Study of the shot peening surface roughness in fretting

Tribology International 193 (2024) 109444 Available online 21 February 2024 0301-679X/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/bync/4.0/). Study of the shot peening surface roughness in fretting M. Moreno-Rubio, D. Erena * , Jesús V´ azquez , Carlos Navarro Universidad de Sevilla, Escuela T´ ecnica Superior de Ingeniería, Departamento de Ingeniería Mec´ anica y Fabricaci´ on, Camino de los Descubrimientos s/n, C.P. 41092, Spain ARTICLE INFO Keywords: Fretting fatigue Surface roughness Shot peening Fatigue model ABSTRACT Fretting fatigue manifests in mechanical systems exposed to contact forces that vary over time, resulting in premature cracks that can lead to component failure. Shot peening treatment is one of the palliatives to mitigate fretting fatigue problems. Recent studies have noted that the fretting fatigue is also extended by exclusively applying SP to the contact pad surface. This suggests that the improvement in the fatigue live produced by SP comes not only from the compressive residual stresses but also due to the characteristic surface roughness from the SP treatment. For this reason, the main goal of this work is to analyse how the shot peened roughness affects the fretting fatigue life. To do so, a confocal microscope was used to measure various profiles of shot peened pads. Using these profiles, and under fretting loads combination, the contact stress and strain field are calculated, followed by the application of a fatigue model. Finally, the estimated fatigue life is contrasted with empirical data derived from a research published by the authors. The study reveals that the fretting fatigue life is influenced by the surface roughness, although to a lesser extent with respect to residual stresses. Besides, the SmithWatson-Toper parameter and stress intensity factor were analysed, suggesting that the rough surface primarily affects the initiation phase rather than the propagation phase, as it was expected. 1. Introduction Fretting fatigue appears in many engineering components where there are parts in contact, for instance cables, bolted joints, bearings and aircraft turbines [1–5], among others. Due to this contact and the loads transferred there is a surface damage and an early initiation of cracks. There are various types of palliatives for fretting fatigue problems, being one of the most popular ones shot peening (SP). SP generates compressive residual stresses beneath the surface, in turn increasing the surface roughness in the material, and it is well known the beneficial effect of compressive residual stresses in plain fatigue as well as in fretting fatigue [6–10]. In a general fretting situation, the effect of contact stresses is mainly restricted to the very near surface material, roughly up to a depth of the order of the surface contact length. This fretting particularity makes that surface treatments producing high compressive residual stresses very close to the contact surface, as is the case of shot peening, have a great -beneficialeffect on the fatigue behaviour, especially in the crack initiation phase. Shot peening in addition hardens the treated surfaces due to cold working, [9], [10], which also leads to a better fatigue behaviour. It has been observed that a high surface roughness in fretting is also beneficial since it helps to maintain the lubrication on the surface, thereby improving the wear properties [11]. Unfortunately, it is not very well analysed in the literature the impact on the fretting behaviour of the shot peening induced surface roughness. The reason for this lack of previous works in this subject is clear: when a shot peened specimen is tested, it is quite difficult to uncouple the effects due to residual stresses and hardening from that produced solely by surface roughness. In that direction, the authors performed a couple of fretting tests campaigns in order to analyse the effect on fretting fatigue by shot peening induced surface roughness [9], [12]. In those fretting tests, and especially for the high cycle regime, it was concluded that a non-negligible beneficial effect is produced by the surface roughness in fretting, in contrary to the well-known negative effect of a high surface roughness on plain fatigue. Most of works modelling the effect of surface roughness in fretting only study the effect produced at the contacting surfaces, but leave aside the near-contact subsurface material [13–16]. The surface roughness in fretting fatigue has been modelled in the literature using three main approaches: •Surface roughness is modelled as a cyclic pattern. Most of works deal with a constant amplitude wavy surface that resembles the summits * Corresponding author. E-mail address: [email protected] (D. Erena). Contents lists available at ScienceDirect Tribology International journal homepage: www.elsevier.com/locate/triboint https://doi.org/10.1016/j.triboint.2024.109444 Received 11 December 2023; Received in revised form 16 February 2024; Accepted 19 February 2024 Tribology International 193 (2024) 109444 2 and valleys of a rough surface [17–19]. Using this approach, and assuming elastic half-plane behaviour, an approximate analytical model –based on the Hertz theory– is possible. •The modelled roughness