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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec

Creep–fatigue endurance of 304 stainless steels Xiancheng Zhang ⇑, Shan-Tung Tu, Fuzhen Xuan Key Laboratory of Pressure Systems and Safety, Ministry of Education, East China University of Science and Technology, Shanghai 200237, PR China

a r t i c l e

i n f o

Article history: Available online xxxx Keywords: Creep fatigue Hold time Strain range Stress relaxation Life prediction model

a b s t r a c t In this paper, the effects of different factors on the combined creep–fatigue endurance of 304 stainless steel as well as the strength and limitation of the different approaches for creep–fatigue life prediction are reviewed. The factors, which inﬂuence on the creep–fatigue endurance, include hold time, strain range, temperature, and stress relaxation behavior. Some relationships between time or number of cycles to failure and the individual factors are summarized from the available data. A phenomenological approach is proposed to describe the synergistic effects of different factors. The approaches used for creep–fatigue life prediction range from purely empirical to mechanistic models based on microstructural observations. Each approach does have some degree of success in dealing with a speciﬁc set of creep–fatigue data. The comparison of the prediction capacities of linear damage summation method, strain-range partitioning method, and damage rate approach is made for the same set of creep–fatigue data through Bayesian information criterion analysis. The damage rate model exhibits higher accuracy than other two models. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Creep–fatigue deformation is expected to be an important damage mode for a lot of high-temperature components as a result of power transients during operation as well as start ups and shut downs, each of which produces cyclic loading, while inherent stress relaxation during steady operation induces creep deformation [1]. Creep–fatigue interaction refers to the situation in which the rate of damage accumulation under complex loading differs from linear summation of damage rates produced by creep component and cyclic component [2], as seen in Fig. 1. Creep–fatigue interactions are often depicted microstructurally as the combined effects of fatigue damage and creep damage. Creep failure is generally manifested as creep voids on interior grain boundaries by cavitation damage while fatigue failure is generally due to by crack damage. However, during fatigue–creep interaction, creep cavitation damage may be found within the material in addition to surface fatigue damage. The linking and interaction of these two damage modes results in an accelerated failure. When such interaction occurs, the failure path would become mixed (transplus intergranular), as schematically shown in Fig. 1.

⇑ Corresponding author. Address: Key Laboratory of Pressure Systems and Safety, Ministry of Education, School of Mechanical and Power Engineering, East China University of Science and Technology, Meilong Road 130, Xuhui District, Shanghai 200237, PR China. Tel.: +86 21 64253513. E-mail address: [email protected] (X. Zhang).

The creep–fatigue interaction behavior of materials is frequently simulated in the laboratory by high-temperature low-cycle fatigue (HTLCF) tests with incorporation of hold time at constant strain or stress. These tests are often used to determine different factors, such as hold time, strain range, strain rate, and temperature, on the creep–fatigue resistance of materials and to develop the design codes for materials at some given conditions. In the past decades, a lot of efforts have been taken to develop the life prediction models and codes for design purposes of high-temperature components. Numerous creep–fatigue life prediction models have been proposed over the past half-century. A survey of the evolution of creep– fatigue life prediction models in 1991 revealed the existence of more than 100 models or variations on these models [3]. Most of the models are phenomenological. Each has enjoyed some degree of success in dealing with a speciﬁc set of creep–fatigue data. However, most models lack the generality that would enable them to achieve widespread acceptance in engineering applications [4]. In this paper, the effects of the different factors, such as hold time, strain range, and temperature, on the creep–fatigue lives of 304 stainless steel (SS) are summarized from the available data. The strength and limitation of the different creep–fatigue life predictions which are usually used for 304 SS are also reviewed on the basis of available creep–fatigue data. The basis and current status of development of the various approaches to the prediction of the combined creep–fatigue endurance are presented. It should be noted that all the data came from tests conducted in air and in strain control with the indicated hold periods at the peak tensile strain point on the hysteresis loop.

http://dx.doi.org/10.1016/j.tafmec.2014.05.001 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.

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Fig. 1. Schematic showing the creep–fatigue interaction and the failure modes due to fatigue, creep and creep–fatigue interaction.

2. Effect of different factors on creep–fatigue endurance of 304 SS 2.1. Hold time effect on creep–fatigue endurance The effect of hold time, th, on the number of cycles to failure, Nf, of 304 SS at 650 °C and 649 °C (1200 F) and at different total strain ranges, Det, can be seen in Fig. 2, where, for reference purposes, the data obtained in the continuous cycling tests have

Fig. 2. The effect of hold time, th, on the number of cycles to failure, Nf, of 304 SS at 650 °C and 649 °C (1200 F) and at total strain ranges of (a) 0.5% and 2.0%, and (b) 1.0% and 4.0% with respect to different strain rates. The open symbols, solid symbols, and half-solid symbols respectively denote the strain rates of 4 103, 4 105, and 3 103 s1. The data are collected from Refs. [5–11].

