Effect of rolling on fretting fatigue assessment of cylindrical contact in partial slip regime

Tribology International 188 (2023) 108772 Available online 7 July 2023 0301-679X/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Effect of rolling on fretting fatigue assessment of cylindrical contact in partial slip regime Sabrina Vantadori a , * , Andrea Zanichelli a , Diego Erena b , Jesús V´ azquez b a Department of Engineering & Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy b Universidad de Sevilla, Escuela T´ ecnica Superior de Ingenieros, Departamento de Ingeniería Mec´ anica y Fabricaci´ on, Camino de los Descubrimientos s/n, C.P., 41092, Spain ARTICLE INFO Keywords: Crack path Fretting fatigue Fatigue lifetime Rolling ABSTRACT The present research work deals with the fatigue assessment of an Al 7075-T651 subjected to fretting fatigue testing characterised by cylindrical contact. A small rolling of the pads is experimentally observed and, consequently, the contact surface stresses are different to the hertzian one. Therefore, the methodology, recently proposed by some of the present authors for the fatigue assessment of fretting-affected metallic components, is here employed for the first time in conjunction with a FE numerical model able to take into account the effect of the rolling of the pads, in order to find out both the crack nucleation orientation and the lifetime for each specimen of the experimental campaign analysed. 1. Introduction Nowadays, a high variety of metallic structural components, such as running cables and overhead conductors, shrink-fitted shafts, spline coupling, dovetail joints, bolt and riveted joints, fuel rods cladding and steam generator tubes of nuclear power plants, are still subjected to failures due to fretting fatigue, thus resulting in catastrophic events [1–7]. In such a context, where high stress-gradients arise near the contact zone [8,9], specific studies have been aimed to determine the stress field within the components in contact, by employing both analytical and numerical tools. Relatively to the analytical tools, Hertz [10] firstly studied the case of frictionless contact between two homogeneous linear-elastic half-spaces in presence of a constant normal load. Under the assumptions that the elasticity theory holds and the contact area is very small with respect to the size and radii of curvature of the bodies in contact, such a solution can be applied to any pair of either spherical or cylindrical shaped bodies characterised by different radii of curvature. Starting from the hertzian solution, many researchers proposed analytical closed-form solutions related to different fretting fatigue conditions. Cattaneo [11] and Mindlin [12] considered the presence of a tangential load in addition to the constant normal load generating a partial slip condition that is, the case of the contact area partialisation into an outermost zone (where relative surface micro-slips occur) and an innermost zone (where no relative motion takes place between the two bodies). This latter solution was subsequently extended to the case of a bulk cyclic load acting on one of the two bodies in contact [13], causing the shift of the inner stick region. Moreover, V´ azquez et al. [14] studied the case of a high value of the applied bulk load, determining two distinct outer slip zones where micro-slips of opposite sign occurred. Other attempts have been done to extend the above solutions, for instance, to more general geometries [15] and problems characterised by varying normal load [16]. Another approach to the fretting fatigue problem was provided by the asymptotic analysis, according to which the stress concentration due to the contact is considered analogously to cracks in fracture mechanics. Such an approach has been mainly employed to analyse conditions where the hertzian theory fails. The case of complete contact problems characterised by sharp contact edges, for instance, was studied by Williams [17] and Bogy [18]. Moreover, Dini and Hills [19] analysed the fretting fatigue condition in partial slip regime, providing a comparison with the results obtained by employing the abovementioned Cattaneo-Mindlin solution, whereas Fleury et al. [20] considered more complex loading histories. Relatively to the numerical tools, many efforts have been made by the researchers in recent decades in order to cover all the situations where the assumptions of the analytical theories are violated. In such a * Corresponding author. E-mail address: [email protected] (S. Vantadori). Contents lists available at ScienceDirect Tribology International journal homepage: www.elsevier.com/locate/triboint https://doi.org/10.1016/j.triboint.2023.108772 Received 29 May 2023; Received in revised form 27 June 2023; Accepted 5 July 2023 Tribology International 188 (2023) 