RA-based fretting fatigue life prediction method of Ni-based single crystal superalloys

Tribology International 134 (2019) 109–117

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RA-based fretting fatigue life prediction method of Ni-based single crystal superalloys

T

Shouyi Sun, Lei Li∗, Weizhu Yang, Zhufeng Yue, Huan Wan P.O. BOX 883m, Department of Engineering Mechanics, Northwestern Polytechnical University (Chang'an Campus), Xi'an, 710129, PR China

ARTICLE INFO

ABSTRACT

Keywords: Ni-based single crystal (NBSX) Fretting fatigue Resolved shear stress Accumulated dissipated energy Finite element analysis

Fretting fatigue damage can significantly reduce the service life of Ni-based single crystal (NBSX) superalloys turbine blades. In this work, we proposed a fretting fatigue life prediction method by considering cracking, wear and their mutual interaction. Specifically, a parameter, denoted as RA, is developed with combination of Resolved shear stress based damage factor and Accumulated dissipated energy based damage factor, by which the contribution from cracking and wear are described respectively. The proposed method is validated with experimental data in the literature. Finite element analyses are performed to study the distribution of stresses and strains. Results show that the predicted crack initiation site, failure plane and life cycles are consistent with the experimental results.

1. Introduction Ni-based single crystal (NBSX) superalloys are favored by aero engine designers for its superior creep and thermal-mechanical fatigue capabilities [1]. Typically, NBSX turbine blades are fixed with the disk through tenon/mortise structure. However, this kind of structure has been found to be highly susceptible to fretting fatigue due to the relatively small slip between the contact surfaces, which results in a substantial increase in maintenance costs and a significant reduction in engine life. In addition, cracking and wear occur simultaneously during the fretting process, making it difficult to characterize the fretting fatigue damage. Therefore, it is crucial to study the fretting fatigue life prediction method of NBSX superalloys. Extensive studies have been conducted on the fretting fatigue behavior of different materials over the past decades [2,3]. It has been recognized that fretting fatigue crack initiation and propagation are affected by as many as 50 different factors [4,5], the key influencing factors are slip amplitude [4], tangential force [6], coefficient of friction (COF) [7], etc. Generally, slip amplitude is considered as one of the primary factors, since it may results in three different fretting conditions depending on its magnitude, i.e. full stick, partial slip and gross sliding [8–10], but the evolution of fretting damage will exhibit quite different characteristics under these three conditions [11,12]. Fretting fatigue process is usually accompanied with cracking and wear. However, in most studies, cracking and wear are studied separately with main focus on the effect of cracking on fretting fatigue [13–15], and

∗

wear damage is often neglected since cracking is assumed to be the predominant damaging factor [16]. In practice, the contributions of cracking and wear to fretting fatigue damage vary under different fretting conditions. Specifically, larger wear volume and higher energy dissipation are observed when the material is subjected to gross sliding condition, whereas the partial slip condition is accompanied by more cracking but less wear, and the damage is smaller in the full stick condition than the other two conditions [7,17–19]. Overall, both cracking and wear will contribute to the fretting fatigue damage. Hence, a combined consideration of cracking and wear is necessary to describe fretting fatigue damage more reasonably. So far, most previous studies were focused on polycrystalline alloys, such as Ti-6Al-4V [7,20], Al7075-T651 [14] and AISI 1034 [21]. Meanwhile, quantitative characterization of fretting fatigue damage and life prediction of fretting fatigue also arouse broad concern [22–24]. Two main methods are frequently used to evaluate fretting fatigue damage and predict fretting fatigue life, i.e. multiaxial fatigue criterion [25] and fracture mechanics approach [26]. To consider the complex stress fields near the fretting contact region, multiaxial fatigue criterion based on critical plane method has been favored for its ability to predict fatigue failure plane and fatigue crack initiation life [27]. In general, the multiaxial fatigue criteria can be divided into three groups according to different damage mechanisms: (i) stress based criteria, (ii) strain based criteria and (iii) strain energy based criteria. Based on the assumption that the fretting fatigue crack initiation was governed by maximum shear stress, Lykins et al. [28] proposed the maximum shear

Corresponding author. E-mail address: [email protected] (L. Li).

