286 Effects of Damage on HCF Properties The complete stress distributions for normal stress, p(x), and shear stress, qx, along the contact boundary can be found in [30]. Of greatest interest in the determination of the FLS for fretting fatigue is the maximum value of the axial stress,  xx , at the interface as shown in Figure 6.21, for example, at the edge of contact. Such stress can be used in a fatigue initiation criterion. In Giannakopoulos et al. [31] the value of this stress is taken as the sum of the contributions due to the contact shear load, the normal pressure, and the bulk applied stress. The endurance limit stress can then be formulated as  end R = max  tot 1 = Q xx + P xx +max  b (6.15) where  end R is the endurance limit stress for a smooth bar at the appropriate stress ratio, R.Ifqx is the shear stress due to a tangential load Q, the maximum tension is given in [32] as  Q xx = 2   b −b qx b −x dx (6.16) For the maximum stress due to the pressure,  P xx , numerical results from [31] are plotted in Figure 6.24 which shows that the value of this stress depends on the ratio of the thickness of the substrate to the width of the contact region (t is the half thickness and b is the half width). For the infinite thick substrate, the tension stress vanishes. Analytical or semi-analytical solutions have been developed for obtaining fretting- fatigue stress fields for more general contact geometries than a cylinder or flat with rounded corners on a half space. Murthy et al. [33] present details of a computationally efficient mechanics-based approach using discrete Fourier transformations to obtain con- tact stresses. The approach is based on the solution to singular integral equations that 0 0.1 0.2 0.3 0.4 0.5 0.01 0.1 1 10 σ xx (πb)/2P t /b P Figure 6.24. Normalized maximum tensile stress for different strip thickness [31]. Fretting Fatigue 287 characterize the contact of two surfaces and takes into account the details of the shape of the contact surfaces. Of particular significance is the ability to account for distortion or irregularities in the contact interface from experiments themselves or from imperfect machining. This particular formulation was eventually incorporated into a computer code at Purdue University that was referred to subsequently in the literature as CAPRI. Later, modifications were made to the code that accounted for the finite thickness of typical specimens as well as accounting for superimposed bulk stresses in the specimen. A description of the mathematics and procedures for determining the surface and internal stresses using semi-analytical procedures and the codes for conducting these analyses is presented in Appendix F. An alternate method for determining the stress fields in a contact region where fretting fatigue takes place is the use of finite element methods. Finite element modeling, in addi- tion to requiring the necessary mesh refinements to capture the steep stress gradients in the contact region, has to represent the boundary conditions appropriately. A comparison of a quasi-analytical approach and FEM results is presented in [34]. There, two different models are used as shown schematically in Figure 6.25. The first, (a), shows the scheme used to represent contact interaction when a normal and shear load are applied to an indentor. In this configuration, no account is taken of the bulk stress but it is used to validate analytical or quasi-analytical solutions to the contact problem which normally do not consider bulk loads. The second model, (b), represents a typical experimental configuration where bulk loads are applied, thereby inducing a tangential load on the indentor through the resistance of the fretting fixture. The displacement boundary con- ditions are represented with rollers as shown schematically in Figure 6.25. The spongy layers represent forces applied with hydraulic fixtures and are represented with elements Q P Specimen Pad P Specimen Pad σ “Spongy” layers (a) (b) “Spongy” layers Compliant spring Figure 6.25. Schematic of finite element configurations for fretting contact, (a) for loading validation, (b) for experimental configuration (after [34]). 288 Effects of Damage on HCF Properties y x 2b h σ Figure 6.26. Schematic of finite element modeling of bridge pad with specimen loaded in tension [23]. with a relatively insignificant modulus. The compliant spring is used to represent the resistance of the experimental fixture and creates the shear load Q which is in-phase with the applied bulk stress, . For a bridge-type fretting-fatigue fixture, shown schematically in Figure 6.34, modeling using the FEM also has to adequately represent the proper boundary conditions. An example of a model for a bridge-type pad is shown in Figure 6.26, after [23], where structural steel JIS SM430A was used for both pad and specimen. The model shown represents the condition where the bulk load, , was 180 MPa in tension and the clamping pressure was 60 