reproduces the actual surface topography, being this approach limited to numerical analyses: finite or boundary element [15,20]. •The surface roughness is modelled using the statistic parameters defining the surface topography. With this approach, only representative values can be obtained [21,22]. In all the above cases, most models consider a linear-elastic material behaviour, that depending on the surface characteristics and contact loads, can be questionable. In any case, it is important to note that contact loading produces very steep stress/strain gradients near the surface, and thus rapidly these quantities reach values below the material yield limit. Therefore, results obtained with a linear-elastic model do shed some light about the effect due to surface roughness on fretting behaviour. The majority of research studies analyse the combined influence of shot peening, acknowledging its benefits on fatigue behaviour. However, when considering its combined influence, both residual stresses and surface roughness are incorporated. The aim of this work is to separate these influences and specifically analyse the influence of shotpeened surface roughness on fatigue behaviour. In view of all the above, the present work provides a deeper knowledge of the fretting fatigue behaviour of a series of tests that have the characteristic shot peening surface roughness but without the compressive residual stresses [10,12]. In these tests, and in order to decouple the effect on the fretting behaviour due to surface roughness from that produced by residual stresses and material hardening, only the contact pads were shot peened. To analyse these tests, a semi-analytical contact formulation of the contact pair is used. In this analyse, in order to resemble the contact conditions between the “smooth” surface (actually machined) of test specimens and the rough surface of shot peened contact pads, the actual topography of shot peened contact pads, measured by a confocal microscope, was introduced into the pad’s surface profile. A total of 20 different profiles of the shot peened contact pad were measured under the microscope. In this work and comparing with the situation of perfectly smooth mating surfaces, we first analyse how the shot peening surface roughness modifies the contact pad and stresses at the surface. Then, the stress/strain fields beneath the test specimen contact surface, which are a key factor for the early initiation of fretting cracks, are analysed. In addition, we study the effect of the roughness in the stress intensity factor (SIF) of physically of small (less than 1 mm) fretting cracks, being this parameter the driving force for the fatigue crack growth. Lastly, and using the results provided by the semi-analytical formulation, we introduce a fatigue model that predicts the fretting fatigue life incorporating the pad surface roughness as if it belonged to the specimen. 2. Materials and experimental data 2.1. Materials The material used to manufacture all the elements, the contact pad and the test specimen was 7075-T651 aluminium alloy, whose chemical composition can be seen in Table 1. In addition, Table 2 shows the mechanical properties of this material. The value of the friction coefficient shown in Table 2 has been obtained in our laboratory [12]. 2.2. Experimental data The experimental fretting fatigue data were obtained using a device developed by [25,26]. The device can be seen schematically in Fig. 1. Firstly, in this setup, a normal load (N) is applied to the contact pads, which are pressed against to the test specimen. The normal load remains constant throughout the entire test, thanks to the installation of two springs specifically for this purpose (see Fig. 1). After that, a cyclic axial load (P) is applied to the sample using a servo-hydraulic actuator. Due to the cyclic axial load and the friction between the test specimen and the contact pads, two tangential loads (Q) appear in-phase with P but in the opposite direction. These tangential loads are measured through load cells. In this setup, the value of the tangential load amplitude can be modified by an adjustable support, and thus allowing the amplitude of Q to be modified regardless of the applied axial load amplitude. The type of contact pair used in these fretting experimental test was a cylindrical contact pair which is very frequent due to its straightforward setup and the availability of analytical expressions for the contact stress [10,27–29]. The shape of the sample is a “dog-bone” type, with a rectangular cross section of 10 mm of width and 8 mm of thickness, and the radius of the contact pad, R, is 100 mm with 8 mm of thickness, see Fig. 1. In this study, seven different load combinations producing fretting fatigue ( σ ,Q,N) were used, it is important to note that these data were extracted from two previously published articles [10,12]. In these fretting tests two different types of contact pads were used: pads having smooth surfaces -actually machinedand pads treated with shot-peening. In all cases the contact surfaces of fretting test specimens are in a machined condition. For each load combination and contact pair, two tests were conducted. Thanks to the tests, the experimental fretting fatigue lives have been obtained for both types of tests: for