been arbitrarily plotted at a hold period of 0.01 min. The testing data were collected from Refs. [5–11]. The strain rate, e_ , used in creep fatigue testing ranges from 3 103 to 4 105 s1. From Fig. 2a, it can be seen that a consistent behavior is indicated for continuous cycling, compression-hold-only, and symmetrical hold testing, namely the number of cycles to failure, Nf, decreases linearly with increasing the hold time in double logarithmic coordinate. Data for the tension-hold-only testing indicate deviations from this graph, and the direction of the deviation is toward reduced fatigue life. The hold time corresponding to the point of this deviation is about 0.1 min for both the 0.5% and 2.0% total strain range. After this point, the number of cycles to failure, Nf, decreases linearly with increasing the tensile hold time in double logarithmic coordinate. However, at a given hold time, the value of Nf in the compression-hold-only or symmetrical hold testing is generally higher than that in the tension-hold-only testing. This result indicates that the tension hold would produce more damage than the compressive hold of 314 SS in creep fatigue testing. Moreover, it is interesting to ﬁnd that the saturation phenomenon of Nf can be observed when the tensile hold time are respectively higher than 30 min and 60 min when the total strain rage are 2.0% and 0.5%. When the total stain ranges are 1% and 4%, only continuous cycling and tension-hold-only testing data can be found, as seen in Fig. 2b. In double logarithmic coordinate, the number of cycles to failure, Nf, decreases linearly with increasing the tensile hold time. Moreover, the saturation behavior of Nf is also found when De was 4.0% and hold time was extremely long, i.e., 180 min. From Fig. 2a and b, a common feature can be found, that is, the number of cycles to failure is almost independent of strain rate. Owing to the limited data available at the extremely low e_ , for instance, 4 106 s1, no deﬁnite conclusions can be made as yet regarding this observation. Fig. 3 shows the variations of Nf of 304 SS along with hold time at 593 °C and at different values of De [9–14], where the data obtained in the continuous cycling tests have been arbitrarily plotted at a hold period of 0.01 min. The strain rate is either 4 103 s1 or 4 105 s1. The testing data show high scattering when De are 1.0% and 2.0%. One main reason for this phenomenon is due to the fact that the specimens may be subjected to different heat treatments prior to testing. A lot of testing data are obtained from Ref. [12], where the experimental results showed that the specimens subjected to aging treatment at 593 °C for 1000 h exhibited high value of Nf. These data have been indicated by the small arrows in Fig. 3. By comparing with Fig. 2, the following common results can be obtained. Firstly, the magnitude of Nf of specimen tested at compression-hold-only or symmetrical hold testing conditions is generally higher than that in the tension-hold-only testing. Secondly, the saturated Nf can be found at high hold time, i.e., higher than 180 min, when De is 1.0% or 2.0%. Thirdly, when the hold time is lower than 1.0 min, the effect of tensile hold on the creep fatigue resistance is relatively low. Fourthly, the value of Nf is almost not dependent of the strain rate at a given total strange range and a given hold time. The effect of tensile hold time on the value of Nf of 304 SS at 550 °C [15,16] and 538 °C [5,7,8,17] can be found in Fig. 4. When the temperature is 550 °C, the tensile hold time on the creep fatigue endurance can be neglected by comparing with the continuous cycling when the hold time is lower than 1.0 min. After hold time of 1.0 min, the number of cycles to failure linearly decreases with increasing the hold time in log–log coordinate. When De is 1.5%, the value of Nf slightly decreases with increasing the strain rate, as seen in Fig. 4a. This phenomenon is also found in the creep–fatigue testing at 538 °C at relatively lower total strain range, namely 0.55% and 1.0%. However, when the total strain range was higher than 4.0%, the effect of strain rate on the value of Nf at a given hold time can be neglected, as seen in Fig. 4b. This result indicates that, at a low testing temperature, the strain rate has an important effect on

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Fig. 3. The effect of hold time, th, on the number of cycles to failure, Nf, of 304 SS at 593 °C and at total strain ranges of (a) 0.5% and 2.0%, and (b) 1.0% and 4.0% with respect to two different strain rates. The solid symbols and open symbols respectively denote the strain rates of 4 103 and 4 105 s1. The data are collected from Refs. [9–14].

the creep fatigue endurance when Det is low. However, when the total strain range is high, the creep fatigue endurance is independent of strain rate. More experiments should be carried out to determine the critical strain rate blow which the strain rate effect on the creep–fatigue endurance should be considered. Conway et al. investigated the unsymmetrical holding on the creep–fatigue endurance of 304 SS at 650 °C [5]. The tests involved a ﬁxed hold period in tension plus a shorter hold period in compression, as seen in Fig. 5, where td and te respectively denote the tensile hold period and compressive hold period. It is shown that the very detrimental effect of a hold period in tension-only on the creep–fatigue can be signiﬁcantly reduced through introducing a short period in the compression portion of the cycle. For instance, when the total strain is set to be around 2.0% and a 3-min compression hold period is introduced, the fatigue life is within 80% of the fatigue life observed in the 30-min symmetrical-holding tests, whereas, without this small hold period in compression, the fatigue life is reduced to about 40% of the 30-min symmetrical holding fatigue life. This phenomenon may be due to healing effect of hold period in compression, which might provide a mechanism to reduce the tendency for internal void formation [5,7]. 2.2. Relationship between hold time and time to fracture The effect of tensile hold time, td, on the time to fracture, tf, of 304 SS specimens tested at three different temperatures is shown

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Fig. 4. The effect of hold time, th, on the number of cycles to failure, Nf, of 304 SS at (a) 550 °C with respect to different strain rates, and (b) 538 °C with respect to different total strain ranges. The solid symbols and open symbols in Fig. 4b respectively denote the strain rates of 4 105 s1 and 4 103 s1. The data at 550 °C and 538 °C are respectively collected from Refs. [15,16] and [5,7,8,17].

Fig. 5. Effect of ratio of compressive hold time and tensile hold time on the number of cycles to failure of 304 SS at 650 °C [5].

in Fig. 6, where the strain ranges are 0.5% and 2.0%, the strain rate is ﬁxed to be 4 103 s1, and data of tf is obtained by experiments or calculated by

tf ¼

Nf þ Nf td f

ð1Þ

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which indicates that the fatigue life measured in cycles to fracture decreases as the length of tensile hold period increases. Actually no contradiction exists and both trends are correct and mutually consistent, since introducing a hold period into the strain cycle increased the cycle time by a factor that is greater than that corresponding to the reduction in Nf [5]. Hence, the introduction of the tensile hold period will lead to the decrement of number of cycles to failure and the increment of time to fracture. If the value for the time to fracture with no hold periods, 2N f De=e_ is subtracted from each time-to-fracture value, a deﬁnite linearity can be obtained, as seen in Fig. 7, where t0 denoted the time to fracture using no hold periods. It can be seen the value for t0 decreased with increasing temperature. The similar phenomena were also found by Dawson et al. for 316 SS [18] and Walker for Cr–Mo steel [19]. Fig. 7 may be used to calculate the time to fracture of 304 SS for any tension-hold-only period when the temperature is given and the values for Det and e_ are respectively ﬁxed to be 2.0% and 4 103 s1. In order to further identify the effect of hold time on the number of cycles and time to failure, the time to failure versus number of cycles to failure of 304 SS tested at 650 °C, 649 °C and 593 °C with respect to various tensile hold times is summarized in Fig. 8, where the strain rate was kept to be 4 103 s1. The data tested at 650 °C (and 649 °C) and 593 °C are respectively collected from Refs. [5–11,7,9–14,20]. A continued degradation in fatigue life with increasing duration of hold time can be found when the testing temperature is 650 °C and the tensile hold period is lower than 60 min. The Log(tf)–Log(Nf) curves have negative slopes since increasing hold periods severely reduce the number of cycles to failure. However, the saturation of number of cycles to failure can be found when the hold-time is higher than 60 min. this phenomenon is also reﬂected in Fig. 2a. In such a case, the value for Nf becomes independent of hold-time and the time to fracture increased with increasing the hold-time. However, when the testing temperature is 593 °C, the linear Log(tf)–Log(Nf) curve with negative slope can be obtained at a given strain range. Moreover, the slops of the Log(tf)–Log(Nf) curves at different strain ranges are almost same. In 1970s, Esztergar investigated the creep–fatigue interaction and cumulative damage evaluations for type 304 SS to develop conﬁdent design criteria [10]. He suggested tf–Nf followed the following equation,

