# Prediction of fretting fatigue damage under variable loading blocks: Effect of plasticity

Mohammed Fartas, Siegfried Fouvry, Pierre Arnaud, Simon Garcin, Fernando Pires

► To cite this version:

Mohammed Fartas, Siegfried Fouvry, Pierre Arnaud, Simon Garcin, Fernando Pires. Prediction of fretting fatigue damage under variable loading blocks: Effect of plasticity. International Journal of Fatigue, 2025, 199, pp.109022. 10.1016/j.ijfatigue.2025.109022. hal-05378013

HAL Id: hal-05378013

https://hal.science/hal-05378013v1

Submitted on 22 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Prediction of fretting fatigue damage under variable loading blocks: effect of plasticity

Mohammed Fartas<sup>a*</sup>, Siegfried Fouvry<sup>a,b*</sup>, Pierre Arnaud<sup>b</sup>, Simon Garcin<sup>a</sup>, Fernando Pires<sup>c</sup>.

a. Ecole Centrale de Lyon, LTDS Laboratory, 36 av Guy de Collongue, 69130 Ecully, France

b. MINES ParisTech, Centre des Matériaux, 63-65 rue Henri-Auguste Desbrueres, F-91003 Evry Cedex, France.

c. TechnipFMC

E-mail addresses: mohammed.fartas@ec-lyon.fr; pierre.arnaud@minesparis.psl.eu; siegfried.fouvry@minesparis.psl.eu; fernando.pires@technipfmc.com

## Abstract:

This research paper investigates the lifetime span and the crack propagation mechanism for a steel cylinder/plane contact under severe variable plastic fretting fatigue conditions that are representative of high pressure dynamic flexible pipe risers. An FEA model was used to predict the total life of the contact by separating the crack nucleation and the crack propagation mechanisms. Good correlation between numerical predictions and experimental results was obtained. Since fretting is well known for generating important stress gradients, a non-local critical distance method is applied to estimate the number of cycles required for crack nucleation. The crack propagation was addressed using Kujawski's fatigue crack driving parameter with Paris law. The variable loading conditions were investigated considering decreasing tangential force and fatigue stress loading blocks and comparing various elastic and elastic-plastic material responses. Since significant plastic deformations are generated during the initial loading blocks (the highly charged block), better predictions of crack extension or fretting fatigue endurance are achieved using elastoplastic hypotheses (monotonic or cyclic hardening). By contrast, elastic assumption which is less accurate provides systematic conservative predictions.

**Keywords:** Fretting fatigue, block loading, FEM, plasticity, flexible pipes, cylinder plane contact

<footnote>*Corresponding authors.</footnote>
## Nomenclature

<table><tr><td>a</td><td>half-width of the contact area (mm)</td><td>R<sub>K</sub></td><td>SIF stress ratio</td></tr><tr><td>b</td><td>crack length (mm)</td><td>R<sub>Q</sub></td><td>fretting stress ratio</td></tr><tr><td>b<sub>max</sub></td><td>maximal crack length at the end of test (mm)</td><td>R<sub>σ</sub></td><td>fatigue stress ratio</td></tr><tr><td>b<sub>φ</sub></td><td>short to long crack transition (μm)</td><td>W</td><td>the fatigue specimens' thickness (mm)</td></tr><tr><td>b<sub>CN</sub></td><td>crack nucleation length (μm)</td><td>S</td><td>deviatoric part of Σ</td></tr><tr><td>D</td><td>crack nucleation damage</td><td>J<sub>2,a</sub></td><td>amplitude of the second invariant of the deviatoric stress amplitude tensor</td></tr><tr><td>E</td><td>Young's modulus (MPa)</td><td></td><td></td></tr><tr><td>K<sub>I</sub></td><td>mode I stress intensity factor (SIF) (</td><td>σ<sub>H,max</sub></td><td>hydrostatic stress gradient</td></tr><tr><td>K<sub>IC</sub></td><td>critical mode I SIF</td><td>ΔK</td><td>nominal SIF</td></tr><tr><td>K̄*</td><td>fatigue crack driving force parameter (</td><td>ΔK<sub>φ</sub></td><td>long crack SIF range threshold</td></tr><tr><td>l<sub>d</sub></td><td>critical distance (μm)</td><td>ΔK<sub>th</sub></td><td>crack arrest condition</td></tr><tr><td>l<sub>opt</sub></td><td>optimal critical distance (μm)</td><td>μ = μ<sub>t</sub></td><td>coefficient of friction at the sliding transition</td></tr><tr><td>N</td><td>number of fretting cycles</td><td>σ</td><td>stres(MPa)</td></tr><tr><td>N<sub>CN</sub></td><td>number of fretting cycles to crack nucleation</td><td>σ<sub>11</sub></td><td>stress in the 11 direction (MPa)</td></tr><tr><td>N<sub>CP</sub></td><td>number of propagating fretting cycles</td><td>σ<sub>y</sub></td><td>cyclic yield stress (MPa)</td></tr><tr><td>N<sub>f</sub></td><td>experimental number of cycles to failure</td><td>σ<sub>m</sub></td><td>mean fatigue stress (MPa)</td></tr><tr><td>N<sub>F,pred</sub></td><td>predicted number of cycles to failure</td><td>σ<sub>a</sub></td><td>fatigue stress amplitude (MPa)</td></tr><tr><td>P</td><td>linear normal force (</td><td>σ<sub>c</sub></td><td>equivalent stress of Crossland's criterion (MPa)</td></tr><tr><td>p</td><td>surface pressure (MPa)</td><td>Σ</td><td>stress tensor</td></tr><tr><td>q</td><td>shear stress (MPa)</td><td>τ<sub>D</sub></td><td>torsion fatigue limite (R<sub>f</sub> = 1) (MPa)</td></tr><tr><td>q<sub>max</sub></td><td>maximum surface shear stress (MPa)</td><td>δ</td><td>fretting displacement (μm)</td></tr></table>

<table><tr><td>Q</td><td>linear tangential fretting force (N/mm)</td><td>CA</td><td>crack arrest</td></tr><tr><td>Q*</td><td>linear tangential fretting force (N/mm)</td><td>CN</td><td>crack nucleation</td></tr><tr><td>Q<sup>*</sup><sub>CN</sub></td><td>crack nucleation threshold (N/mm)</td><td>CP</td><td>crack propagation</td></tr><tr><td>Q<sub>t</sub></td><td>linear tangential fretting force at the transition between partial slip and gross slip regime (N/mm)</td><td>EOT</td><td>End Of Test</td></tr><tr><td>R</td><td>radius of cylinder pad (mm)</td><td>E</td><td>Elastic</td></tr><tr><td>d<sub>c</sub></td><td>crossland risk</td><td>MI</td><td>Monotonic Isotropic</td></tr><tr><td></td><td></td><td>CK</td><td>Cyclic Kinematic</td></tr><tr><td></td><td></td><td>exp</td><td>experimental</td></tr><tr><td></td><td></td><td>pred</td><td>prediction</td></tr><tr><td></td><td></td><td>max</td><td>maximum</td></tr><tr><td></td><td></td><td>FF</td><td>fretting fatigue</td></tr><tr><td></td><td></td><td>PF</td><td>plain fretting</td></tr></table>

# 1. Introduction:

Flexible pipes are used in offshore production to transport and extract oil and gas to surface installations [1]. The flexibility of pipes offers several advantages: they are easy to install and recover and are highly resistant to mechanical stress and corrosion. These benefits were made possible because
of their multilayer structure. The flowlines rest on the seabed and are subject to static stresses, whereas risers are vertical pipes linking the flowlines to the surface and are subject to dynamic stresses.

Flexible pipes are designed to operate for long periods and are subjected to high loadings and various causes of degradation. Thus, it is important to study the phenomena that occur when two layers of the flexible rub against each other. This is a concern specific to Teta/Spiral pressure armor configuration, which is used for high pressure dynamic riser (Figure 1). Due to the wave displacements, flexible pipe risers may undergo cyclic bending, where small oscillations induce a contact force rubbing (fretting) between Teta/Spiral pressure vault wires. Larger cyclic loads from pip weight, operating pressure, tension and bending moments may nucleate and propagate these fretting sites, leading to a mechanical damage phenomenon called fretting fatigue, which is the subject of this paper.

