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Spectrum fatigue life assessment of notched specimens based on the initiation and propagation of short cracks
International Journal of Fatigue 28 (2006) 1011–1021 www.elsevier.com/locate/ijfatigue
Spectrum fatigue life assessment of notched specimens based on the initiation and propagation of short cracks Sascha Laue, Hubert Bomas * Stiftung Institut fu¨r Werkstofftechnik, Badgasteiner Str. 3, D-28359 Bremen, Germany Received 1 October 2004; received in revised form 25 July 2005; accepted 4 August 2005 Available online 28 December 2005
Abstract This contribution deals with the initiation and growth of short fatigue cracks in cyclically loaded notched specimens made of 0.15 wt% carbon steel SAE1017. On the basis of experimentally determined data, a damage model based on cyclic crack growth has been developed which accounts for the anomalous behaviour of short fatigue cracks. In this model the initiation and propagation of a critical, naturally produced crack is regarded as damage. This approach allows the calculation of fatigue life for constant amplitude tests as well as for multi-level tests and irregular loading. Deviations from Miner’s rule, which have often been observed for two-level tests are attributed to the varying fraction of crack initiation and propagation phase for different loads. The impreciseness of Miner’s rule deduced from two-level tests is of secondary importance for service life calculation when compared with the negligence of amplitudes below the fatigue limit. The proposed model yields shorter service life than the elementary version of Miner’s rule. q 2005 Elsevier Ltd. All rights reserved. Keywords: Short cracks; Notches; Fatigue load spectra; J-Integral; Damage accumulation
1. Introduction The calculation of lifetime for cyclically loaded specimens and components is often carried out by use of the well known rule of Miner [1]. According to the linear damage accumulation law DM Z
X Ni NFi
(1)
where Ni denotes the number of cycles and NFi the lifetime on the load level with the subindex i, the component will fail when Miner’s sum DM equals unity. In Eq. (1), the lifetime NF can be taken either from reference stress–life or from strain–life curves, which have been determined in constant amplitude tests. The application of Miner’s rule to the calculation of lifetime under multi-level or irregular loads gives rise to underestimations as well as to overestimations of lifetime. Residual stresses caused by severe overloads and the effects of variable load amplitudes can be regarded as substantial causes for the shortcomings of Miner’s rule. * Corresponding author. Tel.: C49 421 5350; fax: C49 421 5333. E-mail address: [email protected] (H. Bomas).
0142-1123/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2005.08.013
A literature review performed by Buch [2] has shown the inaccuracy of Miner’s rule for two-level program-tests. According to Buch [2], a Miner sum DM larger than unity is ‘usual’ for low–high load sequences, whereas DM is less than unity for high–low sequences. Deviations from this concept have been explained by the beneficial effect of compressive residual stresses. Schu¨tz[3] made an analysis of DM values obtained in various variable-amplitude test series reported in the literature. In the random load tests the average value was near DMZ1, while individual values showed a large amount of scatter [3]. In order to overcome the shortcomings mentioned above, some modifications of Miner’s original rule have been developed to account for the influence of damaging cycles below the fatigue limit SFL on service life. For the elementary version of Miner’s rule, the stress–life curve is extended below the fatigue limit SFL while maintaining the slope k (Fig. 1) [4]. On the basis of a literature review, Zenner and Liu [5] found out that Miner’s sum for irregular loads on average is smaller than unity, even if the elementary version of Miner’s rule is used. An improvement of fatigue life assessment has been achieved by altering the reference stress–life curve. The modified curve deviates with a steeper slope k 0 from the original curve at the maximum value of the load spectrum S a and ends at half of the fatigue limit of the original curve [5].
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S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021
Fig. 1. Reference stress–life curves for different approaches to service life assessment [1,4,5]. Fig. 2. Specimen geometries (all dimensions in mm).
The aim of this contribution is to describe the cumulative damage caused by cyclic loading by use of a damage model based on the initiation and growth of short fatigue cracks. It is to be examined to what extent load sequence effects can be related to short crack growth and what aftermaths will arise for irregular loads. 2. Material, specimen geometry and manufacturing The material used for the experimental work was an unalloyed low carbon steel SAE1017 (DIN short description: Cm15, Material code 1.1140) whose chemical composition is given in Table 1. The material was delivered as flats in the dimensions 3000!100!10 mm3 and thereafter heat treated in order to establish two different but accurately defined initial microstructural states. The normalisation (saltbath 910 8C 20 min/air) resulted in a fine-grained structure, showing a mean grain diameter of 6.7 mm (11 mm according to ASTM E 112-85) determined by means of a digital imaging system. The second heat treatment (vacuum 1200 8C 8 h/furnace 2 K/min 500 8C/air) produced a coarse-grained structure with a mean grain diameter of 56 mm (88.4 mm according to ASTM E 112-85). The fatigue-test specimens shown in Fig. 2 were machined from the heat treated flats by milling and grinding. After grinding the notch root by use of a mounted point, the surface in the notch area has been mechanically and electrochemically polished. The finishing operation was done with the aid of a polishing felt and diamond paste with a grain size in the range of 1–3 mm. After that, compressive residual stresses of about 40–80 MPa were measured in the notch root. Characteristic Table 1 Chemical composition (mass content in %)
DIN 17210 Analysis
C
Si
Mn
P
S
Cr
0.12– 0.18 0.17
%0.40
0.30– 0.60 0.52
%0.035
0.020– 0.040 0.023
–
0.17
0.006
0.14
material properties obtained from standardised tensile specimens are listed in Table 2. 3. Testing techniques and crack monitoring The fatigue tests were carried out on closed loop servohydraulic testing systems. For unnotched specimens an extensometer with a gauge length of 5 mm has been used as sensor element for total strain controlled tests, whereas for notched specimens a strain gauge with a grid length of 1 mm attached to the notch root served as sensor element. The cyclic load was applied in a triangular shape with a constant absolute value of the strain-rate of jd3/dtjZ0.02 sK1 at a strain ratio of R3ZK1. In order to compare the results of strain controlled fatigue test with results obtained from load controlled test, some experiments were also performed under load controlled condition. Service life of notched specimens was determined in load controlled tests using pseudo-random loads (Gaussian distribution). In the case of constant amplitude loading of notched specimens, the fatigue process, which could be directly related to the initiation and propagation of short surface cracks, was observed by surface replicas: Before the crack reached a predefined critical surface length, the testing machine was stopped 30–40 times and surface replicas were taken in the accessible notch. The material used for surface replicas was a two-component silicone of low viscosity which is also used in dentistry (PROVILw novo Light C.D. regular set, Producer Heraeus Kulzer GmbH). After reaching a surface crack length of about 2cz1000 mm in the notch root, the experiment was terminated and the main crack was traced back to its starting point. For that purpose images of the crack were taken by a digital camera mounted on a light-optical microscope. Crack measurements were done by using an image processing software. In order to determine the crack depth a associated to the surface crack length 2c, some specimens had been deepfrozen in liquid nitrogen and broken open. Micrographs taken
S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021
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Table 2 Properties after heat treatment, mean and standard deviation
Fine grain Coarse grain
Rm (MPa)
ReL (MPa)
ReH (MPa)
A (%)
Z (%)
E (GPa)
Grain-: (mm)
444G4 370G9
295G6 183G4
389G16 –
37G2 41G1
62G2 61G3
207G2 195G4
6.7 56
Rm, tensile strength; ReL, lower yield strength; ReH, upper yield strength; A, elongation; Z, reduction of area.
Table 3 Parameters for the Manson–Coffin relationship and the Ramberg–Osgood equation for different specimen geometries and grain sizes
Unnotched (fine grain) Unnotched (coarse grain) Notched (fine grain) Notched (coarse grain)
sf0 (MPa)
b
E (GPa)
3f0
c
K 0 (MPa)
n0
1583 976 678 931
K0.162 K0.129 K0.082 K0.119
207 195 207 195
1.356 6.138 0.265 1.474
K0.662 K0.839 K0.444 K0.645
867 826 – –
0.185 0.178 – –
by a scanning electron microscope were showing almost semicircular fatigue fracture surfaces independent of stress concentration factor and strain amplitude [6]. The following calculations are based on the assumption of a semicircular crack shape.
4. Cyclic properties Incremental step tests have been carried out for unnotched specimens in fine and coarse-grained condition, respectively. The experiments yielded nearly the same cyclic stress–strain curve 3a Z 3ea C 3pa Z
sa sa 1=n0 C E K0
(2)
independent of the grain size, which has been found earlier also by Hoshide et al. [7] as well as by Hatanaka and Yamada [8]. The cyclic strain-hardening exponent n 0 and the cyclic strainhardening coefficient K 0 are therefore nearly the same for both variants. The values for n 0 and K 0 listed in Table 3 maybe slightly biased by the fact that the unnotched specimens show a stress concentration factor of KtZ1.11. Fig. 3 shows the experimental results for unnotched specimens made of material in fine and coarse-grained condition. Continuous lines have been achieved by fitting the experimental data to the Manson– Coffin relationship 3a Z
sf0 ð2NF Þb C 3f0 ð2NF Þc ; E
root needed for the derivation of the constants in Eq. (3) were determined by using Eq. (2). The coarse-grained specimens reach approximately 70% of the cycles to failure of the fine-grained variant. The difference in cycles to failure can be mainly attributed to the influence of grain size because the cyclic stress–strain behaviour is nearly the same for both variants. In Fig. 4 the strain amplitude versus cycles to failure for different specimen geometries is shown. In the case of the fine-grained variant it becomes apparent that the results for notched specimens lie in a narrow scatter band independent of the stress concentration factor. Also specimens that have been stress relieved after machining by annealing can be found in the same scatter band. Some specimens were also tested under load controlled conditions. In these cases the cyclically stabilised strain amplitude was used for the entry into the strain–life diagram. If the strain amplitude approaches the fatigue limit it becomes obvious that the number of cycles to failure for unnotched specimens is much smaller than for notched specimens and the difference rises up to a factor of 10. These results seem to be contradictory to the strain based approach to fatigue that assumes similar cycles to failure for the same strain amplitude.
