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The effects of pre-cycle damage on subsequent material behavior and fatigue resistance of SUS 304 stainless steel
Materials Science & Engineering A 636 (2015) 320–325
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
The effects of pre-cycle damage on subsequent material behavior and fatigue resistance of SUS 304 stainless steel Liang-Hsiung Chou a, Yung-Chuan Chiou b,n, Ying-Jen Huang b a b
Department of Civil and Water Resources Engineering, National Chiayi University, Chiayi 600, Taiwan Department of Biomechatronic Engineering, National Chiayi University, Chiayi 600, Taiwan
art ic l e i nf o
a b s t r a c t
Article history: Received 15 January 2015 Received in revised form 18 March 2015 Accepted 18 March 2015 Available online 31 March 2015
A series of fatigue tests with type I and type II cycle damage histories were carried out to observe the effects of pre-cycle damage on the subsequent cyclic material response and fatigue resistance. It was found that the applied cycle history of damage leads to an increase in the extent of variation in cyclic hardening/softening. Furthermore, the shape of stable hysteresis loop changes with the type of cycle damage history under the same strain amplitude conditions. This is due to the occurrence of additional cyclic hardening/softening. By comparing the strain–life curves with and without a cycle damage history, it is found that the cycle damage history leads to a decrease in the number of cycles to failure. The loss in fatigue resistance is dependent on the type of pre-cycle damage history. For the fatigue test with a precycle history, an equivalent cycle ration C n1 is proposed to predict the residue cycles at the second level in this study. The validity of the modified expression is confirmed by comparing the predicted life results with the corresponding experimental data. It is apparent that the proposed method gives a better prediction than the Miner's rule for the two-step histories. & 2015 Elsevier B.V. All rights reserved.
Keywords: Residual fatigue life prediction Pre-damaged Cyclic behavior
1. Introduction In the course of manufacturing, some pre-damage is induced in the components due to plastic forming, as well as the various mechanical pre-histories on the cyclic stress–strain response lead to a problem of great importance in the fatigue life of components during their service. In several studies [1–3], it has been found that a prior mechanical pretreatment has a strong influence on microstructure and mechanical behavior during cyclic deformation. The extent of variation in stress– strain response, due to the prior pre-deformation, depends on the magnitude of the applied loading, type of pre-deformation and the tested material [4–8]. Based on the above experimental results, it can be concluded that the pre-deformation in the fabrication process leads to the damage in the material. Generally, in the fabrication process, components may receive cyclic pre-damage and are often subject to cyclic loading during later service. Hence, it is interesting to study the stress–strain behavior and predict the residue cycles to failure after cyclic loading at high or low strain level to the specified number of cycles. Generally, the value of summing the cycle ratio at each strain level is close to Miner's proposed value of 1 [9] in most observations. Therefore, the Miner's rule [9] is usually applied to predict a fatigue life of the material subjected to a variable amplitude loading history. It should be pointed out that the rule ignores the sequence effect on
n
Corresponding author. Tel.: þ 886 2717667. E-mail address: [email protected] (Y.-C. Chiou).
http://dx.doi.org/10.1016/j.msea.2015.03.070 0921-5093/& 2015 Elsevier B.V. All rights reserved.
damage accumulation based on the observation so that the estimation of fatigue cycles does not correspond to that with observed fatigue cycles. Consequently, in material with a variable amplitude loading history, the sequence effect on damage accumulation needs to be considered for fatigue life prediction. In the study, both cycle damage histories, which are pre-cycles corresponding to an approximate 20% of cycle ratio at strain amplitudes of 0.35% and 0.70% at R¼ 1, are investigated. The result is compared with the experimental value to assess the influence of cyclic pre-damage on the subsequent cyclic hardening/softening behavior, stable stress–strain response, and the residue cycles to failure of SUS 304 stainless steel. Moreover, for a twostep load history test, an equivalent cycle ratio C n1 corresponding to the second strain level is proposed to estimate the residue cycles during later cyclic straining for the tested material. Simultaneously, the sequence effect on damage accumulation is taken into account on the proposed equivalent cycle ratio. The validity of the proposed method is analyzed in this study.
2. Experiment procedure 2.1. Material and specimen In this study, the test material was SUS 304 stainless steel. The chemical compositions of the materials are given in Table 1. For the measurement on the chemical composition, the repeated times is two. The geometry of the fatigue specimen is shown in
L.-H. Chou et al. / Materials Science & Engineering A 636 (2015) 320–325
Table 1 Chemical compositions of SUS 304 stainless steels (weight %). C
Si
Mn
P
S
Ni
Cr
Cu
Fe
0.04
0.32
1.79
0.027
0.001
8.15
18.18
2.11
bal.
321
a servo-hydraulic Instron 8501 materials testing system with a constant strain rate of 0.01/s and a triangular waveform. A tension–compression extensometer with a 25 mm edge separation was used to control the strain applied in each fatigue test. The cycles to failure, Nf , was defined as the number of cycles performed corresponding to a drop of 10% in load in the fatigue test. The stable stress–strain response was obtained from the hysteresis loop at approximately half-life, where the stress–strain response was stable. In this study, the performed tensile fracture tests and fatigue tests are repeated two or three times under the same experimental condition.
3. Experimental results and discussion 3.1. Effect of cycle damage history on cyclic deformation behavior Fig. 1. Schematic illustration and dimensions of specimens.