tf ¼ nðNf Þm

ð4Þ

Fig. 6. Effect of tensile hold time, td, on the time to fracture, tf, of 304 SS at three different temperatures, (a) 650 °C [5–8,11], (b) 593 °C [7,9,10,12–14], and (c) 538 °C [5,7,8,10].

where f is the continuous cycling frequency and can be written as

f ¼

e_ 2De

ð2Þ

And hence,

tf ¼ Nf

2 De þ td e_

ð3Þ

From Fig. 6, it can be seen that a logarithmic graph of time to fracture vs. the tension-hold period in minutes yielded a curve that is concave upward. With increasing the hold time, the time to fracture increased. This trend might appear to be in contradiction to Fig. 2,

Fig. 7. Graph of tf–t0 vs. hold time in tension only for 304 SS data obtained at different temperature and strain rate of 4 103 s1 with a total strain range of 2.0%.

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Fig. 8. The time to failure versus number of cycles to failure of 304 SS tested at (a) 650 °C, and (b) 593 °C with respect to different total strain ranges. The solid symbols and open symbols in Fig. 8a respectively denote the temperatures of 649 °C and 650 °C.

where n and m are material constants. This relationship can be used to describe tf–Nf relationship for the data at 593 °C and 650 °C when the hold time is lower than 60 min. However, he also found the saturation limits for hold time in cyclic-relaxation tests and for strain rate in continuous cycle-type tests. The saturation limits may be important for design because hold periods in excess of the saturation limits do not contribute to further damage. The number of cycles corresponding to the saturation limits may form a fatigue curve that includes the maximum hold-time effect. Therefore the saturation limits representing time endurance limits can be used for design in a manner analogous to the use of fatigue endurance limits [10]. When the testing temperature is 593 °C, Brinkman also developed the Log(tf)–Log(Nf) diagram for 304 SS [14,20]. The saturation of hold-time effect was also not found. A linear relationship could be established between the above two parameters. However, whether the data points would still fall on these straight lines for very long hold periods should be identiﬁed in the further research.

2.3. Effect of single cycle time on the creep–fatigue endurance Fig. 9 shows the relationship between the time to fracture, tf, and single cycle time, ct, of the data obtained at different testing temperatures, where the continuous cycling data are indicated by the small arrows. The strain rate ranges from 3 105 s1 to 4 103 s1. It can be seen that the value for tf linearly increases with increasing the value for ct in double logarithmic coordinate.

Fig. 9. The relationship between the time to fracture, tf, and single cycle time, ct, of the data obtained at three different testing temperatures, (a) 650 °C [5–10], (b) 593 °C [7,9,10–14], and (c) 550 °C [15,16].

The similar results were also obtained by Coles and Skinner [21] and Conway et al. [5]. An interesting phenomenon found in Fig. 9 is that the relationship between log(tf) and log(Nf) is almost not inﬂuenced by the strain rate used in creep fatigue testing if the total strain range is given. The conclusion from this correlation is that, once the time-to-fracture data are generated at one strain rate in hold-time tests, times to fracture at other strain rates with or without hold periods can be obtained simply by using the appropriate cycle time in conjunction with Fig. 9. For instance, if it is desired to estimate the continuous cycling fatigue behavior at a

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strain rate of 4 105 s1 and a total strain range of 2.0%, the time for one cycle is calculated to be 1000 s. When the temperature is 650 °C, the value for tf is evaluated to be 53 h through Fig. 8a. Using the following equation,

Nf ¼

tr ct

ð5Þ

the value for Nf is estimated to be 191. Hence, Fig. 9 could provide a simple approach to predict the value for Nf with or without hold periods. Fig. 10 shows the comparison between the predicted values and experimental values for Nf obtained from different experimental conditions. These experimental conditions can be divided into three categories. In the ﬁrst category, the continuous cycling fatigue tests were carried out at 650 °C and different total strain ranges with strain rate ranging from 4 106 s1 to 3.6 102 s1. The experimental results were from Ref. [5]. In the second category of experiments, the creep fatigue tests with different hold periods from 0 to 300 min (as indicated by the number in Fig. 10) at 550 °C with the total strain range of 1.0% and strain rate of 4.0 103 s1 were carried by Takahashi et al. [22]. In the third category, the creep fatigue tests with different hold periods from 0.1 to 120 min (as indicated by the number in Fig. 10) at 650 °C with the total strain range of 1.0% and strain rate of 3.0 103 s1 were carried by Schmitt et al. [23]. From Fig. 10, it can be seen that the estimated values of Nf shows fairly good agreement with the experimental data. Hence, the log(tf)–log(ct) relationship in Fig. 9 can be used to predict the fatigue behavior at wide-range strain rates and hold periods. However, more creep–fatigue experiments should be performed at other temperatures and other extremely low strain rates to develop a design criterion for 304 SS. 2.4. Effect of strain range on creep fatigue endurance of 304 SS The effect of the total strain of the number of creep–fatigue cycles to failure, Nf, at different temperatures with respect to different tensile hold periods is shown in Fig. 11, where the strain rate is kept to be 4.0 103 s1. Introducing the hold period in tension would lead to the decrement in Nf at a given total stain range. This effect of tension hold on the value of Nf becomes more obvious at higher temperature. For instance, the gap between Nf and Det line of continuous cycling testing and that of tension-hold-only testing with hold period of 10 min decreases with increasing temperature.