MPa with a value of coefficient of friction  = 08. With values of h = 4 mm and b = 1 mm, a relative displacement of  = 156 m was achieved as shown in the exaggerated deformed profile shown in Figure 6.27. The bulk loading was conducted at R =−1, and under compression the relative displacement was  = −055 m. The corresponding stress distributions obtained by finite element analysis are shown in Figures 6.28 and 6.29 for tension and compression, respectively. It can be seen that, in general, there are three regions on the contact interface: sticking, slipping, and gapping (separation of the two bodies). As the authors point out, it is obvious that there is no stress singularity at the boundary points between the gap and slip or stick and slip regions. In general, the existence of singular stress fields near the edge of contact Δ Δ = 1.56 μm Figure 6.27. Deformation under tension with bridge type fretting pad [23]. Fretting Fatigue 289 –400 –300 –200 –100 0 100 200 –1 –0.5 0 0.5 1 σ y τ xy Stress (MPa) x /b Sticking Slipping Gapping Figure 6.28. Stress distribution on interface under tension [23]. – 400 – 300 – 200 – 100 0 100 200 300 –1 – 0.5 0 0.5 1 Stress (MPa) x /b Slipping Gapping σ y τ xy Figure 6.29. Stress distribution on interface under compression [23]. depends on the geometry, material combinations, and the type of deformation. In the case cited, the singularity appears at the external contact edge under compression loading and at the internal contact edge under tension. Moreover, the singular stress field will dominate crack initiation under fretting fatigue but may not be dominant once the crack tip moves out of the local region. For the bridge-type pad illustrated here, crack initiation and subsequent propagation may take place either at the edge of contact or in the center region of the contact, the latter being the case when there is either no edge singularity or when the singularity disappears due to the wear of the pad and the specimen [23]. The difference in initiation sites is generally a function of the variables mentioned above as well as loading conditions such as stress ratio, R. 290 Effects of Damage on HCF Properties –600 –400 –200 0 200 400 600 800 –0.15 –0.1 –0.05 0 0.05 0.1 0.15 0.2 0.25 2 mm Thick long pad σ f = 350 MPa μ = 0.3 Strees (MPa) x position (mm) σ x σ y τ xy Figure 6.30. Stress distribution along surface at trailing edge of contact [8]. The non-singular, yet steep stress gradients encountered in the contact region between a flat pad with blending radii and a flat specimen are of concern in developing stress-based models for fretting-fatigue initiation or total life. Reasonably accurate stress profiles have been obtained using finite element computations with mesh refinement down to sizes of approximately 65m near the edges of contact where maximum stresses are reached [8]. An example of the nature of the stress distribution near the edge of deformed contact is shown in Figures 6.30 and 6.31 [8]. In both plots, x = 0 corresponds to the edge of deformed contact (EDC) as determined from the finite element computations. The maximum tensile stress occurs just outside the EDC, but at a distance not exceeding about 10 m. Obviously,  y and  xy go to zero outside the contact region. Compared to  x and  y , the values of  xy are relatively small and spread out more while the compressive stresses, of the same order as  x , peak at about 50 m inside the EDC. From Figure 6.30, the axial stress,  x , is seen to decay from a maximum to near zero in a distance of approximately 50 m. The stress gradients into the thickness are plotted in Figure 6.31. Shown is  x at several distances into the thickness direction. The coordinate y = 1 corresponds to the surface of the specimen in contact with the pad and the corresponding stress profile is identical to that shown for  x in Figure 6.30. The stress profiles show that the  x stresses decay rapidly, with stresses at 50 m below the surface being almost the same as those much deeper. Thus, stress gradients into the thickness are of the same order of magnitude as those along the surface. Similar observations were made for other cases involving different pad geometries and pad thicknesses [8]. Given the capability to determine the stress field in a contact region where fretting fatigue takes place, it is tempting to try to develop a parameter that can adequately characterize the degradation of fatigue strength due to fretting fatigue. Unfortunately, Fretting Fatigue 291 0 100 200 300 400 500 600 700 800 –0.15 –0.1 –0.05 0 0.05 0.1 0.15 0.2 0.25 Axial strees (MPa) x position (mm) 2 mm Thick long pad y = 0.919 y = 1.000 y = 0.987 y = 0.957 y = 0.870 Figure 6.31. Stress distribution at various depths at trailing edge of contact [8]. there are many parameters and variables that seem to affect the behavior [28], and many of them can have a synergistic effect with others on the observed behavior. Many parametric studies, too numerous to