Table 2 Mechanical properties for SUS 304 stainless steels. E ðGPaÞ
Sy;0:2% ðMPaÞ
Su ðMPaÞ
E:L: ð%Þ
K ðMPaÞ
n
187.15
490.5
537.5
50.5
895.98
0.1007
Fig. 1. By use of a CNC lathe, the as-received material was machined to form a specimen with a geometry shown in Fig. 1 and the surface was polished with a fine emery paper. Before the fatigue test, the tensile fracture test was performed under stoke controlled condition with a crosshead rate of 0.01 mm/s. The results of the tensile properties are listed in Table 2. It is noted that the values presented in Tables 1 and 2 are obtained by performing a mean value calculation on the measured results. 2.2. Test programs and procedures In this paper, a series of fully reversed fatigue tests, two-stage test with a low-high strain level sequence and two-stage test with a high-low strain level sequence were performed in order to investigate the effects of cyclic pre-damage on the subsequent cyclic deformation behavior. In this fatigue test, the applied strain was maintained at a constant level until failure occurred on the testing specimen, and is referred to as the typical fatigue test. Test data generated from the typical fatigue test is essential in observing the pre-cycle damage effects on the subsequent fatigue behavior. It is worth mentioning that the strain–life curve can provide the estimation of the number of cycles to failure at a given strain amplitude. Hence, the number of cycles at an initial strain level can be calculated on the basis of the typical strain–life curve in the design of cycle pre-damage history. Consequently, the typical fatigue tests had to be carried out first in this study. For the low-high test, the specimen was tested at initial strain amplitude of 0.35% for 8000 cycles ðn1 ¼ 8000Þ. Subsequently, the strain level was changed to a second level until failure occurred. This type of cyclic straining history is defined as type I cycle damage history. With the assistance of the typical strain–life curve, corresponding to type I damage history, the cycle ratio, C 1 ¼ Nf ;at εn1¼ 0:35% , is calculated to be equal to 21.29%. In this proce-
In the strain-controlled model, the variation of peak stress response with the applied cycle is usually observed for the tested material subjected to the repeating loading. As shown in Fig. 2, it has been observed that the typical cyclic-stress response curves at the different strain amplitudes also display significant decreases in the stress response for the first fifteen cycles and then the measured peak stress with cycle number decreases prior to failure. Furthermore, it can be seen from Fig. 2 that the plateau of stress in the cyclic-stress response curve increases as the cyclic strain amplitudes increase from 0.35% to 0.70%. This indicates that cyclic softening occurred during the cyclic deformation for SUS 304 stainless steel. Moreover, midlife hysteresis loop data from the fatigue tests were conventionally used to present the stable stress– strain response for the tested material subjected to the repeating loading. Hence, the scale of stable stress amplitude can be directly measured from the midlife hysteresis loop and the stable plastic Δε Δσ strain amplitude can be estimated from 2 p ¼ Δε 2 2E . In the calculation, the elastic modulus, E, measured from tension tests was used. For the correlation between the stable stress amplitude and stable plastic strain amplitude, it is usually fitted in the form of a power-law type equation and is represented as n Δσ ΔεP n ¼ Kn ð1Þ 2 2 where K n is the cyclic strength coefficient and the exponent nn is the cyclic strain hardening exponent. The two parameters K n and nn represent the stable response of a material to cyclic straining. In Eq. (1), K n and nn are obtained from a least squares fit of stress data
a
dure of determining the cycle ratios, Nf ; at εa ¼ 0:35% is the original cycle to failure life at initial strain amplitude of 0.35%. For the high-low test, the applied initial cyclic straining was at strain amplitude of 0.70% for 500 cycles. The loading history is named as type II cycle damage history. Similarly, the value of the corresponding cycle ratio is 20.64%. All fatigue tests were conducted on
Fig. 2. Superimposed plot of cyclic peak stress with number of cycles that data obtained from typical fatigue test at the strain amplitude ranging from 0.35% to 0.70%.
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L.-H. Chou et al. / Materials Science & Engineering A 636 (2015) 320–325
Δε versus the corresponding data 2 p in log–log scale. For data obtained from the typical fatigue tests, the cyclic strength coefficient K n is calculated to be 616.66 and the cyclic strength exponent nn has a value of 0.0848. Conventionally, the fitted curve is labeled as the cyclic stress–strain curve. In this paper, the curve obtained from the typical fatigue tests is labeled as the typical cyclic stress– strain curve. It is noted that the typical cyclic stress–strain curve can be used to estimate the stable stress amplitude in the range of plastic strain amplitude from 0.1592% to 0.4901% in this study. Moreover, according to Eq. (1), the cyclic stress–strain curves with variable cycle damage histories are also obtained. In order to infer the effect of cycle damage history on strain-hardening capability, the fitted typical cyclic stress–strain curve, cyclic stress–strain curves with variable cycle damage histories, and the tensile stress– strain curve are simultaneously plotted in Fig. 3. As shown in Fig. 3, first, it is found that a good agreement exists between the fitted and experimental data. Furthermore, it can be seen from this figure that the tensile stress–strain curve lies above all plotted cyclic stress–strain curves, indicating that cyclic softening occurs at all applied strain amplitudes for the tested material without and with the cycle damage history effect. From the comparison between the typical cyclic stress–strain curve and cyclic stress– strain curves with variable cycle damage histories shown in Fig. 3, the cyclic stress–strain curve with the type I cycle damage history lies above the typical cyclic stress–strain curve, and the gap between these two curves increases along with the cyclic strain amplitude. The above observations in Fig. 3 are based on the range of plastic strain amplitude from 0.2011% to 0.4901%. It demonstrates that the type I cycle damage history would lead to a variation in cyclic hardening behavior and the degree in variation is more dependent on the magnitude of the followed strain amplitude. It is evident that the cyclic stress–strain curve with the type II cycle damage history is below the typical cyclic stress– strain curve and the extent of differences between both curves decreased with an increase in cyclic strain amplitude. Consequently, it can be concluded that the ratio of the stable plastic amplitude response in the applied two-step load history has a significant effect on the extent of variation in additional cyclic softening/hardening. The fitted typical cyclic stress–strain curve is implemented during the range of plastic strain amplitude from 0.1592% to 0.4901%, and the cyclic stress–strain curve with the type II cycle damage history is conducted up to only 0.44%. It should be pointed out that the above observations in Fig. 3 are based on the range of plastic strain amplitude from 0.1592% to 0.44%. Δσ 2
