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An interaction crack growth model for creep-brittle superalloys with high temperature dwell time
Engineering Fracture Mechanics xxx (2014) xxx–xxx
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An interaction crack growth model for creep-brittle superalloys with high temperature dwell time Hongqin Yang a,⇑, Rui Bao b, Jiazhen Zhang a a b
Beijing Aeronautical Science & Technology Research Institute, Beijing 102211, China Institute of Solid Mechanics, Beihang University, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 29 August 2013 Received in revised form 8 February 2014 Accepted 6 April 2014 Available online xxxx Keywords: Interaction effect Creep–fatigue Crack growth Nickel-based superalloys
a b s t r a c t A three-term interaction crack growth model developed for creep-brittle materials is restudied. Derivation of cycle–time interaction intensity factor from the Gaussian equation is discussed. The model’s prediction ability was verified by test data of a nickel base superalloy under 750 C with 25 s dwell time. Data of two other superalloys namely Alloy 718 and HastelloyÒ X from literature are analyzed to further examine applicability of the model. Satisfactory results are obtained. Parameter fitting procedure and limitation of the model are discussed at the end. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Nickel base powder superalloys are widely used for aerospace engine applications due to high strength under elevated temperatures. New generation of such superalloys are designed to gain improved damage tolerance performance as well as high strength [1]. Studies on crack growth of these superalloys thus receive more attentions, from crack growth behavior to underlying mechanisms and models. A number of studies [2,3] on the low cycle fatigue crack growth of nickel base superalloys have revealed that such materials are so called ‘‘creep-brittle’’ due to their excellent creep resistance. Environmental factors like oxidation contribute most to the time-dependent crack growth during load hold time. Two theories called stress accelerated grain boundary oxidation (SAGBO) [4] and gas phase embrittlement (GPE) [5] were widely used to explain the effect of oxidation. However the mechanisms of crack growth along intergranular boundaries are still not fully understood [6]. To model the crack growth of creep-brittle superalloys, it was found the stress intensity factor (SIF) K max was preferred instead of creep parameters C or C t [7,8]. The simplest equation to model the crack growth is one-term Paris Law [9]. Effects of temperature, frequency and oxidation are all evaluated by model coefficients [3]. Models based on the assumption of competing mechanisms [10,11] were also found good enough for several materials. A more universal model was developed from the idea that the total crack growth rates were the linear superposition of cycle-dependent and time-dependent rates [12,13]. Wei et al. [14] proposed probabilistic creep–fatigue–oxidation crack growth model for creep-brittle materials based on a linear superposition theory. A model using the concept of a damage zone was recently presented by Gustafsson et al. [15] to consider the complex mechanisms of the crack growth during load hold time.
⇑ Corresponding author. Tel.: +86 1057808860. E-mail addresses: [email protected] (H. Yang), [email protected] (R. Bao), [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.engfracmech.2014.04.006 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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Nomenclature a; a0 crack length and initial value A; m parameters in time-dependent crack growth model B thickness of specimens C; n parameters in cycle-dependent crack growth model da=dN; da=dt crack growth rates D; q parameters in interaction crack growth model E material elastic modulus f loading frequency K; DK stress intensity factor (SIF), SIF range N number of loading cycles p1 ; p2 ; p3 parameters in interaction intensity factor equation P; DP load, cyclic load range R nominal stress ratio t time th load dwell time in a cycle tb loading and unloading time in a cycle V force line displacement of specimens W width of specimens g interaction intensity factor
Most of above models does not separate the interaction effects between fatigue, creep and oxidation crack growth from themselves. For the superalloys which were found significantly affected by interaction effects [16,17], accuracy of these models can be noticeably reduced. A study has revealed that interaction induced crack growth usually occurs in the transition stage between pure fatigue and pure creep loading conditions [18]. A three term model was proposed [19] to cover the whole crack growth stage from pure cycle-dependent to extremely long hold times. The work of Grover and Saxena [11] indicates intensity of the creep–fatigue interaction is related to the relative size of creep zone to cyclic plastic zone. But it is only suitable for creep-ductile materials. Another three term model idea was discussed by Prakash et al. [17] where the crack growth