(3)
where sf0 b 3f0 c
fatigue fatigue fatigue fatigue
strength coefficient strength exponent ductility coefficient ductility exponent
The resulting parameters for unnotched and notched specimens are also listed in Table 3. In the case of notched specimens, the elastic and plastic strain amplitudes at the notch
Fig. 3. Strain–life curves of unnotched specimens in fine-grained and coarsegrained condition, R3ZK1.
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S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021
Fig. 4. Strain–life curves of unnotched and notched specimens in fine-grained condition, R3ZK1.
In the case of coarse-grained specimens (Fig. 5) the situation is different. Notched specimens exhibit only a slightly longer lifetime than unnotched specimens. The difference can be described satisfactorily by the statistical size effect that accounts for the different size of highly stressed volume in unnotched and notched specimens [9]. The huge difference in lifetime in the case of fine-grained material can be explained by the initiation and propagation of fatigue Lu¨ders-bands. These bands that can be observed at the surface of cyclically loaded unnotched specimens, cause a significant rise in surface roughness and therefore have a strong influence on the fatigue process [6,9,10]. 5. Modelling of short crack growth in notched specimens 5.1. Constant amplitude tests The growth of a crack under elastic–plastic conditions can be described advantageously by the J-integral that can be calculated by the approximation formula (4) proposed by Dowling [11] s2 1:02 J Z Je C Jp Z 1:24 C pffiffiffi $s$3pl $a (4) n E where
n s 3pl a
strain-hardening exponent stress plastic strain crack depth
For cyclic loading the DK approach to fatigue crack growth is limited to situations where the plastic zone is small compared to the crack length or small compared to the uncracked ligament. But this is not the case for surface cracks in a uniform cyclic plastic strain field, as during lowcycle fatigue tests after crack initiation, and is not also the case for small cracks growing in regions of cyclic plasticity associated with notches. The applicability of the cyclic J-Integral DJ to long crack growth was shown by Dowling and Begley [12] whereas its applicability to cyclic short crack growth in unnotched specimen was shown for example by Vormwald and Seeger [13] and by Schubert [14]. Hoshide and Kusuura [15] used DJ for the simulation of short fatigue cracks in notched specimens whereas Tanaka et al. [16] showed its applicability for short fatigue cracks in specimens subjected to alternating bending. Since, the growth of short fatigue cracks is influenced by crack closure a convention about the consideration of crack closure within the approximation formula for the cyclic
Fig. 5. Strain–life curves of unnotched and notched specimens in coarse-grained condition, R3ZK1.
S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021
J-Integral (5) has to be made ðDseff Þ2 1:02 C pffiffiffiffi0 $Dseff $D3p;eff $a DJeff Z 1:24 E n
(5)
where n 0 cyclic strain-hardening exponent Dseff effective stress range D3p,eff effective plastic strain range Vormwald and Seeger [13] developed a sophisticated method for the determination of crack opening and crack closure levels and thus a method for the determination of effective values of Ds and D3p. Under plane stress condition the resulting DJeff -value can then be used topffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi calculate an equivalent stress intensity factor DKJ;eff Z DJeff $E and furthermore the crack growth rate according to the crack growth Eq. (6). The parameters m and C, usually describing long crack growth, can now be used for the calculation of the growth of physically short cracks da Z CðDKJ;eff Þm ; dN
(6)
where da/dN crack growth rate DK cyclic stress intensity factor C, m parameters describing long crack growth In the case of notched specimens it is necessary to use local values of cyclic stabilised stresses and strains for the evaluation of Eq. (5). This procedure is based upon the assumption that the growth of a crack at the notch root behaves as in an unnotched specimen as long as the crack is small and its growth is not influenced by stress or strain gradients. These local stresses and strains can either be determined by the well-known Neuberformula [17,18] or similar approaches [19–22]. In some cases also an elastic–plastic finite element method [6] may be convenient in order to decide whether plane stress or plane
1015
strain condition exists at the notch root. For a specimen geometry similar to that used in this study Sharpe [19] recommended a combination of Neuber’s rule with Eq. (2). The consideration of crack closure in Eq. (5) for the elastic as well as for the plastic part of DJeff produced values of DKJ,eff that did not coincide with the long crack behaviour shown in Fig. 6. Only the use of full ranges of local stabilised cyclic stresses and strains for the plastic part of Eq. (5) provides DKJ,eff being large enough to coincide with the long crack data. This procedure is in accordance with results of Schubert [14], who found that under cyclic loading in the LCF region, where the plastic part of DJeff is dominant, a proper description of small crack growth could only be achieved if the full ranges of nominal stress and nominal plastic strain were used to calculate the plastic part of DJeff. The elastic part of DJeff was calculated by using Dseff Z UDsZ 0:8Ds, where U describes the fraction of a cycle while the crack is open. The value for U was taken from literature. As mentioned before it is necessary to replace nominal stresses and strains by local stabilised values. Fig. 6 shows the relation between crack growth rate and the effective values of the ranges of the stress intensity factors. The curve for long crack growth was completed with a threshold value DKth,eff, which was taken from literature [23]. The diagram shows the crack growth rate of short cracks oscillating around the crack growth curve for long cracks in the Paris regime. No threshold behaviour can be seen for short cracks. Model calculations have shown that the influence of these oscillations on fatigue life can be neglected. Therefore, it can be assumed that short crack growth can be described by the load parameter DKJ,eff and the long crack constants m and C. In Fig. 7, a comparison of measured and calculated crack growth curves is shown. For the calculation of crack growth the formula da/dNZC(DKJ,eff)m was used. The initial crack length needed for calculation had been chosen in such a way that calculated and measured lifetime coincide for the highest load. For lower loads the calculation considerably underestimates
Fig. 6. Comparison of effective values of the cyclic stress intensity factors for long cracks in compact specimen (DKeff) and short cracks at the notch root (DKJ,eff, DJeff), KtZ1.54.
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S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021
Fig. 7. Comparison of measured and calculated crack growth curves from finegrained specimens (KtZ1.54).
the measured lifetime and a quantitative congruence between measured and calculated crack growth curves can be achieved only if the calculated curve is shifted along the abscissa. These findings may be explained by the fact that the fraction of fatigue life spent in crack initiation increases when the strain amplitude approaches the fatigue limit. As a consequence of this insight, it is necessary to expand the crack growth equation mentioned above. The expansion should be capable to describe crack initiation as well as crack propagation. Investigations concerning the early stages of fatigue have shown that surface damage can be related to shape changes of grains and the associated increase of surface roughness in bcc polycrystals [24]. After a rapid increase of surface damage at the beginning of cyclic loading the plasticity induced surface roughening reaches a steady-state [25]. A continued cyclic loading leads to the initiation of short cracks either in slip bands or at grain boundaries with large altitude difference between neighbouring grains and further damage is concentrated in these sites of crack initiation. Henceforth, the further development of surface roughness is of little importance. Both processes occur more or less at the same time so that a suitable model should be able to describe the process of surface roughening as well as the process of crack growth depending on crack length and local stresses and strains. A simple mathematical model capable of describing the damage process can be established by the addition of two power laws of the Paris-type: da da da Z C ZC1 ðDKJ;eff Þm1 CC2 ðDKJ;eff Þm2 dN dN initiation dN propagation (7)
C1, m1 crack initiation parameters C2, m2 crack growth parameters The effective cyclic stress intensity factor DKJ,eff serves as crack driving force in the crack initiation period as well as in
the crack propagation period. While the constants C2 and m2 for the crack propagation period are known from long-crack experiments, the constants C1 and m1 for the crack initiation period have to be determined by adjustment to measured lifetimes. There is no physical justification for the choice of DKJ,eff as crack driving force in the crack initiation period, where the crack length is in the order of the grain size. Equally, the shear strain range [26] or the shear stress amplitude [15,27] could have been chosen. But the choice of the same crack driving force in the crack initiation period as well as in the crack propagation period allows on the one hand for an easier comparability of damage rates and on the other for a simplified programming of the model. The description of a damage process comparable to the above-mentioned is made possible if a negative exponent m1 in Eq. (7) is used. The first term in Eq. (7) describes a rapid increase of damage due to surface roughening at the beginning of cyclic loading. When the crack becomes larger, the contribution of the first term of Eq. (7) to damage diminishes and further damage can be solely related to crack propagation. The adjustment of the model parameters to measured strain– life curves resulted in the following relationships: fine grain: log C1 ZaCblogð3a ÞCc log2 ð3a Þ
(8)
where a K20.314 b K32.149 c K8.607 m1 K6 coarse grain: C1 Z6:383!1013 35:1411 a
(9)
m1 ZK0:0228530:8053 a In the case of fine-grained specimens it was sufficient to choose m1 independent of the strain amplitude whereas for coarse-grained specimens it was necessary to model m1 load dependent. The parameters were determined by a stepwise iteration of C1 and m1 followed by integration of Eq. (7) for constant strain amplitude. The iteration was terminated when modelled lifetime was in accordance with measured lifetime. To obtain a continuous relationship m1Zf(3a) and C1Zf(3a) it is necessary to repeat this procedure for several different strain amplitudes. Eqs. (8) and (9) were fitted to these discrete results. Fig. 8 illustrates the principle of the model. In the domain of microstructurally short cracks, where the crack length is about the same as the mean grain size, the model predicts higher crack growth rates than for physically short cracks and long cracks, what is a common and often observed feature of short cracks [26]. In this domain the calculated crack growth rate is dominated by the first term in Eq. (7). When the growing crack approaches a dominant microstructural barrier, for example a grain boundary or a pearlite grain, the crack growth rate reaches a minimum. In the domain of physically short and long
S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021 ∆ Jeff [N/mm] 0.1
10-3
∆ Jeff [N/mm] 0.1
1
1017
1
10-3 εa = 0.80 %
10-4 εa = 0.40 %
10-5
10-6
εa = 0.18%
Model εa= 0.40 % εa= 0.24 % εa= 0.18 % εa = 0.16 %
10-7
10-8
εa = 0.16%
da/dN [mm]
da/dN [mm]
10-4
εa= 0.30 %
10-5 εa= 0.12 %
10-6
10-7
10-8
100 200 400 600 800 ∆KJ,eff, ∆Keff [Nmm-3/2]
Model εa = 0.30 % εa = 0.24 % εa = 0.18 % εa = 0.12 % 100
200 400 600 800 ∆KJ,eff, ∆Keff [Nmm-3/2]
Fig. 8. Comparison of measured and modelled crack growth rates for fine (left) and coarse-grained (right) material; KtZ1.54.
