- Email: [email protected]
Notch size effects in the fatigue limit of steel
International Journal of Fatigue 25 (2003) 17–26 www.elsevier.com/locate/ijfatigue
Notch size effects in the fatigue limit of steel M. Makkonen ∗ Andritz Oy Wood Processing, Keskikankaantie 9, 15870 Hollola, Finland Received 20 December 2001; received in revised form 23 April 2002; accepted 13 May 2002
Abstract The fatigue behaviour of grooved test specimens has been investigated in this paper. It is shown that there are two kinds of size effects connected to notches: statistical size effect and geometric size effect. The statistical size effect is calculated based on the distribution of the maximum depth of initiated cracks in specimens with varying stress area. The geometric size effect depends on the stress gradient and can be estimated with the help of the linear elastic fracture mechanics. It is concluded that the notch size effect in blunt notches can be explained with those two factors. When the notch gets sharper, after certain limit the magnitude of the plastic portion of the strain starts to play an important role in the fatigue crack initiation, and the fatigue limit is lower than that predicted by statistical and geometric size effects. Another estimation method shall be used to that kind of notches. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue; Notch size effect; Statistical size effect
1. Introduction Most of the components used in machines have notches like shoulders and holes. The fatigue limit of this kind of specimen is higher than the maximum stress at the notch root, and would indicate. Many attempts have been made to evaluate this phenomenon, which is called the notch size effect. The aim has been to develop an equation, which gives the relationship between the observed fatigue limit and the expectable strength based on tests with smooth specimens. Many empirical formulas have been developed. A collection of the most commonly used methods are presented in Ref. [1]. The magnitude of the notch size effect is material dependent. This is explained with the so-called ‘notch sensitivity’ of the material. All the presented formulas include material parameters, which try to describe this, but no physical background for this phenomenon is given for them. All formulas show considerable scatter when applied to different materials. The goal in this paper is to show that the notch size effect can be explained with two factors: the statistical
size effect and the effect of the stress gradient, which is called the geometric size effect. The statistical size effect can be described as follows: when a component is subjected to an alternating load, a number of microcracks will initiate in its volume. The larger the specimen, the larger initiated cracks will be found. Thus, the higher probability of large initiated cracks appears as a lower fatigue limit for larger specimens. It was shown by Makkonen [1,2] that the size effect in plain specimens results from the statistical size effect alone. The geometric size effect comes into picture with notched specimens. The stress distribution in the vicinity of grooves, shoulders and other discontinuities becomes non-linear, and a high stress peak appears. The stress gradient is steeper in small equally shaped specimens. If equal-sized cracks are initiated in the peak stress area, the stress intensity factor at the crack is higher in a larger specimen. The other aim in this paper is to present a method for estimating the geometric size effect by comparing the stress intensity factors at the initiated crack for two cases: for the actual peak stress distribution and for a linear stress (tension or bending). 2. Statistical methods
Tel.: +358-3-880-3420; fax: +358-3-880-3200. E-mail address: [email protected] (M. Makkonen). ∗
It was shown by Makkonen [1,2] that the distribution of initiated crack depths can be assumed to follow the
0142-1123/02/$ - see front matter. 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 2 ) 0 0 0 5 3 - 1
18
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
Nomenclature a0 a c d dn e f(x) fXn:n(x) kg l0 n n⬘ r sx x xm y z A Ai Aeff D Do F(x) FXn:n(x) K K⬘ Kf KFATIGUE KI Kt Ke Ks m(x,a) PS R Rm s√a0 a b ee ep ep,lim etot g r s s(x) sa ⌬KI ⌬KI,th ⌬Kl ⌬Knl ⌬s ⌬snom ⌬sN ⌬sR,a ⌬sR,Kth ⌬sR,nom ⌬sR,St+G
initiated crack depth crack depth crack length diameter notch depth constant probability density function probability density function of the maximum value of sample geometric size factor material constant sample size cyclic hardening exponent radius standard deviation of a random variable random variable, co-ordinate mean value of a random variable co-ordinate co-ordinate surface area surface area of ith part of the surface effective stress area diameter original bar diameter cumulative distribution function cumulative distribution function of the maximum value of sample stress intensity factor cyclic hardening coefficient notch size factor fatigue concentration factor stress intensity factor stress concentration factor strain concentration factor real (Neuber) stress concentration factor weight function probability of survival stress ratio ultimate strength standard deviation of the initiated crack depth scatter parameter of the lognormal distribution geometry factor elastic strain plastic strain limit value of plastic strain total strain location parameter of the lognormal distribution notch root radius stress stress distribution function stress amplitude stress intensity range stress intensity range threshold stress intensity range from linear stress stress intensity range from non-linear stress stress cycle nominal stress cycle real (Neuber) stress cycle mean fatigue limit fatigue limit based on stress intensity range threshold fatigue limit based on nominal stress cycle fatigue limit calculated using statistical and geometric size effects
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
lognormal distribution. It has the following probability density function: f(x) ⫽
冉
1 ln(x)⫺ γ a a冑2πx 2 1
e
冊
2
(1)
crack in a sample can be defined with the help of Eqs. (1) and (3). The expected value and the standard deviation, respectively, shall be calculated from the following equations:
冕 ⬁
The distribution of the maximum value in a sample can be defined with the help of the order statistics [3]:
xm ⫽ xfXn:ndx
sx ⫽
- probability density function (pdf)
冘
(3)
j
log(PS(⌬si)) Ai log(0.5) i⫽2
(4)
Area A1 has the maximum stress. The term (PS(⌬s1)) presents the probability of survival of the area Ai at stress cycle ⌬si, when the area A1 has probability of survival of 0.5 for fatigue at stress level ⌬s1. When the stress distribution is continuous, the corresponding equation obtains the following form: Aeff ⫽
冉
冊冕
1 log(0.5)
log(PS(⌬s(x,y,z)))dA
(x⫺xm)2f(x)dx
(7)
0
F(x) and f(x) are the cdf and pdf of the parent population of the initiated cracks. Since Eq. (1) cannot integrated in the closed form, numerical methods must be used to solve Eqs. (2) and (3). The sample size n, the number of initiated cracks in the specimen, are needed for the above equations. When notched specimens are in questions, it is clear that the critical cracks will be located at the surface since the maximum stress is always there. Thus, the sample size is the number of cracks on the specimen surface. The exact number of initiated cracks per unit area cannot be known. It was shown in Ref. [2], that possible error in this value does not have significant influence to the calculated results. The value chosen in Ref. [2], 100 cracks per mm2, shall be used in this paper also; see Ref. [2] for further discussion on this matter. If the stress distribution is not constant, an effective stress area must be calculated. This can be done with the method presented by Bo¨ hm [4]. The idea is that, the areas with lower stress level are reduced so that the probability of failure with the calculated effective area with a constant stress is the same as in the original specimen. The effective stress area of a structure consisting of j parts, each of them having a different stress level, can be calculated from the formula: Aeff ⫽ A1 ⫹
冪冕 ⬁
(2)
fXn:n(x) ⫽ nFn⫺1(x)f(x)
(6)
0
- cumulative distribution function (cdf) FXn:n(x) ⫽ Fn(x)
19
(5)
A
The sample size is determined using the area calculated from Eq. (5). The distribution of the maximum
The required connection between the fatigue limit and the initiated crack depth can be established with help of the linear elastic fracture mechanics (LEFM): KI ⫽ bs冑πa0
(8)
Substituting KI with the stress intensity range threshold, ⌬KI,th, and s by the mean fatigue limit, ⌬sR,a, yields [5]:
冑a
0
⫽
冊 冑冉 ⌬KI,th b⌬s R,a π
1
(9)
Since the fatigue limit and the square root of the crack depth are in inverse proportion, the random variable in the above equations will be set x ⫽ 冑a0. The initial crack depth a0 cannot be measured in the experimental tests. Therefore, the mean fatigue limit and its standard deviation shall be determined instead. Eq. (9) is used to convert the results to crack depths. When the mean of the initial crack depth 冑a0and the standard deviation s冑a0 are known, the scatter parameter a and the location parameter g of the lognormal distribution can be defined. The expected maximum crack depth in any sample size can then be calculated from Eqs. (3) and (6). Finally, the expected mean fatigue limit of the target specimen is solved from Eq. (9). It shall be noted that all specimens in question must belong to the same parent population while doing this calculation. This condition is not necessarily fulfilled even if the specimens are made of the same material, see Refs. [1] and [2] for further discussion. The geometry factor, b, depends on the specimen shape, on the crack shape and on the stress distribution. It can be assumed that the crack shape tends to take the shape where the stress intensity factor over the whole crack front is constant.
3. Geometric size effect A schematic presentation of the effect of the stress gradient is given in Fig. 1. There are two shoulders, which have similar shape, but different size. If the nomi-
20
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
Fig. 1.