3.2. Effect of cycle damage history on stable stress–strain response Six actual hysteresis loops from experimental data at approximately half-life with strain amplitudes ranging from 0.35% to 0.70% were plotted with shifted axes such that their compressive loops all fall at the same origin. As shown in Fig. 4(a), it can be seen that the loading curves for all of the hysteresis loops fall on approximately the same locus at controlled strain amplitudes of higher than 0.35%. Some deviation is found between the tensile hysteresis loop and the dashed line, which is obtained by expanding the typical cyclic stress–strain curve with a scale factor of two. The expanded cyclic stress–strain curve is called the Massing curve. The observation indicates that the tested material without a cycle damage history effect has a near Massing behavior [10]. Fig. 4(b) and (c) presents the superimposed stable hysteresis loops with coincident compressive tips for the tested material with types I and II cycles damage history, respectively. As shown in Fig. 4(b) and (c), it is clear that the corresponding Massing curves are very close to all tensile branches of stable hysteresis loops. Therefore, it can be concluded that the tested material after predamage exhibits Massing behavior at all the considered strain amplitudes. Moreover, as shown in Fig. 4(a)–(c), the symmetry between the stable hysteresis loop in tension and compression is also observed. Therefore, based on the results of experimental observation, the shape of stable hysteresis loops shown in Fig. 4(a) –(c) can be simulated with the help of the corresponding expanded Massing curve. In addition, the stable plastic work ΔW p can be calculated by integrating the area within the stable hysteresis loop. A plot of the calculated vs. experimental stable plastic work is shown in Fig. 4(d), in which the diagonal line represents the perfect correlation line. From Fig. 4(d), it is found that data with type II cycles damage history fall within the dash-lines, whereas data with type I cycles damage history are still within the dotlines. Again, it can be seen that most of the data without cycle damage history effect are outside of the dot-lines. The observations indicate that the closeness between the simulated curves and the experimental data with type II cycles damage history is better than the other two groups of experimental data. In other words, the applied cycle damage history could lead to a considerable decrease in the discrepancy between the expanded curve and the hysteresis loop curve in tension. The type II cycle damage history gave somewhat better improvement in the discrepancy than that due to the type I cycle damage history. It was found that most of the data obtained from the typical fatigue tests are closer to the starting point of the X-coordinate axis shown in Fig. 4(d) than data with the cycle damage history under the same strain range condition. Essentially, it is acceptable that the magnitudes in the data with the types I or II damage effect are almost slightly greater in magnitudes than data without damage effect via a mean value calculation on the results under the same experimental conditions. Consequently, the variations in the shape, magnitude of stable hysteresis loop and the improvement on the Massing behavior can be attributed to the cycle damage history effect on the stable stress–strain response. It is pointed that both the applied cycle damage histories are constructed on an approximate 20% of cycle ratio at strain amplitudes of 0.35% and 0.70% in this paper. It is obvious that the effects of the applied strain level in cycle damage history on the subsequent stable stress–strain responses are significant. 3.3. Effect of cycle damage history on the cycles to failure
Fig. 3. Comparison of monotonic stress–strain curve and the simulated cyclic stress–strain curves without and with cycle damage history.
For most metals, it is found that the strain amplitude, Δε=2, versus the cycle number to fracture, Nf , is modeled as a straight line in log–log scale. Thus, the empirical relationship between the
L.-H. Chou et al. / Materials Science & Engineering A 636 (2015) 320–325
900
900
SUS 304 Stainless Steel
C1 = 20.64 % at εa = 0.70 %
700
Stress , σ (MPa)
Stress , σ (MPa)
SUS 304 Stainless Steel
Rε = - 1
700
500
300
100
500
300
100 (n-1)/n 1/n :Δε=(Δσ/E)+2 (Δσ/K)
-100 -0.1
0.2
0.5
0.8
1.1
:Δε=(Δσ/E)+2
-100 -0.1
1.4
0.2
0.5
(n-1)/n
(Δσ/K)
0.8
1.1
1/n
1.4
Strain , ε (%)
Strain , ε (%) 7
900
SUS 304 Stainless Steel
Based on the expended cyclic σ -ε curve
C1 = 21.29 % at εa = 0.35 %
6
, MJ/m3
700
Calculated
500
300
p
(ΔW )
Stress , σ (MPa)
323
100
4 3 : R = -1 : c1 = 21.29 % at ε a = 0.35 %
2 :Δε=(Δσ/E)+2
-100 -0.1
5
0.2
0.5
0.8
(n-1)/n
(Δσ/K)
1.1
:c1 = 20.64 % at ε a = 0.70 %
1/n
1.4
Strain , ε (%)
1
1
2
3
4
5
6
7
( Δ Wp ) Observed , MJ/m3
Fig. 4. Comparison of the expanded cyclic stress–strain curve and stable hysteresis loops obtained from (a) typical fatigue tests, (b) fatigue tests with type I damage history, (c) fatigue tests with type II cycle damage history and (d) plot of calculated vs. observed stable plastic work at various total strain amplitudes.
strain amplitude and the cycle number to fracture is represented as Δε=2 ¼ α ð N f Þβ
ð2Þ
In Eq. (2), the constant, α, and slope, β, are the constants in the best straight-line that can be obtained by linear regression analysis. According to the experimental results from the typical fatigue tests, the fitted value of α is 5.023 and the value of β is 0.2528. Consequently, the typical strain–life curve can be constructed and is plotted in Fig. 5. It can be seen from this figure that a good agreement exists between the solid line and experimental data. Similarly, for the tested material with type I cycles damage history, the strain–life is given by Δε=2 ¼ 4:9485 ð N f Þ 0:2576
ð3Þ
For the tested material with type II cycles damage history, the power law type equation is found to be Δε=2 ¼ 16:0909 ð N f Þ 0:4177
ð4Þ
A superimposed plot of strain–life curves of the tested material without and with type I and type II cycles damage history are shown in Fig. 6. It is interesting to note the effect of cycle damage history on the typical strain–life curve of 304 Stainless Steel. As shown in Fig. 6, it can be seen that the plotted lines are in good agreement with the data. Generally, the typical strain–life curve is superior in the number of cycles to failure under the same strain amplitude condition to both curves with different damage effects.
Fig. 5. Comparison of experimental data and the typical strain–life curve fitted by power function relationship.