<figure><img src="image_2.png" /><figcaption>Figure 1: (a) Identification of the Teta/Spiral steel contact within the flexible [2], (b) elastoplastic pressure profiles obtained with the 2D finite element model, mesh size: $d2D=5\mu m$.</figcaption></figure>

The contact geometry inside the flexible is complex to represent because the wires are helically wound, and they contain multiple contacts, each subjected to a different stress. Therefore, the chosen strategy consisted in selecting the most loaded contact in order to develop a conservative prediction of the flexible pipe resistance against fretting fatigue failure mode. This contact is used to establish a predictive methodology for fretting fatigue crack initiation and propagation behavior. A model considering elastoplastic material laws has been developed to represent the contact between pressure armor layers.

The experimental tests used as reference for the numerical model discussed in this work were performed with a cylinder/plane contact and loads representative of flexible pipe riser pressure armor. This contact has been defined as representative of the one presented in the pressure armor wires of flexible pipe after the analysis with an industrial 3D model and a refined 2D model. In some Tests, the contact is characterized by a very high contact pressure: $p > p_{0,yield}$ causing the local plasticity in the contact (where $p_{0,yield}$ is the hertzian pressure causing the first plasticity under indentation).

Hence, in the frame of this investigation a dedicated cylinder/plane fretting fatigue experiment will be considered to simulate the application. Note that the application of a cylinder/plane contact to replicate the industrial application has been successfully adopted in many literatures for many research work like blade-disk contact of riveted assemblies [3-5].

Most of fretting fatigue research works concern a constant loading condition. Only few studies have been found in the literature that studied the influence of variable loading conditions on the fretting fatigue behavior [6-9]. The objective of this investigation is to investigate the influence of variable loading conditions. The experimental tests were performed with block of decreasing loads. Therefore, the same type of load will be simulated with the numerical model. These blocks are characterized by a
given fatigue stress ($\sigma_{max}$), normal force (P) and tangential force amplitude (Q*). This type of loading sequence (decreasing loads) is known to be the most severe [10]. Indeed, the first blocks which are the most severe are intended to nucleate the crack whereas the following blocks will help propagate the crack until failure. The experimental work is presented in the following sections, where a detailed description of the experimental setup, materials, test procedures and experimental results are exposed. The block of loading (Q* and fatigue bulk loadings) was previously defined to be representative of the wave spectra of a given offshore locations. The only constant load applied was the normal load. Figure 2 illustrates the load blocks that were applied. All of the numerical tools previously detailed in [11, 12] will be used to predict the crack nucleation then the crack propagation. The objective here is to extend this methodology for a more complex loadings and to fill this gap in the literature by studying the impact of block loadings/variable loadings on the fatigue life of materials by combining the following three key aspects which are:

* The application of damage accumulation models regarding crack nucleation and crack propagation,

* The simulation of plastic accommodation within the fretting interface,

* The application of the critical distance method to capture the stress gradient effect in order to achieve reliable predictions.

Hence, using this global approach, it will be investigated in what manner extended complex elastic-plastic simulations can enhance the prediction of fretting damages concerning crack length extension (b) and failure endurance (N) comparatively to a simpler elastic assumption which is classically adopted in literature works.

<figure><img src="image_3.png" /><figcaption>Figure 2: Normal force, fretting loading and fatigue loading for a decreasing block configuration.</figcaption></figure>

# 2. Materials and experimental set-up:

## 2.1 Fretting fatigue contact materials:

The study material was a 32Cr1 industrial low carbon steel, which exhibits low cyclic yield stress $\sigma_{y,flat}$. Its chemical composition is given in Table 1. The material was subjected to fretting fatigue loading using a cylinder/plane contact. The cylinder pad was in FM35 steel and was chosen to have a higher cyclic yield stress compared to the flat material ($\sigma_{y,pad} = 1.25 \times \sigma_{y,flat}$), making it less prone to crack initiation. As a result, crack formation was exclusively observed on the flat specimen. The elastic properties (i.e. young modulus and yield stress values $\sigma_{y,flat%}$) and the fatigue properties (i.e. Paris
law constant and traction-compression fatigue limit $\tau_d$) for the flat and the pad specimen were previously detailed in [11] and the corresponding data are compiled in Table 2.

<table><caption>Table 1: Atomic composition of 32Cr1 steel [13]</caption><thead><tr><th></th><th>C</th><th>Cr</th><th>Mn</th><th>Si</th><th>Cu</th><th>Mo</th><th>Ni</th><th>Al</th><th>P</th><th>S</th></tr></thead><tbody><tr><th>Flat (32Cr1)</th><td>0.32</td><td>0.347</td><td>0.743</td><td>0.201</td><td>0.046</td><td>0.016</td><td>0.047</td><td>0.013</td><td>0.007</td><td>0.002</td></tr></tbody></table>

<table><caption>Table 2: Mechanical properties of 32Cr1 and FM35 (unfortunately due to confidentiality aspects, the material properties ($\sigma_{y,flat}$ and $\tau_d$ cannot be detailed) [11].</caption><thead><tr><th rowspan="2">Material</th><th rowspan="2">E (GPa)</th><th rowspan="2">v</th><th rowspan="2">σ<sub>y,0.2%</sub><br>(MPa)</th><th rowspan="2">τ<sub>d</sub><br>(MPa)</th><th rowspan="2">σ<sub>d</sub><br>(MPa)</th><th colspan="2">Paris law</th><th rowspan="2">K<sub>IC</sub></th><th rowspan="2">K<sub>Th</sub></th></tr><tr><th>C</th><th>m</th></tr></thead><tbody><tr><td>Flat (32Cr1)</td><td>195</td><td>0.3</td><td>σ<sub>y,flat</sub></td><td>τ<sub>d</sub></td><td>1.38 τ<sub>d</sub></td><td>5.35E-10</td><td>3.26</td><td>212</td><td>5.5</td></tr><tr><td>Pad (FM35)</td><td>205</td><td>0.3</td><td>1.25× σ<sub>y,flat</sub></td><td>**</td><td>**</td><td>**</td><td>**</td><td>**</td><td>**</td></tr></tbody></table>

The pad and the flat were subjected to a normal load that was sufficiently high to induce plastic deformations. Therefore, both specimens had to be described using an elastic-plastic law. Lemaitre and Chaboche [14] model with a nonlinear kinematic and isotropic work hardening was used to describe the cyclic behavior of the material:

$$ \dot{\overline{\epsilon^{pl}}} = \epsilon_{eq}^{pl} \cdot \frac{\partial f}{\partial \overline{\sigma}} \qquad (1) $$

Where $f$ is the elastic domain that evolve in the stress space following the expression:

$$ f = J_2(\bar{\sigma} - \bar{X}) - R \qquad (2) $$

$\bar{X}$ and $R$ are hardening parameters that can be written:

$$ \bullet \text{ for the kinematic hardening: } d\bar{X} = C d\overline{\epsilon^{pl}} - \bar{X} |d\overline{\epsilon^{pl}}| \qquad (3) $$

$$ \bullet \text{ for the isotropic hardening: } R = R_0 + Q_{\infty} (1 - e^{-b\epsilon_{eq}^{pl}}) \qquad (4) $$

In order to identify these parameters, tensile tests and strain-controlled cyclic tests were performed on each material [11]. Figure 3(a) shows the extreme values of stabilized stress-strain loop that were plotted as a function of the corresponding deformation level.
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 3: Constitutive steel of flat material, cyclic and monotonic elastic-plastic behaviors. (a) stabilized cyclic conditions compared to monotonic tensile test, (b) stabilized stress-strain loop for 0.5% deformation level, (c) stress softening until stabilization with the applied cycles at 0.5% deformation level, (d) elastic-plastic laws and their bilinear fit used in FEM.</figcaption></figure>

Hence, three material behaviors have been considered for the simulation:

* Elastic law: called E,

* Bilinear **Monotonic Isotropic law**: called **MI**,

* Bilinear **Cyclic Kinematic law**: called **CK**.

