Siyuan CHEN, Dasheng WEI,*, Jialiang WANG, Yanrong WANG,Xianghua JIANG

a School of Energy and Power Engineering, Beihang University, Beijing 100083, China

b Beijing Key Laboratory of Aero-Engine Structure and Strength, Beijing 100083, China

c Jiangxi Research Institute, Beihang University, Nanchang 330096, China

d Science and Technology on Scramjet Laboratory, Beijing Power Machinery Institute, Beijing 100074, China

KEYWORDS Creep-fatigue interaction(CFI);Creep-fatigue test;Life prediction;Low-cycle fatigue (LCF);Superalloy

Abstract The performance of high-temperature components of aero-engines under the Creep-Fatigue Interaction (CFI) behavior gets more attention recently. In this research, the creepfatigue tests of two superalloys of Powder Metallurgy(PM)FGH96 and direct aging GH4169 were performed at 650°C with different types of dwell, and the fracture morphology of FGH96 specimens was observed by Scanning Electron Microscopy (SEM) to analyze the creep-fatigue fracture feature and crack initiation. Additionally, according to phenomenology, the effect of dwell was introduced to develop a new uniaxial fatigue life prediction model based on the total strain equation, which has capability to take dwell time and load ratio into account together. The equations were utilized to model the test data of PM FGH96 and GH4169, together with data of another superalloy PM FGH95 conducted previously. A prominent prediction ability of the model in creep-fatigue life prediction of different superalloys has been manifested. Most data points of test data and estimated data are located within two times scatter band, which is ideal in engineering.?2020 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

With the development of modern industrial technology, the performance of aircraft engine is improving significantly, of which a typical improvement is the increase of gas temperature in turbine inlet. However, accidents related to fracture of turbine disk usually occur, as the work condition of turbines is pretty serious. Turbine disk is an important component of engines, and its work condition with high temperature, high pressure and high rotating speed brings significant challenge to design and manufactory. Under the working cycles of takingoff,cruise and landing,the turbine disks work under the interaction of thermal load and centrifugal load, thus there will be typical Creep-Fatigue Interaction(CFI)cyclic load,which may be the main reason of turbine failure.Thus,in order to choose proper materials for turbine disks and evaluate their serving life and reliability,it is rather necessary or even crucial to take research on the properties of these common used superalloys under creep-fatigue interaction load wave.

When studying a common CFI loading pattern that is based on the triangular load wave, a period of constant peak loads is added to the cycles.The period is usually called dwell,and dwell time refers to the time length of dwell in a single cycle (see Fig. 1). Tensile dwell refers to dwell that is set in the peak of load wave, and compressive dwell means that it is set in the trough. For most materials, when the test was strain-controlled, creep deformation and stress relaxation could be observed during the dwell if the load is high enough,which was be mainly studied in this paper; when it was stresscontrolled, the phenomenon of stress relaxation would be replace by ratcheting behavior.

It has been demonstrated by large amounts of tests that a decline in the fatigue cyclic life could be led by decreasing the load frequency or introducing peak/trough dwell, which is considered to be related to creep. More than 100 CFI life prediction models has been proposed so far,and most of them can be classified into three categories: the generalization of Manson-Coffin equation, the generalization of linear damage summation and the generalization of strain-range partitioning,meanwhile scholars have proposed some methods to evaluate the performance of these models.1At present, some models are been widely using, such as the Manson-Coffin model,2,3Frequency Separation method,3Ostergren model,4strain range partitioning model,5linear damage summation method,6-13generalized hysteresis energy model14and total strain equation.15

where Δεinis inelastic strain range,Nfis fatigue cyclic life, β is cyclic life exponent, β and C are material parameters. The equation is usually used in the continuous fatigue life prediction, but it is also utilized to predict the fatigue-creep life by many engineers. The Manson-Coffin law is simple in form and convenient to use but does not consider more factors(such as strain ratio, CFI and temperature) in fatigue. Thus, the capability is limited.