https://doi.org/10.1016/j.triboint.2019.01.036 Received 19 September 2018; Received in revised form 4 January 2019; Accepted 25 January 2019 Available online 28 January 2019 0301-679X/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

v G K P Q µ

b Burgers vector Edi Dissipated energy during step i RA Fretting damage parameter DR Resolved shear stress based damage factor DE Accumulated dissipated energy based damage factor ai (i = 1,2,3) Coefficients of fretting damage parameter Resolved shear stress on crystal slip system Yield stress s m Material coefficient qi (x ) Local shear stress of step i Local incremental relative slip of step i i (x ) G Energy release rate N Predicted fretting fatigue life in cycles A Fretting fatigue life curve fitting parameter c Fretting fatigue life curve fitting parameter E Elastic modulus

Poisson ratio Shear modulus Fracture toughness Normal contact load Tangential load Frictional coefficient Applied fatigue load Initial contact half width Actual contact half width Position along contact surface Composite stiffness Stress tensor in the crystallographic axial system Schmidt tensor Unit vector of the slip direction in the slip system Unit normal vector of the slip plane Contact pressure Shear stress

bulk

a0 a x E

P m( ) n( ) p (x ) q (x )

stress range (MSSR) criterion to predict crack initiation life and location. However, Navarro [14] considered that the fretting fatigue cracks initiated in shear mode, and applied a strain based criterion, referred to as Fatemi-Socie (FS) criterion, to analyze the crack initiation behavior. Besides, Szolwinski [29] predicted fretting fatigue crack nucleation by using Smith-Watson-Topper (SWT) criterion, which links the cyclic normal stress and strain to the number of cycles required for the initiation of a crack in the critical plane. It can be concluded that all criteria are proposed or chosen based on the failure mode of the material. While the fretting failure mode of NBSX superalloys is distinct that raise difficulties for fretting fatigue life prediction. Different from the above polycrystalline alloys, the fretting fatigue damage of NBSX is closely related to the crystal orientation with slip characteristics due to the unique lattice structure. Many experimental and numerical results indicated that the fretting fatigue cracks of NBSX were found to initiate along the direction of crystal slip lines [15], and the fracture often occurred along the {111} plane during the fretting fatigue process [30–32]. Besides, Huang [33] found that the crack initiated in the direction perpendicular to the contact surface and then grow along the {111} plane. In addition, the fretting direction was considered to have great influence on fretting fatigue behavior, it was found that fretting in the < 110 > direction appears to be more resistant to crack formation [33]. To predict the fretting fatigue life with respect to NBSX superalloys, previously proposed damage criteria were analyzed and new damage criteria were presented recently. Comparison of several existing criteria indicates that Findley parameter outperforms others in the prediction of nucleation life of NBSX superalloys

[31]. Besides, Shi [34] thought that fretting damage was attributed to material properties, fatigue loads, contact loads, etc., and a strain energy based damage parameter was proposed by considering contact width, equivalent stress and slip amplitude, the predicted lives were distributed within 2.8 error bands. However, the correlation between cracking and wear is unclear and the overall damage is difficult to characterize. Besides, the slip characteristics of NBSX superalloys is not considered in the existing life prediction methods. In this paper, we focus on the damage evolution of NBSX superalloys during the fretting fatigue process. As the cracking mechanism of NBSX is mainly sliding along the crystal slip system, the resolve shear stress is usually considered to be the driving force in the crystal slip process. Besides, energy dissipation occurs due to the existence of tangential force and relative slip, which is the key characterization of wear damage. Hence, to estimate the fretting fatigue life of NBSX superalloys, we propose a fretting fatigue damage parameter that combines resolved shear stress based damage factor and accumulated dissipated energy based damage factor, referred to as RA. Subsequently, a series of finite element analyses are conducted to obtain RA by using a simplified model with a flat-rounded pad pressed on a flat substrate, and the fretting fatigue life can be determined via a log-linear function for a given RA. The estimated crack initiation lifetime and orientation are validated by using previous fretting fatigue experimental data [34,35]. The contributions of cracking and wear to the overall damage are analyzed for different fretting conditions. Finally, different damage criteria are compared to show the advantage of the proposed parameter RA. Fig. 1. A contact pair with a rounded edge. Dislocation occurs at the contact edge. (A hollow circle represents an atom; the dashed red line indicates initial connecting state between atoms before dislocation occurs). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Dislocation movement Crack Cra r ck Fretting direction Pad b