tests performed with untreated contact elements having smooth surfaces (NSM f), and for tests performed with contact pads treated with shot peening (NRO f). With this test campaign we tried to understand how the roughness of the shot peened indenter affects life when the contact pad and the test specimen get in contact, so that there are no residual stresses in the test specimen. The results of these tests are presented in Table 3 [12], and it is noticeable how the life improves, in a remarkable manner or slightly, depending on the load combination, but noting that in general the higher the fretting lives the better is the improvement. In Table 1 Chemical Composition for Al7075-T651. [23]. % (weight) Al Zn Mg Cu Fe Si Mn Cr Ti Others Max 91.4 6.1 2.9 2.0 0.5 0.4 0.3 0.28 0.2 0.05 Min 87.1 5.1 2.1 1.2 - - - 0.18 - - Table 2 Material properties. [24]. Material Properties Young’s modulus E 71 GPa Poisson’s ratio ν 0.33 Yield strength σ y 503 MPa Tensile strength σ u 572 MPa Paris’ law coeff. (R=0, m/cyc. and MPa√m) C 8.831‧10 11 Paris’ law exp. (R=0) m 3.322 Mode I SIF threshold (R=0.1) ΔKth 2.2 MPa√m Ramberg-Osgood cyclic hardening coefficient K ′ 712 MPa Ramberg-Osgood cyclic hardening exponent n ′ 0.0410 Fatigue strength coefficient σ ′ f 995.4 MPa Fatigue ductility coefficient ε ′ f 0.0994 Fatigue strength exponent b -0.0941 Fatigue ductility exponent c -0.578 Friction coefficient (Untreated)[12] μ u 0.75 Friction coefficient (Treated with SP)[12] μ SP 0.85 M. Moreno-Rubio et al. Tribology International 193 (2024) 109444 3 these data, it can be observed how the ratio of average fatigue life between NRO f and NSM f depends on fatigue life. For short life, lower than 105 cycles, the value of the ratio is around 1.07, while for values of life larger than 105 cycles, the ratio is around 2.23. This implies that for higher life values, the improvement is more substantial than for short life. In [10], the results of the experimental test with the SP treatment applied to the specimen are presented. There, it is possible to see how the ratio of average fatigue life between shot peened specimens and untreated specimens for lives less than 105 cycles is around 4, and for lives greater than 105 cycles is 14. This suggests that the influence on life due to SP on the specimens is higher at longer life cycles. Actually, through these two sets of experiments, the individual contribution of each phenomenon has been estimated. Assuming that shot peening affects fatigue life only through the surface roughness and the compressive residual stresses, the increasing factor determined in [10] and shown in the previous paragraph must be the multiplication of the factor due to the surface roughness (1.07 for low lives and 2.23 for high lives) by the factor due exclusively to the residual stresses. Therefore, with these set of results we conclude that the residual stresses increment fatigue life by a factor of 3.7 for low lives and 6.3 for high lives. All these values are approximate, to have a good accuracy of these results it is necessary to perform more experimental tests, but these values are enough to have an idea of the importance of each phenomenon separately. Concluding, as it was expected, the improvement attributed to compressive residual stresses is greater than the enhancement produced only by the surface roughness, but this study has quantified the relative importance of each one. 2.3. Surface roughness In a cylindrical contact (cylindrical pad pressed against body having a flat surface) the size of the contact zone depends on the material properties, the applied normal load, and the geometry of the specimen. The hertzian contact half-width [30], aH, for the contact between a cylinder and a half-plane is determined by Eq. (1): aH= 8N ′ R(1− ν 2) π E √(1) where N ′ is the normal load per unit length and R is the pad’s radius. In a fretting test, if the surface of one of the mating elements is topographically modified, that issue could notably influence the contact stresses generated and, consequently, the initiation of cracks, thereby affecting the fretting fatigue life. Specifically, the shot peening surface treatment causes a rough surface with diverse peaks and valleys throughout the entire contact surface, affecting the homogenization of the contact stresses. This is because in areas with peaks, the contact surface stresses, normal and shear, will be higher compared to those produced along the valleys. Fig. 2 shows a pictorial representation of a rough surface, representing the pad contact surface of specimens depicted in Table 3. The rectangle represents the contact zone, that is the area that would come into contact after being pressed against the test specimen. Furthermore, Fig. 1. Scheme of the fretting fatigue test setup and dimensions (mm) of the test specimen and the contact pad. Table 3 Load combinations and experimental lives [12]. Load Comb. σ (MPa) Q (N) N (N) N f SM (Cycles) N f RO (Cycles) 1 70 971 6629 316603/ 165696 714702/ 361128 2 150 2113 3006 41002/ 34904 28782/ 28596 3 150 2113 5429 36431/ 32339 34031/ 33181 4 175 971 3006 26587/ 31815 34312/ 33421 5 175 971 5429 35171/ 29100 69466/ 43882 6 175 2113 3006 21669/ 21207 21846/ 24030 7 175 2113 5429 28178/ 28112 19130/ 20662 Fig. 2. Pictorial representation of a rough