3D contact effects in fretting fatigue tests

Theoretical and Applied Fracture Mechanics 118 (2022) 103260 Available online 8 February 2022 0167-8442/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents lists available at ScienceDirect Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec 3D contact effects in fretting fatigue tests J. Vázquez, D. Erena∗, C. Navarro, J. Domínguez Departamento de Ingeniería Mecánica y Fabricación, Universidad de Sevilla, Camino de los Descubrimientos s/n, C.P 41092, Sevilla, Spain ARTICLE INFO Keywords: Fretting fatigue Cylindrical contact 3D geometrical effects FEM ABSTRACT Fretting fatigue is a material damage phenomenon produced in mechanical contacts. This phenomenon mainly produces surface cracking, and depending on the fretting regime, surface wear. In the bibliography many different types of fretting tests, with different geometries have been proposed. This paper numerically analyses, via FEM models, some fretting tests with cylindrical pads and test specimens having different geometries that are frequently used in the literature. In such cases, non-negligible 3D effects are produced. The differences between the contact stress field obtained with different approaches are analysed and discussed. 1. Introduction Fretting fatigue is a material damage phenomenon produced when two mechanical elements, that are pressed together, are subjected to time variable bulk loads, generally cyclic. The mismatch of the strains produced between both contacting surfaces leads to a relative displacement (slip) between them, at least partially. This relative displacement, in junction with the friction, produce tangential tractions on the surface which generate high stresses and strains close to the contacting surfaces [1]. The time-varying nature of these stresses and strains fields produces fatigue crack initiation and crack growth. The crack growth rate of these cracks depends on the bulk loads applied to the components, materials and many other parameters [2]. If cracks stop growing away from the surface or rotate to grow towards the surface, producing some kind of pitting, the phenomenon is called fretting wear. If the bulk stresses make the crack to continue growing from the surface until the final fracture of the component, the phenomenon is called fretting fatigue. Fretting fatigue is produced in many different systems. There are many examples of fretting fatigue failures in mechanical joints, such as rotor-blade connections in gas or vapour turbines, or in bolted, riveted and in shaft-hub connections [3]. It can also be produced in cables for electric conduction or structural applications [3]. Fretting fatigue depends on different factors, some common to the majority of fatigue processes and others typical of fretting. These include the contact pressure between the elements, the friction coefficient, tangential forces on surfaces, surfaces roughness or the amplitude of relative slip between them. In addition, the stress state near the initiation point has a high gradient, is strongly multiaxial and their components vary out of phase. An idea of the complexity of the ∗Corresponding author. E-mail address: [email protected] (D. Erena). phenomenon and the effect of some parameters can be found in the literature [2,4–7]. To analyse and understand the fretting fatigue phenomenon, many different types of tests have been proposed [8,9]. The types of tests can be divided into two groups: those trying to reproduce real situations, considering real geometries and loads, and other group of tests with simple loads and contacting geometries. In these last type of tests, the surfaces of the contacting bodies are usually spherical, cylindrical, plane and plane with rounded edges. The first group is used normally to check the capability of a real system to support the expected loads during the required working time and also to improve the fretting fatigue strength of actual designs. The complexity of the contact conditions and geometries in those cases makes very difficult to relate the failure process to the local stress and strain conditions