Specimen

Contact region

110

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2. Fretting fatigue life prediction model

cracks and promote the formation of new wear debris. Therefore, slight wear can mitigate the threat of crack propagation and increase fretting fatigue life. Nevertheless, a larger slip amplitude will cause excessive wear, and cracks are more likely to initiate and propagate (Fig. 2 (c)), Due to the large slip amplitude between the contact surfaces, the local temperature of the contact region will be high, that reduces the fatigue resistance of the material, and thus the life of the structure is rapidly reduced. Generally, the effect of wear on fretting fatigue is relevant to many variables, such as contact pressure, slip amplitude, COF, etc. Besides, the wear-induced pressure redistribution have a great influence on the structural damage [4,20]. To simulate the wear process, Archard model [40] and energy model [41] are usually used, especially in combination with the finite element method [42,43]. Dissipated energy Edi is directly or indirectly included in both models, which plays a key role in describing fretting fatigue damage. Energy model is favored by many researchers since it can easily calculate the dissipated energy of different contact regions [43]. Dissipated energy Edi in energy model is defined as follows:

2.1. Fretting fatigue damage mechanism of NBSX superalloys Different from polycrystalline alloys, the fatigue fracture mechanism of NBSX superalloys is mainly crystallographic dislocation slip at temperatures below 750 °C. As mentioned above, the fretting fatigue damage of NBSX superalloys exhibits the similar characteristics related to the crystal orientation. During the fretting fatigue process, stress concentration occurs at the contact edge, resulting in large RSS, which leads to the occurrence of dislocations. Fretting fatigue failure plane is closely related to the crystal slip systems. Therefore, the RSS on the crystal slip system is considered to be a key factor in controlling the fretting fatigue damage [32]. Once the resolved shear stress reaches a critical value, dislocations will occur and then slide along the slip plane to the free surface [15], as shown in Fig. 1. As the dislocations accumulate, the crack will initiate and propagate. It has also been shown in various studies that cracks usually initiate at the contact edge and grow along the crystal slip plane [30,31,33]. RSS, as the surface traction component in the slip plane, is considered to be critical for the fatigue life of NBSX superalloys [36,37]. Arakere used the maximum magnitude of RSS to characterize the fatigue damage, which is in good agreement with the test data of NBSX superalloys [38]. Levkovitch established the relationship between RSS and the accumulated damage of each slip system to predict fatigue life, wherein the functional form is adopted from the Satoh and Krempl equation [39]. In addition, Matlik provided another insight into the crack location and orientation by utilizing the range of amplitudes of RSS and normal stress to describe fretting damage [32]. Hence, RSS is supposed to be an essential factor in characterizing the fretting fatigue damage of NBSX superalloys. Apart from RSS, fretting fatigue is accompanied by fretting wear, however, the effect of wear on fretting damage was often neglected. Many studies have shown that wear affects fretting fatigue damage and is therefore important for predicting the fretting fatigue life [20]. Vingsbo et al. studied the relationship among slip amplitude, wear and fretting fatigue life [11], and concluded that the fretting fatigue damage mechanism varies with the slip amplitude. When the slip amplitude is small (i.e. partial slip condition), there is nearly no wear in the contact surface (Fig. 2(a)), and the high stress at the contact edge will promote crack initiation. As the slip amplitude increases (Fig. 2 (b)), wear will occur at the contact surface, especially at the contact edge where cracks are likely to initiate. However, this slight wear helps to eliminate micro

(1)

Edi = qi (x ) i (x ), where qi (x ) is the local shear stress, and relative slip.,

i (x )

is the local incremental

2.2. Fretting fatigue life prediction model for NBSX superalloys As discussed in Section 2.1, to describe the fretting fatigue damage of NBSX superalloys, the damage parameter, denoted as RA, can be defined by considering both RSS and ADE, where RSS characterizes crack initiation and early propagation along the crystal slip system and ADE describes the surface wear damage, including the generation of surface damage and elimination of micro crack. Furthermore, the interaction between cracking and wear can be indicated by an additional cross term. Finally, RA can be expressed as

RA = a1 DR2 + a2 DE (DE

a3 DR ),

(2)

where ai (i = 1,2,3) (ai > 0 ) are coefficients that can be determined by fitting experimental data. DR and DE are dimensionless damage factor based on RSS and ADE, respectively. The expression of DR is given by

DR = max

1 s

min

m

max

,

(3)

the value of which is different for different crystal slip systems. In Eq.