surface. M. Moreno-Rubio et al. Tribology International 193 (2024) 109444 4 it is possible to see how there are several peaks and valleys at the contact trailing edge along the specimen thickness. According to this, the initiation of fretting cracks may occur at different positions along the contact trailing edge, where axial stress (or any other suitable fatigue parameter) reaches high values. To consider in the simulations the influence of the actual topography along and across the contact zone, the actual 3D contact pair is simplified in a series of 2D contact pair problems, each of them with different rough contact profiles. For example, for a fixed load combination, instead of solving a 3D problem with a specific topography, we solve a series of 2D contact problems with different surface profiles subtracted from the 3D topography (See Fig. 3). In this way, in addition to simplifying the 3D model into several 2D model, we also conduct a statistical analysis. This is necessary because surface roughness does not adhere to a specific pattern; it can vary between different areas or specimens. The results shown from now on have been done for this work. To achieve this, and with the aim of understanding the behaviour of the contact pair with a shot peened indenter under fretting fatigue conditions, the shot peened contact elements were carefully measured using a confocal microscope. This allowed for a detailed examination of the surface characteristics. In Fig. 3a, a photograph of the shot peened contact pad surface is presented, providing a visual representation of the surface topography. Additionally, in Fig. 3b, the surface roughness of one of these contact pads was measured using the confocal microscope, enabling a quantitative analysis of the roughness features. Through these measurements, different 2D surface profiles were acquired, which were subsequently used in the numerical model shown in the appendix of this paper. Two of these profiles can be observed in Fig. 3c. In total, 20 different profiles were measured. In a 2D contact case, the relative position between the rough surface (the contact pad) and the test specimen in which the contact occurs is important. For example, for a specific profile it is not the same if the contact trailing edge matches with a peak, near it, or with a valley. Therefore, in order to make the statistical analysis even more complete, 41 more profiles have been obtained from each of the measured profiles, simply by shifting them in 10 µm increments (equivalent to rotating the cylindrical pad by 0.0057º in each increment). In this way for each of the measured profiles it is guaranteed that the edge of the contact zone will be at different positions between a peak and a valley of the same profile thanks to these sub-profiles. Fig. 4 represents 3 different sub-profiles (Sub. 1, Sub. 2 and Sub. 3) as an example. Therefore, a total of 820 different profiles will subsequently be analysed. As previously mentioned, this study will conduct a comparison between the fretting fatigue behaviour obtained using untreated surface contact pads (i.e., after machining) and treated surface contact pads (with shot peening). As a first step in this study, and with the aim of simplifying the modelling, we investigate if in the surface contact stress simulations, the effect due to the actual surface profile due to machining is relevant when compared with the ideal smooth one. For this reason, various roughness measurements along the machined surface were carried out. Fig. 5 compares the profiles measured in a shot peened pad and in machined surface with an ideal smooth profile. In view of these results, it can be said that the difference between the machined surface and the theoretical or smooth profile is negligible when compared with the rough surface. Then these machined profiles were incorporated to the test specimen profile in the numerical contact model and the contact Fig. 3. Process to acquire 2D profiles using a confocal microscope: a) photograph of a shot peened contact pad, b) surface morphology of the contact pad roughness and c) cross-sectional of 2 of the measured profiles. Fig. 4. Example of three sub-profiles obtained from a single profile (Sub. 1, Sub. 2 and Sub. 3). Fig. 5. Machined, rough and smooth profiles. M. Moreno-Rubio et al. Tribology International 193 (2024) 109444 5 stresses produced by the contact pairs smooth–SP roughness and machined–SP roughness were compared, noticing negligible differences. So that, in order to simplify the formulation, from now on the machined surfaces will be considered as ideally smooth. The stress and strain resulting from these profiles, including an estimation of their life, will be analysed in the next sections of the current work. The model used to obtain the stresses have been developed in the Appendix, while the results of the estimation life will be presented in Section 4. 