close to the crack initiation points. That is the reason why, to better understand the phenomenon and to analyse the effect of different parameters on the fretting fatigue strength and life, tests with simple geometries and loads are carried out for research. In these tests, by using simple contact geometries, it is possible to know more precisely the stress and strain fields produced close to the contact zone, where cracks initiate and so to relate these fields with the crack initiation and propagation lives. These last types of tests are standardised [8,9] and have been carried out by many researchers [4,10–17]. Although there have been some research groups doing tests with spherical contact [4,16–19] which is the simplest and easiest type of test, chiefly due to assembly and alignment issues, most researchers made tests with cylindrical, plane or plane with rounded edges [20,21], which are closer to real contact conditions, but simpler. In these three cases, most researchers relate the crack initiation and growth process with the stresses and strains fields by using analytical or numerical 2D https://doi.org/10.1016/j.tafmec.2022.103260 Received 15 October 2021; Received in revised form 14 January 2022; Accepted 16 January 2022 Theoretical and Applied Fracture Mechanics 118 (2022) 103260 2 J. Vázquez et al. Fig. 1. (a) Scheme of an ideal fretting bridge, (b) Rotation due to tangential loading in the pads of the fretting bridge. (b) Rotation in a cylindrical pad. solutions, in most cases assuming plane strain behaviour, in order to reproduce these fields. Mutoh et al. [22], used pads with a flat surface mounted in fretting bridges to analyse the fretting fatigue process, and related the crack initiation and propagation with the stresses obtained with a 2D model [23,24]. Similar analyses have been made by other researchers [25,26]. Hills and Nowell, and their group [10,11,13,27], as well as Szolwinski and Farris [12] and other researchers [14,15,28– 30] carried out many fretting fatigue tests with cylindrical contact. They used mainly a 2D analytical solution [31] to relate stresses and fretting fatigue process. Approaching an actual contact pair, under normal and tangential loads, by means of a 2D model is a simple and accurate approach, but some cautions should be taken in the experimental procedure in order to be sure that the conditions to apply the analytical or numerical 2D approach are given. Otherwise, the theoretical or numerical 2D stress/strain solution may not reproduce the actual stress states. One important aspect in the experimental setup is that, as far as possible, the rotations of the pads must be avoided, because this issue would change the stress distributions and the position of the contact zone. Fig. 1a depicts schematically a fretting bridge with flat pads and the theoretical symmetric stress distribution usually assumed for 2D calculations (Fig. 1a). On the other hand, Fig. 1b shows a possible stress distribution in case of rotation of the pad produced by the tangential contact loads in a flexible bridge. In such situation, the stress distribution will be quite different to the symmetrical one (no rotation) and therefore the results obtained with the model without considering the possible rotation may not be useful. In case of cylindrical contact, and assuming that the normal and tangential forces do not change with the rotation of the contact pad, there is no variation of the stress distribution in the contact zone, but the rotation of the pad will produce rolling of it and a motion of the contact zone, and thus modifying the contact zone location in the specimen (Fig. 1c). Usually, the stiffness of the fretting bridges used for testing is high enough to guarantee that the rotation is small, and thus producing small stress variations relative to the theoretical ones. Anyway, an analysis before testing to check that the rotations in a certain fretting bridge are small enough, should be carried out. In order to properly model a contact pair using a two-dimensional model, it is important to analyse and compare the contact stress distributions