To reduce the computing time and to simplify the model, only the bilinear fits were used to describe the plastic behavior (Figure 3 (d)), for either an isotropic or kinematic hardening model. The used parameters in the FEM model are listed in Table 3.
<table><caption>Table 3: Plastic parameters used in the linear kinematic strain-hardening models.</caption><thead><tr><th></th><th>Material</th><th>Yield strength/ σ<sub>y,flat</sub></th><th>Slope γ</th></tr></thead><tbody><tr><td rowspan="2">Monotonic law (MI)</td><td>Pad</td><td>1.17</td><td>2.368</td></tr><tr><td>Flat</td><td>0.99</td><td>1.704</td></tr><tr><td rowspan="2">Cyclic law (CK)</td><td>Pad</td><td>1</td><td>14.170</td></tr><tr><td>Flat</td><td>0.75</td><td>16</td></tr></tbody></table>

## 2.2 Fretting fatigue rig:

The fretting fatigue rig used in this study is shown in Figure 4(a). The fretting specimen consisted of a cylindrical punch with radius R=25, 17, 15 or 10mm. These pad radiuses have been optimized to reproduce as close as possible the “quasi” elliptical pressure profile related to the application. The fretting and fatigue specimen geometry are detailed in Figure 4(b). The cylinder pad is moved by the upper actuator, by applying a purely alternating sinusoidal cyclic displacement δ(t) in the fretting contact. This generates an alternating tangential force Q(t) in the contact.

The lower actuator was controlled in force and a fatigue load σ(t) was applied to the flat specimen. Fretting and fatigue cyclic loading were controlled in phase (i.e. both fretting and fatigue actuator move in the same direction) as detailed by Montalvo et al [15]. This means that the maximum fretting load Q*max and the maximum fatigue load σmax were applied simultaneously.

The normal force P was constant throughout the test, and it was applied to the contact using a mechanical spring system. Tests were carried out to 2x10⁶ cycles for short tests, and 10⁷ cycles for long tests to better evaluate the crack propagation. The frequency was constant and was set to 12Hz. Fretting loading, which is made up of an average stress and amplitude, and fatigue load were changed from one block to another in a decreasing manner. The fretting ratio was kept constant at RQ* = Q*min/Q*max = -1, while high loading ratios were chosen because they represent better the flexible service conditions (Rσ = σmin/σmax > 0.5).
<figure><img src="image_8.png" /><figcaption>Figure 4: (a) photo of the fretting fatigue test set-up at the LTDS laboratory, (b) pad specimen geometry and flat specimen geometry.</figcaption></figure>

## 2.3 Test procedure:

Before starting the fretting fatigue tests, the contact surfaces are cleaned with ethanol. The pad had been carefully aligned to be perpendicular to the specimens to ensure a uniform pressure field and allow a linear and homogeneous pressure distribution. In this way, crack nucleation begins, at the same time, along all the contact edges and the crack propagates through the specimen up to the final rupture. After aligning the assembly, first the mean bulk fatigue load $\sigma_m$ then the normal force P are applied. During the first 2000 fretting cycles, the alternated bulk fatigue load $\sigma_a$ and the displacement amplitude are increased simultaneously to reach the target fatigue stress amplitude $\sigma_a$ and the maximum tangential force amplitude $Q_t$ just before the gross slip transition. It is important to clarify that the tests were systematically performed under partial slip conditions to avoid gross slip surface wear. However, if the contact is too close to the gross slip transition, the interface becomes highly unstable, preventing the control of the fretting load through force-based control of the fretting actuator. Indeed, if the contact shifts to full sliding, this results in a displacement overshoot that destroys the contact. To address this issue, the chosen strategy consists in continuously adjusting the fretting actuator displacement in order to reach and subsequently maintain the desired tangential force amplitude. This approach allows for the reproduction of tangential loading blocks while ensuring the stability of the tests. In addition to controlling the tangential force amplitude, the evolution of the mean stress and the amplitude of the fatigue loading are also adjusted to reproduce as close as possible the target fretting fatigue block loadings.
## 2.4 Post test expertise/Examination of fretting scars and cracks at the end of the test:

All fretting fatigue tests were stopped at 2 million cycles if failure of the specimens did not occur. In such Test, the crack length was studied using a destructive method. The sample was cut in half, then imbedded in epoxy to be polished up to mirror surface state. Afterwards, cracks were observed using an optical microscope. The polishing and observation procedure was duplicated at least three times to obtain up to six cross section observations. An average value of the maximum projected crack length is then estimated. From each cross-section observation, the maximum projected crack length (i.e. along the z direction perpendicular to the surface) is determined, then an average value of these six crack lengths is computed to estimate a representative value of the maximum crack length extension of the test.

Figure 5(a) displays a crack observed after analyzing the fatigue specimens. The tests were repeated three times, and each time the same surface analysis was performed, and the corresponding crack length was measured. In the Test when crack arrest conditions were not reached and the crack propagates until failure, the fracture surface (Figure 5(b)) is then observed with a binocular microscope.

<figure><img src="image_9.png" /><figcaption>Figure 5: Methodology for examining crack length in unbroken specimen, (b) optical image of a broken specimen.</figcaption></figure>
### 3. Experimental results:

Previous investigations [16,17] have shown that the most critical fretting fatigue loadings are related to partial slip conditions, that is because under gross slip, wear tends to mitigate the cracking process. The block loading configurations were then selected to get as many failures as possible, that is why partial slip conditions have been prioritized. All tests were stopped after 2 million cycles (sometimes 10 million cycles to better quantify the crack propagation mechanism). We can distinguish between three different outcomes: Either no cracks are nucleated, or a crack nucleates but the fatigue loading is too low to propagate it and stops after a certain number of cycles, or the fatigue loading is large enough for the nucleated cracks to propagate until the specimen reaches failure. All tests were duplicated three times, and the average values of projected crack length or failure cycles are compiled in Table 4a and Table 4b respectively. In addition, the corresponding standard deviation was indicated. $N_f^*/b^*(\mu m)$: each test has been duplicated three times, and the given value is an average value of the failure cycles/crack length at the end of the test of the three tests.