mention or cite, have been conducted over many years. In addition to the effect of environment on the fretting-fatigue behavior of materials, parameters such as slip amplitude, magnitude and distribution of contact stresses, and friction are widely accepted as major influences on the observed behavior. The earliest and simplest approaches for quantifying the degradation of fretting-fatigue lives or fatigue strengths is with the use of S–N curves obtained under fretting-fatigue conditions and comparing them with those obtained on smooth specimens. The resulting fretting-fatigue strength reduction factor, k ff , is analogous to the fatigue notch factor (see Chapter 8) and is probably equally useful (useless) because the data themselves are necessary to validate any formulas developed to quantify such a parameter. Further, the S–N curves are normally neither parallel nor proportional in stress amplitude over a range of values in N . For the purposes of HCF design, the value of k ff is needed at very high values of N where fretting-fatigue data are normally not obtained because of the time required to perform such experiments. Generally speaking, fretting exerts its maximum effect in HCF since its main effect is the initiation of a propagating crack at stresses well below the normal fatigue limit [19]. A typical example of such a reduction is shown in Figure 6.32, taken from [19], for an annealed austenitic stainless steel tested at R =0. In the HCF regime, corresponding to 10 7 cycles, the knockdown factor is approximately two whereas at 10 5 cycles, plain and fretting fatigue have nearly identical strengths under the specific test conditions. Similar results are shown in Figure 6.33, taken from [23], for a structural steel JIS SM430A tested in a bridge-type fixture at R =−1. There, the knockdown factor at 10 7 cycles is about one-third and the fretting and plain fatigue behavior is the same at about 10 4 cycles. While 292 Effects of Damage on HCF Properties 100 150 200 250 300 350 10 4 10 5 10 6 10 7 Plain fatigue Fretting fatigue Alternating stress (MPa) Cycles to failure Figure 6.32. S–N curves for austenitic stainless steel (R =0 ) in plain and fretting fatigue (after [19]). 0 50 100 150 200 250 300 350 10 3 10 4 10 5 10 6 10 7 10 8 Plain fatigue Fretting fatigue Strees amplitude (MPa) Cycles to failure Figure 6.33. S–N data for structural steel JIS SM430A (R =−1) in plain and fretting fatigue (after [23]). these two examples provide evidence that fretting fatigue is primarily a HCF problem, the results of these types of tests are dependent on many variables and a general conclusion about fretting fatigue being strictly an HCF problem should not be drawn. 6.9. ROLE OF SLIP AMPLITUDE Of the many parameters that influence fretting fatigue, slip amplitude has been considered to be a primary driver in reducing fatigue strength in fretting fatigue [35]. In order to assess the influence of slip amplitude, the appropriate experiments must be performed. Fretting Fatigue 293 S P Q S P Q Figure 6.34. Schematic of bridge type pad for fretting fatigue experiments. A bridge-type pad shown schematically in Figure 6.34 has been used extensively because it is particularly convenient for studying the effects of slip amplitude [35]. For very small slip amplitudes, however, where stick–slip conditions may occur under the pad, the variability of slip from one pad to another under nominally identical conditions makes it difficult to determine the actual conditions under each pad, either experimentally or computationally. Slip amplitude has been incorporated into a fretting-fatigue parameter developed by Ruiz et al. to quantify the extent of fretting-fatigue damage [36]. They postulated that both slip amplitude, , and interface shear stress, , in combination, cause the damage that leads to reduction of fatigue strength. The amount of work done in overcoming friction per unit surface area in the slip zone can be characterized by the product, . If taken at the point where the product is a maximum, they postulated that the fretting damage is a maximum and is given by the product. Arguing also that the crack development is influenced by the stress,  t , tangential to the surface, they ended up with a fretting-fatigue damage parameter (FFDP) that combines aspects of initiation as well as propagation ∗ in the form FFDP = t  (6.17) A crack is predicted to initiate at the point where the parameter FFDP reaches a critical value. While it is difficult to establish the values of the variables in the parameter in a real component, it may provide guidelines in design for mitigating the effects of fretting fatigue by attempting to minimize the value of FFDP by tweaking the values of the individual terms. Doing this, however, has no physical basis and although trends may be established for some combinations of material and geometry, the general applicability of this parameter has not been established. ∗ This type of formulation has little, if any, mechanics-based justification. 