108772 2 context, the Finite Element (FE) method has been largely employed, providing satisfactory results in analysing fretting fatigue problems characterised by complex geometry and loading history, presence of voids in the material and specific physical phenomena (i.e., rolling, debris formation, mass-eccentricity) that could not be considered by the theoretical formulations for reasons of simplicity and generality of the solution [21–29]. Moreover, FE models have been developed in order to consider the effect of wear [30] or to overcome the limitations of an elastic analysis by including plasticity [31]. Further, the extended FE method has recently provided the opportunity to enrich the solution of standard FE method, by implementing discontinuous functions. Mo¨ es et al. [32] developed an extended FE model to simulate, without any remeshing, the fatigue crack growth. In order to estimate the stress intensity factors in complete contact problems by employing a coarse mesh in favour of a low computational effort, Giner et al. [33] enriched the FE formulation by employing singular expressions. Moreover, Cardoso et al. [34] have used enrichment functions to achieve accurate contact stress solutions even when using coarse meshes. Recently, some of the present authors have proposed a methodology for the fatigue assessment of fretting-affected metallic structures [35–38]. According to such a methodology, whose peculiarity is also to incorporate a material microstructure-related parameter, the fretting fatigue assessment is performed through the multiaxial fatigue criterion by Carpinteri et al. [39–42], employed in conjunction with the Critical Direction Method [43,44]. Nomenclature aH theoretical hertzian contact half-width. aR average experimental value of the contact half-width. Ca amplitude of the stress component lying on the critical plane. d average grain size of the material. E elastic modulus. E* elastic modulus for plain strain condition. Hcal estimated crack nucleation location (hot-spot). Hexp experimental crack nucleation location. m slope of the S-N curve under fully reversed normal loading. m* slope of the S-N curve under fully reversed shear loading. Na amplitude of the stress component normal to the critical plane. Ncal number of loading cycles to failure estimated neglecting the rolling of the pads in the FE model. NR cal number of loading cycles to failure estimated incorporating the rolling of the pads in the FE model. Neq,a equivalent normal stress amplitude. Nexp experimental number of loading cycles to failure. Nm mean value of the stress component normal to the critical plane. ¯ Na average value of Na. ¯ Nm average value of Nm. P constant normal force applied to the pads. Q(t)cyclic tangential force applied to the pads. Qa amplitude of the cyclic tangential force. R radius of the pads. t time. Scr verification point. α orientation of the critical plane candidate. α cal crack nucleation orientation estimated neglecting the rolling of the pads in the FE model. α R cal crack nucleation orientation estimated incorporating the rolling of the pads in the FE model. α exp experimental crack orientation. f friction coefficient. ν Poisson coefficient. θ(t)cyclic rotation of the pads. θmax maximum value of rotation of the pads. σ af,−1 material fatigue strength under fully reversed normal loading, at a reference number N0 of loading cycles. σ B(t)cyclic axial stress applied to the specimen. σ B,a amplitude of the cyclic axial stress. σ u ultimate tensile strength. τ af,−1 material fatigue strength under fully reversed shear loading, at a reference number N0 of loading cycles. Fig. 1. Schematisation of the contact problem analysed. S. Vantadori et al. Tribology International 188 (2023) 108772 3 In the present research work, the abovementioned methodology is employed for the first time, together with the FE numerical model recently proposed by Erena et al. [23,45], for the analysis of the fretting fatigue tests presented in Ref. [46], characterised by a cylindrical contact and rolling of the contact pads. Both a FE model neglecting and a FE model incorporating the rolling of the pads are considered. Subsequently, by exploiting the stress field coming from the above FE analyses, the methodology for fretting fatigue assessment is employed in order to determine both the crack nucleation orientation and the fatigue life of the tested specimens. 