<table><thead><tr><th rowspan="2">Test name</th><th>Pad radius</th><th rowspan="2">Block number</th><th rowspan="2">cycles per<br>block</th><th rowspan="2">Cumulated cycles</th><th>Normalized fretting<br>loading</th><th>Normalized mean<br>fatigue loading</th><th>Normalized alternated<br>fatigue loading</th><th rowspan="2">Average projected crack length<br>at the end of the test</th></tr><tr><th>R(mm)</th><th>Q*/P</th><th>σ<sub>m</sub>/σ<sub>y,flat</sub></th><th>σ<sub>a</sub>/σ<sub>y,flat</sub></th><th>b(μm)*</th></tr></thead><tbody><tr><td rowspan="7">Test A</td><td rowspan="7">17</td><td>1</td><td>621</td><td>621</td><td>0.65</td><td>0.77</td><td>0.080</td><td rowspan="7">15 ± 1</td></tr><tr><td>2</td><td>2 139</td><td>2 760</td><td>0.62</td><td>0.74</td><td>0.046</td></tr><tr><td>3</td><td>5 704</td><td>8 464</td><td>0.46</td><td>0.73</td><td>0.032</td></tr><tr><td>4</td><td>44 804</td><td>53 268</td><td>0.41</td><td>0.72</td><td>0.038</td></tr><tr><td>5</td><td>88 849</td><td>142 117</td><td>0.27</td><td>0.72</td><td>0.032</td></tr><tr><td>6</td><td>839 891</td><td>982 008</td><td>0.21</td><td>0.72</td><td>0.034</td></tr><tr><td>7</td><td>1 017 992</td><td>2 000 000</td><td>0.12</td><td>0.71</td><td>0.020</td></tr><tr><td rowspan="7">Test B</td><td rowspan="7">15</td><td>1</td><td>621</td><td>621</td><td>0.76</td><td>0.77</td><td>0.110</td><td rowspan="7">21 ± 15</td></tr><tr><td>2</td><td>2 139</td><td>2 760</td><td>0.61</td><td>0.74</td><td>0.073</td></tr><tr><td>3</td><td>5 704</td><td>8 464</td><td>0.46</td><td>0.73</td><td>0.052</td></tr><tr><td>4</td><td>44 804</td><td>53 268</td><td>0.41</td><td>0.72</td><td>0.056</td></tr><tr><td>5</td><td>88 849</td><td>142 117</td><td>0.27</td><td>0.72</td><td>0.043</td></tr><tr><td>6</td><td>839 891</td><td>982 008</td><td>0.20</td><td>0.72</td><td>0.042</td></tr><tr><td>7</td><td>1 017 992</td><td>2 000 000</td><td>0.12</td><td>0.71</td><td>0.024</td></tr><tr><td rowspan="7">Test C</td><td rowspan="7">15</td><td>1</td><td>621</td><td>621</td><td>0.70</td><td>0.77</td><td>0.090</td><td rowspan="7">60 ± 5</td></tr><tr><td>2</td><td>2 139</td><td>2 760</td><td>0.61</td><td>0.74</td><td>0.055</td></tr><tr><td>3</td><td>5 704</td><td>8 464</td><td>0.46</td><td>0.72</td><td>0.038</td></tr><tr><td>4</td><td>44 804</td><td>53 268</td><td>0.41</td><td>0.72</td><td>0.043</td></tr><tr><td>5</td><td>88 849</td><td>142 117</td><td>0.27</td><td>0.72</td><td>0.036</td></tr><tr><td>6</td><td>839 891</td><td>982 008</td><td>0.20</td><td>0.72</td><td>0.036</td></tr><tr><td>7</td><td>1 017 992</td><td>2 000 000</td><td>0.11</td><td>0.71</td><td>0.021</td></tr><tr><td rowspan="7">Test D</td><td rowspan="7">10</td><td>1</td><td>621</td><td>621</td><td>0.63</td><td>0.77</td><td>0.090</td><td rowspan="7">20 ± 2</td></tr><tr><td>2</td><td>2 139</td><td>2 760</td><td>0.61</td><td>0.74</td><td>0.055</td></tr><tr><td>3</td><td>5 704</td><td>8 464</td><td>0.46</td><td>0.72</td><td>0.038</td></tr><tr><td>4</td><td>44 804</td><td>53 268</td><td>0.41</td><td>0.72</td><td>0.043</td></tr><tr><td>5</td><td>88 849</td><td>142 117</td><td>0.27</td><td>0.72</td><td>0.036</td></tr><tr><td>6</td><td>839 891</td><td>982 008</td><td>0.20</td><td>0.72</td><td>0.036</td></tr><tr><td>7</td><td>1 017 992</td><td>2 000 000</td><td>0.11</td><td>0.71</td><td>0.021</td></tr><tr><td rowspan="7">Test E</td><td rowspan="7">10</td><td>1</td><td>621</td><td>621</td><td>0.75</td><td>0.77</td><td>0.090</td><td rowspan="7">79 ± 40</td></tr><tr><td>2</td><td>2 139</td><td>2 760</td><td>0.61</td><td>0.74</td><td>0.055</td></tr><tr><td>3</td><td>5 704</td><td>8 464</td><td>0.46</td><td>0.72</td><td>0.038</td></tr><tr><td>4</td><td>44 804</td><td>53 268</td><td>0.41</td><td>0.72</td><td>0.043</td></tr><tr><td>5</td><td>88 849</td><td>142 117</td><td>0.27</td><td>0.72</td><td>0.036</td></tr><tr><td>6</td><td>839 891</td><td>982 008</td><td>0.20</td><td>0.72</td><td>0.036</td></tr><tr><td>7</td><td>1 017 992</td><td>2 000 000</td><td>0.11</td><td>0.71</td><td>0.021</td></tr><tr><td rowspan="7">Test F</td><td rowspan="7">10</td><td>1</td><td>621</td><td>621</td><td>0.63</td><td>0.78</td><td>0.049</td><td rowspan="7">91 ± 4</td></tr><tr><td>2</td><td>2 139</td><td>2 760</td><td>0.61</td><td>0.78</td><td>0.049</td></tr><tr><td>3</td><td>5 704</td><td>8 464</td><td>0.46</td><td>0.77</td><td>0.042</td></tr><tr><td>4</td><td>44 804</td><td>53 268</td><td>0.41</td><td>0.77</td><td>0.042</td></tr><tr><td>5</td><td>118 849</td><td>172 117</td><td>0.27</td><td>0.77</td><td>0.042</td></tr><tr><td>6</td><td>839 891</td><td>1 012 008</td><td>0.20</td><td>0.77</td><td>0.035</td></tr><tr><td>7</td><td>987 992</td><td>2 000 000</td><td>0.11</td><td>0.77</td><td>0.035</td></tr><tr><td rowspan="7">Test G</td><td rowspan="7">25</td><td>1</td><td>719</td><td>719</td><td>0.56</td><td>0.56</td><td>0.043</td><td rowspan="7">13 ± 1</td></tr><tr><td>2</td><td>3 214</td><td>3 933</td><td>0.52</td><td>0.54</td><td>0.020</td></tr><tr><td>3</td><td>8 297</td><td>12 230</td><td>0.41</td><td>0.54</td><td>0.015</td></tr><tr><td>4</td><td>54 245</td><td>66 475</td><td>0.33</td><td>0.54</td><td>0.014</td></tr><tr><td>5</td><td>137 976</td><td>204 451</td><td>0.23</td><td>0.54</td><td>0.013</td></tr><tr><td>6</td><td>1 013 317</td><td>1 217 768</td><td>0.19</td><td>0.53</td><td>0.010</td></tr><tr><td>7</td><td>782 233</td><td>2 000 000</td><td>0.10</td><td>0.53</td><td>0.007</td></tr><tr><td rowspan="7">Test H</td><td rowspan="7">17</td><td>1</td><td>554</td><td>554</td><td>0.53</td><td>0.59</td><td>0.092</td><td rowspan="7">15 ± 4</td></tr><tr><td>2</td><td>2 185</td><td>2 739</td><td>0.51</td><td>0.54</td><td>0.041</td></tr><tr><td>3</td><td>5 391</td><td>8 130</td><td>0.40</td><td>0.53</td><td>0.032</td></tr><tr><td>4</td><td>42 202</td><td>50 332</td><td>0.33</td><td>0.53</td><td>0.025</td></tr><tr><td>5</td><td>79 378</td><td>129 710</td><td>0.23</td><td>0.53</td><td>0.018</td></tr><tr><td>6</td><td>831 387</td><td>961 097</td><td>0.18</td><td>0.53</td><td>0.015</td></tr><tr><td>7</td><td>1 038 903</td><td>2 000 000</td><td>0.11</td><td>0.53</td><td>0.011</td></tr></tbody></table>

