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Stress intensity factors for elliptical surface cracks in round bars with different stress concentration coefficient
International Journal of Fatigue 25 (2003) 733–741 www.elsevier.com/locate/ijfatigue
Stress intensity factors for elliptical surface cracks in round bars with different stress concentration coefficient W. Guo ∗, H. Shen, H. Li Department of Aircraft Engineering and Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China Received 15 July 2002; received in revised form 11 October 2002; accepted 22 January 2003
Abstract Stress intensity factors (SIFs) of elliptical surface cracks in notched tensile round bars are calculated by using three-dimensional finite element analysis (FEA) models with singular 20-node elements arranged around the crack tip. Systematic analyses are performed for semi-circular type, V-type and U-type annular notches with theoretical stress concentration coefficient Kt ranging from 1– 5. It is shown that the SIFs are strongly dependent on Kt, and the influence of notch geometry is negligibly weak for a given stress concentration coefficients in the analyzed range. An empirical expression for the SIF as a function of crack geometry and Kt is then obtained by fitting the numerical results. Comparison of present results with available numerical results in the literature shows good agreement. Application of the empirical expression to a screw bolt yields good coincidence with experimental results of SIFs by using a modified James-Anderson method. Therefore, the empirical expression of SIFs can be used conveniently in integrity assessment of various notched bars, at least in the analyzed range in this paper. 2003 Elsevier Science Ltd. All rights reserved. Keywords: Stress intensity factor; Surface crack; Stress concentration coefficient; Notch; Three-dimensional finite element analysis
1. Introduction Surface cracks emanating from stress concentrating locations are the most common phenomena of fatigue failure. Screw bolts, dimple rivets, bars with variable cross-section and so on are a category of cylindrical parts and components extensively used in engineering mechanisms. Surface fatigue cracks are frequently initiated in such components at the stress concentrating locations, then propagate into the interior of the parts and can cause final fracture abruptly. In addition, smooth and notched round bars have been used as standard specimens to obtain the fatigue property of materials for safe design and assessment. Some of the widely used round bars are shown in Fig. 1. Many numerical analyses, theoretical studies and experimental investigations have been conducted to obtain stress intensity factors (SIFs) for three-dimen-
∗
Corresponding author. Tel.: +86-25-4891370; fax: +86-4890513. E-mail address: [email protected] (W. Guo).
Fig. 1. bars.
Surface crack from stress concentrating location of round
sional (3D) cracked bodies. Explicit solutions or empirical expressions have been achieved for surface and corner cracks on smooth strips or straight round bars and at circular holes in finite thickness plates [1–5]. For surface
0142-1123/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0142-1123(03)00050-1
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cracks in round bars or bolts, SIFs are studied mainly for very low stress concentration (with stress concentration coefficient Kt⬵1) and a few typical screws with relatively high stress concentration. However, SIFs for 3D cracks emanating from general stress concentrating locations are very limited. To obtain universal SIF formula of surface cracks in notched bodies with different notch geometry and Kt is significant for application of 3D fracture mechanics in engineering. The purpose of the present paper is to investigate the dependence of SIFs of surface cracks in notched round bars upon stress concentration coefficient and notch geometry. Three-dimensional finite element (FE) models with 20-node singular elements arranged around the crack tip will be used to calculate the SIFs of elliptical surface cracks in round bars with different notches and theoretical stress concentration coefficients. A wide range of notch and crack geometry parameters are considered in the numerical analysis to yield empirical formula for the SIFs of 3D notched bars of engineering interest. Comparison with available numerical as well as experimental results is made to show the effectiveness of the present results.
2. Modeling of the problem 2.1. Notch geometry and crack configuration In order to investigate the influences of notch geometry and theoretical stress concentration coefficient upon the SIFs of surface cracks at the root of notches, round bars without notch, with semicircular annular notch, U-type annular notch and V-type annular notch of 60° opening angle are considered in the analyses as shown in Fig. 2. Semi-elliptical surface cracks are arranged on the minimum cross-section of the notched bar as shown in Fig. 2(b) and (c). Six sets of geometric parameters are assumed as listed in Table 1. Theoretical stress concentration coefficients of the notched bars without a crack in the six cases can be found to be 1, 2.5, 2.5, 2.5, 4 and 5, respectively [6,7]. For convenience,
Fig. 2.
the diameters d of the minimum cross-section in all the investigated cases are set at 7.5 mm. The main purpose for choosing the straight round bar (case 1 in Table 1) is to examine the validity of the algorithms for calculating SIFs in this work by comparison with those in Ref. [5] and provide a keynote to developing the empirical expression of SIFs for notched bars. Different notch geometries in cases 2, 3 and 4 with the same Kt = 2.5 are arranged to show the effect of notch geometry on SIFs. The surface crack has a semi-elliptical geometry with semi-major axis c and semi-minor axis a as shown in Fig. 2(c), where a is also the depth of surface crack. The length of the arc s measured from the center L of the crack front to any point P on the crack front is defined to describe the position considered. 2.2. Finite element modeling As symmetry, only a quarter of each specimen is considered in the FE model, as shown in Fig. 3. Twentynode hexahedron elements are used in the FE model to provide high precision of the notch root radius and the crack shape. In order to simulate the theoretical inverse square root singularity of stress and strains near the crack border, singular elements with mid-side node relocated to their 1/4-point position are arranged around the crack front, as shown by the amplified region B in Fig. 3. The 20-node hexahedron element and the singular element come to being by virtue of the FE software ansys. A total about 9000 elements and 130,000 freedoms are used in the modeling. 2.3. Calculation of the SIFs For a surface crack in a round bar as shown in Fig. 4, a local coordinate set (X,Y) is defined at any interesting point P on the crack front to calculate the SIF at the point P. With coordinate origin at P, the Y axis is perpendicular to the crack surface at P, the X axis is perpendicular to the crack front at P and the Y axis, and (r, f) are the polar coordinates in the X–Y plane. Assuming the expression of crack opening displacement
Dimensions of round bar specimens and elliptical surface crack.