produced by 2D and 3D models. The authors’ previous work [32], showed that a non-negligible difference in fretting fatigue life predictions are obtained between 2D and 3D approaches. This paper analyses several fretting tests with cylindrical pads and test specimens having typical geometries used in the literature. In such cases, 3D effects are produced which are not considered when a 2D approach is used. To understand these effects, 3D finite element models (FEM) are carried out over several geometries used by different authors and then compared with the frequently assumed 2D approach used in order to analyse the effect of different stress/strain parameters on the fretting fatigue behaviour. The differences between different approaches are analysed and discussed. 2. Fretting fatigue tests configurations From the beginning of testing with fretting bridges different padspecimen geometries have been used. Mutoh et al. [33] used a flat pad with a contact width usually longer than the specimen width (Fig. 2a) and a specimen having a cross section that was not rectangular. Nowell, Hills et al. [11] made many of their tests with cylindrical contact and a pad-specimen geometry as represented schematically in Fig. 2b. Szolwinski and Farris [12], used cylindrical pads with the same width as the test specimens, Fig. 2c. Mall et al. [6] as well as Noraphaiphipaksa et al. [34] used cylindrical pads whose thickness was longer than that of test specimens (Fig. 2d). In tests configurations shown in Fig. 2a and 2d, the width of contact zone is smaller than that of contact pad and thus leading to a notable stress raiser in the borders of the contact zone. Table 1 shows the summary of a review made about the different types of geometries and analysis used for fretting fatigue test in the bibliography. Schemes for the pads and test specimens geometries used in these tests are shown in Fig. 2f1 to f7. Theoretical and Applied Fracture Mechanics 118 (2022) 103260 3 J. Vázquez et al. Fig. 2. Different test configurations frequently used in the bibliography. Fig. 3. Test configurations analysed. The ASTM Standard E2789 [8] gives some recommendations for fretting fatigue testing, but does not include any recommendation about the pad-specimen geometries to be used. But, depending on the geometry considered, 3D effects make the stress distribution produced along the contact width to be different, so that test results may not be comparable between different configurations. Also those 3D stresses are different to that obtained with a 2D numerical or analytical solution. For instance, in the case of cylindrical contact, for the same pad radius and the same loads applied, the stress distributions produced in the specimen are different in the three cases represented in Fig. 2b to 2d. So the test results obtained with these geometries are not directly comparable if a 2D model is used in all cases to analyse the stress and strain fields. 3. Cases analysed The cases analysed in the present paper are shown in Fig. 3a to c, which correspond to the most usual test configurations found in the bibliography. These configurations can be grouped into two general cases: a first case (Figs. 3a.1, 3a.2 and 3c) in which the fretting fatigue test specimen and contact pad have the same width at the contact zone, and a second case in which at the contact zone the test specimen’s width is shorter than contact pad’s width, i.e., 𝑤𝑡< 𝑤𝑝(Fig. 3b). A fillet is included in cases shown in Figs. 3a.2 and 3b; in the former it is included to analyse a geometry that resembles that obtained after a typical machining process (a sharp edge is not a real feature), whereas in the latter, and in addition to consider a real (machined) corner, the Theoretical and Applied Fracture Mechanics 118 (2022) 103260 4 J. Vázquez et al. Fig. 4. (a) Details of FEM model used, (b) Fretting loads application in the model. Table 1 A compilation of different tests configurations used in the bibliography. Ref. Geom.aCont. PadbType anal.cRef. Geom.aCont. PadbType anal.c [35] f1 Flat Num. & Exp. [36] f1 Cyl. Anal. & Exp. [24] f1 Flat