Fig. 2. Fretting fatigue damage for different slip amplitudes. 111

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(3), is the RSS on crystal slip system , min and max are the minimum and maximum values of during the loading process, and the ratio of min to max represents the fatigue effects. s is the yield stress of the material, m is a coefficient which can be obtained through experimental fitting. DR is different for different loading condition, and the bigger DR indicates that the components withstand more severe damage. ADE is induced by the frictional work. During one fretting fatigue cycle, the ADE based damage factor DE can be written as

DE =

1 G

2

Edi = i=1

1 G

Table 1 Material parameters of DD3 along [001] crystallographic orientation.

qi (x ) i (x ),

(4)

where G denotes energy release rate. One fatigue cycle is divided into two steps, that is the fatigue load changes from the minimum value to the maximum value, and then changes back to the minimum value. In Eq. (2), a2 DE (DE a3 DR ) represents the effect of wear on fretting fatigue damage. In particular, a positive value of a2 DE (DE a3 DR ) means that wear will accelerate the damage of structure due to more severe surface damage in the contact area, while a negative value a2 DE (DE a3 DR ) means that the slight wear will eliminate the initial micro crack. It should be noted that the value of a2 DE (DE a3 DR ) depends on both the value of DR and DE , which are loading-condition dependent. Generally, the fretting fatigue life of components can be obtained via a log-linear function [31,34]:

RA = A

G (GPa)

σs(MPa)

K (MPa·m1/2)

115

0.313

137

930

94

No.

P (MPa)

Stress ratio

Life cycles

FF-1

65.45

300

0.1

65.45

400

0.1

FF-3

130.9

400

0.1

FF-4

229.07

300

0.1

77529 61450 67190 49236 35358 30546 34318 35562 30067 73204 39076 28687 102011

FF-2

bulk (MPa )

specimen. This structure has similar contact form to the simplified model of dovetail structure [34,44]. The fretting fatigue test configuration is shown in Fig. 3(a). Firstly, a constant normal load P is applied on both pads of which two sides are constrained with all degree freedoms except the normal direction, and then the periodic fatigue load bulk is applied on one end of the specimen with the other end fixed. The tangential load Q between the pads and specimen is generated due to friction. Initial contact length and fillet radius of both pads are 5 mm and 3.5 mm, respectively. The heights of 9 mm and 10 mm are used to model the pads and the specimen (x-direction), respectively. Both the specimen and the pads are 5 mm thick along the out-of-plane direction. The test material is DD3 NBSX superalloy and the specific material parameters are given in the Table 1 [45,46]. Four different loading conditions were studied and the specific test parameters are listed in

(5)

c ln N ,

v

Table 2 Loading conditions of fretting fatigue experiment.

2 i=1

E (GPa)

where A and c are fitting parameters, which can be calculated by using the least square method under given RA and N . 3. Experimental data and finite element analysis 3.1. Instruction of the experiment Here, we introduce the experiment conducted by Shi et al. [34,35] that will be used to verify the proposed RA based fretting fatigue life prediction method. In the experiment, a classical fretting fatigue model was used that consisted of flat-rounded pads pressed onto a flat test

Fig. 3. (a) Schematic of the fretting fatigue test model. (b) Loading sequence of retting fatigue test.

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Table 2. The experimental fretting fatigue life was defined as the cycles the specimen experienced when the visible crack length reaches about 1 mm, which is used to define crack initiation stage [29,47]. 3.2. Finite element analysis of the experimental model Finite element (FE) method is frequently used for its ability to obtain accurate stress and strain data for complex structures and loading conditions. Taking into account the symmetry of the geometry, contact configuration and the crystal anisotropy, a 3D axisymmetric contact model is created. The eight-node linear brick, reduced integration, hourglass control element (C3D8R) is chosen for the mesh in the contact region, as shown in Fig. 4. The general purpose, non-linear finite element analysis (FEA) code, ABAQUS, are used to solve the problem. All the numerical analyses are conducted based on the assumption of linear elasticity, which were available for NBSX superalloys contact problems [31,32,38,48]. Lagrange multiplier contact algorithm could strictly enforce the stick condition, but it is difficult to converge if many points are iterating between sticking and slipping conditions [49], hence, the penalty method is applied. The master-slave algorithm is used with surface-to-surface discretization, in which the flat-rounded surface of the pad is defined as a master surface and the top surface of the specimen acts as a slave surface. A constant COF is used to investigate the fretting fatigue behavior for all loading conditions. The value of COF used in the FEM model is set as 0.8 referred to the previous experiments [33,48,50]. Mesh refinement is performed around the contact region to accurately capture the stress and strain distribution, as shown in Fig. 4. To validate the mesh-size independence, a series of numerical calculations based on the plane strain contact model are conducted and compared with analytical results, the meshing of the plane strain contact model is coincident with the 3D model shown in Fig. 4. Ciavarella [51,52] presented that the contact pressure distribution of a flat-rounded pad pressed on an infinite flat could be expressed as:

