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An approach to life prediction for a nickel-base superalloy under isothermal and thermo-mechanical loading conditions
International Journal of Fatigue 53 (2013) 49–57
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
An approach to life prediction for a nickel-base superalloy under isothermal and thermo-mechanical loading conditions F. Vöse ⇑, M. Becker, A. Fischersworring-Bunk, H.-P. Hackenberg MTU Aero Engines GmbH, Dachauer Straße 665, 80995 München, Germany
a r t i c l e
i n f o
Article history: Available online 6 December 2011 Keywords: Life prediction Cyclic fatigue Thermo-mechanical fatigue Nickel-base superalloy
a b s t r a c t Scope of the present work is the development of a unified approach for the life prediction under isothermal and thermo-mechanical loading conditions. The basis for the life prediction is the stress–strain response of the material for one selected cycle obtained in the context of a Finite Element Analysis. Here the application of an appropriate inelastic constitutive model is essential. A phenomenological fatigue model is proposed which is able to describe mean-stress effects, as well as dependencies on hold time and cycle duration with reasonable accuracy. The model performance is compared with that of a mechanism based literature model. The model accuracy is evaluated on a statistical basis through an evaluation of the variance in the ratio between predicted life and actual life. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The design of jet engine components is generally based on specimen testing results. The challenge in the design phase is to predict component life as exact as possible to avoid expensive redesign. Typically isothermal low-cyclic fatigue (LCF) tests, as well as more realistic thermo-mechanical fatigue (TMF) tests are therefore available. In the present paper the criterion defining initiation life Ni is set to the occurrence of a 0.4 mm deep crack. For the alloy IN718 under investigation a close correlation of Ni and the number of cycles to failure Nf can be expected. One major difference between specimen tests and fatigue of real components is the typical duration of the performed load cycles. Under service conditions jet engines can experience about 1200–10,000 startstop cycles within 10,000 h of flight operation. In order to design parts for a certain target life including safety margin usually 20,000–40,000 cycles have to be demonstrated. The only possibility to perform tests at such low load levels is to increase the test frequency to 1–60 cycles per minute. Historically these tests have been performed primarily under isothermal conditions. To predict initiation life on this basis a Manson–Coffin approach [1,2] can be used. It correlates the number of cycles to initiation Ni to the inelastic strain range Dein.
Dein ¼ e0f ð2N i ÞC 2
ð1Þ
Since time-dependent damage mechanisms become more and more dominant at higher temperatures, frequency effects have great
⇑ Corresponding author. E-mail address: [email protected] (F. Vöse). 0142-1123/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2011.10.018
impact on life prediction of hot section parts. The so called frequency modified or frequency separation models proposed by Coffin and others [3,4] have been successfully used to model high-temperature isothermal tests with different strain rates and hold times. In addition real engine parts will be exposed to non-isothermal load histories. An important characteristic is the TMF phase angle which describes the phasing between mechanical strain and temperature during strain-controlled TMF tests (see Fig. 1). Depending on the actual TMF phase conditions load cycles can lead to different damage mechanisms. This has been accounted for by Neu and Sehitoglu [5] when they proposed their so called ‘damage rate model’. The idea is to differentiate between three different damage mechanisms, namely cyclic fatigue, in-phase (IP) TMF and out-ofphase (OP) TMF. While cyclic fatigue caused by small fatigue crack growth is assumed to take place under any cyclic loading condition, enhanced intergranular creep damage occurs mainly during IP-TMF, and oxidation induced brittle surface cracking occurs predominantly during OP-TMF. Neu and Sehitoglu [5] model these IP and OP phase effects by introducing empirical phase factors. Other authors like Nissley [6], Schmitt et al. [7] consider microcrack growth to be the dominant damage mechanism. Their models correlate the cyclic crack growth rate to a strain energy density parameter similar to that proposed for isothermal conditions by Dowling [8]. Within the present work an empirical approach will be developed, allowing a unified description of cyclic fatigue tests under isothermal and TMF conditions. The predictive capabilities of the newly developed model will be compared with the mechanismbased literature model from [5]. To show the applicability of both models for a larger set of fatigue tests, experimental data taken from several literature sources is considered.
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F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
Nomenclature A B b
coefficient in creep damage equation coefficient in oxide growth equation parameter describing the strain ratesensitivity of environmental damage b exponent on time in oxidation law C Manson–Coffin exponent C1, C2 elastic life curve parameters C3, C4 inelastic life curve parameters C5 coefficient in frequency modified term Dfat, Dcr, Denv damage portions due to fatigue, creep, and environmental degradation D0 diffusion coefficient d0 ductility of brittle surface layer Dein inelastic strain range Demech mechanical strain range Dr stress range e0f fatigue ductility coefficient e_ mechanical strain rate e_ s secondary creep rate e_ th thermal strain rate fcyc test frequency (=1/tcyc) critical oxide growth when oxidation assisted cracks hcr start to grow faster Kp,eff effective parabolic oxidation constant k, k0 exponent in frequency modified term
L m Na Nf Ni NP⁄
t0 pi
Uenv Q Qenv Q0 R Rr
r ramp rstatistic rUTS T Tmax/min t tcyc tref nenv, ncr
maximum likelihood function walker exponent actual fatigue life from experiment number of cycles to failure number of cycles to initiation predicted fatigue life intrinsic activation volume probability of occurrence phase factor for environmental damage nominal activation energy for time-dependent damage contributions activation energy for oxidation intrinsic activation energy universal gas constant stress ratio stress stress amplitude standard deviation to determine L ultimate tensile strength temperature (in Kelvin) maximum/minimum temperature time cycle time reference time parameters of phasing functions
Fig. 1. Visualization of several TMF load cycles with different phase angles.
2. Experimental database
3. Constitutive model calculations
The precipitate strengthened nickel-base superalloy IN718 is one of the most common structural alloys used in aero engines. Therefore a large number of publications reporting its chemical and physical properties exist. The present paper uses cyclic fatigue test data reported in [9–12]. A comparison of the chemical compositions of the different specimens is given in Appendix A (see Table A.1). Heat treatments and processing are similar. This paper is limited to smooth uniaxial conditions only. All valid tests used for model regression are given in Appendix B. Most tests have been performed under isothermal conditions. In addition 16 TMF tests complete the selection of 88 cyclic fatigue tests. A large number of tests with different loading conditions, including different test frequencies, R-ratios, hold times, test temperatures, and TMF phase conditions (c.p. Table B.1), are considered. Since the tests also range about several orders of magnitude in life (102. . .106) it is expected that creep- and oxidation-dominated failure mechanisms will occur at least in some specimens.
Becker and Hackenberg [13] have proposed a constitutive model, which captures the relevant effects observed for IN718, such as cyclic plasticity, creep, and ratcheting behavior. The constitutive model parameters have been obtained after regression [13] using special constitutive tests involving a complex load history. Even though the model parameters have not been re-adjusted to the experiments considered in the present study, overall a sufficient correlation between measured stress amplitudes and the stress amplitudes predicted by the constitutive model has been found for Ni/2 (see Fig. 2). Since raw data on stress–strain behavior is not available for all tests within the present work, constitutive model calculations have been performed. Therefore the stress–strain–temperature history for each test has been simulated cycle by cycle by Finite Element Analysis. The results were stored only for the cycle at Ni/2. The full hysteresis loop data of this cycle is later on needed to determine integral quantities during the fatigue life calculation. Since all
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F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
Experimental stress amplitude [MPa]
1200
1100 1000
K p:eff ¼
900
1 tcyc
800
Uenv ¼
700
Z
ð4Þ
D0 expðQ env =RTÞdt
ð5Þ
t cyc
0
Z
1 t cyc
1=b
2ðDemech Þ2=bþ1 ðe_ mech Þ1b=b
hcr d0 BUenv K p:eff
Denv ¼
"
t cyc
exp
0
2 # 1 e_ th =e_ mech þ 1 dt 2 nenv
ð6Þ
600 500 400 300
Model stress amplitude Fig. 2. Comparison of measured stress amplitude versus predicted stress amplitude incl. line of best fit.
specimen tests are smooth no geometry dependency is expected. A single 20 node brick element is used for the analysis of the deformation behavior during the test. The experimental database given in Appendix B includes LCF and TMF specimen tests from four different publications [9–12]. Clearly, both, the small differences in the chemical composition of the alloys (see Appendix A), as well as slight differences in the heat treatments, have an impact on stress–strain behavior and fatigue life. In this context the scatter observed in Fig. 2 when comparing measured and predicted stress amplitudes might be explained by the variations between the four different materials under investigation.
Here the cyclic oxidation parameters b = 0.75 and b = 1.5 will be applied. The same values have been successfully used to model environmental damage in Mar-M-247 [14] and 1070 steel [5]. The remaining parameters can be treated as four independent parameters hcrd0/B, D0, Qenv, nenv. During IP-TMF and high temperature tensile dwell intergranular cracking is expected to further decrease fatigue life. For IN718 this was confirmed by [9] where a mixture of transgranular and intergranular crack growth was observed for an IP-TMF test experiencing a high cyclic load level. The creep damage formulation (7) can be regarded as a mathematically identical formulation to what was proposed in [14], if e_ s takes the form e_ s ¼ expðQ cr =RTÞ ½ðjrj þ rÞ=ð3KÞm . For reasons of consistency it is reasonable to apply the same formulation within both, the constitutive and the fatigue model. In our case the constitutive model from [13] using a different formulation for e_ s is applied. Its general form can be described by e_ s ¼ f ðhri; TÞ. An advantage of the creep law from [13] is that it has been fit to a large number of proprietary creep tests for IN718 and is expected to be more accurate than the approach from [5,14].