2. Experimental fretting tests analysed The experimental campaign carried out by some of the present authors [46] at the University of Seville is analysed in the present paper. In particular, fretting fatigue tests were performed in partial slip regime. The testing setup consisted in two pads, characterised by a cylindrical shape of the contact surface with a radius, R, equal to 100 mm, placed in contact with a flat specimen, characterised by a dog-bone shape with a rectangular cross section in the central part (being the width equal to 10 mm, whereas the thickness equal to 8 mm). The schematisation of the contact problem involving both pads and specimen is shown in Fig. 1. Both specimens and pads were made of Al 7050-T651 aluminium alloy, which is widely employed due to its high strength-to-weight ratio in a variety of applications where catastrophic fretting fatigue failures still occur (such as aircraft structural components, gears, fuse parts, connections, highly stressed parts, and structural components used in the automotive and aerospace industries). Such an alloy is characterised by the mechanical [45] and fatigue [47] properties that have been listed in Table 1, an average grain size, d, equal to 50 μ m [45] and a coefficient of friction, f, of 0.75 (in a cylindrical contact configuration) [46]. Three different fretting loading conditions were examined, as listed in Table 2. During each test, a constant normal force, P, was firstly applied to the pads, in order to push them against the specimen, as schematically shown in Fig. 1. Subsequently, a cyclic axial stress, σ B(t) (loading ratio equal to −1), was imposed on the specimen, with a frequency equal to 10 Hz (Fig. 1). Consequently, due to the compliance of the fretting device, a cyclic tangential load, Q(t)(in phase with the cyclic axial stress), arose in the pads with opposite direction with respect to σ B(t)(see Fig. 1). The amplitude, Qa, of such a tangential load depended on the amplitude, σ B,a, of the axial stress applied on the specimen, and could be adjusted by modifying the stiffness of the adjustable supports of the pads. Details may be found in Ref. [46]. The experimental number of loading cycles to failure, Nexp, is listed in Table 2 for each tested specimen. The contact zone was measured in correspondence of the fretting scars by means of an optical microscope. Then, such images were analysed through a CAD software to compute the average value of the contact half-width, aR, listed in Table 2. More details related to such a procedure can be found in Ref. [45]. It was observed that, for each loading condition analysed, the value of aR was significantly higher than the theoretical hertzian contact half-width, aH, given by: aH= 4PR π E* √(1) As it is proved in Ref. [23], this was the result of the rolling phenomenon, that is, due to the stiffness of the supports, when the cyclic axial stress was imposed on the specimen, the pads experienced a rotation that promoted the contact zone to move in the direction opposite to that of the cyclic axial load applied (see Fig. 2). In other words, if an entire loading cycle was considered, although the contact width was instantaneously equal to 2 aH, the overall observable contact width (i.e., the width of the scars at the end of the cycle) was greater than 2 aH and equal to 2 aR. The maximum value of the above rotation, θmax, was computed by employing the pad radius and the arc length of the pad surface that came into contact due to rolling, obtained as aR−aH [23]. Such a maximum value was equal to 0.19º for tests No. T1, T2, T5 and T6, and 0.15º for tests No. T3 and T4. Moreover, the crack nucleation location on the contact surface, Hexp, was found by employing the abovementioned procedure described for Table 1 Mechanical [45] and fatigue [47] properties of the Al 7050-T651 aluminium alloy examined. Material E[GPa] ν [ − ] σ u[MPa] σ y[MPa] σ af,−1[MPa]m[ − ] τ af,−1[MPa]m*[ − ] N0[cycles] Al 7050-T651 71 0.33 572 503 193 -0.12 111 -0.12 2 . 10 6 Table 2 Fretting loading conditions applied (P, Qa and σ B,a), theoretical and experimental contact half-width (aH and aR), and experimental both crack nucleation location (xexp), crack orientation ( α exp) and fatigue life (Nexp). Test No. P[N/mm]Qa[N/mm] σ B,a[MPa]aR[mm]aH[mm]xexp[mm] α exp[◦]Nexp[cycles] μ [◦] σ [◦] T1 527 193 110 1.64 1.30 1.41 32.0 13.9 88216 T2 89376 T3 679 121 150 1.74 1.47 1.54 15.9 4.5 47737 T4 51574 T5 679 193 150 1.80 1.47 1.48 28.5 7.5 50268 T6 39202 Fig. 2. Schematisation of the rolling phenomenon (only one pad is represented in the Figure). S. Vantadori et al. Tribology International 188 (2023) 108772 4 the contact width determination. More precisely, an average value xexp (see the reference system shown in Fig. 1) of the crack nucleation location was obtained for each tested specimen. The average crack nucleation location experimentally observed, xexp, is listed in Table 2 for each loading condition. Furthermore, in order to analyse the crack profiles experimentally observed, a three-dimensional map of the crack surface was obtained through a confocal microscope for each tested specimen [45]. Subsequently, from such 3D maps, different experimental crack profiles were extracted for each specimen analysed and reported in Fig. 3. The cracks were observed to be characterised by an experimental crack orientation, α exp, equal to 13◦and 39◦for the tests No. T1 and T2, 9◦and 18◦for the tests No. T3 and T4, and 24◦and 31◦for the tests No. T5 and T6. Note that such values have been obtained by linearly interpolating each experimental crack profile up to 100 μ m depth. The angle measured from the obtained line and the line perpendicular to the material surface has been assumed as α exp. The mean values, μ , and corresponding standard deviations, σ , of the experimental crack orientation, α exp, are listed in Table 2. 