cracks where the crack is about the size of a few grains, fatigue crack growth is governed by the second term of Eq. (7). Fig. 8 makes clear as well, that the method chosen for crack monitoring does not allow for the measurement of crack growth rate, when crack length is smaller than the grain size. If the normalised crack length, which is defined as actual surface crack length 2c divided by the criterion for a critical crack length of 1 mm, is taken as damage D, the integration of Eq. (7) yields the evolution of damage for different cyclic loads. Fig. 9 makes clear that the established model provides a reasonable description of damage evolution in cyclically loaded specimens. What can also be seen in Fig. 9 is that different strain amplitudes lead to different damage curves. The load dependence of the damage curve is a vital requirement for the description of load sequence effects [28]. 5.2. Two-level tests A magnified cutout of Fig. 9 showing the evolution of damage for small cycle ratios N/NF, is shown in Fig. 10. Starting
at the same initial damage represented by an initial crack depth aZ1 mm, different strain amplitudes lead to different damage curves. The evolution of damage proposed by Miner’s rule is shown as well and clarifies the difference to the recent model. The following example shows that the model is capable of describing the effect of load sequence on fatigue life: A specimen is cycled with a local strain amplitude of 3aZ0.16% until the number of cycles reaches N/NFZ0.5. The local strain is subsequently increased to 3aZ0.4% which means changing the damage curve on a horizontal line DZconst. It can be seen from Fig. 10 that the remaining life is about N/NFZ0.85. Thus, the model yields a Miner sum DMZ1.35. If the strain sequence is started on a high level until N/NFZ0.1 and then lowered to a low level, a remaining life of N/NFZ0.6 can be deduced from Fig. 10. The corresponding Miner sum is DMZ0.7. These results are in good agreement with numerous former experiments dealing with load sequence effects and Miner’s damage function [2]. Further on, Weber and Dengel [29] have shown for high–low load sequences that the inaccuracy of Miner’s rule can be attributed to the fact that the transition from
1.0
1.0 εa = 0.30 %
εa = 0.40 % εa = 0.24 %
εa = 0.18 %
εa= 0.16 %
0.6
Miner Model 0.4
0.2
0.0 0.0
εa = 0.24 %
0.8
εa = 0.18 %
DMiner, D = 2c/1mm
DMiner, D = 2c/1mm
0.8
εa = 0.12 % 0.6
Miner Model
0.4
0.2
0.2
0.4
0.6 N/NF
0.8
1.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
N/NF
Fig. 9. Modelled and measured relationship between normalised crack length and normalised life (KtZ1.54); left: fine-grained material, right: coarse-grained material.
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S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021 0.15
1.0 coarse grain 0.8
LH
εa = 0.24 %
0.10
εa = 0.18 %
N2/NF2
DMiner, D = 2c/1mm
εa = 0.40 %
εa = 0.16 % Miner
Short crack model: 0.10 % <-> 0.30 % 0.12 % <-> 0.30 % 0.24 % <-> 0.30 % Miner:
0.6 0.4
0.05
Experiment (Kt = 2.57): 0.12 % <-> 0.30 %
HL
0.2 0.0 0.0
0.00 0.0
0.1
0.2
0.3
0.4
0.2
0.5
0.4 0.6 N1/NF1
0.8
1.0
N/NF
Fig. 10. Magnified cutout of the modelled curves in Fig. 9 (fine-grained material, KtZ1.54).
an undamaged state into a damaged condition occurs at different cycle ratios N/NF if different load levels are involved. The proposed short crack model describes a load dependent transition point as well. The minimum of the crack growth rate in Fig. 8, which can be found as inflection point in the damage curve in Fig. 10 can be regarded as transition from minor damage to severe damage. Figs. 11 and 12 show the influence of load sequence on remaining life fraction for the fine-grained material and the coarse-grained material, respectively. In case of low–high (LH) load sequences, the calculation was started with a low strain amplitude 3a until a certain life fraction N1/NF1 was reached. After that the calculation was continued with a higher strain amplitude until the final crack length of 2cZ1 mm was reached. In the case of high–low (HL) load sequences the calculation was started with a high amplitude and then continued with a low strain amplitude. From Figs. 11 and 12 it can be deduced that LH-sequences always lead to Miner sums larger than unity whereas HL-sequences always lead to Miner sums smaller than unity. The resulting curves are quite similar to those which can be achieved by the application of the Damage Curve Approach (DCA) or the Double Linear Damage Rule (DLDR) [30]. The predicted results are in reasonable 1.0 fine grain Short crack model: 0.16 % <-> 0.18 % 0.16 % <-> 0.24 % 0.16 % <-> 0.40 % Miner:
0.8
N2/NF2
LH 0.6
Experiment (Kt = 2.57): Two-level tests 0.16 % <-> 0.24 %
0.4
0.2
0.2
agreement with experimental results although there is a certain amount of scatter, especially for the LH-sequence. The calculation for two closely neighbouring strain amplitudes (e.g. 3aZ0.24% and 3aZ0.30% in Fig. 12) leads to Miner’s rule again, which is in agreement with the insight, that a monotonically increasing non-linear damage function still leads to Miner’s rule. 5.3. Irregular loads To overcome the inaccuracy that will be caused by the insufficient transferability of cycles to failure from unnotched to notched specimens (Fig. 4) the short crack model was only applied for spectrum fatigue life assessment of notched specimens. First of all the strain–life curve of notched specimens in fine and coarse-grained conditions using Eq. (7) were calculated for several discrete values of the cyclic strain amplitude. A relationship of the form 3a Za1 ð2NF Þa2 Cb1 ð2NF Þb2
(10)
was then adjusted to these discrete results in order to obtain a continuous description of the calculation results, which is needed for a calculation of service life based on Miner’s rule. The corresponding constants are given in Table 4. A comparison between measured and modelled strain versus life curves for notched specimens subjected to constant amplitude loading is shown in Figs. 13 and 14 for fine and coarse-grained material, respectively. The proposed model can now also be used to calculate lifetime for randomised cyclic loading. The calculation is based on a randomly generated period of amplitudes that can be Table 4 Constants used for the strain–life relationship of notched specimens
HL 0.0 0.0
Fig. 12. Calculation of the remaining life in two-level tests for high–low (HL) and low–high (LH) load sequences (coarse-grained material). Scatter bars indicate the scatter of experimentally determined remaining life for 10 and 90% probability of crack initiation, respectively (five specimens).
0.4 0.6 N1/NF1
0.8
a1 (%)
1.0
Fig. 11. Calculation of the remaining life in two-level tests for high–low (HL) and low–high (LH) load sequences (fine-grained material). Scatter bars indicate the scatter of experimentally determined remaining life for 10 and 90% probability of crack initiation, respectively (five specimens).
Notched (fine grain) Notched (coarse grain)
a2
b1 (%)
b2
37
K0.485
0.694
K0.134
5474
K1.141
3.1
K0.237
S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021
Strain amplitude εa, ε a [%]
Notched specimensKKt = 1.54 and Kt= 2.57: measured strain vs. life curve calculated strain vs. life curve (short crack model) calculated prestrain strain vs. life curve with εa = 0.40 % until 0.02 NF Notched specimens Kt = 1.54: measured service life calculated service life (original Miner, εFL = 0.12 %) calculated service life (elementary Miner, OL = 0.5*εFL ) calculated service life (short crack model, OL = 0.5*εFL)
0.8 0.7 0.6 0.5 0.4 0.3 0.2
εFL
0.1
0.1 104
105
106
107
Cycles to failure NF Fig. 13. Comparison of measured data and model calculation for constant-level and pseudo-random loads (Gaussian distribution) for fine-grained material (KtZ1.54 and 2.57). FL, fatigue limit; OL, omission level.
described by a gaussian load spectrum of the form H ZH0 eKðSa =Sa Þ
2
ln H0
with H0 Z106 ; I Z1; RZK1
(11)
where H cumulative frequency H0 size of period I irregularity factor R stress ratio.