Dcr ¼ 4. Modified literature model In the following the governing equations are presented for the chosen literature model proposed by Neu and Sehitoglu [5,14]. A linear damage rule (2) describes the summation of creep damage (Dcr), environmental damage (Denv), and temperature-independent cyclic fatigue damage (Dfat).
1=Ni ¼ Dfat þ Dcr þ Denv
m1
tcyc
Z
t cyc 0
"
2 # 1 e_ th =e_ mech 1 Ae_ s dt exp 2 ncr
ð7Þ
In (7) ncr denotes the creep phase parameter which takes values between 0 and 0.5 to account for the observation that Dcr dominates primarily for IP-TMF conditions. The additional parameter A is a linear factor which scales the amount of creep damage. In total the model as described here incorporates 12 temperature independent and one temperature-dependent parameter (m).
ð2Þ
Dfat is modeled by a four parameter life curve approach (3). The parameters C1, C2, C3, and C4 define an upper boundary life curve which describes the low-temperature fatigue behavior of the alloy. To account for R-ratio effects a Walker-like correction term [15] is added to the damage parameter Demech. The temperature-dependent parameter m takes values between 0 and 1.
Demech ð1 Rr Þm1 21m ¼ C 1 ð1=Dfat ÞC 2 þ C 3 ð1=Dfat ÞC4
1
ð3Þ
The term (1 Rr) in (3) is used to describe the mean stress dependency for Rr between 1 and values of about 0.6. For tests experiencing negative mean stresses no additional benefit in life compared to Rr = 1 tests is expected. In this case Rr is set to 1. Nucleation of critical surface cracks especially during OP-TMF loading has been identified as an additional damage contribution [5,14]. The assumed mechanism is growth of oxide layer and depleted zone near the surface during periods of high temperature exposure and subsequent cracking of brittle oxide layers during tensile loading. Neu and Sehitoglu [5] have proposed the relations (4–6) to model oxidation-assisted cracking and the observed dependencies on strain rate and mechanical strain range. According to the model Denv becomes unity if a critical crack length hcr is reached. Final failure will occur shortly afterwards, since it has been observed [5] that such cracks accelerate rapidly once the critical size is reached.
5. Empirical model Microcrack growth parameters like in [8] were successfully applied to model LCF life at ambient temperatures. This approach can be extended for high temperatures by accounting for viscous deformations around the crack tip. The resulting creep-induced acceleration of cyclic crack growth has been modeled in [7] by introducing the semi-empirical F-function. However, for cycle numbers exceeding 50,000 a threshold concept had to be incorporated in order to improve the model performance of microcrack growth approaches in the long-life regime. Instead a phenomenological life curve approach will be used here to describe cyclic fatigue life for a wide range of load levels. Similar to the idea of [3,4] a frequency modified life curve approach (8) is proposed for isothermal high temperature conditions. From a phenomenological standpoint microcrack growth can still be regarded as the relevant damage mechanism since life curves described by (8) can also be predicted based on the models from [7,8], at least for LCF conditions (Ni < 104). 0
Demech ð1 Rr Þm1 21m ð1 þ C 5 t cyc Þk ¼ C 1 ðNi ÞC 2 þ C 3 ðNi ÞC 4 k1
ð8Þ
In (8) the classic frequency modification term (fcyc) has been 0 replaced by the function (1 + C5tcyc)k , which is similar to the Ffunction from [7]. The advantage of this formulation is that for very
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F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
short cycle durations a time-independent upper boundary life estimate can be given by setting tcyc to zero. The parameter C5 has the unit of inverse time. Assuming that viscous deformation behavior at different temperatures can be modeled by an Arrhenius term as proposed in [7], a time integral is introduced in (9) defining an effective time analog to tcyc in (8). No difference is made between hold times and the time spent during cycling. The integral extends over the entire cycle. The creep activation energy can be used as a first estimate for Q. The parameters C1, . . ., C5 are expected to stay constant for all temperatures.
Demech ð1 Rr Þm1 21m ð1 þ C 5 ¼ C 1 ðNi ÞC2 þ C 3 ðNi ÞC4
Z
exp
0 Q dtÞk RT ð9Þ
For increasing temperatures the isothermal life curves based on (9) can model continuously increasing degradation. But several authors [16,17] have reported that life curves for IN718 at temperatures between 300 and 540 °C show a crossover behavior. For instance, isothermal tests at 500 °C were observed to have the largest fatigue strength [16], whereas tests at about 300 °C were most damaging [16,17]. In [17] this effect has been attributed to an assumed mechanism called ‘thermal activation’ that reduces the generation of substructure defects and thus decelerates fatigue. According to the authors stress concentrations between grains can be lowered by diffusion processes. These processes can occur either along the grain boundaries or by lattice diffusion. A modeling approach is presented in [17] where a stress-based life curve is corrected by an Arrhenius function. The apparent activation energy Q of the proposed mechanism appears to be stress-assisted and is therefore modeled as a function of the stress amplitude ramp (10). Here t0 denotes the intrinsic activation volume.
Q ¼ Q 0 t0 ramp 1
ramp 2rUTS
ð10Þ
In the following the formulation (10) will be applied to (9) to describe the crossover behavior on a phenomenological basis. The application to isothermal and TMF conditions is straight forward. Only during thermal cycling the temperature dependent quantities rUTS, t0 , and m have to be determined for a well defined temperature. Which temperature has been chosen, is explained below. Fractographic investigations of specimens tested under isothermal and different TMF phase conditions in [9] revealed no fundamental change in failure mechanisms. Transgranular crack growth was dominant. Therefore, cyclic crack growth by alternating slip mechanism [18] can be assumed for most loading conditions. In case of OP-TMF, cracks blunting by alternating slip will probably experience the largest amount of crack propagation during the rising branch of the hysteresis loop up to maximum tensile loading at Tmin. The same holds for maximum tensile loading at Tmax during IP-TMF. So the idea is to evaluate the parameters rUTS, t0 , and m are for T(rmax) and to treat TMF cracks like isothermal cracks at the given temperature. The model Eqs. (9) and (10) have to be interpreted in view of the stress–strain response predicted by the constitutive model. For isothermal tests with different cycle times rather small differences can be expected for the cyclic fatigue load parameters Demech and Rr. However, especially during OP-TMF loading, low test frequencies or compressive hold times will strongly affect the R-ratio and possibly Demech as well. Therefore, not only the frequency correction will cause OP-TMF tests to be increasingly harmful for lower test frequencies. A second effect is due to the development of significant mean stresses which are accounted for by the Walkerlike approach. On the other hand IP-TMF tests will cause the largest
values for the proposed frequency modification term. High temperatures and tensile stresses acting simultaneously will drive this term. The final model contains 7 temperature dependent and the temperature independent parameters t0 and m.
6. Parameter identification In view of the large number of parameters a hybrid strategy is chosen to identify meaningful parameters and to optimize the model performance. If parameters can be estimated based on physical measurements or experiences from other publications they will be fixed to the appropriate values. The remaining parameters will be regressed using a maximum likelihood function L as the optimization criterion. In order to ensure a fair comparison between the two models, the adjustable parameters are treated in a unified manner. All parameters are defined at five distinct temperatures (20 °C, 316 °C, 482 °C, 649 °C, and 732 °C). In between linear interpolation is applied. In case of temperature independent parameters a constant value is applied for all temperatures. A FORTRAN based tool has been set up allowing semi-automatic parameter optimization by using several optimization algorithms. The final loop was performed using a modified version [20] of the algorithm described in [19]. The overall likelihood L which is minimized during regression is defined as follows.
L¼
Y
pi ðNa =Np Þ
8 > > >
ð11Þ
2 N =N p 1 exp 12 rastatistic 8; specimens failed pi ¼ 2 R > N a =N p > x1 > exp 12 rstatistic dx 8; runout tests : 1 1 r 1 pffiffiffiffi 2p 1 pffiffiffiffi statistic 2p
statistic
ð12Þ
By treating the experimental specimen life Na for each test as a random sample and assuming that the predicted life Np is a perfect average model, the probability of occurrence pi can be calculated for each test. If a constant standard deviation rstatistic is defined (c.p. Fig. 3) the individual probabilities for each test can be computed (12) depending if the test failed or it was stopped prematurely. During the regression of the modified literature model only the two cyclic oxidation parameters b and b were fixed to the values given in Section 4. All other parameters were initially set to the values given for the cast nickel alloy Mar-M-247 in [14]. After the optimization had converged the values for m at various temperatures were smoothened and kept constant during the final optimization loop.
Fig. 3. Example tests including life curve approach to visualize the maximum likelihood approach.
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F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
99.9% 99.5% 99%
Cumulated probability [-]
Regression of the frequency modified model was done in three steps. During the first optimization loop C5 was set to zero and room temperature tests were used to determine initial values for C1, . . ., C4, and m. During the second loop the activation energy Q0 in the frequency modified model was set to the value of 240 kJ/mol given in [17]. The regression was then started using the average value for t0 from [17]. Again smoothing was applied to the temperature dependent parameters. For the final optimization m and t0 were fixed and Q0 was no longer kept constant. Values of Q0 close to 230 kJ/mol and t0 between 104 and 103 were finally obtained.
B50=1.001
95% 90% 80% 70% 60% 50% 40% 30% 20% 10% 5%
B0.1=0.1764
1% 0.5%
Experiments Log-Normal distribution
0.1%
7. Model evaluation
10-1
100
101
Actual life / Predicted life [-] Fig. 5. A/P probability diagram for the new frequency modified approach.
106
Scatter band of factor 2
Predicted life [-]
10
5
104
103
0° TMF 180° TMF 90° TMF -90° TMF LCF
102
101 10 1
10 2
10 3
10 4
Cumulated probability [-]
95% 90% 80% 70% 60% 50% 40% 30% 20% 10% 5%
B50=0.991
10 6
Fig. 6. Actual versus predicted life diagram for the modified literature model.