Num. [37] Unspec. Cyl. Num. & Exp. [38] f1 Flat Num. & Exp. [39] f5 Cyl. & Flat Anal. & Exp. [40] f1 Flat Num. & Exp. [41] f5 Cyl. & Flat Num. & Exp. [42] f6 Flat Exp. [43] f5 F.R. Num. & Exp. [44] f5 Cyl. Num. [45] f5 F.R. Num. & Exp. [46] f5 Cyl. Num. [47] f5 F.R. Num. [48] f5 Cyl. Num. & Exp. [49] f5 F.R. Num. [50] Unspec. Cyl. Anal. & Exp. [20] f5 F.R. Exp. [51] f4 Cyl. Num. & Exp. [14] f4 Cyl. Num. & Exp. [52] f4 Cyl. Num. & Exp. [26] f4 Cyl. Exp. [53] Unspec. Cyl. & F.R. Num. & Exp. [54] f5 Cyl. Exp. [55] Unspec. Cyl. & F.R. Num. & Exp. [56] f5 Cyl. Num. & Exp. [57] f4 Cyl. Anal. & Exp. [58] f5 Cyl. Exp. [59] f4 Cyl. Exp. [60] f4 Flat Num. & Exp. [12] f4 Cyl. Anal. & Exp. [61] f5 Cyl. Num. & Exp. [62] f4 Cyl. Num. & Exp. [63] f3 Cyl. & Flat Anal. & Exp. [64] f5 Flat Num. & Exp. [65] f5 Flat Exp. [66] f5 Flat Num. & Exp. [21] f5 Flat Exp. [67] f5 Flat Num. & Exp. [38] f2 Flat Num. & Exp. [68] f5 Flat Exp. [69] f3 Flat Exp. [70] f1 Cyl. Anal. & Exp. [71] f5 Flat Exp. [72] f4 Cyl. Anal. & Exp. [73] f5 Flat Exp. [74] f4 Cyl. Anal. & Exp. [75] f3 Flat Exp. [27] f5 Cyl. Anal., Num. & Exp aSee Figure 2f1 to 2f6 for an explanation (Unspec.=unspecified). bType of contact pad: Cyl.=cylindrical, F.R.=Flat with rounded ends. cType of analysis of the contact: Exp.=experimental, Anal.=analytic, Num.=numeric. Theoretical and Applied Fracture Mechanics 118 (2022) 103260 5 J. Vázquez et al. Fig. 5. Contact pressure distribution, |||𝜎𝑦𝑦 (𝑥, 0, 𝑧)|||, over test specimen in configuration 1. Table 2 Geometric dimensions for the FEM models. 𝑎 𝑅 𝑙 ℎ𝑝ℎ𝑡𝑤𝑡 1.8mm 100mm 20𝑎10𝑎10𝑎4𝑎 fillet is also included to avoid a theoretically, and unrealistic, infinite stress raiser. 3.1. FEM models Even for the simplest case analysed here, case shown in Fig. 3a.1, there is no 3D analytic solution, therefore a numerical model is required in order to study the behaviour of the fretting tests considered here. In this paper all analyses have been performed without a bulk stress, because in the present work we are focused on the effects produced by the geometry of test specimen and contact pad on the surface stress field, and it is thought that the effect due to this bulk stress is not relevant. In all cases the applied total normal load, 𝑁, produces an average load per unit length (line load), 𝑁∗= 1000 N∕mm, and the total tangential load is always 𝑄= 0.75 μN; values that are representative of usual fretting tests. Regarding the geometry, in all cases the pad radius is 𝑅= 100 mm, the height of test specimen and contact pad, ℎ𝑡and ℎ𝑝 respectively (see Fig. 3), are set to ten times (10𝑎) the Hertzian contact semi-width under plane strain conditions; 𝑎= 1.80 mm in the present case, and the width of the test specimen in the contact zone, 𝑤𝑡, is set equal to 4𝑎. In addition, and to avoid size effects in the modelling, the length of test specimen, 𝑙, and contact pad is set to 20𝑎. As a summary, Table 2 shows the leading geometric dimensions used in the FEM models. It is important to note that the geometric dimensions has been chosen taking into account that the specimen width is sufficiently large in comparison with the contact width. In this sense, the use of larger dimensions should not influence the results. This is not the situation for cases having a specimen width similar or smaller than the contact width, since in these cases results would tend to a stress state similar to that of plane stress. The commercial software ANSYS has been used to simulate the mechanical behaviour of all contact pairs. As stated above, we are focused on the effects produced by the geometry of test specimen, and for this reason a linear elastic material behaviour is considered for both, the contact pad and test specimen; although an elastic–plastic model can be considered, the different results between geometries are better analysed with a linear elastic behaviour. Furthermore, it is quite common for fretting tests to be carried out under high cycle fatigue regime, and thus under elastic behaviour. The elastic modulus (Young’s modulus) and Poisson’s ratio are 𝐸= 70 GPa and 𝜈= 0.33 respectively, and thus resembling the case in which