Strain amplitude εa,εa [%]
Notched specimens Kt = 1.54 and Kt = 2.57: measured strain vs. life curve calculated strain vs. life curve (short crack model) calculated prestrain strain vs. life curve with εa = 0.40 % until 0.1 NF Notched specimens Kt = 1.54: measured service life calculated service life (original Miner, εFL = 0.1 %) calculated service life (elementary Miner, OL = 0.5*εFL) calculated service life (short crack model, OL = 0.5*εFL)
0.80 0.70 0.60 0.50 0.40 0.30 0.20
εFL
0.10 0.09 0.08 0.07 0.06
104
105
106
0.1
107
Cycles to failure NF
Fig. 14. Comparison of measured data and model calculation for constant-level and pseudo-random loads (Gaussian distribution) for coarse-grained material (KtZ1.54 and 2.57). FL, fatigue limit; OL, omission level.
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To calculate the incremental damage caused by a single cycle, the local stress and strain amplitudes that correspond to the random nominal stress amplitude Sa have to be calculated. This can be done either by a modified Neuber-formula [19] or an elastic–plastic finite element method [6]. Starting with a crack length of aZ1 mm the evolution of damage can be calculated using Eq. (7) in connexion with Eq. (8) or Eq. (9). The calculation was terminated when the crack reached a critical length of 2cZ1 mm and aZ0.5 mm, respectively. A similar procedure was adopted for the calculation of service life based on the original and elementary rule of Miner. The local strain amplitude was used to calculate the incremental damage DZ1/NF of a single cycle by numerical evaluation of Eq. (10). The calculation was terminated when Miner’s sum equalled unity. Figs. 13 and 14 give a comparison of service lives for notched specimen calculated on the basis of short crack growth, the elementary and the original version of Miner’s rule. The calculation results are completed with experimentally determined service lives. Since the service life of notched specimens has been measured under load controlled conditions it is necessary to convert the maximum stress S a of the Gaussian load spectrum into the corresponding maximum local strain amplitude 3a . This was done by use of the approximation formula mentioned before. In the case of fine-grained material, the original version of Miner’s rule overestimates service life, in particular when the maximum strain approaches the fatigue limit. When the elementary version of Miner’s rule is used, an improved calculation of service life can be achieved (Fig. 13). The convergence of calculated service life for higher strains 3a indicates that the inaccuracy of ‘original Miner’ is mainly caused by the negligence of damaging cycles below the fatigue limit. The use of the proposed short crack model gives rise to a further improvement of service life assessment, which may be attributed to the incorporation of the load sequence effect into the model. This effect may be explained as follows: the early occurrence of high amplitudes during irregular cycling loading causes a severe reduction of the remaining life for lower amplitudes. The result is an increased damage per cycle compared with constant-level tests at the same low amplitude. This effect is demonstrated in Fig. 13 by the calculation of a prestrain strain–life curve. Short cycling at a high strain level (3aZ 0.4%) causes a severe reduction of remaining life, if the experiment is continued on a lower load level. The presented results agree well with the investigations of Zenner and Liu [5], who proposed a reference curve with drastically reduced lifetimes in order to improve service lifeassessment based on Miner’s rule. That empirically determined curve may be interpreted as a stress–life curve, whose crack initiation portion has been cut off. The application of the original variant of Miner’s rule overestimates service life also in the case of coarse-grained material. The use of the elementary version of Miner’s rule leads to an improved calculation of service life (Fig. 14). It can be seen from Fig. 14 that the application of the proposed short
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S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021
crack model gives the same results as the elementary version of Miner’s rule. Both approaches overestimate the measured service life for high strain amplitudes 3a whereas for low strain amplitudes the calculation provides a reasonable agreement with experimental results. From the considerations above it can be assumed that under pseudo-random loading the initiation of a fatal crack in the that coarse-grained material takes place at an early stage of fatigue life. This means that the damage evolution can be attributed mainly to crack propagation, whereas crack initiation is of minor significance. This in turn means that the evolution of crack growth can be described by a crack growth law. The application of a crack growth equation of the Paris-type for the description of damage evolution yields the same result as using Miner’s rule [31]. This may be the explanation for the congruence of service life calculation based on Miner’s rule as well as on the short crack model in the case of coarse-grained material.
even when the elementary version of Miner’s rule is used. Deviations from this concept may be attributed to crack retardation effects due to overloads or situations where crack initiation and growth is influenced by residual stresses.
Acknowledgements The financial support of this work by the Deutsche Forschungsgemeinschaft DFG in the framework of the priority programme ‘Fatigue life-time prediction of metals based on microstructural behaviour’ is gratefully acknowledged. Special thanks go to Dr J. Hu¨necke and Dr D. Scho¨ne of Bundesanstalt fu¨r Materialforschung und -pru¨fung for the experimental determination of service life of notched specimens and strain–life curves of unnotched specimens.
6. Conclusions
References
1. In the case of constant amplitude tests the local strain approach to fatigue underestimates the number of cycles to failure for notched specimens up to a factor of ten in the case of steel SAE1017 in fine-grained condition whereas for coarse-grained material the local strain approach yields reasonable results. The huge difference in lifetime in the former case can be explained by the initiation and propagation of fatigue Lu¨ders-bands. 2. The growth of small cracks in notched specimens of low carbon steel can satisfactorily be described by the cyclic J-integral in the crack propagation phase, but at lower amplitudes in the high cycle fatigue region the calculation of lifetime based on a Paris-equation and the cyclic J-integral underestimates the measured lifetimes. In order to obtain a more realistic description of the damage process, it was necessary to incorporate the evolution of damage during the crack initiation period into the model. This can be done by the combination of two power laws of the Paristype. The proposed model describes a strong increase of damage at the beginning of cyclic loading due to high growth rates of short cracks. 3. The model provides the opportunity to relate the deviations from Miner’s rule in two-level tests to the load dependent portions of crack initiation phase on different strain levels. That sequence effect increases with increasing distance between load amplitudes and decreasing grain size. 4. The sequence effect deduced from two-level tests allows for the following assumptions. High loads, occurring at the beginning of fatigue cycling under random loading, cause early crack initiation. This crack can now be advanced by low load amplitudes near and even below the fatigue limit meaning there is no initiation period any longer. A Wo¨hler curve determined by constant amplitude tests always includes a certain load dependent portion of crack initiation period. Therefore spectrum fatigue life assessment based on Miner’s rule should normally overestimate service life,
[1] Miner MA. Cumulative damage in fatigue. J Appl Mech 1945;12: A159–A64. [2] Buch A. Loading sequence effects in two-level program-tests. Materialpru¨fung 1975;17:307–12. [3] Schu¨tz W. The prediction of fatigue life in the crack initiation and propagation stages. A state of the art survey. Eng Fract Mech 1979;11: 405–21. [4] Schijve J. Fatigue of structures and materials. Dordrecht: Kluwer Academic Publishers; 2001. p. 258. [5] Zenner H, Liu J. Vorschlag zur Verbesserung der Lebensdauerabscha¨tzung nach dem Nennspannungskonzept. Konstruktion 1992;44: 9–17. [6] Laue S. Experimentelle und theoretische Untersuchungen zur Ausbreitung kurzer Ermu¨dungsrisse in gekerbten Proben des Stahls Cm15. PhD Thesis, Universita¨t Bremen; 2004. [7] Hoshide T, Yamada T, Fujimura S, Hayashi T. Short crack growth and life prediction in low-cycle fatigue of smooth specimens. Eng Fract Mech 1985;21:85–101. [8] Hatanaka K, Yamada T. The effect of grain size on low-cycle fatigue strength in mild steel. Trans JSME 1981;A47:123–30. [9] Laue S, Bomas H, Mayr P. Untersuchungen zur Anrisslebensdauervorhersage gekerbter Proben mit dem o¨rtlichen Konzept am Beispiel des Stahls Cm15. Z Metallkd 2003;94(5):499–504. [10] Laue S, Bomas H, Mayr P. The effect of Lu¨ders bands onto the fatigue strength of samples of different geometries made of a Cm15 normalised steel. Pract Metallogr 2005;42:62–73. [11] Dowling NE. J-Integral estimates for cracks in infinite bodies. Eng Fract Mech 1987;26:333–48. [12] Dowling B, Begley JA. Fatigue crack growth during gross plasticity and the J-Integral. Mech Crack Growth 1976:82–103 [ASTM STP 590]. [13] Vormwald M, Seeger T. The consequences of short crack closure on fatigue crack growth under variable amplitude loading. Fatigue Fract Eng Mater Struct 1991;14:205–25. [14] Schubert T. Verformungsverhalten und Risswachstum bei low cycle fatigue. Fortschr.-Ber. VDI Reihe 18 Nr. 73. Du¨sseldorf: VDI-Verlag; 1989 p. 158–61. [15] Hoshide T, Kusuura K. Life prediction by simulation of crack growth in notched components with different microstructures under multiaxial fatigue. Fatigue Fract Eng Mater Struct 1998;21:201–13. [16] Tanaka K, Hoshide T, Maekawa O. Surface-crack propagation in planebending fatigue of smooth specimen of low-carbon steel. Eng Fract Mech 1982;16:207–20.