10 6
Scatter band of factor 2 10
5
10 4
10 3
0° TMF 180° TMF 90° TMF -90° TMF LCF
99.9% 99.5% 99%
10 5
Actual life [-]
Predicted life [-]
In the following we will assess the capability of the models to describe the experimental data discussed in Section 2. Fatigue is regarded as a statistical phenomenon since repetition tests will most probably lead to a different initiation life. A common assumption is that the same log-normal probability distribution applies for tests at various load levels and loading conditions. In this case the ratio actual life (Na) to predicted life (Np) for an ideal model, i.e. a hypothetical model which does not introduce any additional scatter other than the scatter in the test data alone, can be regarded as a statistical quantity. Standard deviation, median and average values can be determined. To check if the assumed log-normal distribution is valid, a diagram of the cumulated probability versus log(Na/Np) is done for all sorted ratios Na/Np (see Fig. 4). Here the y-scale is transformed according to the quantiles of the standard normal distribution. If the majority of points can be approximated by a straight line, the assumed log-normal distribution is valid. By looking at Figs. 4 and 5 one can see that a log-normal distribution gives a good description of the cumulated A/P probability distributions for both models. Two statistical values are of particular interest to quantify the model accuracy. The 0.1% quantile (B0.1) gives the minimum ratio Na/NP compared to which 99.9% of all tests show higher values Na/NP. Analogously B50 defines the ratio Na/NP for which 50% of the tests have lower and also 50% of the tests have higher values of Na/NP. The respective values for B0.1 and B50 are visualized in Figs. 4 and 5. After model regression B50 usually takes values close to one. The ratio B50/B0.1 should be as low as possible, where the lowest possible value is given by the scatter contained in the test data itself. Therefore, the values 5.68, for the newly proposed model, and 8.59, for the modified literature model, show that the frequency modified approach performs better. This can also be observed by comparing Figs. 6 and 7 where the frequency modified model predicts more tests within a scatter band of factor two.
10 2
10 1 10 1
102
10 3
10 4
105
106
Actual life [-]
B0.1=0.1153
Fig. 7. Actual versus predicted life diagram for the new frequency modified approach.
1% 0.5%
10 -1
8. Discussion
Experiments Log-Normal distribution
0.1% 10 0
Actual life / Predicted life [-] Fig. 4. A/P probability diagram for the modified literature model.
10 1
A final validation of the model parameters is somehow difficult since a truly unique global optimum does probably not exist. Therefore, after regression the validity of the model parameters identified by the optimization routine shall be investigated. In case
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F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
LCF (20°C) LCF (315°C)
Dfat=100%
LCF (427-593°C) LCF (649°C) LCF (732°C) IP-TMF (649°C) non IP-TMF (649°C) IP-TMF (732°C) non IP-TMF (732°C)
Dcr=100%
Denv=100%
Fig. 8. Triangle plot of the relative Sehitoglu damage contributions due to fatigue, creep, and oxidation analyzed with respect to temperatures.
LCF (t<3s) LCF (t<10s) LCF (t<30s) LCF (t<100s) LCF (t<300s) LCF (t<1000s) -90°-TMF 0°-TMF 90°-TMF 180°-TMF
Dcr=100%
Dfat=100%
oxidation occurs primarily during OP-TMF. In our study an intermediate value of 1.02 was determined by regression. Based on the previous considerations it is expected that the optimization described in Section 6 gives reasonable parameters for the modified literature model. Unfortunately the newly proposed modified frequency model cannot be rationalized based on plots similar to Figs. 8 and 9. On the other hand the new approach can be analyzed much easier due to the simple structure of the approach. Fatigue life is determined based on an empirical life curve approach and the damaging effects of positive mean stresses and time dependent creep deformations are modeled by multiplicative correction of the damage parameter Demech. Meaningful life predictions are expected since the dimensions of the parameters Q0 and t0 are within the range of values given in [17] and the Walker exponent takes values between 0 and 1. An investigation of the frequency modification term in (9) has shown that values between 1 and 4 are obtained. It also revealed that primarily those tests have largest contributions from frequency correction, which experienced significant creep damage according to the modified literature model. The proposed frequency modification term dominates for IP-TMF and slow LCF tests. This implies that OP-TMF tests, which are described fairly well (see Fig. 10), are modeled by a superposition of the frequency and the Walker-like mean stress correction. Both terms attribute to the increasingly damaging effect of OP-TMF for longer cycle times. The application of an appropriate constitutive model giving realistic predictions for Rr and ramp is therefore considered to be essential for modeling TMF life properly. 9. Conclusions
Denv=100%
Fig. 9. Triangle plot of the relative Sehitoglu damage contributions due to fatigue, creep, and oxidation analyzed with respect to test frequencies and TMF phase.
of the modified literature model Figs. 8 and 9 provide the relative percentages of the three damage mechanisms. Here linear scaling applies to the triangle plots. As already mentioned in Section 4, tests at room temperature are expected to experience almost 100% of their damage due to pure cyclic fatigue. This can be confirmed by looking at the upper edge of the triangle shown in Fig. 8. Tests having the largest relative amount of oxidation damage can be identified in both figures to be primarily OP-TMF load cases. In case of creep-dominated cycles two trends can be observed. Whether the tests were performed under IP-TMF conditions and high maximum temperatures or they were slowly cycled under isothermal conditions, both can lead to significant amounts of creep damage. The difference between oxidation and creep dominance is that the latter case requires high temperatures and high stresses acting simultaneously, whereas oxidation primarily relies to high temperature exposure and some small amount of cyclic mechanical loading. The trend that OP-TMF load cases experience the largest relative amounts of oxidation damage depends on the actual value of the phase factor nenv. In [5] this parameter has been set to 2, allowing isothermal tests to accumulate significant portions of Denv. In [14] a rather small value of nenv = 0.44 has been found implying that
A new model able to describe cyclic fatigue tests under isothermal and thermo-mechanical conditions has been proposed. The approach incorporates a Walker-like mean stress correction as well as a formulation capturing the dependencies on cycle time and hold times. The frequency modification approach has been transformed to a time integral formulation in which temperature dependencies are described by an Arrhenius term. The statistical evaluation of the new life prediction model in line with the modified literature model from [5,14] reveals that the frequency modified approach derived in Section 5 performs at least as good. Appendix A See Table A.1. Table A.1 Chemical composition of the specimens used for cyclic fatigue tests. Source Material/ specification Element
NASA-CR189221 Inconel718/ (AMS 5663) Weight%
NASA-TM106881 Inconel718/ (AMS5663D) Weight%
Socie et al. [11] Inconel718/ AMS 5663B Weight%
NASA-CR182247 Inconel718/-
S B P C Cu Si Mn Co Al Ti Mo Nb + Ta Cr Ni Fe
– – – – – – – – – – – – – – –
0.002 0.004 0.006 0.034 0.05 0.07 0.12 0.39 0.57 0.95 2.87 5.19 17.52 53.58 Balance
0.001 0.0035 0.007 0.034 0.005 0.06 0.09 0.28 0.54 1.01 2.89 5.03 18.23 52.12 19.473
0.002 0.002 0.006 0.06 0.03 0.17 0.17 0.1 0.45 1.12 3.03 5.11 17.92 53.63 18.32
Weight%
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F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
Appendix B See Table B.1.