(a)
<table><thead><tr><th rowspan="2">Test name</th><th>Pad radius</th><th rowspan="2">Block number</th><th rowspan="2">cycles per<br>block</th><th rowspan="2">Cumulated cycles</th><th colspan="3">Normalized</th><th rowspan="2">Average failure cycle<br>N<sub>f</sub>*</th></tr><tr><th>R(mm)</th><th>d fretting</th><th>Q*/P</th><th>Normalized mean<br>fatigue loading</th><th>Normalized<br>alternated</th></tr></thead><tbody><tr><td rowspan="5">Test I</td><td rowspan="5">17</td><td>1</td><td>7650</td><td>7650</td><td>0.82</td><td>0.90</td><td>0.083</td><td rowspan="5">876,640 ± 35,000</td></tr><tr><td>2</td><td>12660</td><td>20310</td><td>0.79</td><td>0.89</td><td>0.070</td></tr><tr><td>3</td><td>600000</td><td>620310</td><td>0.69</td><td>0.88</td><td>0.059</td></tr><tr><td>4</td><td>100000</td><td>1620310</td><td>0.59</td><td>0.87</td><td>0.049</td></tr><tr><td>5</td><td>379690</td><td>2000000</td><td>0.42</td><td>0.86</td><td>0.039</td></tr><tr><td rowspan="5">Test J</td><td rowspan="5">10</td><td>1</td><td>7650</td><td>7650</td><td>0.89</td><td>0.90</td><td>0.083</td><td rowspan="5">140, 200 ± 3,000</td></tr><tr><td>2</td><td>12660</td><td>20310</td><td>0.79</td><td>0.89</td><td>0.070</td></tr><tr><td>3</td><td>600000</td><td>620310</td><td>0.68</td><td>0.88</td><td>0.059</td></tr><tr><td>4</td><td>100000</td><td>1620310</td><td>0.59</td><td>0.87</td><td>0.049</td></tr><tr><td>5</td><td>379690</td><td>2000000</td><td>0.42</td><td>0.86</td><td>0.039</td></tr><tr><td rowspan="5">Test K</td><td rowspan="5">10</td><td>1</td><td>7650</td><td>7650</td><td>0.71</td><td>0.90</td><td>0.083</td><td rowspan="5">181,807 ± 40,000</td></tr><tr><td>2</td><td>12660</td><td>20310</td><td>0.79</td><td>0.89</td><td>0.070</td></tr><tr><td>3</td><td>600000</td><td>620310</td><td>0.68</td><td>0.88</td><td>0.059</td></tr><tr><td>4</td><td>100000</td><td>1620310</td><td>0.59</td><td>0.87</td><td>0.049</td></tr><tr><td>5</td><td>379690</td><td>2000000</td><td>0.42</td><td>0.86</td><td>0.039</td></tr><tr><td rowspan="5">Test L</td><td rowspan="5">10</td><td>1</td><td>450</td><td>450</td><td>0.72</td><td>0.84</td><td>0.031</td><td rowspan="5">636,916 ± 200,000</td></tr><tr><td>2</td><td>7540</td><td>7990</td><td>0.93</td><td>0.85</td><td>0.032</td></tr><tr><td>3</td><td>19450</td><td>27440</td><td>0.90</td><td>0.84</td><td>0.032</td></tr><tr><td>4</td><td>37175</td><td>64615</td><td>0.79</td><td>0.84</td><td>0.031</td></tr><tr><td>5</td><td>165380</td><td>229995</td><td>0.66</td><td>0.84</td><td>0.029</td></tr><tr><td rowspan="5">Test M</td><td rowspan="5">10</td><td>1</td><td>7650</td><td>7650</td><td>0.80</td><td>0.90</td><td>0.083</td><td rowspan="5">171,201 ± 90,000</td></tr><tr><td>2</td><td>12660</td><td>20310</td><td>0.79</td><td>0.89</td><td>0.070</td></tr><tr><td>3</td><td>600000</td><td>620310</td><td>0.68</td><td>0.88</td><td>0.059</td></tr><tr><td>4</td><td>100000</td><td>1620310</td><td>0.59</td><td>0.87</td><td>0.049</td></tr><tr><td>5</td><td>379690</td><td>2000000</td><td>0.42</td><td>0.86</td><td>0.039</td></tr><tr><td rowspan="5">Test N</td><td rowspan="5">10</td><td>1</td><td>450</td><td>450</td><td>0.80</td><td>0.84</td><td>0.031</td><td rowspan="5">124,934 ± 25,000</td></tr><tr><td>2</td><td>1253</td><td>1703</td><td>0.94</td><td>0.85</td><td>0.032</td></tr><tr><td>3</td><td>3241</td><td>4944</td><td>0.90</td><td>0.84</td><td>0.032</td></tr><tr><td>4</td><td>6196</td><td>11140</td><td>0.79</td><td>0.84</td><td>0.031</td></tr><tr><td>5</td><td>27563</td><td>38703</td><td>0.66</td><td>0.84</td><td>0.029</td></tr><tr><td rowspan="5">Test O</td><td rowspan="5">10</td><td>1</td><td>7650</td><td>7650</td><td>0.80</td><td>0.90</td><td>0.083</td><td rowspan="5">1,005,471 ± 35,000</td></tr><tr><td>2</td><td>6425</td><td>7420</td><td>0.79</td><td>0.81</td><td>0.024</td></tr><tr><td>3</td><td>9965</td><td>17385</td><td>0.65</td><td>0.81</td><td>0.021</td></tr><tr><td>4</td><td>7438</td><td>24823</td><td>0.58</td><td>0.81</td><td>0.022</td></tr><tr><td>5</td><td>15972</td><td>40795</td><td>0.51</td><td>0.80</td><td>0.022</td></tr><tr><td rowspan="5"></td><td rowspan="5"></td><td>4</td><td>7438</td><td>24823</td><td>0.58</td><td>0.81</td><td>0.022</td><td rowspan="5"></td></tr><tr><td>4</td><td>15972</td><td>40795</td><td>0.51</td><td>0.80</td><td>0.022</td></tr><tr><td>4</td><td>35520</td><td>76315</td><td>0.43</td><td>0.80</td><td>0.021</td></tr><tr><td>4</td><td>100605</td><td>176920</td><td>0.36</td><td>0.80</td><td>0.020</td></tr><tr><td>4</td><td>192200</td><td>369120</td><td>0.29</td><td>0.80</td><td>0.020</td></tr><tr><td rowspan="5"></td><td rowspan="5"></td><td>4</td><td>309137</td><td>678257</td><td>0.23</td><td>0.80</td><td>0.018</td><td rowspan="5"></td></tr><tr><td>5</td><td>7438</td><td>685695</td><td>0.58</td><td>0.81</td><td>0.022</td></tr><tr><td>5</td><td>15972</td><td>701667</td><td>0.51</td><td>0.80</td><td>0.022</td></tr><tr><td>5</td><td>35520</td><td>737187</td><td>0.43</td><td>0.80</td><td>0.021</td></tr><tr><td>5</td><td>100605</td><td>837792</td><td>0.36</td><td>0.80</td><td>0.020</td></tr><tr><td rowspan="5"></td><td rowspan="5"></td><td>5</td><td>192200</td><td>1029992</td><td>0.29</td><td>0.80</td><td>0.020</td><td rowspan="5"></td></tr><tr><td>5</td><td>309137</td><td>1339129</td><td>0.23</td><td>0.80</td><td>0.018</td></tr><tr><td>6</td><td>7438</td><td>1346567</td><td>0.58</td><td>0.81</td><td>0.022</td></tr><tr><td>6</td><td>15972</td><td>1362539</td><td>0.51</td><td>0.80</td><td>0.022</td></tr><tr><td>6</td><td>35520</td><td>1398059</td><td>0.43</td><td>0.80</td><td>0.021</td></tr><tr><td rowspan="5"></td><td rowspan="5"></td><td>6</td><td>100605</td><td>1498664</td><td>0.36</td><td>0.80</td><td>0.020</td><td rowspan="5"></td></tr><tr><td>6</td><td>192200</td><td>1690864</td><td>0.29</td><td>0.80</td><td>0.020</td></tr><tr><td>6</td><td>309137</td><td>2000000</td><td>0.23</td><td>0.80</td><td>0.018</td></tr></tbody></table>

(b)

<caption>Table 4: list of fretting-fatigue block conditions tested experimentally and simulated, (a) tests with crack arrest conditions, (b) tests that experienced failure (the cumulated cycles at the ith block corresponds to the sum of the cycles of the previous blocks. Example marked in red in (a) and (b): the red value in the column “cumulated cycles” is the sum of the red values compiled in the column “cycles per block”).</caption>
### 3.1 fracture analysis:

#### 3.2.1 Analysis of block test conditions inducing no nucleation (i.e. $b < b_{CN} = 50\mu m$):

For the loading Test A, fretting and fatigue stressing are rather small such that the average value of the maximum projected crack remains inferior to 15µm. Previous studied [5] have shown that for this specific type of loading and this specific material couple, the crack nucleation length $b_{CN}$ was optimized to 50 µm. Since the projected length of the cracks issued from test-A was significantly smaller than this value, we can say that the test-A loading configuration induced no crack nucleation at least in mechanical point of view. The small observed cracks may be related to surface defects or small cracks caused by plasticity on the surface. They are not considered dangerous so that this loading conditions can be considered as no nucleation situation.

Figure 6(a) shows the evolution of the normal force P, tangential force Q* and the fatigue loading $\sigma_{max}$ (mean fatigue + amplitude) during test A. The normal force was kept constant during the test, whereas Q* and $\sigma_{max}$ decreased following a bloc sequence. Figure 6(b) shows an elliptic fretting cycle that proves that the partial slip conditions were respected. Looking at the morphology of the fretting scar Figure 6(c) we can distinguish two zones: a central zone corresponding to the stick zone, and two sliding zones that surround the stick zone. An asymmetry of the stick zone can be observed. It is due to the stick zone eccentricity induced by the fatigue loading imposed to the fatigue specimen but not to the pad. This aspect has been formalized by Nowell et al in [18] and it will be taken into account by the FEA simulation which leads to a dissymmetric profile of the surface shear distribution generated by the contact.
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

(c)

<caption>Figure 6: evolution of (a) normal load, tangential force and fatigue loading during the test, (b) fretting cycle during the test and (c) fretting scar and a crack for test_A.</caption>

### 3.2.2 Analysis of block test condition inducing failure:

Figure 7(a) shows the loading conditions for test M. Compared to test A, the loading conditions were more severe, the normal force was high enough to activate the plasticity. The tangential force and the fatigue loading were important, causing the specimens to break at 81 000 cycles. Looking at the fracture surface, we can distinguish three different zones: the first corresponds to a mixed fretting fatigue crack propagation, the second corresponds to propagation by fatigue alone, the last represents ultimate failure surface that refers to a high plastic deformation zone. Since the tangential force is controlled via the displacement of the fretting actuator, slight differences arise between the experimentally applied tangential force and the target theoretical loading block, especially during the initial test sequence (Figure 7a). This deviation remains negligible in magnitude and duration, ensuring the theoretical block accurately represents the imposed contact loading.
<figure><img src="image_14.png" /><figcaption>Figure 7: (a) evolution of normal load, tangential force and fatigue loading during the test, (b) fretting cycle during the test and (c) optical front view of the fracture surface.</figcaption></figure>
# 4.Predictive methodology for variable loading fretting fatigue test:

## 4.1 Finite Elements Analysis

The Finite Elements model used in this study was similar to that one developed by Gandiolle [11,12]. A 2D plane strain model was used to represent the fretting-fatigue test (Figure 8). The simulation closely matched the parameters of the physical experiment, including the dimensions, boundary conditions, the application of normal force and contact accommodation. The rolling radius is the same as the one used in the experimental tests. The loading order application was respected and was kept like the experimental tests:

* the average fatigue load is applied to the flat specimens;

* the contact between the punch and the test piece and between the rolling and the test piece is then created by displacement (indentation) of the punch and the bearing;

* the normal force is applied;

* the cyclic fretting loading and, the amplitude fatigue loading is finally applied.