ap ( ) = P

2/ 2

0

sin 2

× 0

(

2 0)cos

+ ln

sin( + sin(

0)

Fig. 5. Contact pressure distributions along the contact direction under different mesh sizes.

2PR = E a2 2

1 1 = (1 E E1

a0 1 a

a0 a

2

,

(8)

2 1

)+

1 (1 E2

2 2 ),

(9)

where Ei is Young's modulus and i is Poisson's ratio of body i . Fig. 5 plots the contact pressure distributions obtained from different mesh sizes, indicating that as the mesh size decreases, the calculated contact pressure is close to the analytical results. It can be seen that the contact pressures converge when the mesh size decreases to 0.02 mm and the error of peak stress is less than 5% compared with the analytical solution. The slight difference between FEA and analytical results may attribute to the finite height of the specimen in FE model. Besides, the allowable elastic slip used in penalty method is 2.24 × 10

× tan

+ 2

0

tan

0

sin 0

2

−4

where P represents the normal load applied on the top of pad. sin 0 is the ratio of the initial contact half width a0 to the actual contact halfwidth a . At a specified location x , can be obtained through Eq. (7).

,

(6)

mm when the mesh size is 0.02 mm. DD3 NBSX superalloy has three sets of slip systems that may be activated, i.e. octahedral slip systems ({111} < 110 >), cube slip systems ({100} < 110 >) and dodecahedron slip systems ({111} < 112 >). As mentioned above, the calculation of RSS on each crystal slip system is the key to obtain fretting fatigue damage. The

(7)

= arcsin(x /a)

a0 a

where E is the composite stiffness of the contact bodies defined as

sin

0)

arcsin

The actual contact half-width a is given by:

Fig. 4. Finite element model with load and boundary conditions. 113

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resolved shear stress ( )

= :

( )

can be expressed as

where is stress tensor in the crystallographic axial system. P Schmidt tensor defined as

P( ) = where and n(

1 ( ) ( (m n 2

)T

+ n( ) m(

) T ),

is

(11)

is the unit vector of the slip direction in the slip system is the unit normal vector of the slip plane.

m( ) )

the model are extracted from the FEA results to calculate RSS on each slip system. Subsequently, the value of DR is calculated by Eq. (3). Besides, the accumulated dissipated energy damage factor DE during one fretting fatigue cycle can be calculated through an incremental accumulation method. Then the value of RA can then be obtained by Eq. (2). Given RA, the fretting fatigue initiation life can be predicted via Eq. (5). The crack initiates at the node where the maximum RA occurs, and the corresponding DR determines which slip system would be activated. By means of the least square method, the coefficients of the fretting fatigue damage parameter could be fitted by experimental data. The detailed fitting process is as shown in Fig. 8. First of all, the relationship between fatigue life and damage parameter could be deduced according to Eq. (2) and Eq. (5), as shown in Eq. (12):

(10)

P ( ),

,

4. Results and discussion 4.1. Contact traction analysis The fretting condition can be determined by comparing the shear stress q (x ) and theoretical friction stress µp (x ) . When q (x ) < µp (x ) at a node, it indicates that the area around the node is in stick condition. Otherwise, relative slip will occur between the master surface and the slave surface, resulting in energy dissipation. Fig. 6 shows the distribution of shear stress q (x ) and theoretical friction stress µp (x ) of test FF-1 along the contact surface. Obviously, as the fatigue load changes, the fretting condition also changes. When the bulk load is increased from 30 MPa to 300 MPa, the fretting condition changes from full stick to gross slip. It is noted that when the fatigue load is only 84 MPa, the fretting condition changes to gross slip, as shown in Fig. 6 (c). This indicates that the full stick stage only accounts for a small fraction of the fretting fatigue process. In addition, as the fatigue load changes, the change of shear stress at the right contact edge is more pronounced at the left.

