Fretting Fatigue Life Estimations Based on Fretting Mechanisms

Tribology International 44 (2011) 1389–1393

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Tribology International

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Fretting fatigue life estimations based on fretting mechanisms

Toshio Hattori a,n, Vu Trung Kien b, Minoru Yamashita c a Department of Mechanical and System Engineering, Gifu University, Gifu, Japan b Graduate School of Engineering, Gifu University, Gifu, Japan c Department of Mechanical and System Engineering, Gifu University, Gifu, Japan article info abstract

Article history: Generally the fretting fatigue S–N curve has two regions: one is the high cycle (low stress) region and the Received 14 May 2010 second is the low cycle (high stress) region. In a previous paper we introduced the fretting fatigue life Received in revised form estimation methods in high cycle region by considering the wear process; with this estimation method 20 October 2010 the fretting fatigue limit can be estimated to be the crack initiation limit at the contact edge. In this paper Accepted 21 October 2010 we estimate the low cycle fretting fatigue life based on a new critical distance theory, modiﬁed for a high Available online 14 January 2011 stress region using ultimate tensile strength sB and fracture toughness KIC. The critical distance for Keywords: estimating low cycle fretting fatigue strength was calculated by interpolation of the critical distance on Stress singularity the fretting fatigue limit (estimated from sw0 and DKth) with critical distance on static strength (estimated High cycle fatigue from s and K ). By unifying this low cycle fretting fatigue life estimation method with the high cycle Low cycle fatigue B IC fretting fatigue life estimation method, which was presented in the previous paper, we can estimate the Critical distance theory total fretting life easily. And to conﬁrm the availability of this estimation method we perform the fretting fatigue test using Ni–Mo–V steel. & 2010 Elsevier Ltd. All rights reserved.

1. Introduction during the 1970s as a result of fretting fatigue cracking as shown in Fig. 1 [14]. In this case the number of loading cycles in just one year Fretting can occur when a pair of structural elements are in contact was about 1.6 109, and this problem was observed after many years under a normal load while cyclic stress and relative displacement are of operation. This very-high-cycle fatigue life cannot be explained forced along the contact surface. This condition can be seen in bolted using only initial stress analysis results. Then in a previous paper we or riveted joints [1,2], in shrink-fitted shafts [3,4], in the blade dovetail presented the very-high-cycle fatigue life estimation method con- region of turbo machinery [5,6], etc. During fretting the fatigue sidering fretting wear [15] and change in the contact surface [16,17]. strength decreases to less than one-third of that without fretting [7,8]. In the case of designing rotational machinery such as turbo The strength is reduced because of concentrations of contact stresses, machinery, we must estimate the low cycle fatigue strength or the such as contact pressure and tangential stress at the contact edge, life of the blade/disk that connects structures in the start/stop process where fretting fatigue cracks initiate and propagate. (as shown in Fig. 2). Here, in this paper we introduced the low cycle This concentration of stress can be calculated using the finite fatigue life method using the critical distance theory, and finally we element method [9] or the boundary element method. Methods of can estimate the S–N curve of the entire process from low cycle fatigue estimation of the strength of fretting fatigue have been developed, to very high cycle fatigue. To confirm the availability of this estimation which use the values of this stress concentration on a contact surface method we perform the fretting fatigue test using Ni–Mo–V steel. [3,5]. However, the stress fields near the contact edges show singularity behavior, where the stresses at contact edges are infinite. Thus, maximum stresses cannot be used to evaluate fretting fatigue strength. 2. Fretting fatigue process So, in our previous papers we presented the fretting crack initiation estimation method, using stress singularity parameters at In our previous paper [16,17] we presented a fretting fatigue contact edges [10,11,13], and fretting fatigue limit or life estimation process model as illustrated in Fig. 3. Cracking due to fretting methods using fracture mechanics [7,12,13]. Using these fretting fatigue starts very early in fretting fatigue life. We used stress fatigue strength or life estimation methods we could not predict singularity parameters at the contact edge to estimate the initiation the super-high-cycle fretting fatigue problems in the industrial field. of these cracks [10,11,13]. During this early period, fretting fatigue For instance, a 660 MW turbo-generator rotor failed in England cracks tend to close and propagate very slowly especially in a low stress amplitude range due to the high contact pressure acting near n Corresponding author. Tel.: +58 293 2503. this contact edge. But wear on the contact surface reduces the E-mail address: [email protected] (T. Hattori). contact pressure near the contact edge, and cracks gradually start

Pressure

Crack initiation

K Stress singularity Stress 0 parameters H,λ

σ 0

Pressure

Wear extension Wear Stress K Fig. 1. Fretting fatigue failure example in turbo-generator rotor. After Lindley and Nix 0 Archard’s eq. (1991) [14]. W=A x p x s σ 0

Pressure

Crack propagation

Stress K Fracture mechanics Δ m 0 ΔK da/dN=C ( K) σ 0

Fig. 3. Fretting fatigue mechanisms of various processes.