<figure><img src="image_15.png" /><figcaption>Figure 8: Finite elements model of the fretting fatigue test and Crossland evolution under the contact surface.</figcaption></figure>

The results revealed that material's local accommodation and plasticity decreased after each loading cycle, leading to a stable elastic shakedown state for both the monotonic and kinematic hardening hypothesis. An indirect illustration of the convergence of the contact stability consists in measuring the deformed surface area—quantifiable via the half-width of the contact zone. The simulations illustrated in Figure 9 show that both isotropic and kinematic hardening allow us to reach the stabilization of the contact very rapidly, 3 cycles for CK and 17 for MI. The accelerated stabilization observed with kinematic hardening can be attributed to the greater compliance of its plastic constitutive law, as illustrated in Figure 3. Fatigue analysis for predicting crack nucleation and propagation in variable loading test sequences were carried out on this stable state. The size of the mesh under the contact was 15 µm and allows a reasonable computing time.
<figure><img src="image_16.png" /><figcaption>Figure 9: Evolution of the half-width of the contact area as a function of the simulated number of cycles for the different hypothesis: elastic, monotonic and kinematic. (Q*/P=0.89, σ<sub>m</sub>/σ<sub>y,flat</sub>=0.90, σ<sub>a</sub>/σ<sub>y,flat</sub>=0.083)</figcaption></figure>

## 4.2 crack nucleation life

The objective of this model is to estimate the crack-nucleation risk using a multiaxial fatigue criterion. The criterion used for this study was proposed by Crossland which depends on the traction/compression fatigue limit σ<sub>d</sub> as well as the torsion fatigue limit τ<sub>d</sub>. Crossland equivalent stress σ<sub>c</sub> is expressed as [19]:

$$ \sigma_c = \sqrt{J_{2,a}} + \alpha \cdot \sigma_{H,Max} \quad (5) $$

$$ \text{Where:} \qquad \sigma_{H,max} = \max_{t \in T} \left( \frac{1}{3} \text{trace} \left( \Sigma(t) \right) \right) \qquad (6) $$

$$ \sqrt{J_{2,a}} = \frac{1}{2} \max_{t \in T} \left\{ \max_{t \in T} \left[ \frac{1}{2} (\underline{S}(t) - \underline{S}(t_0)) : (\underline{S} - \underline{S}(t_0)) \right]^{1/2} \right\} \qquad (7) $$

$$ \alpha = \frac{\tau_d - \frac{\sigma_d}{\sqrt{3}}}{\frac{\sigma_d}{3}} \qquad (8) $$

In these expressions, $J_{2,a}$ represents the second invariant of the deviatoric stress amplitude tensor during a loading cycle, while $\sigma_{H,max}$ is the maximum value of the hydrostatic pressure $\sigma_H$ during a cycle. According to Crossland, crack nucleation occurs when $\sigma_c \ge \tau_d$.

The subsurface distribution of the Crossland equivalent stress related to the fretting crack nucleation condition at test I is illustrated in Figure 10(a). The graph shows severe stress gradients at the contact borders. The stress gradient leads to an over-conservative estimation of the cracking risk at the hot spot, thus the importance of using non-local fatigue stress analysis like stress averaging over a process volume [20] or the critical distance method [21,22]. In the present analysis, we consider the critical method consisting of evaluating Crossland criterion not directly at the surface of the material, but beneath it at a so called the critical distance $l_d$ [20,23]. For a specific value of x called $x_f$, the critical distance can be defined as the distance in which we have $\frac{\sigma_c(x_f, l_d)}{\tau_d} = 1$.

Gandiolle et al. demonstrated that the highest stability of the critical method (i.e. Displaying the smallest dispersion versus experiments) was achieved when the crack length related to the crack
nucleation threshold is equal to the short-to-long crack transition such that $b_{CN} = b_\emptyset$ (which will be considered in the following of this sections). For such default flaw size, the optimal critical distance was found to be confirming here Taylor's theory [21].

Another important aspect concerns the x coordinate at which the crack nucleates. To address this, two mathematical criteria were introduced:

* $x_{Max Surf}$ that represents the position where Crossland is maximum in the surface ($z = 0$):

$$x_{Max \ surf} \quad \text{where} \quad \frac{\sigma_c(x_{Maxsurf}; z)}{\tau_d} = \max\left(\frac{\sigma_c(x; z = 0)}{\tau_d}\right) \qquad (9)$$

* $x_{Max b_\emptyset/2}$ that represents the position where Crossland is maximum for ($z = \frac{b_\emptyset}{2}$):

$$x_{Max \ b_\emptyset/2} \quad \text{where} \quad \frac{\sigma_c(x_{Max \ b_\emptyset/2}; z = b_\emptyset/2)}{\tau_d} = \max\left(\frac{\sigma_c(x; z = b_\emptyset/2)}{\tau_d}\right) \qquad (10)$$

<figure><img src="image_17.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(b)</figcaption></figure>

<caption>Figure 10: (a) Crossland cracking risk at the plane surface, (b) Crossland distribution in the subsurface for the 2 different criteria.</caption>

Figure 10(b) illustrates the evolution of the Crossland risk in the subsurface for the two criteria. The results indicate that different values have been found at the same critical distance $l_d = 25\mu m$, with a Crossland risk higher when calculated using the critical distance approach compared to when calculated on the surface. This difference becomes particularly evident when the stress gradient is significant, especially in the Test of elastic simulations with high loadings. For the elastic-plastic model, the same value was observed for both criteria. To ensure a conservative approach, Crossland risk was calculated along the path corresponding to the max value at $z = b_\emptyset/2$ the vertical of $x_{Max b_\emptyset/2}$.

### 4.3. crack nucleation endurance:

The fretting fatigue endurance of the studied interface was established by maintaining the contact loading conditions (R=10, P) and varying the applied tangential force amplitude ($Q_{CN}^*$) for different number of cycles to identify the ($d_c = \frac{\sigma_c}{\tau_d}, N_{CN}$) conditions that induce a crack nucleation length of
$b_{CN} = b_{\emptyset}$. This analysis allows us to express the crack nucleation endurance using a Basquin-Coffin-Manson model. The first half of the equation, usually with a higher slope, represents the low cycle fatigue (LCF), therefore describes the plastic behavior whereas the second half with a lower slope represents the high cycle fatigue (HCF) and thus the elastic regime.

$$N_{CN} = a_1 * e^{(d_c-1)*b_1} + a_2 * e^{(d_c-1)*b_2} \qquad (11)$$

With $a_1$, $a_2$, $b_1$ and $b_2$, constants that depends on the plastic law and $d_c = \frac{\sigma_c}{\tau_d}$ the crossland cracking risk (i.e. cracking is activated when $d_c > 1$).

Variable loading block test requires a damage accumulation rule to take into consideration the damage history if the crack does not nucleate in the first block. Assuming a linear damage Miner approach [24], each load level is assigned to a partial damage increment $\Delta D_i$ which represents the relation between the loading increment $\Delta N_i$ and the crack nucleation endurance $N_{CNi}$ related to the fretting fatigue loading conditions of block $i$:

$$\Delta D_i = \frac{\Delta N_i}{N_{CN,i}} \qquad (12)$$

Where

$$N_{CN,i} = a_1 * e^{(d_{c,i}-1)*b_1} + a_2 * e^{(d_{c,i}-1)*b_2} \\ \text{and } d_{c,i} = \left(\frac{\sigma_c}{\tau_d}\right)_i \text{ the crossland cracking risk} \qquad (13)$$

A crack of length $b_{CN} = b_{\emptyset}$ nucleates when the sum of the partial damages equals 1:

$$D = \sum \Delta D_i = 1 \qquad (14)$$

The critical distance and the parameters of the endurance law depends heavily on the assumed elastic-plastic law used. Hence, for each elastic-plastic law, a post-processing analysis of plain fretting and fretting fatigue test related to the crack nucleation (i.e. $b = b_{CN} = 50\mu m$) have been performed to identify the optimal critical distance [25]. The loading conditions used are compiled in Table 5.