S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021 [17] Neuber H. Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress–strain law. J Appl Mech 1961;28:544–50. [18] Topper TH, Wetzel RM, Morrow JD. Neuber’s rule applied to fatigue of notched specimens. J Mater JMLSA 1969;4:200–9. [19] Sharpe Jr WN. ASME 1993 Nadai lecture—elastoplastic stress and strain concentrations. J Eng Mater Technol 1995;117:1–7. [20] Glinka G, Ott W, Nowack H. Elastoplastic plane strain analysis of stresses and strains at the notch root. ASME J Eng Mater Technol 1988;110: 195–204. [21] Walker EK. Multiaxial stress–strain approximations for notch fatigue behaviour. J Test Eval 1977;5:106–13. [22] Seeger T, Beste A. Zur Weiterentwicklung von Na¨herungsformeln fu¨r die Berechnung von Kerbbeanspruchungen im elastisch-plastischen Bereich. VDI-Fortschrittsberichte Reihe 18, 2, Du¨sseldorf. Du¨sseldorf: VDIVerlag; 1977. [23] Taylor D, Jianchun L. Sourcebook on fatigue crack propagation: thresholds and crack closure. Warley: Engineering Materials Advisory Services Ltd. 1985.
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[24] Mughrabi H. Introduction to the viewpoint set on: surface effects in cyclic deformation and fatigue. Scripta Metall Mater 1992;26:1499–504. [25] Kato A, Hayashi M. Fatigue life estimation of steel using laser speckle sensor. NDT E Int 1999;32:763–74. [26] Miller KJ. Initiation and propagation of short fatigue cracks. In: Lu¨tjering G, Nowack H, editors. Fatigue’96: Proceedings of the sixth international fatigue congress, vol.1. Berlin, Germany; 1996. p. 253–64. [27] Tanaka K, Mura T. A dislocation model for fatigue crack initiation. J Appl Mech 1981;48:97–103. [28] Marco SM, Starkey WL. A concept of fatigue damage. Trans ASME 1954;76:627–32. [29] Weber H, Dengel D. Eine effiziente Methode zur Ermittlung des Scha¨digungsbereiches bei Wo¨hleruntersuchungen. Mat-Wiss u Werkstofftech 1988;19:418–24. [30] Manson SS, Halford GR. Re-examination of cumulative fatigue damage analysis—an engineering perspective. Eng Fract Mech 1986; 25:539–71. [31] Haibach E. Betriebsfestigkeit. Berlin: Springer; 2002.
Spectrum fatigue life assessment of notched specimens based on the initiation and propagation of short cracks Sascha Laue, Hubert Bomas * Stiftung Institut fu¨r Werkstofftechnik, Badgasteiner Str. 3, D-28359 Bremen, Germany Received 1 October 2004; received in revised form 25 July 2005; accepted 4 August 2005 Available online 28 December 2005
Abstract This contribution deals with the initiation and growth of short fatigue cracks in cyclically loaded notched specimens made of 0.15 wt% carbon steel SAE1017. On the basis of experimentally determined data, a damage model based on cyclic crack growth has been developed which accounts for the anomalous behaviour of short fatigue cracks. In this model the initiation and propagation of a critical, naturally produced crack is regarded as damage. This approach allows the calculation of fatigue life for constant amplitude tests as well as for multi-level tests and irregular loading. Deviations from Miner’s rule, which have often been observed for two-level tests are attributed to the varying fraction of crack initiation and propagation phase for different loads. The impreciseness of Miner’s rule deduced from two-level tests is of secondary importance for service life calculation when compared with the negligence of amplitudes below the fatigue limit. The proposed model yields shorter service life than the elementary version of Miner’s rule. q 2005 Elsevier Ltd. All rights reserved. Keywords: Short cracks; Notches; Fatigue load spectra; J-Integral; Damage accumulation
1. Introduction The calculation of lifetime for cyclically loaded specimens and components is often carried out by use of the well known rule of Miner [1]. According to the linear damage accumulation law DM Z
X Ni NFi
(1)
where Ni denotes the number of cycles and NFi the lifetime on the load level with the subindex i, the component will fail when Miner’s sum DM equals unity. In Eq. (1), the lifetime NF can be taken either from reference stress–life or from strain–life curves, which have been determined in constant amplitude tests. The application of Miner’s rule to the calculation of lifetime under multi-level or irregular loads gives rise to underestimations as well as to overestimations of lifetime. Residual stresses caused by severe overloads and the effects of variable load amplitudes can be regarded as substantial causes for the shortcomings of Miner’s rule. * Corresponding author. Tel.: C49 421 5350; fax: C49 421 5333. E-mail address: [email protected] (H. Bomas).
0142-1123/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2005.08.013
A literature review performed by Buch [2] has shown the inaccuracy of Miner’s rule for two-level program-tests. According to Buch [2], a Miner sum DM larger than unity is ‘usual’ for low–high load sequences, whereas DM is less than unity for high–low sequences. Deviations from this concept have been explained by the beneficial effect of compressive residual stresses. Schu¨tz[3] made an analysis of DM values obtained in various variable-amplitude test series reported in the literature. In the random load tests the average value was near DMZ1, while individual values showed a large amount of scatter [3]. In order to overcome the shortcomings mentioned above, some modifications of Miner’s original rule have been developed to account for the influence of damaging cycles below the fatigue limit SFL on service life. For the elementary version of Miner’s rule, the stress–life curve is extended below the fatigue limit SFL while maintaining the slope k (Fig. 1) [4]. On the basis of a literature review, Zenner and Liu [5] found out that Miner’s sum for irregular loads on average is smaller than unity, even if the elementary version of Miner’s rule is used. An improvement of fatigue life assessment has been achieved by altering the reference stress–life curve. The modified curve deviates with a steeper slope k 0 from the original curve at the maximum value of the load spectrum S a and ends at half of the fatigue limit of the original curve [5].
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Fig. 1. Reference stress–life curves for different approaches to service life assessment [1,4,5]. Fig. 2. Specimen geometries (all dimensions in mm).
The aim of this contribution is to describe the cumulative damage caused by cyclic loading by use of a damage model based on the initiation and growth of short fatigue cracks. It is to be examined to what extent load sequence effects can be related to short crack growth and what aftermaths will arise for irregular loads. 2. Material, specimen geometry and manufacturing The material used for the experimental work was an unalloyed low carbon steel SAE1017 (DIN short description: Cm15, Material code 1.1140) whose chemical composition is given in Table 1. The material was delivered as flats in the dimensions 3000!100!10 mm3 and thereafter heat treated in order to establish two different but accurately defined initial microstructural states. The normalisation (saltbath 910 8C 20 min/air) resulted in a fine-grained structure, showing a mean grain diameter of 6.7 mm (11 mm according to ASTM E 112-85) determined by means of a digital imaging system. The second heat treatment (vacuum 1200 8C 8 h/furnace 2 K/min 500 8C/air) produced a coarse-grained structure with a mean grain diameter of 56 mm (88.4 mm according to ASTM E 112-85). The fatigue-test specimens shown in Fig. 2 were machined from the heat treated flats by milling and grinding. After grinding the notch root by use of a mounted point, the surface in the notch area has been mechanically and electrochemically polished. The finishing operation was done with the aid of a polishing felt and diamond paste with a grain size in the range of 1–3 mm. After that, compressive residual stresses of about 40–80 MPa were measured in the notch root. Characteristic Table 1 Chemical composition (mass content in %)
DIN 17210 Analysis
C
Si
Mn
P
S
Cr
0.12– 0.18 0.17
%0.40
0.30– 0.60 0.52
%0.035
0.020– 0.040 0.023
–
0.17
0.006
0.14
material properties obtained from standardised tensile specimens are listed in Table 2. 3. Testing techniques and crack monitoring The fatigue tests were carried out on closed loop servohydraulic testing systems. For unnotched specimens an extensometer with a gauge length of 5 mm has been used as sensor element for total strain controlled tests, whereas for notched specimens a strain gauge with a grid length of 1 mm attached to the notch root served as sensor element. The cyclic load was applied in a triangular shape with a constant absolute value of the strain-rate of jd3/dtjZ0.02 sK1 at a strain ratio of R3ZK1. In order to compare the results of strain controlled fatigue test with results obtained from load controlled test, some experiments were also performed under load controlled condition. Service life of notched specimens was determined in load controlled tests using pseudo-random loads (Gaussian distribution). In the case of constant amplitude loading of notched specimens, the fatigue process, which could be directly related to the initiation and propagation of short surface cracks, was observed by surface replicas: Before the crack reached a predefined critical surface length, the testing machine was stopped 30–40 times and surface replicas were taken in the accessible notch. The material used for surface replicas was a two-component silicone of low viscosity which is also used in dentistry (PROVILw novo Light C.D. regular set, Producer Heraeus Kulzer GmbH). After reaching a surface crack length of about 2cz1000 mm in the notch root, the experiment was terminated and the main crack was traced back to its starting point. For that purpose images of the crack were taken by a digital camera mounted on a light-optical microscope. Crack measurements were done by using an image processing software. In order to determine the crack depth a associated to the surface crack length 2c, some specimens had been deepfrozen in liquid nitrogen and broken open. Micrographs taken
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Table 2 Properties after heat treatment, mean and standard deviation
Fine grain Coarse grain
Rm (MPa)
ReL (MPa)
ReH (MPa)
A (%)
Z (%)
E (GPa)
Grain-: (mm)
444G4 370G9
295G6 183G4
389G16 –
37G2 41G1
62G2 61G3
207G2 195G4
6.7 56
Rm, tensile strength; ReL, lower yield strength; ReH, upper yield strength; A, elongation; Z, reduction of area.