rates consisted of cycle-dependent, pure creep and environmentally enhanced rates. Due to complex mechanism of oxidation or lack of supporting data, these three term models have not gained widely application. Realizing the interaction between cycle and time induced crack growth mainly relies on SIF and load frequency, the authors presented a novel factor [7] to represent the interaction intensity. The factor has a form derived from the Gaussian equation, which reaches its maximum value at an intermediate frequency and vanishes to zero at low or high frequencies for a given SIF value. A three term model consisting of cycle-dependent, time-dependent and their interaction was developed based on the factor for creep-brittle materials. The capacity of the model was experimentally verified on a nickel base superalloy FGH97. The study presented here is a continuation of that work. Derivation of the model is reviewed based on creep-brittle crack growth mechanism. More experimental data from FGH97 is provided to give evidence for creep-brittle behavior of the material and the prediction ability of the model. To examine applicability of the model on other materials, data of Alloy 718 and Hastelloy X from literature are analyzed. Application procedure of the model is also discussed to get a better results when there is a lack of sufficient test data. 2. Crack growth model 2.1. Linear superposition model The linear superposition equation is preferentially considered to model the creep–fatigue crack growth rates [20]:
da da da ¼ þ dN dN fatigue dN creep
ð1Þ
where the subscript ‘‘fatigue’’ and ‘‘creep’’ denote the crack growth rates caused by fatigue and creep loading, respectively. For creep-brittle materials, the crack growth under hold load is mainly caused by environmental oxidation instead of large scale creep deformation. Therefore it is more reasonable to rewrite the model as follows [14,15]:
da da da ¼ þ dN dN cycle dN time
ð2Þ
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
H. Yang et al. / Engineering Fracture Mechanics xxx (2014) xxx–xxx
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where the subscript ‘‘cycle’’ and ‘‘time’’ indicate the crack growth rates under load cycle and hold time, respectively. The cyclic growth rates can be presented by Eq. (3) due to the Paris Law [9]. A formula based on the maximum SIF was proposed and validated for the time-dependent crack growth of nickel base superalloys [21], which was shown in Eq. (4). The models do not separate the environment effect due to the difficulty of tests in vacuum or inert gas and the fact that most superalloys serve under normal atmosphere conditions.
da dN
da dN
¼ CðDKÞn
ð3Þ
¼ AðK max Þm t h
ð4Þ
cycle
time
where DK and K max are SIF range and maximum value in a fatigue load cycle with the relationship K max ¼ DK=ð1 RÞ. C and n are temperature related parameters yielded by pure cyclic crack growth data. A and m can be obtained by pure load hold test or trapezoidal waveshape tests with sufficient dwell time. th is the load hold time in a single load cycle. The crack growth rates can also be expressed in the form of da=dt by applying the following two equations, which is necessary for pure load hold tests.
da da ¼ f dt dN
ð5Þ
f ¼ ðt b þ t h Þ1
ð6Þ
where f is the frequency of the load cycle and tb the load period of one cycle excluding the load hold time. A three-term crack growth model consisting of fatigue, creep and environment effects was proposed by Bain and Pelloux [22]. Ghonem and Zheng [23] discussed another model with an environmental interaction term. Considering to add an cycle– time interaction term to Eq. (2), a linear superposition model is obtained:
da da da da ¼ þ þ dN dN cycle dN time dN interaction
ð7Þ
where ‘‘interaction’’ denotes the crack growth rate caused by the interaction between cyclic and dwell load. 2.2. The interaction intensity factor Grover and Saxena [11] suggested that the domination of creep–fatigue crack growth depended on the relative size of the cyclic plastic zone to the creep zone at the crack tip. Under extremely short or long load hold time condition, the crack growth is dominated by pure fatigue or creep respectively. The creep–fatigue interaction is considered existing only when the cyclic plastic zone size and creep zone size are numerically comparable. However, for creep-brittle nickel base materials, the interaction become more complex as oxidation plays a significant role. Realizing that the fatigue, creep and oxidation interaction effects on the crack growth are linked to the crack tip governing parameter DK and the load frequency f, Yang et al. [7] proposed an exponential equation to characterize the intensity of the interaction as a function of the two parameters, shown in Eq. (8).
g ¼ ep1 ðln f þp2 DKþp3 Þ
2
ð8Þ
where g is the interaction intensity factor. p1 ; p2 and p3 are experimentally determined constants. The formula was derived from the Gaussian function [24] thus has the feature to reach the maximum value of 1 at a medium frequency and approach 0 while the frequency is extremely high or low. This relationship was illustrated in Fig. 1.