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Table 1 Dimensions and theoretical stress concentration coefficients of specimens Specimen number
Case Case Case Case Case Case
Notch type
1 2 3 4 5 6
Without notch Semi-circular annular notch ‘U’ annular notch ‘V’ annular notch ‘V’ annular notch ‘V’ annular notch
Dimension of specimen (mm) D d
r
Stress concentration coefficient Kt
– 8.532 12 12 12 8.969
– 0.516 0.79 0.39 0.26 0.135
1 2.5 2.5 2.5 4 5
7.5 7.5 7.5 7.5 7.5 7.5
where n is Poisson’s ratio, E the elastic modulus and KI is the SIF at point P. λ is a function of Poisson’s ratio and stress state which will be reflected by E⬘ in Eq. (3). According to Eq. (1), the opening displacement V of the semi-elliptical crack surface of a round bar under remote tensile stress s can be obtained as V兩f=180° ⫽
(1 ⫹ n)(l ⫹ 1)KI E
冪2π. r
(2)
When V兩f = 180° is provided by the 3D FE analysis, Eq. (2) leads to the relationship between KI and r. Fitting the linear part of the KI–r curve and extrapolating the line to r = 0, SIF at the point on the surface crack border can then be obtained [9], KI ⫽
冑2πE⬘V兩 4 冑r
|
f=180°
(3)
r→0
where E⬘ = E in plane stress state and E⬘ = E / (1⫺n2) in plane strain state. For general 3D state, it can be obtained by using a constraint factor [10] that E⬘ = E / (1⫺Tzn), where Tz = szz / (sxx + syy) changes between 0 and the Poisson’s ratio n. In this paper, Tz = 0 is assumed at the free surface and Tz = n is assumed for the interior points for convenience. The possible error caused by this assumption will be less than 11% when n = 0.3.
Fig. 3. Finite element model and element mesh.
3. Results of analysis 3.1. Validity of the SIF calculation Fig. 4.
Local coordinates at elliptical crack front.
given by the r-1/2 singular solution is valid for the surface crack [8,9], the displacement perpendicular to the crack surface can be written as V⫽
(1 ⫹ n)KI 2E
冪2π冋(2l ⫹ 1) sin2⫺sin 2 册, r
f
3f
(1)
To check the validity of the above method for SIF evaluation by the FE model, comparison between the calculated SIF results of surface cracks in straight round bar and the solution obtained by Li and Chen [5] is made in Fig. 5. In the figure g is the relative location of the interesting point P on the crack front, which is expressed as the ratio of s to half of the length of the whole crack
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SIF of surface cracks in notched bars is dominated by the stress concentration coefficient Kt, and it seems to be independent of notch geometry, at least in the range of analysis. 3.3. Influence of Kt on SIFs of surface crack of round bar
Fig. 5.
Normalized SIFs of surface crack of straight bar.
front arc S as shown in Fig. 2(c), and Y is the normalized SIF which will be defined in Eq. (4). The aspect ratio of the surface crack is a / c = 0.792 in the figure. It is shown that the present FE results of SIF are in pretty good coincidence with that of Ref. [5]. 3.2. Influence of notch geometry on SIFs of surface cracked round bars The SIFs for cracks emanating from the semi-circular notch, V-notch and U-notch which have the same stress concentration coefficient Kt = 2.5 are drawn together in Fig. 6 as a function of crack depth ratio a/d at different locations along the crack border. Although the notch geometry changes significantly, the corresponding change in SIFs is less than 10%, nearly within the scatter of the FE results. Therefore, it can be concluded that the
The influences of stress concentration coefficient upon the SIFs for surface crack with different depth ratio a/d are shown in Fig. 7. It can be seen that the influence of Kt is very strong for small surface crack as shown in Fig. 7(a) and becomes weaker as crack growth, see Fig. 7(b)–(c). For small surface crack with depth a / d = 0.08, Fig. 7(a) shows that the SIF increases in the whole crack border with increasing Kt, but for crack with a / d = 0.333, the influence of Kt on SIF in the interior of the surface crack becomes very weak(Fig. 7(c)). For larger crack, the SIF in the interior of the crack decreases with increasing Kt when Kt ⬍ 2.5, and remains unchanged when Ktⱖ2.5, see Fig. 7(d). Near the intersection of the crack front with the notch root, the SIF is always a increasing function of Kt. The influence of Kt on average SIF along the whole crack front is shown in Fig. 8. It is obvious that the overall influence of Kt on the SIF decreases as crack grows. This is easy to understand as stress concentration is a local effect near the notch root. It is well-known that, under the same fatigue loading, the bar with greater Kt always has shorter life, and the life of small crack growth often accounts for more than 70–80% of the whole life of the structure. So the strong influence of Kt on the average SIF of small surface crack of round bar is worthy of attention. 3.4. SIF analysis for surface cracks at stress concentrators For each of the cases 1, 4, 5 and 6 in Table 1, four crack depth ratios are adopted: a = a / d = 0.08, 0.1733, 0.3333 and 0.48. For each a/d, three aspect ratios b = a / c = 0.6114, 0.792 and 0.9118 are calculated as many experiments yield crack aspect ratio within this range [8,11]. As the SIF is nearly independent of notch geometry for a given Kt, the expression of the SIF for notched bars can be written as a function of Kt and crack geometry and size: KI(a,b,g,Kt) ⫽ Y(a,b,g,Kt)s冑πa / 冑⌽,
Fig. 6.
Influence of notch geometry on SIFs for Kt = 2.5.
(4)
where s is the nominal stress on the minimum cross-section of the bar without a crack ⌽ = 1+1.464(a/ c)1.65.
W. Guo et al. / International Journal of Fatigue 25 (2003) 733–741
Fig. 7.
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Influence of Kt on SIFs of surface cracked annularly notched round bars.
Fitting all of the FE results of SIFs, it can be obtained that A(x) ⫽ 2.4125 ⫹ 0.5502x⫺0.1725x2 ⫹ 0.0169x3, B(x) ⫽ ⫺1.4832⫺4.9667x ⫹ 1.5739x2⫺0.14713x3, C(x) ⫽ ⫺3.8407 ⫹ 0.1951x⫺0.0338x2 ⫹ 0.0018x3, D(x) ⫽ 1.5336⫺1.6162x ⫹ 0.5256x2⫺0.0537x3, E(x) ⫽ ⫺17.1369 ⫹ 25.0059x⫺8.0307x2 ⫹ 0.7747x3, F(x) ⫽ ⫺1.8702 ⫹ 0.0154x ⫹ 0.0059x2⫺0.0013x3, G(x) ⫽ ⫺2.8833 ⫹ 3.5737x⫺1.5945x2 ⫹ 0.1864x3, Fig. 8. Influence of Kt on average SIFs along the surface crack front.