Table 3 Parameters for the Manson–Coffin relationship and the Ramberg–Osgood equation for different specimen geometries and grain sizes
Unnotched (fine grain) Unnotched (coarse grain) Notched (fine grain) Notched (coarse grain)
sf0 (MPa)
b
E (GPa)
3f0
c
K 0 (MPa)
n0
1583 976 678 931
K0.162 K0.129 K0.082 K0.119
207 195 207 195
1.356 6.138 0.265 1.474
K0.662 K0.839 K0.444 K0.645
867 826 – –
0.185 0.178 – –
by a scanning electron microscope were showing almost semicircular fatigue fracture surfaces independent of stress concentration factor and strain amplitude [6]. The following calculations are based on the assumption of a semicircular crack shape.
4. Cyclic properties Incremental step tests have been carried out for unnotched specimens in fine and coarse-grained condition, respectively. The experiments yielded nearly the same cyclic stress–strain curve 3a Z 3ea C 3pa Z
sa sa 1=n0 C E K0
(2)
independent of the grain size, which has been found earlier also by Hoshide et al. [7] as well as by Hatanaka and Yamada [8]. The cyclic strain-hardening exponent n 0 and the cyclic strainhardening coefficient K 0 are therefore nearly the same for both variants. The values for n 0 and K 0 listed in Table 3 maybe slightly biased by the fact that the unnotched specimens show a stress concentration factor of KtZ1.11. Fig. 3 shows the experimental results for unnotched specimens made of material in fine and coarse-grained condition. Continuous lines have been achieved by fitting the experimental data to the Manson– Coffin relationship 3a Z
sf0 ð2NF Þb C 3f0 ð2NF Þc ; E
root needed for the derivation of the constants in Eq. (3) were determined by using Eq. (2). The coarse-grained specimens reach approximately 70% of the cycles to failure of the fine-grained variant. The difference in cycles to failure can be mainly attributed to the influence of grain size because the cyclic stress–strain behaviour is nearly the same for both variants. In Fig. 4 the strain amplitude versus cycles to failure for different specimen geometries is shown. In the case of the fine-grained variant it becomes apparent that the results for notched specimens lie in a narrow scatter band independent of the stress concentration factor. Also specimens that have been stress relieved after machining by annealing can be found in the same scatter band. Some specimens were also tested under load controlled conditions. In these cases the cyclically stabilised strain amplitude was used for the entry into the strain–life diagram. If the strain amplitude approaches the fatigue limit it becomes obvious that the number of cycles to failure for unnotched specimens is much smaller than for notched specimens and the difference rises up to a factor of 10. These results seem to be contradictory to the strain based approach to fatigue that assumes similar cycles to failure for the same strain amplitude.
(3)
where sf0 b 3f0 c
fatigue fatigue fatigue fatigue
strength coefficient strength exponent ductility coefficient ductility exponent
The resulting parameters for unnotched and notched specimens are also listed in Table 3. In the case of notched specimens, the elastic and plastic strain amplitudes at the notch
Fig. 3. Strain–life curves of unnotched specimens in fine-grained and coarsegrained condition, R3ZK1.
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Fig. 4. Strain–life curves of unnotched and notched specimens in fine-grained condition, R3ZK1.
In the case of coarse-grained specimens (Fig. 5) the situation is different. Notched specimens exhibit only a slightly longer lifetime than unnotched specimens. The difference can be described satisfactorily by the statistical size effect that accounts for the different size of highly stressed volume in unnotched and notched specimens [9]. The huge difference in lifetime in the case of fine-grained material can be explained by the initiation and propagation of fatigue Lu¨ders-bands. These bands that can be observed at the surface of cyclically loaded unnotched specimens, cause a significant rise in surface roughness and therefore have a strong influence on the fatigue process [6,9,10]. 5. Modelling of short crack growth in notched specimens 5.1. Constant amplitude tests The growth of a crack under elastic–plastic conditions can be described advantageously by the J-integral that can be calculated by the approximation formula (4) proposed by Dowling [11] s2 1:02 J Z Je C Jp Z 1:24 C pffiffiffi $s$3pl $a (4) n E where
n s 3pl a
strain-hardening exponent stress plastic strain crack depth
For cyclic loading the DK approach to fatigue crack growth is limited to situations where the plastic zone is small compared to the crack length or small compared to the uncracked ligament. But this is not the case for surface cracks in a uniform cyclic plastic strain field, as during lowcycle fatigue tests after crack initiation, and is not also the case for small cracks growing in regions of cyclic plasticity associated with notches. The applicability of the cyclic J-Integral DJ to long crack growth was shown by Dowling and Begley [12] whereas its applicability to cyclic short crack growth in unnotched specimen was shown for example by Vormwald and Seeger [13] and by Schubert [14]. Hoshide and Kusuura [15] used DJ for the simulation of short fatigue cracks in notched specimens whereas Tanaka et al. [16] showed its applicability for short fatigue cracks in specimens subjected to alternating bending. Since, the growth of short fatigue cracks is influenced by crack closure a convention about the consideration of crack closure within the approximation formula for the cyclic
Fig. 5. Strain–life curves of unnotched and notched specimens in coarse-grained condition, R3ZK1.
S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021
J-Integral (5) has to be made ðDseff Þ2 1:02 C pffiffiffiffi0 $Dseff $D3p;eff $a DJeff Z 1:24 E n
(5)
where n 0 cyclic strain-hardening exponent Dseff effective stress range D3p,eff effective plastic strain range Vormwald and Seeger [13] developed a sophisticated method for the determination of crack opening and crack closure levels and thus a method for the determination of effective values of Ds and D3p. Under plane stress condition the resulting DJeff -value can then be used topffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi calculate an equivalent stress intensity factor DKJ;eff Z DJeff $E and furthermore the crack growth rate according to the crack growth Eq. (6). The parameters m and C, usually describing long crack growth, can now be used for the calculation of the growth of physically short cracks da Z CðDKJ;eff Þm ; dN
(6)
where da/dN crack growth rate DK cyclic stress intensity factor C, m parameters describing long crack growth In the case of notched specimens it is necessary to use local values of cyclic stabilised stresses and strains for the evaluation of Eq. (5). This procedure is based upon the assumption that the growth of a crack at the notch root behaves as in an unnotched specimen as long as the crack is small and its growth is not influenced by stress or strain gradients. These local stresses and strains can either be determined by the well-known Neuberformula [17,18] or similar approaches [19–22]. In some cases also an elastic–plastic finite element method [6] may be convenient in order to decide whether plane stress or plane
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strain condition exists at the notch root. For a specimen geometry similar to that used in this study Sharpe [19] recommended a combination of Neuber’s rule with Eq. (2). The consideration of crack closure in Eq. (5) for the elastic as well as for the plastic part of DJeff produced values of DKJ,eff that did not coincide with the long crack behaviour shown in Fig. 6. Only the use of full ranges of local stabilised cyclic stresses and strains for the plastic part of Eq. (5) provides DKJ,eff being large enough to coincide with the long crack data. This procedure is in accordance with results of Schubert [14], who found that under cyclic loading in the LCF region, where the plastic part of DJeff is dominant, a proper description of small crack growth could only be achieved if the full ranges of nominal stress and nominal plastic strain were used to calculate the plastic part of DJeff. The elastic part of DJeff was calculated by using Dseff Z UDsZ 0:8Ds, where U describes the fraction of a cycle while the crack is open. The value for U was taken from literature. As mentioned before it is necessary to replace nominal stresses and strains by local stabilised values. Fig. 6 shows the relation between crack growth rate and the effective values of the ranges of the stress intensity factors. The curve for long crack growth was completed with a threshold value DKth,eff, which was taken from literature [23]. The diagram shows the crack growth rate of short cracks oscillating around the crack growth curve for long cracks in the Paris regime. No threshold behaviour can be seen for short cracks. Model calculations have shown that the influence of these oscillations on fatigue life can be neglected. Therefore, it can be assumed that short crack growth can be described by the load parameter DKJ,eff and the long crack constants m and C. In Fig. 7, a comparison of measured and calculated crack growth curves is shown. For the calculation of crack growth the formula da/dNZC(DKJ,eff)m was used. The initial crack length needed for calculation had been chosen in such a way that calculated and measured lifetime coincide for the highest load. For lower loads the calculation considerably underestimates
Fig. 6. Comparison of effective values of the cyclic stress intensity factors for long cracks in compact specimen (DKeff) and short cracks at the notch root (DKJ,eff, DJeff), KtZ1.54.