Table B.1 Experiment database including cyclic fatigue tests from [9–12]. #
Test ID
Test control
Source
Tmin (°C)
Tmax (°C)
TMFphase (°)
Dwell type
Cycle time (s)
Dwell time (s)
R-ratio
Amplitude (%/MPa)
Discontinued (0)/ rupture (1)
Cycles to 50% load drop N50
1
N1–33
Strain
427
427
0
–
170.00
0
1.00
0.8500%
1
1075
2
N1–34
Strain
538
538
0
–
6.80
0
1.00
0.8500%
1
1100
3
N1–35
Strain
593
593
0
–
350.00
0
1.00
1.7500%
1
20
4
N1–38
Strain
649
649
0
–
4.60
0
1.00
0.5750%
1
1850
5
N1–39
Strain
427
427
0
–
6.80
0
1.00
0.8500%
1
1350
6
N1–40
Strain
593
593
0
–
4.60
0
1.00
0.5750%
1
2375
7
N1–41
Strain
649
649
0
–
170.00
0
1.00
0.8500%
1
200
8
N1–42
Strain
427
427
0
–
350.00
0
1.00
1.7500%
1
110
9
N1–43
Strain
538
538
0
–
4.60
0
1.00
0.5750%
1
3110
10
N1–44
Strain
593
593
0
–
170.00
0
1.00
0.8500%
1
325
11
N1–45
Strain
649
649
0
–
14.00
0
1.00
1.7500%
1
35
12
N1–46
Strain
538
538
0
–
350.00
0
1.00
1.7500%
1
60
13
N3–1
Strain
427
427
0
–
4.60
0
1.00
0.5750%
1
3710
14
N3–11
Strain
538
538
0
–
115.00
0
1.00
0.5750%
1
2125
15
N3–12
Strain
593
593
0
–
6.80
0
1.00
0.8500%
1
725
16
N3–13
Strain
649
649
0
–
350.00
0
1.00
1.7500%
1
33
17
N3–2
Strain
538
538
0
–
170.00
0
1.00
0.8500%
1
650
18
N3–3
Strain
593
593
0
–
14.00
0
1.00
1.7500%
1
80
19
N3–4
Strain
649
649
0
–
115.00
0
1.00
0.5750%
1
830
20
N3–5
Strain
427
427
0
–
115.00
0
1.00
0.5750%
0
1725
21
N3–7
Strain
593
593
0
–
115.00
0
1.00
0.5750%
1
1820
22
N3–8
Strain
649
649
0
–
6.80
0
1.00
0.8500%
1
515
23
307
Strain
649
649
0
–
2.00
0
1.00
0.3290%
1
8903
24
576
Strain
732
732
0
–
2.00
0
1.00
0.2705%
1
101702
25
583
Strain
732
732
0
–
2.00
0
0.99
0.3365%
1
6500
26
587
Strain
732
732
0
–
4.80
0
0.17
0.2715%
1
5586
27
592
Strain
732
732
0
–
4.80
0
0.20
0.2705%
1
16237
28
7181
Strain
732
732
0
–
60.00
0
1.01
0.3915%
1
2939
29
7182
Strain
732
732
0
–
60.00
0
1.00
0.3230%
1
6506
30
7183
Strain
649
649
0
–
60.00
0
0.99
0.3975%
1
2868
31
7184
Strain
649
649
0
–
60.00
0
0.99
0.3230%
1
13332
32
7185
Strain
649
649
0
peak
68.97
60
0.94
0.3177%
1
13668
33
7186
Strain
732
732
0
peak
69.04
60
0.94
0.3170%
1
2219
34
7187
Strain
NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR-
732
732
0
valley
75.00
60
1.00
0.3231%
1
3012 (continued on next page)
56
F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
Table B.1 (continued) #
Test ID
Test control
35
7188
Strain
36
7189
Strain
37
71810
Strain
38
71811
Strain
39
71812
Strain
40
71813
Strain
41
71814
Strain
42
71815
Strain
43
71816
Strain
44
71817
Strain
45
71818
Strain
46
71819
Strain
47
71820
Strain
48
71821
Strain
49
71822
Strain
50
71823
Strain
51
71826
Strain
52
71827
Strain
53
71828
Strain
54
71830
Strain
55
71810T
Strain
56
71811T
Strain
57
71812T
Strain
58
71813T
Strain
59
71814T
Strain
60
71815T
Strain
61
71816T
Strain
62
71817T
Strain
63
71819T
load
64
71820T
load
65
7182T
Strain
66
7183T
Strain
67
7184T
Strain
68
7185T
Strain
69
7187T
Strain
Source 189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221
Tmin (°C)
Tmax (°C)
TMFphase (°)
Dwell type
Cycle time (s)
Dwell time (s)
R-ratio
Amplitude (%/MPa)
Discontinued (0)/ rupture (1)
Cycles to 50% load drop N50
649
649
0
peak
909.09
900
0.95
0.3910%
1
689
649
649
0
valley
909.09
900
1.00
0.3982%
1
1751
732
732
0
–
59.94
0
617.27
0.3215%
1
7260
649
649
0
–
2.00
0
1.00
0.3279%
1
13762
649
649
0
–
2.00
0
1.00
0.4020%
1
4323
482
482
0
–
2.00
0
1.00
0.3270%
1
20234
482
482
0
–
2.00
0
1.00
0.5034%
1
4020
316
316
0
–
2.00
0
1.01
0.3278%
1
21999
316
316
0
–
2.00
0
1.00
0.4018%
1
11824
316
316
0
–
2.00
0
1.01
0.5068%
1
5401
649
649
0
–
2.00
0
0.99
0.5065%
1
1785
732
732
0
–
2.00
0
1.01
0.3269%
1
9159
732
732
0
–
2.00
0
1.00
0.4025%
1
3612
732
732
0
–
2.00
0
0.99
0.5055%
1
1115
649
649
0
–
60.00
0
1.00
0.5000%
1
1072
732
732
0
–
60.00
0
1.00
0.5005%
1
590
732
732
0
–
59.94
0
0.00
0.3215%
1
4521
649
649
0
–
2.00
0
0.00
0.3224%
1
10011
649
649
0
–
2.00
0
3390.69
0.3225%
1
31056
316
316
0
–
2.00
0
0.00
0.3229%
1
18023
316
649
0
–
60.00
0
1.00
0.4975%
1
740
316
732
180
–
60.00
0
1.00
0.2755%
1
9601
316
732
0
–
60.00
0
0.99
0.2720%
1
20586
316
732
180
–
60.00
0
1.01
0.4000%
1
2852
316
732
0
–
60.00
0
1.00
0.3982%
1
744
316
732
180
–
60.00
0
0.00
0.3255%
1
4084
316
732
180
–
60.00
0
224.19
0.3275%
1
4152
316
732
90
–
60.00
0
1.00
0.3225%
1
7821
316
649
0
–
60.00
0
0.67
513.8
1
3427
316
732
0
–
60.00
0
1.00
445.4
1
8484
316
649
180
–
60.00
0
1.00
0.4000%
1
4487
316
649
0
–
60.00
0
1.00
0.3225%
1
17001
316
649
0
–
60.00
0
1.00
0.3950%
1
4983
316
732
180
–
60.00
0
1.01
0.3255%
1
2779
316
649
-90
–
60.00
0
1.00
0.3225%
1
13297
57
F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57 Table B.1 (continued)
a b
#
Test ID
Test control
Source
Tmin (°C)
Tmax (°C)
TMFphase (°)
Dwell type
Cycle time (s)
Dwell time (s)
R-ratio
Amplitude (%/MPa)
Discontinued (0)/ rupture (1)
Cycles to 50% load drop N50
70
7189T
Strain
316
649
180
–
60.00
0
1.01
0.4990%
1
2434
71
IN23
Strain
20
20
0
–
1.00
0
1.00
0.3950%
1
290429b
72
IN24
Strain
20
20
0
–
1.00
0
1.00
0.4420%
1
76253b
73
IN26
Strain
20
20
0
–
5.00
0
1.00
0.5625%
1
12812b
74
IN28
Strain
20
20
0
–
5.00
0
1.00
0.5100%
1
27550b
75
IN29
Strain
20
20
0
–
10.00
0
1.00
0.6365%
1
9333b
76
IN31
Strain
20
20
0
–
10.00
0
1.00
1.1150%
1
975b
77
IN32
Strain
20
20
0
–
10.00
0
1.00
0.7015%
1
6264b
78
IN33
Strain
20
20
0
–
10.00
0
1.00
0.8085%
1
3357b
79
IN34
Strain
20
20
0
–
10.00
0
1.00
0.9160%
1
2484b
80
IN35
Strain
20
20
0
–
10.00
0
1.00
1.0660%
1
1353b
81 82 83 84 85 86 87 88
B-12 B-14 B-15 B-33 B-36 B-5 B-6 B-9
Strain Strain Strain Strain Strain Strain Strain Strain
NASA-CR189221 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 Socie et al. Socie et al. Socie et al. Socie et al. Socie et al. Socie et al. Socie et al. Socie et al.
20 20 20 20 20 20 20 20
20 20 20 20 20 20 20 20
0 0 0 0 0 0 0 0
– – – – – – – –
6.00a 6.00a 6.00a 6.00a 6.00a 6.00a 6.00a 6.00a
0 0 0 0 0 0 0 0
1.00 1.00 0.00 1.00 0.00 0.00 1.00 0.00
0.5000% 1.0000% 0.5000% 1.0000% 0.5000% 1.0000% 0.5000% 1.0000%
1 1 1 1 1 1 1 1
10703b 1058b 6388b 978b 5612b 743b 11342b 761b
Cycle time for room temperature test of Socie et al. [11] was set to 6s. N50 was guessed based on linear correlation between N50 and Nf derived from test data provided in [12].
References [1] Manson SS. Behaviour of materials under conditions of thermal stresses. In: Proc. Heat Transfer Symposium, University of Michigan Engineering Research Institute, 1953. p. 9–75. [2] Coffin Jr LF. A study of cyclic thermal stresses on a ductile metal. Trans ASME 1954;76:931–50. [3] Coffin L Jr. The concept of frequency separation in life prediction for timedependent fatigue. In: 1976 ASME-MPC symposium on creep-fatigue interaction, ASME, New York; 1976. p. 349–64. [4] Ostergren WJ. A damage function and associated failure equations for predicting hold time and frequency effects in elevated temperature – Low cycle fatigue. J Test Eval 1976;4:327–39. [5] Neu RW, Sehitoglu H. Thermomechanical fatigue, oxidation and creep part II – life prediction. Metall Trans A 1989;20A:1769–83. [6] Nissley DM. Thermomechanical fatigue life prediction in gas turbine superalloys a fracture mechanics approach. AIAA J 1995;33:1114–20. [7] Schmitt W, Mohrmann R, Riedel H, Dietsche A, Fischersworring Bunk A. Modelling of the fatigue life of automobile exhaust components fatigue. In: Blom AF, editor, Engineering materials advisory services Ltd. Cradley Heath, UK; 2002. p. 781–8. [8] Dowling NE. Crack growth during low-cycle fatigue of smooth axial specimens cyclic stress-strain and plastic deformation aspects of fatigue crack growth. ASTM STP 1977;637:97–121. [9] Nelson RS, Levan GW, Harvey PR. Creep fatigue life prediction for engine hot section materials (Isotropic) – Final Report, NASA Contractor Report 189221; 1992.
