296 Effects of Damage on HCF Properties or near the edge of contact where stresses are normally at a maximum. Because of the steep stress gradients that are present in a contact region, parameters that are used for notch fatigue (see Chapter 5) should also have merit in fretting-fatigue life prediction. Parameters used for notches such as a modified SWT, Walker equivalent stress, modified shear stress range, or Findley parameter have been tried in an attempt to consolidate data from different fretting-fatigue experiments or from a range of conditions in a single experiment. These parameters are either unidirectional, representing a maximum stress in a direction normally parallel to the contact surface, or take into account the stresses in different orientations with respect to the contact plane and require consideration of values on all planes in order to find the maximum value. Naboulsi and Mall [38] investigated three critical-plane-based parameters for crack initiation by computing their values over a process volume and compared the results with those obtained by averaging stress or strain over some region and then computing the parameter based on the average value. They used a modified SWT parameter on a plane where the value is maximum. The parameter, proposed in [21], had the form SWT = max a (6.18) where max is the maximum normal stress and a is the normal strain range. A second parameter, the Shear Stress Range (SSR), proposed in [39] has the form SSR = crit = max 1−R m (6.19) where max is the maximum shear stress on the critical plane where is maximum, R is the shear stress ratio on the critical plane, and m a fitting parameter obtained from smooth bar fatigue data. The third parameter used is that of Findley [40] in the form FP = +k max n (6.20) where is the shear stress amplitude on a critical plane, max n is the maximum stress normal to the maximum shear plane, and k is a fitting parameter. Numerical results were obtained based on averaging the parameter over a critical pro- cess zone and averaging the stresses over the same volume and computing the parameter based on the averaged stress. These did not differ significantly when obtained from fretting-fatigue experiments on Ti-6Al-4V using cylindrical pads. It was found, however, that the SWT parameter (based on normal stresses) had a steeper gradient than either the SSR or the FP parameters (based on shear stresses). For the SSR or the FP parameters, the values decreased less than 10% from their peak values at the surface when averaged over a process volume that extends 50m from the peak. This corresponds to approximately 4 grain sizes for this material. The use of a process volume, in this case, is somewhat dependent on the specific parameter being used if attempts are made to consolidate data with smooth bar fatigue data. Fretting Fatigue 297 To add further fidelity to a fretting-fatigue parameter, Namjoshi et al. [41] introduced a modified shear-stress-range (MSSR) parameter that contains the better aspects of the SSR parameter, equation (6.19), and the Findley parameter, equation (6.20). The MSSR parameter considers both the shear stress range on a critical plane and the normal stress on that same plane. The critical plane is defined as the one where the MSSR parameter has its maximum value. The MSSR parameter has the form MSSR =A B crit +C D max (6.21) where crit is as defined in Equation (6.19) and represents the maximum shear stress range on any plane, and max is the maximum stress in the direction normal to the critical plane. It is of interest to note that the coefficients B and D have been chosen as 0.5 resulting in a parameter that is less sensitive to stress amplitude than one where stress is not raised to a power. ∗ In [41], this MSSR parameter was found to be able to represent data better than the earlier SSR parameter by including effects of normal stresses on a critical plane. While it will be shown later that crack propagation can also be an important part of fretting-fatigue life prediction, even in the HCF regime, critical plane- based initiation parameters such as MSSR have been shown to be able to predict both the location of crack initiation as well as the angle of inclination of the crack formed at the surface [41, 42]. Neu et al. [42] point out that although fretting-fatigue life involves both crack initiation as well as crack propagation, the formation of a crack (initiation) is often desirable to avoid in design of complex structures. In a series of experiments on PH 13-8 Mo stainless steel, they observed that maximum shear strain amplitude predicted the location and orientation of the nucleated crack, but was unable to separate differences in life due to mean stress. Several other parameters investigated showed deficiencies in predicting all aspects of crack nucleation. For their material and test conditions, the Fatemi-Socie parameter (see below), Equation (6.22), was found to be the only parameter that predicted