Stress gradients and cracks at shoulders with same Kt.
nal stress cycle is same in both specimens, the peak stress Kt⌬snom is also the same. Let us now assume that there is an initiated crack of depth a0 at the peak stress area of both notches. It can easily be seen that the stress cycle at the crack tip is higher in the larger specimen (⌬s2) than in the smaller one (⌬s1). This means, the stress intensity factor range also is higher in the larger specimen. Thus, it can be concluded, that the critical crack depth is smaller in the larger specimen. This leads to lower fatigue limit. This phenomenon is called the geometric size effect. It can be concluded that the effect of the stress gradient to the fatigue limit can be estimated by calculating the stress intensity factor range in both specimens. Similarly, comparison to smooth specimen is also possible. In fact, the geometric size effect can be estimated as follows: 1. the value of the stress intensity factor range, ⌬Knl, is calculated for the non-linear stress distribution using the assumed initiated crack depth a0; 2. the stress intensity factor range, ⌬Kl, is calculated for a constant stress Kt⌬snom; 3. the geometric size effect is obtained as the ratio kg ⫽ ⌬Knl / ⌬K1. The following things shall be noted: 앫 The crack size used in the calculation above is the depth a0 after the crack initiation phase. An estimate of this crack depth can be obtained from Eq. (9). 앫 The best assumption for the aspect ratio a/c is the one which gives constant stress intensity factor along the crack front. This means that the aspect ratio is different in the non-linear stress distribution and in the assumed constant stress field. 앫 There is a slight geometric size effect in a smooth specimen subjected to bending load. Anyway, the
stress intensity factors in bending and in constant tension vary so little that this can be neglected [1]. The calculation of the geometric size effect requires that the stress intensity factor for a non-linear stress can be calculated. Solutions for this kind of stress distributions are normally not available. Fortunately, values for very small cracks are needed, in the situation, where the initiated crack is about to change the mode to stable crack growth (Paris law). It can be shown that the influence of the specimen shape is negligible for very small cracks [1], when the crack size gets smaller and smaller, the stress intensity factor approaches the value for surface crack in a semi-infinite body. Therefore, solutions for surface cracks for plates can always be used. Wang and Lambert [6] have developed a calculation method, which is based on the use of weight functions. The stress intensity factor can be calculated with the integral:
冕 a
K ⫽ s(x)m(x,a)dx
(10)
0
Weight functions m(x,a) are given for the deepest point as well as for the surface point. This makes it possible to adjust the crack aspect ratio a/c so that the stress intensity factor is equal in both points. It must be possible to present the stress distribution in the form of a function s(x). The actual stress distribution can be defined e.g. by using the finite element method. A suitable function is then fitted to the calculated stress points. This function must be accurate from the specimen surface to the depth of a0. The reference value of the stress intensity factor for a surface crack in constant stress can be found in the paper of Isida et al. [7]. The stress intensity factor has same value in the deepest point and in the surface point with the aspect ratio of approximately a / c ⫽ 0.8.The geometry factor is then b ⫽ 0.735.
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
Fig. 2.
21
The statistical parameters of the parent population of the initial cracks must be defined with the help of one test specimen. It is most convenient to use a smooth specimen as this kind of reference test, since it is free of the geometric size effect. As explained in Section 2, different size of specimens cannot without doubt be assumed to belong to the same parent population. As a matter of fact, some results of Magin show that they do not belong to the same parent population [2]. Therefore, the statistical constants are determined separately for each specimen diameter, using the same size of smooth bar as reference. Note that Magin’s specimens 7/1 and 10/1 are made of same size of original bar and thus, belong to the same parent population. The applied loading mode was alternating tension– compression in Bo¨ hm’s tests and rotating bending in Magin’s tests. Thus, the stress ratio in both works was R ⫽ ⫺1.The number of cycles at the endurance limit chosen by Bo¨ hm was 2 × 106, Magin used the value 107. The endurance limit was defined with the two-point strategy. The test specimens were divided into two groups. The first one was tested with a stress value just below the assumed endurance limit, the other group with a value little bit higher than that. The mean endurance limit, or actually the mean of the maximum crack depth, and the standard deviation can be determined with the help of the failure probabilities of the two specimen groups. The main difficulty now is that it was assumed that the initiated crack size follows a distribution given by
Test specimen shapes.
4. Experimental results The experimental results from two German doctoral theses are utilised here, Bo¨ hm’s [4] and Magin’s [13]. They both used similar specimen shapes, see Fig. 2. Even the material was the same: 34CrNiMo8. The heat treatment for each original bar diameter was chosen individually in order to achieve the same ultimate strength. In Bo¨ hm’s test, specimens the ultimate strength varied from 898 to 966 MPa, in Magin’s specimens from 884 to 936 MPa (Table 1). The surface was carefully polished in order to avoid the influence of varying surface roughness. Table 1 Test specimens
Bo¨ hm
Magin
Test series
Kt
D (mm)
d (mm)
L (mm)
r (mm)
DB (mm) r (mm)
Rm (MPa)
Reference specimen
X1 X2 X3 Y1 Y2 Y3 Z1 Z2 Z3 7/1 10/1 20/1 38/1 80/1 7/2 20/2 38/2 80/2 10/5 20/5 35/5 80/5
1 1 1 2.24 2.25 2.25 5.11 5.75 5.86 1 1 1 1 1 2.16 2.25 2.25 2.25 5.38 5.38 6.10 5.71
7 20 38 9.6 29 53.6 11 32.9 60.3 7 10 20 38 80 9.2 27.6 52.1 108.8 16 32 56 128.3
– – – 7 20.8 38.6 7 20.9 38.3 – – – – – 6.8 20.4 38.5 80.4 10 20 35 80.3
16.0 48.5 90.0 – – – – – – 30.1 58.9 47.5 102.0 149.6 – – – – – – – –
– – – 0.92 2.75 5.05 0.15 0.34 0.60 – – – – – 0.66 1.8 3.43 7.1 0.12 0.22 0.31 0.85
16 38 76 16 38 76 16 38 76 10 16 32 58 128 16 32 58 128 16 32 58 128
? ? ? ? ? ? ? ? ? 938 938 899 884 896 938 899 884 896 938 899 884 896
– – – X1 X2 X3 X1 X2 X3 7/1 20/1 38/1 80/1 7/1 20/1 38/1 80/1
20 60 120 20 60 120 20 60 120 75 75 125 125 200 75 125 125 200 75 125 125 200
22
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
(2). The distribution parameters are needed in order to calculate the mean crack size from Eq. (4). In order two avoid a tedious iterative process, it is assumed that the fatigue limit is approximately normally distributed. With this simplification, the mean endurance limit and the standard deviation can be determined with the help of probability paper or numerically. The distribution parameters a and g of the parent population for each specimen size is defined with the help of the reference specimen belonging to the same parent population. The expected value of the crack depth can now be calculated using Eqs. (1)–(7). When the prediction of the crack depth is available, the estimate of the mean fatigue limit is obtained from the following equation:
冉 冊
1 ⌬KI,th ⌬sR,a ⫽ 2 冑π b冑a0
was calculated with FEM. An example of the stress distribution is presented in Fig. 3. The applied load was tension in Bo¨ hm’s test specimens and bending moment in Magin’s specimens. The effective stress Aeff area is calculated using the following procedure: - the groove area is divided into three degrees (Y1– Y3, 7/2–80/2) or six degrees (Z1–Z5, 10/5–80/5) segments; - the surface area for each segment is calculated from Eq. (12); - the stresses for the segments are taken from the FE-models, the applied value is the value at the centre of the segment; - the probability of survival corresponding to the stress level is obtained from a normal probability table; - effective stress area of the segment is calculated from Eq. (4).
(11)
The left hand side is divided by 2 because only the tension part of the stress cycle is assumed effective, when calculating the stress intensity range. The exact value of the stress intensity range threshold ⌬KI,th for the material in question is not known. The value ⌬KI,th ⫽ 6MPa冑m is used. The experimental results (re-analysed by the author) and the calculated estimates are shown in Table 2. The stress distribution along the surface of the grooves
The equation of the surface area of a groove segment is:
冋 冉冊
A ⫽ 2πry0 asin
冉冊 册
x1 x2 x2 x1 ⫺ ⫺asin ⫹ r y0 r y0
(12)
where, x1 and x2 are the coordinates of the beginning and end of each segmenty0 ⫽ d / 2 ⫹ r.