H(x) ⫽ 24.5604⫺6.0402x ⫹ 1.6236x2⫺0.1508x3, I(x) ⫽ ⫺8.8774 ⫹ 1.6565x⫺0.0944x2⫺0.0288x3,
The normalized SIF Y(a, b, g, Kt) is related to crack geometry and the theoretical stress concentration coefficient Kt of the specimen. It is assumed that
J(x) ⫽ 0.4757 ⫹ 2.3452x⫺0.569x2 ⫹ 0.0479x3,
Y(a,b,g,Kt) ⫽ A(Kt) ⫹ B(Kt)a ⫹ C(Kt)b ⫹ D(Kt)g
L(x) ⫽ 17.346⫺7.3113x ⫹ 1.1086x2⫺0.0387x3, M(x) ⫽ ⫺16.2690 ⫹ 4.6985x⫺1.5545x2 ⫹ 0.1610x3,
⫹ E(Kt)a2 ⫹ F(Kt)b2 ⫹ G(Kt)g2 ⫹ H(Kt)ab ⫹ I(Kt)ag ⫹ J(Kt)bg ⫹ K(Kt)a2b ⫹ L(Kt)a2g
K(x) ⫽ ⫺6.2255⫺2.7665x ⫹ 1.9194x2⫺0.2466x3,
(5)
N(x) ⫽ ⫺0.4188⫺1.8031x ⫹ 0.5777x2⫺0.0538x3,
⫹ M(Kt)b2a ⫹ N(Kt)b2g ⫹ O(Kt)g2a ⫹ P(Kt)g2b
O(x) ⫽ 1.6233 ⫹ 3.1095x⫺0.7301x2 ⫹ 0.0758x3,
⫹ Q(Kt)a3 ⫹ R(Kt)b3 ⫹ S(Kt)g3.
P(x) ⫽ 0.5967⫺0.8797x ⫹ 0.0972x2⫺0.0069x3,
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Q(x) ⫽ 28.2456⫺27.7861x ⫹ 8.1856x2⫺0.7485x3, R(x) ⫽ 3.5086⫺0.0646x⫺0.0028x ⫹ 0.0014x , 2
2
S(x) ⫽ 0.6139⫺1.3383x ⫹ 0.8547x ⫺0.1058x . 2
3
Eq. (5) is also drawn in Fig. 7(a) in dashed lines. Fig. 9 provides further comparison between the fitting equation and the FE results for surface cracks with smaller aspect ratio a / c = 0.6114 and 0.792 for V-notch of Kt = 4. The discrepancy between the FE results and Eq. (5) is less than 8% in all the cases studied in this paper. For an elliptical surface crack in a straight round bar with Kt = 1, Eq. (5) can be simplified to the following expression: s冑πa , K ⫽ Y(a,b,g) 冑⌽ Y(a,b,g) ⫽ 2.8070⫺5.0233a⫺3.6777b
Fig. 9. SIFs of ‘V’-notched bars with Kt = 4.
⫹ 0.3893g ⫹ 0.6130a2⫺1.8501b2 ⫺0.7178g2 ⫹ 19.9930ab⫺7.3442ag⫹ 2.3000bg (6) ⫺7.3193a2b ⫹ 11.1045a2g⫺12.964b2a ⫺1.6981b2g ⫹ 4.0786g2a ⫺0.1928g2b ⫹ 7.8967a3 ⫹ 3.4425b3 ⫹ 0.0246g3. 4. Application to cracked screw bolts Screw bolts are the most important mechanical components. When the pitch of the screw is small compared to the diameter of the bolt, space axisymmetrical FE model as shown in Fig. 10 can be used to obtain the theoretical stress concentration coefficients Kt. The geometry and size of the screw bolt simulated here are taken from the standard in Ref. [11] for a coarse plain thread as SIFs of the same bolt have been measured experimentally in the literature [12]. We also calculate Kt for other series of threads, such as trapezoidal threads and saw-tooth threads as shown in Table 2. In the FE models, the radius of the guide corners of the coarse plain threads, trapezoidal threads and saw-tooth threads are taken as 0.13, 0.09 and 0.0124p respectively, where p is the pitch of the screws [11]. The calculated theoretical stress concentration coefficients are listed in Table 2. Calculated results show that the theoretical stress concentration coefficients of these threads are distributed over a range of 1.9–2.92. For the coarse plain thread M22 × 1.5 which will be discussed in the following,
Fig. 10. Axis-symmetric FE model to calculate stress concentration coefficient of screw.
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Table 2 Theoretical stress concentration coefficients of tensile screw bolts Coarse plain thread Trapezoidal thread Saw-tooth threads (3°, 30°)
Type Kt Type Kt Type Kt
M4 2.60 Tr10 × 2 1.91 B10 × 2 2.10
M8 2.92 Tr14 × 2 2.05 B14 × 2 2.23
M18 2.23 Tr18 × 2 2.16 B18 × 2 2.37
Kt = 2.48. It is near the average value of all the threads in Table 2. Serial fatigue experimental investigations on M22 × 1.5 coarse plain thread bolts made of 20CrMo steel have been conducted by Wei and Zhong [12]. By using the James-Anderson method (JAM), they evaluated the SIFs of surface cracks in the bolts. However, as the strong influence of stress state upon fatigue crack growth has not been taken into account in their method, the SIFs near the free surface are significantly underestimated. In the experimental investigation [12,13], 15 × 30 mm2 three-point-bending (TPB) specimens are used to produce the crack growth rate to SIF range curve da / dN–⌬K. The da / dN–⌬K curve is used directly to evaluate SIFs of surface cracks in the cracked screw bolt just as the stress ratio is the same. In fact, the stress state at the center of the surface crack is dominated by plane strain state and is close to that of the TPB specimen, but at the free surface, plane stress state takes precedence and the da / dN–⌬K relation obtained from the TPB specimen cannot be used without modification. To overcome this difficulty, a modified JAM was developed by Shen and Guo [14] by using the 3D crack closure theory. The capability of the modified JAM has been proved by 3D numerical and experimental data of surface cracks in round bars. By using the modified JAM, SIFs of the cracked 20CrMo steel threads are reevaluated and the results are listed in Table 3. For comparison, SIFs given by Eq. (5) with Kt = 2.48 are presented in Table 3 as
M22 2.63 Tr22 × 3 2.23 B22 × 3 2.27
M22 × 1.5 2.48 Tr22 × 5 2.32 B22 × 5 2.06
M36 2.30 Tr36 × 3 2.40 B36 × 3 2.54
M52 2.79 Tr52 × 3 2.57 B52 × 3 2.78
well, where, Y = Y(a,b,g,Kt)兩Kt = 2.48,g = 0 for the center points and Y = Y(a,b,g,Kt)兩Kt = 2.48,g = 1 for the surface points. From Table 3 it can be seen that the SIF results given by the empirical Eq. (5) agree well with the experimental data by the modified JAM. The discrepancy between the experimental data and Eq. (5) is less than 9.7% at the center of the crack and 22% at the surface point. A very important reason for the large discrepancy at the surface point is the undetermined growth of the fatigue crack. It has been found in Ref. [12] that the crack aspect ratio a/c does not decrease linearly with increasing a/d as an overall tendency. But it can be seen from Table 3 that under the same test condition, at some points a/c is abnormally high and just at these points the experimental data obtained by the modified JAM drop suddenly and cause large discrepancy with Eq. (5). This abnormality can be shown more clearly in Fig. 11. Except these abnormal experimental data, the discrepancy is less than 13.5%.