Fig 1 Fatigue-creep loading pattern (tensile dwell).

To revise the effect of loading frequency,Coffin introduced the tensile and compressive frequency to Eq.(5)and developed the Frequency-Separation model.3The expression is

where σTis maximum tensile stress, v is frequency, and others are material parameters.

Later, Halford et al. proposed the Strain Range Partition(SRP) model.5In the SRP model, inelastic strain is divided into three separate parts: completely reserved fatigue plastic strain,completely reserved creep strain and fatigue-creep interactive strain. The damage accumulation method will then be used to accumulate the respective damage of the three strain partitions and calculate the total damage and cyclic life. This method has been accepted by many scholars and is widely used in the field of industry.

Taira6proposed the well-known creep-fatigue linear damage summation equation in 1962 based on Miner’s law,7which was also described in A.S.M.E. Code Case N47. The primary goal of this method is to calculate the fatigue damage and creep damage, superimpose both of them on the overall damage and predict the service term. The basic formula is

The first term of Eq.(4)is the Miner linear damage summation law, where n is the cyclic number under a specific load,and Dcrefers to the creep damage in every cycle. A basic hypothesis of this model is that creep damage does not interact with fatigue damage.Fatigue damage is obtained by a reciprocal continuous fatigue cyclic life, and in the description of creep damage, more types of theory have been developed.There are three common utilized types of model:

(1) Defining creep damage in every cycle as the ratio of dwell time to the creep rupture time.8

(2) Defining creep damage in every cycle as the ratio of the creep strain to the rupture strain.9,10

(3) Strain Energy Density Exhaustion (SEDE) model.11

An obvious defect of the time-ratio definition is the neglect of creep hardening/softening.Creep is regarded as a linear process, but it is actually a hardening/softening procedure. A downward trend in the changing rate of many creep indicators(creep strain rate and damage rate)could be seen in the beginning of creep (named as creep hardening), of which the length depends on the load and temperature.This trend may lead to a nonlinear relationship between the fatigue-creep cyclic life and dwell time.However,sometimes the accuracy of models based on time-ratio definition could also meet the engineering requirement.

Many scholars and engineers have been using linear damage summation method to predict the fatigue-creep cyclic life and trying to introduce more factors. Many new models for CFI life prediction based on linear damage summation method has been developed by scholars in recent years. For example,Pons et al.developed a new model12based on the combination of the Manson-Haford creep-rupture parameter13and Manson-Coffin equation. The model predicts all the ranges of CFI life from pure fatigue to pure creep under different temperatures. The formula is

Prof. Wang et al.22proposed a useful method to determine the parameters in the total strain equation in 2013,which built a relationship between the equation parameters and strength limit and section reduction of monotonic stretching based on the physical meaning of fatigue strength and the ductility coefficient.The method is strongly convenient for the usage of the total strain equation in engineering.

Overall,previous CFI life prediction models have been able to give useful results based on test data,but their validation is usually limited to some specific load conditions and materials.In recent years,new developed CFI life prediction models tried to include more factors in engineering. However, with the increasing of calculation volume,the accuracy of these models is still limited by the dispersivity of tests data and highly depend on the volume of data.

In this paper,based on the total strain equation with Walker’s strain ratio correction21and parameter determination method,22a new efficient, accurate and applicable uniaxial CFI life prediction model was developed for turbine disk superalloys by introducing the effect of dwell and creep. The purpose is to reduce calculation volume and use the data in material manuals to the maximum extent while maintaining its calculation accuracy in turbine disk superalloys. Meanwhile,a set of CFI tests were conducted to validate this model and illustrate the CFI behavior of these materials (FGH96,FGH95 and GH4169). The following work has been done in this paper: 1) perform fatigue-creep tests of PM FGH96 and GH4169 superalloys using smooth bar specimens; 2) develop a uniaxial CFI life prediction model; 3) validate the model by using test results conducted in this research and Ref. 23.