3. FE numerical models employed The FE numerical models, employed for the analysis of the fretting fatigue tests summarised in Section 2 , are hereafter described. In particular, a FE model that neglects the rolling of the pads and a FE model that incorporates its effect (both of them recently proposed by Erena et al. [23,45]) are employed in the present work for the determination of the stress field in the tested specimens. 3.1. Numerical model neglecting rolling A FE numerical model has been recently proposed by Erena et al. [23, 45] for the analysis of fretting fatigue tests characterised by cylindrical contact. In particular, by employing such a model, the experimental campaign described in Section 2 may be simulated by neglecting the rolling of the pads. The commercial software ANSYS® is employed for the numerical simulations. As shown in Fig. 4 (a), only half a specimen is considered Fig. 3. Experimental crack profiles related to the tests No.: (a) T1-T2, (b) T3- T4, and (c) T5-T6. Fig. 4. FE models employed: (a) neglecting and (b) incorporating the rolling of the pad. S. Vantadori et al. Tribology International 188 (2023) 108772 5 and the contact with one of the two fretting pads is modelled, since the contact problem analysed is characterised by a symmetry condition. The boundary conditions used to reproduce the testing setup are shown in the same figure too. A constant normal force and a cyclic tangential force are applied to a master node (the rotation of such a node is restricted), which is able to distribute such loading conditions to all the nodes that are located at the top of the pad, whereas a cyclic axial stress is directly applied to the specimen. The constant normal force, P, is applied in the first load step in order to bring the specimen and the pad into contact. Subsequently, Q(t) and σ B(t)are simultaneously applied with opposite directions and made to vary, while P is kept constant. Each load is applied by means of small increments according to the loading history shown in Fig. 5 (a). Note that, in the load step No. 4 the maximum value of σ B(t)is applied to the specimen, whereas the minimum one is attained in the load steps No. 2 and 6. A plane strain condition is assumed in the numerical model, due to the values of both the pad/specimen thickness and the contact length. Therefore, 2D plane linear elements, named PLANE182, characterised by two degrees of freedom at each node, are used with triangular or quadrilateral shape in both the specimen and the pad. A total number of 315000 finite elements has been used. In order to accurately capture the high stress/strain gradient due to the contact problem, a very refined mesh is defined after a convergence analysis at the contact interface, with 5 µm-size quadrilateral elements. Moreover, CONTA171 and TARGE169 elements are used to model the contact pair on the specimen and pad surfaces, respectively. In this region of the model, the augmented Lagrangian algorithm is set for the contact, and a Coulomb friction law is used with a coefficient of friction equal to 0.75. The default values for the normal, FKN, and tangential, FKT, stiffness factors and for the slip tolerance factor, SLTO, are assumed. 3.2. Numerical model incorporating rolling A FE numerical model has been recently proposed by Erena et al. [23, 45] for the analysis of fretting fatigue tests characterised by cylindrical contact, by also taking into account the rolling of the pads. In particular, by employing such a model, the experimental campaign described in Section 2 may be simulated incorporating the rolling of the pads. The commercial software ANSYS® is employed for the numerical simulations. As shown in Fig. 4 (b), only a half of the specimen and one pad are considered in the model due to symmetry condition. The boundary conditions used to reproduce the testing setup are shown in the same Figure too. A constant normal force, a cyclic tangential force and a cyclic inplane rotation are applied to a master node, which is able to transfer them to all the nodes that are located at the top of the pad, whereas a cyclic axial stress is directly applied to the specimen. The constant normal force, P, is applied in the first load step in order to bring