both the location and the orientation of a nucleated crack and captured the correct trend in life with change in mean stress. While many theories exist for fretting-fatigue crack nucleation, and some are able to consolidate data from selected experiments on specific materials, most of them fall short of quantitatively predicting the cycles to crack nucleation, however that condition is defined [21]. “Since the fretting damage process involves a myriad of interacting mechanical and chemical phenomena – wear, corrosion and fatigue – the notion of an all-inclusive model for fretting fatigue crack nucleation borders on the absurd.” [21]. For this and other reasons, attention has been directed toward the role that crack propagation plays in determining total life under fretting-fatigue conditions. ∗ See the discussion in Chapter 8 under Section 8.1, “Factors of Safety” for the implications of using a stress-based parameter that is in units of stress raised to a power. 298 Effects of Damage on HCF Properties Of particular concern in developing a fatigue initiation model is the combined effect of a local biaxial stress field with the steep stress gradients typically found in the contact region where cracks are likely to form. Noting that multiaxial criteria normally do not account for stress gradient effects, Araújo and Nowell [43] investigated the applicability of two critical plane models to rapidly decaying contact stress fields. Using a modified version of the SWT parameter, Equation (6.18) above, as well as a fatigue parameter suggested by Fatemi and Socie [44], they found that direct application of the models using the peak stresses at the edge of contact produced highly conservative predictions of initiation life compared to experimental data. The FS parameter used has the form FS = 2 1+ max y (6.22) where is the difference between maximum and minimum values of the shear strain in a cycle, max is the maximum stress normal to the critical plane, y is the yield strength, and is a constant that approaches unity at long lives. The experimental data for total life were corrected for initiation life by subtracting out a computed crack propagation life. The predictions were worse for small contacts than for large contacts because of the steeper stress gradients at smaller contacts which also correspond to the HCF (long life) regime. To get better correlation with experimental results, the gradient effect was addressed by either averaging the stress over some volume or averaging the parameters over a volume, similar to the procedure used in [38] described above. The differences between the two methods were small. The most critical aspect of this approach was the size of the averaging volume which was determined not to be a material constant for consolidating data under several different test conditions on two different alloys, Al-4% Cu and Ti-6Al-4V. However, the best values for a critical volume were of the order of the grain size of the material to give the best estimates of fatigue initiation life as observed experimentally. For life prediction, [33] used an equivalent stress parameter, equiv , to consolidate stresses at various values of R as equiv =05 psu w max 1−w (6.23) where psu is the alternating pseudo-stress range defined as psu = 1 √ 2 xx − yy 2 + yy − zz 2 zz − xx 2 +6 2 xy + 2 yz + 2 zx 1/2 (6.24) and ij defines the pseudo-stress range for each stress component based on maximum and minimum points in the fatigue cycle. The Walkerized equivalent stress, Equation (6.23), was used with the definition of pseudo-stress, Equation (6.24) in a multiaxial model to predict the cycles to crack initi- ation in their fretting-fatigue experiments. This approach had been previously used for Fretting Fatigue 299 predicting notch fatigue behavior and was found to work only if the values of psu were averaged over some volume to account for the stress gradients typical of notched speci- mens. However, even averaging over a few elements of a FEM model over the surface was not sufficiently accurate to consolidate results with smooth bar fatigue data. This led to the use of a weakest link model which was referred to as the stressed surface area, or Fs approach, described in Appendix E. The inability of this approach to consolidate fretting-fatigue data with smooth bar fatigue data was attributed to both the inability to account for local plasticity in the fretting region near the edge of contact and the neglect of crack propagation life when comparing results from an initiation model to total life experimental values. One additional consideration in trying to predict crack initiation under a contact stress field with steep stress gradients is surface finish. Whereas the surface finish is smooth and can be controlled somewhat in notch fatigue, the surface where maximum stresses occur in fretting fatigue is in the slip regime where material damage and surface irregularities can be produced. In order to compare