Table 2 Experimental results and calculated estimates Test series
X1 X2 X3 Y1 Y2 Y3 Z1 Z2 Z3 7/1 10/1 20/1 38/1 80/1 7/2 20/2 38/2 80/2 10/5 20/5 35/5 80/5
Aeff (mm)2
352 3047 10739 6.82 64.8 235.5 1.41 10.1 32.7 662 1822 2985 12177 37599 6.15 48.4 131.5 536.4 1.53 3.67 26.7 93.2
D (mm) Experimental
7 20 38 9.6 29 53.6 11 32.9 60.3 7 10 20 38 80 9.2 27.6 52.1 108.8 16 32 56 128.3
Calculated
⌬sR,a/2 (MPa)
√am mm0.5
s√a
⌬sR,a/2 (MPa) √am mm0.5
s√a
a/c
kg
a
g
468 400 389 564 530 510 652 597 570 462 440 425 370 394 642 581 525 514 660 668 496 555
0.311 0.362 0.374 0.267 0.279 0.288 0.278 0.268 0.265 0.313 0.334 0.345 0.394 0.368 0.240 0.254 0.279 0.289 0.326 0.251 0.357 0.271
0.015 0.022 0.025 0.015 0.023 0.023 0.033 0.024 0.018 0.017 0.017 0.027 0.033 0.013 0.027 0.024 0.016 0.016 0.035 0.009 0.099 0.019
Ref. Ref. Ref. 573 496 486 748 604 573 Ref. Ref. Ref. Ref. Ref. 614 565 517 451 947 794 717 494
Ref. Ref. Ref. 0.016 0.022 0.024 0.017 0.022 0.024 Ref. Ref. Ref. Ref. Ref. 0.018 0.025 0.029 0.013 0.019 0.025 0.028 0.014
0.8 0.8 0.8 0.67 0.73 0.75 0.41 0.54 0.62 0.8 0.8 0.8 0.8 0.8 0.62 0.71 0.73 0.73 0.30 0.47 0.42 0.64
1 1 1 0.97 0.99 1.0 0.81 0.91 0.96 1 1 1 1 1 0.96 0.98 0.99 0.99 0.68 0.86 0.82 0.96
0.167 0.230 0.265 0.167 0.228 0.262 0.167 0.228 0.262 0.193 0.193 0.238 0.275 0.336 0.193 0.238 0.275 0.336 0.193 0.238 0.275 0.336
⫺1.86 ⫺2.08 ⫺2.28 ⫺1.86 ⫺2.07 ⫺2.27 ⫺1.86 ⫺2.07 ⫺2.27 ⫺1.99 ⫺1.99 ⫺2.14 ⫺2.29 ⫺2.56 ⫺1.99 ⫺2.14 ⫺2.29 ⫺2.56 ⫺1.99 ⫺2.14 ⫺2.29 ⫺2.56
Ref. Ref. Ref. 0.262 0.296 0.301 0.241 0.265 0.265 Ref. Ref. Ref. Ref. Ref. 0.248 0.263 0.285 0.327 0.228 0.213 0.250 0.308
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
Fig. 3.
Stress distribution at the notch root of specimen Y2.
This procedure is actually presented by Eq. (4). The area A1 is the first segment at the notch root. The number of initiated cracks per unit area is assumed to be 100 per mm2 [1,2]. Thus, the sample size is Aeff × 100. The stress distribution below the surface was also calculated with FEM using a very fine element mesh. In order to be able to calculate the stress intensity factor with Eq. (10), the stress distribution must be presented in the form of a function. The stress was first divided to constant tension and peak stress. The peak stress part can be described with sufficient accuracy with the following kind of polynomial:
冉 冊 冉 冊 冉 冊
s(x) ⫽ a⫺b
23
2 3 d d d ⫺y ⫹ c ⫺y ⫺d ⫺y y 2 2 2
(13)
⬍ d/2 The factors a to d are determined so that the curve gives the best fit to the calculated points deeper than the initiated crack depth a0. Fig. 4 shows an example of the stress distribution of the FEM model and the fitted function with values a ⫽ 525.2, b ⫽ 8903, c ⫽ 87070 and d ⫽ 347000. The total stress intensity factor range can be calculated
as the sum of the constant part and the peak part with the help of Eqs. (8) and (10):
冕
a0
⌬Knl ⫽ b⌬snom冑πa0 ⫹ ⌬s(y)m(y,a)dy
(14)
0
The stress intensity factor range for a constant stress Kt⌬snom would be: ⌬Kl ⫽ bKt⌬snom冑πa0
(15)
The geometric size is obtained from the formula: kg ⫽
⌬Knl ⌬Kl
(16)
As stated in Section 3, the geometry factor for a constant stress is b ⫽ 0.735 corresponding aspect ratio a / c ⫽ 0.8. The geometry factors of the non-linear stress vary depending on the stress gradient. The value of the aspect ratio a/c used in the calculation is indicated in Table 2.
5. Discussion Numerous methods to the estimation of the notch size effect have been presented in the past. Results from some of them have been collected into Table 3. The Bo¨ hm method is another statistical method, which is based on weakest link principle. Interested reader should see Ref. [4] for details of the method. The traditional Peterson formula [8] is: Kf ⫽ 1 ⫹
Fig. 4. Stress distribution of the Bo¨ hm specimen Z1.
(Kt⫺1) 1 ⫹ (a / r)
(17)
The value of a ⫽ 0.0635mm (0.0025⬙) for tempered steels was used for the material constant. The last three methods have been developed based on the use of fracture mechanics. Topper and El Haddad [9] have proposed the following kind of equation:
24
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
Table 3 Notch size factors Kf Test series
Kt
Y1 Y2 Y3 Z1 Z2 Z3 7/2 20/2 38/2 80/2 10/5 20/5 35/5 80/5
2.24 2.25 2.25 5.11 5.75 5.86 2.16 2.25 2.25 2.25 5.38 5.38 6.10 5.71
冉
Kf ⫽ 1 ⫹
Bo¨ hm
Exp. Kf
Author
Kf
Kf
Error %
Kf
Error % Kf
Error % –
–
KFATIGUE
Error Kf %
Error %
1.86 1.99 2.06 3.66 4.50 4.78 1.56 1.79 1.98 2.02 3.76 3.73 5.76 4.68
1.83 2.12 2.17 3.72 4.55 4.79 1.66 1.86 2.04 2.33 3.80 3.91 3.98 5.41
⫺1.6 6.5 5.3 1.6 1.1 0.2 6.4 3.9 3.0 15.3 0.3 4.8 ⫺30.9 15.6
1.83 2.06 2.21 3.78 4.76 5.17 1.73 1.96 2.07 2.20 4.09 4.33 5.54 5.19
⫺1.6 3.5 7.3 3.3 5.8 8.2 10.9 9.5 4.5 8.9 8.8 16.1 ⫺3.8 10.9
16.1 11.6 8.7 6.6 11.1 13.0 32.7 23.5 12.6 10.9 3.5 17.0 ⫺9.0 15.0
– – – ⫺18.9 ⫺8.2 10.0 – – – – ⫺33.2 ⫺12.3 ⫺28.6 25.6
3.18 3.22 3.22 5.39 5.77 5.82 3.45 3.45 3.45 3.45 6.57 6.57 6.57 6.57
71.0 61.8 56.3 47.3 28.2 21.8 121 92.7 74.2 70.8 74.7 76.1 14.1 40.4
59.7 50.3 45.1 ⫺18.6 ⫺33.6 ⫺37.2 89.7 67.0 51.0 48.5 ⫺20.7 ⫺19.8 ⫺28.1 ⫺35.9
冪l 冊 dn
Peterson
2.16 2.22 2.24 3.90 5.00 5.40 2.07 2.21 2.23 2.24 3.89 4.40 5.24 5.38
(18)
0
variable dn is the notch depth. The material parameter l0 is calculated from a formula similar to (9): l0 ⫽
冉 冊
1 ⌬Kth π ⌬sR0
2
(19)
Topper and El Haddad [9] conclude that Eq. (18) should be used for sharp notches and that for blunt notches Kf is about the same as Kt. Smith and Miller [10] state that the minimum stress required to initiate a crack can be calculated with help of fatigue concentration factor given by:
冋
KFATIGUE ⫽ 1 ⫹ 7.69
冪 册 dn r
冉
2⌬sR,nom
dn 2⫹ dn ⫹ l0 / 2
冊 冉 2
– – – 2.97 4.13 5.26 – – – – 2.51 3.27 4.11 5.88
(20)
dn ⫹3 dn ⫹ l0 / 2
冊
4
(21)
It can be seen from Table 3 that the statistical methods are the only ones that give satisfactory results. The methods developed on the basis of fracture mechanics do not work at all to the specimen shape in question. Table 2 shows that author’s prediction of the fatigue limit of notched specimens with the presented method
Smith and Miller
Taylor
2.97 2.99 2.99 2.98 2.99 3.00 2.96 2.99 2.99 3.00 2.98 2.99 3.00 3.00
is good for most of the specimens. The most remarkable error is in the sharpest specimens, which have the steepest stress gradient. The most probable reason to the different behaviour of the specimens with sharp notches is the increased amount of plastic deformation at the notch root. Therefore, the strain components at the notch roots are estimated in the following. The elastic–plastic behaviour of the specimens is assumed to follow the Neuber hyperbole: K2t ⫽ KsKe
(22)
The total strain amplitude in cyclic loading can be estimated with the Ramberg–Osgood equation:
0.5
Taylor [14] has developed a method, which is based on analogy of stress fields at cracked bodies and notched components. He has given three choices: point, line and area method. All methods give results that are quite close together. The following equation applies to the point method: ⌬sR ⫽
Topper and El Haddad
etot ⫽ ee ⫹ ep ⫽
冉冊
sa sa ⫹ ⬘ E K
1 n
(23)
Bo¨ hm and Magin did not define the constants in above equation. Therefore, values for similar material from the handbook of Boller and Seeger [11] are used: E ⫽ 206000N / mm2, K⬘ ⫽ 972N / mm2, n⬘ ⫽ 0.085. It shall be noted, that the real values probably vary depending on the specimen diameter. The stress state at the notch root of the notched specimen becomes multiaxial. It was therefore assumed that no yielding occurs before the equivalent stress exceeds the cyclic yield limit according to von Mises criteria. With the help of Eqs. (22) and (23) it is now possible to calculate the estimates of the elastic and plastic strain components as well as the real Neuber stress at the notch root. The results are shown in Table 3. It can easily be seen that the plastic deformation is highest in the specimens, where the estimated fatigue limit deviates most