2. Materials and experiments

2.1. Materials

The compositions of PM FGH96, PM FGH95 and GH4169 are listed in Table 1.24Tests of GH4169 (strain ratio R equals to 0.1)and FGH96 were conducted in this research,while tests of GH4169(R=-1)and FGH95 have been published in Ref.23 previously.

With high-strength and low-ductility,PM FGH95 is manufactured by hot isostatic pressing. FGH95 consists of γ and γ′phases and carbides, and the volume percentage of the strengthening phase γ′is approximately 50%.24FGH96 is a second-generation damage-tolerant powder metallurgy superalloy, and both FGH95 and FGH96 is commonly used in high-temperature components of engines such as turbine disks because of their prominent performance under 600°C-700°C.Comparing with the previous generation of the powder superalloy material FGH95, FGH96′s content of metallic elements such as titanium, chromium, cobalt, tungsten and molybdenum was increased,while that of niobium,aluminum and carbon was decreased; the content of γ′phases was reduced, and the gain size was adjusted.25The overall strength of theFGH96 alloy was reduced by approximately 10% compared with FGH95,but the increase in cobalt and tungsten improved its mechanism behavior at high temperature, leading to a better creep resistance.24

Table 1 Chemical compositions (in %) of experiment materials (Ni - the rest) (Ref. 24).

GH4169 is an age-hardenable superalloy exhibiting good performance in the temperature range from -253°C to 700°C. Comparing with powder metallurgy superalloy FGH95 and FGH96, GH4169 has a relatively lower strength and higher ductility. In general, the microstructure of GH4169 consists of γ matrix, face-centered cubic γ′phase and body-centered cubic γ′′phase.The γ′and γ′′phases precipitate after the aging treatment to enhance the matrix, and the γ′′phase is the primary strengthening phase.23

Fig 2 Drawing of the specimens used in FGH96 and GH4169 tests.

The preparation of FGH95 includes powder pretreatment,HIP,and heat treatment.After performing the HIP procedure for 3-5 h at 1120°C-1170°C, the following heat treatment was performed: 1120°C-1160°C for 2-4 h+(870±10)°C for 1 h+ air cooling + (650±5) °C for 2 h+ air cooling.

The preparation of FGH96 is similar to that of FGH95.The heat treatment was performed: 1140°C-1150°C for 2 h+760°C for 8 h+air cooling.

The procedure of heat treatment used for GH4169 was direction aging,which was performed as follows:(720±10)°C for 8 h+furnace cooling+(620±10)°C for 8 h+air cooling.

2.2. Test procedure

The specimens used for both materials in the tests were equalsection smooth bars of the same size, with a gauge length and diameter of 12 mm and 6 mm, respectively. The specimens were prepared as the instrument of standard ASTM-E2714-13.26The specimen drawing is given as Fig. 2.

For FGH96 tests, it was strain-controlled, the strain ratio included -1 and 0.1, and the employed temperature was 650°C. Four types of dwell were used: no dwell (continuous fatigue), 60 s tensile dwell, 120 s tensile dwell and 60 s compressive dwell. The test data are shown in Fig. 3(a).

Fig 3 CFI life respect with total strain range of PM FGH96 and GH4169 under 650°C.

For GH4169, the tests were also strain-controlled, the strain ratio for all specimens was 0.1, and the employed temperature was also 650°C. Three types of dwell were used: no dwell, 60 s tensile dwell and 120 s tensile dwell. The test data are shown in Fig. 3(b). As the authors had implemented the GH4169 creep-fatigue tests (including tensile, compressive and symmetric dwell) under the strain ratio of -1 which had been published in Ref. 23, the tests of GH4169 with R=-1 and compressive dwell were not implemented in the research of this paper.