Fig. 1. Illustration of g as a function of load frequency and SIF range.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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By making the partial derivative of g with respect to f equal to zero, a linear relationship between the logarithmic frequency and the SIF range was obtained, when satisfied the maximum interaction effect occurs. Equations are shown as follows:
@g ¼0 @f
ð9Þ
ln f þ p2 DK þ p3 ¼ 0
ð10Þ
The interaction crack growth rates were expressed as the product of interaction intensity factor and a crack growth base term:
da dN
¼ DðDKÞq g
ð11Þ
interaction
where D and q are constants fitted by experimental data satisfying Eq. (10). 3. Experiment on FGH97 3.1. Specimen and test Compact tension (CT) specimens cut from a turbine disk made by a nickel base PM superalloy named FGH97 were tested. Details could be found in the authors’ previous publications [25,26]. Material and test are briefly summarized as follows. The material was strengthened by the formation of c0 precipitates based on Ni3 Al and received both solid-solution-strengthening and age-hardening treatment. The width and thickness of CT specimen were chosen to be 50 mm and 25 mm respectively according to ASTM standard [27,28]. All tests were conducted under a typical service temperature of 750 C. An 1.5–1.5 s triangular wave with the load ratio R ¼ 0:05 was used for pre-crack and fatigue tests. Different dwell time t h ¼ 90 s, 450 s and 1500 s at the peak load were superimposed to the baseline fatigue cycle to obtain a trapezoidal waveshape. 3.2. Compliance result The compliance of CT specimens was measured and compared with the theoretical elastic value to see how significant the creep deformation was. The normalized compliance was calculated and presented in Fig. 2. It suggested that the creep induced opening displacements were quite limited if existed. The normalized elastic compliance was calculated according to the following equation [29]:
2 EVB 1þa ¼ f ðaÞ P elastic 1a
ð12Þ
f ðaÞ ¼ 2:16 þ 12:2a 20:1a2 0:992a3 þ 20:6a4 9:91a5
ð13Þ
where
a ¼ a=W is the normalized crack length. V is the force line displacement.
Fig. 2. Normalized compliance vs. elastic values.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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The material’s creep-brittle fracture behavior was further confirmed by the side surface scanning electron microscope (SEM). Fig. 3(a) gives a clear image that the crack grew along the intergranular path. The spectral analysis results by energy-dispersive X-ray spectroscopy (EDS) from the two points located on the grain boundary and inside a grain were shown in Fig. 3(b) and (c) respectively. The weight percentage of oxygen on the boundary was found increased from about 1% to more than 25%, revealing the contribution of oxidation to the intergranular crack growth. 3.3. Crack growth rates The previously developed interaction model was adopted for data reduction and the crack growth rates results were presented in Fig. 4. A good agreement was achieved between experiments and the model under 0 s, 90 s, 450 s and 1500 s dwell conditions with the fitted parameters given in the figure. To further verify the capability of the model, an additional test was conducted with dwell time t h ¼ 25 s. The results were also included in Fig. 4 and excellent correlation was obtained. To have a clear image of the interaction effect, the rates data was transformed to the plot of da=dN vs. the frequency f in log–log scale. According to Eqs. (3), (4) and (6), the pure cycle-dependent and time-dependent crack growth rates were plotted as dashed and dotted lines in Fig. 5 with the slopes of 0 and 1 respectively. Thus the total rates can be obtained by the superposition of them as shown by the dash-dotted lines in the figure if no interaction occurred. The experimental data from FGH97 indicated by the symbols in Fig. 5 showed noticeable increases of crack growth rates compared to the superposition model. It implied significant interaction effect existed, especially under medium frequency. The solid lines in the figure plotted by the interaction model are in excellent accordance with experiments. It should be noticed the data indicated by the open symbols were not included in the model’s parameter fitting and still accurately predicted by the interaction model.
Fig. 3. (a) SEM image from the side view of specimen tested under 750 C with 90 s dwell time; (b)&(c) spectral analysis results located on grain boundary and inside a grain respectively.
Fig. 4. Crack growth rates vs. SIF of superalloy FGH 97.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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Fig. 5. Plots of crack growth rates vs. frequency of superalloy FGH 97.
4. Model applicability to other materials 4.1. Alloy 718 In order to examine the applicability of the model on other nickel base materials, experimental data of Alloy 718 from literature [30] was analyzed. Alloy 718 is a wrought polycrystalline nickel base superalloy with a large amount of Fe and Cr. The crack growth tests were conducted on Kb-type specimens with rectangular cross sections of 4:2 10:2 mm. Two temperatures of 550 C and 650 C with three or four dwell times were involved. Details of the experiments can be found in reference [30]. Fig. 6(a) gives the plot of the crack growth rates under 550 C vs. SIF range. The interaction model fits the data well except for 90 s dwell time at high DK values where the experimental rates are lower than predicted. When it turns to the plot of da=dN vs. f in Fig. 6(b), it is quite clear that the increase of crack growth rates under low frequencies compared to the linear superposition model can be characterized by the interaction model. The interaction effect vanished under high frequencies pffiffiffiffiffi pffiffiffiffiffi or when the K max rose to 36 MPa m. The tested rates under 90 s dwell time at K max ¼ 36 MPa m are below the interaction model values and even the pure time-dependent rates, which is in correspondence with Fig. 6(a). The mechanism remains unclear if not caused by experimental uncertainties. The crack growth rates under 650 C were plotted in Fig. 7. It should be noticed that the specimen applied 21,600 s dwell load reached its critical crack length before the finish of first load cycle. So the data should be recognized as pure time-dependent. To include that data, crack growth rates were calculated in the form of da=dt. Both the plots of da=dt vs. DK in Fig. 7 and da=dt vs. f in Fig. 7(b) show good agreement between experiment and interaction model. It’s also worth mentioning that the accuracy of the shape of interaction intensity factor g revealed in Fig. 7(b) could be improved. Due to the lack of data from more load frequencies, the current g function was actually determined by one dwell time test, which is the only one in the significant interaction effect region.