the specimen and the pad into contact. Subsequently, the cyclic tangential force, Q(t), the cyclic in-plane rotation, θ(t), and the cyclic axial stress, σ B(t), are simultaneously applied and made to vary, while P is kept constant, according to the loading history shown in Fig. 5 (b). Note that, in the load step No. 4 the maximum tensile stress is achieved in the specimen, whereas both a tangential load (acting in the direction corresponding to −x) and a counter clockwise rotation are applied to the master node. The maximum value for the cyclic in-plane rotation, used in the present numerical model incorporating rolling, was computed as the average value of the maximum rotation experimentally observed for each loading condition (see Section 2), that is 0.19º for the simulations of the tests No. T1, T2, T5 and T6, and 0.15º for that of the tests No. T3- T4. The element types, contact formulation and parameters are the same described in Section 3.1. 4. Methodology for fretting fatigue assessment The methodology here applied for the fatigue assessment of frettingaffected metallic structures [35–38] needs some input data. More in detail, it is necessary to know: – the geometric sizes of the bodies in contact (i.e., the specimen and the pads); – the loading and the boundary conditions of the contact problem; – the material mechanical properties (for both the specimen and the pads), that is, the elastic modulus, E, the Poisson coefficient, ν , and the ultimate tensile strength, σ u; – the material fatigue properties for the specimen, that is, both the fatigue strengths, σ af,−1 and τ af,−1, and the slope of the S-N curve, m and m*, under fully reversed normal and shear loading, respectively, at a reference number N0 of loading cycles; – the material average grain size, d, for the specimen; – the coefficient of friction, f, between the specimen and the pads. First of all, the stress field in the specimen needs to be computed. In the present research work, the FE numerical models described in Section 3 are used to this aim. Fig. 5. Loading histories applied in FE models: (a) neglecting and (b) incorporating the rolling of the pad. Fig. 6. Scheme for determining the critical plane orientation. Positive values of α are considered when the segment is towards the centre of the contact zone. S. Vantadori et al. Tribology International 188 (2023) 108772 6 Subsequently, the crack nucleation location, Hcal, (that is, the hot-spot location on the material surface) is found: it is the point on the material surface achieving the maximum value of σ 1 in one fretting cycle, being σ 1 the maximum among the principal stresses. Subsequently, the crack nucleation orientation is determined. This orientation is assumed to coincide with that of the critical plane, α cr, which is found according to the Critical Direction Method (CDM) proposed by Araujo et al. [43,44]. By assuming a plane strain condition for the considered contact problem, each critical plane candidate appears as a segment of given length, l, which starts from Hcal, as shown in Fig. 6. According to the present methodology, the physical size l is assumed equal to 2d. Moreover, the orientation, α , of such a segment is identified as shown in Fig. 6. A fatigue parameter is computed for each segment obtained by varying α (−90◦≤ α ≤ +90◦and Δ α =1◦). The orientation identifying the maximum value of the fatigue parameter is considered as that corresponding to the critical plane one, α cr. According to the present methodology, the fatigue parameter is Neq,a, and it is given by [39–42]: Neq,a=¯ Na+ σ af ,−1(¯ Nm σ u)(2) being ¯ Na and ¯ Nm the average values, computed for each α −oriented segment, of the amplitude, Na, and of the mean value, Nm, of the normal stress component perpendicular to the critical plane candidate. Note that, Na and Nm are computed in many equally-spaced points along each critical plane candidate, from the hot-spot up to 2d (see Fig. 6). Finally, the fatigue life, Ncal, is estimated according to the Carpinteri et al. criterion [39–42], by employing the stress components acting on the critical plane at the verification point, Scr, located at a distance 2d from the hot-spot along a segment with an orientation α cr, by employing the following equation:  [Na(2d, α cr) + σ af ,−1(Nm(2d, α cr) σ u)]2 +( σ ′ af ,−1 τ ′ af ,−1)2 [Ca(2d, α cr) ]2 √ = σ ′ af ,−1(3) where Ca is the amplitude of the shear stress component lying on the critical plane, and σ ′ af,−1 and τ ′ af,−1 are given by: Fig. 7. Maximum principal stress, σ 1, along the contact surface, obtained with the FE model neglecting rolling at different time steps between the load steps No. 4 and 6 (see Fig. 5(a)), for the loading conditions of the tests No.: (a) T1-T2, (b) T3-T4, and (c) T5-T6. Fig. 8. Tests No. T1-T2: (a) amplitude and (b) mean value of the normal