In Ref.23,we had published creep-fatigue tests data of two engine-using superalloy GH4169 and PM FGH95. The tests were strain-controlled, with strain ratio of -1 and four different types of dwell employed, including no dwell, 60 s tensile dwell, 60 s compressive dwell and 30 s dwell in both tensile and compressive peak). In order to make a comparing of the CFI properties of these three superalloys (FGH96, GH4169 and FGH95), we give the tests data of Ref. 23 in the form of Δεt-Nffigures.

The waveform of no dwell was triangular, and the others were trapezoidal (shown in Fig. 2).

2.3. Test results

In this section,some results of recent CFI tests of FGH96 and GH4169 (R=0.1) are listed, including the data of specimens’test life(Fig.3),typical hysteresis loops(Fig.4)as well as cyclic hardening/softening curves of FGH96(Fig.5).Some of our previous test data of GH4169(R=0.1)and FGH95 in Ref.23 were also put in this section (Fig. 6) to make comparison and analysis. The fracture observation photos of FGH96 is shown in Figs. 7 and 8.

Fig. 3 shows the relationship between fatigue cyclic life of specimens and total strain range under different strain ratios and dwell types of Fig. 3(a) FGH96 and Fig. 3(b) GH4169.As shown in Fig.3(a),the fatigue performance of FGH96 is sensitive to both tensile and compressive dwell, and all types of dwell seem to decrease the fatigue cyclic life. Additionally, the data revealed a slightly more significant effect of compressive dwell on fatigue life than that of tensile dwell. Fig. 3(b) gives the cyclic life data of GH4169 under R=0.1.It is shown that the cyclic life of GH4169 specimens with 60 s tensile dwell was close to or even longer than those with no dwell under a same load level, while the 120 s tensile dwell reduced the cyclic life greatly,except for one data point that seems to be abnormal.

Fig 4 Schematic diagram of different types of strain in static hysteresis loops.

Fig 5 FGH96 cyclic hardening/softening curve.

Fig 6 Life data of creep-fatigue tests of FGH95 and GH4169 in Ref. 23.

Fig 7 SEM observation of fractography of specimens under different loads and dwell types.

Comparing with continuous fatigue cycles, those cycles with dwell have larger inelastic strain amplitude caused by creep when the load conditions are the same. This increase of inelastic strain amplitude in every cycle is suspected to be the mean reason of the decrease of specimens’ life (see Fig.4).In the total strain Eq.(7),the first and second items in the right refer to the elastic strain amplitude and inelastic strain amplitude, respectively, which are shown in Fig. 4(a)(subscript a refers to amplitude). A typical strain-controlled static cyclic curve with 60 s tensile dwell is shown in Fig. 4(b). During the dwell process, small stress relaxation occurs in the specimen, producing creep strain. Thus, compared with the hysteresis loop of continuous fatigue under the same condition (load and temperature), the loop with 60 s tensile dwell has a larger inelastic amplitude and value of the increase equal to half of the creep strain in dwell,as shown in Fig.4(b).Considering the total strain amplitude in test is constant,the elastic strain amplitude was reduced by the same amount.

Fig. 5 shows the cyclic hardening/softening curve of FGH96 under the waveform of Fig. 5(a) continuous fatigue(no dwell)and Fig.5(b)120 s tensile dwell,respectively.Transitory cyclic hardening occurs in the initial creep-fatigue tests,followed by continuous cyclic softening.Overall,the lower the test loads were, the more obvious and longer the hardening stage. Additionally, for specimens with 120 s tensile dwell (as shown in Fig. 5(b)), the cyclic hardening stage was not obvious, high-load tests (1.2% total strain range and above) proceeded without cyclic hardening, and the softening was much fiercer. By contrast, the cyclic hardening period of specimens with no dwell (see Fig. 5(a)) typically lasted for tens to hundreds of cycles, while the softening stage was comparatively gentle.

Fig 8 Fracture SEM images of two magnifications showing the a specimen under R=0.1, Δεt =1.6%, 60 s tensile dwell.

