6958 1.8152 1.8958 2.0257
Table 3 SIFs for the concerned case at the maximum depth (2h/p = 1)
5.0 4.0 3.0 2.0 1.6 1.4 1.2
W/a 5.0
4.0
3.0
2.0
1.6
1.4
1.2
1.0435 1.0494 1.0660 1.1231 1.1821 1.2317 1.3036
1.0465 1.0520 1.0683 1.1242 1.1829 1.2324 1.3052
1.0548 1.0585 1.0739 1.1311 1.1874 1.2369 1.3077
1.0843 1.0854 1.0946 1.1507 1.2135 1.2635 1.3340
1.1180 1.1190 1.1264 1.1740 1.2370 1.2914 1.3709
1.1488 1.1506 1.1586 1.2018 1.2597 1.3133 1.3967
1.2052 1.2069 1.2168 1.2621 1.3134 1.3623 1.4415
2.0
1.8
η
H/a
W/a=1.2
1.6
1.4
W/a= 2.0 W/a= 5.0
1.2
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
H/a
Fig. 4. Variation of the normalized SIFs g with W/a and H/a (2h/p = 0).
542
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
1.4
η
1.3
W/a=1.2 1.2
W/a=2.0
1.1
W/a=5.0 1.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
H/a
Fig. 5. Variation of normalized SIFs g g with W/a and H/a (2h/p = 1).
(1) As H/a is large enough, for example, H/a > 5, the effect of the boundaries parallel to the crack on the SIFs is negligible. (2) With decrease of H/a, the normalized SIFs gradually increase. As H/a approaches to 2, the normalized SIFs increase sharply, especially at the free surface intersection point. Two quantity data are given as follows: (1) As W/a = 5, the normalized SIFs of the free surface intersection point when H/a = 1.2 is 34.1% more than that when H/a = 5.0. (2) The normalized SIFs of the free surface intersection point when H/a = 1.2 and W/a = 1.2 is 72.6% more than that when H/a = 5.0 and W/a = 5.0. For case 2, the maximum normalized SIFs also occur at the intersection point where the crack front curve intersect the free surface (i.e., h = 0), while the minimum normalized SIFs also occur at the maximum depth (i.e., h = p/2). Tables 4 and 5 list these SIFs. For further illustrating the effectiveness of the present numerical results, the calculated SIFs corresponding to H/a = 5 in Table 4 is compared with Newman’s results, as shown in Fig. 6, from which it is found that the agreement is very good. For conveniently observing the effect of the boundaries parallel to the crack the SIFs, some of the numerical results given in Tables 4 and 5 are pictured in Figs. 7 and 8.
Table 4 SIFs for the concerned case at near the free surface (2h/p = 0) H/a
5.0 4.0 3.0 2.0 1.6 1.4 1.2
T/a 5.0
4.0
3.0
2.0
1.6
1.4
1.2
1.1733 1.1807 1.2034 1.2905 1.3725 1.4572 1.5731
1.1774 1.1933 1.2051 1.2958 1.3848 1.4578 1.5795
1.1996 1.2116 1.2393 1.3025 1.3998 1.4829 1.6075
1.2884 1.2984 1.3214 1.3992 1.4790 1.5559 1.6662
1.352 1.3641 1.3903 1.4823 1.5505 1.6382 1.7436
1.4077 1.4218 1.4528 1.5519 1.6367 1.7075 1.8356
1.4856 1.5034 1.5386 1.6260 1.7327 1.8280 1.9433
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
543
Table 5 SIFs for the concerned case at the maximum depth (2h/p = 1) H/a
5.0 4.0 3.0 2.0 1.6 1.4 1.2
T/a 5.0
4.0
3.0
2.0
1.6
1.4
1.2
1.0435 1.0494 1.0660 1.1231 1.1830 1.2327 1.3054
1.0500 1.0558 1.0711 1.1254 1.1847 1.2341 1.3070
1.0658 1.0684 1.0835 1.1350 1.1907 1.2375 1.3088
1.0874 1.0952 1.1114 1.1654 1.2240 1.2721 1.3419
1.1006 1.1099 1.1275 1.1837 1.2441 1.2952 1.3723
1.1064 1.1169 1.1363 1.1951 1.2549 1.3064 1.3844
1.1203 1.1330 1.1557 1.2175 1.2790 1.3312 1.4075
1.5
present results Newman's results
η
1.4
1.3
1.2
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
t/a
Fig. 6. The present results and Newman’s result (H/a = 5.0).