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Fig. 7. Comparison of measured and calculated crack growth curves from finegrained specimens (KtZ1.54).
the measured lifetime and a quantitative congruence between measured and calculated crack growth curves can be achieved only if the calculated curve is shifted along the abscissa. These findings may be explained by the fact that the fraction of fatigue life spent in crack initiation increases when the strain amplitude approaches the fatigue limit. As a consequence of this insight, it is necessary to expand the crack growth equation mentioned above. The expansion should be capable to describe crack initiation as well as crack propagation. Investigations concerning the early stages of fatigue have shown that surface damage can be related to shape changes of grains and the associated increase of surface roughness in bcc polycrystals [24]. After a rapid increase of surface damage at the beginning of cyclic loading the plasticity induced surface roughening reaches a steady-state [25]. A continued cyclic loading leads to the initiation of short cracks either in slip bands or at grain boundaries with large altitude difference between neighbouring grains and further damage is concentrated in these sites of crack initiation. Henceforth, the further development of surface roughness is of little importance. Both processes occur more or less at the same time so that a suitable model should be able to describe the process of surface roughening as well as the process of crack growth depending on crack length and local stresses and strains. A simple mathematical model capable of describing the damage process can be established by the addition of two power laws of the Paris-type: da da da Z C ZC1 ðDKJ;eff Þm1 CC2 ðDKJ;eff Þm2 dN dN initiation dN propagation (7)
C1, m1 crack initiation parameters C2, m2 crack growth parameters The effective cyclic stress intensity factor DKJ,eff serves as crack driving force in the crack initiation period as well as in
the crack propagation period. While the constants C2 and m2 for the crack propagation period are known from long-crack experiments, the constants C1 and m1 for the crack initiation period have to be determined by adjustment to measured lifetimes. There is no physical justification for the choice of DKJ,eff as crack driving force in the crack initiation period, where the crack length is in the order of the grain size. Equally, the shear strain range [26] or the shear stress amplitude [15,27] could have been chosen. But the choice of the same crack driving force in the crack initiation period as well as in the crack propagation period allows on the one hand for an easier comparability of damage rates and on the other for a simplified programming of the model. The description of a damage process comparable to the above-mentioned is made possible if a negative exponent m1 in Eq. (7) is used. The first term in Eq. (7) describes a rapid increase of damage due to surface roughening at the beginning of cyclic loading. When the crack becomes larger, the contribution of the first term of Eq. (7) to damage diminishes and further damage can be solely related to crack propagation. The adjustment of the model parameters to measured strain– life curves resulted in the following relationships: fine grain: log C1 ZaCblogð3a ÞCc log2 ð3a Þ
(8)
where a K20.314 b K32.149 c K8.607 m1 K6 coarse grain: C1 Z6:383!1013 35:1411 a
(9)
m1 ZK0:0228530:8053 a In the case of fine-grained specimens it was sufficient to choose m1 independent of the strain amplitude whereas for coarse-grained specimens it was necessary to model m1 load dependent. The parameters were determined by a stepwise iteration of C1 and m1 followed by integration of Eq. (7) for constant strain amplitude. The iteration was terminated when modelled lifetime was in accordance with measured lifetime. To obtain a continuous relationship m1Zf(3a) and C1Zf(3a) it is necessary to repeat this procedure for several different strain amplitudes. Eqs. (8) and (9) were fitted to these discrete results. Fig. 8 illustrates the principle of the model. In the domain of microstructurally short cracks, where the crack length is about the same as the mean grain size, the model predicts higher crack growth rates than for physically short cracks and long cracks, what is a common and often observed feature of short cracks [26]. In this domain the calculated crack growth rate is dominated by the first term in Eq. (7). When the growing crack approaches a dominant microstructural barrier, for example a grain boundary or a pearlite grain, the crack growth rate reaches a minimum. In the domain of physically short and long
S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021 ∆ Jeff [N/mm] 0.1
10-3
∆ Jeff [N/mm] 0.1
1
1017
1
10-3 εa = 0.80 %
10-4 εa = 0.40 %
10-5
10-6
εa = 0.18%
Model εa= 0.40 % εa= 0.24 % εa= 0.18 % εa = 0.16 %
10-7
10-8
εa = 0.16%
da/dN [mm]
da/dN [mm]
10-4
εa= 0.30 %
10-5 εa= 0.12 %
10-6
10-7
10-8
100 200 400 600 800 ∆KJ,eff, ∆Keff [Nmm-3/2]
Model εa = 0.30 % εa = 0.24 % εa = 0.18 % εa = 0.12 % 100
200 400 600 800 ∆KJ,eff, ∆Keff [Nmm-3/2]
Fig. 8. Comparison of measured and modelled crack growth rates for fine (left) and coarse-grained (right) material; KtZ1.54.
cracks where the crack is about the size of a few grains, fatigue crack growth is governed by the second term of Eq. (7). Fig. 8 makes clear as well, that the method chosen for crack monitoring does not allow for the measurement of crack growth rate, when crack length is smaller than the grain size. If the normalised crack length, which is defined as actual surface crack length 2c divided by the criterion for a critical crack length of 1 mm, is taken as damage D, the integration of Eq. (7) yields the evolution of damage for different cyclic loads. Fig. 9 makes clear that the established model provides a reasonable description of damage evolution in cyclically loaded specimens. What can also be seen in Fig. 9 is that different strain amplitudes lead to different damage curves. The load dependence of the damage curve is a vital requirement for the description of load sequence effects [28]. 5.2. Two-level tests A magnified cutout of Fig. 9 showing the evolution of damage for small cycle ratios N/NF, is shown in Fig. 10. Starting
at the same initial damage represented by an initial crack depth aZ1 mm, different strain amplitudes lead to different damage curves. The evolution of damage proposed by Miner’s rule is shown as well and clarifies the difference to the recent model. The following example shows that the model is capable of describing the effect of load sequence on fatigue life: A specimen is cycled with a local strain amplitude of 3aZ0.16% until the number of cycles reaches N/NFZ0.5. The local strain is subsequently increased to 3aZ0.4% which means changing the damage curve on a horizontal line DZconst. It can be seen from Fig. 10 that the remaining life is about N/NFZ0.85. Thus, the model yields a Miner sum DMZ1.35. If the strain sequence is started on a high level until N/NFZ0.1 and then lowered to a low level, a remaining life of N/NFZ0.6 can be deduced from Fig. 10. The corresponding Miner sum is DMZ0.7. These results are in good agreement with numerous former experiments dealing with load sequence effects and Miner’s damage function [2]. Further on, Weber and Dengel [29] have shown for high–low load sequences that the inaccuracy of Miner’s rule can be attributed to the fact that the transition from
1.0
1.0 εa = 0.30 %
εa = 0.40 % εa = 0.24 %
εa = 0.18 %
εa= 0.16 %
0.6
Miner Model 0.4
0.2
0.0 0.0
εa = 0.24 %
0.8
εa = 0.18 %
DMiner, D = 2c/1mm
DMiner, D = 2c/1mm
0.8
εa = 0.12 % 0.6
Miner Model
0.4
0.2
0.2
0.4
0.6 N/NF
0.8
1.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
N/NF
Fig. 9. Modelled and measured relationship between normalised crack length and normalised life (KtZ1.54); left: fine-grained material, right: coarse-grained material.
1018
S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021 0.15
1.0 coarse grain 0.8
LH
εa = 0.24 %
0.10
εa = 0.18 %
N2/NF2
DMiner, D = 2c/1mm
εa = 0.40 %
εa = 0.16 % Miner
Short crack model: 0.10 % <-> 0.30 % 0.12 % <-> 0.30 % 0.24 % <-> 0.30 % Miner:
0.6 0.4
0.05
Experiment (Kt = 2.57): 0.12 % <-> 0.30 %
HL
0.2 0.0 0.0
0.00 0.0
0.1
0.2
0.3
0.4
0.2
0.5
0.4 0.6 N1/NF1
0.8
1.0
N/NF
Fig. 10. Magnified cutout of the modelled curves in Fig. 9 (fine-grained material, KtZ1.54).
an undamaged state into a damaged condition occurs at different cycle ratios N/NF if different load levels are involved. The proposed short crack model describes a load dependent transition point as well. The minimum of the crack growth rate in Fig. 8, which can be found as inflection point in the damage curve in Fig. 10 can be regarded as transition from minor damage to severe damage. Figs. 11 and 12 show the influence of load sequence on remaining life fraction for the fine-grained material and the coarse-grained material, respectively. In case of low–high (LH) load sequences, the calculation was started with a low strain amplitude 3a until a certain life fraction N1/NF1 was reached. After that the calculation was continued with a higher strain amplitude until the final crack length of 2cZ1 mm was reached. In the case of high–low (HL) load sequences the calculation was started with a high amplitude and then continued with a low strain amplitude. From Figs. 11 and 12 it can be deduced that LH-sequences always lead to Miner sums larger than unity whereas HL-sequences always lead to Miner sums smaller than unity. The resulting curves are quite similar to those which can be achieved by the application of the Damage Curve Approach (DCA) or the Double Linear Damage Rule (DLDR) [30]. The predicted results are in reasonable 1.0 fine grain Short crack model: 0.16 % <-> 0.18 % 0.16 % <-> 0.24 % 0.16 % <-> 0.40 % Miner:
0.8
N2/NF2
LH 0.6
Experiment (Kt = 2.57): Two-level tests 0.16 % <-> 0.24 %
0.4
0.2
0.2
agreement with experimental results although there is a certain amount of scatter, especially for the LH-sequence. The calculation for two closely neighbouring strain amplitudes (e.g. 3aZ0.24% and 3aZ0.30% in Fig. 12) leads to Miner’s rule again, which is in agreement with the insight, that a monotonically increasing non-linear damage function still leads to Miner’s rule. 5.3. Irregular loads To overcome the inaccuracy that will be caused by the insufficient transferability of cycles to failure from unnotched to notched specimens (Fig. 4) the short crack model was only applied for spectrum fatigue life assessment of notched specimens. First of all the strain–life curve of notched specimens in fine and coarse-grained conditions using Eq. (7) were calculated for several discrete values of the cyclic strain amplitude. A relationship of the form 3a Za1 ð2NF Þa2 Cb1 ð2NF Þb2
(10)
was then adjusted to these discrete results in order to obtain a continuous description of the calculation results, which is needed for a calculation of service life based on Miner’s rule. The corresponding constants are given in Table 4. A comparison between measured and modelled strain versus life curves for notched specimens subjected to constant amplitude loading is shown in Figs. 13 and 14 for fine and coarse-grained material, respectively. The proposed model can now also be used to calculate lifetime for randomised cyclic loading. The calculation is based on a randomly generated period of amplitudes that can be Table 4 Constants used for the strain–life relationship of notched specimens
HL 0.0 0.0
Fig. 12. Calculation of the remaining life in two-level tests for high–low (HL) and low–high (LH) load sequences (coarse-grained material). Scatter bars indicate the scatter of experimentally determined remaining life for 10 and 90% probability of crack initiation, respectively (five specimens).