crack initiation in fretting fatigue to crack initiation under a similar gradient stress field at a notch, Nowell and Dini [45] proposed using notch fatigue tests. Using a notch such as depicted in Figure 6.37, they demonstrated that a notch geometry could be found that has a stress gradient very similar to that obtained in several fretting-fatigue contact stress geometries. While their analysis was for complete sliding, any conditions could be modeled. Also, while py in Figure 6.37 implies a stress that varies in the y-direction, the analysis for comparing stress gradients in fretting with those from a notch was conducted for a uniform stress field and considered the stresses at x =0 starting at the root of the notch. These stresses, in turn, were compared with fretting stresses starting at the contact surface. For a best fit notch angle of 2 = 60 , best fit notch root radii of 0.03 to 0.12 mm were found, indicating that such dimensions could 2 ψ y p (y ) ρ d x Figure 6.37. Geometry of notch to simulate fretting fatigue [45]. 300 Effects of Damage on HCF Properties be achieved without any special machining necessary. Notch and fretting experiments with equivalent stress gradients are therefore possible and these would provide a possible means of separating out surface damage and stress gradient effects. 6.11. FRACTURE MECHANICS APPROACHES One of the early applications of fracture mechanics to fretting fatigue was that of Nix and Lindley [46] who computed the critical size of a defect that could eventually propagate under fretting-fatigue conditions. For 2014A aluminum specimens with 3.5NiCrMoV steel fretting contact pads, they obtained the results shown in Figure 6.38 for two contact pressures. No attempt was made to account for possible small crack effects (probably not important in their case). For a non-fretting condition, the curves shown represent the constant threshold stress intensity (denoted by K 0 ) and the applied stress intensity for a smooth specimen as a function of crack depth, a. For the fretting case, the applied stress intensity is derived from the combination of the local contact stress field as well as the applied bulk stress in the specimen. Because of the complexity of the local contact stress field, the stress ratio changes with increasing crack depth as reflected in the figure. This, in turn, causes a variation in the threshold stress intensity because K 0 is a function of R. The intersection of the applied K and the threshold K defines the crack depth above which a crack or defect could propagate whereas for smaller crack depths, the driving force (applied value of K = K a is below the crack growth threshold. An alternate way in which data of this type can be interpreted would be to suggest that a fretting-fatigue crack of that critical size would have to be initiated or nucleated without growing as a crack. This would normally be the case for very small cracks, probably on the order of the microstructural features of the material. Attempts to use the stress states in the contact region of fretting fatigue in models for the initiation and subsequent propagation of cracks have generally been less than successful. In particular, similar stress states have not been found using different specimen geometries that produce the same fretting-fatigue life of 10 7 cycles [8]. Further, these stress states have been found to depend highly on the assumed COF, [48]. Finally, the stress states generally are multiaxial and involve steep stress gradients, making it difficult to predict the values that might lead to crack initiation based on smooth bar fatigue data. This is essentially the same problem as that in notches, discussed in Chapter 5. An alternate approach has been to look at the problem from a fracture mechanics viewpoint, with an attempt to see if crack driving forces exceed the threshold for crack propagation [47, 49]. The primary questions to be answered in this approach are what constitutes an initiated crack length and along what direction should the crack be expected to propagate. To help answer some of these questions, a detailed analysis of the fracture-mechanics parameters in the contact region was conducted by Nicholas et al. [47]. There, several Fretting Fatigue 301 R (fretting) R (no fretting) R (no fretting) Critical defect size (fretting) ΔK 0 (fretting) ΔK a (fretting) ΔK a (no fretting) ΔK 0 (no fretting) ΔK 0 (no fretting) ΔK 0 (fretting) ΔK a (fretting) ΔK a (no fretting) Critical defect size (no fretting) Critical defect size (no fretting) Critical defect size (fretting) Applied stress = 125 ± 15 MPa Contact pressure = 31 MPa Defect shape = extended (a) (b) Crack depth, a (mm) Crack depth, a (mm) 0 3 2 1 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0 1 2 3 Stress ratio, R Stress ratio, R 0.8 0.6 0.4 0.2 0 ΔK MPa√m ΔK MPa√m Applied stess = 125 ± 15 MPa Contact pressure = 139 MPa Defect shape extended R (fretting) Figure 6.38. Applied and threshold stress intensity factors with and without fretting. (a) Contact pressure =31 MPa, (b) contact pressure =139 MPa [46]. 