3. Life prediction model Nowadays, there are many fatigue life prediction models. Some of them focus on the initiation phase, while others only consider the propagation phase. Moreover, certain models use a combination of both phases. The life prediction model applied in this work considers the last one, where the total life combines separately the initiation and propagation phases, and it is not necessary to define where the initiation phase ends, and the propagation phase begins. The boundary between these two phases is obtained during the process. This model has already been applied in former works, concluding with very satisfactory results [24, 31–34]. In the initiation phase analysis, the number of cycles necessary to produce a crack of a certain length, a i , is obtained. In this phase, thanks to the Coffin-Manson curve, it is possible to obtain a curve (a i -N i ), see Fig. 6, which represents the relationship between the number of cycles, N, and different discrete values of a i . Furthermore, the propagation phase is considered as the number of cycles to propagate a crack of an initial length a i , until failure of the component. The curve (a i -N p ) is obtained integrating the crack growth from the different values of a i up to failure, see Fig. 6. In this part, fracture mechanics-based methods were used. Therefore, after analysing both phases separately, two curves are obtained which addition gives the total life curve, representing the total cycles as a function of the initiation crack length, (a i -N i ) +(a i -N p ) =(a i - N T ), see Fig. 6. The minimum of the curve determines the fatigue life estimation N t *=N i *+N p *; furthermore, this total life value provides the initiation crack length value. This value can be considered as the proper crack initiation length, indicating the point at which the initiation phase concludes, and the propagation phase begins (see Fig. 6 and Fig. 7). It can be proved that the crack growth rate based on the curve (a i -N i ) is higher than the one from the curve (a i -N p ) for crack lengths below the initiation length, a i *, and vice versa above this length [35], which is an explanation with more physical meaning. To develop this model, it is essential to establish a location on the surface where a crack could potentially initiate. This point is referred to as the critical point (see Fig. 7). The conventional approach often involves selecting the contact trailing edge as the critical point. However, this usually occurs with a smooth surface. In the case of a rough surface, it is uncertain whether this approach is accurate or not. For this reason, in this study, the critical point is considered to be where the axial stress is highest at the surface. Following that, the crack path will be assumed to be perpendicular to the contact surface. This assumption has been verified in [33], although in a real situation, the crack path is not entirely perpendicular; rather, it exhibits a slight inclination [36]. Next, various crack lengths along the path will be evaluated. For each length, as previously discussed, the initiation and propagation phases will be analysed according to [3,33]. From these analyses, the total life will be determined, and consequently, the initiation crack length will be obtained, see Fig. 7. It is important to note that there is a difference between [33] and the current work. In this study, the damage parameter used is the Smith-Watson-Topper (SWT) parameter [37], Eq. 2, and is evaluated at the depth corresponding to each crack length, instead of using the average of the stresses from the critical point to crack length. SWT = σ maxΔ ε 2(2) Where σ max and Δ ε are the normal stress and the range of the normal strain to the material plane and along the loading cycle. 4. Numerical results 4.1. Fretting fatigue predictions The described procedure to obtain the fatigue life is applied using the stress/strain fields obtained with the numerical formulation developed in the Appendix. At first, in order to validate the fatigue model, the experimental results obtained with the completely smooth specimens are compared with the predicted ones in Fig. 8. In this representation, Fig. 6. Scheme of life prediction model [24]. Fig. 7. Fatigue crack estimation and stress distribution. Fig. 8. Experimental vs estimated life for smooth specimens. M. Moreno-Rubio et al. Tribology International 193 (2024) 109444 6 the estimated lives are plotted on the y-axis, and the experimental are plotted on the x-axis. Thus, if the estimated and experimental life are equal, the red circles coincide with the central dashed line. However, if the estimated lives exceed the experimental ones, the circles would lie above this line; conversely, they would fall below it if the estimated lives were shorter. The lines “x2 ″ and “x3 ″ indicate where an estimated life would be double or triple the experimental one. The other two lines symmetric with respect the central one indicate estimated lives 1/2 and 1/3 of the experimental one. These lines help assess how good or bad are the fatigue estimations. The results show that almost all the estimations lie inside the scatter band of 2. Nevertheless, this is not applicable for large lives, as the model predicts infinite lives. Likely this bad behaviour can be improved with a better knowledge of the crack growth threshold, ΔKth, especially under fretting situations. As the fatigue model is accurate enough, specifically for lives ranging between 10 4 to 10 5 cycles, it is applied to all the shot peened profiles aforementioned and to all of the fretting loading states here considered. Thus, as mentioned earlier, 20 profiles were analysed, each with their