c ln N = A

a1 DR2

a2 DE (DE

a3 DR )

(12)

where c is assumed to be 1. Selecting the initial maximum damage position where the maximum DR occurred, then the coefficients of the damage parameter could be obtained by using the least square method. Based on the calculated coefficient, the maximum damage position would be reselected as the position at which the RA was maximum, and the new coefficients will then be re-fitted. When the relative fitting errors of the coefficients between the ith fitted coefficients and the (i + 1)th fitting coefficients are less than 1.0E-6, the fitting process will A= finish. The obtained coefficients are: 11.5260, a1 = 0.2789, a2 = 0.5650, a3 = 0.9088. 4.3. Predicting results by RA Table 3 lists the values of DR , DE and RA of four loading conditions, and the activated slip systems are given in the last column. The results show that for all cases, the maximum RA occurs at the right contact edge, but the crack initiation plane is slightly different for each loading condition. The fretting fatigue damage parameters under different loading conditions are shown in Fig. 9. The maximum RA appears in the

4.2. Calculation of RA On the basis of the FE method, RA can be calculated through the procedure given by Fig. 7. Firstly, the numerical model of fretting fatigue is established and solved by using FEA. The stresses and strains of

Fig. 6. Distribution of shear stress and friction stress on the contact surface for FF-1: (a) bulk load = 30 MPa; (b) bulk load = 57 MPa; (c) bulk load = 84 MPa; (b) bulk load = 300 MPa. 114

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Fig. 7. Flow chart of fretting fatigue life prediction model.

Fig. 9. The value of RA on different crystal slip systems. Fig. 8. The flow chart of damage parameter coefficients fitting.

SEM analyses for FF-1, FF-3 and FF-4, the morphologies of fretting fatigue crack initiation region are consistent with the predicted results. As mentioned earlier, there is a competitive mechanism between cracking and wear. On the one hand, wear accelerates the surface damage of component, on the other hand, slight wear helps to eliminate the initiated micro crack. The value of DR are almost the same for FF-1 and FF-2, while the bigger DE of FF-2 accelerate the surface damage, which results in a bigger RA and lower fretting fatigue life of FF-2. Besides, it can be observed that the value of DR for FF-4 is nearly five times that for FF-1, but the value of DE for FF-1 is larger than FF-4. Since an appropriate DE will help to reduce the material damage, we can obtain approximately equal RA for these two loading conditions. The performance of the proposed RA is compared with other five popular damage parameters, i.e. SSR [53], MSSR [54], FS [55], SWT [56] and RSS. The fretting fatigue lives predicted by different parameters are obtained via a log-linear function(Eq. (5)), the results are plotted in Fig. 11. The maximum values of all damage parameters occurs near the right contact edge (i.e. x / a 1), where crack initiates. Almost all predicted results lie within ± 2Ni scatter band, but the life cycles predicted by previous damage parameters are more scattered compared with life cycles predicted by RA. This is due to the wear

Table 3 Results of simulation for different loading conditions. Loading condition

DR

DE

RA

Activated slip system

FF-1 FF-2 FF-3 FF-4

0.3638 0.4073 0.5689 1.6010

0.9710 1.4776 1.6281 0.3157

0.3881 0.9707 1.1122 0.5115

[10-1](-11-1) [101](1-1-1) [10-1](-11-1) [100](011)

octahedral slip systems for FF-1, FF-2 and FF-3, that indicates the crack initiating along the octahedral slip plane, and the predicted fracture surface is shown in Fig. 10(a). This fracture feature is consistent with previous results [30–32]. Noting that the maximum RA of FF-4 appears on the cube slip system, which is different from other three loading conditions. In this case, the crack would initiate perpendicular to the contact surface, then grow along {111} plane. This phenomenon is consistent with the Huang's results [33], as shown in Fig. 10(b). It is because that the larger contact load causes the specimen undergo greater deformation on the {001} slip plane. Shi [35] had conducted 115

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Fig. 10. Schematic of fracture surface of fretting fatigue: (a) crack initiates along {111} slip plane; (b) crack initiates along {001} slip planes.