stress component computed in equally-spaced points along each α −oriented segment, from the hot-spot (point 1) up to 2d (point 10), by neglecting the rolling of the pad. S. Vantadori et al. Tribology International 188 (2023) 108772 7 σ ′ af ,−1= σ af ,−1(Ncal N0)m (4a) τ ′ af ,−1= τ af ,−1(Ncal N0)m* (4b) 5. Results and discussion The experimental campaign described in Section 2, is herein analysed by employing the methodology for fretting fatigue assessment summarised in Section 4, where the stress state which serves as an input for this methodology is determined through the FE numerical models described in Section 3. In particular, the results obtained without considering the rolling of the pads (that is, by employing the FE numerical model described in sub-Section 3.1) are reported in sub-Section 5.1, whereas sub-Section 5.2 is devoted to the results obtained by taking into account the effect of rolling (that is, by employing the FE numerical model described in sub-Section 3.2). 5.1. Results obtained neglecting rolling Firstly, the hot-spot, Hcal, is determined. In particular, such a point is found at xcal =1.25mm, 1.42mm, and 1.42mm for the simulations of the tests No. T1-T2, T3-T4, and T5-T6, respectively. In particular, it can be noted that the hot-spot is found inside the slip zone for all the simulations performed, in agreement with the experimental results. By considering the loading conditions of the tests No. T1-T2, T3-T4 and T5- T6, the maximum principal stress, σ 1, along the contact surface is plotted in Fig. 7(a), (b) and (c), respectively, for 14 different equally-spaced time steps between the load steps No. 4 and 6 (see Fig. 5(a)). Subsequently, the crack nucleation orientation, α cal, is determined for each loading condition analysed. Na and Nm are computed in 10 equally-spaced points along each α −oriented segment (see Fig. 6), from the hot-spot (point 1) up to 2d (point 10). Such values are plotted in Figs. 8–10 for the loading conditions of the tests No. T1-T2, T3-T4 and T5-T6, respectively. Then, the average values, ¯ Na and ¯ Nm, of the above stress components are determined, in order to compute the fatigue parameter, Neq,a (see Eq. (2)), corresponding to each considered orientation. By maximizing the fatigue parameter (see Fig. 11 (a)), the crack nucleation orientation is evaluated for each loading condition. Such values are listed in Table 3, where it can be observed that they correspond to cracks nucleating toward the centre of the contact zone, in agreement with the experimental ones. However, it can be highlighted that the present methodology seems to fail in estimating the crack nucleation orientation when the rolling phenomenon is neglected, since the results do not fall within the range of the experimental standard deviation (see Table 2). Ncal is listed in Table 3 for each examined loading condition. Moreover, such values are compared to the corresponding experimental ones in Fig. 12(a), where the dashed lines represent the scatter band equal to 2. It is possible to observe that only one slightly non-conservative estimation has been obtained (i.e., that related to the test No. T3), and all the results were inside the scatter band 3. Therefore, a quite good correlation in terms of fatigue life has been obtained between the experimental findings and the results estimated by employing the present methodology. In order to further check the accuracy of the present methodology, the root mean square error [48] is computed as: Fig. 9. Tests No. T3-T4: (a) amplitude and (b) mean value of the normal stress component computed in equally-spaced points along each α −oriented segment, from the hot-spot (point 1) up to 2d (point 10), by neglecting the rolling of the pad. Fig. 10. Tests No. T5-T6: (a) amplitude and (b) mean value of the normal stress component computed in equally-spaced points along each α −oriented segment, from the hot-spot (point 1) up to 2d (point 10), by neglecting the rolling of the pad. S. Vantadori et al. Tribology International 188 (2023) 108772 8 TRMS =10  ∑6 j=1Log2(Ncal/Nexp )j 6 √(5) In the present case, a TRMS value equal to 1.55 has been found, thus confirming the fairly good accuracy of the results obtained. 