the raw material of the contact pair is an aluminium alloy. A linear formulation for the solid elements has been considered. The submodelling technique is used to obtain a very precise contact solution over the contacting areas. In the submodel the mesh density, especially at the contact zone, is fine enough to capture the stress gradients in the contact region; the minimum element size is approximately 4.0 μm. With this mesh size, a convergence in the peak value for the direct stress 𝜎𝑥𝑥 at the contact surface was achieved. We decided to use the direct stress 𝜎𝑥𝑥 because it presents a very steep gradient at the contact trailing edge, and thus is a good benchmark for the convergence. For all test configurations two submodels were used: one to analyse the near front part (𝑧=𝑤𝑡∕2) and another for the analysis of near middle-width section part (𝑧= 0). A scheme of the model, for the case in which 𝑤𝑡=𝑤𝑝, and a view of the meshes, coarse and fine models (front part submodel in the present figure), is shown in Fig. 4a. Due to symmetry considerations (about 𝑧= 0 plane), only half of the complete geometry is modelled. Fretting loads, 𝑁and 𝑄, are applied remotely on the pad top surface via a master node. The test specimen bottom surface is clamped, and because the height of this part (10𝑎) is large enough, this boundary condition does not affect the results obtained at the contact zone. In all cases, first the normal load is sequentially applied up to its maximum value, then this value is kept constant, and finally the tangential load is sequentially applied up to its maximum value. As shown in Fig. 4b, both normal and tangential load are applied by consecutive steps. Each step load is divided in 10 substeps, so that the load is increased by a factor of 1∕10 of its final value between consecutive substeps. In addition, the time increment in each substep is equal to 0.0025. The contact algorithm used in the simulations is the Augmented Lagrangian Method, which produces a good ratio between accuracy and CPU times. According to experimental values [28], the coefficient of friction, 𝜇, has been fixed equal to 0.7, a high value that makes necessary the use of an unsymmmetric iterative solver in order to achieve numerical convergence. Finally, in all the FEM models the CPU time was around 2 h in a computer with 12 cores (3.7 GHz) and 64 GB of RAM. Theoretical and Applied Fracture Mechanics 118 (2022) 103260 6 J. Vázquez et al. Fig. 6. (a) Surface contact stress, 𝜎𝑦𝑦 (𝑥), for 2D models and at two sections (𝑧= 0 and 𝑧=𝑤𝑡∕2) of the test specimen (3D FEM), (b) Surface contact stress, 𝜎𝑥𝑦 (𝑥), for 2D models and at two sections (𝑧= 0 and 𝑧=𝑤𝑡∕2) of the test specimen (3D FEM), (c) Surface direct stress, 𝜎𝑥𝑥 (𝑥), for 2D models and at two sections (𝑧= 0 and 𝑧=𝑤𝑡∕2) of the test specimen (3D FEM). Table 3 Specific FEM parameters. Configuration 1 Configuration 1.1 Configuration 2 Configuration 3 𝑤𝑝4𝑎4𝑎8𝑎4𝑎 𝑁1(𝑁)𝑁∗𝑤𝑡 2𝑁∗(𝑤𝑡 2−𝑟)𝑁∗(𝑤𝑡 2−𝑟)𝑁∗𝑤𝑡 2 𝑁∗(𝑁∕𝑚𝑚)1000 1000 1000 1000 𝑟00.1𝑎0.1𝑎0 𝑁1Normal load applied to the FEM model considering symmetry Following, the analysis and discussion of the configurations here considered are shown. Finally, and although in next sections a complete explanation of the different configurations will be shown, Table 3 shows the specific parameters, loading and geometric, for each of the cases here analysed, with the aim to easily compare the differences between the simulated cases. 3.2. Configuration 1. Contact pad and test specimen having the same width (𝑤𝑝=𝑤𝑡) The geometries of the contact pad and test specimen considered in this section are shown in Fig. 3a.1. In this case, and to have a complete visualisation of the stress field on the test specimen contact area, a complete model with a fine mesh at the contact zone has been solved; the features of this model are the same to that mentioned above for the coarse model and submodel. Fig. 5 shows the absolute value of the contact pressure distribution, |||𝜎𝑦𝑦 (𝑥, 0, 𝑧)|||, obtained on the test specimen surface with the complete model at step 2; unless stated otherwise, this is the time step used along all the text. In