considered in RA through a2 DE (DE a3 DR ) . Besides, DR can be utilized to predict which crystal slip system to activate. The value of RA reflects the structural damage, and can be used to predict fretting fatigue life of NBSX superalloys via a log-linear function. Finite element analysis is conducted to calculate the stresses and strains of a flat-rounded contact model. Mesh size of 0.02 mm is applied near the contact region to ensure accurate stresses and strains are obtained. Then, DR and DE are calculated for each loading condition, the related constants ai (i = 1,2,3) in Eq. (2) are obtained through an iterative analysis. The fretting fatigue life predicted by RA and other 5 popular damage parameters are compared against the experimental data, the results indicate that RA performs the best among all the studied parameters. It is found that fretting fatigue cracks tend to initiate along the octahedral slip systems under most loading conditions, whereas for FF-4, the crack tends to initiate along the cube slip systems and then turns to the octahedral slip systems due to the effect of wear and wear-cracking interaction. With comprehensive consideration of both wear and cracking, the RA based method is an effective tool for predicting fretting fatigue behavior of NBSX superalloys. Moreover, plasticity and deformation hardening have influence on the fretting fatigue behavior [57], relevant research will be carried out in the future.

Fig. 11. Comparison between experimental and predicted life.

effects on fretting fatigue life are not considered, which results in poor predictions. The normal load of FF-4 is 229.07 MPa, which is the greatest among all of the four loading conditions. Based on the numerical calculation, we obtain that the maximum resolved shear stress damage factor DR appear at the contact edge, and the maximum value of DR is 1.6010 which is much higher than other three loading conditions. Due to the large DR , there may be multiple micro cracks randomly distributed at the contact edge. However, considering the random distribution of micro cracks, some of the micro cracks could be eliminated and others will continue to grow. So, it is assumed that the elimination of crack affected the fretting fatigue life of the specimen. When only a part of the micro cracks of the specimen were eliminated by wear, the specimen will experience a relatively short fretting fatigue life. Otherwise, the life cycles will be longer if more micro cracks were eliminated. Hence, the experimental life cycles show a large scatter.

Acknowledgement National Natural Science Foundation of China(Grant No. 51575444), Aviation Power Foundation of China (Grant No. 6141B090319) and Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018JM5173) support this work. References [1] Yi JZ, Torbet CJ, Feng Q, Pollock TM, Jones JW. Ultrasonic fatigue of a single crystal Ni-base superalloy at 1000 degrees C. Mat Sci Eng a-Struct 2007;443:142–9. [2] Li X, Yang JW, Li MH, Zuo ZX. An investigation on fretting fatigue mechanism under complex cyclic loading conditions. Int J Fatig 2016;88:227–35. [3] Xue F, Wang ZX, Zhao WS, Zhang XL, Qu BP, Wei L. Fretting fatigue crack analysis of the turbine blade from nuclear power plant. Eng Fail Anal 2014;44:299–305. [4] Madge JJ, Leen SB, Mccoll IR, Shipway PH. Contact-evolution based prediction of fretting fatigue life: effect of slip amplitude. Wear 2007;262:1159–70. [5] Tobi ALM, Ding J, Bandak G, Leen SB, Shipway PH. A study on the interaction between fretting wear and cyclic plasticity for Ti-6Al-4V. Wear 2009;267:270–82. [6] Jin O, Mall S. Shear force effects on fretting fatigue behavior of Ti-6Al-4V. Metall Mater Trans A 2004;35A:131–8. [7] Fouvry S, Duo P, Perruchaut P. A quantitative approach of Ti-6Al-4V fretting damage: friction, wear and crack nucleation. Wear 2004;257:916–29. [8] Jin O, Mall S. Effects of slip on fretting behavior: experiments and analyses. Wear 2004;256:671–84. [9] Jin O, Mall S. Effects of independent pad displacement on fretting fatigue behavior of Ti-6Al-4V. Wear 2002;253:585–96. [10] Vingsbo O, Odfalk M, Shen NE. Fretting maps and fretting behavior of some F.C.C.

5. Conclusion and future work Slipping along the crystal slip system and wear are the common feature of fretting fatigue for NBSX superalloys. In our study, the RAbased fretting fatigue damage parameter is developed by considering the common failure mechanism of NBSX superalloys, including cracking, wear, and their competitive relationship. The proposed damage parameter RA is applicable for a wide range of slip amplitude. Thereinto, DR and DE are used to describe the damage of cracking and wear, respectively, and the mutual effect between wear and cracking is 116

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