5. Conclusions SIFs of surface cracks in notched round bars with a large range of stress concentration coefficient and notch geometry are studied systemically by using the 3D FE method with 20-node singular elements arranged around the crack border. From the numerical results, an empiri-
Table 3 SIFs of cracked screw thread bolt M22 × 1.5 under tension a/d
0.1675 0.2105 0.2505 0.251 0.293 0.3145 0.3245 0.362 0.3795 0.413
a/c
0.764 0.706 0.688 0.712 0.687 0.725 0.687 0.669 0.683 0.637
Normalized SIFs at center (g = 0)
Normalized SIFs at surface (g = 1)
Experimental Y
Eq. (5)
Error (%)
Experimental Y
Eq. (5)
Error (%)
0.836 0.977 1.094 1.064 1.214 1.255 1.301 1.394 1.398 1.523
0.817 0.961 1.045 1.009 1.117 1.19 1.175 1.274 1.292 1.416
2.3 1.8 4.5 5.2 7.9 5.2 9.7 8.6 7.6 7.1
1.345 1.687 1.911 1.583 2.056 1.893 2.223 2.306 2.398 2.62
1.494 1.779 1.983 1.934 2.19 2.224 2.361 2.617 2.702 3.004
11.1 5.4 3.7 22 6.5 17 6.2 13.4 12.6 14.7
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Acknowledgements This work is supported by the National Science Foundation of China (No. 50275073), the National Distinguished Young Scientist Fund of China, the Cheung Kong Scholars Programme, and the Aeronautical Fund of China (No. 01B52012).
References
Fig. 11. Comparison of Eq. (5) with experimental results yielded by the modified JAM.
cal formula of engineering interest for the SIF of surface cracks in notched bars is obtained. Comparison with available numerical as well as experimental results is made to validate the present results. The following conclusions can be drawn from the study. 1. The 3D FE model with singular 20-node elements arranged around the crack border is effective to yield reliable SIFs for surface cracks at notches in round bars. By using the 3D FE model, surface cracks with depth ratio a / d = 0.08–0.48 and aspect ratio a / c = 0.61–0.91 emanating from smooth bars, semi-circular notched bars and V- and U-notched bars with a range of theoretical stress concentration coefficients Kt = 1–5 are analyzed and the corresponding SIFs are obtained. 2. It is shown that the SIFs are strongly influenced by the theoretical stress concentration coefficient Kt, especially near the notch root. However, the SIFs are nearly independent of the notch geometry for given theoretical stress concentration coefficient Kt in this paper. 3. An empirical expression for the SIFs as a function of crack geometry and Kt is obtained by fitting the numerical results. Direct application of the empirical expression of SIFs to screw bolts in engineering yields good coincidence with the experimental results obtained by using a modified JAM from fatigue crack growth data. Therefore, the empirical expression can be used conveniently in life prediction of notched bars with various notch geometry and stress concentration coefficients at least within the range of parameters studied in this work.
[1] Nishioka T, Alturi SN. Analytical solution for elliptical cracks and finite element alternating method for elliptical surface cracks subjected to arbitrary loadings. Engineering Fracture Mechanics 1983;17:247–68. [2] Shivakumar KN, Newman Jr JC. Stress intensity factors for large aspect ratio surface and corner cracks at a semicircular notch in a tension specimen. Engineering Fracture Mechanics 1991;38:467–73. [3] Chen H, Li Z. Stress intensity factor of sucker rods by using the domain-decomposition and the finite element method. Journal of Engineering Mathematics 1994;11:81–6 [in Chinese]. [4] James LA, Anderson WE. A simple experimental procedure for stress intensity factor calibration. Engineering Fracture Mechanics 1969;1:565–8. [5] Li Z, Chen H. The 3D finite analysis of stress intensity factor of elliptical surface crack in sucker rods. China Petroleum Machinery 1994;22:20–4 [in Chinese]. [6] Heywood RB. In: Designing against fatigue. London: Chapman and Hall; 1962. p. 114–26. [7] Aviation Material Institution of China. Handbook of fatigue properties of aircraft metal material. Peking: Aviation Material Institution of China, 1981 [in Chinese]. [8] Lin XB. Fatigue growth simulation for cracks in notched and unnotched round bars. International Journal of Mechanical Science 1998;40:405–19. [9] Stanley P. In: Fracture mechanics in engineering practice. London: Applied Science Publishers; 1977. p. 62. [10] Guo W, Wang C, Rose F. The influence of cross-sectional thickness on fatigue crack growth. Fatigue and Fracture of Engineering Materials and Structures 1999;22:437–44. [11] Wu Z. In: Practical handbook for mechanism design. Peking: Chemical Industry Press; 1999. p. 456–8 [in Chinese]. [12] Wei J, Zhong B. Experimental research on stress intensity factor for surface crack at external thread root of threaded connection. Chinese Journal of Mechanical Engineering 1994;30:37–43 [in Chinese]. [13] Wang G, Wei J. Probability analysis on crack propagation of sucker rod material. Journal of Daqing Petroleum Institute 1993;17:78–83 [in Chinese]. [14] Shen H, Guo W. Modified James-Anderson Method with 3d effects of fatigue crack propagation considered. Journal of Engineering Material and Technology, in press.
Further reading Reference is made to the literature in preparing the finite element model and the empirical expression (4) for fitting the calculated stress intensity factors. [15] Schijve J. Comparison between empirical and calculated stress intensity factors of hole edge cracks. Engineering Fracture Mechanics 1985;32:48–9.
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[16] Newman Jr JC. Stress intensity factor equations for crack in three-dimensional finite bodies. ASTM STP791; 1983. p. 238– 256. [17] Raju IS, Newman Jr JC. Stress intensity factors for internal and external surface cracks in cylindrical vessels. ASME Journal of Pressure Vessel Technology 1982;104:293–8. [18] Newman Jr JC, Raju IS. Stress intensity factor equation for in three-dimensional bodies subjected to tension and bending loads. In: Atluri SN, editor. Computational method in the mechanics of fracture. Amsterdam: Elsevier; 1986. p. 311–34. [19] Nuller B, Karapetian E, Kachanov M. On the stress intensity fac-
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tor for the elliptical crack. International Journal of Fracture 1998;92:17–20. [20] Fabrikant VI. Stress intensity factor for an external elliptical crack. International Journal of Solids and Structures 1987;23:465–7. [21] Barsoum RS. On the use of isoparametric finite elements in linear fracture mechanics. International Journal of Numerical Methods in Engineering 1976;10:25–37. [22] Lin XB, Smith RA. Stress intensity factors for semielliptical surface cracks in semicircularly notched tension plates. Journal of Strain Analysis 1997;32:229–36.