2.0
1.8
η
1.6
T/a=1.2
1.4
T/a=2.0 T/a=5.0
1.2
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
H/a
Fig. 7. Variation of normalized SIFs g with T/a and H/a (2h/p = 0).
From Tables 4 and 5 (also see Figs. 7 and 8), it can be seen that the variations of the effect of the boundaries parallel to the crack on the SIFs with T/a are basically same as those listed in Tables 2 and 3 with T/a instead of W/a.
544
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
1.4
η
1.3
1.2
T/a=1.2 T/a=2.0
1.1
T/a=5.0 1.0 1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
H/a
Fig. 8. Variation of normalized SIFs g with T/a and H/a (2h/p = 1).
Acknowledgement Special thanks are due to the National Natural Science Foundation of China (No.10272037 and No.10672046) for supporting the present work. References [1] Raju IS, Newman Jr Jc. Stress intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Eng Fract Mech 1979;11(4):817–29. [2] Newman Jr Jc, Raju IS. An empirical stress-intensity factor equation for the surface crack. Eng Fract Mech 1981;15(1–2):185–92. [3] Newman Jr JC, Raju IS. Stress intensity factors equations for cracks in three-dimensional finite bodies. NASA TM 1981;83200:1–49. [4] Raju IS, Newman Jr JC. Stress-intensity factors for two symmetric corner cracks. ASTM Special Technical Publication 1979;677:411– 430. [5] Tan PW, Newman Jr JC, Bigelow CA. Three-dimensional finite-element analyses of corner cracks at stress concentrations. Eng Fract Mech 1966;55(3):505–12. [6] Zhao W, Newman Jr JC, Sutton MA, Wu XR, Shivakumar KN. Stress intensity factors for corner cracks at a hole by a 3-D weight function method with stresses from the finite element method. Fatigue Fract Eng Mater Struct 1997;20(9):1255–67. [7] Shivakumar KN, Newman Jr JC. Stress intensity factors for large aspect ratio surface and corner cracks at a semi-circular notch in a tension specimen. Eng Fract Mech 1991;38(6):467–73. [8] Newman Jr JC, Reuter WG, Aveline Jr CR. Stress and fracture analyses of semi-elliptical surface cracks. ASTM Special Technical Publication 1999;1360:403–423. [9] Zhao W, Newman Jr JC, Sutton MA, Shivakumar KN, Wu XR. Stress intensity factors for surface cracks at a hole by a threedimensional weight function method with stresses from the finite element method. Fatigue Fract Eng Mater Struct 1998;21(2):229–39. [10] Isisa M, Noguchi H, Yoshida T. A semi-elliptical surface crack in finite-thickness plates under tension and bending. Int J Frac 1984;26:157–88. [11] Isisa M, Yoshida T, Noguchi H. A plate with a pair of semi-elliptical surface cracks under tension. Trans Jpn Soc Mech Engrs 1983;49(448):1572–80. [12] Isisa M, Noguchi H. Tension of a plate containing an embedded elliptical crack. Eng Frac Mech 1984;20(3):387–408. [13] Isida M. Effect of width and length on stress intensity factors of internally cracked plates under various boundary conditions. Int J Frac 1971;7:301–16. [14] Yan X. Cracks emanating from circular hole or square hole in rectangular plate in tension. Eng Fract Mech 2006;73(12):1743–54. [15] Lutz ED. Numerical methods for hyper-singular and near-singular boundary integrals in fracture mechanics. Ph.D. Thesis, Cornell University, 1991. [16] Chan SK, Tuba IS, Wilson WK. On the finite element method in linear fracture mechanics. Eng Fract Mech 1970;2:1–17.