Fig. 11. Effect of the total strain of the number of creep–fatigue cycles to failure, Nf, with respect to different tensile hold periods and different temperatures, (a) 650 °C [5–11], (b) 593 °C [7,9,10–14,24–26], and (c) 538 °C [5,7,10].

Fig. 10. Comparison of experimental and estimated fatigue behavior of 304 SS at different temperatures and total strain ranges.

When the testing temperature is 650 °C, the value of Nf decreases with increasing the hold period when the hold period is lower than 30 min. However, with further increasing the hold period, the Nf– Det is almost not inﬂuenced by value of hold period, which also identiﬁes the saturation effect of hold-time in Fig. 8a. The relationship between the plastic strain range, Dep, and the number of cycles to failure at 650 °C and 593 °C with respect to different tensile hold periods can be seen in Fig. 12. The strain rate is kept to be 4.0 103 s1. At a given hold time, a linear relationship between Dep and Nf, can be found. This linearity is identical to the Cofﬁn–Manson relation [27], namely

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Fig. 13. A temperature correlation of the low-cycle-fatigue data for 304SS. The strain rate is kept to be 4.0 103 s1 [5].

strain range, the line describing hold-time data has a slope that is deﬁnitely steeper than that describing the no-hold-time data. Using the previously reported low-cycle-fatigue data for 304 SS, the linear relationship between log(Nf) and temperature at a given hold time can be found, as seen in Fig. 14. With increasing the tensile hold time, the slop of the line becomes steeper. In any case the trend in Fig. 14 is worthy of further study. Such a study could lead to an identiﬁcation of the temperature below which hold periods would not affect the fatigue life. For instance, when the total strain range is kept to be 1.0% and the temperature was 450 °C, the value of Nf is almost not depended on the hold time in the range from 0.1 min to 10 min. Fig. 12. Relationship between the plastic strain range, Dep, and the number of cycles to failure at (a) 650 °C [5,9,28,29], and (b) 593 °C [9,12,13], with respect to different tensile hold periods.

Dep ¼ Að2Nf Þc 2

ð6Þ

where A and c denote the fatigue-ductility coefﬁcient and fatigueductility exponent, respectively. The values of A with a positive sign and c with a negative sign increase with increasing the tensile hold time. Hence, the Dep–Nf curve becomes steeper as the length of hold time increased. The curves in Fig. 12 provide a method of extrapolating results from relatively short hold time tests at high plastic strain range to much low plastic strain ranges. At a given value of Dep, the difference in Nf between the testing with a given hold time and the continuous cycling testing is relatively small at 593 °C. This result indicates that the effect of hold time on the value of Nf becomes more detrimental at higher temperature. Moreover, more experiments should be carried out to determine the Dep–Nf curve at different temperatures. Then, the values of A and c would be ﬁtted with respect to hold time and temperature. A phenomenological model, which can be considered as the modiﬁcation of Cofﬁn– Manson equation, could be developed to predict the value of Nf at a given Dep. 2.5. Effect of temperature on the creep–fatigue endurance of 304 SS Conway et al. found the cyclic fatigue life to vary with temperature in the manner shown in Fig. 13 [5]. For a given strain range, the graph of log(Nf) versus temperature is linear over a fairly wide temperature range. For the continuous cycling, the slop of the line between log(Nf) versus temperature corresponding to strain range of 0.5% is obviously higher than that corresponding to strain range higher than 1.0%. This result indicates that temperature effect on the value of Nf at low stain range is relatively strong. At a given

Fig. 14. Variations of number of cycles to failure of 304SS along with temperature at the total strange of (a) 2.0% [5,7,8,13,14,17,24], and (b) 1.0% [16,23].

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2.6. Stress relaxation behavior Considering the creep–fatigue interaction in which the tension and compression with a hold period at a constant peak strain, the different stress values observed in the cyclic regime near half-life can be schematically deﬁned in Fig. 15. The correlations of stress-amplitude, Dr, at Nf/2 with time-to-fracture, tf, obtained from tension-hold-only testing at two different temperatures are shown in Fig. 16. There is a linear relation between Dr and tf on logarithmic graph. With increasing the value of Dr, the time-tofracture decreases. It is also noteworthy that the lines described for the two temperatures are essentially parallel (slope of 0.043). The comparison between the fatigue data involving hold periods in tension only with typical stress-rupture data for 304 SS tested at 650 °C and 593 °C is shown in Fig. 17, where the fatigue data are plotted as average tensile stress, ravg, vs. the total time under tensile stress to fracture. Here ravg is taken as the arithmetic mean of rt,max and rt,min, where rt,max is the maximum tensile stress imposed to attain the desired tensile-strain amplitude and rt,min is the tensile stress after relaxation during the hold period, as deﬁned in Fig. 15. When the temperature is 650 °C, fairly good agreement between the fatigue data with the published stress-rupture data is indicated. The data at 593 °C shown in Fig. 17 exhibited high scattering, since the data obtained from the heat-treated specimens are also involved. The heat treatment would have an important inﬂuence on the stress to rupture curve for 304 SS [30]. The stress-rupture curves for as-received and heat-treated 304 SS at 593 °C were determined by Shih [30] and shown in Fig. 17. It can be seen that almost all fatigue data involving tensile hold periods at 593 °C fall between two stress–rupture curves. The variations of tensile stress relaxation, rr,tension, along with the hold time at two different temperatures and two strain ranges are shown in Fig. 18. The deﬁnition of rr,tension can be seen in Fig. 15. It is obvious that the creep–fatigue tests of 304 SS exhibited cyclic-hardening characteristics. The shape of the relaxation curve is strongly dependent on strain range. At a given hold time, the stress relaxation would be high at high strain range. When the testing temperature is 650 °C, the conﬁguration of the relaxation curve is independent of hold time when the hold time is respectively longer than 180 min and 60 min at the strain ranges of 2.0% and 0.5%. This trend is not found when the testing temperature was 593 °C due to the limited stress relaxation data at long hold time. The variation of stress relaxation rate which is deﬁned as log(rr,tension/rmax) along with the hold time can be seen in Fig. 19. The negative value of stress relaxation rate decreases with increasing the hold time. Moreover, the effect of strain rate used in creep–fatigue testing on the stress relaxation rate is not obvious. In double logarithmic coordinate, a linear relationship

Fig. 16. Stress-amplitude vs. time-to-fracture graph for 304 SS tested at (a) 650 °C [5,9], and (b) 593 °C [9]. The numbers denoted by arrows represented the tensionhold period in minutes, and the open symbols and solid symbols respectively denote the strain rate of 4.0 103 s1 and 4.0 105 s1.