302 Effects of Damage on HCF Properties different fretting-fatigue conditions were analyzed that produced long lives using a step- loading procedure. This type of experiment was analogous to determining the FLS under uniaxial fatigue using a step-loading procedure corresponding to a chosen HCF life. From analysis of these experiments, both multiaxial stress fields and mixed-mode stress intensity factors for specific cases were obtained. The results were analyzed in order to find some commonality which could lead to development of parameters having broad application to fretting fatigue. Variations in COF, , were also evaluated. Stresses in the region of contact were computed using the 2-dimensional FEM assuming linear-elastic material behavior and plane-stress conditions. The FEM analysis of stresses in the contact region was made for an uncracked body. The development of a crack would influence the local contact stresses which, in this case, are treated as “far-field” for the purpose of the stress analysis and in the subsequent K analysis. The stress analysis for the contact region was made using the common assumption that the development of a crack and accounting for its compliance does not alter the computed stress fields. It is noteworthy that such a procedure was accomplished and showed, somewhat surprisingly, that the K analysis of the crack emanating from a contact region does not change appreciably if the crack is accounted for in the FEM analysis [50]. The approach taken to establish the threshold for propagation of a crack in the fretting region was similar to that of Chan et al. [49] where they employed a worst case notch (WCN) concept and applied it to a fretting-fatigue problem. For the biaxial plane-stress state, the Mode I and Mode II stress intensity factors were calculated along a crack having an arbitrary orientation with respect to the plane of the contact as depicted in Figure 6.39. From the computed stresses, the weight function method was used to calculate the Mode I and Mode II stress intensities along the crack. To be able to represent conditions at different values of RR= K Imin /K Imax with a single parameter, an equivalent Mode I stress intensity was used in the form K Ieq =K I 1−R m−1 (6.25) The value of m was assumed to be identical for the threshold and low growth rates, and was found to be m = 056. For R<0, the equivalent K was taken as the maximum value of K and when K Imax was negative, K was taken to be zero. σ x Pad x r,a –y θ Figure 6.39. Nomenclature for orientation in K analysis. Fretting Fatigue 303 For mixed-mode behavior, the effective stress intensity factor was defined as K eff = K 2 Ieq +K 2 IIeq (6.26) where K Ieq is determined from the stress field and corrected for different values of R using equeation (6.25) while K IIeq is taken simply as K II with no correction for R. The effective threshold was modified to account for small crack effects by introducing the El Haddad parameter, a 0 , as shown in previous sections. This produces an ever- decreasing threshold as the crack length, a, gets smaller as seen on a Kitagawa diagram. K eff =K LC a a +a 0 (6.27) where K LC is the long crack threshold and a 0 is defined as a 0 = 1 K th e 2 (6.28) In Equation (6.28), K th is the effective long crack threshold, corrected for R while e is the endurance limit stress, also corrected for R. The values of K along a straight line inclined at an arbitrary angle to the line normal to the contact surface (see Figure 6.39) were obtained through the equations: K I a = a 0 h 11 r a r dr + a 0 h 12 r a r r dr (6.29) K II a = a 0 h 21 r a qq r dr + a 0 h 22 r a r r dr (6.30) where r and r r represent the normal and shear stress on the crack surface, respectively, and h 11 h 12 h 21 , and h 22 are the mixed-mode weight functions developed by Beghini et al. [51] for an edge crack in an infinite medium. One of several cases analyzed was a 4mm thick specimen with a bulk stress of 200 MPa. From the FE analysis the stresses at the maximum and minimum applied load were found to stabilize within the first complete cycle. For this case, the stresses were computed along lines at angles of 0 10 30 , and 50 . The normal and shear stresses along the crack at an angle of 0 are shown in Figure 6.40. For this and all angles evaluated, the computations showed that the shear stress amplitude is small. Further, the stress ratio for the stresses normal to the crack surface, , is nominally about 0.5 for the crack oriented at = 0 . Normal stresses were also seen to decay rapidly with crack length, a, and became compressive for larger angles, = 30 and = 50 . From these stresses, values of K I and K II were determined using the weight-function method, and the short-crack-corrected threshold for the value of R at each point was 304 Effects of Damage on HCF Properties 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 