(a)
(b)
Fig. 6. The crack growth rates of Alloy 718 under 550 C (a) vs. SIF range; (b) vs. frequency.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
H. Yang et al. / Engineering Fracture Mechanics xxx (2014) xxx–xxx
(a)
7
(b)
Fig. 7. The crack growth rates of Alloy 718 under 650 C (a) vs. SIF range; (b) vs. frequency.
4.2. Hastelloy X HastelloyÒ X alloy is a solid-solution strengthened nickel base superalloy that combines exceptional oxidation resistance, good fabricability and excellent high temperature strength. The data is from the experiments conducted on CT specimens with thickness of 3.2 mm, height of 61.0 mm and width of 63.5 mm[31]. Two temperatures of 816 C and 917 C were involved. The crack growth rates and fitted curves by interaction model were presented in Fig. 8(a) and (b) with good agreements. Due to the plot of da=dN vs. f in Fig. 9, there is hardly any interaction effect for the HastelloyÒ X alloy under the two temperatures. The crack growth was dominated by a competing mechanism between pure time-dependent and pure cycle-dependent. In these cases, the interaction model degraded to the linear superposition model. 4.3. Discussion As shown in above analysis, the capability of interaction model was verified by the experiments on FGH 97 and data of Alloy 718 and HastelloyÒ X from literature. However, as up to five parameters require to be fitted for applying the model, it’s therefore worth determining whether significant interaction effect occurred first. This can be ensured by plotting the crack growth rates vs. frequency in log–log scale, as we did for each above analysis. The crack growth behavior of Haynes 230 superalloy [32] plotted in Fig. 10 in the form of da=dN vs. f present exactly three different typical crack growth phenomenons and mechanisms. The experiments were performed under constant DK control pffiffiffiffiffi with a value of 27:5 MPa m. When tested under 649 C, the interaction effect increased the crack growth rates of the material up to more than five times. If ignored, considerable underestimation of crack growth rates can be made. The interaction disappeared when the temperature rose to 816 C where the data points were well predicted by the linear superposition model. The crack growth behavior was fully dominated by pure time-dependent mechanism under 927 C, leaving no chance for cycle-dependent mechanism or interaction.
(a)
(b)
Fig. 8. The crack growth rates vs. SIF range of Hastelloy X under 816 C and 927 C.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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Fig. 9. The crack growth rates vs. frequency of Hastelloy X under 816 C and 927 C at DK ¼ 28 MPa
Fig. 10. The crack growth behavior of Haynes 230 under three different temperatures at DK ¼ 27:5 MPa
pffiffiffiffiffi m.
pffiffiffiffiffi m [32].
As conducted under constant DK control, the data under 649 C in Fig. 10 drew a clear image of the interaction induced crack growth and were sufficient to fit g. For constant DP control tests, however, there may be a lack of frequencies involved. The data shown in Fig. 7 is one case of that. It is suggested to manually adjust the shape parameter p1 in Eq. (8) to simplify the g fitting process in those cases. Thus the application procedure of the interaction model could be summarized as follows: (a) Fit the cycle-dependent parameters in Eq. (3) and time-dependent parameters in Eq. (4) by pure cyclic and load hold tests respectively. (b) Plot the crack growth rates vs. frequency to determine whether interaction exists. (c) If it does, fit the interaction parameters in Eqs. (8) and (11). (d) Manually adjust the shape parameter p1 in Eq. (8) during fitting if no sufficient frequencies involved. 5. Conclusion The superposition model consisting of cycle-dependent and time-dependent terms was widely used to evaluate the high temperature crack growth of superalloys under fatigue–creep loading conditions. A cycle–time interaction model developed based on the concept of interaction intensity factor for creep-brittle materials was restudied. The prediction ability and applicability were examined by experiments and data from literature. Key results and conclusions drawn from this investigation were summarized below: 1. The interaction intensity factor g derived from the Gaussian equation has the ability to characterize the interaction effects varying with frequency and SIF range. 2. The applicability of the model was verified by test on FGH 97 and experimental data of Alloy 718 and HastelloyÒ X from literature. 3. As up to five parameters to be fitted, it is recommended to examine whether significant interaction occurred by plotting da=dN vs. frequency f in log–log scale before applying the model. Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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4. Once all parameters obtained by a set of test data, crack growth under other load holding time can be predicted. The effect of varied temperatures, however, is not explicitly included. This could be one of the improvements of the model in future studies.