[10] Halford GR, McGaw MA. Prestraining and Its Influence on Subsequent Fatigue Life, NASA Technical Memorandum 106881 1995 55. [11] Socie DF, Waill LA, Dittmer DF. Biaxial fatigue of inconel 718 including mean stress effects. ASTM STP 1985;853:463–81. [12] Kim KS, Vanstone RH, Malik SN, Laflen JH. Elevated temperature crack growth – Final Report NASA Contract Report 182247; 1988. [13] Becker M, Hackenberg H-P. A constitutive model for rate dependent and rate independent inelasticity application to IN718. Int J Plasticity 2010;27:596–619. [14] Sehitoglu H, Boismier DA. Thermo-mechanical fatigue of Mar-M247 part 2 – life prediction. J. Eng. Mater. – Trans ASME 1990;112:80–9. [15] Walker K. The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminium. ASTM STP 1970;462:1–14. [16] Chen Q, Kawagoishi N, Nisitani H. Evaluation of fatigue crack growth rate and life prediction of Inconel 718 at room and elevated temperatures. Mater Sci Eng 2000;A277:250–7. [17] Warren J, Wei DY. Thermally activated fatigue mechanism and life prediction. Int J Fatigue 2008;30:1699–707. [18] Pelloux RMN. Mechanism of formation of ductile fatigue striations. Trans ASME 1969;62:281–5. [19] Boender CGE, Kan AHGR, Timmer GT, Stougie L. A stochastic method for global optimization. Math Program 1982;22:125–40. [20] Csendes T. Global, Jozsef Attila University, Szeged, Hungary,
Contents lists available at SciVerse ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
An approach to life prediction for a nickel-base superalloy under isothermal and thermo-mechanical loading conditions F. Vöse ⇑, M. Becker, A. Fischersworring-Bunk, H.-P. Hackenberg MTU Aero Engines GmbH, Dachauer Straße 665, 80995 München, Germany
a r t i c l e
i n f o
Article history: Available online 6 December 2011 Keywords: Life prediction Cyclic fatigue Thermo-mechanical fatigue Nickel-base superalloy
a b s t r a c t Scope of the present work is the development of a unified approach for the life prediction under isothermal and thermo-mechanical loading conditions. The basis for the life prediction is the stress–strain response of the material for one selected cycle obtained in the context of a Finite Element Analysis. Here the application of an appropriate inelastic constitutive model is essential. A phenomenological fatigue model is proposed which is able to describe mean-stress effects, as well as dependencies on hold time and cycle duration with reasonable accuracy. The model performance is compared with that of a mechanism based literature model. The model accuracy is evaluated on a statistical basis through an evaluation of the variance in the ratio between predicted life and actual life. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The design of jet engine components is generally based on specimen testing results. The challenge in the design phase is to predict component life as exact as possible to avoid expensive redesign. Typically isothermal low-cyclic fatigue (LCF) tests, as well as more realistic thermo-mechanical fatigue (TMF) tests are therefore available. In the present paper the criterion defining initiation life Ni is set to the occurrence of a 0.4 mm deep crack. For the alloy IN718 under investigation a close correlation of Ni and the number of cycles to failure Nf can be expected. One major difference between specimen tests and fatigue of real components is the typical duration of the performed load cycles. Under service conditions jet engines can experience about 1200–10,000 startstop cycles within 10,000 h of flight operation. In order to design parts for a certain target life including safety margin usually 20,000–40,000 cycles have to be demonstrated. The only possibility to perform tests at such low load levels is to increase the test frequency to 1–60 cycles per minute. Historically these tests have been performed primarily under isothermal conditions. To predict initiation life on this basis a Manson–Coffin approach [1,2] can be used. It correlates the number of cycles to initiation Ni to the inelastic strain range Dein.
Dein ¼ e0f ð2N i ÞC 2
ð1Þ
Since time-dependent damage mechanisms become more and more dominant at higher temperatures, frequency effects have great
⇑ Corresponding author. E-mail address: [email protected] (F. Vöse). 0142-1123/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2011.10.018
impact on life prediction of hot section parts. The so called frequency modified or frequency separation models proposed by Coffin and others [3,4] have been successfully used to model high-temperature isothermal tests with different strain rates and hold times. In addition real engine parts will be exposed to non-isothermal load histories. An important characteristic is the TMF phase angle which describes the phasing between mechanical strain and temperature during strain-controlled TMF tests (see Fig. 1). Depending on the actual TMF phase conditions load cycles can lead to different damage mechanisms. This has been accounted for by Neu and Sehitoglu [5] when they proposed their so called ‘damage rate model’. The idea is to differentiate between three different damage mechanisms, namely cyclic fatigue, in-phase (IP) TMF and out-ofphase (OP) TMF. While cyclic fatigue caused by small fatigue crack growth is assumed to take place under any cyclic loading condition, enhanced intergranular creep damage occurs mainly during IP-TMF, and oxidation induced brittle surface cracking occurs predominantly during OP-TMF. Neu and Sehitoglu [5] model these IP and OP phase effects by introducing empirical phase factors. Other authors like Nissley [6], Schmitt et al. [7] consider microcrack growth to be the dominant damage mechanism. Their models correlate the cyclic crack growth rate to a strain energy density parameter similar to that proposed for isothermal conditions by Dowling [8]. Within the present work an empirical approach will be developed, allowing a unified description of cyclic fatigue tests under isothermal and TMF conditions. The predictive capabilities of the newly developed model will be compared with the mechanismbased literature model from [5]. To show the applicability of both models for a larger set of fatigue tests, experimental data taken from several literature sources is considered.
50
F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
Nomenclature A B b
coefficient in creep damage equation coefficient in oxide growth equation parameter describing the strain ratesensitivity of environmental damage b exponent on time in oxidation law C Manson–Coffin exponent C1, C2 elastic life curve parameters C3, C4 inelastic life curve parameters C5 coefficient in frequency modified term Dfat, Dcr, Denv damage portions due to fatigue, creep, and environmental degradation D0 diffusion coefficient d0 ductility of brittle surface layer Dein inelastic strain range Demech mechanical strain range Dr stress range e0f fatigue ductility coefficient e_ mechanical strain rate e_ s secondary creep rate e_ th thermal strain rate fcyc test frequency (=1/tcyc) critical oxide growth when oxidation assisted cracks hcr start to grow faster Kp,eff effective parabolic oxidation constant k, k0 exponent in frequency modified term
L m Na Nf Ni NP⁄
t0 pi
Uenv Q Qenv Q0 R Rr
r ramp rstatistic rUTS T Tmax/min t tcyc tref nenv, ncr
maximum likelihood function walker exponent actual fatigue life from experiment number of cycles to failure number of cycles to initiation predicted fatigue life intrinsic activation volume probability of occurrence phase factor for environmental damage nominal activation energy for time-dependent damage contributions activation energy for oxidation intrinsic activation energy universal gas constant stress ratio stress stress amplitude standard deviation to determine L ultimate tensile strength temperature (in Kelvin) maximum/minimum temperature time cycle time reference time parameters of phasing functions
Fig. 1. Visualization of several TMF load cycles with different phase angles.
2. Experimental database
3. Constitutive model calculations
The precipitate strengthened nickel-base superalloy IN718 is one of the most common structural alloys used in aero engines. Therefore a large number of publications reporting its chemical and physical properties exist. The present paper uses cyclic fatigue test data reported in [9–12]. A comparison of the chemical compositions of the different specimens is given in Appendix A (see Table A.1). Heat treatments and processing are similar. This paper is limited to smooth uniaxial conditions only. All valid tests used for model regression are given in Appendix B. Most tests have been performed under isothermal conditions. In addition 16 TMF tests complete the selection of 88 cyclic fatigue tests. A large number of tests with different loading conditions, including different test frequencies, R-ratios, hold times, test temperatures, and TMF phase conditions (c.p. Table B.1), are considered. Since the tests also range about several orders of magnitude in life (102. . .106) it is expected that creep- and oxidation-dominated failure mechanisms will occur at least in some specimens.
Becker and Hackenberg [13] have proposed a constitutive model, which captures the relevant effects observed for IN718, such as cyclic plasticity, creep, and ratcheting behavior. The constitutive model parameters have been obtained after regression [13] using special constitutive tests involving a complex load history. Even though the model parameters have not been re-adjusted to the experiments considered in the present study, overall a sufficient correlation between measured stress amplitudes and the stress amplitudes predicted by the constitutive model has been found for Ni/2 (see Fig. 2). Since raw data on stress–strain behavior is not available for all tests within the present work, constitutive model calculations have been performed. Therefore the stress–strain–temperature history for each test has been simulated cycle by cycle by Finite Element Analysis. The results were stored only for the cycle at Ni/2. The full hysteresis loop data of this cycle is later on needed to determine integral quantities during the fatigue life calculation. Since all
51
F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
Experimental stress amplitude [MPa]
1200
1100 1000
K p:eff ¼
900
1 tcyc
800
Uenv ¼
700
Z
ð4Þ
D0 expðQ env =RTÞdt
ð5Þ
t cyc
0
Z
1 t cyc
1=b
2ðDemech Þ2=bþ1 ðe_ mech Þ1b=b
hcr d0 BUenv K p:eff
Denv ¼
"
t cyc
exp
0
2 # 1 e_ th =e_ mech þ 1 dt 2 nenv
ð6Þ
600 500 400 300
Model stress amplitude Fig. 2. Comparison of measured stress amplitude versus predicted stress amplitude incl. line of best fit.
specimen tests are smooth no geometry dependency is expected. A single 20 node brick element is used for the analysis of the deformation behavior during the test. The experimental database given in Appendix B includes LCF and TMF specimen tests from four different publications [9–12]. Clearly, both, the small differences in the chemical composition of the alloys (see Appendix A), as well as slight differences in the heat treatments, have an impact on stress–strain behavior and fatigue life. In this context the scatter observed in Fig. 2 when comparing measured and predicted stress amplitudes might be explained by the variations between the four different materials under investigation.