respective 41 sub-profiles, resulting in a total of 820 fatigue lives per loading combination. Fig. 9 compares the results between the estimated fatigue lives and the experimental values. In this plot, the minimum (among the 41 sub-profiles) fatigue lives obtained for each profile and for each load combination are represented. The boxplots represent the value of the median, as a line inside the box, the upper and lower limit of the box itself represent the percentiles 75% and 25% respectively, and the values outside this range are depicted as a cruciform point. It observed that, the median value falls within a scatter band of 2. Nevertheless, with load combination 1 the model is not able to accurately predict the fatigue life as occurs with the smooth surface. These results show that the procedure here considered for transforming a 3D analysis into many 2D analysis, and choosing from all of the different profiles (including also the sub-profiles) that one leading to the most unfavourable result, is adequate and yields very good results in terms of life prediction. Besides, it is also remarkable that all the median values are inside the scatter bands of two. It is true that taking the minimum value is less accurate, but more conservative. After a more in-depth analysis of this graph, it was determined that, in order to achieve an accurate fatigue life prediction, it is necessary to study at least 6 different profiles. With this number of profiles, it can be ensured that the life results will fall within a scatter band of 2. Lastly it is noteworthy that the predictions tend to be of the order of or greater than the experimental results. A proper method to analyse quantitatively the displayed graph (Fig. 9) is to calculate the ratio of the average fatigue life between NT est and NT exp. This ratio is represented by the parameter RNf : RNf =10∑ n i=1..n log10NT est /NT exp n(3) It is important to note that the value used in the estimated life is the median value obtained. This parameter provides us with the average distance value between the medians of the x1 line. If RNf =1, we are on the x1 line, with NT est equal to NT exp. When RNf >1, NT est >NT exp, and when RNf <1, we find that NT est <NT exp. For Fig. 8, the RNf value was 0.95, this value indicates a good life estimation with the model developed. For Fig. 9, we obtain a RNf value of 1.07, indicating that the estimated life (considering the median value) is almost equal to the experimental life for rough surface, the standard deviation value for the RNf is 1.5. This also validates the model used. The same remark can be addressed based on Fig. 10 where the estimated fatigue lives for smooth specimens are represented vs. the minimum values estimated for each profile in boxplot form (Fig. 9). It is observed that the median value for the rough specimens is always larger than the value of the smooth specimen. All the results, inside the box, lie in or over the median line, which means that larger life predictions are obtained for the rough specimens, as in the experiments. This increase in life is not pronounced, which is in agreement with the experimental results obtained from Table 3. There is a possibility of an improvement in life due to surface roughness, although this benefit may be overshadowed by the dispersion observed in both experimental and estimated data. To conclude this section, Fig. 11 depicts the estimated life for smooth surface versus the experimental life for a rough surface, in order to evaluate how shot peened roughness influences a real experimental test with respect to the estimated life of an ideally smooth surface. Upon analysing the RNf parameter for the values in this figure, we obtain a value of 0.77, suggesting that the experimental life for the rough surface is slightly higher than the estimated life for an ideal smooth surface. In other words, when considering roughness, the estimated results are more accurate. Nevertheless, if roughness is not considered the error assumed is not that high. In this case, the standard deviation value for the RNf is 1.5, matching that of Fig. 9. Fig. 9. Experimental vs estimated life for rough specimens. M. Moreno-Rubio et al. Tribology International 193 (2024) 109444 7 4.2. Subsurface analysis In view of the above fatigue predictions, a more exhaustive analysis has been performed to understand the origin of these results. To do so, first the SWT damage parameter and the SIF in mode I have been represented in Fig. 12 as a function of the depth below the contact surface. These results have been obtained for a vertical path emanating from the critical surface point depicted in Fig. 7, point where the maximum value of axial stress is found. Among all possible results, due to the large number of profiles/sub-profiles, the evolutions shown are those corresponding to the minimum value in terms of life obtained for each loading combination. Fig. 12a compares the results for the SWT damage parameter as we move beneath the contact surface of the test specimen. Continuous lines represent the evolution of the SWT obtained with pads having smooth contact surfaces, and dashed lines for those with a shot peened surface. Here it is possible to note that, independently of the loading state, the damage parameter at the surface is