The results may be attributed to the fact that the formation of fatigue crack initiation is accelerated on the fatigued specimen after the applied cyclic pre-damage. The typical strain–life curve lies a little above the curve with type I damage effect, and they are almost parallel to each other. It reveals that the decrease in fatigue life is almost the same whatever the strain amplitude is. For the
324
L.-H. Chou et al. / Materials Science & Engineering A 636 (2015) 320–325
n2;
at Δε2
is estimated from
n2; at Δε2 ¼ ð1 C n1 Þ N f ; at Δε2
3.4. Fatigue life prediction with cycle damage history effect The testing condition of cyclic pre-damage is composed of the strain level, Δε1 =2, and the corresponding cycle ratio C 1 ¼ n1 =Nf ; at Δε1 . For the cycles damage history used in this study, an approximate 20% of cycle ratio is adapted. Based on the experimental results as shown in Fig. 6, it is found that the degree of reduction in fatigue life is dependent on the initial strain level (Δε1 =2) and the subsequent strain level (Δε2 =2). Meanwhile, the variation in the residue fatigue life is more sensitive to the ratio of the stable plastic strain range (Δεp;2 ) corresponding to the second strain level to that measured in the initial strain level. In this study, for a material subjected a two-step cyclic loading, an equivalent cycle ratio C n1 that modifies the number of cycles at the strain level Δε2 =2 is proposed and expressed as follows: C n1 ¼ eω C 1
ð5Þ
where
ω ¼ 1
Δεp;2 Δεp;1
ð6Þ
Clearly, in Eq. (5), the consumed fatigue cycles at strain level Δε1 =2 are translated into the consumed fatigue cycles at strain level Δε2 =2 via the specific factor ω. In other words, the cycle ratio C1 at the strain level Δε1 =2 is equivalent to the cycle ratio C n1 at the specified strain level (Δε2 =2). Since the sum of the cycle ratio is 1 in a fully reversed cyclic straining condition, therefore, the cycle ratio C n2 at strain level Δε2 =2 is the value of (1 C n1 ). Further, the corresponding residue cycles to failure
where Δi represents the ratio of the predicted result to the experimental value, and Δn is the average value of all calculated Δi . In this study, the value for n is 23. The calculated value of Π for Miner's rule is 1.1886 and the value for the proposed method is 0.8586. Based on the calculated results for Π , it can be inferred that the proposed method provides a more accurate prediction than Miner's rule method in making residue life predictions.
4. Conclusions An experimental investigation into cyclic fatigue behavior of SUS 304 stainless steel with two type cycles damage history has been presented. The experimental data obtained from typical fatigue tests provides the basis for recognizing the significances of variation in the subsequent stable stress–strain response and the residue cycles to failure with pre-cycles damage. By taking the effect of cycle damage history into consideration, a new method is proposed to estimate the residue cycles to failure for the tested material with pre-cycles damage subjected to symmetrical cyclic straining. Both the proposed method and Miner's method are also used to
10
5
: C1* + C2* = 1 : C1 + C2 = 1
f
strain-curve with type II damage effect, the typical cyclic stress– strain curve is much higher than the curve. It can be seen that the extent of the gap between the strain life curve with type I damage effect and that with type II damage effect decreases with an increase in strain amplitude. Consequently, it can be concluded that the loss in fatigue resistance caused by type II cycles damage history is higher than that due to type I cycles damage history. It is attributed to the fact that the residue plastic strain at initial high strain level markedly increased more than that at initial low strain level. On the other hand, it can be inferred that the sequence effect on damage accumulation is significant. Based on the above observation, it is concluded that the residue plastic strain due to the pre-cycle load history must be taken into account in the prediction of the residue cycles to failure.
In Eq. (7), the value of N f ; at Δε2 corresponding to strain level Δε2 =2 can be determined with the assistance of the typical strain–life curve. In order to confirm the applicability of the proposed method, the residue cycles to failure for 304 Stainless Steel with cycle damage history effect are predicted by using Eqs. (5)–(7). Moreover, the Miner's rule that the value of summing the cycle ratio at each strain level is 1 is also used to perform the residue fatigue life prediction on the tested material with cycle damage history effect in this study. The calculated cycles using the proposed method and Miner's rule, versus the corresponding observed cycles, are also plotted and shown in Fig. 7. It was found in this figure that almost all data points fell within the bounds of factor 3. Therefore, it is confirmed that both the methods used can provide good predictions of the residue cycles to failure for 304 Stainless Steel under two-step loading condition. Moreover, in order to identify which of the two used methods provides the better prediction performance, the following formula is applied: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u X ð Δi Δn Þ2 Π¼t ð8Þ n1 i¼1
Predicted Residue Life, N (Cycles)
Fig. 6. Comparison of experimental data and the fitted strain–life curves without and with the cycle damage history.
ð7Þ
10
10
4
3
w = 1 - ( Δ εp,2 / Δ ε
p,1
)
w
C1* = e C1
2
10 2 10
10
3
10
4
10
5
Observed Residue Life, N (Cycles) f
Fig. 7. Comparison of the observed residue cycles and the predicted residue cycles by the proposed method and Miner's rule .
L.-H. Chou et al. / Materials Science & Engineering A 636 (2015) 320–325
cycles to failure were found to be within the bound of factor 3. However, via a simple statistical procedure, the proposed method gives better predictions than Miner's rule in making residue life predictions.
estimate the residual cycle to failure. Based on the discussions in the preceding sections, the following conclusions can be drawn: 1. The extent of variation in additional cyclic softening/hardening is dependent upon the type of cycle damage history and the subsequent stable plastic strain amplitude response. 2. The effects of the applied strain level in cycle damage history on the subsequent stable stress–strain response, such as the shape of stable hysteresis loop and the improvement on the Massing behavior, are significant. 3. The loss in fatigue resistance caused by type II cycles damage history is higher than that due to type I cycles damage history. Simultaneously, the variation in the residue fatigue life is more sensitive to the ratio of the plastic strain range response, Δεp;2 , at second strain level to that at the initial level. 4. For the tested material with pre-cycle-damage, a transformable expression in the cycle ratio is proposed to estimate the residue cycles to failure via the specific ω factor that is dependent on the ratio of Δεp;2 to Δεp;1 . In addition, the sequence effect on the fatigue life is also involved in the proposed ω factor. 5. Based on the fatigue life prediction result using Miner's rule and the proposed method, good predictions of the residue
325
Acknowledgment The author would like to extend his gratitude to the financial support of NSC 101-2221-E-415-001. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
M. Zheng, J.H. Luo, X.W. Zhao, Z.Q. Bai, R. Wang, Int. J. Press. Ves. Pip. 82 (2005) 546. G. Waterloo, V. Hansen, J. Gjønnes, S.R. Skjervold, Mater. Sci. Eng. A 303 (2001) 226. K. Schoeler, H.-J. Christ, Int. J. Fatigue 23 (2001) 767. S. Mall, T. Nicholas, T.-W. Park, Int. J. Fatigue 25 (2003) 1109. H.-J. Christ, G. Hoffmann, O Ottinger, Mater. Sci. Eng. A (1995) 1. L. Dietrich, L. Zbigniew Kowalewski, Int. J. Fatigue 13 (1997) 87. M.C. Yip, Y.M. Jen, Fatigue Fract. Eng. Mater. Struct. 18 (1995) 1323. H. Ni, Z. Wang, Mater. Sci. Eng. A 314 (2001) 12. M.A. Miner, J. Appl. Mech. (1945) 159. G.R. Halford, J. Mater. 1 (1) (1966) 3.