The observation of fracture morphology of FGH96 specimens was conducted. The fractured part was cleaned in acetone solution to remove the oxide layer and impurities on the fracture surface, and the cleaned sample was placed under a Scanning Electron Microscope(SEM)to observe the fracture morphology. Fig. 7 shows the fracture morphology of some specimens tested under different loads and dwell types of Fig. 7(a) R=0.1, Δεt=1.0%, no dwell, Fig. 7(b) R=0.1,Δεt=1.0%, 60 s tensile dwell, (c) R= -1, Δεt=1.0%, 60 s tensile dwell and (d) R= -1, Δεt=1.6%, 60 s compressive dwell. All fractographies under different strain ratios, total strain ranges and dwell types showed typical fatigue fracture morphology, where the crack source and crack propagation zone with fatigue striations can be clearly observed. Some of the fatigue source zones have been marked on the pictures,surrounded by the crack propagation zones with dense beachwear striations.Considering that the grain of PM FGH96 was facecentered cubic structured and had good slip properties, a very noticeable macroscopic flat rough region could be seen on images.

SEM images revealed that fatigue cracks are more likely to initiate from surface or inclusions. Low- and highmagnification images of the fracture of a typical specimen(R=0.1, Δεt=1.6%, 60 s tensile dwell) are shown in Fig. 8 in which the fatigue cracks initiate from the inclusions located at the subsurface.Apparent fatigue striations appeared around the inclusion (see Fig. 8(a)). Some research studies27,28have been performed to reveal the effect of defects (inclusions or microcavities) on the fatigue behavior of PM materials, holding the view that the microdefects near the surface accelerate the initiation of fatigue cracks and cause the decrease in fatigue life. Similarly, the fatigue characteristic of FGH96 is sensitive to defects. The enlarged SEM images shown in Fig. 8(b) indicate that the inclusions are intended to become the initiation zone of cracks, leading to further propagation with fatigue striations.

2.4. Discussion about the test results

Comparing and analyzing the test data (including our recent work of FGH96 and GH4169 alloys, and published test data in Ref. 23), it can been found that the fatigue performance of the three superalloys were roughly at the same level, and the cyclic life of FGH96 and GH4169 specimens were slightly longer than that of FGH95 specimens under the same total strain ranges(for R=-1 and no dwell).Meanwhile,as shown by most of those data points, keeping the total strain range,strain ratio and temperature unchanged, the introduction of dwell was going to shorten the cyclic life of specimens. However, the differences in dwell length and types (tensile or compressive) had bring different degree of effect.

Firstly,comparing the no dwell,60 s and 120 s tensile dwell data for the same material and strain ratio,it can be found that the longer the dwell in every cycle,the sharper was the decrease of cyclic life of the specimens.This can be easily explained that a longer dwell will have larger creep strain and introduce more damage in every cycle, as a result, the cyclic life will drop.However, the data in Fig. 3(b) shows some exception. For GH4169 under R=0.1,as shown in the figure,the data points of 60 s tensile dwell and no dwell were mixed with each other,some specimens with 60 s tensile dwell even had longer cyclic life than those with no dwell. Analyzing the GH4169 data of R=0.1(Fig.3(b))and R=-1(Fig.6(b))together,we could find that GH4169 only shown slight sensitivity to short tensile dwell (this phenomenon has also been observed in the tests of other metals like P91 steel11,29), while other types of dwell(such as compressive dwell, symmetric dwell and 120 s longer tensile dwell)affected the cyclic life of GH4169 much more significantly, which may relate to creep property of the alloy.Additionally,the creep-fatigue test data itself have obvious dispersity.Thus,we judge that the phenomenon in Fig.3(b)does not indicate that the 60 s tensile dwell will prolong GH4169′s cyclic life under R=0.1. The reason for this phenomenon is that GH4169 is not sensitive to such a loading pattern, meanwhile the test data is dispersive.

