Thermo-mechanical fatigue failure of a single crystal Ni-based superalloy

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International Journal of Fatigue 30 (2008) 318–323

International Journalof Fatigue www.elsevier.com/locate/ijfatigue

Thermo-mechanical fatigue failure of a single crystal Ni-based superalloy M. Okazaki *, M. Sakaguchi Nagaoka University of Technology, Tomioka, Nagaoka 940-2188, Japan Accepted 15 January 2007 Available online 20 March 2007

Abstract Behavior of thermo-mechanical fatigue (TMF) of a single crystal Ni-based superalloy, CMSX-4, was studied, compared with that of isothermal low cycle fatigue (LCF). Strain-controlled TMF and LCF tests of CMSX-4 were carried out under various test conditions, where the experimental variables were strain rates, strain ratio, temperature range, and strain/temperature phase angle. It was shown experimentally that the TMF and LCF failures took places, associated with some noteworthy characteristics which were rarely seen in the traditional polycrystalline heat-resistant alloys. They could not be explained reasonably, based on the historical approaches. A new micromechanics model was proposed to predict the TMF and LCF lives, applying the Eshelby’s theory. The model enabled us to successfully estimate the characteristics in the TMF and LCF failures.  2007 Elsevier Ltd. All rights reserved. Keywords: Single crystal Ni-based superalloy; Low cycle fatigue; Thermo-mechanical fatigue; c/c 0 microstructure; Life prediction model

1. Introduction Ni-based superalloys, especially single crystal superalloys, have been received special interests for blades and vanes in advanced industrial gas turbines [1,2], since they have superior creep strength at elevated temperatures. One of the most attractive properties of superalloys as heat resting alloys is their unique temperature dependence in strength under monotonic and creep loadings, i.e., the strength increases with increasing temperature up to a certain temperature. In general Ni-based superalloys have naturally developed composite microstructure consisting of a solid solution matrix, c, with Ni3(Al, Ti) intermetallics precipitates, c 0 , ordered and coherent with matrix [1,2]. Many researchers have been pointed out that this type of microstructure may play an essential role in the above attractive properties [1–4,8,10,14]. During the gas turbine operation period, on the other hand, *

Corresponding author. Tel.: +81 258 47 9722; fax: +81 258 47 9770. E-mail address: [email protected] (M. Okazaki).

0142-1123/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.01.044

the hot section components are subjected to many types of damages. Nowadays, thermo-mechanical fatigue (TMF) failure has been one of the critical issues to be concerned. Many types of efforts have been made to understand the TMF failure mechanisms and the life criteria for design, reliability, and long term durability [1–14]. One of traditional methods from an engineering point of view is a trial to estimate the TMF failure life from an empirical correlation with the isothermal low cycle fatigue (LCF) life [12,13]. While this type of approach has been successful for many kinds of polycrystalline alloys. However, as will be also shown in this work, superalloys, especially single crystal superalloys, exhibit many types of inconceivable TMF failure behaviors which are not well understood from these historical understandings. For an example, the TMF failure life under the out-of-phase condition often exhibits a life significantly lower than supposed [3]. The effect of strain ratio on LCF lives is also an article which has not been well understood [14]. More or less, a series of these unique characteristics in superalloys may be related to

M. Okazaki, M. Sakaguchi / International Journal of Fatigue 30 (2008) 318–323

the c/c 0 composite microstructures [1–4,9,10]. Nevertheless, no quantitative models have been established to provide reasonable explanations. It is the first object of this work to investigate the role of microstructure in TMF failure of single crystal superalloy. The second is to explore a quantitative life prediction method, taking account of microscopic failure mechanisms and the role of microstructures. In earlier part of this paper, some experimental results on the LCF and TMF tests in a single crystal superalloy, CMSX-4 are shown. Then, a new micromechanics model is proposed to estimate the LCF and TMF lives. 2. Experimental procedures The material tested in this work is a second generation single crystal superalloy, CMSX-4, which has the following chemical compositions in weight percent: 6.4Cr, 9.7Co, 0.6Mo, 6.4W, 1.0Ti, 6.5Ta, 2.9Re, 0.1Hf, 5.7Al, and bal. Ni. For this materials the heat treatments were given as follows: the eight stages solution treatments by 1277 C · 1 h + 1288 C · 2 h + 1296 C · 3 h + 1304 C · 3 h + 1313 C · 2 h + 1316 C · 2 h + 1318 C · 2 h + 1321 C · 2 h in Argon atmosphere, and then the two stages aging treatments by 1140 C · 6 h + 870 C · 20 h in air. As the result of the heat treatments the CMSX-4 revealed very regular microstructure, involving of cubic c 0 precipitates. The volume fraction and the size of cubic c 0 precipitates are about 0.5 lm and 63%, respectively. The solid cylindrical specimens, of which gauge section has 6.5 mm in diameter and 20 mm in length, were machined for the LCF and TMF tests. All specimen’s axes lie within 5 from Æ0 0 1æ crystallographic orientation. The LCF and TMF tests were carried out according to the test program summarized in Table 1, where the experimental variables were strain range, strain wave-shape and strain ratio. These tests were performed under strain-controlled condition in air, utilizing a servo–electro hydraulic test system. The number of cycles to failure of the TMF and LCF lives was defined by the number of strain cycles at which the tensile stress was reduced by 50% from the stationary value.