Stress intensity factors for elliptical surface cracks in round bars with different stress concentration coefficient W. Guo ∗, H. Shen, H. Li Department of Aircraft Engineering and Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China Received 15 July 2002; received in revised form 11 October 2002; accepted 22 January 2003
Abstract Stress intensity factors (SIFs) of elliptical surface cracks in notched tensile round bars are calculated by using three-dimensional finite element analysis (FEA) models with singular 20-node elements arranged around the crack tip. Systematic analyses are performed for semi-circular type, V-type and U-type annular notches with theoretical stress concentration coefficient Kt ranging from 1– 5. It is shown that the SIFs are strongly dependent on Kt, and the influence of notch geometry is negligibly weak for a given stress concentration coefficients in the analyzed range. An empirical expression for the SIF as a function of crack geometry and Kt is then obtained by fitting the numerical results. Comparison of present results with available numerical results in the literature shows good agreement. Application of the empirical expression to a screw bolt yields good coincidence with experimental results of SIFs by using a modified James-Anderson method. Therefore, the empirical expression of SIFs can be used conveniently in integrity assessment of various notched bars, at least in the analyzed range in this paper. 2003 Elsevier Science Ltd. All rights reserved. Keywords: Stress intensity factor; Surface crack; Stress concentration coefficient; Notch; Three-dimensional finite element analysis
1. Introduction Surface cracks emanating from stress concentrating locations are the most common phenomena of fatigue failure. Screw bolts, dimple rivets, bars with variable cross-section and so on are a category of cylindrical parts and components extensively used in engineering mechanisms. Surface fatigue cracks are frequently initiated in such components at the stress concentrating locations, then propagate into the interior of the parts and can cause final fracture abruptly. In addition, smooth and notched round bars have been used as standard specimens to obtain the fatigue property of materials for safe design and assessment. Some of the widely used round bars are shown in Fig. 1. Many numerical analyses, theoretical studies and experimental investigations have been conducted to obtain stress intensity factors (SIFs) for three-dimen-
∗
Corresponding author. Tel.: +86-25-4891370; fax: +86-4890513. E-mail address: [email protected] (W. Guo).
Fig. 1. bars.
Surface crack from stress concentrating location of round
sional (3D) cracked bodies. Explicit solutions or empirical expressions have been achieved for surface and corner cracks on smooth strips or straight round bars and at circular holes in finite thickness plates [1–5]. For surface
0142-1123/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0142-1123(03)00050-1
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cracks in round bars or bolts, SIFs are studied mainly for very low stress concentration (with stress concentration coefficient Kt⬵1) and a few typical screws with relatively high stress concentration. However, SIFs for 3D cracks emanating from general stress concentrating locations are very limited. To obtain universal SIF formula of surface cracks in notched bodies with different notch geometry and Kt is significant for application of 3D fracture mechanics in engineering. The purpose of the present paper is to investigate the dependence of SIFs of surface cracks in notched round bars upon stress concentration coefficient and notch geometry. Three-dimensional finite element (FE) models with 20-node singular elements arranged around the crack tip will be used to calculate the SIFs of elliptical surface cracks in round bars with different notches and theoretical stress concentration coefficients. A wide range of notch and crack geometry parameters are considered in the numerical analysis to yield empirical formula for the SIFs of 3D notched bars of engineering interest. Comparison with available numerical as well as experimental results is made to show the effectiveness of the present results.
2. Modeling of the problem 2.1. Notch geometry and crack configuration In order to investigate the influences of notch geometry and theoretical stress concentration coefficient upon the SIFs of surface cracks at the root of notches, round bars without notch, with semicircular annular notch, U-type annular notch and V-type annular notch of 60° opening angle are considered in the analyses as shown in Fig. 2. Semi-elliptical surface cracks are arranged on the minimum cross-section of the notched bar as shown in Fig. 2(b) and (c). Six sets of geometric parameters are assumed as listed in Table 1. Theoretical stress concentration coefficients of the notched bars without a crack in the six cases can be found to be 1, 2.5, 2.5, 2.5, 4 and 5, respectively [6,7]. For convenience,
Fig. 2.
the diameters d of the minimum cross-section in all the investigated cases are set at 7.5 mm. The main purpose for choosing the straight round bar (case 1 in Table 1) is to examine the validity of the algorithms for calculating SIFs in this work by comparison with those in Ref. [5] and provide a keynote to developing the empirical expression of SIFs for notched bars. Different notch geometries in cases 2, 3 and 4 with the same Kt = 2.5 are arranged to show the effect of notch geometry on SIFs. The surface crack has a semi-elliptical geometry with semi-major axis c and semi-minor axis a as shown in Fig. 2(c), where a is also the depth of surface crack. The length of the arc s measured from the center L of the crack front to any point P on the crack front is defined to describe the position considered. 2.2. Finite element modeling As symmetry, only a quarter of each specimen is considered in the FE model, as shown in Fig. 3. Twentynode hexahedron elements are used in the FE model to provide high precision of the notch root radius and the crack shape. In order to simulate the theoretical inverse square root singularity of stress and strains near the crack border, singular elements with mid-side node relocated to their 1/4-point position are arranged around the crack front, as shown by the amplified region B in Fig. 3. The 20-node hexahedron element and the singular element come to being by virtue of the FE software ansys. A total about 9000 elements and 130,000 freedoms are used in the modeling. 2.3. Calculation of the SIFs For a surface crack in a round bar as shown in Fig. 4, a local coordinate set (X,Y) is defined at any interesting point P on the crack front to calculate the SIF at the point P. With coordinate origin at P, the Y axis is perpendicular to the crack surface at P, the X axis is perpendicular to the crack front at P and the Y axis, and (r, f) are the polar coordinates in the X–Y plane. Assuming the expression of crack opening displacement
Dimensions of round bar specimens and elliptical surface crack.