There are also some differences between the sensitivity to tensile and compressive dwell for the three alloys. Comparing the test data with 60 s tensile and compressive dwell (R=-1,same total strain ranges),it can be found that the cyclic life of specimens with 60 s compressive dwell were markedly shorter than those with 60 s tensile dwell for all three alloys(as shown in Fig. 3 (a) and Fig. 6(a) and 6(b)), and FGH95 shown most significant difference in the sensitivity to tensile and compressive dwell. This phenomenon seems difficult to be explain by fracture morphology, as the fracture features of specimens with tensile and compressive dwell dose not shown any obvious differences. The reason for the difference in sensitivity might be due to the fact that compressive loading could cause shearing and slipping of free surfaces more easily, and bring lager damage into the specimens than tensile dwell of same periods. Additionally, it can be found that the sensitivity of FGH95 and GH4169 to 30 s symmetric dwell is also obvious,which is more significant than that to 60 s tensile dwell. This also indicate that the compressive dwell in every cycle could bring more damage than tensile dwell, with other test conditions unchanged.

Different strain ratios also affected the cyclic life of FGH96 and GH4169 specimens. Comparing the data in Fig. 3(a),Fig. 3(b) and Fig. 6(b), it can be found that the cyclic life of specimens under R=-1 usually slightly longer than those under R=0.1, with the total strain ranges unchanged. This result is reasonable, since lager strain ratio will bring higher stress peak value, which will accurate the process of crack propagation and decrease the cyclic life of specimens. Meanwhile, there are also difference between the sensitivity of GH4169 and FGH96 to the strain ratio. GH4169 was shown to be more sensitive to the change of strain ratio (comparing data in Fig. 3(b) and Fig. 6(b)), while FGH96 was almost not sensitive to strain ratio (see Fig. 3(a)). This difference may be due to the reason that strength of FGH96 is better than that of GH4169 under 650°C (both yield limit and strength limit of FGH96 are higher than those of GH4169,as was given in Ref. 24). As a result, for FGH96 alloy, slightly increasing the stress peak value in every cycle may not be able to bring obvious effect to its life. Furthermore, as shown in the test data, materials’ sensitivity to the strain ratio seem do not change due to the introduction of dwell.

3. Theory model

In this section, the ability of the traditional life prediction models (Manson-Coffin equation, Frequency-Separation model and Ostergren model)on FGH96 fatigue life prediction will be shown first. Then, a new uniaxial CFI life prediction model will be derived from Walker total strain equation.

3.1. Cyclic life prediction of the traditional model

Manson-Coffin Eq.(1),Frequency-Separation method(2)and Ostergren model (3) were utilized to predict the fatigue life of FGH96, as shown in Fig. 9. Manson-coffin equation did not take dwell or frequency into account so this model cannot reflect the effect of them. Thus, the prediction is not ideal.Comparatively, Frequency-Separation method and Ostergren model are able to consider the dwell factor. Thus, the prediction result of these two models is better than that of Manson-Coffin equation. However, overall, as shown in Fig. 9,the prediction result of those traditional methods mentioned above is still limited to some extent, as many data fall outside the two times scatter band. Additionally, the models above did not consider the stress ratio(strain ratio).Thus,they lack ability to address data points under R=0.1 in FGH96 and GH4169 tests.As a result,it is necessary to build an accurate and useful fatigue life prediction method considering both the CFI and stress ratio (strain ratio) for engine-used superalloys.

3.2. Continuous fatigue life prediction model

Fig 9 Cyclic life prediction of FGH96 modelled by three traditional models.

The strength limit and section shrinkage can be found in the material manual.P is the rupture load,and A0and Afare section areas of initiation and rupture,respectively.Adopting the transformation above and fitting the parameters b and c by creep-fatigue test data, we are able to obtain the total strain equation of specific material much more easily. Additionally,in this way, the information in material manuals can be fully used, and the physical meaning of parameters can be better guaranteed.