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3. Results 3.1. Cyclic stress–strain relationships Some of the typical stress–strain hysteresis loops in the LCF and TMF tests are given in Fig. 1. The hysteresis loops revealed the characteristic shapes when the strain ratio Re, was changed in the LCF tests at 900 C (compare among Fig. 1c, d and e). The mean stress was built up depending on the test condition: it was in tensile under Re = 0, about zero under Re = 1, and compressive under Re = 1, respectively. However, the mean stress level shifted towards zero by repeating strain cycles, and the hysteresis loops resulted in the stable shape. 3.2. LCF and TMF lives Fig. 2 summarizes the LCF and TMF lives on the basis of mechanical strain range (Demech), where each symbol follows Table 1, and a dotted line expresses an overall trend of LCF life at 600 C [4]. For an easily understanding, the LCF and TMF lives are compared in Fig. 3 under a fixed strain range, Demech = 1.0%. The following characteristics should be pointed out from Figs. 2 and 3: (i) Regarding the effects of test temperature on the LCF lives; the f–f type LCF lives at 400 C were longer than those at 900 C. (ii) Regarding the effects of strain rate on the LCF lives; the LCF lives under f–f type of strain wave-shape were longer than those under the s–s type of strain wave-shape, at 900 C. (iii) Regarding the effects of strain ratio on LCF lives; the CMSX-4 failed even when it was subjected to 0-compression strain cycling. Note that the LCF life under Re = 1 was almost comparable to that under Re = 0. Thus, the LCF lives were not significantly influenced by the strain ratio at 900 C. (iv) Regarding the effects of temperature/strain phase angle on the TMF lives; the TMF lives under the diamond conditions were the longest of all the TMF lives. The out-of-phase TMF lives were almost comparable

Table 1 LCF and TMF tests programs employed Test type

Strain waveform

Frequency (Hz)

Temp-strain phase difference

Max. Temp. (C)

Min. Temp. (C)

Strain ratio

Symbol

LCF

f–f

1/30

–

1/600

400 900 900

1

s–s

400 900 900

0 1 1

d j n h ,

In-phase Out-of-phase Diamondphase

1/600

900

400

1

}

TMF

0 180 90

320

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Fig. 1. Hysteresis loops obtained from LCF and TMF tests under the condition of Demech = 1.2%. (a) LCF at 400 C (f–f, Re = 1), (b) LCF at 900 C (f–f, Re = 1), (c) LCF at 900 C (s–s, Re = 1), (d) LCF at 900 C (s–s, Re = 0), (e) LCF at 900 C (s–s, Re = 1), (f) TMF in-phase (Re = 1), (g) TMF out-of-phase (Re = 1), and (h) TMF diamond phase (Re = 1).

to the in-phase TMF lives, and they were noticeably lower than those under the diamond phase condition. This kind of phase angle dependence has been explained by the mean stress effect [7]. (v) Regarding the relation between LCF and TMF lives; The empirical law: the TMF lives under in-phase and out-of-phase conditions are almost comparable to the LCF lives which are measured at the maximum and at the intermediate temperatures of the TMF tests, respectively; was not satisfied in the CMSX-4 (Figs. 2 and 3).

graphic planes. The former was almost perpendicular to the stress axis, and was formed at the crack nucleation process and the subsequent early growth process during which

These essential behaviors in the TMF and LCF failures of the CMSX-4 were too hard to be reasonably understood, so far as we relied on the macroscopic parameters: e.g., in-elastic strain range, stress range, or the Ostergren’s energy parameter. 3.3. Fracture surfaces The SEM micrographs of fracture surfaces showed the following features [4,20]: the most of fracture surface was more or less composed of {1 0 0} and {1 1 1} crystallo-

Fig. 2. LCF and TMF lives of CMSX-4.