5.2. Results obtained incorporating rolling In this case, the hot-spot, HR cal, is found at xR cal =0.94mm, 1.16mm, and 1.10mm for the simulations of the tests No. T1-T2, T3-T4, and T5-T6, respectively. In particular, it can be noted that the hot-spot is found inside the slip zone for all the simulations performed, in agreement with the experimental results. However, it has to be noted that the crack nucleation locations predicted by neglecting the rolling are closer to the experimental observations. More precisely, although with both FE models the hot-spot is found more towards the centre of the contact zone with respect to the experimental one, the horizontal coordinate computed when the rolling is incorporated in the numerical simulation, xR cal, is about 0.3mm lower than the corresponding one obtained neglecting this phenomenon, xcal, for each loading condition. Such a finding is related to the fact that the maximum value of σ 1 achieved in the fretting cycle is located more towards the centre of the contact zone when rolling is considered, than in the case of neglecting it. This can be noted by comparing respectively Fig. 7(a), (b) and (c) to Fig. 13(a), (b) and (c), where the maximum principal stress, σ 1, along the contact surface is plotted for 14 different equally-spaced time steps between the load steps No. 4 and 6 (see Fig. 5(b)), by considering the loading conditions of the tests No. T1-T2, T3-T4 and T5-T6, respectively. Subsequently, the crack nucleation orientation, α cal, is determined for each loading condition analysed. The values of Na and Nm, computed Fig. 11. Amplitude of the equivalent normal stress, N eq,a, acting on the critical plane candidate as a function of the critical plane candidate orientation, α , computed with the FE model: (a) neglecting rolling and (b) incorporating rolling. Table 3 Crack nucleation orientations, α cal and α R cal, and fatigue lives, Ncal and NR cal, obtained with the present methodology, by neglecting and incorporating the rolling of the pads, respectively. Test No. α cal[◦] α R cal[◦]Ncal[cycles]Ncal/Nexp NR cal[cycles]NR cal/Nexp T1 12 24 67948 0.77 85629 0.97 T2 0.76 0.96 T3 11 20 48653 1.02 76421 1.60 T4 0.94 1.48 T5 11 21 22130 0.44 31567 0.63 T6 0.56 0.81 Fig. 12. Experimental fatigue lives against the corresponding values calculated with the presented methodology and exploiting the FE model: (a) neglecting and (b) incorporating the rolling of the pads. S. Vantadori et al. Tribology International 188 (2023) 108772 9 along each α −oriented segment (see Fig. 6), from the hot-spot (point 1) up to 2d (point 10), are plotted in Figs. 14–16 for the loading conditions of the tests No. T1-T2, T3-T4 and T5-T6, respectively. Then, the crack nucleation orientation, that is the orientation that maximizes the fatigue parameter, Neq,a (see Fig. 11 (b)), computed according to Eq. (2) by employing the average values, ¯ Na and ¯ Nm, of the above stress components, is evaluated for each loading condition. Such values are listed in Table 3, where it can be observed that they agree with the experimental ones, since both of them indicate cracks nucleating toward the centre of the contact. Moreover, it is possible to highlight that all the estimations are in good agreement with the experimental findings, falling within the range of the experimental standard deviations (see Table 2). Accordingly, it can be stated that the present methodology is able to correctly estimate the crack nucleation orientation when the rolling phenomenon is considered. Moreover, it can be highlighted that the estimations provided by the present methodology in terms of crack nucleation orientation are more accurate when the rolling phenomenon is considered with respect to the case of neglecting it. Furthermore, NR cal is listed in Table 3 for each examined loading condition. Moreover, in Fig. 12(b) such values are compared to the corresponding experimental ones. It is possible to observe that all the results fall into scatter band 2, with 67% of conservative estimations. Therefore, a good correlation in terms of fatigue life has been obtained between the experimental findings and the results estimated by employing the present methodology. The TR RMS value is equal to 1.39, thus confirming the high level of accuracy of the results obtained. Moreover, it is interesting to observe that such a value is 10% lower than the corresponding TRMS value computed neglecting the rolling (see Sub-Section 5.1). Therefore, taking the rolling into account when applying the present methodology improves the fatigue life estimations with respect to the case of neglecting such a phenomenon. 6. Conclusions In the present paper, the fretting fatigue assessment of an aluminium alloy has been performed through a methodology based on both a multiaxial fatigue criterion and the CDM, in conjunction with FE numerical simulations. Both a FE model that neglects the rolling of the pads Fig. 13. Maximum principal stress, σ 1, along the contact surface, obtained with the FE model neglecting rolling at different time steps between the load steps No. 4 and 6 (see Fig. 5(b)), for the loading conditions of the tests No.: (a) T1-T2, (b) T3-T4, and (c) T5-T6. Fig. 14. Tests No. T1-T2: (a) amplitude and (b) mean value of the normal stress component computed in equally-spaced points along each α −oriented segment, from the hot-spot (point 1) up to 2d (point 10), by incorporating the rolling of the pad. S. Vantadori et al.