this picture we can observe the notable variation of the contact pressure with the 𝑧-coordinate, showing that the maximum contact pressure is obtained at 𝑧= 0 (middle-width plane). This fact by itself could justify a different fretting fatigue behaviour between the outer plane (𝑧=𝑤𝑡∕2) and the middle-width plane. The physical insight of this assertion, lies in the above results which indicate a notable different normal stiffness at 𝑧= 0 and 𝑧=𝑤𝑡∕2, and thus a similar variation of the tangential stresses is expected. Fig. 6a to c shows the contact stress distribution, |||𝜎𝑦𝑦 (𝑥)|||and 𝜎𝑥𝑦 (𝑥), and the direct stress, 𝜎𝑥𝑥 (𝑥), obtained via submodels at test specimen’s sections 𝑧= 0 and 𝑧=𝑤𝑡∕2. In addition, and for comparison purposes, these figures also show the corresponding theoretical (Hertzian) 2D stress distribution (assuming plane stress and plane strain behaviour) obtained with 𝑁∗= 1000 N∕mm and 𝑄∗= 0.75 μN∗; the analytical equations for these stresses can be obtained elsewhere [31]. A remarkable fact in this figure, although already shown in Fig. 5, is the difference between the contact pressure distributions obtained at 𝑧= 𝑤𝑡∕2 and 𝑧= 0; the ratio between the maximum values found at 𝑧= 0 and 𝑧=𝑤𝑡∕2 is about 1.6. In addition, and when comparing results between Figs. 5 and 6a, it is noteworthy that both the complete model and submodels produce nearly the same peak values in the contact normal stress 𝜎𝑦𝑦. On the other hand, it is observed that for 𝑧= 0 the contact pressure distribution is quite similar to that obtained with the theoretical 2D plane strain model, which is in line with expectations. Meanwhile, a quite different behaviour is obtained at 𝑧=𝑤𝑡∕2, being Theoretical and Applied Fracture Mechanics 118 (2022) 103260 7 J. Vázquez et al. the 3D pressure distribution far below its expected counterpart 2D solution (plane stress). These results show that the normal load is not uniformly distributed along the test specimen’s width (𝑧-coordinate), being this proportion higher in the middle section (𝑧= 0) and lower at the surface (𝑧=𝑤𝑡∕2). In addition, Fig. 6a shows that the contact sizes obtained with the numerical model are very close to that obtained analytically, especially when plane stress is assumed; the contact semiwidth assuming plane stress behaviour is 𝑎∗= 1.90 mm. In relation to the 3D surface shear stress, 𝜎𝑥𝑦,Fig. 6b shows the distributions obtained at 𝑧= 0 and 𝑧=𝑤𝑡∕2. It is important to note that 𝜎𝑥𝑦 distributions for plane strain and 3D FEM at 𝑧= 0 are quite similar, although a wider stick zone is predicted by FEM. On the contrary, 3D FEM 𝜎𝑥𝑦 distribution at 𝑧=𝑤𝑡∕2 shows the lowest values and the smallest stick zone. Such a shear stress distribution is expected in view of the normal pressure distribution obtained at 𝑧=𝑤𝑡∕2 with the FEM model. The small stick zone predicted by FEM at 𝑧=𝑤𝑡∕2 may be attributed to the manner in which the tangential load is distributed along the 𝑧-coordinate, which should be different to that of the normal load and thus producing a 𝑄∗∕ μN∗ratio different from 0.75. Finally, Fig. 6c shows the surface direct stress 𝜎𝑥𝑥; distributions obtained via 3D FEM and those produced with 2D models are plotted together for comparison. This figure shows that 3D results at 𝑧= 0 are very close to those obtained with 2D models, especially assuming plane strain behaviour. On the other hand, the results at 𝑧=𝑤𝑡∕2 are far from those obtained with bi-dimensional models, but in agreement with the normal and shear contact stress obtained via FEM at 𝑧=𝑤𝑡∕2; see for example that at 𝑧=𝑤𝑡∕2 the extension of the nearly flat portion in 𝜎𝑥𝑥 coincides with the stick zone obtained in the 𝜎𝑥𝑦 distribution. In all numerical cases the peak direct stress is produced at the contact trailing edge, place where the value of direct stress 𝜎𝑥𝑥 is only due to the surface shear stress distribution, as no bulk stress is considered in the simulations. Note that 𝜎𝑥𝑥 peak values at 𝑧= 0 and 𝑧=𝑤𝑡∕2 are 388 MPa and 259 MPa respectively, and based on this crack initiation is likely to occur at 𝑧= 0. However, the normal pressure distribution shown in Fig. 5, suggests