<table><caption>Table 5: loading conditions for the calibration of $l_d$ (FF: fretting fatigue, PF: plane fretting)[25]</caption><thead><tr><th></th><th>Test type</th><th>R</th><th>Q<sup>*</sup><sub>CN</sub>/P</th><th>σ<sub>m</sub>/σ<sub>y,flat</sub></th><th>σ<sub>a</sub>/σ<sub>y,flat</sub></th><th>Number of cycles</th><th>l<sub>d E</sub></th><th>l<sub>d MI</sub></th><th>l<sub>d CK</sub></th></tr></thead><tbody><tr><td rowspan="10">Critical<br>distance<br>calibration</td><td>FF</td><td>10</td><td>0.44</td><td>0.59</td><td>0.15</td><td>1 000 000</td><td>36</td><td>36</td><td>-</td></tr><tr><td>FF</td><td>10</td><td>0.16</td><td>0.66</td><td>0.07</td><td>1 000 000</td><td>35</td><td>31</td><td>28</td></tr><tr><td>FF</td><td>10</td><td>0.17</td><td>0.71</td><td>0.04</td><td>1 000 000</td><td>36</td><td>29</td><td>28</td></tr><tr><td>FF</td><td>10</td><td>0.19</td><td>0.74</td><td>0</td><td>1 000 000</td><td>39</td><td>30</td><td>28</td></tr><tr><td>PF</td><td>10</td><td>0.23</td><td>0</td><td>0</td><td>1 000 000</td><td>31</td><td>29</td><td>26</td></tr><tr><td>PF</td><td>10</td><td>0.33</td><td>0</td><td>0</td><td>1 000 000</td><td>22</td><td>22</td><td>14</td></tr><tr><td>PF</td><td>10</td><td>0.20</td><td>0</td><td>0</td><td>1 000 000</td><td>34</td><td>36</td><td>30</td></tr><tr><td colspan="2"></td><td colspan="2">Average</td><td colspan="2"></td><td>33</td><td>30</td><td>26</td></tr><tr><td colspan="2"></td><td colspan="2">%Std</td><td colspan="2"></td><td>5.5</td><td>4.6</td><td>5.8</td></tr></tbody></table>
<figure><img src="image_19.png" /><figcaption>Figure 11: Crack lengths as a function of tangential fretting force for plain fretting test conditions (R = 4.6 mm, P, f = 12 Hz, N =10⁶ cycles)[5]</figcaption></figure>

The procedure is the following: various tests have been performed to bracket the threshold $Q_{CN}^*$ tangential force inducing a $b = b_{CN} = 50\mu m$ crack length after $10^6$ cycles (Figure 11). Previous work has shown that the $Q_{CN}^*$ initiation threshold no longer evolves for $N>10^6$ cycles, particularly for steels [26]. The critical distance $l_d$ associated with each test was determined. Then, the average value of the critical distance $l_d \text{ avg}$ and its standard deviation were calculated for each material law (Table 5). We found that $l_{d(E)} = 33\mu m$, $l_{d(MI)} = 30\mu m$ and $l_{d(CK)} = 26\mu m$. These values are close to the one identified by Gandiolle [5] (i.e. $l_{opt} = 28\mu m$) for a similar alloy assuming a monotonic plastic laws and a pad with the radius of 4.6 mm while still considering $b_{CN} = 50\mu m$. A posteriori, this stability regarding $l_d$ and related crack nucleation predictions associated to the crack nucleation is optimal and can be related to the short-long crack transition $b_0$ so that ($b = b_{CN} = 50\mu m$).

However, the decision was taken to consider for each elastic-plastic law the corresponding optimal critical distance so that: $l_{d(E)} = 33\mu m$, $l_{d(MI)} = 30\mu m$ and $l_{d(CK)} = 26\mu m$.

<table><caption>Table 6: loading conditions for the calibration of the endurance parameters $d_c = \frac{\sigma_c}{\tau_d}$ (PF: plain fretting)[25]</caption><thead><tr><th></th><th>Test type</th><th>R</th><th>Q<sup>*</sup><sub>CN</sub>/P</th><th>Number of cycles</th><th>d<sub>c</sub> (E)</th><th>d<sub>c</sub> (BI)</th><th>d<sub>c</sub> (CK)</th></tr></thead><tbody><tr><td rowspan="5">Endurance calibration</td><td>PF</td><td>10</td><td>0.44</td><td>20 000</td><td>2.25</td><td>1.72</td><td>1.38</td></tr><tr><td>PF</td><td>10</td><td>0.28</td><td>50 000</td><td>1.43</td><td>1.34</td><td>1.16</td></tr><tr><td>PF</td><td>10</td><td>0.25</td><td>100 000</td><td>1.25</td><td>1.23</td><td>1.0845</td></tr><tr><td>PF</td><td>10</td><td>0.23</td><td>500 000</td><td>1.185</td><td>1.103</td><td>1.0532</td></tr><tr><td>PF</td><td>10</td><td>0.24</td><td>1 000 000</td><td>1.181</td><td>1.102</td><td>1.0524</td></tr></tbody></table>

The second phase involved defining the constants of the endurance law (Eq 8). To accomplish this, series of simulations were performed for each material law. Table 6 shows the loading conditions associated with the nucleation of a crack of $50\mu m$ for various numbers of cycles. For each configuration, the average critical distance previously obtained was used to calculate the crack nucleation risk $d_c = \frac{\sigma_c}{\tau_d}$ for each material law hypothesis. The corresponding ($N_{CN}, \frac{\sigma_c}{\tau_d} - 1$) are plotted in Figure 12. The power law adopted for this fit takes the following form:

$$N_{CN} = a_1 * e^{(d_c - 1) * b_1} + a_2 * e^{(d_c - 1) * b_2} \quad (15)$$

The constants for each material law can be found in Table 7.
<figure><img src="image_20.png" /><figcaption>Figure 12: Endurance law for the different elastic-plastic laws studied.</figcaption></figure>

<table><caption>Table 7: Endurance law parameters for the different elastic-plastic models</caption><thead><tr><th></th><th>E</th><th>MI</th><th>CK</th></tr></thead><tbody><tr><th>a<sub>1</sub></th><td>1,00E+09</td><td>6,00E+07</td><td>4,00E+07</td></tr><tr><th>a<sub>2</sub></th><td>7,00E+04</td><td>1,35E+05</td><td>1,50E+05</td></tr><tr><th>b<sub>1</sub></th><td>-40</td><td>-45</td><td>-90</td></tr><tr><th>b<sub>2</sub></th><td>-1</td><td>-2,7</td><td>-5,5</td></tr></tbody></table>
### 4.4 Crack propagation prediction:

To estimate the crack propagation, the studied conditions were computed using finite element modelling (FEM) to extract the normal stress fields at the minimum and maximum loading states along the crack path. The crack path is located at the trailing contact borders along the z direction perpendicular to the contact surface to be consistent with the experimental projected crack length analysis.

Then the mode I Stress Intensity Factor (SIF) $K_I$ was calculated using Bueckner weight function approach [27] as detailed in Figure 13(a) (b):

$$K_I = \sqrt{\frac{2}{\pi}} \int_{0}^{h} M(t) \cdot \sigma_x(t) dt \qquad (16)$$

$$\text{With} \qquad (17) \\ M(t) = t - 1/2 \left[ 1 + m_1 \cdot \frac{t}{h} + m_2 \cdot \left( \frac{t}{h} \right)^2 \right]$$

$$m_i = A_i + B_i r^2 + C_i r^6$$

<figure><img src="image_21.png" /><figcaption>Figure 13: The evolution in the subsurface of: (a) maximum and minimal principal stress S11, (b) maximal and minimal SIF, (c) $R_k = K_{min}/K_{max}$</figcaption></figure>

The mode II contribution is neglected since it only affects crack propagation for the first microns below the surface. Fouvry et al. [28] showed that the $K_I$ value derived from a crack normal to the contact surface using mode I approximation yields results comparable to those obtained from computation of mixed mode I and II stress intensity factor, which suggests that the mode II contribution, can be neglected. Crack arrest conditions are reached when the effective stress intensity factor range (i.e. $\Delta K_{eff}$) operating at the crack tip falls below a threshold known as crack arrest threshold $\Delta K_{th}$.