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
from the experimental values, except Magin’s specimen 35/5. Since plastic deformation plays an important role in the fatigue behaviour of very sharp notches, it becomes obvious that no single formula can predict the fatigue limit of all kind of notches. Another rule shall be developed for those specimens, where the fatigue limit exceeds considerably, the cyclic yield strength of the material. Developing a simple engineering approximation is tested in the following. As proved before, the upper bound of the fatigue limit can be estimated with the combination of statistical and geometric size effect. On the other hand, Smith and Miller [10] have shown that the lower bound is given by the stress cycle, that corresponds the stress intensity factor range threshold of the material. This would be valid when the notch root radius approaches 0. The crack depth shall be taken equal to the notch depth in this case. The real components fall between these two cases. It is now assumed that the fatigue limit of this kind of specimens can be estimated with an interpolation based on the magnitude of the plastic deformation. An equation that satisfies the above conditions is for example: ⌬sR ⫽ ⌬sR,St+G when epⱕep,lim ⌬sR,nom ⫽
冉
冉 冊
⌬sR,St+G ep,lim ⫹ ⌬sR,Kth Kt ep
冊
(24)
ep,lim 1⫺ when ep ⬎ ep,lim ep
25
The latter part of the equation approaches the stress cycle, that is calculated based on the stress intensity factor range threshold when plastic strain approaches infinity:
冉 冊
1 ⌬KI,th ⌬sR,Kth ⫽ 2 冑π b冑dn
(25)
This is the same equation as (11) except that initial crack depth a0 has been replaced by notch depth dn. Note that, here is also assumed that only the tension part of the stress cycle is effective in the stress intensity factor range. In addition to the threshold KI,th, the value for the geometry factor b is needed. Formulas for axially symmetric grooves can be found in Tada et al. [12]. The values of b for the specimen shapes in question vary from approximately 1.3 to 1.5. The value of ep,lim shall be estimated with help of estimated strain values in Table 4. It seems that below the value etop / ep ⫽ 1.07 the plastic strain does not have a significant influence on the fatigue limit. Therefore, ep,lim was calculated based on this value. It shall be noted that Ramberg–Osgood Eq. (23) gives a small portion of plastic deformation at any stress level. There is not enough data available to define a reliable value for the exponent e. The value of e ⫽ 1 / 3 was chosen in this paper since it gives good results compared to the experimental values. Eq. (24) is used to calculate an estimate of notch size factor Kf for the specimens, where the plastic strain at the calculated fatigue limit
Table 4 Elastic–plastic behaviour of the test specimens Test series Bohm X1 X2 X3 Y1 Y2 Y3 Z1 Z2 Z3 Magin 7/1 10/1 20/1 38/1 80/1 7/2 20/2 38/2 80/2 10/5 20/5 35/5 80/5
Kt
D (mm)
d (mm)
r (mm)
⌬sR,a/2 (MPa)
⌬sN/2 (MPa)
etot/ep
1 1 1 2.24 2.25 2.25 5.11 5.75 5.86
7 20 38 9.6 29 53.6 11 32.9 60.3
– – – 7 20.8 38.6 7 20.9 38.3
– – – 0.92 2.75 5.05 0.15 0.34 0.60
468 400 389 564 530 510 652 597 570
455 398 387 523 498 483 579 547 530
1.060 1.013 1.010 1.068 1.041 1.028 1.141 1.069 1.049
1 1 1 1 1 2.16 2.25 2.25 2.25 5.38 5.38 6.10 5.71
7 10 20 38 80 9.2 27.6 52.1 108.8 16 32 56 128.3
– – – – – 6.8 20.4 38.5 80.4 10 20 35 80.3
– – – – – 0.6 1.8 3.4 7.1 0.1 0.2 0.35 0.8
462 440 425 373 394 647 581 525 514 660 668 496 555
454 431 419 372 392 573 537 498 482 588 593 469 521
1.058 1.033 1.024 1.009 1.013 1.166 1.087 1.038 1.029 1.138 1.148 1.010 1.035
26
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
exceeds the limit value ep,lim, these values are indicated by bold characters in Table 3. Finally, it shall be noted that the calculation of the initiated crack depth from Eq. (9) is approximate by nature. Furthermore, the exact value of the stress intensity threshold is not known, and it is questionable if a value normally defined for long cracks is applicable for very small initial cracks. It was shown in Ref. [1] that the possible error in the estimated crack size is ignorable in the calculation of the statistical size effect. The influence to the geometric size effect is naturally larger. This was tested to Bo¨ hm’s specimen Z3 by increasing the estimated stress intensity factor range threshold from 6 to 9 MPa √m. The obtained estimate of the fatigue limit was decreased by 9.7%. As far as notched specimens are in question, the shape of the initiated crack has a great influence in the obtained value of the stress intensity factor also. It was assumed in this work that the initiated crack tends to take the shape, which gives constant stress intensity factor along the crack front. This may be questionable for very small cracks. 6. Conclusions 1. One single method cannot predict the fatigue limit of blunt notches and sharp notches. In very small details the plastic portion of strain even at the fatigue limit is considerable, and this shall be accounted for. 2. The fatigue limit of blunt notches (notches with insignificant portion of plastic strain) can be accurately estimated with the combination of statistical and geometric size effects. For engineering purposes, the statistical size effect alone gives adequately accurate results. 3. The fatigue limit of ‘sharp’ notches (notches where the plastic strain exceeds a limit value) shall be estimated with a method, which takes into account that, when the notch root radius get smaller, the fatigue limit approaches a value, which is obtained as follows:
앫 The notch is assumed to act as an initial crack; 앫 The depth of the crack equals the depth of the notch; 앫 The fatigue limit is calculated to this crack with the help of linear elastic fracture mechanics and the stress intensity factor range threshold.
References [1] Makkonen M. Size effect and notch size effect in metal fatigue. Thesis for the degree of Doctor of Science (Technology), Lappeenranta University of Technology. Acta Universitatis Lappeenrantaensis 83; 1999. [2] Makkonen M. Statistical size effect in the fatigue limit of steel. Int J Fatigue 2001;23:395–402. [3] Castillo E. Extreme value theory in engineering. New York: Academic Press Inc., 1988. [4] Bo¨ hm J. Zur Vorhersage von Dauerscwingfestigkeiten ungekerbter und gekerbter Bauteile unter Beru¨ cksichtigung des statistischen Gro¨ sseneinflu¨ sses. Dissertation, Technische Universita¨ t Mu¨ nchen; 1979. [5] Cameron AD, Smith RA. Fatigue life prediction for notched members. Int J Pres Ves Pip 1982;10:205–17. [6] Wang X, Lambert SB. Stress intensity factors for low aspect ratio semi-elliptical surface cracks in finite-thickness plates subjected to nonuniform stresses. Eng Fract Mech 1995;51(4): 517–32. [7] Isida M, Noguchi H, Yoshida T. Tension and bending of finite thickness plates with a semi-elliptical surface cracks. Int J Fract 1984;26:157–88. [8] Peterson RE. Notch sensitivity. In: Sines G, Waisman JL, editors. Metal fatigue. New York: McGraw Hill; 1959. p. 293–306. [9] Topper TH, El Haddad MH. Fatigue strength prediction of notches based on fracture mechanics. Fatigue Thresh 1982;2:777–97 [EMAS, Warley, UK]. [10] Smith RA, Miller KJ. Prediction of fatigue regimes in notched components. Int J mech Sci 1978;20:201–6 [Pergamon Press]. [11] Boller Chr, Seeger T. Materials data for cyclic loading, Part A: Unalloyed steels. Netherlands: Elsevier, 1987. [12] Tada H, Paris P, Irwin R. The stress analysis of cracks handbook. Hellerton, Pennsylvania: Del Research Corporation, 1973. [13] Magin W. Untersuchung des geometrischen Gro¨ sseeinflu¨ sses bei Umlaufbiegeanspruchung unter besonderer beru¨ cksichtigung technologische Einflu¨ sse. Dissertation, Technische Hochschule Darmstadt, 1981. [14] Taylor D. Geometrical effects in fatigue: a unifying theoretical model. Int J Fatigue 1999;21:413–20.