very large for the rough surface compared with the smooth surface, which is in accordance with the elastic behaviour considered and the large stresses produced at the peaks of the profile. However, due to the steep gradient, quickly the damage Fig. 10. Estimated life for smooth surface vs estimated life for rough surface. Fig. 11. Estimated life for smooth surface vs experimental life for rough surface. Fig. 12. SWT damage parameter and FIT mode I for smooth specimens vs rough specimens. M. Moreno-Rubio et al. Tribology International 193 (2024) 109444 8 parameter decreases to values smaller than those of the smooth specimen, and approximately coalescing for depths larger than 0.5 millimetres. This observation is directly related to the experimental results because it is well known that the crack nucleation is a volumetric process. That is why, although very high values are reached for the rough surface cases, these are quickly compensated as we advance through the depth of the specimen, taking place all this for depths less than 200 µm. On the other hand, if the SIF, which is directly related to fatigue crack propagation rate, is analysed, something similar is observed, Fig. 12b. Near the surface, higher values are reached for the rough surface cases than for the smooth surface cases. However, both curves tend to coalesce even faster than those of the SWT damage parameter. Therefore, the effect of roughness on crack propagation is only important at the very near surface material. Therefore, as already mentioned in previous paragraphs, the results suggest that roughness may be favourable for the increase of life, specifically that this improvement is more associated with the crack initiation phase and the early crack propagation phase. However, this possible improvement is not clear, since it is smaller than the dispersion of the results. In order to better understand what happens under the contact, the SIF in mode I and mode II were analysed for different crack lengths and orientations in order to cover the possible actual crack orientation, although in previous results it was assumed that the crack grows perpendicular to the contact surface. The SIF analysis was conducted for one profile per load combination, and similarly to the previous analyses, specifically the one that provides the lowest estimated life for each load combination. To begin, the stress field inside the test specimen and the critical point must be known. It is assumed that the critical point is located where the superficial axial stress is at its maximum. In the case of smooth profiles, as it is an ideal surface profile, the maximum axial stress matches with the contact trailing edge. From the critical point, different crack lengths will be considered, from S 1 , S 2 , S 3 up to S n . For each crack length, the SIF will be analysed for different orientations of the crack, with angles ranging from −75◦to 75◦, and 0◦representing a crack perpendicular to the surface. For each of these critical points, the SIF will be calculated by tracing a straight line from the critical point to the new point and integrating the stresses along this line using a weight function for inclined cracks [19,36,38]. These data will be used to generate a polar contour plot. A scheme of this procedure can be seen in the Fig. 13. The mode I SIF (KI) has been calculated using only the positive stress range, since we assume that the crack is closed when KI is negative. In contrast, for the mode II SIF, KII, both the positive and negative SIF have been taken into account. This is because under mode II loading conditions both the positive and negative SIF play a role for the crack propagation. In Fig. 14, two types of figures can be observed: those depicting the relationship between KI and those illustrating the relationship of KII. In these, it is possible to see that at 0º, the SIF in mode I for a smooth surface, ΔKSM I, compared to the rough surface, ΔKRO I, is slightly higher. Usually, in actual tests, fretting cracks have a small inclination between 16º and 25º [24], depending on the test. If we take this consideration, we can observe how in KI appears a great difference at 25º for both loads, where the ΔKSM Itends to be much larger than ΔKRO I, around 2–5 times greater. At this angle, the KII for rough surface, ΔKRO II , behaves differently depending on the load level. For load 3, close to the surface, there is a small area where the ΔKRO II is less than ΔKSM II , while for the load 5, we can observe this difference not only close to the surface but also at a greater depth. Considering the value of KII to determine the angle at which the crack is more likely to grow, a more significant difference is observed between ΔKRO II and ΔKSM II at 15º, and this difference is more pronounced under load 5. In contrast, the difference for KI at 15º is lower than at 25º. Nevertheless, it is appreciable how the ΔKRO I is consistently lower than ΔKSM I, approaching 1 under load 3. This disparity is also evident in experimental tests, Table 3, where the fatigue lives under load 3 for rough surface are similar to those of the smooth surface. Conversely, for load 5, the rough surface shows a longer life than smooth surface. This suggests that if an inclined crack had been considered, the fatigue life would