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
The effects of pre-cycle damage on subsequent material behavior and fatigue resistance of SUS 304 stainless steel Liang-Hsiung Chou a, Yung-Chuan Chiou b,n, Ying-Jen Huang b a b
Department of Civil and Water Resources Engineering, National Chiayi University, Chiayi 600, Taiwan Department of Biomechatronic Engineering, National Chiayi University, Chiayi 600, Taiwan
art ic l e i nf o
a b s t r a c t
Article history: Received 15 January 2015 Received in revised form 18 March 2015 Accepted 18 March 2015 Available online 31 March 2015
A series of fatigue tests with type I and type II cycle damage histories were carried out to observe the effects of pre-cycle damage on the subsequent cyclic material response and fatigue resistance. It was found that the applied cycle history of damage leads to an increase in the extent of variation in cyclic hardening/softening. Furthermore, the shape of stable hysteresis loop changes with the type of cycle damage history under the same strain amplitude conditions. This is due to the occurrence of additional cyclic hardening/softening. By comparing the strain–life curves with and without a cycle damage history, it is found that the cycle damage history leads to a decrease in the number of cycles to failure. The loss in fatigue resistance is dependent on the type of pre-cycle damage history. For the fatigue test with a precycle history, an equivalent cycle ration C n1 is proposed to predict the residue cycles at the second level in this study. The validity of the modified expression is confirmed by comparing the predicted life results with the corresponding experimental data. It is apparent that the proposed method gives a better prediction than the Miner's rule for the two-step histories. & 2015 Elsevier B.V. All rights reserved.
Keywords: Residual fatigue life prediction Pre-damaged Cyclic behavior
1. Introduction In the course of manufacturing, some pre-damage is induced in the components due to plastic forming, as well as the various mechanical pre-histories on the cyclic stress–strain response lead to a problem of great importance in the fatigue life of components during their service. In several studies [1–3], it has been found that a prior mechanical pretreatment has a strong influence on microstructure and mechanical behavior during cyclic deformation. The extent of variation in stress– strain response, due to the prior pre-deformation, depends on the magnitude of the applied loading, type of pre-deformation and the tested material [4–8]. Based on the above experimental results, it can be concluded that the pre-deformation in the fabrication process leads to the damage in the material. Generally, in the fabrication process, components may receive cyclic pre-damage and are often subject to cyclic loading during later service. Hence, it is interesting to study the stress–strain behavior and predict the residue cycles to failure after cyclic loading at high or low strain level to the specified number of cycles. Generally, the value of summing the cycle ratio at each strain level is close to Miner's proposed value of 1 [9] in most observations. Therefore, the Miner's rule [9] is usually applied to predict a fatigue life of the material subjected to a variable amplitude loading history. It should be pointed out that the rule ignores the sequence effect on
n
Corresponding author. Tel.: þ 886 2717667. E-mail address: [email protected] (Y.-C. Chiou).
http://dx.doi.org/10.1016/j.msea.2015.03.070 0921-5093/& 2015 Elsevier B.V. All rights reserved.
damage accumulation based on the observation so that the estimation of fatigue cycles does not correspond to that with observed fatigue cycles. Consequently, in material with a variable amplitude loading history, the sequence effect on damage accumulation needs to be considered for fatigue life prediction. In the study, both cycle damage histories, which are pre-cycles corresponding to an approximate 20% of cycle ratio at strain amplitudes of 0.35% and 0.70% at R¼ 1, are investigated. The result is compared with the experimental value to assess the influence of cyclic pre-damage on the subsequent cyclic hardening/softening behavior, stable stress–strain response, and the residue cycles to failure of SUS 304 stainless steel. Moreover, for a twostep load history test, an equivalent cycle ratio C n1 corresponding to the second strain level is proposed to estimate the residue cycles during later cyclic straining for the tested material. Simultaneously, the sequence effect on damage accumulation is taken into account on the proposed equivalent cycle ratio. The validity of the proposed method is analyzed in this study.
2. Experiment procedure 2.1. Material and specimen In this study, the test material was SUS 304 stainless steel. The chemical compositions of the materials are given in Table 1. For the measurement on the chemical composition, the repeated times is two. The geometry of the fatigue specimen is shown in
L.-H. Chou et al. / Materials Science & Engineering A 636 (2015) 320–325
Table 1 Chemical compositions of SUS 304 stainless steels (weight %). C
Si
Mn
P
S
Ni
Cr
Cu
Fe
0.04
0.32
1.79
0.027
0.001
8.15
18.18
2.11
bal.
321
a servo-hydraulic Instron 8501 materials testing system with a constant strain rate of 0.01/s and a triangular waveform. A tension–compression extensometer with a 25 mm edge separation was used to control the strain applied in each fatigue test. The cycles to failure, Nf , was defined as the number of cycles performed corresponding to a drop of 10% in load in the fatigue test. The stable stress–strain response was obtained from the hysteresis loop at approximately half-life, where the stress–strain response was stable. In this study, the performed tensile fracture tests and fatigue tests are repeated two or three times under the same experimental condition.
3. Experimental results and discussion 3.1. Effect of cycle damage history on cyclic deformation behavior Fig. 1. Schematic illustration and dimensions of specimens.