0.4 0.6 N1/NF1
0.8
a1 (%)
1.0
Fig. 11. Calculation of the remaining life in two-level tests for high–low (HL) and low–high (LH) load sequences (fine-grained material). Scatter bars indicate the scatter of experimentally determined remaining life for 10 and 90% probability of crack initiation, respectively (five specimens).
Notched (fine grain) Notched (coarse grain)
a2
b1 (%)
b2
37
K0.485
0.694
K0.134
5474
K1.141
3.1
K0.237
S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021
Strain amplitude εa, ε a [%]
Notched specimensKKt = 1.54 and Kt= 2.57: measured strain vs. life curve calculated strain vs. life curve (short crack model) calculated prestrain strain vs. life curve with εa = 0.40 % until 0.02 NF Notched specimens Kt = 1.54: measured service life calculated service life (original Miner, εFL = 0.12 %) calculated service life (elementary Miner, OL = 0.5*εFL ) calculated service life (short crack model, OL = 0.5*εFL)
0.8 0.7 0.6 0.5 0.4 0.3 0.2
εFL
0.1
0.1 104
105
106
107
Cycles to failure NF Fig. 13. Comparison of measured data and model calculation for constant-level and pseudo-random loads (Gaussian distribution) for fine-grained material (KtZ1.54 and 2.57). FL, fatigue limit; OL, omission level.
described by a gaussian load spectrum of the form H ZH0 eKðSa =Sa Þ
2
ln H0
with H0 Z106 ; I Z1; RZK1
(11)
where H cumulative frequency H0 size of period I irregularity factor R stress ratio.
Strain amplitude εa,εa [%]
Notched specimens Kt = 1.54 and Kt = 2.57: measured strain vs. life curve calculated strain vs. life curve (short crack model) calculated prestrain strain vs. life curve with εa = 0.40 % until 0.1 NF Notched specimens Kt = 1.54: measured service life calculated service life (original Miner, εFL = 0.1 %) calculated service life (elementary Miner, OL = 0.5*εFL) calculated service life (short crack model, OL = 0.5*εFL)
0.80 0.70 0.60 0.50 0.40 0.30 0.20
εFL
0.10 0.09 0.08 0.07 0.06
104
105
106
0.1
107
Cycles to failure NF
Fig. 14. Comparison of measured data and model calculation for constant-level and pseudo-random loads (Gaussian distribution) for coarse-grained material (KtZ1.54 and 2.57). FL, fatigue limit; OL, omission level.
1019
To calculate the incremental damage caused by a single cycle, the local stress and strain amplitudes that correspond to the random nominal stress amplitude Sa have to be calculated. This can be done either by a modified Neuber-formula [19] or an elastic–plastic finite element method [6]. Starting with a crack length of aZ1 mm the evolution of damage can be calculated using Eq. (7) in connexion with Eq. (8) or Eq. (9). The calculation was terminated when the crack reached a critical length of 2cZ1 mm and aZ0.5 mm, respectively. A similar procedure was adopted for the calculation of service life based on the original and elementary rule of Miner. The local strain amplitude was used to calculate the incremental damage DZ1/NF of a single cycle by numerical evaluation of Eq. (10). The calculation was terminated when Miner’s sum equalled unity. Figs. 13 and 14 give a comparison of service lives for notched specimen calculated on the basis of short crack growth, the elementary and the original version of Miner’s rule. The calculation results are completed with experimentally determined service lives. Since the service life of notched specimens has been measured under load controlled conditions it is necessary to convert the maximum stress S a of the Gaussian load spectrum into the corresponding maximum local strain amplitude 3a . This was done by use of the approximation formula mentioned before. In the case of fine-grained material, the original version of Miner’s rule overestimates service life, in particular when the maximum strain approaches the fatigue limit. When the elementary version of Miner’s rule is used, an improved calculation of service life can be achieved (Fig. 13). The convergence of calculated service life for higher strains 3a indicates that the inaccuracy of ‘original Miner’ is mainly caused by the negligence of damaging cycles below the fatigue limit. The use of the proposed short crack model gives rise to a further improvement of service life assessment, which may be attributed to the incorporation of the load sequence effect into the model. This effect may be explained as follows: the early occurrence of high amplitudes during irregular cycling loading causes a severe reduction of the remaining life for lower amplitudes. The result is an increased damage per cycle compared with constant-level tests at the same low amplitude. This effect is demonstrated in Fig. 13 by the calculation of a prestrain strain–life curve. Short cycling at a high strain level (3aZ 0.4%) causes a severe reduction of remaining life, if the experiment is continued on a lower load level. The presented results agree well with the investigations of Zenner and Liu [5], who proposed a reference curve with drastically reduced lifetimes in order to improve service lifeassessment based on Miner’s rule. That empirically determined curve may be interpreted as a stress–life curve, whose crack initiation portion has been cut off. The application of the original variant of Miner’s rule overestimates service life also in the case of coarse-grained material. The use of the elementary version of Miner’s rule leads to an improved calculation of service life (Fig. 14). It can be seen from Fig. 14 that the application of the proposed short
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S. Laue, H. Bomas / International Journal of Fatigue 28 (2006) 1011–1021
crack model gives the same results as the elementary version of Miner’s rule. Both approaches overestimate the measured service life for high strain amplitudes 3a whereas for low strain amplitudes the calculation provides a reasonable agreement with experimental results. From the considerations above it can be assumed that under pseudo-random loading the initiation of a fatal crack in the that coarse-grained material takes place at an early stage of fatigue life. This means that the damage evolution can be attributed mainly to crack propagation, whereas crack initiation is of minor significance. This in turn means that the evolution of crack growth can be described by a crack growth law. The application of a crack growth equation of the Paris-type for the description of damage evolution yields the same result as using Miner’s rule [31]. This may be the explanation for the congruence of service life calculation based on Miner’s rule as well as on the short crack model in the case of coarse-grained material.
even when the elementary version of Miner’s rule is used. Deviations from this concept may be attributed to crack retardation effects due to overloads or situations where crack initiation and growth is influenced by residual stresses.
Acknowledgements The financial support of this work by the Deutsche Forschungsgemeinschaft DFG in the framework of the priority programme ‘Fatigue life-time prediction of metals based on microstructural behaviour’ is gratefully acknowledged. Special thanks go to Dr J. Hu¨necke and Dr D. Scho¨ne of Bundesanstalt fu¨r Materialforschung und -pru¨fung for the experimental determination of service life of notched specimens and strain–life curves of unnotched specimens.
6. Conclusions
References
1. In the case of constant amplitude tests the local strain approach to fatigue underestimates the number of cycles to failure for notched specimens up to a factor of ten in the case of steel SAE1017 in fine-grained condition whereas for coarse-grained material the local strain approach yields reasonable results. The huge difference in lifetime in the former case can be explained by the initiation and propagation of fatigue Lu¨ders-bands. 2. The growth of small cracks in notched specimens of low carbon steel can satisfactorily be described by the cyclic J-integral in the crack propagation phase, but at lower amplitudes in the high cycle fatigue region the calculation of lifetime based on a Paris-equation and the cyclic J-integral underestimates the measured lifetimes. In order to obtain a more realistic description of the damage process, it was necessary to incorporate the evolution of damage during the crack initiation period into the model. This can be done by the combination of two power laws of the Paristype. The proposed model describes a strong increase of damage at the beginning of cyclic loading due to high growth rates of short cracks. 3. The model provides the opportunity to relate the deviations from Miner’s rule in two-level tests to the load dependent portions of crack initiation phase on different strain levels. That sequence effect increases with increasing distance between load amplitudes and decreasing grain size. 4. The sequence effect deduced from two-level tests allows for the following assumptions. High loads, occurring at the beginning of fatigue cycling under random loading, cause early crack initiation. This crack can now be advanced by low load amplitudes near and even below the fatigue limit meaning there is no initiation period any longer. A Wo¨hler curve determined by constant amplitude tests always includes a certain load dependent portion of crack initiation period. Therefore spectrum fatigue life assessment based on Miner’s rule should normally overestimate service life,
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