Fig. 17. Correlation of fatigue data involving hold periods in tension only with typical stress-rupture data for 304 SS tested in air at 650 °C [5] and 593 °C [9,12] and at different total strain ranges.

between log(rr,tension/rmax) and hold time can be plotted, as seen in Fig. 20. The slops of the lines are dependent of strain ranges. 3. Life-prediction methods for combined creep–fatigue cycling

Fig. 15. Deﬁnition of various stress values observed in hold-time tests.

A lot of life prediction methods have been proposed to assess fatigue lives of the components under creep fatigue conditions.

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Fig. 18. Effect of tensile hold time on the tensile stress relaxation of 304 SS tested at 650 °C [5] and 593 °C [9,12] with respect to two different total strain ranges.

Fig. 19. Effect of tensile hold time on the tensile stress relaxation rate, log(rr,tension/ rmax) of 304 SS tested at 650 °C [5] and 593 °C [9,12] with respect to different total strain ranges. The open symbols and solid symbols respectively denote the strain rate of 4.0 103 s1 and 4.0 105 s1.

For 304 SS, the following methods have been usually used, namely, linear damage summation method, frequency separation method, strain range partitioning method, damage function method based on hysteresis energy, damage rate model by Majumder and Maiya [13,25,31], Cavitation model by Nam et al. [24,29], extended strain-life equation [32]. Detailed reviews of most of the models mentioned above are available in the literature [33–36]. Rodriguez and Mannan [15] discussed the salient features of various life prediction methods. Also, the basis of development of the various approaches to the prediction of the combined creep–fatigue endurance was reviewed by Lloyd and Wareing [33]. It was concluded that an inadequate materials data base made it difﬁcult to draw sensible conclusions about the prediction capability of each of the available methods.

3.1. Linear damage summation method The linear damage summation (LDS) method was developed on the basis of the assumption that the creep and fatigue damage mechanisms were independent in nature. This method has been incorporated into Appendix of Code Case N47 of the ASME Boiler and Pressure Vessel Code [37]. This approach involves the linear summation of time and cycle fractions, wherein time fraction is used as a measure of creep damage and cycle fraction is used as a measure of fatigue damage. The total failure was expressed as the sum of these life fractions reaching 100%, i.e.,

Fig. 20. Linear relationship between log(rr,tension/rmax) and tensile hold time of 304 SS tested at (a) 650 °C [5], and (b) 593 °C [9,12] in double logarithmic coordinate. The open symbols and solid symbols respectively denote the strain rate of 4.0 103 s1 and 4.0 105 s1.

X Nf Xt þ ¼1 tr Npf

ð7Þ

where Nf denotes the number of cycles at a strain range, Npf is the pure fatigue endurance at the same strain range, t is the time at a given stress and tr is the time to rupture under pure monotonic creep loading at the same stress. The ASME design criterion stemming from this was altered slightly to a limit for allowable damage D. Further, Npf and tr are replaced by Nd and Td respectively,

p X Nf j¼1

Nd

j

þ

q X t D Td k k¼1

ð8Þ

where D is total allowable creep–fatigue damage value, Nf is number of cycles of loading condition j, Nd is number of designallowable cycles of loading condition j, t is time duration of load condition k and Td is allowable time at a given stress intensity. The values for D are different for different materials. The value of D is in the range from 0.6 to 1.0 for types 304 stainless steel [37]. A creep–fatigue interaction diagram based on LDS approach for 304 SS at 649 °C obtained by Campbell [6] is provided in Fig. 21. It should be noted that the creep damage and fatigue damage are calculated at 5% load drop off during the creep–fatigue tests. The creep–fatigue data for 304 SS at 538 °C obtained by Jaske et al. [17] is also involved in Fig. 21. Fig. 22 give the creep–fatigue interaction diagram with the hold time in the range from 30 to 600 min for 304 SS at different temperature by Takahashi et al. [22]. It can

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X. Zhang et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

Fig. 23. Creep–fatigue life prediction by using LDS method for 304 SS at 593 °C [13].

Fig. 21. Creep–fatigue damage interaction for tensile hold time tests of 304 SS at 649 °C [6] and 538 °C [17].

be seen that the damage values for type 304 steel with hold periods up to 30 min fall close to or much below the D = 1 curve, as seen in Fig. 21. Moreover, with decreasing the temperature, the creep damage fraction decreases. When the temperature is 500 °C, the total damage values is much lower than the D = 1 curve. However, when the temperatures are 550 °C and 600 °C, the ratio of the creep damage to the fatigue damage in Takahashi et al.’s testing data varies between 0.4 and 2.5, with the average value being about unity. The fatigue-life prediction results for various heats of Type 304 SS at 593 °C obtained by the LDS method was given by Maiya [13], as seen in Fig. 23. The dashed lines indicate deviation of the predicted life from the experimental life by a factor of two. The LDS method generates nonconservative results for loading case of continuous cycling and conservative results for loading case of compression-hold-only. Takahashi et al. [22] compared the experimental lives and predictions through damage summation method for 304 SS at different temperatures. The comparisons are summarized in Fig. 24. At 500 °C, predicted lives tend to be longer than the test results, while the opposite tendency can be found for the

Fig. 22. Creep–fatigue damage interaction for tensile hold time tests of 304 SS at three different temperatures [22].

results at 550 °C and 600 °C. Hence, temperature dependence of creep–fatigue life is not well represented by the LDS method, perhaps because of ignorance of the temperature dependence of ductility [22]. Although, the damage summation method was extensively employed in design codes in view of its simplicity for life prediction, this approach has some limitations. Firstly, there is an inherent assumption of load-path independence to each of the fatigue and creep damage processes in this method. This would become evident from isothermal tests designed to simulate the thermally induced creep–fatigue deformation mode commonly encountered in high-temperature plant operation. Here, cyclic strain is induced by thermal transients while creep strain accumulates during the time between the transients [33]. Secondly, the method assumes a compressive hold to be equally as damaging as a tensile hold. In practice, compression holds are found to have a healing effect in austenitic steel [38]. Thirdly, the strain rate dependence of creep damage accumulation is not incorporated in this model. Hence, this method cannot be used for waveforms without tensile hold periods. 3.2. Frequency separation methods This approach is essentially a modiﬁcation of the Cofﬁn–Manson relationship for pure fatigue. The time parameter is introduced

Fig. 24. Creep–fatigue life prediction by using LDS method for 304 SS at 593 °C [22].