Here the cyclic oxidation parameters b = 0.75 and b = 1.5 will be applied. The same values have been successfully used to model environmental damage in Mar-M-247 [14] and 1070 steel [5]. The remaining parameters can be treated as four independent parameters hcrd0/B, D0, Qenv, nenv. During IP-TMF and high temperature tensile dwell intergranular cracking is expected to further decrease fatigue life. For IN718 this was confirmed by [9] where a mixture of transgranular and intergranular crack growth was observed for an IP-TMF test experiencing a high cyclic load level. The creep damage formulation (7) can be regarded as a mathematically identical formulation to what was proposed in [14], if e_ s takes the form e_ s ¼ expðQ cr =RTÞ ½ðjrj þ rÞ=ð3KÞm . For reasons of consistency it is reasonable to apply the same formulation within both, the constitutive and the fatigue model. In our case the constitutive model from [13] using a different formulation for e_ s is applied. Its general form can be described by e_ s ¼ f ðhri; TÞ. An advantage of the creep law from [13] is that it has been fit to a large number of proprietary creep tests for IN718 and is expected to be more accurate than the approach from [5,14].
Dcr ¼ 4. Modified literature model In the following the governing equations are presented for the chosen literature model proposed by Neu and Sehitoglu [5,14]. A linear damage rule (2) describes the summation of creep damage (Dcr), environmental damage (Denv), and temperature-independent cyclic fatigue damage (Dfat).
1=Ni ¼ Dfat þ Dcr þ Denv
m1
tcyc
Z
t cyc 0
"
2 # 1 e_ th =e_ mech 1 Ae_ s dt exp 2 ncr
ð7Þ
In (7) ncr denotes the creep phase parameter which takes values between 0 and 0.5 to account for the observation that Dcr dominates primarily for IP-TMF conditions. The additional parameter A is a linear factor which scales the amount of creep damage. In total the model as described here incorporates 12 temperature independent and one temperature-dependent parameter (m).
ð2Þ
Dfat is modeled by a four parameter life curve approach (3). The parameters C1, C2, C3, and C4 define an upper boundary life curve which describes the low-temperature fatigue behavior of the alloy. To account for R-ratio effects a Walker-like correction term [15] is added to the damage parameter Demech. The temperature-dependent parameter m takes values between 0 and 1.
Demech ð1 Rr Þm1 21m ¼ C 1 ð1=Dfat ÞC 2 þ C 3 ð1=Dfat ÞC4
1
ð3Þ
The term (1 Rr) in (3) is used to describe the mean stress dependency for Rr between 1 and values of about 0.6. For tests experiencing negative mean stresses no additional benefit in life compared to Rr = 1 tests is expected. In this case Rr is set to 1. Nucleation of critical surface cracks especially during OP-TMF loading has been identified as an additional damage contribution [5,14]. The assumed mechanism is growth of oxide layer and depleted zone near the surface during periods of high temperature exposure and subsequent cracking of brittle oxide layers during tensile loading. Neu and Sehitoglu [5] have proposed the relations (4–6) to model oxidation-assisted cracking and the observed dependencies on strain rate and mechanical strain range. According to the model Denv becomes unity if a critical crack length hcr is reached. Final failure will occur shortly afterwards, since it has been observed [5] that such cracks accelerate rapidly once the critical size is reached.
5. Empirical model Microcrack growth parameters like in [8] were successfully applied to model LCF life at ambient temperatures. This approach can be extended for high temperatures by accounting for viscous deformations around the crack tip. The resulting creep-induced acceleration of cyclic crack growth has been modeled in [7] by introducing the semi-empirical F-function. However, for cycle numbers exceeding 50,000 a threshold concept had to be incorporated in order to improve the model performance of microcrack growth approaches in the long-life regime. Instead a phenomenological life curve approach will be used here to describe cyclic fatigue life for a wide range of load levels. Similar to the idea of [3,4] a frequency modified life curve approach (8) is proposed for isothermal high temperature conditions. From a phenomenological standpoint microcrack growth can still be regarded as the relevant damage mechanism since life curves described by (8) can also be predicted based on the models from [7,8], at least for LCF conditions (Ni < 104). 0
Demech ð1 Rr Þm1 21m ð1 þ C 5 t cyc Þk ¼ C 1 ðNi ÞC 2 þ C 3 ðNi ÞC 4 k1
ð8Þ
In (8) the classic frequency modification term (fcyc) has been 0 replaced by the function (1 + C5tcyc)k , which is similar to the Ffunction from [7]. The advantage of this formulation is that for very
52
F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
short cycle durations a time-independent upper boundary life estimate can be given by setting tcyc to zero. The parameter C5 has the unit of inverse time. Assuming that viscous deformation behavior at different temperatures can be modeled by an Arrhenius term as proposed in [7], a time integral is introduced in (9) defining an effective time analog to tcyc in (8). No difference is made between hold times and the time spent during cycling. The integral extends over the entire cycle. The creep activation energy can be used as a first estimate for Q. The parameters C1, . . ., C5 are expected to stay constant for all temperatures.
Demech ð1 Rr Þm1 21m ð1 þ C 5 ¼ C 1 ðNi ÞC2 þ C 3 ðNi ÞC4
Z
exp
0 Q dtÞk RT ð9Þ
For increasing temperatures the isothermal life curves based on (9) can model continuously increasing degradation. But several authors [16,17] have reported that life curves for IN718 at temperatures between 300 and 540 °C show a crossover behavior. For instance, isothermal tests at 500 °C were observed to have the largest fatigue strength [16], whereas tests at about 300 °C were most damaging [16,17]. In [17] this effect has been attributed to an assumed mechanism called ‘thermal activation’ that reduces the generation of substructure defects and thus decelerates fatigue. According to the authors stress concentrations between grains can be lowered by diffusion processes. These processes can occur either along the grain boundaries or by lattice diffusion. A modeling approach is presented in [17] where a stress-based life curve is corrected by an Arrhenius function. The apparent activation energy Q of the proposed mechanism appears to be stress-assisted and is therefore modeled as a function of the stress amplitude ramp (10). Here t0 denotes the intrinsic activation volume.
Q ¼ Q 0 t0 ramp 1
ramp 2rUTS
ð10Þ
In the following the formulation (10) will be applied to (9) to describe the crossover behavior on a phenomenological basis. The application to isothermal and TMF conditions is straight forward. Only during thermal cycling the temperature dependent quantities rUTS, t0 , and m have to be determined for a well defined temperature. Which temperature has been chosen, is explained below. Fractographic investigations of specimens tested under isothermal and different TMF phase conditions in [9] revealed no fundamental change in failure mechanisms. Transgranular crack growth was dominant. Therefore, cyclic crack growth by alternating slip mechanism [18] can be assumed for most loading conditions. In case of OP-TMF, cracks blunting by alternating slip will probably experience the largest amount of crack propagation during the rising branch of the hysteresis loop up to maximum tensile loading at Tmin. The same holds for maximum tensile loading at Tmax during IP-TMF. So the idea is to evaluate the parameters rUTS, t0 , and m are for T(rmax) and to treat TMF cracks like isothermal cracks at the given temperature. The model Eqs. (9) and (10) have to be interpreted in view of the stress–strain response predicted by the constitutive model. For isothermal tests with different cycle times rather small differences can be expected for the cyclic fatigue load parameters Demech and Rr. However, especially during OP-TMF loading, low test frequencies or compressive hold times will strongly affect the R-ratio and possibly Demech as well. Therefore, not only the frequency correction will cause OP-TMF tests to be increasingly harmful for lower test frequencies. A second effect is due to the development of significant mean stresses which are accounted for by the Walkerlike approach. On the other hand IP-TMF tests will cause the largest
values for the proposed frequency modification term. High temperatures and tensile stresses acting simultaneously will drive this term. The final model contains 7 temperature dependent and the temperature independent parameters t0 and m.
6. Parameter identification In view of the large number of parameters a hybrid strategy is chosen to identify meaningful parameters and to optimize the model performance. If parameters can be estimated based on physical measurements or experiences from other publications they will be fixed to the appropriate values. The remaining parameters will be regressed using a maximum likelihood function L as the optimization criterion. In order to ensure a fair comparison between the two models, the adjustable parameters are treated in a unified manner. All parameters are defined at five distinct temperatures (20 °C, 316 °C, 482 °C, 649 °C, and 732 °C). In between linear interpolation is applied. In case of temperature independent parameters a constant value is applied for all temperatures. A FORTRAN based tool has been set up allowing semi-automatic parameter optimization by using several optimization algorithms. The final loop was performed using a modified version [20] of the algorithm described in [19]. The overall likelihood L which is minimized during regression is defined as follows.
L¼
Y
pi ðNa =Np Þ
8 > > >
ð11Þ
2 N =N p 1 exp 12 rastatistic 8; specimens failed pi ¼ 2 R > N a =N p > x1 > exp 12 rstatistic dx 8; runout tests : 1 1 r 1 pffiffiffiffi 2p 1 pffiffiffiffi statistic 2p
statistic
ð12Þ
By treating the experimental specimen life Na for each test as a random sample and assuming that the predicted life Np is a perfect average model, the probability of occurrence pi can be calculated for each test. If a constant standard deviation rstatistic is defined (c.p. Fig. 3) the individual probabilities for each test can be computed (12) depending if the test failed or it was stopped prematurely. During the regression of the modified literature model only the two cyclic oxidation parameters b and b were fixed to the values given in Section 4. All other parameters were initially set to the values given for the cast nickel alloy Mar-M-247 in [14]. After the optimization had converged the values for m at various temperatures were smoothened and kept constant during the final optimization loop.
Fig. 3. Example tests including life curve approach to visualize the maximum likelihood approach.
53
F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
99.9% 99.5% 99%
Cumulated probability [-]
Regression of the frequency modified model was done in three steps. During the first optimization loop C5 was set to zero and room temperature tests were used to determine initial values for C1, . . ., C4, and m. During the second loop the activation energy Q0 in the frequency modified model was set to the value of 240 kJ/mol given in [17]. The regression was then started using the average value for t0 from [17]. Again smoothing was applied to the temperature dependent parameters. For the final optimization m and t0 were fixed and Q0 was no longer kept constant. Values of Q0 close to 230 kJ/mol and t0 between 104 and 103 were finally obtained.