Notch size effects in the fatigue limit of steel M. Makkonen ∗ Andritz Oy Wood Processing, Keskikankaantie 9, 15870 Hollola, Finland Received 20 December 2001; received in revised form 23 April 2002; accepted 13 May 2002
Abstract The fatigue behaviour of grooved test specimens has been investigated in this paper. It is shown that there are two kinds of size effects connected to notches: statistical size effect and geometric size effect. The statistical size effect is calculated based on the distribution of the maximum depth of initiated cracks in specimens with varying stress area. The geometric size effect depends on the stress gradient and can be estimated with the help of the linear elastic fracture mechanics. It is concluded that the notch size effect in blunt notches can be explained with those two factors. When the notch gets sharper, after certain limit the magnitude of the plastic portion of the strain starts to play an important role in the fatigue crack initiation, and the fatigue limit is lower than that predicted by statistical and geometric size effects. Another estimation method shall be used to that kind of notches. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue; Notch size effect; Statistical size effect
1. Introduction Most of the components used in machines have notches like shoulders and holes. The fatigue limit of this kind of specimen is higher than the maximum stress at the notch root, and would indicate. Many attempts have been made to evaluate this phenomenon, which is called the notch size effect. The aim has been to develop an equation, which gives the relationship between the observed fatigue limit and the expectable strength based on tests with smooth specimens. Many empirical formulas have been developed. A collection of the most commonly used methods are presented in Ref. [1]. The magnitude of the notch size effect is material dependent. This is explained with the so-called ‘notch sensitivity’ of the material. All the presented formulas include material parameters, which try to describe this, but no physical background for this phenomenon is given for them. All formulas show considerable scatter when applied to different materials. The goal in this paper is to show that the notch size effect can be explained with two factors: the statistical
size effect and the effect of the stress gradient, which is called the geometric size effect. The statistical size effect can be described as follows: when a component is subjected to an alternating load, a number of microcracks will initiate in its volume. The larger the specimen, the larger initiated cracks will be found. Thus, the higher probability of large initiated cracks appears as a lower fatigue limit for larger specimens. It was shown by Makkonen [1,2] that the size effect in plain specimens results from the statistical size effect alone. The geometric size effect comes into picture with notched specimens. The stress distribution in the vicinity of grooves, shoulders and other discontinuities becomes non-linear, and a high stress peak appears. The stress gradient is steeper in small equally shaped specimens. If equal-sized cracks are initiated in the peak stress area, the stress intensity factor at the crack is higher in a larger specimen. The other aim in this paper is to present a method for estimating the geometric size effect by comparing the stress intensity factors at the initiated crack for two cases: for the actual peak stress distribution and for a linear stress (tension or bending). 2. Statistical methods
Tel.: +358-3-880-3420; fax: +358-3-880-3200. E-mail address: [email protected] (M. Makkonen). ∗
It was shown by Makkonen [1,2] that the distribution of initiated crack depths can be assumed to follow the
0142-1123/02/$ - see front matter. 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 2 ) 0 0 0 5 3 - 1
18
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
Nomenclature a0 a c d dn e f(x) fXn:n(x) kg l0 n n⬘ r sx x xm y z A Ai Aeff D Do F(x) FXn:n(x) K K⬘ Kf KFATIGUE KI Kt Ke Ks m(x,a) PS R Rm s√a0 a b ee ep ep,lim etot g r s s(x) sa ⌬KI ⌬KI,th ⌬Kl ⌬Knl ⌬s ⌬snom ⌬sN ⌬sR,a ⌬sR,Kth ⌬sR,nom ⌬sR,St+G
initiated crack depth crack depth crack length diameter notch depth constant probability density function probability density function of the maximum value of sample geometric size factor material constant sample size cyclic hardening exponent radius standard deviation of a random variable random variable, co-ordinate mean value of a random variable co-ordinate co-ordinate surface area surface area of ith part of the surface effective stress area diameter original bar diameter cumulative distribution function cumulative distribution function of the maximum value of sample stress intensity factor cyclic hardening coefficient notch size factor fatigue concentration factor stress intensity factor stress concentration factor strain concentration factor real (Neuber) stress concentration factor weight function probability of survival stress ratio ultimate strength standard deviation of the initiated crack depth scatter parameter of the lognormal distribution geometry factor elastic strain plastic strain limit value of plastic strain total strain location parameter of the lognormal distribution notch root radius stress stress distribution function stress amplitude stress intensity range stress intensity range threshold stress intensity range from linear stress stress intensity range from non-linear stress stress cycle nominal stress cycle real (Neuber) stress cycle mean fatigue limit fatigue limit based on stress intensity range threshold fatigue limit based on nominal stress cycle fatigue limit calculated using statistical and geometric size effects
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
lognormal distribution. It has the following probability density function: f(x) ⫽
冉
1 ln(x)⫺ γ a a冑2πx 2 1
e
冊
2
(1)
crack in a sample can be defined with the help of Eqs. (1) and (3). The expected value and the standard deviation, respectively, shall be calculated from the following equations:
冕 ⬁
The distribution of the maximum value in a sample can be defined with the help of the order statistics [3]:
xm ⫽ xfXn:ndx
sx ⫽
- probability density function (pdf)
冘
(3)
j
log(PS(⌬si)) Ai log(0.5) i⫽2
(4)
Area A1 has the maximum stress. The term (PS(⌬s1)) presents the probability of survival of the area Ai at stress cycle ⌬si, when the area A1 has probability of survival of 0.5 for fatigue at stress level ⌬s1. When the stress distribution is continuous, the corresponding equation obtains the following form: Aeff ⫽
冉
冊冕
1 log(0.5)
log(PS(⌬s(x,y,z)))dA
(x⫺xm)2f(x)dx
(7)
0
F(x) and f(x) are the cdf and pdf of the parent population of the initiated cracks. Since Eq. (1) cannot integrated in the closed form, numerical methods must be used to solve Eqs. (2) and (3). The sample size n, the number of initiated cracks in the specimen, are needed for the above equations. When notched specimens are in questions, it is clear that the critical cracks will be located at the surface since the maximum stress is always there. Thus, the sample size is the number of cracks on the specimen surface. The exact number of initiated cracks per unit area cannot be known. It was shown in Ref. [2], that possible error in this value does not have significant influence to the calculated results. The value chosen in Ref. [2], 100 cracks per mm2, shall be used in this paper also; see Ref. [2] for further discussion on this matter. If the stress distribution is not constant, an effective stress area must be calculated. This can be done with the method presented by Bo¨ hm [4]. The idea is that, the areas with lower stress level are reduced so that the probability of failure with the calculated effective area with a constant stress is the same as in the original specimen. The effective stress area of a structure consisting of j parts, each of them having a different stress level, can be calculated from the formula: Aeff ⫽ A1 ⫹
冪冕 ⬁
(2)
fXn:n(x) ⫽ nFn⫺1(x)f(x)
(6)
0
- cumulative distribution function (cdf) FXn:n(x) ⫽ Fn(x)
19
(5)
A
The sample size is determined using the area calculated from Eq. (5). The distribution of the maximum
The required connection between the fatigue limit and the initiated crack depth can be established with help of the linear elastic fracture mechanics (LEFM): KI ⫽ bs冑πa0
(8)
Substituting KI with the stress intensity range threshold, ⌬KI,th, and s by the mean fatigue limit, ⌬sR,a, yields [5]:
冑a
0
⫽
冊 冑冉 ⌬KI,th b⌬s R,a π
1
(9)
Since the fatigue limit and the square root of the crack depth are in inverse proportion, the random variable in the above equations will be set x ⫽ 冑a0. The initial crack depth a0 cannot be measured in the experimental tests. Therefore, the mean fatigue limit and its standard deviation shall be determined instead. Eq. (9) is used to convert the results to crack depths. When the mean of the initial crack depth 冑a0and the standard deviation s冑a0 are known, the scatter parameter a and the location parameter g of the lognormal distribution can be defined. The expected maximum crack depth in any sample size can then be calculated from Eqs. (3) and (6). Finally, the expected mean fatigue limit of the target specimen is solved from Eq. (9). It shall be noted that all specimens in question must belong to the same parent population while doing this calculation. This condition is not necessarily fulfilled even if the specimens are made of the same material, see Refs. [1] and [2] for further discussion. The geometry factor, b, depends on the specimen shape, on the crack shape and on the stress distribution. It can be assumed that the crack shape tends to take the shape where the stress intensity factor over the whole crack front is constant.
3. Geometric size effect A schematic presentation of the effect of the stress gradient is given in Fig. 1. There are two shoulders, which have similar shape, but different size. If the nomi-
20
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
Fig. 1.