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to account for other time-dependent factors, by incorporating a frequency term as follow a

Dep ¼ A1 ðNf v k1 Þ

ð9Þ

where A1, k and a are temperature-dependent constants and Nfvk1 is the frequency-modiﬁed life. In Eq. (9), v denotes the frequency which is calculated from v = 1/(t + th), where t is the time for continuous cycling and th is the hold time in each cycle. Subsequently, modiﬁcations were made to include the effects of wave shape by postulating that these effects are associated with tension going time and hysteresis loop imbalance [39]. Frequency separation equation can be written as,

! A1 v t 1k De0p 2

Nf ¼

ð10Þ

where De0p is equivalent plastic strain range, which can be expressed as

De0p ¼ Dep

" #a=a0 ðv c =v t Þk1 þ 1 2

ð11aÞ

and 0

k1 ¼ k1 þ aðk 1Þ

ð11bÞ

where vt is the tension-going frequency, vc is the compression-going 0 0 frequency, and a and k1 are the appropriate constants derived from the Basquin law. The frequency separation method has been found to give a good correlation between the experimental and the predicted lives for a variety of wave shapes and dwell times, as seen in Fig. 25. In this method, subdivision of inelastic strains into various components is not made. Hence, it can be directly used by designers. However, an assumption is used in this method, namely signiﬁcant damage for a cycle occurred near the peak tensile strain and is affected by strain rate in reaching this maximum. No distinction is made between the effects of tension and compression dwell periods on the damage. This assumption may be applicable to 304 SS. Moreover, an interesting feature of frequency separation is that it predicts fatigue lives which decrease indeﬁnitely with decreasing frequency. For type 304 SS, this is contrary to observation, since thermal ageing could improve the fatigue life [41].

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Research Center as an attempt to overcome many of the limitations of other life prediction methods to the creep–fatigue problem then available [42,43]. It was founded on the belief that two major changes in the damage summation method would improve its accuracy considerably. The ﬁrst is to use cyclic rupture data in place of the static data used then (and now). The second is to eliminate the double counting of the ‘creep effect’. The mechanistic basis of the method lies in its recognition of at least two different types of strains (plasticity by crystallographic slip and creep strain by grain-boundary sliding). Thus, SRP recognition is made of the fact that the total inelastic strain range is partitioned into four possible components depending on the directions of straining (tension or compression) and the type of inelastic strain (creep or time independent plasticity). The damage fractions resulting from each of the partitioned strain range components are summed up by an interaction damage rule to predict the creep–fatigue lives. The failure is assumed to occur when the summation of the damage fractions equaled unity. The detained introduction of this method can be seen in Ref. [4]. For 304 SS, SRP has been found to satisfactorily predict life within a factor of 2. For instance, Kuwabara and Nitta [44] performed the isothermal low-cycle fatigue tests of 304 SS at 550 °C and 600 °C. In some tests, the tensile holds with the periods ranging from 1 min to 60 min were used. By using this method, it is possible to predict the isothermal fatigue life within a factor of 1.5, as seen in Fig. 26. Maiya et al. compared the experimental lives and predictions through SRP method for 304 SS at 593 °C with respect to different tensile hold periods [9], as seen in Fig. 27. It can be seen that this method leads to non-conservative results for loading cases with extremely long tensile hold periods, i.e., 180 min and 600 min. the data with extremely short hold period of 1 min tend to fall outside the factor of 2 scatter band on life. Although the SRP method has been widely used, it has some inherent limitations. For instance, this method is unable to deal with elastic strains, hence making it most useful for large plastic strains. Accurate knowledge of cyclic history was required when this method is used. Complex loops naturally become difﬁcult to partition.

3.4. Damage functions based on hysteresis energy

Strain-range partitioning (SRP) method was introduced in 1970s by Manson, Halford, and Hirschberg of the NASA-Lewis

Ostergren has proposed a damage function based on net tensile hysteresis energy to predict fatigue life under different frequencies and hold time conditions [45]. The mean stress concept was introduced to take into account the detrimental effects of compression holds in some nickel base superalloys [46]. By analogy with crack

Fig. 25. Fatigue life predictions for various heats of type 304 SS at 593 °C by frequency-separation method [40].

Fig. 26. Fatigue life predictions for type 304 SS at 550 °C and 600 °C by SRP method [44].

3.3. Strain-range partitioning method

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X. Zhang et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

Fig. 27. Creep–fatigue life prediction by using SRP method for 304 SS at 593 °C [13].

closure, the tensile hysteresis energy is approximated as DWT = rTDep, where rT is the tensile stress range. It was then assumed that the universality of the Cofﬁn–Manson law applies to hysteresis energy and hence

rT Dep Nbf ¼ constant

ð12Þ

Fig. 28. Correlation of number of cycles to creep–fatigue failure, Nf, of type 304 SS at 650 °C with damage function rT Debp v bðk1Þ . The data are collected from Ref. [5].