B50=1.001
95% 90% 80% 70% 60% 50% 40% 30% 20% 10% 5%
B0.1=0.1764
1% 0.5%
Experiments Log-Normal distribution
0.1%
7. Model evaluation
10-1
100
101
Actual life / Predicted life [-] Fig. 5. A/P probability diagram for the new frequency modified approach.
106
Scatter band of factor 2
Predicted life [-]
10
5
104
103
0° TMF 180° TMF 90° TMF -90° TMF LCF
102
101 10 1
10 2
10 3
10 4
Cumulated probability [-]
95% 90% 80% 70% 60% 50% 40% 30% 20% 10% 5%
B50=0.991
10 6
Fig. 6. Actual versus predicted life diagram for the modified literature model.
10 6
Scatter band of factor 2 10
5
10 4
10 3
0° TMF 180° TMF 90° TMF -90° TMF LCF
99.9% 99.5% 99%
10 5
Actual life [-]
Predicted life [-]
In the following we will assess the capability of the models to describe the experimental data discussed in Section 2. Fatigue is regarded as a statistical phenomenon since repetition tests will most probably lead to a different initiation life. A common assumption is that the same log-normal probability distribution applies for tests at various load levels and loading conditions. In this case the ratio actual life (Na) to predicted life (Np) for an ideal model, i.e. a hypothetical model which does not introduce any additional scatter other than the scatter in the test data alone, can be regarded as a statistical quantity. Standard deviation, median and average values can be determined. To check if the assumed log-normal distribution is valid, a diagram of the cumulated probability versus log(Na/Np) is done for all sorted ratios Na/Np (see Fig. 4). Here the y-scale is transformed according to the quantiles of the standard normal distribution. If the majority of points can be approximated by a straight line, the assumed log-normal distribution is valid. By looking at Figs. 4 and 5 one can see that a log-normal distribution gives a good description of the cumulated A/P probability distributions for both models. Two statistical values are of particular interest to quantify the model accuracy. The 0.1% quantile (B0.1) gives the minimum ratio Na/NP compared to which 99.9% of all tests show higher values Na/NP. Analogously B50 defines the ratio Na/NP for which 50% of the tests have lower and also 50% of the tests have higher values of Na/NP. The respective values for B0.1 and B50 are visualized in Figs. 4 and 5. After model regression B50 usually takes values close to one. The ratio B50/B0.1 should be as low as possible, where the lowest possible value is given by the scatter contained in the test data itself. Therefore, the values 5.68, for the newly proposed model, and 8.59, for the modified literature model, show that the frequency modified approach performs better. This can also be observed by comparing Figs. 6 and 7 where the frequency modified model predicts more tests within a scatter band of factor two.
10 2
10 1 10 1
102
10 3
10 4
105
106
Actual life [-]
B0.1=0.1153
Fig. 7. Actual versus predicted life diagram for the new frequency modified approach.
1% 0.5%
10 -1
8. Discussion
Experiments Log-Normal distribution
0.1% 10 0
Actual life / Predicted life [-] Fig. 4. A/P probability diagram for the modified literature model.
10 1
A final validation of the model parameters is somehow difficult since a truly unique global optimum does probably not exist. Therefore, after regression the validity of the model parameters identified by the optimization routine shall be investigated. In case
54
F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
LCF (20°C) LCF (315°C)
Dfat=100%
LCF (427-593°C) LCF (649°C) LCF (732°C) IP-TMF (649°C) non IP-TMF (649°C) IP-TMF (732°C) non IP-TMF (732°C)
Dcr=100%
Denv=100%
Fig. 8. Triangle plot of the relative Sehitoglu damage contributions due to fatigue, creep, and oxidation analyzed with respect to temperatures.
LCF (t<3s) LCF (t<10s) LCF (t<30s) LCF (t<100s) LCF (t<300s) LCF (t<1000s) -90°-TMF 0°-TMF 90°-TMF 180°-TMF
Dcr=100%
Dfat=100%
oxidation occurs primarily during OP-TMF. In our study an intermediate value of 1.02 was determined by regression. Based on the previous considerations it is expected that the optimization described in Section 6 gives reasonable parameters for the modified literature model. Unfortunately the newly proposed modified frequency model cannot be rationalized based on plots similar to Figs. 8 and 9. On the other hand the new approach can be analyzed much easier due to the simple structure of the approach. Fatigue life is determined based on an empirical life curve approach and the damaging effects of positive mean stresses and time dependent creep deformations are modeled by multiplicative correction of the damage parameter Demech. Meaningful life predictions are expected since the dimensions of the parameters Q0 and t0 are within the range of values given in [17] and the Walker exponent takes values between 0 and 1. An investigation of the frequency modification term in (9) has shown that values between 1 and 4 are obtained. It also revealed that primarily those tests have largest contributions from frequency correction, which experienced significant creep damage according to the modified literature model. The proposed frequency modification term dominates for IP-TMF and slow LCF tests. This implies that OP-TMF tests, which are described fairly well (see Fig. 10), are modeled by a superposition of the frequency and the Walker-like mean stress correction. Both terms attribute to the increasingly damaging effect of OP-TMF for longer cycle times. The application of an appropriate constitutive model giving realistic predictions for Rr and ramp is therefore considered to be essential for modeling TMF life properly. 9. Conclusions
Denv=100%
Fig. 9. Triangle plot of the relative Sehitoglu damage contributions due to fatigue, creep, and oxidation analyzed with respect to test frequencies and TMF phase.
of the modified literature model Figs. 8 and 9 provide the relative percentages of the three damage mechanisms. Here linear scaling applies to the triangle plots. As already mentioned in Section 4, tests at room temperature are expected to experience almost 100% of their damage due to pure cyclic fatigue. This can be confirmed by looking at the upper edge of the triangle shown in Fig. 8. Tests having the largest relative amount of oxidation damage can be identified in both figures to be primarily OP-TMF load cases. In case of creep-dominated cycles two trends can be observed. Whether the tests were performed under IP-TMF conditions and high maximum temperatures or they were slowly cycled under isothermal conditions, both can lead to significant amounts of creep damage. The difference between oxidation and creep dominance is that the latter case requires high temperatures and high stresses acting simultaneously, whereas oxidation primarily relies to high temperature exposure and some small amount of cyclic mechanical loading. The trend that OP-TMF load cases experience the largest relative amounts of oxidation damage depends on the actual value of the phase factor nenv. In [5] this parameter has been set to 2, allowing isothermal tests to accumulate significant portions of Denv. In [14] a rather small value of nenv = 0.44 has been found implying that
A new model able to describe cyclic fatigue tests under isothermal and thermo-mechanical conditions has been proposed. The approach incorporates a Walker-like mean stress correction as well as a formulation capturing the dependencies on cycle time and hold times. The frequency modification approach has been transformed to a time integral formulation in which temperature dependencies are described by an Arrhenius term. The statistical evaluation of the new life prediction model in line with the modified literature model from [5,14] reveals that the frequency modified approach derived in Section 5 performs at least as good. Appendix A See Table A.1. Table A.1 Chemical composition of the specimens used for cyclic fatigue tests. Source Material/ specification Element
NASA-CR189221 Inconel718/ (AMS 5663) Weight%
NASA-TM106881 Inconel718/ (AMS5663D) Weight%
Socie et al. [11] Inconel718/ AMS 5663B Weight%
NASA-CR182247 Inconel718/-
S B P C Cu Si Mn Co Al Ti Mo Nb + Ta Cr Ni Fe
– – – – – – – – – – – – – – –
0.002 0.004 0.006 0.034 0.05 0.07 0.12 0.39 0.57 0.95 2.87 5.19 17.52 53.58 Balance
0.001 0.0035 0.007 0.034 0.005 0.06 0.09 0.28 0.54 1.01 2.89 5.03 18.23 52.12 19.473
0.002 0.002 0.006 0.06 0.03 0.17 0.17 0.1 0.45 1.12 3.03 5.11 17.92 53.63 18.32
Weight%
55
F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
Appendix B See Table B.1.