294 Effects of Damage on HCF Properties While much attention has been paid to the role of slip in fretting fatigue, stress analysis of the contact region in conjunction with a fracture mechanics analysis can shed light on criteria such as the growth or arrest of a crack that initiates in the contact region. Observation of a distinct transition from short to long life in fretting-fatigue experiments on aluminum using cylindrical pads led to a fracture mechanics-based analysis to explain the effect [3]. Results from a series of experiments using different cylindrical radii and different contact widths were plotted as shown in Figure 6.35 where fatigue life is seen to decrease suddenly when the contact width is increased. In this figure, a 0 is the critical contact size in one series of experiments where fatigue lives transition from in excess of 10 7 cycles (run-outs are shown schematically with arrows in the figure) to shorter lives (<10 6 cycles). Individual data points are not shown, but the transition from run-out to short lives occurs over a small range in contact length. In the various experiments, the peak stress in the Hertzian (cyclindrical) contact was maintained as a constant. The local stress field, including the gradient, changed as contact width was changed. By comparing the crack driving force due to the combination of contact stresses and applied stresses, K, to the threshold for crack propagation, K 0 , corrected for short crack effects, Araújo and Nowell [3] were able to demonstrate the existence of a transition from crack arrest to crack growth depending on the particular stress field. Their numerical values based on contact widths agreed quite closely with those observed experimentally. A plot very similar to Figure 6.35 was obtained by Magaziner et al. [37] in a similar set of fretting-fatigue experiments using cylindrical pads and a titanium alloy. However, their explanation was quite different. Whereas the experiments reported in [4] involved stick– slip conditions, those in [37] were found to have gross slip under certain combinations of loading. The data reported in [37] are plotted in Figure 6.36 and show that small contact 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 Fatigue life (million cycles) Contact half width (mm) a 0 range Figure 6.35. Schematic of observed fretting fatigue life behavior for Hertzian contact [7]. Fretting Fatigue 295 0 50 100 150 200 250 300 0 0.5 1 1.5 2 Failure Run-out δ (μm) 2a (mm) Figure 6.36. Comparison of slip distance with contact width in fretting fatigue. Data from [37] widths are associated not only with run-out conditions but with large slip distances that involve total slip rather than stick–slip. In the data reported, run-out was taken as having lives greater than 10 6 cycles whereas the shorter lives, denoted as “failure” in Figure 6.36, involved failure in less than 10 5 cycles. While the cycle count range and slip amplitudes in the two cited studies differed somewhat, both studies related the observed phenomenon of a jump in life with size of contact as a size effect. The individual analyses show that size is not the effect, it is either the stress field in the first study or the change from gross slip to slip–stick in the latter. The differences in test type (the latter controlling slip amplitude directly), material, and COF may have a greater effect on the observed behavior than just a “size effect.” As for a stress analysis, as the authors of the latter study point out, there are concerns when trying to conduct a stress analysis when gross slip occurs. Both the possible wearing away of cracks as well as the exact location of the maximum stress (which may vary from cycle to cycle) make it difficult to apply a fracture-mechanics type of analysis to establish the conditions where crack arrest may occur. 6.10. STRESS-AT-A-POINT APPROACHES Many approaches to life prediction under fretting-fatigue conditions have dealt with stress at a point, the value of a parameter at a point, or the stress or parameter at some distance from the surface, or an average value over some surface area or volume. In HCF, the notion that most of the life of a material is consumed in the nucleation or initiation stage has led to the widespread use of the maximum value of a parameter at . 0.870 Figure 6 .31. Stress distribution at various depths at trailing edge of contact [8]. there are many parameters and variables that seem to affect the behavior [28], and many of them can have a synergistic. fretting-fatigue conditions have dealt with stress at a point, the value of a parameter at a point, or the stress or parameter at some distance from the surface, or an average value over some surface area. from cycle to cycle) make it difficult to apply a fracture -mechanics type of analysis to establish the conditions where crack arrest may occur. 6.10. STRESS-AT -A- POINT APPROACHES Many approaches