It should be noted that Eq. (12) contains the inﬂuence of mean stress rm as rT = rm + Dr/2, where Dr is the stress range associated with a given Dep, as denoted in Fig. 15. Generally, austenitic stainless steels at elevated temperatures do not exhibit signiﬁcant cyclic hardening or softening [5]. In such a case, mean stresses do not develop and equation (12) would be equivalent to the Cofﬁn–Manson law. The above equation has been modiﬁed considering frequency and hold time conditions during which, time-dependent damages become signiﬁcant as,

rT Dep Nbf v bðk1Þ ¼ constant

ð13Þ

Frequency is suitably deﬁned depending upon the damaging effects of compression and tension holds. In this way, the beneﬁcial and harmful effects of compression holds could be taken into account. For austenitic type 304 SS, since the compression holds almost do not have an important inﬂuence on the damage, the frequency can be expressed as

v¼

1 tc þ td te

ð14Þ

where tc is the time for continuous cycling, and td and te respectively denote the tensile hold period and compressive hold period. Using the creep–fatigue testing data for 304 SS at 650 °C in Ref. [5], the frequency-modiﬁed damage function, rT Debp v bðk1Þ is plotted against number of cycles to failure Nf in Fig. 28. It can be seen that the value of parameter b is close to 1 and the value of k is lower than 1. Fig. 29a shows the ﬁtting relation between rT Debp v bðk1Þ and Nf for 304 SS at 593 °C. The data are collected from Refs. [9] and [13]. Both the values of b and k are lower than 1. However, for the 304 SS with different heat treatments investigated by Brinkman and Korth [12], although the ﬁtting line relationship between rT Debp v bðk1Þ and Nf can be ﬁtted in double logarithmic coordinate, the value of k is higher than 1, as seen in Fig. 29b. Hence, the microstructure could have an important inﬂuence on the parameter ﬁtting when this method this method was used.

Fig. 29. Correlation of number of cycles to creep–fatigue failure, Nf, of type 304 SS at 593 °C with damage function rT Debp v bðk1Þ . The data are respectively collected from (a) Refs. [9,13], and (b) Ref. [12].

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3.5. Damage rate model The damage rate (DR) model was developed by Majumadar and Maiya in 1980s [13,25,31]. In this approach, it was assumed that there were two types of damage namely cracks (fatigue) and cavities (creep) in creep–fatigue situation. The creep–fatigue damage was measured by the current crack length, a, and cavity size, c. The initial crack length and cavity size were set to be a0 and c0, respectively. The crack growth rate due to fatigue and cavity growth rate due to creep can be respectively written as [25]

1 da ¼ a dt 1 dc ¼ c dt

T ð1 þ a lnðc=c0 ÞÞjep jm je_ p jk C

G G

jep jm je_ p jkc

ð15aÞ

ð15bÞ

where T, C, G, m, k, and kc are material parameters depending on temperature, environment and metallurgical state of the material, ep and e_ p are values of plastic strain and strain rate respectively. In Eq. (15), T and G are used in presence of tension, and C and G are used in the presence of compression. In fact, Eq. (15a) reﬂects the fatigue crack growth due to the interactive fatigue and creep damages through the term (1 + a ln (c/c0)). Hence, when the fatigue crack growth is due to the interactive fatigue and creep damages, the creep–fatigue life can be evaluated through integration of Eqs. (15a) and (15b). If the failure is only due to creep damage by the cavity growth, the creep–fatigue life can be evaluated through integration of Eq. (15b). Using these two approaches, Majumder [25] predicted the cycles to failure of 304 SS due to tensile-hold fatigue at 593 °C and different total strain ranges, as respectively seen in Fig. 30a and b. It can be seen that the DR model using interactive damage approach could satisfactorily predict fatigue lives for 304 SS. The number of cycles predicted by the creep damage approach is lower than that predicted only by creep damage approach. However, when the hold time is extremely long, i.e., 600 min, the number of cycles predicted by creep damage is close to experimental data, as seen in Fig. 30b. Moreover, the interactive DR model is found to satisfactorily predict fatigue lives for 304 SS under creep–fatigue tests at different temperatures and with different loading conditions, as seen in Fig. 31 [31]. Some parameters identiﬁed in Eq. (15) changes along with temperature. The main advantage of this model is that it considers the microstructural damage parameters and their evolution with time. Also, it takes into account the acceleration effect of creep damage due to cavity growth on the fatigue damage due to crack growth. Moreover, strain history of components can be calculated more reliably than their stress history, making this method more useful to designers [15]. However, the parameters involved in Eq. (15) might change along with testing condition, as seen in Fig. 31. These parameters would inﬂuence the interaction of creep and fatigue damages and ﬁnally inﬂuence the predicted results. Determination of these parameters needs a lot of elaborate experiments. Moreover, how to quantitatively determine the initial crack length and cavity size is also a problem to be solved. 3.6. Cavitation model Nam et al. developed the creep–fatigue life prediction model on the basis of the concept of round-type cavity nucleation factor [24,29]. In their model, creep–fatigue failure was assumed to be controlled by general creep damage rather than by a process of fatigue crack initiation and propagation. In this model, it was assumed that cavities were formed in every cycle and that the number of cavities in one cycle was proportional to the plastic strain range. Fatigue life was deﬁned by unstable crack advance,

Fig. 30. Prediction of the cycles to failure of 304 SS due to tensile-hold fatigue at 593 °C and at different total strain ranges by using DR model, (a) creep–fatigue interactive damage approach through integration of Eqs. (15a) and (15b), and (b) creep damage approach through integration of Eq. (15b) [25].

Fig. 31. Comparison between the predicted results and experimental results for 304 SS at different temperatures and under different loading conditions using DR model [31].

which happened if the crack tip opening displacement became equal to the spacing of the nucleated intergranular cavities. However, the cavity nucleation factor is found to be closely related with the density of grain boundary precipitates. When this method is used, the generalized relationship between the value of cavity nucleation factor and the characteristics of grain boundary precipitates of a given material should be known. The creep fatigue life is then predicted by measuring the density of grain boundary precipitates of a material. The method is capable of predicting lives for 203 SS even for conditions of long hold periods, as seen in Fig. 32.