be greater in the rough case than in the smooth case. 5. Conclusions In this study, we investigated how the rough surface generated by shot peening affects the prediction of fatigue life in fretting fatigue. To achieve this, various shot-peened contact pads were analysed, and stress/strain fields were determined for each analysis to estimate subsequent fatigue life. The major conclusions are as follows: •The fatigue model was validated using smooth profiles, followed by analysis and comparison of both smooth and rough cases. This comparison reveals that surface roughness affects stress/strain distribution, initiation and propagation phases, SWT parameter, and mode I SIF. •The SWT parameter and the mode I SIF, in most cases, tend to be marginally lower when a shot peened rough surface is considered, especially for depths below the first tenths microns. This suggests that the rough surface primarily affects the initiation phase rather than the propagation phase. •It is possible to conclude that not only the compressive residual stress affects the fretting fatigue life when the shot peening treatment is used, but also the modification of the surface roughness, albeit to much lesser extent. The experimental tests have quantified this effect. •The theoretical model developed is able to separate the beneficial effect of roughness from residual stresses of the SP. Although the difference between high and low cycle fatigue could not be observed in the theoretical model developed. •It has been observed that using the actual surface instead of the ideal one in the theoretical model yields more accurate results. Statement Originality The authors, Maria Moreno Rubio, Diego Erena, Jesús V´ azquez and Carlos Navarro affirm that the article entitled as “STUDY OF THE SHOT PEENING SURFACE ROUGHNESS IN FRETTING” has not been published previously; is not under consideration elsewhere; has the full consent of all authors; and that, if accepted, will not be published elsewhere in the same form, in English or in any other language, without the written consent of the Publisher. CRediT authorship contribution statement M. Moreno-Rubio: Writing – original draft, Validation, Fig. 13. Procedure to obtain de SIF. M. Moreno-Rubio et al. Tribology International 193 (2024) 109444 9 Methodology, Investigation. D Erena: Writing – review & editing, Writing – original draft, Validation, Software, Investigation, Formal analysis, Data curation. Jesús V´ azquez: Writing – review & editing, Supervision, Software, Project administration, Investigation, Funding acquisition, Data curation, Conceptualization. Carlos Navarro: Writing – review & editing, Supervision, Project administration, Funding acquisition. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability Data will be made available on request. Acknowledgements The authors wish to thank the Ministry of Science and Innovation for funding the research through the project RTI2018–096059-B-I00. Appendix This appendix shows the basic equations and the numerical procedure used to obtain the surface normal and shear stresses, σ yy(x)and σ xy(x) respectively, in a two-dimensional elastic rough contact. In this procedure, both bodies are assumed to be of the same material and behave as elastic half-planes, thus the classical formulae for hertzian contacts are herein used. Note that in the case of a body having a rough surface the half-plane assumption is quite questionable, especially in the very near surface material. Nonetheless, this weak point in the formulation can be circumvented due to the extremely high gradient with depth of the contact stress and strain fields, producing a rapidly homogenisation of these fields and thus leading to a behaviour quite similar to that of a smooth contact, but taking into account the effect of the rough surface. The first equation is that relating the surface normal pressure, σ yy(x), the relative profile between the contacting bodies, h(x), and the vertical rigid body displacement, δ [39]: δ=4(1− ν 2) π E∫contact σ yy(s)ln|x−s|ds +h(x)(A-1) In a rough contact it is expected that both bodies touch at multiple, and non-connected, contact zones, and thus a discretized version of the above integral is required. For this, initially we need to assume the total contact length, i.e. the minimum and maximum x-coordinate of all the multiple contact zones. In order to achieve convergency, the initial assumed total contact length must be greater than that obtained after applying the numerical procedure. For the time being, assume that the multiple contacts are produced along the segment [a,b]. This segment is discretized into n smaller segments/elements of constant length Δ, and then we assume that the normal stress, σ yy, is constant along each of these smaller elements. In that manner, along the i-element of length Δ, and extending from si to si+1, the normal stress is constant and equal to σ i yy. With all these, for a point xj (collocation point) located at the centre of the j-element, the integral appearing in equation A.1 along the i-element can be calculated as: ∫si+1 si σ yy(s)lnxj−sds = σ i yy ∫a+Δi a+Δ(i−1) ln Δ(j−1 2)−s ds = σ i yyKi,j(A-2) where: Fig. 14. Comparison of SIF between smooth and rough surface (K SM/KRO), in mode I (a, c) and mode II (b, d). M. Moreno-Rubio et al.