M. Okazaki, M. Sakaguchi / International Journal of Fatigue 30 (2008) 318–323

d ¼ 2ðac0  ac Þ=ðac0 þ ac Þ

Fig. 3. Bar graph of LCF and TMF lives under the condition of Demech = 1.2%. The symbol represents the data interpolated from the best fit S–N curve.

the cracks propagated in the narrow c channel in all the test conditions. Meanwhile the latter planes were formed during the final fracture failure process. While there were some differences depending on the test conditions, the following was common in this work: the preferential crack nucleation and propagation site was inside the narrow c channels.

4.1. A life prediction model One of main reason(s) why the traditional parameters and approaches could not be always applied to the present results, must be attributed to that they neglect the existing of internal stress resulting from a lattice misfit between c and c 0 precipitates. In order to consider the effect of internal stress, the microstructure of single crystal Ni-based superalloys is modeled by a composite material system in which spherical c 0 precipitates distribute uniformly with a volume fraction of Vf in the c matrix as shown in Fig. 4. For a simplicity, both the c and c 0 phases are assumed to be elastic-fully-plastic solids in this work. An important factor to be taken into account in the model is that the both phases have different elastic constants in general. The other important factor is a misfit strain between them. For the latter, it is assumed that there is an isotropic lattice misfit strain, d, between them, which is defined by misfit strain, δ, with γ and volume fraction, Vf γ’ γ

Fig. 4. Model of superalloy microstructure containing ellipsoidal c 0 precipitates.

ð1Þ

where ac and ac0 are the lattice constants of c 0 and c, respectively. The misfit strain, d, produces misfit stress even under no external load, which are denoted by Ærij(d)c in the c matrix and hrij ðdÞic0 in the c 0 precipitates, respectively [15]. When the geometry of precipitates is a type of ellipsoids, these stresses are estimated by applying both the Eshelby’s equivalent inclusion theory [15,16], and the averaging stress field approximation by Mori and Tanaka [17]. When the composite material system is subjected to external stresses, rA additional stresses, Ærij(inh)æc, ij , hrij ðinhÞic0 , should be produced in each material element (i.e., c and c 0 phase) to compensate an inhomogeneous elastic deformation [15]. Furthermore, when the material system undergoes plastic deformation, new additional internal stresses, Ærij(p)æc, hrij ðpÞic0 , should be built up by inhomogeneous plastic strain between c and c 0 [15]. Thus, when the composite material system in Fig. 4 undergoes a plastic deformation, the c and c 0 material element is supposed to carry the sum of each stress component as follows: hrij ic ¼ rA ij þ hrij ðinhÞic þ hrij ðdÞic þ hrij ðpÞic hrij ic0 ¼

4. Discussion

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rA ij

þ hrij ðinhÞic0 þ hrij ðdÞic0 þ hrij ðpÞic0

ð2-1Þ ð2-2Þ

Each stress term in Eq. (2) is calculated by solving the simultaneous equations of the Eshelby’s theory, the averaging stress field approximation, and the stable state condition of energy of the composite system. The details of the calculation method have been given in Ref. [18]. It is worth noting that the calculation can predict a linear kinematic work hardening behavior of the composite material system under monotonic loading [15,18]. It is not hard to extend the above procedure to cyclic loading. The numerical calculation will be shown in the next section. In order to combine the stress/strain response thus estimated with fatigue failure, it is necessary to express ‘‘fatigue damage’’ by an appropriate parameter. Since damage is irreversible process, it should be represented by a parameter which can reasonably denote energy dissipation process. Plastic work density, Wp, should be such a candidate. Here, Wp is defined by Z ð3-1Þ W p ¼ rij dep;ij where dep,ij is a plastic strain increment. The Wp under a cyclic loading should be represented by a value per one cycle: I W p ¼ rij  dep;ij ð3-2Þ It is worth noting that the Manson–Coffin’s law has been derived in terms of Wp by this type of concept [19]. From these backgrounds, the following relations are assumed in this work:

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Table 2 Material properties of c matrix and c 0 precipitates used for the numerical calculations

c c0

Young’s modulus (GPa)

Poisson’s ratio

Volume fraction

150 120

0.3 0.3

0.35 0.65

Yield stress (MPa) 400 C

650 C

900 C

700 1000

600 1100

500 750

Lattice misfit of c 0 with c matrix (%) 0.1

Fig. 5. Calculated hysteresis loops under Demech = 1.2%. (a) LCF at 900 C (Re = 1), (b) TMF in-phase (Re = 1), and (c) TMF out-of-phase (Re = 1).