Stress gradients and cracks at shoulders with same Kt.
nal stress cycle is same in both specimens, the peak stress Kt⌬snom is also the same. Let us now assume that there is an initiated crack of depth a0 at the peak stress area of both notches. It can easily be seen that the stress cycle at the crack tip is higher in the larger specimen (⌬s2) than in the smaller one (⌬s1). This means, the stress intensity factor range also is higher in the larger specimen. Thus, it can be concluded, that the critical crack depth is smaller in the larger specimen. This leads to lower fatigue limit. This phenomenon is called the geometric size effect. It can be concluded that the effect of the stress gradient to the fatigue limit can be estimated by calculating the stress intensity factor range in both specimens. Similarly, comparison to smooth specimen is also possible. In fact, the geometric size effect can be estimated as follows: 1. the value of the stress intensity factor range, ⌬Knl, is calculated for the non-linear stress distribution using the assumed initiated crack depth a0; 2. the stress intensity factor range, ⌬Kl, is calculated for a constant stress Kt⌬snom; 3. the geometric size effect is obtained as the ratio kg ⫽ ⌬Knl / ⌬K1. The following things shall be noted: 앫 The crack size used in the calculation above is the depth a0 after the crack initiation phase. An estimate of this crack depth can be obtained from Eq. (9). 앫 The best assumption for the aspect ratio a/c is the one which gives constant stress intensity factor along the crack front. This means that the aspect ratio is different in the non-linear stress distribution and in the assumed constant stress field. 앫 There is a slight geometric size effect in a smooth specimen subjected to bending load. Anyway, the
stress intensity factors in bending and in constant tension vary so little that this can be neglected [1]. The calculation of the geometric size effect requires that the stress intensity factor for a non-linear stress can be calculated. Solutions for this kind of stress distributions are normally not available. Fortunately, values for very small cracks are needed, in the situation, where the initiated crack is about to change the mode to stable crack growth (Paris law). It can be shown that the influence of the specimen shape is negligible for very small cracks [1], when the crack size gets smaller and smaller, the stress intensity factor approaches the value for surface crack in a semi-infinite body. Therefore, solutions for surface cracks for plates can always be used. Wang and Lambert [6] have developed a calculation method, which is based on the use of weight functions. The stress intensity factor can be calculated with the integral:
冕 a
K ⫽ s(x)m(x,a)dx
(10)
0
Weight functions m(x,a) are given for the deepest point as well as for the surface point. This makes it possible to adjust the crack aspect ratio a/c so that the stress intensity factor is equal in both points. It must be possible to present the stress distribution in the form of a function s(x). The actual stress distribution can be defined e.g. by using the finite element method. A suitable function is then fitted to the calculated stress points. This function must be accurate from the specimen surface to the depth of a0. The reference value of the stress intensity factor for a surface crack in constant stress can be found in the paper of Isida et al. [7]. The stress intensity factor has same value in the deepest point and in the surface point with the aspect ratio of approximately a / c ⫽ 0.8.The geometry factor is then b ⫽ 0.735.
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
Fig. 2.
21
The statistical parameters of the parent population of the initial cracks must be defined with the help of one test specimen. It is most convenient to use a smooth specimen as this kind of reference test, since it is free of the geometric size effect. As explained in Section 2, different size of specimens cannot without doubt be assumed to belong to the same parent population. As a matter of fact, some results of Magin show that they do not belong to the same parent population [2]. Therefore, the statistical constants are determined separately for each specimen diameter, using the same size of smooth bar as reference. Note that Magin’s specimens 7/1 and 10/1 are made of same size of original bar and thus, belong to the same parent population. The applied loading mode was alternating tension– compression in Bo¨ hm’s tests and rotating bending in Magin’s tests. Thus, the stress ratio in both works was R ⫽ ⫺1.The number of cycles at the endurance limit chosen by Bo¨ hm was 2 × 106, Magin used the value 107. The endurance limit was defined with the two-point strategy. The test specimens were divided into two groups. The first one was tested with a stress value just below the assumed endurance limit, the other group with a value little bit higher than that. The mean endurance limit, or actually the mean of the maximum crack depth, and the standard deviation can be determined with the help of the failure probabilities of the two specimen groups. The main difficulty now is that it was assumed that the initiated crack size follows a distribution given by
Test specimen shapes.
4. Experimental results The experimental results from two German doctoral theses are utilised here, Bo¨ hm’s [4] and Magin’s [13]. They both used similar specimen shapes, see Fig. 2. Even the material was the same: 34CrNiMo8. The heat treatment for each original bar diameter was chosen individually in order to achieve the same ultimate strength. In Bo¨ hm’s test, specimens the ultimate strength varied from 898 to 966 MPa, in Magin’s specimens from 884 to 936 MPa (Table 1). The surface was carefully polished in order to avoid the influence of varying surface roughness. Table 1 Test specimens
Bo¨ hm
Magin
Test series
Kt
D (mm)
d (mm)
L (mm)
r (mm)
DB (mm) r (mm)
Rm (MPa)
Reference specimen
X1 X2 X3 Y1 Y2 Y3 Z1 Z2 Z3 7/1 10/1 20/1 38/1 80/1 7/2 20/2 38/2 80/2 10/5 20/5 35/5 80/5
1 1 1 2.24 2.25 2.25 5.11 5.75 5.86 1 1 1 1 1 2.16 2.25 2.25 2.25 5.38 5.38 6.10 5.71
7 20 38 9.6 29 53.6 11 32.9 60.3 7 10 20 38 80 9.2 27.6 52.1 108.8 16 32 56 128.3
– – – 7 20.8 38.6 7 20.9 38.3 – – – – – 6.8 20.4 38.5 80.4 10 20 35 80.3
16.0 48.5 90.0 – – – – – – 30.1 58.9 47.5 102.0 149.6 – – – – – – – –
– – – 0.92 2.75 5.05 0.15 0.34 0.60 – – – – – 0.66 1.8 3.43 7.1 0.12 0.22 0.31 0.85
16 38 76 16 38 76 16 38 76 10 16 32 58 128 16 32 58 128 16 32 58 128
? ? ? ? ? ? ? ? ? 938 938 899 884 896 938 899 884 896 938 899 884 896
– – – X1 X2 X3 X1 X2 X3 7/1 20/1 38/1 80/1 7/1 20/1 38/1 80/1
20 60 120 20 60 120 20 60 120 75 75 125 125 200 75 125 125 200 75 125 125 200
22
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
(2). The distribution parameters are needed in order to calculate the mean crack size from Eq. (4). In order two avoid a tedious iterative process, it is assumed that the fatigue limit is approximately normally distributed. With this simplification, the mean endurance limit and the standard deviation can be determined with the help of probability paper or numerically. The distribution parameters a and g of the parent population for each specimen size is defined with the help of the reference specimen belonging to the same parent population. The expected value of the crack depth can now be calculated using Eqs. (1)–(7). When the prediction of the crack depth is available, the estimate of the mean fatigue limit is obtained from the following equation:
冉 冊
1 ⌬KI,th ⌬sR,a ⫽ 2 冑π b冑a0
was calculated with FEM. An example of the stress distribution is presented in Fig. 3. The applied load was tension in Bo¨ hm’s test specimens and bending moment in Magin’s specimens. The effective stress Aeff area is calculated using the following procedure: - the groove area is divided into three degrees (Y1– Y3, 7/2–80/2) or six degrees (Z1–Z5, 10/5–80/5) segments; - the surface area for each segment is calculated from Eq. (12); - the stresses for the segments are taken from the FE-models, the applied value is the value at the centre of the segment; - the probability of survival corresponding to the stress level is obtained from a normal probability table; - effective stress area of the segment is calculated from Eq. (4).
(11)
The left hand side is divided by 2 because only the tension part of the stress cycle is assumed effective, when calculating the stress intensity range. The exact value of the stress intensity range threshold ⌬KI,th for the material in question is not known. The value ⌬KI,th ⫽ 6MPa冑m is used. The experimental results (re-analysed by the author) and the calculated estimates are shown in Table 2. The stress distribution along the surface of the grooves
The equation of the surface area of a groove segment is:
冋 冉冊
A ⫽ 2πry0 asin
冉冊 册
x1 x2 x2 x1 ⫺ ⫺asin ⫹ r y0 r y0
(12)
where, x1 and x2 are the coordinates of the beginning and end of each segmenty0 ⫽ d / 2 ⫹ r.