Table 2 Mechanical properties for SUS 304 stainless steels. E ðGPaÞ
Sy;0:2% ðMPaÞ
Su ðMPaÞ
E:L: ð%Þ
K ðMPaÞ
n
187.15
490.5
537.5
50.5
895.98
0.1007
Fig. 1. By use of a CNC lathe, the as-received material was machined to form a specimen with a geometry shown in Fig. 1 and the surface was polished with a fine emery paper. Before the fatigue test, the tensile fracture test was performed under stoke controlled condition with a crosshead rate of 0.01 mm/s. The results of the tensile properties are listed in Table 2. It is noted that the values presented in Tables 1 and 2 are obtained by performing a mean value calculation on the measured results. 2.2. Test programs and procedures In this paper, a series of fully reversed fatigue tests, two-stage test with a low-high strain level sequence and two-stage test with a high-low strain level sequence were performed in order to investigate the effects of cyclic pre-damage on the subsequent cyclic deformation behavior. In this fatigue test, the applied strain was maintained at a constant level until failure occurred on the testing specimen, and is referred to as the typical fatigue test. Test data generated from the typical fatigue test is essential in observing the pre-cycle damage effects on the subsequent fatigue behavior. It is worth mentioning that the strain–life curve can provide the estimation of the number of cycles to failure at a given strain amplitude. Hence, the number of cycles at an initial strain level can be calculated on the basis of the typical strain–life curve in the design of cycle pre-damage history. Consequently, the typical fatigue tests had to be carried out first in this study. For the low-high test, the specimen was tested at initial strain amplitude of 0.35% for 8000 cycles ðn1 ¼ 8000Þ. Subsequently, the strain level was changed to a second level until failure occurred. This type of cyclic straining history is defined as type I cycle damage history. With the assistance of the typical strain–life curve, corresponding to type I damage history, the cycle ratio, C 1 ¼ Nf ;at εn1¼ 0:35% , is calculated to be equal to 21.29%. In this proce-
In the strain-controlled model, the variation of peak stress response with the applied cycle is usually observed for the tested material subjected to the repeating loading. As shown in Fig. 2, it has been observed that the typical cyclic-stress response curves at the different strain amplitudes also display significant decreases in the stress response for the first fifteen cycles and then the measured peak stress with cycle number decreases prior to failure. Furthermore, it can be seen from Fig. 2 that the plateau of stress in the cyclic-stress response curve increases as the cyclic strain amplitudes increase from 0.35% to 0.70%. This indicates that cyclic softening occurred during the cyclic deformation for SUS 304 stainless steel. Moreover, midlife hysteresis loop data from the fatigue tests were conventionally used to present the stable stress– strain response for the tested material subjected to the repeating loading. Hence, the scale of stable stress amplitude can be directly measured from the midlife hysteresis loop and the stable plastic Δε Δσ strain amplitude can be estimated from 2 p ¼ Δε 2 2E . In the calculation, the elastic modulus, E, measured from tension tests was used. For the correlation between the stable stress amplitude and stable plastic strain amplitude, it is usually fitted in the form of a power-law type equation and is represented as n Δσ ΔεP n ¼ Kn ð1Þ 2 2 where K n is the cyclic strength coefficient and the exponent nn is the cyclic strain hardening exponent. The two parameters K n and nn represent the stable response of a material to cyclic straining. In Eq. (1), K n and nn are obtained from a least squares fit of stress data
a
dure of determining the cycle ratios, Nf ; at εa ¼ 0:35% is the original cycle to failure life at initial strain amplitude of 0.35%. For the high-low test, the applied initial cyclic straining was at strain amplitude of 0.70% for 500 cycles. The loading history is named as type II cycle damage history. Similarly, the value of the corresponding cycle ratio is 20.64%. All fatigue tests were conducted on
Fig. 2. Superimposed plot of cyclic peak stress with number of cycles that data obtained from typical fatigue test at the strain amplitude ranging from 0.35% to 0.70%.
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Δε versus the corresponding data 2 p in log–log scale. For data obtained from the typical fatigue tests, the cyclic strength coefficient K n is calculated to be 616.66 and the cyclic strength exponent nn has a value of 0.0848. Conventionally, the fitted curve is labeled as the cyclic stress–strain curve. In this paper, the curve obtained from the typical fatigue tests is labeled as the typical cyclic stress– strain curve. It is noted that the typical cyclic stress–strain curve can be used to estimate the stable stress amplitude in the range of plastic strain amplitude from 0.1592% to 0.4901% in this study. Moreover, according to Eq. (1), the cyclic stress–strain curves with variable cycle damage histories are also obtained. In order to infer the effect of cycle damage history on strain-hardening capability, the fitted typical cyclic stress–strain curve, cyclic stress–strain curves with variable cycle damage histories, and the tensile stress– strain curve are simultaneously plotted in Fig. 3. As shown in Fig. 3, first, it is found that a good agreement exists between the fitted and experimental data. Furthermore, it can be seen from this figure that the tensile stress–strain curve lies above all plotted cyclic stress–strain curves, indicating that cyclic softening occurs at all applied strain amplitudes for the tested material without and with the cycle damage history effect. From the comparison between the typical cyclic stress–strain curve and cyclic stress– strain curves with variable cycle damage histories shown in Fig. 3, the cyclic stress–strain curve with the type I cycle damage history lies above the typical cyclic stress–strain curve, and the gap between these two curves increases along with the cyclic strain amplitude. The above observations in Fig. 3 are based on the range of plastic strain amplitude from 0.2011% to 0.4901%. It demonstrates that the type I cycle damage history would lead to a variation in cyclic hardening behavior and the degree in variation is more dependent on the magnitude of the followed strain amplitude. It is evident that the cyclic stress–strain curve with the type II cycle damage history is below the typical cyclic stress– strain curve and the extent of differences between both curves decreased with an increase in cyclic strain amplitude. Consequently, it can be concluded that the ratio of the stable plastic amplitude response in the applied two-step load history has a significant effect on the extent of variation in additional cyclic softening/hardening. The fitted typical cyclic stress–strain curve is implemented during the range of plastic strain amplitude from 0.1592% to 0.4901%, and the cyclic stress–strain curve with the type II cycle damage history is conducted up to only 0.44%. It should be pointed out that the above observations in Fig. 3 are based on the range of plastic strain amplitude from 0.1592% to 0.44%. Δσ 2