Engineering Failure Analysis 16 (2009) 538–544 www.elsevier.com/locate/engfailanal
Short communication
Effect of boundaries parallel to crack on the SIFs of a semi-circular surface crack – A numerical study Yonggang Zhao, Xiangqiao Yan * Research Laboratory on Composite Materials, Harbin Institute of Technology, Harbin 150001, PR China Received 6 January 2008; accepted 24 January 2008 Available online 6 February 2008
Abstract This note is concerned with a surface semi-circular crack by using finite element method. Attention is specifically paid to the effect of the boundaries parallel to the crack on the stress intensity factors (SIFs). The numerical results illustrate that when the boundaries parallel to the crack are closer to the crack surface the effect of the boundaries parallel to the crack on the SIFs must be considered. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Stress intensity factor; Surface crack; Finite element
1. Introduction An elastic surface crack problem is very important subject in fracture mechanics. Many results have been reported in the literature [1–12]. The numerical results obtained by Newman and his coworkers [1,2] by using finite element method, especially, have been widely used in practical applications. It is found that from the previous investigation results [1–12] that the effect of the boundaries parallel to the crack on the SIFs is not considered. From the investigations of two-dimensional crack problems reported in Refs [13,14], it is found that the effect of the boundaries parallel to the crack on the SIFs is not ignored when H/a (see Fig. 1) is less than a certain value. For a center crack problem (see Figs. 1 and 2) in rectangular plate in tension, for example, it is found from the numerical results of Ref. [13] that (1) even that at a/W = 0.2, the SIFs when H/W = 0.4 is 22.4% more than that when H/W = 2; (2) at a/W = 0.5, the SIFs when H/W = 0.4 is 89.3% more than that when H/W = 2. Based on the above consideration, in this note, we specially study the effect of the boundaries parallel to the crack on the SIFs for an elastic surface crack problem. Here, stress analysis is performed by using finite element analysis software Ansys. The SIFs are computed using the displacement correlation technique [15,16]. *
Corresponding author. E-mail address: [email protected] (X. Yan).
1350-6307/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2008.01.005
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
539
σ
H
2a
H 2W σ Fig. 1. Schematic of a center crack in rectangular plate in tension.
4.0
H/W=0.4
3.5
Normalized SIFs
3.0
2.5
0.7 2.0
1.0 1.5
2.0 1.0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
a/W Fig. 2. Normalized SIFs of center crack in rectangular plate in tension.
2. Numerical results and discussion For the purpose of illustrating the accuracy of the numerical results obtained in this section, first a surface semi-circular crack problem (see Fig. 3) in a finite elastic body is considered: W =a ¼ 5:0;
T =a ¼ 5:0;
H =a ¼ 5:0
The calculated SIFs is normalized by rffiffiffiffiffiffi pa Þ ðQ ¼ p2 =4Þ g ¼ K 1 =ðr Q and is given in Table 1. For the comparison purpose, Table 1 also lists those reported in Ref. [1], from which it is found that the agreement is very good. In order to study the effect of the boundaries parallel to the crack on the SIFs, then the following two cases are considered:
540
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
Fig. 3. The schematic of a surface semi-circular crack in a finite plate.
Table 1 Normalized SIFs of a semi-circular surface crack in the case of W/a = 5.0, T/a = 5.0, H/a = 5.0 2h/p
0
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1
Present Raju-Newman [1] Relative error (%)
1.1733 1.174 0.6
1.1226 1.145 1.96
1.0813 1.105 2.14
1.0702 1.082 1.09
1.0596 1.067 0.69
1.052 1.058 0.56
1.047 1.053 0.57
1.0444 1.05 0.53
1.0435 1.049 0.52
Case 1: T/a = 5.0, W/a = 5.0, 4.0, 3.0, 2.0, 1.6, 1.4, 1.2, H/a = 5.0, 4.0, 3.0, 2.0, 1.6, 1.4, 1.2. Case 2: W/a = 5.0, T/a = 5.0, 4.0, 3.0, 2.0, 1.6, 1.4, 1.2, H/a = 5.0, 4.0, 3.0, 2.0, 1.6, 1.4, 1.2. For case 1, the maximum normalized SIFs occur at the intersection point where the crack front curve intersects the free surface (i.e., h = 0), while the minimum normalized SIFs occur at the maximum depth (i.e.,