that a significant portion of the test specimen width must be subjected to a direct stress distribution similar to that obtained at 𝑧= 0, suggesting that fretting cracks would initiate along a big portion of the contact trailing edge. These results are in agreement with the experimentally observed behaviour in a previous authors’ work [76], in which almost simultaneously many cracks initiate along the contact trailing edge. All the results shown above invite to analyse the manner in which the normal and tangential load are distributed along the test specimen width. It is possible to obtain the variation for 𝑁∗and 𝑄∗over the test specimen semi-width by integrating the normal and shear contact stress distribution, 𝜎𝑦𝑦 (𝑥)and 𝜎𝑥𝑦 (𝑥), with respect to the 𝑥-coordinate at different 𝑧values. These results are shown in Fig. 7 as 𝑁∗(𝑧)and 𝑄∗(𝑧). Firstly, notice how different is the variation of 𝑁∗(𝑧)and 𝑄∗(𝑧) with respect to 𝑧; the first one presents an important decrease for 𝑧∕(𝑤𝑡∕2)>0.5, meanwhile 𝑄∗(𝑧)is roughly constant up to a value of 𝑧∕(𝑤𝑡∕2)= 0.94, indicating that the contact normal stiffness is strongly affected by the 3D behaviour but not so much the tangential stiffness. In view of these curves, it is expected that crack initiation would likely occur first in the contact trailing edge between 𝑧∕(𝑤𝑡∕2)= 0.5and 𝑧∕(𝑤𝑡∕2)= 0.9, approximately (region with a grey crossing pattern); area where 𝑁∗(𝑧)decreases rapidly but 𝑄∗(𝑧)maintains its value. It is known that, for the same 𝑄∗, a higher value of 𝑁∗delays the fretting crack initiation process [12,57,76]. Also, note that close to 𝑧≈ 0, 𝑁∗(𝑧)>1000 N∕mm (the reference value), and it explains by itself the differences obtained in Fig. 6a between the theoretical (plane strain behaviour) and FEM (at 𝑧= 0) stress distributions. In addition, and in view of the value of 𝑁∗(𝑤𝑡∕2), a similar explanation can be given for the differences at 𝑧=𝑤𝑡∕2 between the theoretical (plane strain or plain stress behaviour) and FEM normal pressure distributions. Regarding the tangential loading, the value for 𝑄∗(𝑤𝑡∕2)and 𝑁∗(𝑤𝑡∕2) explain the surface contact shear stress distribution obtained via FEM at 𝑧=𝑤𝑡∕2. In addition, it is found that for 𝑧= 0 the tangential line load value is 𝑄∗= 530.0 N∕mm, a very similar value to the reference value of 525 N∕mm (0.75 μN∗). To complete the analysis, the fretting load ratio, 𝑄∗(𝑧)∕(μN∗(𝑧)), is shown in Fig. 7, which depicts that this ratio is very close to the reference value of 0.75 up to a width of 𝑧∕(𝑤𝑡∕2)≈ 0.75. Then this ratio increases rapidly with 𝑧up to a value very close 1.0, and thus explaining the small stick zone obtained by FEM at 𝑧=𝑤𝑡∕2. Finally, to deepen in the knowledge of the 3D behaviour and how it affects the contact stress field, the values obtained from the FEM analysis for 𝑁∗(𝑧)and 𝑄∗(𝑧)at 𝑧= 0 and 𝑧=𝑤𝑡∕2 have been used to recalculate the corresponding 2D analytical surface contact stress distributions. These distributions have been obtained assuming plane strain for 𝑧= 0 and plane stress for 𝑧=𝑤𝑡∕2. The results are depicted in Fig. 8, and show that now 3D FEM distribution obtained at 𝑧= 0 and those assuming plane strain conditions, are nearly the same and with only a slight difference at the contact edges. On the other hand, FEM results obtained at 𝑧=𝑤𝑡∕2 differ notably with those obtained assuming plane stress behaviour; in view of the plots shown in Fig. 8 similar results are expected if plane strain behaviour is considered. Nonetheless, and surprisingly, both models, plane stress and FEM at 𝑧=𝑤𝑡∕2, virtually produce the same stick zone length. Finally, the above results indicate that in the middle-width section (𝑧= 0), and when properly selected the line loads 𝑁∗and 𝑄∗, the surface contact stress distributions can be accurately predicted assuming the theoretical plane strain solution. In contrast, the mechanical behaviour at 𝑧=𝑤𝑡∕2 (and likely at sections near the free surface defined by 𝑧=𝑤𝑡∕2) greatly differ to that obtained by using plane stress or plane strain theoretical solutions. 3.2.1. Configuration 1.1. 