3.3. Fatigue-creep life prediction model

The previous research given above has revealed the relationship between the total strain amplitude εaand fatigue cyclic life Nfin double log-linear form. In this section, we try to introduce the effect of dwell from the perspective of strain change.Defining Nwcand Nwas the cyclic life with and without dwell under same test conditions(subscript w means the cyclic life has been reserved by the strain ratio mentioned in Eqs.(8)and (9)). As discussed above, the strain amplitudes of fatigue with tensile dwell can be expressed as follows

Now, there are only two uncertain parameters, bnewand cnew,in the prediction equation,and they are function of dwell time.When the dwell time equals 0,bnewand cnewwill degenerate to b and c. Additionally, as the dwell time increases, creep strain introduced per cycle will also increases (other variables are keeping unchanged), and the ratios of bnew/b and cnew/c will be further away from 1.

According to the FGH96 test data of the no dwell and 60 s and 120 s tensile dwell waveforms, we optimized the corresponding bnewand cnewparameters using Eqs. (19) and (20)and tried to find the relationships between bnew/b and cnew/c and the dwell time. The optimization method is given as follows. First, the material parameters b, c, εfand σfshould be obtained from continuous fatigue tests and material manuals.Next,several groups of CFI cyclic life data under different Δεtwere used to calculate Nwcand Nw, and the value of εcin the dwell of the static cycle should be extracted from the test data.Finally, Eqs. (18) and (19) will be used to optimize bnewand cnew, corresponding to the dwell time by the logarithmic method.

The relative parameters of FGH96 and GH4169 optimized by the method above are listed in Table 2. As shown in the data, the relationships between dwell time and the ratios of bnew/b and cnew/c of each material can be described in the form of a power function. The expression can be written as

Table 2 bnew and cnew parameters of FGH96 and GH4169 fitted by Eqs. (19) and (20).

Fig 10 Effect of bnew and cnew revision on the total strain equation of FGH96 under 650°C and two strain ratios.

where tcreepis dwell time in every single dwell,and the material parameters wb, wcand wtcan be fitted by the relationship of bnew, cnewand tcreepin Eqs. (23) and(24). From the perspective of mathematics,bnewand cnewin Eqs.(19)and(20)are actually the slopes of the double log-linear relationship between εa，eand εa，inand cyclic life, respectively. If we draw these two double log-linear relationships in ε-Nwcfigures, as shown in Fig. 10, it will be more intuitive to describe the revising of bnewand cnewparameters.

4. Model validation

In summary,the procedures of modeling and life predicting of specific materials using the above CFI life prediction model are given as follows:

(1) Obtain material parameters E, σband ψ from material manuals, calculate σfand εfusing Eqs. (12) and (13).

(2) Obtain material parameters b,c and γ in the Walker Eqs.(8) and (9) by fitting the continuous fatigue test data.

(3) Obtain material parameters wb, wcand wtin Eqs. (19),(20), (23) and (24) by fitting the fatigue-creep test data.

(4) After obtaining the parameters above,in the case where the strain amplitude εa, strain ratio R and dwell time tcreepare known,users can predict the uniaxial CFI cyclic life by Eqs. (21) and (24).

The CFI cyclic life of FGH96, GH4169 and FGH95 has been predicted based on the CFI prediction model.A comparison of the prediction results and test data is shown in Fig.11,and the parameters are given in Table 3.The model can ideally predict the fatigue cycle life of these materials with the effect of CFI. For FGH96 and FGH95, all the data points fall within the two times scatter band except for some abnormal event.For GH4169, because of the reason that the raw materials of the two group of tests (R=0.1 and R=-1) were not the same batch, leading to some slight differences in the fatigue and creep properties, the modeling result is not as ideal as FGH96 and FGH95. Some GH4169 data points fall outside the two times scatter band but inside the three times scatter band.