Table 2 Experimental results and calculated estimates Test series
X1 X2 X3 Y1 Y2 Y3 Z1 Z2 Z3 7/1 10/1 20/1 38/1 80/1 7/2 20/2 38/2 80/2 10/5 20/5 35/5 80/5
Aeff (mm)2
352 3047 10739 6.82 64.8 235.5 1.41 10.1 32.7 662 1822 2985 12177 37599 6.15 48.4 131.5 536.4 1.53 3.67 26.7 93.2
D (mm) Experimental
7 20 38 9.6 29 53.6 11 32.9 60.3 7 10 20 38 80 9.2 27.6 52.1 108.8 16 32 56 128.3
Calculated
⌬sR,a/2 (MPa)
√am mm0.5
s√a
⌬sR,a/2 (MPa) √am mm0.5
s√a
a/c
kg
a
g
468 400 389 564 530 510 652 597 570 462 440 425 370 394 642 581 525 514 660 668 496 555
0.311 0.362 0.374 0.267 0.279 0.288 0.278 0.268 0.265 0.313 0.334 0.345 0.394 0.368 0.240 0.254 0.279 0.289 0.326 0.251 0.357 0.271
0.015 0.022 0.025 0.015 0.023 0.023 0.033 0.024 0.018 0.017 0.017 0.027 0.033 0.013 0.027 0.024 0.016 0.016 0.035 0.009 0.099 0.019
Ref. Ref. Ref. 573 496 486 748 604 573 Ref. Ref. Ref. Ref. Ref. 614 565 517 451 947 794 717 494
Ref. Ref. Ref. 0.016 0.022 0.024 0.017 0.022 0.024 Ref. Ref. Ref. Ref. Ref. 0.018 0.025 0.029 0.013 0.019 0.025 0.028 0.014
0.8 0.8 0.8 0.67 0.73 0.75 0.41 0.54 0.62 0.8 0.8 0.8 0.8 0.8 0.62 0.71 0.73 0.73 0.30 0.47 0.42 0.64
1 1 1 0.97 0.99 1.0 0.81 0.91 0.96 1 1 1 1 1 0.96 0.98 0.99 0.99 0.68 0.86 0.82 0.96
0.167 0.230 0.265 0.167 0.228 0.262 0.167 0.228 0.262 0.193 0.193 0.238 0.275 0.336 0.193 0.238 0.275 0.336 0.193 0.238 0.275 0.336
⫺1.86 ⫺2.08 ⫺2.28 ⫺1.86 ⫺2.07 ⫺2.27 ⫺1.86 ⫺2.07 ⫺2.27 ⫺1.99 ⫺1.99 ⫺2.14 ⫺2.29 ⫺2.56 ⫺1.99 ⫺2.14 ⫺2.29 ⫺2.56 ⫺1.99 ⫺2.14 ⫺2.29 ⫺2.56
Ref. Ref. Ref. 0.262 0.296 0.301 0.241 0.265 0.265 Ref. Ref. Ref. Ref. Ref. 0.248 0.263 0.285 0.327 0.228 0.213 0.250 0.308
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
Fig. 3.
Stress distribution at the notch root of specimen Y2.
This procedure is actually presented by Eq. (4). The area A1 is the first segment at the notch root. The number of initiated cracks per unit area is assumed to be 100 per mm2 [1,2]. Thus, the sample size is Aeff × 100. The stress distribution below the surface was also calculated with FEM using a very fine element mesh. In order to be able to calculate the stress intensity factor with Eq. (10), the stress distribution must be presented in the form of a function. The stress was first divided to constant tension and peak stress. The peak stress part can be described with sufficient accuracy with the following kind of polynomial:
冉 冊 冉 冊 冉 冊
s(x) ⫽ a⫺b
23
2 3 d d d ⫺y ⫹ c ⫺y ⫺d ⫺y y 2 2 2
(13)
⬍ d/2 The factors a to d are determined so that the curve gives the best fit to the calculated points deeper than the initiated crack depth a0. Fig. 4 shows an example of the stress distribution of the FEM model and the fitted function with values a ⫽ 525.2, b ⫽ 8903, c ⫽ 87070 and d ⫽ 347000. The total stress intensity factor range can be calculated
as the sum of the constant part and the peak part with the help of Eqs. (8) and (10):
冕
a0
⌬Knl ⫽ b⌬snom冑πa0 ⫹ ⌬s(y)m(y,a)dy
(14)
0
The stress intensity factor range for a constant stress Kt⌬snom would be: ⌬Kl ⫽ bKt⌬snom冑πa0
(15)
The geometric size is obtained from the formula: kg ⫽
⌬Knl ⌬Kl
(16)
As stated in Section 3, the geometry factor for a constant stress is b ⫽ 0.735 corresponding aspect ratio a / c ⫽ 0.8. The geometry factors of the non-linear stress vary depending on the stress gradient. The value of the aspect ratio a/c used in the calculation is indicated in Table 2.
5. Discussion Numerous methods to the estimation of the notch size effect have been presented in the past. Results from some of them have been collected into Table 3. The Bo¨ hm method is another statistical method, which is based on weakest link principle. Interested reader should see Ref. [4] for details of the method. The traditional Peterson formula [8] is: Kf ⫽ 1 ⫹
Fig. 4. Stress distribution of the Bo¨ hm specimen Z1.
(Kt⫺1) 1 ⫹ (a / r)
(17)
The value of a ⫽ 0.0635mm (0.0025⬙) for tempered steels was used for the material constant. The last three methods have been developed based on the use of fracture mechanics. Topper and El Haddad [9] have proposed the following kind of equation:
24
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
Table 3 Notch size factors Kf Test series
Kt
Y1 Y2 Y3 Z1 Z2 Z3 7/2 20/2 38/2 80/2 10/5 20/5 35/5 80/5
2.24 2.25 2.25 5.11 5.75 5.86 2.16 2.25 2.25 2.25 5.38 5.38 6.10 5.71
冉
Kf ⫽ 1 ⫹
Bo¨ hm
Exp. Kf
Author
Kf
Kf
Error %
Kf
Error % Kf
Error % –
–
KFATIGUE
Error Kf %
Error %
1.86 1.99 2.06 3.66 4.50 4.78 1.56 1.79 1.98 2.02 3.76 3.73 5.76 4.68
1.83 2.12 2.17 3.72 4.55 4.79 1.66 1.86 2.04 2.33 3.80 3.91 3.98 5.41
⫺1.6 6.5 5.3 1.6 1.1 0.2 6.4 3.9 3.0 15.3 0.3 4.8 ⫺30.9 15.6
1.83 2.06 2.21 3.78 4.76 5.17 1.73 1.96 2.07 2.20 4.09 4.33 5.54 5.19
⫺1.6 3.5 7.3 3.3 5.8 8.2 10.9 9.5 4.5 8.9 8.8 16.1 ⫺3.8 10.9
16.1 11.6 8.7 6.6 11.1 13.0 32.7 23.5 12.6 10.9 3.5 17.0 ⫺9.0 15.0
– – – ⫺18.9 ⫺8.2 10.0 – – – – ⫺33.2 ⫺12.3 ⫺28.6 25.6
3.18 3.22 3.22 5.39 5.77 5.82 3.45 3.45 3.45 3.45 6.57 6.57 6.57 6.57
71.0 61.8 56.3 47.3 28.2 21.8 121 92.7 74.2 70.8 74.7 76.1 14.1 40.4
59.7 50.3 45.1 ⫺18.6 ⫺33.6 ⫺37.2 89.7 67.0 51.0 48.5 ⫺20.7 ⫺19.8 ⫺28.1 ⫺35.9
冪l 冊 dn
Peterson
2.16 2.22 2.24 3.90 5.00 5.40 2.07 2.21 2.23 2.24 3.89 4.40 5.24 5.38
(18)
0
variable dn is the notch depth. The material parameter l0 is calculated from a formula similar to (9): l0 ⫽
冉 冊
1 ⌬Kth π ⌬sR0
2
(19)
Topper and El Haddad [9] conclude that Eq. (18) should be used for sharp notches and that for blunt notches Kf is about the same as Kt. Smith and Miller [10] state that the minimum stress required to initiate a crack can be calculated with help of fatigue concentration factor given by:
冋
KFATIGUE ⫽ 1 ⫹ 7.69
冪 册 dn r
冉
2⌬sR,nom
dn 2⫹ dn ⫹ l0 / 2
冊 冉 2
– – – 2.97 4.13 5.26 – – – – 2.51 3.27 4.11 5.88
(20)
dn ⫹3 dn ⫹ l0 / 2
冊
4
(21)
It can be seen from Table 3 that the statistical methods are the only ones that give satisfactory results. The methods developed on the basis of fracture mechanics do not work at all to the specimen shape in question. Table 2 shows that author’s prediction of the fatigue limit of notched specimens with the presented method
Smith and Miller
Taylor
2.97 2.99 2.99 2.98 2.99 3.00 2.96 2.99 2.99 3.00 2.98 2.99 3.00 3.00
is good for most of the specimens. The most remarkable error is in the sharpest specimens, which have the steepest stress gradient. The most probable reason to the different behaviour of the specimens with sharp notches is the increased amount of plastic deformation at the notch root. Therefore, the strain components at the notch roots are estimated in the following. The elastic–plastic behaviour of the specimens is assumed to follow the Neuber hyperbole: K2t ⫽ KsKe
(22)
The total strain amplitude in cyclic loading can be estimated with the Ramberg–Osgood equation:
0.5
Taylor [14] has developed a method, which is based on analogy of stress fields at cracked bodies and notched components. He has given three choices: point, line and area method. All methods give results that are quite close together. The following equation applies to the point method: ⌬sR ⫽
Topper and El Haddad
etot ⫽ ee ⫹ ep ⫽
冉冊
sa sa ⫹ ⬘ E K
1 n
(23)
Bo¨ hm and Magin did not define the constants in above equation. Therefore, values for similar material from the handbook of Boller and Seeger [11] are used: E ⫽ 206000N / mm2, K⬘ ⫽ 972N / mm2, n⬘ ⫽ 0.085. It shall be noted, that the real values probably vary depending on the specimen diameter. The stress state at the notch root of the notched specimen becomes multiaxial. It was therefore assumed that no yielding occurs before the equivalent stress exceeds the cyclic yield limit according to von Mises criteria. With the help of Eqs. (22) and (23) it is now possible to calculate the estimates of the elastic and plastic strain components as well as the real Neuber stress at the notch root. The results are shown in Table 3. It can easily be seen that the plastic deformation is highest in the specimens, where the estimated fatigue limit deviates most