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Table 1 Dimensions and theoretical stress concentration coefficients of specimens Specimen number
Case Case Case Case Case Case
Notch type
1 2 3 4 5 6
Without notch Semi-circular annular notch ‘U’ annular notch ‘V’ annular notch ‘V’ annular notch ‘V’ annular notch
Dimension of specimen (mm) D d
r
Stress concentration coefficient Kt
– 8.532 12 12 12 8.969
– 0.516 0.79 0.39 0.26 0.135
1 2.5 2.5 2.5 4 5
7.5 7.5 7.5 7.5 7.5 7.5
where n is Poisson’s ratio, E the elastic modulus and KI is the SIF at point P. λ is a function of Poisson’s ratio and stress state which will be reflected by E⬘ in Eq. (3). According to Eq. (1), the opening displacement V of the semi-elliptical crack surface of a round bar under remote tensile stress s can be obtained as V兩f=180° ⫽
(1 ⫹ n)(l ⫹ 1)KI E
冪2π. r
(2)
When V兩f = 180° is provided by the 3D FE analysis, Eq. (2) leads to the relationship between KI and r. Fitting the linear part of the KI–r curve and extrapolating the line to r = 0, SIF at the point on the surface crack border can then be obtained [9], KI ⫽
冑2πE⬘V兩 4 冑r
|
f=180°
(3)
r→0
where E⬘ = E in plane stress state and E⬘ = E / (1⫺n2) in plane strain state. For general 3D state, it can be obtained by using a constraint factor [10] that E⬘ = E / (1⫺Tzn), where Tz = szz / (sxx + syy) changes between 0 and the Poisson’s ratio n. In this paper, Tz = 0 is assumed at the free surface and Tz = n is assumed for the interior points for convenience. The possible error caused by this assumption will be less than 11% when n = 0.3.
Fig. 3. Finite element model and element mesh.
3. Results of analysis 3.1. Validity of the SIF calculation Fig. 4.
Local coordinates at elliptical crack front.
given by the r-1/2 singular solution is valid for the surface crack [8,9], the displacement perpendicular to the crack surface can be written as V⫽
(1 ⫹ n)KI 2E
冪2π冋(2l ⫹ 1) sin2⫺sin 2 册, r
f
3f
(1)
To check the validity of the above method for SIF evaluation by the FE model, comparison between the calculated SIF results of surface cracks in straight round bar and the solution obtained by Li and Chen [5] is made in Fig. 5. In the figure g is the relative location of the interesting point P on the crack front, which is expressed as the ratio of s to half of the length of the whole crack
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SIF of surface cracks in notched bars is dominated by the stress concentration coefficient Kt, and it seems to be independent of notch geometry, at least in the range of analysis. 3.3. Influence of Kt on SIFs of surface crack of round bar
Fig. 5.
Normalized SIFs of surface crack of straight bar.
front arc S as shown in Fig. 2(c), and Y is the normalized SIF which will be defined in Eq. (4). The aspect ratio of the surface crack is a / c = 0.792 in the figure. It is shown that the present FE results of SIF are in pretty good coincidence with that of Ref. [5]. 3.2. Influence of notch geometry on SIFs of surface cracked round bars The SIFs for cracks emanating from the semi-circular notch, V-notch and U-notch which have the same stress concentration coefficient Kt = 2.5 are drawn together in Fig. 6 as a function of crack depth ratio a/d at different locations along the crack border. Although the notch geometry changes significantly, the corresponding change in SIFs is less than 10%, nearly within the scatter of the FE results. Therefore, it can be concluded that the
The influences of stress concentration coefficient upon the SIFs for surface crack with different depth ratio a/d are shown in Fig. 7. It can be seen that the influence of Kt is very strong for small surface crack as shown in Fig. 7(a) and becomes weaker as crack growth, see Fig. 7(b)–(c). For small surface crack with depth a / d = 0.08, Fig. 7(a) shows that the SIF increases in the whole crack border with increasing Kt, but for crack with a / d = 0.333, the influence of Kt on SIF in the interior of the surface crack becomes very weak(Fig. 7(c)). For larger crack, the SIF in the interior of the crack decreases with increasing Kt when Kt ⬍ 2.5, and remains unchanged when Ktⱖ2.5, see Fig. 7(d). Near the intersection of the crack front with the notch root, the SIF is always a increasing function of Kt. The influence of Kt on average SIF along the whole crack front is shown in Fig. 8. It is obvious that the overall influence of Kt on the SIF decreases as crack grows. This is easy to understand as stress concentration is a local effect near the notch root. It is well-known that, under the same fatigue loading, the bar with greater Kt always has shorter life, and the life of small crack growth often accounts for more than 70–80% of the whole life of the structure. So the strong influence of Kt on the average SIF of small surface crack of round bar is worthy of attention. 3.4. SIF analysis for surface cracks at stress concentrators For each of the cases 1, 4, 5 and 6 in Table 1, four crack depth ratios are adopted: a = a / d = 0.08, 0.1733, 0.3333 and 0.48. For each a/d, three aspect ratios b = a / c = 0.6114, 0.792 and 0.9118 are calculated as many experiments yield crack aspect ratio within this range [8,11]. As the SIF is nearly independent of notch geometry for a given Kt, the expression of the SIF for notched bars can be written as a function of Kt and crack geometry and size: KI(a,b,g,Kt) ⫽ Y(a,b,g,Kt)s冑πa / 冑⌽,
Fig. 6.
Influence of notch geometry on SIFs for Kt = 2.5.
(4)
where s is the nominal stress on the minimum cross-section of the bar without a crack ⌽ = 1+1.464(a/ c)1.65.
W. Guo et al. / International Journal of Fatigue 25 (2003) 733–741
Fig. 7.
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Influence of Kt on SIFs of surface cracked annularly notched round bars.
Fitting all of the FE results of SIFs, it can be obtained that A(x) ⫽ 2.4125 ⫹ 0.5502x⫺0.1725x2 ⫹ 0.0169x3, B(x) ⫽ ⫺1.4832⫺4.9667x ⫹ 1.5739x2⫺0.14713x3, C(x) ⫽ ⫺3.8407 ⫹ 0.1951x⫺0.0338x2 ⫹ 0.0018x3, D(x) ⫽ 1.5336⫺1.6162x ⫹ 0.5256x2⫺0.0537x3, E(x) ⫽ ⫺17.1369 ⫹ 25.0059x⫺8.0307x2 ⫹ 0.7747x3, F(x) ⫽ ⫺1.8702 ⫹ 0.0154x ⫹ 0.0059x2⫺0.0013x3, G(x) ⫽ ⫺2.8833 ⫹ 3.5737x⫺1.5945x2 ⫹ 0.1864x3, Fig. 8. Influence of Kt on average SIFs along the surface crack front.