Table B.1 Experiment database including cyclic fatigue tests from [9–12]. #
Test ID
Test control
Source
Tmin (°C)
Tmax (°C)
TMFphase (°)
Dwell type
Cycle time (s)
Dwell time (s)
R-ratio
Amplitude (%/MPa)
Discontinued (0)/ rupture (1)
Cycles to 50% load drop N50
1
N1–33
Strain
427
427
0
–
170.00
0
1.00
0.8500%
1
1075
2
N1–34
Strain
538
538
0
–
6.80
0
1.00
0.8500%
1
1100
3
N1–35
Strain
593
593
0
–
350.00
0
1.00
1.7500%
1
20
4
N1–38
Strain
649
649
0
–
4.60
0
1.00
0.5750%
1
1850
5
N1–39
Strain
427
427
0
–
6.80
0
1.00
0.8500%
1
1350
6
N1–40
Strain
593
593
0
–
4.60
0
1.00
0.5750%
1
2375
7
N1–41
Strain
649
649
0
–
170.00
0
1.00
0.8500%
1
200
8
N1–42
Strain
427
427
0
–
350.00
0
1.00
1.7500%
1
110
9
N1–43
Strain
538
538
0
–
4.60
0
1.00
0.5750%
1
3110
10
N1–44
Strain
593
593
0
–
170.00
0
1.00
0.8500%
1
325
11
N1–45
Strain
649
649
0
–
14.00
0
1.00
1.7500%
1
35
12
N1–46
Strain
538
538
0
–
350.00
0
1.00
1.7500%
1
60
13
N3–1
Strain
427
427
0
–
4.60
0
1.00
0.5750%
1
3710
14
N3–11
Strain
538
538
0
–
115.00
0
1.00
0.5750%
1
2125
15
N3–12
Strain
593
593
0
–
6.80
0
1.00
0.8500%
1
725
16
N3–13
Strain
649
649
0
–
350.00
0
1.00
1.7500%
1
33
17
N3–2
Strain
538
538
0
–
170.00
0
1.00
0.8500%
1
650
18
N3–3
Strain
593
593
0
–
14.00
0
1.00
1.7500%
1
80
19
N3–4
Strain
649
649
0
–
115.00
0
1.00
0.5750%
1
830
20
N3–5
Strain
427
427
0
–
115.00
0
1.00
0.5750%
0
1725
21
N3–7
Strain
593
593
0
–
115.00
0
1.00
0.5750%
1
1820
22
N3–8
Strain
649
649
0
–
6.80
0
1.00
0.8500%
1
515
23
307
Strain
649
649
0
–
2.00
0
1.00
0.3290%
1
8903
24
576
Strain
732
732
0
–
2.00
0
1.00
0.2705%
1
101702
25
583
Strain
732
732
0
–
2.00
0
0.99
0.3365%
1
6500
26
587
Strain
732
732
0
–
4.80
0
0.17
0.2715%
1
5586
27
592
Strain
732
732
0
–
4.80
0
0.20
0.2705%
1
16237
28
7181
Strain
732
732
0
–
60.00
0
1.01
0.3915%
1
2939
29
7182
Strain
732
732
0
–
60.00
0
1.00
0.3230%
1
6506
30
7183
Strain
649
649
0
–
60.00
0
0.99
0.3975%
1
2868
31
7184
Strain
649
649
0
–
60.00
0
0.99
0.3230%
1
13332
32
7185
Strain
649
649
0
peak
68.97
60
0.94
0.3177%
1
13668
33
7186
Strain
732
732
0
peak
69.04
60
0.94
0.3170%
1
2219
34
7187
Strain
NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR182247 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR-
732
732
0
valley
75.00
60
1.00
0.3231%
1
3012 (continued on next page)
56
F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57
Table B.1 (continued) #
Test ID
Test control
35
7188
Strain
36
7189
Strain
37
71810
Strain
38
71811
Strain
39
71812
Strain
40
71813
Strain
41
71814
Strain
42
71815
Strain
43
71816
Strain
44
71817
Strain
45
71818
Strain
46
71819
Strain
47
71820
Strain
48
71821
Strain
49
71822
Strain
50
71823
Strain
51
71826
Strain
52
71827
Strain
53
71828
Strain
54
71830
Strain
55
71810T
Strain
56
71811T
Strain
57
71812T
Strain
58
71813T
Strain
59
71814T
Strain
60
71815T
Strain
61
71816T
Strain
62
71817T
Strain
63
71819T
load
64
71820T
load
65
7182T
Strain
66
7183T
Strain
67
7184T
Strain
68
7185T
Strain
69
7187T
Strain
Source 189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221 NASA-CR189221
Tmin (°C)
Tmax (°C)
TMFphase (°)
Dwell type
Cycle time (s)
Dwell time (s)
R-ratio
Amplitude (%/MPa)
Discontinued (0)/ rupture (1)
Cycles to 50% load drop N50
649
649
0
peak
909.09
900
0.95
0.3910%
1
689
649
649
0
valley
909.09
900
1.00
0.3982%
1
1751
732
732
0
–
59.94
0
617.27
0.3215%
1
7260
649
649
0
–
2.00
0
1.00
0.3279%
1
13762
649
649
0
–
2.00
0
1.00
0.4020%
1
4323
482
482
0
–
2.00
0
1.00
0.3270%
1
20234
482
482
0
–
2.00
0
1.00
0.5034%
1
4020
316
316
0
–
2.00
0
1.01
0.3278%
1
21999
316
316
0
–
2.00
0
1.00
0.4018%
1
11824
316
316
0
–
2.00
0
1.01
0.5068%
1
5401
649
649
0
–
2.00
0
0.99
0.5065%
1
1785
732
732
0
–
2.00
0
1.01
0.3269%
1
9159
732
732
0
–
2.00
0
1.00
0.4025%
1
3612
732
732
0
–
2.00
0
0.99
0.5055%
1
1115
649
649
0
–
60.00
0
1.00
0.5000%
1
1072
732
732
0
–
60.00
0
1.00
0.5005%
1
590
732
732
0
–
59.94
0
0.00
0.3215%
1
4521
649
649
0
–
2.00
0
0.00
0.3224%
1
10011
649
649
0
–
2.00
0
3390.69
0.3225%
1
31056
316
316
0
–
2.00
0
0.00
0.3229%
1
18023
316
649
0
–
60.00
0
1.00
0.4975%
1
740
316
732
180
–
60.00
0
1.00
0.2755%
1
9601
316
732
0
–
60.00
0
0.99
0.2720%
1
20586
316
732
180
–
60.00
0
1.01
0.4000%
1
2852
316
732
0
–
60.00
0
1.00
0.3982%
1
744
316
732
180
–
60.00
0
0.00
0.3255%
1
4084
316
732
180
–
60.00
0
224.19
0.3275%
1
4152
316
732
90
–
60.00
0
1.00
0.3225%
1
7821
316
649
0
–
60.00
0
0.67
513.8
1
3427
316
732
0
–
60.00
0
1.00
445.4
1
8484
316
649
180
–
60.00
0
1.00
0.4000%
1
4487
316
649
0
–
60.00
0
1.00
0.3225%
1
17001
316
649
0
–
60.00
0
1.00
0.3950%
1
4983
316
732
180
–
60.00
0
1.01
0.3255%
1
2779
316
649
-90
–
60.00
0
1.00
0.3225%
1
13297
57
F. Vöse et al. / International Journal of Fatigue 53 (2013) 49–57 Table B.1 (continued)
a b
#
Test ID
Test control
Source
Tmin (°C)
Tmax (°C)
TMFphase (°)
Dwell type
Cycle time (s)
Dwell time (s)
R-ratio
Amplitude (%/MPa)
Discontinued (0)/ rupture (1)
Cycles to 50% load drop N50
70
7189T
Strain
316
649
180
–
60.00
0
1.01
0.4990%
1
2434
71
IN23
Strain
20
20
0
–
1.00
0
1.00
0.3950%
1
290429b
72
IN24
Strain
20
20
0
–
1.00
0
1.00
0.4420%
1
76253b
73
IN26
Strain
20
20
0
–
5.00
0
1.00
0.5625%
1
12812b
74
IN28
Strain
20
20
0
–
5.00
0
1.00
0.5100%
1
27550b
75
IN29
Strain
20
20
0
–
10.00
0
1.00
0.6365%
1
9333b
76
IN31
Strain
20
20
0
–
10.00
0
1.00
1.1150%
1
975b
77
IN32
Strain
20
20
0
–
10.00
0
1.00
0.7015%
1
6264b
78
IN33
Strain
20
20
0
–
10.00
0
1.00
0.8085%
1
3357b
79
IN34
Strain
20
20
0
–
10.00
0
1.00
0.9160%
1
2484b
80
IN35
Strain
20
20
0
–
10.00
0
1.00
1.0660%
1
1353b
81 82 83 84 85 86 87 88
B-12 B-14 B-15 B-33 B-36 B-5 B-6 B-9
Strain Strain Strain Strain Strain Strain Strain Strain
NASA-CR189221 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 NASA-TM106881 Socie et al. Socie et al. Socie et al. Socie et al. Socie et al. Socie et al. Socie et al. Socie et al.
20 20 20 20 20 20 20 20
20 20 20 20 20 20 20 20
0 0 0 0 0 0 0 0
– – – – – – – –
6.00a 6.00a 6.00a 6.00a 6.00a 6.00a 6.00a 6.00a
0 0 0 0 0 0 0 0
1.00 1.00 0.00 1.00 0.00 0.00 1.00 0.00
0.5000% 1.0000% 0.5000% 1.0000% 0.5000% 1.0000% 0.5000% 1.0000%
1 1 1 1 1 1 1 1
10703b 1058b 6388b 978b 5612b 743b 11342b 761b
Cycle time for room temperature test of Socie et al. [11] was set to 6s. N50 was guessed based on linear correlation between N50 and Nf derived from test data provided in [12].
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[10] Halford GR, McGaw MA. Prestraining and Its Influence on Subsequent Fatigue Life, NASA Technical Memorandum 106881 1995 55. [11] Socie DF, Waill LA, Dittmer DF. Biaxial fatigue of inconel 718 including mean stress effects. ASTM STP 1985;853:463–81. [12] Kim KS, Vanstone RH, Malik SN, Laflen JH. Elevated temperature crack growth – Final Report NASA Contract Report 182247; 1988. [13] Becker M, Hackenberg H-P. A constitutive model for rate dependent and rate independent inelasticity application to IN718. Int J Plasticity 2010;27:596–619. [14] Sehitoglu H, Boismier DA. Thermo-mechanical fatigue of Mar-M247 part 2 – life prediction. J. Eng. Mater. – Trans ASME 1990;112:80–9. [15] Walker K. The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminium. ASTM STP 1970;462:1–14. [16] Chen Q, Kawagoishi N, Nisitani H. Evaluation of fatigue crack growth rate and life prediction of Inconel 718 at room and elevated temperatures. Mater Sci Eng 2000;A277:250–7. [17] Warren J, Wei DY. Thermally activated fatigue mechanism and life prediction. Int J Fatigue 2008;30:1699–707. [18] Pelloux RMN. Mechanism of formation of ductile fatigue striations. Trans ASME 1969;62:281–5. [19] Boender CGE, Kan AHGR, Timmer GT, Stougie L. A stochastic method for global optimization. Math Program 1982;22:125–40. [20] Csendes T. Global, Jozsef Attila University, Szeged, Hungary,