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Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
An interaction crack growth model for creep-brittle superalloys with high temperature dwell time Hongqin Yang a,⇑, Rui Bao b, Jiazhen Zhang a a b
Beijing Aeronautical Science & Technology Research Institute, Beijing 102211, China Institute of Solid Mechanics, Beihang University, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 29 August 2013 Received in revised form 8 February 2014 Accepted 6 April 2014 Available online xxxx Keywords: Interaction effect Creep–fatigue Crack growth Nickel-based superalloys
a b s t r a c t A three-term interaction crack growth model developed for creep-brittle materials is restudied. Derivation of cycle–time interaction intensity factor from the Gaussian equation is discussed. The model’s prediction ability was verified by test data of a nickel base superalloy under 750 C with 25 s dwell time. Data of two other superalloys namely Alloy 718 and HastelloyÒ X from literature are analyzed to further examine applicability of the model. Satisfactory results are obtained. Parameter fitting procedure and limitation of the model are discussed at the end. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Nickel base powder superalloys are widely used for aerospace engine applications due to high strength under elevated temperatures. New generation of such superalloys are designed to gain improved damage tolerance performance as well as high strength [1]. Studies on crack growth of these superalloys thus receive more attentions, from crack growth behavior to underlying mechanisms and models. A number of studies [2,3] on the low cycle fatigue crack growth of nickel base superalloys have revealed that such materials are so called ‘‘creep-brittle’’ due to their excellent creep resistance. Environmental factors like oxidation contribute most to the time-dependent crack growth during load hold time. Two theories called stress accelerated grain boundary oxidation (SAGBO) [4] and gas phase embrittlement (GPE) [5] were widely used to explain the effect of oxidation. However the mechanisms of crack growth along intergranular boundaries are still not fully understood [6]. To model the crack growth of creep-brittle superalloys, it was found the stress intensity factor (SIF) K max was preferred instead of creep parameters C or C t [7,8]. The simplest equation to model the crack growth is one-term Paris Law [9]. Effects of temperature, frequency and oxidation are all evaluated by model coefficients [3]. Models based on the assumption of competing mechanisms [10,11] were also found good enough for several materials. A more universal model was developed from the idea that the total crack growth rates were the linear superposition of cycle-dependent and time-dependent rates [12,13]. Wei et al. [14] proposed probabilistic creep–fatigue–oxidation crack growth model for creep-brittle materials based on a linear superposition theory. A model using the concept of a damage zone was recently presented by Gustafsson et al. [15] to consider the complex mechanisms of the crack growth during load hold time.
⇑ Corresponding author. Tel.: +86 1057808860. E-mail addresses: [email protected] (H. Yang), [email protected] (R. Bao), [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.engfracmech.2014.04.006 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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Nomenclature a; a0 crack length and initial value A; m parameters in time-dependent crack growth model B thickness of specimens C; n parameters in cycle-dependent crack growth model da=dN; da=dt crack growth rates D; q parameters in interaction crack growth model E material elastic modulus f loading frequency K; DK stress intensity factor (SIF), SIF range N number of loading cycles p1 ; p2 ; p3 parameters in interaction intensity factor equation P; DP load, cyclic load range R nominal stress ratio t time th load dwell time in a cycle tb loading and unloading time in a cycle V force line displacement of specimens W width of specimens g interaction intensity factor
Most of above models does not separate the interaction effects between fatigue, creep and oxidation crack growth from themselves. For the superalloys which were found significantly affected by interaction effects [16,17], accuracy of these models can be noticeably reduced. A study has revealed that interaction induced crack growth usually occurs in the transition stage between pure fatigue and pure creep loading conditions [18]. A three term model was proposed [19] to cover the whole crack growth stage from pure cycle-dependent to extremely long hold times. The work of Grover and Saxena [11] indicates intensity of the creep–fatigue interaction is related to the relative size of creep zone to cyclic plastic zone. But it is only suitable for creep-ductile materials. Another three term model idea was discussed by Prakash et al. [17] where the crack growth rates consisted of cycle-dependent, pure creep and environmentally enhanced rates. Due to complex mechanism of oxidation or lack of supporting data, these three term models have not gained widely application. Realizing the interaction between cycle and time induced crack growth mainly relies on SIF and load frequency, the authors presented a novel factor [7] to represent the interaction intensity. The factor has a form derived from the Gaussian equation, which reaches its maximum value at an intermediate frequency and vanishes to zero at low or high frequencies for a given SIF value. A three term model consisting of cycle-dependent, time-dependent and their interaction was developed based on the factor for creep-brittle materials. The capacity of the model was experimentally verified on a nickel base superalloy FGH97. The study presented here is a continuation of that work. Derivation of the model is reviewed based on creep-brittle crack growth mechanism. More experimental data from FGH97 is provided to give evidence for creep-brittle behavior of the material and the prediction ability of the model. To examine applicability of the model on other materials, data of Alloy 718 and Hastelloy X from literature are analyzed. Application procedure of the model is also discussed to get a better results when there is a lack of sufficient test data. 2. Crack growth model 2.1. Linear superposition model The linear superposition equation is preferentially considered to model the creep–fatigue crack growth rates [20]:
da da da ¼ þ dN dN fatigue dN creep
ð1Þ
where the subscript ‘‘fatigue’’ and ‘‘creep’’ denote the crack growth rates caused by fatigue and creep loading, respectively. For creep-brittle materials, the crack growth under hold load is mainly caused by environmental oxidation instead of large scale creep deformation. Therefore it is more reasonable to rewrite the model as follows [14,15]:
da da da ¼ þ dN dN cycle dN time
ð2Þ
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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where the subscript ‘‘cycle’’ and ‘‘time’’ indicate the crack growth rates under load cycle and hold time, respectively. The cyclic growth rates can be presented by Eq. (3) due to the Paris Law [9]. A formula based on the maximum SIF was proposed and validated for the time-dependent crack growth of nickel base superalloys [21], which was shown in Eq. (4). The models do not separate the environment effect due to the difficulty of tests in vacuum or inert gas and the fact that most superalloys serve under normal atmosphere conditions.