Three FGH96 data points fall outside the two times scatter band obviously, as shown in Fig. 11(b). This is related to the fact that load applied in these tests is relatively low,making life of these specimens obviously higher than that of the others.For the FGH96 alloy, this condition should be taken into the category of HCF, so total strain equation is difficult to describe with these three points together with others which belong to LCF.One GH4169 data point falls outside the three times scatter band(120 s tensile dwell),as shown in Fig.11(d),which should be regarded as an abnormal event.

Additionally,from the test data,we found that the sensitivity to compressive dwell of all three alloys was slightly more serious than that to tensile dwell. In this paper, the difference between tensile and compressive dwell was not considered in the model derivation because of the lack of test data. Thus,it is difficult to predict the cyclic life of these two types of dwell by the same group of parameters. As a result, only the condition with tensile dwell was applied in this paper to make the life prediction. Generally, the dangerous zone in turbine disks of aircraft engines work under the load waveform with tensile dwell,thus,the model is still applicable even if the compressive dwell was not considered.Materials’difference in sensitivity to compressive and tensile dwell will be studied in the future,and be considered in the model based on more sufficient test data.

The new CFI life prediction model have some advantages:

(1) The model is developed from the Walker total strain equation with strain ratio correction, and is suitable to solve the fatigue problems with different loads, stress(strain) ratios and dwell times. The application can be very wide.

Fig 11 Comparison of test data and predicted cyclic life of FGH96, GH4169 and FGH95.

(2) The data in material manuals can be fully utilized in the model establishment and parameter optimization, and the prediction accuracy is ideal.

(3) No additional creep test data, theoretical models and damage calculations are required, and parameter optimization is convenient, making the application much easier.

The model possesses some limitations:

(1) The predicted results have tolerance, mainly because of the dispersion of test data.

(2) The difference of sensitivity to tensile and compressive dwell was not considered in the life prediction model;compressive and symmetrical dwell have not yet beenincluded.However,generally the dangerous zone in turbine disks of aircraft engines work under the load waveform with tensile dwell,thus,the model is still applicable even if the this difference was not considered.

Table 3 Material parameters of the new CFI life prediction model (Eqs. (12), (13), (21)-(24)).

5. Conclusion

This research developed a uniaxial fatigue life prediction model by introducing the CFI effect into total strain equation.Creep-fatigue tests of PM FGH96 and GH4169 superalloys were performed under 650 with different types of dwell. Combined with the data in previous work Ref. 23, some phenomena can be found from the test results. The research conclusions can be drawn as follows:

(1) As shown in test results, the cyclic life of FGH96,GH4169 and FGH95 under 650°C and same total strain range were found to be in the same level approximately,meanwhile the fatigue and CFI performance of FGH96 and GH4169 are better than that of FGH95 slightly.

(2) The existing of dwell was found to have non-ignorable effect on the fatigue cyclic life of all three superalloys.It has been found that the longer the dwell time, the shorter is the fatigue cyclic life under the same test conditions (total strain range and temperature). For all three alloys, the sensitivity to compressive dwell was found to be more serious than that to tensile dwell.

(3) The new developed model has been applied to life prediction of three Ni-based superalloys PM FGH95, PM FGH96 and GH4169. As shown in the prediction results,most data points of test data and estimated data are located within two times scatter band,which is ideal in engineering.It shows that the model has a strong prediction ability in creep-fatigue life prediction of different superalloys.

(4) This model has capability to takes load level,dwell time and load ratio into account together,thus it can be very applicable. However, because of the lack of further test data and scientific research,the engineering factors such as the dwell type (tensile and compressive), stress gradient and thermomechanical fatigue have not been considered in the model yet.These factors will be studied in the future, and be considered in the model based on more sufficient test data.

Acknowledgement

The authors gratefully acknowledge the financial supports from National Natural Science Foundation of China (NSFC,No. 51475024)