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
from the experimental values, except Magin’s specimen 35/5. Since plastic deformation plays an important role in the fatigue behaviour of very sharp notches, it becomes obvious that no single formula can predict the fatigue limit of all kind of notches. Another rule shall be developed for those specimens, where the fatigue limit exceeds considerably, the cyclic yield strength of the material. Developing a simple engineering approximation is tested in the following. As proved before, the upper bound of the fatigue limit can be estimated with the combination of statistical and geometric size effect. On the other hand, Smith and Miller [10] have shown that the lower bound is given by the stress cycle, that corresponds the stress intensity factor range threshold of the material. This would be valid when the notch root radius approaches 0. The crack depth shall be taken equal to the notch depth in this case. The real components fall between these two cases. It is now assumed that the fatigue limit of this kind of specimens can be estimated with an interpolation based on the magnitude of the plastic deformation. An equation that satisfies the above conditions is for example: ⌬sR ⫽ ⌬sR,St+G when epⱕep,lim ⌬sR,nom ⫽
冉
冉 冊
⌬sR,St+G ep,lim ⫹ ⌬sR,Kth Kt ep
冊
(24)
ep,lim 1⫺ when ep ⬎ ep,lim ep
25
The latter part of the equation approaches the stress cycle, that is calculated based on the stress intensity factor range threshold when plastic strain approaches infinity:
冉 冊
1 ⌬KI,th ⌬sR,Kth ⫽ 2 冑π b冑dn
(25)
This is the same equation as (11) except that initial crack depth a0 has been replaced by notch depth dn. Note that, here is also assumed that only the tension part of the stress cycle is effective in the stress intensity factor range. In addition to the threshold KI,th, the value for the geometry factor b is needed. Formulas for axially symmetric grooves can be found in Tada et al. [12]. The values of b for the specimen shapes in question vary from approximately 1.3 to 1.5. The value of ep,lim shall be estimated with help of estimated strain values in Table 4. It seems that below the value etop / ep ⫽ 1.07 the plastic strain does not have a significant influence on the fatigue limit. Therefore, ep,lim was calculated based on this value. It shall be noted that Ramberg–Osgood Eq. (23) gives a small portion of plastic deformation at any stress level. There is not enough data available to define a reliable value for the exponent e. The value of e ⫽ 1 / 3 was chosen in this paper since it gives good results compared to the experimental values. Eq. (24) is used to calculate an estimate of notch size factor Kf for the specimens, where the plastic strain at the calculated fatigue limit
Table 4 Elastic–plastic behaviour of the test specimens Test series Bohm X1 X2 X3 Y1 Y2 Y3 Z1 Z2 Z3 Magin 7/1 10/1 20/1 38/1 80/1 7/2 20/2 38/2 80/2 10/5 20/5 35/5 80/5
Kt
D (mm)
d (mm)
r (mm)
⌬sR,a/2 (MPa)
⌬sN/2 (MPa)
etot/ep
1 1 1 2.24 2.25 2.25 5.11 5.75 5.86
7 20 38 9.6 29 53.6 11 32.9 60.3
– – – 7 20.8 38.6 7 20.9 38.3
– – – 0.92 2.75 5.05 0.15 0.34 0.60
468 400 389 564 530 510 652 597 570
455 398 387 523 498 483 579 547 530
1.060 1.013 1.010 1.068 1.041 1.028 1.141 1.069 1.049
1 1 1 1 1 2.16 2.25 2.25 2.25 5.38 5.38 6.10 5.71
7 10 20 38 80 9.2 27.6 52.1 108.8 16 32 56 128.3
– – – – – 6.8 20.4 38.5 80.4 10 20 35 80.3
– – – – – 0.6 1.8 3.4 7.1 0.1 0.2 0.35 0.8
462 440 425 373 394 647 581 525 514 660 668 496 555
454 431 419 372 392 573 537 498 482 588 593 469 521
1.058 1.033 1.024 1.009 1.013 1.166 1.087 1.038 1.029 1.138 1.148 1.010 1.035
26
M. Makkonen / International Journal of Fatigue 25 (2003) 17–26
exceeds the limit value ep,lim, these values are indicated by bold characters in Table 3. Finally, it shall be noted that the calculation of the initiated crack depth from Eq. (9) is approximate by nature. Furthermore, the exact value of the stress intensity threshold is not known, and it is questionable if a value normally defined for long cracks is applicable for very small initial cracks. It was shown in Ref. [1] that the possible error in the estimated crack size is ignorable in the calculation of the statistical size effect. The influence to the geometric size effect is naturally larger. This was tested to Bo¨ hm’s specimen Z3 by increasing the estimated stress intensity factor range threshold from 6 to 9 MPa √m. The obtained estimate of the fatigue limit was decreased by 9.7%. As far as notched specimens are in question, the shape of the initiated crack has a great influence in the obtained value of the stress intensity factor also. It was assumed in this work that the initiated crack tends to take the shape, which gives constant stress intensity factor along the crack front. This may be questionable for very small cracks. 6. Conclusions 1. One single method cannot predict the fatigue limit of blunt notches and sharp notches. In very small details the plastic portion of strain even at the fatigue limit is considerable, and this shall be accounted for. 2. The fatigue limit of blunt notches (notches with insignificant portion of plastic strain) can be accurately estimated with the combination of statistical and geometric size effects. For engineering purposes, the statistical size effect alone gives adequately accurate results. 3. The fatigue limit of ‘sharp’ notches (notches where the plastic strain exceeds a limit value) shall be estimated with a method, which takes into account that, when the notch root radius get smaller, the fatigue limit approaches a value, which is obtained as follows:
앫 The notch is assumed to act as an initial crack; 앫 The depth of the crack equals the depth of the notch; 앫 The fatigue limit is calculated to this crack with the help of linear elastic fracture mechanics and the stress intensity factor range threshold.
References [1] Makkonen M. Size effect and notch size effect in metal fatigue. Thesis for the degree of Doctor of Science (Technology), Lappeenranta University of Technology. Acta Universitatis Lappeenrantaensis 83; 1999. [2] Makkonen M. Statistical size effect in the fatigue limit of steel. Int J Fatigue 2001;23:395–402. [3] Castillo E. Extreme value theory in engineering. New York: Academic Press Inc., 1988. [4] Bo¨ hm J. Zur Vorhersage von Dauerscwingfestigkeiten ungekerbter und gekerbter Bauteile unter Beru¨ cksichtigung des statistischen Gro¨ sseneinflu¨ sses. Dissertation, Technische Universita¨ t Mu¨ nchen; 1979. [5] Cameron AD, Smith RA. Fatigue life prediction for notched members. Int J Pres Ves Pip 1982;10:205–17. [6] Wang X, Lambert SB. Stress intensity factors for low aspect ratio semi-elliptical surface cracks in finite-thickness plates subjected to nonuniform stresses. Eng Fract Mech 1995;51(4): 517–32. [7] Isida M, Noguchi H, Yoshida T. Tension and bending of finite thickness plates with a semi-elliptical surface cracks. Int J Fract 1984;26:157–88. [8] Peterson RE. Notch sensitivity. In: Sines G, Waisman JL, editors. Metal fatigue. New York: McGraw Hill; 1959. p. 293–306. [9] Topper TH, El Haddad MH. Fatigue strength prediction of notches based on fracture mechanics. Fatigue Thresh 1982;2:777–97 [EMAS, Warley, UK]. [10] Smith RA, Miller KJ. Prediction of fatigue regimes in notched components. Int J mech Sci 1978;20:201–6 [Pergamon Press]. [11] Boller Chr, Seeger T. Materials data for cyclic loading, Part A: Unalloyed steels. Netherlands: Elsevier, 1987. [12] Tada H, Paris P, Irwin R. The stress analysis of cracks handbook. Hellerton, Pennsylvania: Del Research Corporation, 1973. [13] Magin W. Untersuchung des geometrischen Gro¨ sseeinflu¨ sses bei Umlaufbiegeanspruchung unter besonderer beru¨ cksichtigung technologische Einflu¨ sse. Dissertation, Technische Hochschule Darmstadt, 1981. [14] Taylor D. Geometrical effects in fatigue: a unifying theoretical model. Int J Fatigue 1999;21:413–20.