H(x) ⫽ 24.5604⫺6.0402x ⫹ 1.6236x2⫺0.1508x3, I(x) ⫽ ⫺8.8774 ⫹ 1.6565x⫺0.0944x2⫺0.0288x3,
The normalized SIF Y(a, b, g, Kt) is related to crack geometry and the theoretical stress concentration coefficient Kt of the specimen. It is assumed that
J(x) ⫽ 0.4757 ⫹ 2.3452x⫺0.569x2 ⫹ 0.0479x3,
Y(a,b,g,Kt) ⫽ A(Kt) ⫹ B(Kt)a ⫹ C(Kt)b ⫹ D(Kt)g
L(x) ⫽ 17.346⫺7.3113x ⫹ 1.1086x2⫺0.0387x3, M(x) ⫽ ⫺16.2690 ⫹ 4.6985x⫺1.5545x2 ⫹ 0.1610x3,
⫹ E(Kt)a2 ⫹ F(Kt)b2 ⫹ G(Kt)g2 ⫹ H(Kt)ab ⫹ I(Kt)ag ⫹ J(Kt)bg ⫹ K(Kt)a2b ⫹ L(Kt)a2g
K(x) ⫽ ⫺6.2255⫺2.7665x ⫹ 1.9194x2⫺0.2466x3,
(5)
N(x) ⫽ ⫺0.4188⫺1.8031x ⫹ 0.5777x2⫺0.0538x3,
⫹ M(Kt)b2a ⫹ N(Kt)b2g ⫹ O(Kt)g2a ⫹ P(Kt)g2b
O(x) ⫽ 1.6233 ⫹ 3.1095x⫺0.7301x2 ⫹ 0.0758x3,
⫹ Q(Kt)a3 ⫹ R(Kt)b3 ⫹ S(Kt)g3.
P(x) ⫽ 0.5967⫺0.8797x ⫹ 0.0972x2⫺0.0069x3,
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Q(x) ⫽ 28.2456⫺27.7861x ⫹ 8.1856x2⫺0.7485x3, R(x) ⫽ 3.5086⫺0.0646x⫺0.0028x ⫹ 0.0014x , 2
2
S(x) ⫽ 0.6139⫺1.3383x ⫹ 0.8547x ⫺0.1058x . 2
3
Eq. (5) is also drawn in Fig. 7(a) in dashed lines. Fig. 9 provides further comparison between the fitting equation and the FE results for surface cracks with smaller aspect ratio a / c = 0.6114 and 0.792 for V-notch of Kt = 4. The discrepancy between the FE results and Eq. (5) is less than 8% in all the cases studied in this paper. For an elliptical surface crack in a straight round bar with Kt = 1, Eq. (5) can be simplified to the following expression: s冑πa , K ⫽ Y(a,b,g) 冑⌽ Y(a,b,g) ⫽ 2.8070⫺5.0233a⫺3.6777b
Fig. 9. SIFs of ‘V’-notched bars with Kt = 4.
⫹ 0.3893g ⫹ 0.6130a2⫺1.8501b2 ⫺0.7178g2 ⫹ 19.9930ab⫺7.3442ag⫹ 2.3000bg (6) ⫺7.3193a2b ⫹ 11.1045a2g⫺12.964b2a ⫺1.6981b2g ⫹ 4.0786g2a ⫺0.1928g2b ⫹ 7.8967a3 ⫹ 3.4425b3 ⫹ 0.0246g3. 4. Application to cracked screw bolts Screw bolts are the most important mechanical components. When the pitch of the screw is small compared to the diameter of the bolt, space axisymmetrical FE model as shown in Fig. 10 can be used to obtain the theoretical stress concentration coefficients Kt. The geometry and size of the screw bolt simulated here are taken from the standard in Ref. [11] for a coarse plain thread as SIFs of the same bolt have been measured experimentally in the literature [12]. We also calculate Kt for other series of threads, such as trapezoidal threads and saw-tooth threads as shown in Table 2. In the FE models, the radius of the guide corners of the coarse plain threads, trapezoidal threads and saw-tooth threads are taken as 0.13, 0.09 and 0.0124p respectively, where p is the pitch of the screws [11]. The calculated theoretical stress concentration coefficients are listed in Table 2. Calculated results show that the theoretical stress concentration coefficients of these threads are distributed over a range of 1.9–2.92. For the coarse plain thread M22 × 1.5 which will be discussed in the following,
Fig. 10. Axis-symmetric FE model to calculate stress concentration coefficient of screw.
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Table 2 Theoretical stress concentration coefficients of tensile screw bolts Coarse plain thread Trapezoidal thread Saw-tooth threads (3°, 30°)
Type Kt Type Kt Type Kt
M4 2.60 Tr10 × 2 1.91 B10 × 2 2.10
M8 2.92 Tr14 × 2 2.05 B14 × 2 2.23
M18 2.23 Tr18 × 2 2.16 B18 × 2 2.37
Kt = 2.48. It is near the average value of all the threads in Table 2. Serial fatigue experimental investigations on M22 × 1.5 coarse plain thread bolts made of 20CrMo steel have been conducted by Wei and Zhong [12]. By using the James-Anderson method (JAM), they evaluated the SIFs of surface cracks in the bolts. However, as the strong influence of stress state upon fatigue crack growth has not been taken into account in their method, the SIFs near the free surface are significantly underestimated. In the experimental investigation [12,13], 15 × 30 mm2 three-point-bending (TPB) specimens are used to produce the crack growth rate to SIF range curve da / dN–⌬K. The da / dN–⌬K curve is used directly to evaluate SIFs of surface cracks in the cracked screw bolt just as the stress ratio is the same. In fact, the stress state at the center of the surface crack is dominated by plane strain state and is close to that of the TPB specimen, but at the free surface, plane stress state takes precedence and the da / dN–⌬K relation obtained from the TPB specimen cannot be used without modification. To overcome this difficulty, a modified JAM was developed by Shen and Guo [14] by using the 3D crack closure theory. The capability of the modified JAM has been proved by 3D numerical and experimental data of surface cracks in round bars. By using the modified JAM, SIFs of the cracked 20CrMo steel threads are reevaluated and the results are listed in Table 3. For comparison, SIFs given by Eq. (5) with Kt = 2.48 are presented in Table 3 as
M22 2.63 Tr22 × 3 2.23 B22 × 3 2.27
M22 × 1.5 2.48 Tr22 × 5 2.32 B22 × 5 2.06
M36 2.30 Tr36 × 3 2.40 B36 × 3 2.54
M52 2.79 Tr52 × 3 2.57 B52 × 3 2.78
well, where, Y = Y(a,b,g,Kt)兩Kt = 2.48,g = 0 for the center points and Y = Y(a,b,g,Kt)兩Kt = 2.48,g = 1 for the surface points. From Table 3 it can be seen that the SIF results given by the empirical Eq. (5) agree well with the experimental data by the modified JAM. The discrepancy between the experimental data and Eq. (5) is less than 9.7% at the center of the crack and 22% at the surface point. A very important reason for the large discrepancy at the surface point is the undetermined growth of the fatigue crack. It has been found in Ref. [12] that the crack aspect ratio a/c does not decrease linearly with increasing a/d as an overall tendency. But it can be seen from Table 3 that under the same test condition, at some points a/c is abnormally high and just at these points the experimental data obtained by the modified JAM drop suddenly and cause large discrepancy with Eq. (5). This abnormality can be shown more clearly in Fig. 11. Except these abnormal experimental data, the discrepancy is less than 13.5%.