3.2. Effect of cycle damage history on stable stress–strain response Six actual hysteresis loops from experimental data at approximately half-life with strain amplitudes ranging from 0.35% to 0.70% were plotted with shifted axes such that their compressive loops all fall at the same origin. As shown in Fig. 4(a), it can be seen that the loading curves for all of the hysteresis loops fall on approximately the same locus at controlled strain amplitudes of higher than 0.35%. Some deviation is found between the tensile hysteresis loop and the dashed line, which is obtained by expanding the typical cyclic stress–strain curve with a scale factor of two. The expanded cyclic stress–strain curve is called the Massing curve. The observation indicates that the tested material without a cycle damage history effect has a near Massing behavior [10]. Fig. 4(b) and (c) presents the superimposed stable hysteresis loops with coincident compressive tips for the tested material with types I and II cycles damage history, respectively. As shown in Fig. 4(b) and (c), it is clear that the corresponding Massing curves are very close to all tensile branches of stable hysteresis loops. Therefore, it can be concluded that the tested material after predamage exhibits Massing behavior at all the considered strain amplitudes. Moreover, as shown in Fig. 4(a)–(c), the symmetry between the stable hysteresis loop in tension and compression is also observed. Therefore, based on the results of experimental observation, the shape of stable hysteresis loops shown in Fig. 4(a) –(c) can be simulated with the help of the corresponding expanded Massing curve. In addition, the stable plastic work ΔW p can be calculated by integrating the area within the stable hysteresis loop. A plot of the calculated vs. experimental stable plastic work is shown in Fig. 4(d), in which the diagonal line represents the perfect correlation line. From Fig. 4(d), it is found that data with type II cycles damage history fall within the dash-lines, whereas data with type I cycles damage history are still within the dotlines. Again, it can be seen that most of the data without cycle damage history effect are outside of the dot-lines. The observations indicate that the closeness between the simulated curves and the experimental data with type II cycles damage history is better than the other two groups of experimental data. In other words, the applied cycle damage history could lead to a considerable decrease in the discrepancy between the expanded curve and the hysteresis loop curve in tension. The type II cycle damage history gave somewhat better improvement in the discrepancy than that due to the type I cycle damage history. It was found that most of the data obtained from the typical fatigue tests are closer to the starting point of the X-coordinate axis shown in Fig. 4(d) than data with the cycle damage history under the same strain range condition. Essentially, it is acceptable that the magnitudes in the data with the types I or II damage effect are almost slightly greater in magnitudes than data without damage effect via a mean value calculation on the results under the same experimental conditions. Consequently, the variations in the shape, magnitude of stable hysteresis loop and the improvement on the Massing behavior can be attributed to the cycle damage history effect on the stable stress–strain response. It is pointed that both the applied cycle damage histories are constructed on an approximate 20% of cycle ratio at strain amplitudes of 0.35% and 0.70% in this paper. It is obvious that the effects of the applied strain level in cycle damage history on the subsequent stable stress–strain responses are significant. 3.3. Effect of cycle damage history on the cycles to failure
Fig. 3. Comparison of monotonic stress–strain curve and the simulated cyclic stress–strain curves without and with cycle damage history.
For most metals, it is found that the strain amplitude, Δε=2, versus the cycle number to fracture, Nf , is modeled as a straight line in log–log scale. Thus, the empirical relationship between the
L.-H. Chou et al. / Materials Science & Engineering A 636 (2015) 320–325
900
900
SUS 304 Stainless Steel
C1 = 20.64 % at εa = 0.70 %
700
Stress , σ (MPa)
Stress , σ (MPa)
SUS 304 Stainless Steel
Rε = - 1
700
500
300
100
500
300
100 (n-1)/n 1/n :Δε=(Δσ/E)+2 (Δσ/K)
-100 -0.1
0.2
0.5
0.8
1.1
:Δε=(Δσ/E)+2
-100 -0.1
1.4
0.2
0.5
(n-1)/n
(Δσ/K)
0.8
1.1
1/n
1.4
Strain , ε (%)
Strain , ε (%) 7
900
SUS 304 Stainless Steel
Based on the expended cyclic σ -ε curve
C1 = 21.29 % at εa = 0.35 %
6
, MJ/m3
700
Calculated
500
300
p
(ΔW )
Stress , σ (MPa)
323
100
4 3 : R = -1 : c1 = 21.29 % at ε a = 0.35 %
2 :Δε=(Δσ/E)+2
-100 -0.1
5
0.2
0.5
0.8
(n-1)/n
(Δσ/K)
1.1
:c1 = 20.64 % at ε a = 0.70 %
1/n
1.4
Strain , ε (%)
1
1
2
3
4
5
6
7
( Δ Wp ) Observed , MJ/m3
Fig. 4. Comparison of the expanded cyclic stress–strain curve and stable hysteresis loops obtained from (a) typical fatigue tests, (b) fatigue tests with type I damage history, (c) fatigue tests with type II cycle damage history and (d) plot of calculated vs. observed stable plastic work at various total strain amplitudes.
strain amplitude and the cycle number to fracture is represented as Δε=2 ¼ α ð N f Þβ
ð2Þ
In Eq. (2), the constant, α, and slope, β, are the constants in the best straight-line that can be obtained by linear regression analysis. According to the experimental results from the typical fatigue tests, the fitted value of α is 5.023 and the value of β is 0.2528. Consequently, the typical strain–life curve can be constructed and is plotted in Fig. 5. It can be seen from this figure that a good agreement exists between the solid line and experimental data. Similarly, for the tested material with type I cycles damage history, the strain–life is given by Δε=2 ¼ 4:9485 ð N f Þ 0:2576
ð3Þ
For the tested material with type II cycles damage history, the power law type equation is found to be Δε=2 ¼ 16:0909 ð N f Þ 0:4177
ð4Þ
A superimposed plot of strain–life curves of the tested material without and with type I and type II cycles damage history are shown in Fig. 6. It is interesting to note the effect of cycle damage history on the typical strain–life curve of 304 Stainless Steel. As shown in Fig. 6, it can be seen that the plotted lines are in good agreement with the data. Generally, the typical strain–life curve is superior in the number of cycles to failure under the same strain amplitude condition to both curves with different damage effects.
Fig. 5. Comparison of experimental data and the typical strain–life curve fitted by power function relationship.