da dN
da dN
¼ CðDKÞn
ð3Þ
¼ AðK max Þm t h
ð4Þ
cycle
time
where DK and K max are SIF range and maximum value in a fatigue load cycle with the relationship K max ¼ DK=ð1 RÞ. C and n are temperature related parameters yielded by pure cyclic crack growth data. A and m can be obtained by pure load hold test or trapezoidal waveshape tests with sufficient dwell time. th is the load hold time in a single load cycle. The crack growth rates can also be expressed in the form of da=dt by applying the following two equations, which is necessary for pure load hold tests.
da da ¼ f dt dN
ð5Þ
f ¼ ðt b þ t h Þ1
ð6Þ
where f is the frequency of the load cycle and tb the load period of one cycle excluding the load hold time. A three-term crack growth model consisting of fatigue, creep and environment effects was proposed by Bain and Pelloux [22]. Ghonem and Zheng [23] discussed another model with an environmental interaction term. Considering to add an cycle– time interaction term to Eq. (2), a linear superposition model is obtained:
da da da da ¼ þ þ dN dN cycle dN time dN interaction
ð7Þ
where ‘‘interaction’’ denotes the crack growth rate caused by the interaction between cyclic and dwell load. 2.2. The interaction intensity factor Grover and Saxena [11] suggested that the domination of creep–fatigue crack growth depended on the relative size of the cyclic plastic zone to the creep zone at the crack tip. Under extremely short or long load hold time condition, the crack growth is dominated by pure fatigue or creep respectively. The creep–fatigue interaction is considered existing only when the cyclic plastic zone size and creep zone size are numerically comparable. However, for creep-brittle nickel base materials, the interaction become more complex as oxidation plays a significant role. Realizing that the fatigue, creep and oxidation interaction effects on the crack growth are linked to the crack tip governing parameter DK and the load frequency f, Yang et al. [7] proposed an exponential equation to characterize the intensity of the interaction as a function of the two parameters, shown in Eq. (8).
g ¼ ep1 ðln f þp2 DKþp3 Þ
2
ð8Þ
where g is the interaction intensity factor. p1 ; p2 and p3 are experimentally determined constants. The formula was derived from the Gaussian function [24] thus has the feature to reach the maximum value of 1 at a medium frequency and approach 0 while the frequency is extremely high or low. This relationship was illustrated in Fig. 1.
Fig. 1. Illustration of g as a function of load frequency and SIF range.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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By making the partial derivative of g with respect to f equal to zero, a linear relationship between the logarithmic frequency and the SIF range was obtained, when satisfied the maximum interaction effect occurs. Equations are shown as follows:
@g ¼0 @f
ð9Þ
ln f þ p2 DK þ p3 ¼ 0
ð10Þ
The interaction crack growth rates were expressed as the product of interaction intensity factor and a crack growth base term:
da dN
¼ DðDKÞq g
ð11Þ
interaction
where D and q are constants fitted by experimental data satisfying Eq. (10). 3. Experiment on FGH97 3.1. Specimen and test Compact tension (CT) specimens cut from a turbine disk made by a nickel base PM superalloy named FGH97 were tested. Details could be found in the authors’ previous publications [25,26]. Material and test are briefly summarized as follows. The material was strengthened by the formation of c0 precipitates based on Ni3 Al and received both solid-solution-strengthening and age-hardening treatment. The width and thickness of CT specimen were chosen to be 50 mm and 25 mm respectively according to ASTM standard [27,28]. All tests were conducted under a typical service temperature of 750 C. An 1.5–1.5 s triangular wave with the load ratio R ¼ 0:05 was used for pre-crack and fatigue tests. Different dwell time t h ¼ 90 s, 450 s and 1500 s at the peak load were superimposed to the baseline fatigue cycle to obtain a trapezoidal waveshape. 3.2. Compliance result The compliance of CT specimens was measured and compared with the theoretical elastic value to see how significant the creep deformation was. The normalized compliance was calculated and presented in Fig. 2. It suggested that the creep induced opening displacements were quite limited if existed. The normalized elastic compliance was calculated according to the following equation [29]:
2 EVB 1þa ¼ f ðaÞ P elastic 1a
ð12Þ
f ðaÞ ¼ 2:16 þ 12:2a 20:1a2 0:992a3 þ 20:6a4 9:91a5
ð13Þ
where
a ¼ a=W is the normalized crack length. V is the force line displacement.