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
541
h = p/2). The numerical results are listed in Tables 2 and 3, respectively. For conveniently observing the effect of the boundaries parallel to the crack the SIFs, some of the numerical results given in Tables 2 and 3 are pictured in Figs. 4 and 5, respectively. From Tables 2 and 3 (also see Figs. 4 and 5), it can be seen the effect of the boundaries parallel to the crack on the SIFs has the following characterizations:
Table 2 SIFs for the concerned case at near the free surface (2h/p = 0) H/a
5.0 4.0 3.0 2.0 1.6 1.4 1.2
W/a 5.0
4.0
3.0
2.0
1.6
1.4
1.2
1.1733 1.1807 1.2034 1.2905 1.3725 1.4572 1.5731
1.1869 1.1930 1.2075 1.2921 1.3886 1.4753 1.6043
1.1921 1.1966 1.2204 1.3085 1.3966 1.4888 1.6145
1.2397 1.2407 1.2555 1.3452 1.4438 1.5248 1.6362
1.3164 1.3192 1.3246 1.4058 1.5073 1.5974 1.7336
1.3911 1.3921 1.4090 1.4916 1.6004 1.6853 1.8237
1.5592 1.5624 1.5848 1.6958 1.8152 1.8958 2.0257
Table 3 SIFs for the concerned case at the maximum depth (2h/p = 1)
5.0 4.0 3.0 2.0 1.6 1.4 1.2
W/a 5.0
4.0
3.0
2.0
1.6
1.4
1.2
1.0435 1.0494 1.0660 1.1231 1.1821 1.2317 1.3036
1.0465 1.0520 1.0683 1.1242 1.1829 1.2324 1.3052
1.0548 1.0585 1.0739 1.1311 1.1874 1.2369 1.3077
1.0843 1.0854 1.0946 1.1507 1.2135 1.2635 1.3340
1.1180 1.1190 1.1264 1.1740 1.2370 1.2914 1.3709
1.1488 1.1506 1.1586 1.2018 1.2597 1.3133 1.3967
1.2052 1.2069 1.2168 1.2621 1.3134 1.3623 1.4415
2.0
1.8
η
H/a
W/a=1.2
1.6
1.4
W/a= 2.0 W/a= 5.0
1.2
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
H/a
Fig. 4. Variation of the normalized SIFs g with W/a and H/a (2h/p = 0).
542
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
1.4
η
1.3
W/a=1.2 1.2
W/a=2.0
1.1
W/a=5.0 1.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
H/a
Fig. 5. Variation of normalized SIFs g g with W/a and H/a (2h/p = 1).
(1) As H/a is large enough, for example, H/a > 5, the effect of the boundaries parallel to the crack on the SIFs is negligible. (2) With decrease of H/a, the normalized SIFs gradually increase. As H/a approaches to 2, the normalized SIFs increase sharply, especially at the free surface intersection point. Two quantity data are given as follows: (1) As W/a = 5, the normalized SIFs of the free surface intersection point when H/a = 1.2 is 34.1% more than that when H/a = 5.0. (2) The normalized SIFs of the free surface intersection point when H/a = 1.2 and W/a = 1.2 is 72.6% more than that when H/a = 5.0 and W/a = 5.0. For case 2, the maximum normalized SIFs also occur at the intersection point where the crack front curve intersect the free surface (i.e., h = 0), while the minimum normalized SIFs also occur at the maximum depth (i.e., h = p/2). Tables 4 and 5 list these SIFs. For further illustrating the effectiveness of the present numerical results, the calculated SIFs corresponding to H/a = 5 in Table 4 is compared with Newman’s results, as shown in Fig. 6, from which it is found that the agreement is very good. For conveniently observing the effect of the boundaries parallel to the crack the SIFs, some of the numerical results given in Tables 4 and 5 are pictured in Figs. 7 and 8.
Table 4 SIFs for the concerned case at near the free surface (2h/p = 0) H/a
5.0 4.0 3.0 2.0 1.6 1.4 1.2
T/a 5.0
4.0
3.0
2.0
1.6
1.4
1.2
1.1733 1.1807 1.2034 1.2905 1.3725 1.4572 1.5731
1.1774 1.1933 1.2051 1.2958 1.3848 1.4578 1.5795
1.1996 1.2116 1.2393 1.3025 1.3998 1.4829 1.6075
1.2884 1.2984 1.3214 1.3992 1.4790 1.5559 1.6662
1.352 1.3641 1.3903 1.4823 1.5505 1.6382 1.7436
1.4077 1.4218 1.4528 1.5519 1.6367 1.7075 1.8356
1.4856 1.5034 1.5386 1.6260 1.7327 1.8280 1.9433
Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
543
Table 5 SIFs for the concerned case at the maximum depth (2h/p = 1) H/a
5.0 4.0 3.0 2.0 1.6 1.4 1.2
T/a 5.0
4.0
3.0
2.0
1.6
1.4
1.2
1.0435 1.0494 1.0660 1.1231 1.1830 1.2327 1.3054
1.0500 1.0558 1.0711 1.1254 1.1847 1.2341 1.3070
1.0658 1.0684 1.0835 1.1350 1.1907 1.2375 1.3088
1.0874 1.0952 1.1114 1.1654 1.2240 1.2721 1.3419
1.1006 1.1099 1.1275 1.1837 1.2441 1.2952 1.3723
1.1064 1.1169 1.1363 1.1951 1.2549 1.3064 1.3844
1.1203 1.1330 1.1557 1.2175 1.2790 1.3312 1.4075
1.5
present results Newman's results
η
1.4
1.3
1.2
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
t/a
Fig. 6. The present results and Newman’s result (H/a = 5.0).