The results may be attributed to the fact that the formation of fatigue crack initiation is accelerated on the fatigued specimen after the applied cyclic pre-damage. The typical strain–life curve lies a little above the curve with type I damage effect, and they are almost parallel to each other. It reveals that the decrease in fatigue life is almost the same whatever the strain amplitude is. For the
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n2;
at Δε2
is estimated from
n2; at Δε2 ¼ ð1 C n1 Þ N f ; at Δε2
3.4. Fatigue life prediction with cycle damage history effect The testing condition of cyclic pre-damage is composed of the strain level, Δε1 =2, and the corresponding cycle ratio C 1 ¼ n1 =Nf ; at Δε1 . For the cycles damage history used in this study, an approximate 20% of cycle ratio is adapted. Based on the experimental results as shown in Fig. 6, it is found that the degree of reduction in fatigue life is dependent on the initial strain level (Δε1 =2) and the subsequent strain level (Δε2 =2). Meanwhile, the variation in the residue fatigue life is more sensitive to the ratio of the stable plastic strain range (Δεp;2 ) corresponding to the second strain level to that measured in the initial strain level. In this study, for a material subjected a two-step cyclic loading, an equivalent cycle ratio C n1 that modifies the number of cycles at the strain level Δε2 =2 is proposed and expressed as follows: C n1 ¼ eω C 1
ð5Þ
where
ω ¼ 1
Δεp;2 Δεp;1
ð6Þ
Clearly, in Eq. (5), the consumed fatigue cycles at strain level Δε1 =2 are translated into the consumed fatigue cycles at strain level Δε2 =2 via the specific factor ω. In other words, the cycle ratio C1 at the strain level Δε1 =2 is equivalent to the cycle ratio C n1 at the specified strain level (Δε2 =2). Since the sum of the cycle ratio is 1 in a fully reversed cyclic straining condition, therefore, the cycle ratio C n2 at strain level Δε2 =2 is the value of (1 C n1 ). Further, the corresponding residue cycles to failure
where Δi represents the ratio of the predicted result to the experimental value, and Δn is the average value of all calculated Δi . In this study, the value for n is 23. The calculated value of Π for Miner's rule is 1.1886 and the value for the proposed method is 0.8586. Based on the calculated results for Π , it can be inferred that the proposed method provides a more accurate prediction than Miner's rule method in making residue life predictions.
4. Conclusions An experimental investigation into cyclic fatigue behavior of SUS 304 stainless steel with two type cycles damage history has been presented. The experimental data obtained from typical fatigue tests provides the basis for recognizing the significances of variation in the subsequent stable stress–strain response and the residue cycles to failure with pre-cycles damage. By taking the effect of cycle damage history into consideration, a new method is proposed to estimate the residue cycles to failure for the tested material with pre-cycles damage subjected to symmetrical cyclic straining. Both the proposed method and Miner's method are also used to
10
5
: C1* + C2* = 1 : C1 + C2 = 1
f
strain-curve with type II damage effect, the typical cyclic stress– strain curve is much higher than the curve. It can be seen that the extent of the gap between the strain life curve with type I damage effect and that with type II damage effect decreases with an increase in strain amplitude. Consequently, it can be concluded that the loss in fatigue resistance caused by type II cycles damage history is higher than that due to type I cycles damage history. It is attributed to the fact that the residue plastic strain at initial high strain level markedly increased more than that at initial low strain level. On the other hand, it can be inferred that the sequence effect on damage accumulation is significant. Based on the above observation, it is concluded that the residue plastic strain due to the pre-cycle load history must be taken into account in the prediction of the residue cycles to failure.
In Eq. (7), the value of N f ; at Δε2 corresponding to strain level Δε2 =2 can be determined with the assistance of the typical strain–life curve. In order to confirm the applicability of the proposed method, the residue cycles to failure for 304 Stainless Steel with cycle damage history effect are predicted by using Eqs. (5)–(7). Moreover, the Miner's rule that the value of summing the cycle ratio at each strain level is 1 is also used to perform the residue fatigue life prediction on the tested material with cycle damage history effect in this study. The calculated cycles using the proposed method and Miner's rule, versus the corresponding observed cycles, are also plotted and shown in Fig. 7. It was found in this figure that almost all data points fell within the bounds of factor 3. Therefore, it is confirmed that both the methods used can provide good predictions of the residue cycles to failure for 304 Stainless Steel under two-step loading condition. Moreover, in order to identify which of the two used methods provides the better prediction performance, the following formula is applied: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u X ð Δi Δn Þ2 Π¼t ð8Þ n1 i¼1
Predicted Residue Life, N (Cycles)
Fig. 6. Comparison of experimental data and the fitted strain–life curves without and with the cycle damage history.
ð7Þ
10
10
4
3
w = 1 - ( Δ εp,2 / Δ ε
p,1
)
w
C1* = e C1
2
10 2 10
10
3
10
4
10
5
Observed Residue Life, N (Cycles) f
Fig. 7. Comparison of the observed residue cycles and the predicted residue cycles by the proposed method and Miner's rule .
L.-H. Chou et al. / Materials Science & Engineering A 636 (2015) 320–325
cycles to failure were found to be within the bound of factor 3. However, via a simple statistical procedure, the proposed method gives better predictions than Miner's rule in making residue life predictions.
estimate the residual cycle to failure. Based on the discussions in the preceding sections, the following conclusions can be drawn: 1. The extent of variation in additional cyclic softening/hardening is dependent upon the type of cycle damage history and the subsequent stable plastic strain amplitude response. 2. The effects of the applied strain level in cycle damage history on the subsequent stable stress–strain response, such as the shape of stable hysteresis loop and the improvement on the Massing behavior, are significant. 3. The loss in fatigue resistance caused by type II cycles damage history is higher than that due to type I cycles damage history. Simultaneously, the variation in the residue fatigue life is more sensitive to the ratio of the plastic strain range response, Δεp;2 , at second strain level to that at the initial level. 4. For the tested material with pre-cycle-damage, a transformable expression in the cycle ratio is proposed to estimate the residue cycles to failure via the specific ω factor that is dependent on the ratio of Δεp;2 to Δεp;1 . In addition, the sequence effect on the fatigue life is also involved in the proposed ω factor. 5. Based on the fatigue life prediction result using Miner's rule and the proposed method, good predictions of the residue
325
Acknowledgment The author would like to extend his gratitude to the financial support of NSC 101-2221-E-415-001. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
M. Zheng, J.H. Luo, X.W. Zhao, Z.Q. Bai, R. Wang, Int. J. Press. Ves. Pip. 82 (2005) 546. G. Waterloo, V. Hansen, J. Gjønnes, S.R. Skjervold, Mater. Sci. Eng. A 303 (2001) 226. K. Schoeler, H.-J. Christ, Int. J. Fatigue 23 (2001) 767. S. Mall, T. Nicholas, T.-W. Park, Int. J. Fatigue 25 (2003) 1109. H.-J. Christ, G. Hoffmann, O Ottinger, Mater. Sci. Eng. A (1995) 1. L. Dietrich, L. Zbigniew Kowalewski, Int. J. Fatigue 13 (1997) 87. M.C. Yip, Y.M. Jen, Fatigue Fract. Eng. Mater. Struct. 18 (1995) 1323. H. Ni, Z. Wang, Mater. Sci. Eng. A 314 (2001) 12. M.A. Miner, J. Appl. Mech. (1945) 159. G.R. Halford, J. Mater. 1 (1) (1966) 3.