Fig. 2. Normalized compliance vs. elastic values.
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The material’s creep-brittle fracture behavior was further confirmed by the side surface scanning electron microscope (SEM). Fig. 3(a) gives a clear image that the crack grew along the intergranular path. The spectral analysis results by energy-dispersive X-ray spectroscopy (EDS) from the two points located on the grain boundary and inside a grain were shown in Fig. 3(b) and (c) respectively. The weight percentage of oxygen on the boundary was found increased from about 1% to more than 25%, revealing the contribution of oxidation to the intergranular crack growth. 3.3. Crack growth rates The previously developed interaction model was adopted for data reduction and the crack growth rates results were presented in Fig. 4. A good agreement was achieved between experiments and the model under 0 s, 90 s, 450 s and 1500 s dwell conditions with the fitted parameters given in the figure. To further verify the capability of the model, an additional test was conducted with dwell time t h ¼ 25 s. The results were also included in Fig. 4 and excellent correlation was obtained. To have a clear image of the interaction effect, the rates data was transformed to the plot of da=dN vs. the frequency f in log–log scale. According to Eqs. (3), (4) and (6), the pure cycle-dependent and time-dependent crack growth rates were plotted as dashed and dotted lines in Fig. 5 with the slopes of 0 and 1 respectively. Thus the total rates can be obtained by the superposition of them as shown by the dash-dotted lines in the figure if no interaction occurred. The experimental data from FGH97 indicated by the symbols in Fig. 5 showed noticeable increases of crack growth rates compared to the superposition model. It implied significant interaction effect existed, especially under medium frequency. The solid lines in the figure plotted by the interaction model are in excellent accordance with experiments. It should be noticed the data indicated by the open symbols were not included in the model’s parameter fitting and still accurately predicted by the interaction model.
Fig. 3. (a) SEM image from the side view of specimen tested under 750 C with 90 s dwell time; (b)&(c) spectral analysis results located on grain boundary and inside a grain respectively.
Fig. 4. Crack growth rates vs. SIF of superalloy FGH 97.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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Fig. 5. Plots of crack growth rates vs. frequency of superalloy FGH 97.
4. Model applicability to other materials 4.1. Alloy 718 In order to examine the applicability of the model on other nickel base materials, experimental data of Alloy 718 from literature [30] was analyzed. Alloy 718 is a wrought polycrystalline nickel base superalloy with a large amount of Fe and Cr. The crack growth tests were conducted on Kb-type specimens with rectangular cross sections of 4:2 10:2 mm. Two temperatures of 550 C and 650 C with three or four dwell times were involved. Details of the experiments can be found in reference [30]. Fig. 6(a) gives the plot of the crack growth rates under 550 C vs. SIF range. The interaction model fits the data well except for 90 s dwell time at high DK values where the experimental rates are lower than predicted. When it turns to the plot of da=dN vs. f in Fig. 6(b), it is quite clear that the increase of crack growth rates under low frequencies compared to the linear superposition model can be characterized by the interaction model. The interaction effect vanished under high frequencies pffiffiffiffiffi pffiffiffiffiffi or when the K max rose to 36 MPa m. The tested rates under 90 s dwell time at K max ¼ 36 MPa m are below the interaction model values and even the pure time-dependent rates, which is in correspondence with Fig. 6(a). The mechanism remains unclear if not caused by experimental uncertainties. The crack growth rates under 650 C were plotted in Fig. 7. It should be noticed that the specimen applied 21,600 s dwell load reached its critical crack length before the finish of first load cycle. So the data should be recognized as pure time-dependent. To include that data, crack growth rates were calculated in the form of da=dt. Both the plots of da=dt vs. DK in Fig. 7 and da=dt vs. f in Fig. 7(b) show good agreement between experiment and interaction model. It’s also worth mentioning that the accuracy of the shape of interaction intensity factor g revealed in Fig. 7(b) could be improved. Due to the lack of data from more load frequencies, the current g function was actually determined by one dwell time test, which is the only one in the significant interaction effect region.