K I,max K I,min K II,max K II,min a (mm) θ = 0° K (MPa√m) Figure 6.40. K distribution as function of crack depth for Modes I and II [47]. computed for comparison. The mixed-mode driving force and threshold for each of the four angles were determined for crack lengths up to 200 m. The results showed that the threshold is exceeded at crack lengths that are fairly large compared to the value of a 0 which tends to be about 30m for these computations. From the computations, crack growth would occur only at crack lengths above 185m for the crack at 0 as shown in Figure 6.41. Experimental observations of crack orientations were made and indicate that crack nucleation angles are typically around 45 as shown in Figure 6.42. These cracks turn towards a direction normal to the contact surface within crack lengths of the order of tens of microns where they become primarily Mode I cracks. Based on these observations, –500 0 500 1000 1500 0 0.05 0.1 0.15 0.2 0.25 Stress (MPa) Location along crack line, r (mm) θ = 0° τ rθ,max τ rθ,min σ θθ,min σ θθ,max Figure 6.41. Shear and normal stress distribution as function of depth [47]. Fretting Fatigue 305 the analysis only of what happens along a line = 0 , which is perpendicular to the contact surface, is deemed important. From Figure 6.41 at 0 , the maximum and minimum values of the Mode I stress intensity show that the stress ratio, R, used in the equations for KR= K Imin /K Imax , is very close to the applied stress ratio for the bulk load which was 0.5. Figure 6.41 also shows that at 0 , the Mode II component is smaller than that for Mode I but, more importantly, that K II is quite small. Comparisons were made to evaluate the effects of specimen thickness, clamping load, and COF on the tendency of a crack to grow or not grow under the experimentally observed loading conditions corresponding to a fatigue life of 10 7 cycles. In one case the crack growth was close to occurring for all crack lengths even though the growth condition does not occur until the crack is over 100m long while in another case, the driving force was well below the threshold until the crack reached a length of almost 200 m. There appeared to be more of a tendency for crack growth to occur at either a higher clamping stress or a higher shear stress. Overall, it was found that the fracture mechanics methodology holds more promise for predicting failure or FLSs than analysis of the stress field. The reason for this is that a threshold stress intensity is available for long cracks either under Mode I only or under mixed-mode conditions and can be easily corrected for short crack effects. On the other hand, knowledge of the stress field involving steep gradients requires a gradient stress failure model which is not readily attainable from smooth bar failure data. Further, the stress fields were very sensitive to the value of the COF, , whereas the K solutions were less sensitive other than for extremely short crack lengths. The role of in fretting-fatigue analysis is discussed in the next subsection. The length of an inclined crack before it turns 90 to the fretting surface varies widely with material, pad geometry, and fretting fixture and loading conditions. Whereas Figure 6.42 shows a crack that is normal to the surface within 10m for Ti-6Al-4V, the same alloy at 260 C is reported to develop cracks that do not turn perpendicular to the loading direction before achieving a length from 200 to 500 m [52]. An example of such a crack is shown in Figure 6.43 which was obtained using different loading conditions and a somewhat different fretting-fatigue fixture than that for which the crack in Figure 6.42 was obtained. In yet another fretting-fatigue fixture, that of the bridge type, cracks are reported to initiate at an angle of 60 to the surface and then to gradually turn normal to the bulk loading direction. For a case where the pad and specimen material were SM430 steel, the crack did not become perpendicular to the surface until a depth of over 700 m which the authors attribute to where the fretting effect is eliminated [23]. Mutoh and Xu [23] propose that an accurate assessment of crack propagation life must take into account the mixed-mode loading conditions and the combination of Mode I and Mode II loading on a crack, even when the crack is normal to the surface in a fretting- . of Damage on HCF Properties or near the edge of contact where stresses are normally at a maximum. Because of the steep stress gradients that are present in a contact region, parameters that are. which was referred to as the stressed surface area, or Fs approach, described in Appendix E. The inability of this approach to consolidate fretting-fatigue data with smooth bar fatigue data was attributed. El Haddad parameter, a 0 , as shown in previous sections. This produces an ever- decreasing threshold as the crack length, a, gets smaller as seen on a Kitagawa diagram. K eff =K LC a a +a 0 (6.27) where