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X. Zhang et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

3.7. Extended strain-life equation approach A procedure developed by Zhang et al. to predict the fatigue life over a wide range of temperatures and strain rates [32]. The isothermal fatigue life was expressed in terms of the total strain range through a continuous damage concept. A strain-life correlation was suggested in terms of three variables, namely the total strain range, the temperature, and the strain rate. When the hold periods are introduced in the fatigue cycling, an equivalent strain rate is used to determine the effect of hold period. The detailed introduction of this approach can be found in Ref. [32]. The predicted lives by using this approach are compared with test data of AISI 304 at 650 °C with different hold periods, as shown in Fig. 33. For tensile hold periods only, this method predicts fatigue lives greater than the test results in most cases and lead to non-conservative results. This situation may be explained by the fact that the equivalent strain rate could not correctly describe the evolution of the stress state in the material during the hold time. In fact, the creep effect during the hold period would be larger than that produced by the loading with the equivalent strain rate (triangular wave-form). The equivalent strain rate concept may be used only when the hold time was very short. In contrast, for symmetrical tensile and compression hold periods, the predicted lives are smaller than the test data. This is due to the fact that, the compression hold time effect is supposed to be additive to that of tensile hold time in this method. This assumption seems to be unrealistic, since the healing effect due to an additional compression hold time for 304 SS is not considered [5].

Fig. 33. Prediction of the cycles to failure of 304 SS by using extended strain-life equation approach [32].

3.8. Comparison of prediction capacities of LDS, SRP, and DR methods Maiya predicted the creep–fatigue life for different heats of type 304 SS LDS, SRP, and DR methods [13]. The predicted results from LDS and SRP methods are respectively shown in Figs. 23 and 27. When the DR method was used, he used two values for kc, i.e.,0.55 and 0.60, in Eq. (15) to obtain two different set of predicted data. In order to estimate the prediction capacities of LDS, SRP, and DR methods, two error analysis methods are used. The ﬁrst method is based on calculation of standard deviation (SD) and the second method is on the basis of Bayesian information criterion (BIC). Generally, for a given set of data, the model which gives the minimum value of SD or BIC is selected as the best model with best prediction capacity. The values of SD and BIC can be expressed as

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pn 2 i¼1 ðlg N p lg N e Þ SD ¼ n1

ð16aÞ

Fig. 34. Comparison of prediction capacities of LDS, SRP, and DR methods through SD and BIC analyses.

BIC ¼ m lnðnÞ 2 ln½lnð^hÞ

ð16bÞ

where n is the number of the experimental data, m is the number of the estimated parameters which are depended on the loading condition used in creep–fatigue testing, lnð^ hÞ is the maximum likelihood, Np is the predict result and Ne is the experimental result. The value of maximum likelihood can be obtained using the trust region method in the optimization toolbox of MATLAB 6.5. The detailed introduction of BIC method can be seen elsewhere [47,48]. Using the data by Maiya [13], the calculated values of BIC and SD for LDS, SRP, and DR methods can be seen in Fig. 34, where DR-a and DR-b are obtained from two different predicted results by using DR model with kc = 0.55 and 0.60. It can be seen that, the values of SD and BIC are lowest for DR model. The SRP method generates the largest values of BIC and SD. This result indicates that the damage rate model exhibites highest accuracy. However, more experiments with the wide range of experimental conditions should be used to support this result. 4. Conclusions

Fig. 32. Comparison between the predicted results and experimental results for 304 SS at 593 °C and at different total strain ranges. The numbers denote the tension-hold period in minutes [24,29].

The complex nature of creep–fatigue phenomenon of 304 SS is addressed in this review. The effects of different factors, such as hold time, strain range, temperature on the creep–fatigue duration are discussed. The stress relaxation behavior during the creep fatigue testing is also discussed. Then, some life prediction techniques, which have been used for 304 SS under creep fatigue conditions,

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X. Zhang et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

are brieﬂy reviewed. The prediction capacities of these techniques are discussed on the basis of the collection of the data from the open literature. The following conclusions are drawn from this work:

15

Science and Technology Commission of Shanghai Municipality (13DJ1400202 and 12JC1403200).

References (1) When other experimental conditions are given, the tension hold produces more damage than the compressive hold of 304 SS in creep fatigue testing. The introduction of the tensile hold period would lead to the decrement of number of cycles to failure and the increment of time to failure. For creep–fatigue testing with tensile hold time, if the value for the time to fracture with no hold periods, t0, is subtracted from time-to-fracture value, tf, a linear relationship between tf and t0 and hold time can be found. When the temperature is relatively high (i.e., 650 °C), the saturation of number of cycles to failure can be found when the hold-time is higher than 60 min. (2) When the tensile hold time is same, the strain rate may have an important effect on the creep fatigue endurance at low temperature, especially when the total strain range is relatively low. Generally, the number of cycles to failure would increase with decreasing the temperature. When the temperature is lower a certain value, the effect of hold time on the creep–fatigue endurance might be not obvious. (3) A linear relationship between the time to fracture, tf, and single cycle time, ct, in double logarithmic coordinate is existed on the basis of analyses on the data obtained at different conditions. By using this relationship, a phenomenological approach is proposed to predict the creep–fatigue lives for 304 SS. (4) When the temperature and strain rate are given, the relationship between the plastic strain range, Dep, and number of cycles to failure, Nf, would follow the Cofﬁn–Manson relation. Hence, the Dep–Nf curve becomes steeper as the length of hold time increases. More experiments should be carried out to determine the Dep–Nf curve at different temperature to develop a phenomenological model. (5) The creep–fatigue tests of 304 SS exhibits cyclic-hardening characteristics. The shape of the relaxation curve is strongly dependent on strain range. In double logarithmic coordinate, a linear relationship between log(rr,tension/rmax) and hold time can be plotted. The value of log(rr,tension/rmax) is used to characterize the stress relaxation rate. (6) The basis and current status of development of the various approaches, which have been usually used to predict the combined creep–fatigue endurance of 304 SS are reviewed. These range from purely empirical to mechanistic models based on microstructural observations. Each approach has some degree of success in dealing with a speciﬁc set of creep–fatigue data. (7) By using Bayesian information criterion analysis, the comparison of the prediction capacities of linear damage summation method, strain-range partitioning method, and damage rate approach is made for the same set of creep– fatigue data. The damage rate model exhibits highest accuracy. However, the parameters used in damage rate approach are dependent of the testing condition, which would inﬂuence the predicted results.

Acknowledgements The authors are grateful for the support by National Natural Science Foundations of China (51322510, 11172102 and 51175177). The author Xiancheng Zhang is also grateful for the support by Special Programs for Key Basic Research of

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