W p ðcÞN f ðcÞ ¼ CðcÞ 0

0

ð4-1Þ 0

W p ðc ÞN f ðc Þ ¼ Cðc Þ

ð4-2Þ 0

0

where Nf(c) and Nf(c ) are the fatigue life of c and c phases, and C(c) and C(c 0 ) are the material constants, respectively. The fatigue life of the composite material system is supposed to be governed by a shorter value: either Nf(c) or Nf(c 0 ). 4.2. Numerical calculations 4.2.1. Estimation of stress–strain curve The geometry of c 0 precipitate is approximated to be spherical in this work. This factor is taken into account by the value of Eshelby’s tensor, when Eq. (2) is calculated. The material constants used for the numerical calculation are summarized in Table 2 [1]. The method presented in Section 4.1 can be directly applied for the isothermal LCF tests of this work. On the other hand, the TMF test was approximated by ‘‘bi-thermal’’ fatigue test; apparent LCF tests at two step temperatures: the maximum and minimum temperatures of the TMF test. Fig. 5 shows some of the predicted hysteresis loops, where the symbols, through , denote some key points in deformation. From 0 to the material deforms elastically during the 1st monotonic loading period. During the periods between –, – and –, the material system deforms inelastically associated with a plastic deformation only in c phase, and with no plastic strains in c 0 phase. At least in this work, these hysteresis loops did not change after the 2nd fatigue cycles, and no ratcheting phenomena were calculated. It seems from these figures that both the yield stress level and the characteristics in hysteresis loops under the respective test conditions seem to be well reproduced by the present method.

4.2.2. Life prediction As shown in Section 3.2, the fatigue fracture occurred in the c matrix, thus, we assumed that the fatigue life of the composite material system is dominated by Eq. (4-1). By this approximation, it is possible for us to make a relative comparison of life by comparing the values of Wp(c) depending on the test condition, without knowing the actual value of C(c). Fig. 6 shows a summary of the calculated result of 1/Wp(c), a parameter corresponding to fatigue life. This figure can be directly compared with Fig. 3. It is found by the comparison that the characteristics (i), (iii), and (iv) in the LCF and TMF failures can be successfully estimated. It is worth noting that the features which the traditional approach failed to explain (e.g., effect of strain ratio in the LCF life, and the correlation between TMF and LCF lives) are successfully predicted in Fig. 6. The similar predicted results were seen in other strain ranges. A consideration of internal stress resulting from c/c 0 misfit may greatly contribute to this good result. However, there is a

Fig. 6. Predicted TMF and LCF lives under Demech = 1.0%.

M. Okazaki, M. Sakaguchi / International Journal of Fatigue 30 (2008) 318–323

following discrepancy in Fig. 6: the calculation predicts that the TMF life under the in-phase condition is longer than the LCF life at 900 C. This must be attributed to some assumptions employed in this work. For life prediction more exactly, it is necessary to know the material properties of c and c 0 more precisely. It is also necessary to extend the present model so that time dependence phenomena, or creep or relaxation, can be taken into account. 5. Conclusions It was shown from the experiments that a single crystal Ni-based superalloy, CMSX-4, revealed the unique TMF and LCF lives which were not always interpreted well by the traditional macroscopic mechanical parameters and approaches. New micromechanics model was proposed to estimate the TMF and LCF lives, taking into account the c/c 0 composite microstructure in superalloys. The proposed method enabled us to estimate some unique characteristics in the TMF and LCF failure, although some quantitative hurdles should be overcome further. Acknowledgements Financial supports by the Ministry of Education, Japan, as Grant-in-Aid for Scientific Research (Nos. 1705454 and 15360046) are greatly acknowledged. A support from the Nano-Coating Project (directed by Prof. T. Yoshida, The University of Tokyo) by NEDO is also acknowledged.

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