2.0
1.8
η
1.6
T/a=1.2
1.4
T/a=2.0 T/a=5.0
1.2
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
H/a
Fig. 7. Variation of normalized SIFs g with T/a and H/a (2h/p = 0).
From Tables 4 and 5 (also see Figs. 7 and 8), it can be seen that the variations of the effect of the boundaries parallel to the crack on the SIFs with T/a are basically same as those listed in Tables 2 and 3 with T/a instead of W/a.
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Y. Zhao, X. Yan / Engineering Failure Analysis 16 (2009) 538–544
1.4
η
1.3
1.2
T/a=1.2 T/a=2.0
1.1
T/a=5.0 1.0 1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
H/a
Fig. 8. Variation of normalized SIFs g with T/a and H/a (2h/p = 1).
Acknowledgement Special thanks are due to the National Natural Science Foundation of China (No.10272037 and No.10672046) for supporting the present work. References [1] Raju IS, Newman Jr Jc. Stress intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Eng Fract Mech 1979;11(4):817–29. [2] Newman Jr Jc, Raju IS. An empirical stress-intensity factor equation for the surface crack. Eng Fract Mech 1981;15(1–2):185–92. [3] Newman Jr JC, Raju IS. Stress intensity factors equations for cracks in three-dimensional finite bodies. NASA TM 1981;83200:1–49. [4] Raju IS, Newman Jr JC. Stress-intensity factors for two symmetric corner cracks. ASTM Special Technical Publication 1979;677:411– 430. [5] Tan PW, Newman Jr JC, Bigelow CA. Three-dimensional finite-element analyses of corner cracks at stress concentrations. Eng Fract Mech 1966;55(3):505–12. [6] Zhao W, Newman Jr JC, Sutton MA, Wu XR, Shivakumar KN. Stress intensity factors for corner cracks at a hole by a 3-D weight function method with stresses from the finite element method. Fatigue Fract Eng Mater Struct 1997;20(9):1255–67. [7] Shivakumar KN, Newman Jr JC. Stress intensity factors for large aspect ratio surface and corner cracks at a semi-circular notch in a tension specimen. Eng Fract Mech 1991;38(6):467–73. [8] Newman Jr JC, Reuter WG, Aveline Jr CR. Stress and fracture analyses of semi-elliptical surface cracks. ASTM Special Technical Publication 1999;1360:403–423. [9] Zhao W, Newman Jr JC, Sutton MA, Shivakumar KN, Wu XR. Stress intensity factors for surface cracks at a hole by a threedimensional weight function method with stresses from the finite element method. Fatigue Fract Eng Mater Struct 1998;21(2):229–39. [10] Isisa M, Noguchi H, Yoshida T. A semi-elliptical surface crack in finite-thickness plates under tension and bending. Int J Frac 1984;26:157–88. [11] Isisa M, Yoshida T, Noguchi H. A plate with a pair of semi-elliptical surface cracks under tension. Trans Jpn Soc Mech Engrs 1983;49(448):1572–80. [12] Isisa M, Noguchi H. Tension of a plate containing an embedded elliptical crack. Eng Frac Mech 1984;20(3):387–408. [13] Isida M. Effect of width and length on stress intensity factors of internally cracked plates under various boundary conditions. Int J Frac 1971;7:301–16. [14] Yan X. Cracks emanating from circular hole or square hole in rectangular plate in tension. Eng Fract Mech 2006;73(12):1743–54. [15] Lutz ED. Numerical methods for hyper-singular and near-singular boundary integrals in fracture mechanics. Ph.D. Thesis, Cornell University, 1991. [16] Chan SK, Tuba IS, Wilson WK. On the finite element method in linear fracture mechanics. Eng Fract Mech 1970;2:1–17.