Structural and load condition

Fig. 2. Assembled gas-turbine compressor rotor and blade dovetail joint. Surface condition Surface modification to propagate. Hence, the fretting fatigue life in the low stress Stress analysis amplitude range will be dominated by the propagation of these small cracks that initiate at the contact edge. To estimate the Fracture mechanics analysis fretting fatigue strength or life in this low stress region, the precise estimation of the fretting wear progress is indispensable. The Wear analysis propagation life of a long crack length region can be estimated Δ K > Δ K using fracture mechanics. In our previous paper [16,17] we TH discussed the method of estimation of wear extension on contact surfaces near the contact edge, and presented the fretting fatigue Fretting fatigue strength and life crack propagation estimation method considering the fretting wear extension process. Fig. 4. Flow chart of fretting fatigue life analysis. On the other hand, in the case when the stress range is high, the crack initiation will lead to failure easily without wear. In this paper This fretting wear amount is estimated using the classic Archard’s we estimate the fretting fatigue life in this high stress range using equation as follows [15]: the critical distance theory. W ¼ KPS ð1Þ where W is the wear depth, K the wear coefficient, P is the contact 3. Fretting fatigue life analysis considering fretting wear in pressure and S the accumulative slippage. high cycle region In this equation the slippage S was calculated from the deformation of each node on the contact surface, and the frictional In Fig. 4 the flow of fretting fatigue life analysis, considering the coefficient on this contact surface was set at 0.7. extension of fretting wear, is shown. Firstly the fretting wear Then, the fretting fatigue life under each loading condition, amount is estimated using contact pressure and relative slippage considering the wear process, can be estimated by comparing the under each loading condition [16,17], and then the shapes of operating stress intensity factor range DK with the threshold stress contact surfaces are modified following the fretting wear amount. intensity factor range DKth. If the operating DK is higher than the T. Hattori et al. / Tribology International 44 (2011) 1389–1393 1391

threshold stress intensity factor range DKth we can estimate this threshold conditions we can estimate the fretting fatigue S–N load cycle to be a fretting life, and if the operating DK is lower than curve considering the wear process. the threshold stress intensity factor range DKth the fretting wear amount will be calculated using a new contact pressure and a new 4. Fatigue life analysis using critical distance theory in low relative slippage; these processes will be repeated until operating cycle region DK reaches the threshold stress intensity factor range DKth. In these comparisons, the threshold stress intensity factor ranges were Even under the fretting conditions, the fretting wear will be estimated considering the crack length and stress ratio, derived in neglected in a high stress region. In these cases the fatigue life will the previous paper [7,12,13]. By connecting these fretting be estimated to be the crack initiation condition. In this paper we also estimated the fretting fatigue life using the critical distance stress theory (point method and line method). In this method the Δ Kth fatigue strength limit can be obtained from typical material σ σ (r) = 2r strength parameters, such as the fatigue limit of smooth specimens

sw0 and the threshold stress intensity factor range DKth of the cracked specimens as shown in Figs. 5 and 6. In the case of the point σw0 method, the fatigue failure is supposed to occur when the stress

range at speciﬁc length rP from the maximum stress point reaches Dsw0, and in the case of line method the fatigue failure is supposed to occur when the mean stress range between the maximum stress rP r point and speciﬁc length point r reaches Ds (¼s 2); r and r crack L w0 w0 P L can be derived as follows:

Fig. 5. Derivation of critical distance r . 2 P For point method, rP ¼ðDKth=DswoÞ =2p ð2Þ

2 ∆Κ And for line method, rL ¼ 2ðDKth=DswoÞ =p ð3Þ σ σ (r) = th 2πr S2 S1 In this paper we extended this method to the low cycle fatigue regions. First the critical distance in the low cycle fatigue region is derived through the interpolation between the critical distance in the fatigue limit and the critical distance in static strength. This static strength critical distance can be derived using ultimate

strength of smooth specimen sB and the fracture toughness KIC of the cracked specimen as shown in Eqs. (4) and (5): crack rL r 2 For point method, rPu ¼ðKIC=sBÞ =2p ð4Þ