5. Conclusions SIFs of surface cracks in notched round bars with a large range of stress concentration coefficient and notch geometry are studied systemically by using the 3D FE method with 20-node singular elements arranged around the crack border. From the numerical results, an empiri-
Table 3 SIFs of cracked screw thread bolt M22 × 1.5 under tension a/d
0.1675 0.2105 0.2505 0.251 0.293 0.3145 0.3245 0.362 0.3795 0.413
a/c
0.764 0.706 0.688 0.712 0.687 0.725 0.687 0.669 0.683 0.637
Normalized SIFs at center (g = 0)
Normalized SIFs at surface (g = 1)
Experimental Y
Eq. (5)
Error (%)
Experimental Y
Eq. (5)
Error (%)
0.836 0.977 1.094 1.064 1.214 1.255 1.301 1.394 1.398 1.523
0.817 0.961 1.045 1.009 1.117 1.19 1.175 1.274 1.292 1.416
2.3 1.8 4.5 5.2 7.9 5.2 9.7 8.6 7.6 7.1
1.345 1.687 1.911 1.583 2.056 1.893 2.223 2.306 2.398 2.62
1.494 1.779 1.983 1.934 2.19 2.224 2.361 2.617 2.702 3.004
11.1 5.4 3.7 22 6.5 17 6.2 13.4 12.6 14.7
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Acknowledgements This work is supported by the National Science Foundation of China (No. 50275073), the National Distinguished Young Scientist Fund of China, the Cheung Kong Scholars Programme, and the Aeronautical Fund of China (No. 01B52012).
References
Fig. 11. Comparison of Eq. (5) with experimental results yielded by the modified JAM.
cal formula of engineering interest for the SIF of surface cracks in notched bars is obtained. Comparison with available numerical as well as experimental results is made to validate the present results. The following conclusions can be drawn from the study. 1. The 3D FE model with singular 20-node elements arranged around the crack border is effective to yield reliable SIFs for surface cracks at notches in round bars. By using the 3D FE model, surface cracks with depth ratio a / d = 0.08–0.48 and aspect ratio a / c = 0.61–0.91 emanating from smooth bars, semi-circular notched bars and V- and U-notched bars with a range of theoretical stress concentration coefficients Kt = 1–5 are analyzed and the corresponding SIFs are obtained. 2. It is shown that the SIFs are strongly influenced by the theoretical stress concentration coefficient Kt, especially near the notch root. However, the SIFs are nearly independent of the notch geometry for given theoretical stress concentration coefficient Kt in this paper. 3. An empirical expression for the SIFs as a function of crack geometry and Kt is obtained by fitting the numerical results. Direct application of the empirical expression of SIFs to screw bolts in engineering yields good coincidence with the experimental results obtained by using a modified JAM from fatigue crack growth data. Therefore, the empirical expression can be used conveniently in life prediction of notched bars with various notch geometry and stress concentration coefficients at least within the range of parameters studied in this work.
[1] Nishioka T, Alturi SN. Analytical solution for elliptical cracks and finite element alternating method for elliptical surface cracks subjected to arbitrary loadings. Engineering Fracture Mechanics 1983;17:247–68. [2] Shivakumar KN, Newman Jr JC. Stress intensity factors for large aspect ratio surface and corner cracks at a semicircular notch in a tension specimen. Engineering Fracture Mechanics 1991;38:467–73. [3] Chen H, Li Z. Stress intensity factor of sucker rods by using the domain-decomposition and the finite element method. Journal of Engineering Mathematics 1994;11:81–6 [in Chinese]. [4] James LA, Anderson WE. A simple experimental procedure for stress intensity factor calibration. Engineering Fracture Mechanics 1969;1:565–8. [5] Li Z, Chen H. The 3D finite analysis of stress intensity factor of elliptical surface crack in sucker rods. China Petroleum Machinery 1994;22:20–4 [in Chinese]. [6] Heywood RB. In: Designing against fatigue. London: Chapman and Hall; 1962. p. 114–26. [7] Aviation Material Institution of China. Handbook of fatigue properties of aircraft metal material. Peking: Aviation Material Institution of China, 1981 [in Chinese]. [8] Lin XB. Fatigue growth simulation for cracks in notched and unnotched round bars. International Journal of Mechanical Science 1998;40:405–19. [9] Stanley P. In: Fracture mechanics in engineering practice. London: Applied Science Publishers; 1977. p. 62. [10] Guo W, Wang C, Rose F. The influence of cross-sectional thickness on fatigue crack growth. Fatigue and Fracture of Engineering Materials and Structures 1999;22:437–44. [11] Wu Z. In: Practical handbook for mechanism design. Peking: Chemical Industry Press; 1999. p. 456–8 [in Chinese]. [12] Wei J, Zhong B. Experimental research on stress intensity factor for surface crack at external thread root of threaded connection. Chinese Journal of Mechanical Engineering 1994;30:37–43 [in Chinese]. [13] Wang G, Wei J. Probability analysis on crack propagation of sucker rod material. Journal of Daqing Petroleum Institute 1993;17:78–83 [in Chinese]. [14] Shen H, Guo W. Modified James-Anderson Method with 3d effects of fatigue crack propagation considered. Journal of Engineering Material and Technology, in press.
Further reading Reference is made to the literature in preparing the finite element model and the empirical expression (4) for fitting the calculated stress intensity factors. [15] Schijve J. Comparison between empirical and calculated stress intensity factors of hole edge cracks. Engineering Fracture Mechanics 1985;32:48–9.
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[16] Newman Jr JC. Stress intensity factor equations for crack in three-dimensional finite bodies. ASTM STP791; 1983. p. 238– 256. [17] Raju IS, Newman Jr JC. Stress intensity factors for internal and external surface cracks in cylindrical vessels. ASME Journal of Pressure Vessel Technology 1982;104:293–8. [18] Newman Jr JC, Raju IS. Stress intensity factor equation for in three-dimensional bodies subjected to tension and bending loads. In: Atluri SN, editor. Computational method in the mechanics of fracture. Amsterdam: Elsevier; 1986. p. 311–34. [19] Nuller B, Karapetian E, Kachanov M. On the stress intensity fac-
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tor for the elliptical crack. International Journal of Fracture 1998;92:17–20. [20] Fabrikant VI. Stress intensity factor for an external elliptical crack. International Journal of Solids and Structures 1987;23:465–7. [21] Barsoum RS. On the use of isoparametric finite elements in linear fracture mechanics. International Journal of Numerical Methods in Engineering 1976;10:25–37. [22] Lin XB, Smith RA. Stress intensity factors for semielliptical surface cracks in semicircularly notched tension plates. Journal of Strain Analysis 1997;32:229–36.