(a)
(b)
Fig. 6. The crack growth rates of Alloy 718 under 550 C (a) vs. SIF range; (b) vs. frequency.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
H. Yang et al. / Engineering Fracture Mechanics xxx (2014) xxx–xxx
(a)
7
(b)
Fig. 7. The crack growth rates of Alloy 718 under 650 C (a) vs. SIF range; (b) vs. frequency.
4.2. Hastelloy X HastelloyÒ X alloy is a solid-solution strengthened nickel base superalloy that combines exceptional oxidation resistance, good fabricability and excellent high temperature strength. The data is from the experiments conducted on CT specimens with thickness of 3.2 mm, height of 61.0 mm and width of 63.5 mm[31]. Two temperatures of 816 C and 917 C were involved. The crack growth rates and fitted curves by interaction model were presented in Fig. 8(a) and (b) with good agreements. Due to the plot of da=dN vs. f in Fig. 9, there is hardly any interaction effect for the HastelloyÒ X alloy under the two temperatures. The crack growth was dominated by a competing mechanism between pure time-dependent and pure cycle-dependent. In these cases, the interaction model degraded to the linear superposition model. 4.3. Discussion As shown in above analysis, the capability of interaction model was verified by the experiments on FGH 97 and data of Alloy 718 and HastelloyÒ X from literature. However, as up to five parameters require to be fitted for applying the model, it’s therefore worth determining whether significant interaction effect occurred first. This can be ensured by plotting the crack growth rates vs. frequency in log–log scale, as we did for each above analysis. The crack growth behavior of Haynes 230 superalloy [32] plotted in Fig. 10 in the form of da=dN vs. f present exactly three different typical crack growth phenomenons and mechanisms. The experiments were performed under constant DK control pffiffiffiffiffi with a value of 27:5 MPa m. When tested under 649 C, the interaction effect increased the crack growth rates of the material up to more than five times. If ignored, considerable underestimation of crack growth rates can be made. The interaction disappeared when the temperature rose to 816 C where the data points were well predicted by the linear superposition model. The crack growth behavior was fully dominated by pure time-dependent mechanism under 927 C, leaving no chance for cycle-dependent mechanism or interaction.
(a)
(b)
Fig. 8. The crack growth rates vs. SIF range of Hastelloy X under 816 C and 927 C.
Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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Fig. 9. The crack growth rates vs. frequency of Hastelloy X under 816 C and 927 C at DK ¼ 28 MPa
Fig. 10. The crack growth behavior of Haynes 230 under three different temperatures at DK ¼ 27:5 MPa
pffiffiffiffiffi m.
pffiffiffiffiffi m [32].
As conducted under constant DK control, the data under 649 C in Fig. 10 drew a clear image of the interaction induced crack growth and were sufficient to fit g. For constant DP control tests, however, there may be a lack of frequencies involved. The data shown in Fig. 7 is one case of that. It is suggested to manually adjust the shape parameter p1 in Eq. (8) to simplify the g fitting process in those cases. Thus the application procedure of the interaction model could be summarized as follows: (a) Fit the cycle-dependent parameters in Eq. (3) and time-dependent parameters in Eq. (4) by pure cyclic and load hold tests respectively. (b) Plot the crack growth rates vs. frequency to determine whether interaction exists. (c) If it does, fit the interaction parameters in Eqs. (8) and (11). (d) Manually adjust the shape parameter p1 in Eq. (8) during fitting if no sufficient frequencies involved. 5. Conclusion The superposition model consisting of cycle-dependent and time-dependent terms was widely used to evaluate the high temperature crack growth of superalloys under fatigue–creep loading conditions. A cycle–time interaction model developed based on the concept of interaction intensity factor for creep-brittle materials was restudied. The prediction ability and applicability were examined by experiments and data from literature. Key results and conclusions drawn from this investigation were summarized below: 1. The interaction intensity factor g derived from the Gaussian equation has the ability to characterize the interaction effects varying with frequency and SIF range. 2. The applicability of the model was verified by test on FGH 97 and experimental data of Alloy 718 and HastelloyÒ X from literature. 3. As up to five parameters to be fitted, it is recommended to examine whether significant interaction occurred by plotting da=dN vs. frequency f in log–log scale before applying the model. Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006
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4. Once all parameters obtained by a set of test data, crack growth under other load holding time can be predicted. The effect of varied temperatures, however, is not explicitly included. This could be one of the improvements of the model in future studies.
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Please cite this article in press as: Yang H et al. An interaction crack growth model for creep-brittle superalloys with high temperature dwell time. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.04.006