2 Fig. 6. Derivation of critical distance rL (point method and line method). And for line method, rLu ¼ 2ðKIC=sBÞ =p ð5Þ

a S-N Curve of pane specimen σ

= 2 σ B σ

Δ σ

σ w0 Specific distance and stress σ σ w0 Kth B Estimated cycles to failure

2 Stress amplitude Δ K K 100 102 104 ……….. 107 1 th σ IC Δσ Number of cycles to failure Nf 2 w0 σ Crack propagation rate w0 σ B KIC Stress distributions Δ of objectives KIC Kth Δ K 2 Δ 2 Δ K 2 1 th 2 1 KIC 1 IC σ 2 Δσ 2 B 2 σB w0 Distancer

(Δσw0 = 2σw0)

Δ Kth

Crack propagation rate da/dN Stress intensity factor range Δ K

Fig. 7. Derivation of speciﬁc distance in low cycle fatigue region and estimation of low cycle fatigue life. 1392 T. Hattori et al. / Tribology International 44 (2011) 1389–1393

S-N Curve of plane specimen Ni-Mo-V Steel 1000 σ B=705MPa (Mpa) a σ 500 σ =360MPa W0 Specific distance and stress σ W0 Kth σ Δ 2 B Estimated cycle to failure 2 Kth K π Δσ IC 100 w0 (Mpa)

Stress amplitude 3 4 5 6 7 8 10 10 10 10 10 10 σ σ w0 Number of cycles to failure N σ K σ =200MPa f B IC Δ a Kth

Stress Stress distributions Crack propagation rate 2 2 in fretting model 2 KIC π σ B Distance r (mm)

Δσ σ ( w0 = 2 w0)

Fig. 8. Derivation of speciﬁc distance in low cycle fatigue region and estimation of low cycle fretting fatigue life.

In this section we illustrate the approach using the point method. The critical distance at each stress level is calculated by interpolation of critical distance on fretting fatigue limit (rP, estimated from sw0 and DKth) with critical distance on static 0 strength (rP , estimated from sB and KIC ) as shown by chain line in Fig. 7(right). The critical distance under objective conditions (structure, load) can be estimated by reﬂecting the stress distributions of the objective structure as shown by dotted line in Fig. 7(right). The low cycle fatigue life in this objective condition can be estimated by applying this reference stress s at critical distance r on S–N curve of smooth specimens as shown in Fig. 7(left Fig. 9. FEM fretting model. upper).

shown in Fig. 10. The mean contact pressure sp and mean axial stress sa in this case are 200 and 100 MPa, respectively. 5. Application on low cycle fretting fatigue life analysis The critical distance r on each loading condition can be estimated by reﬂecting these stress distributions in Fig. 8(right) The extended critical distance theory will then be applied in the as shown by the dotted line. As an example we can estimate the fretting fatigue life prediction. In Fig. 8(left upper side) the S–N critical distance r just 2 times higher than that of loading condition curve of the Ni–Mo–V steel smooth specimen under complete shown in Fig. 10. As a cross point of stress distribution (dotted line) reversed loading conditions (R¼1), and in Fig. 8(left lower) the with solid line, r and s can be derived as 0.12 mm and 490 MPa, crack propagation characteristics of the cracked specimen are respectively. The low cycle fretting fatigue life in the loading shown. From these material characteristics we found that the condition, with nominal axial stress sa 200 MPa, can then be 0 critical distance rP is 0.011 mm and rP is 2.13 mm as shown in estimated by applying the stress (s¼490 MPa) on the S–N curve of Fig. 8(right). The stress distributions under the fretting conditions smooth specimens as shown in Fig. 8(left upper). By connecting were calculated using the FEM model as shown in Fig. 9. The these fretting fatigue life at each stress level we can estimate the calculated example of stress distribution near the contact edge is fretting fatigue S–N curve as shown in solid line in Fig. 12. T. Hattori et al. / Tribology International 44 (2011) 1389–1393 1393

fretting fatigue life using the critical distance theory are shown by the dotted line in Fig. 12. The estimated results of high cycle fretting fatigue life considering fretting wear process, which was presented in previous paper [16,17], is shown by the dashed line in Fig. 12. The estimated fretting fatigue limit (142 MPa) without considering fretting wear, which was presented in previous paper [12,13],is shown by two points on the dot–dash line in Fig. 12. We can see that these three kinds of fretting fatigue strength and life prediction results coincided well with the experimental results at each stress and life level, and we can conﬁrm the validity of these fretting Fig. 10. Calculated result of stress distributions. fatigue strength and life estimation methods.

7. Conclusions

1. Low cycle fretting fatigue strength was estimated using critical distance theory. 2. These fretting fatigue strength and life estimated results coincided well with the fretting fatigue test results at each stress and life level, and we can conﬁrm the availability of these fretting fatigue strength and life estimation methods through the standardized fretting fatigue S–N curve estimation method.

References

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