𝑤𝑝=𝑤𝑡but test specimen having a fillet Now the pad and test specimen geometries are those shown in Fig. 3a.2. In the case analysed here test specimen and contact pad have a fillet with a radius, 𝑟𝑓𝑖𝑙𝑙𝑒𝑡 = 0.1𝑎= 0.18 mm, which is a feasible value after a typical machining process. In order to apply a total normal load, 𝑁, producing a value of 𝑁∗close to 1000 N∕mm, and thus making this analysis comparable with that shown above, 𝑁 has a value equal to 𝑁∗(𝑤𝑡∕2 − 𝑟𝑓𝑖𝑙𝑙𝑒𝑡); we selected this value because FEM results showed that only a minor area of the fillets come into contact. In the present geometry, a complete model having a resolution high enough to properly capture the stress gradient present at the fillets is out of our CPU capabilities. In any case, Fig. 9a shows the contact pressure distribution at test specimen’s surface obtained with the coarse (complete) model, in which it is observed that a maximum value of 366.38 MPa is produced close to the fillet edge. For a better analysis of the contact stress distributions at the fillet, more precise results have been obtained via submodelling. For completeness, Fig. 9b shows the surface normal pressure obtained near the filleted zone. Fig. 10 shows the |||𝜎𝑦𝑦 (𝑥, 0, 𝑧)|||,𝜎𝑥𝑦 (𝑥, 0, 𝑧)and 𝜎𝑥𝑥 (𝑥, 0, 𝑧)distributions obtained with the submodels at 𝑧= 0 mm and 𝑧≈ 0.94𝑤𝑡∕2 = 3.39 mm, being this last location the place where the maximum normal contact pressure (602.71 MPa) has been found. Additionally to the FEM results, and for the sake of clarity, in this figure only theoretical stress distributions corresponding to plane strain behaviour have been included. Similarly to the previous case, stress distributions obtained at 𝑧= 0 are very close to those obtained assuming plane strain behaviour. On the other hand, the contact stress distributions |||𝜎𝑦𝑦 (𝑥)|||and 𝜎𝑥𝑦 (𝑥)obtained at 𝑧= 3.39 mm, are far from those at the middle-width section; see for example the big difference between the peak values for 𝜎𝑦𝑦 at 𝑧= 3.39 and 𝑧= 0. Regarding the direct stress, 𝜎𝑥𝑥, values near the filleted zone are lower than values at 𝑧= 0. We can see that the difference between the maximum peak values at 𝑧= 3.39 and 𝑧= 0 is roughly 100 MPa, a difference similar to that previously found in the case without fillets. In view of the shear stress distribution obtained at 𝑧= 3.39 mm, a higher difference between these 𝜎𝑥𝑥 peak values would be expected, a fact that can be only attributed to the especial 3D behaviour provided by the fillet. Theoretical and Applied Fracture Mechanics 118 (2022) 103260 8 J. Vázquez et al. Fig. 7. Variation of 𝑁∗(𝑧)and 𝑄∗(𝑧)and fretting load ratio 𝑄∗(𝑧)∕(𝜇𝑁∗(𝑧)) along the test specimen’s width. Fig. 8. Comparison of 3D FEM contact stress distribution with the corresponding 2D theoretical ones (plane stress and plane strain) but using the values of 𝑁∗(𝑧)and 𝑄∗(𝑧) obtained at 𝑧= 0 and 𝑧=𝑤𝑡∕2. Fig. 9. (a) Surface contact stress |||𝜎𝑦𝑦 (𝑥, 0, 𝑧)|||obtained with the coarse model in the case of geometries with fillets, (b) Surface contact stress |||𝜎𝑦𝑦 (𝑥, 0, 𝑧)|||obtained with the submodel near the fillet. Theoretical and Applied Fracture Mechanics 118 (2022) 103260 9 J. Vázquez et al. Fig. 10. (a) Surface contact stress, 𝜎𝑦𝑦 (𝑥), for 2D model (plane strain) and at two sections (𝑧= 0 and 𝑧= 3.39 mm) of the test specimen (3D FEM), (b) Surface contact stress, 𝜎𝑥𝑦 (𝑥), for 2D model (plane strain) and at two sections (𝑧= 0 and 𝑧=𝑧= 3.39 mm) of the test specimen (3D FEM), (c) Surface direct stress, 𝜎𝑥𝑥 (𝑥), for 2D model (plane strain) and at two sections (𝑧= 0 and 𝑧=𝑧= 3.39 mm) of the test specimen (3D FEM). Fig. 11. Variation of 𝑁∗(𝑧)and 𝑄∗(𝑧)and fretting load ratio 𝑄∗(𝑧)∕(𝜇𝑁∗(𝑧)) along test specimen‘s width in the case with a fillet. Theoretical and Applied Fracture Mechanics 118 (2022) 103260 16 J. Vázquez et al. Acknowledgement All authors approved the version of the manuscript to be published. References [1] D.A. Hills, D. Nowell, Mechanics of Fretting Fatigue, Kluber Academic Publishers, Dordrech/Boston/London, ISBN: 0-7923-2866-3, 1994. [2] J.M. Dobromirski, Variables of fretting process: are there 50 of them? in: M.H. Attia, R.B. 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