It is important to note that the SIF ratio $R_k = K_{I,min}/K_{I,max}$ is not constant below the subsurface of the specimens even though the fretting stress ratio and fatigue stress ratio are constant: $R_Q = -1$ and $R_\sigma = cste$ (Figure 13(c)). This justifies the need to evaluate the SIF at every point below the surface to describe the crack propagation. In the present situation, $R_k$ was negative until $z=0.6mm$ because the calculated virtual $K_{min}$ was negative which have no physical meaning. To capture the $R_k$ fluctuation, the Kujawski's fatigue crack driving parameter (i.e. $\bar{K}^*$) was considered [29]:
$$ \bar{K}^* = (\Delta K^+)^{0.5} \cdot K_{Imax}^{0.5} \qquad (18) $$

With

$$ \Delta K^+ = \Delta K = K_{Imax} - K_{Imin} \text{ if } R_k > 0 $$

$$ \Delta K^+ = K_{Imax} \text{ if } R_k < 0 $$

The crack propagation was described using Paris law [11]:

$$ N_{CP} = \int_{b=b_0}^{b=failure} \frac{db}{C\bar{K}^*m} \qquad (19) $$

With

$$ C = 3.35 \times 10^{-10} \text{ and } m = 3.26 $$

Note that $\bar{K}^*$ corresponds presently to $\Delta K_{eff}$.

The threshold crack arrest condition is defined using El Haddad's continuous model [30]:

$$ \Delta K_{th} = \Delta K_{\emptyset} \sqrt{\frac{b}{b + b_{\emptyset}}} \qquad (20) $$

With $b_{\emptyset} = 50 \mu m$ the short to long crack transition and

$$ \Delta K_{\emptyset} = 2MPa \cdot m^{1/2} \text{ for the elastic model and } 5.5 \text{ for MI and CK [12]} $$

The evolution of the $\bar{K}^*$ parameters beneath the surface for two different scenarios is illustrated in (a) (b)

Figure 14(a). If the fatigue stress is sufficiently low, $\bar{K}^*$ tends to increase initially due to the crack extension, however below a critical depth, after reaching a maximum, it sharply decreases due to the reduction of the fretting stress beneath the surface. This $\bar{K}^*$ falls below the threshold SIF of the material and the crack arrests (see Test (a) on (a) (b))

Figure 14(a)) (i.e. $\bar{K}^* < \Delta K_{th}$). In contrast, if the fretting-fatigue loading is sufficiently high, the $\bar{K}^*$ rises again deeper below the surface due to the crack length extension, and the crack continues to propagate until the failure (see Test (b) on (a) (b))

Figure 14(a)). To predict the number of propagation cycles until failure, an equivalent Paris law based on Kujawski's parameter $\bar{K}^*$ is adopted.

<figure><img src="image_22.png" /><figcaption>Figure 14: (a) Illustration of the crack arrest and non-crack arrest situation, (b) equivalent K Paris law to integrate the number of propagation cycles before failure.</figcaption></figure>
## 4.5 Global predictive method:

The global algorithm for predicting the lifetime of a contact under block loading sequences starts by simulating fretting fatigue loading using finite elements analysis on Abaqus 6.16 (Figure 15). The Crossland cracking risk was then evaluated at the end of each loading sequence. The crack nucleation endurance $N_{CN,i}$ was estimated using Eq (5) and a damage $D$ was calculated using Miner linear damage law. If at the last loading block $D < 1$, then no crack is nucleated under the applied conditions and the test is estimated to have an infinite endurance. On the other hand, if $D > 1$, a crack is nucleated with a crack length of $b_0 = 50\mu m$ and $N_{CN} = N(D = 1)$ (Figure 16). The $K^*$ parameter was then calculated using Bueckner weight function, and the crack propagation is predicted following Paris law. If the crack arrest condition is reached, the maximum number of cycles provided will be reached and the crack length ($b_{pred}$) at the end of the test will be given.

If the crack arrest condition is not reached, the algorithm moves directly to the next block and the finite lifetime is estimated as follows:

$$ \text{If } K_{Imax} > K_{IC}, \text{ failure is occurring and } N_{F,pred} = N_{CN} + N_{CP} \qquad (21) $$

<figure><img src="image_23.png" /><figcaption>Figure 15: Algorithm for calculating the total lifetime of block tests.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Figure 16: (a) Cumulative damage evolution during a simulation as a function of the loading block(b) Illustration of the propagation crack length after nucleation depending on the decreasing loading block conditions, in this example the crack is nucleated in the middle of block 2, the subsequent process is governed by propagation: first within the second part of Block 2 and then into the successive blocks.</figcaption></figure>

Figure 17 shows the graphic user interface was developed on Matlab to automate this algorithm. The material parameters and the loading conditions were input into the algorithm, then at the end of the session a datasheet report was automatically generated containing all the useful information on the crack nucleation and crack propagation process.

<figure><img src="image_25.png" /><figcaption>Figure 17: Graphic User Interface on MATLAB for the lifetime prediction algorithm.</figcaption></figure>
# 5. Comparison between experiments and simulations

## 5.1 Influence of the plastic law

The various tests (Table 4) were simulated using either a pure elastic law E, plastic law with an isotropic hardening MI, or with a kinematic hardening CK. The simulations have been compared to the experimental result considering either the number of cycles to failure ($N_f$), or the crack length after $2.10^6$ cycles ($b_{max}$).

Using a plastic law instead of an elastic law introduces plastic strain in the model. These strains lead to an increase in the contact area, which reduces the contact pressure and consequently reduces the maximum contact shear stress $q_{max}$. This indirectly controls the fretting stressing and crack nucleation condition [23]. Upon closer examination of Figure 18(a)(b) it can be concluded that the softer CK hypothesis leads to a slightly larger contact area and consequently a smaller contact pressure compared to the MI law. However, both CK and MI hypothesis predict lower maximum shear stresses $q_{max}$ compared to the elastic hypothesis (Figure 18(b)), suggesting that fretting damages would be more severe when using the elastic hypothesis than the CK and MI hypothesis.

The most striking aspect concerns the comparison of MI and CK plastic simulations regarding the elastic assumption. In fact, taking into account plastic deformation makes it possible to predict a doubling of the contact area and conversely a reduction by half of the contact pressure and especially of the maximum shear stress. We show here the effect of plasticity with respect to the estimation of fretting contact stressing and ultimately the prediction of fretting fatigue damages.

An indirect validation of the plastic deformation consists in comparing the contact size (half-width of the contact area “a”) predicted by the elastic (E), Kinematic Cyclic (KC) and Monotonic Isotropic (MI) simulations with the experimental measurements on the counter-face fretting scar (Figure 19). In this comparison, only Test A, G and H with a projected crack length of 14 µm, 11 µm and 15µm respectively have been considered as these tests represent crack lengths inferior to 15 µm. In fact, as the crack propagates in the subsurface, the compliance of the contact reduces which will lead to an over extension of the contact surface. And since our model does not physically simulate the crack nucleation and propagation, a gap between the experimental radius and the predicted one is to be expected. Examining the results of these tests where the crack extension and therefore its influence are negligible, we observe an almost perfect correlation between the measured contact radius and the CK predictions. On the other hand, the elastic predictions fell short and underestimated the contact extension because the plastic deformations were not considered.
<figure><img src="image_26.png" /><figcaption>Figure 18: (a) fretting scar, comparison with each material law hypothesis elastic model E: —, Monotonic law MI: — and cyclic law CK: —, (b) pressure profiles, (c) shear profiles (Test_H, Q/P = 0.53).</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>Figure 19: Analysis of the contact radius: comparison between experimental values and predicted.</figcaption></figure>

Figure 20 (a) depicts the development of the Crossland criteria at the surface at the end of the first block. We observe concentrated stress at the edge of the contact leading to an overestimation of the fretting crack. This is attributed to the fretting stress gradient situation highlighting the necessity to consider the critical distance approach to palliate such overestimation. Crossland was then evaluated at this critical distance, which depends on the material law used as previously established. The evolution of the Crossland criteria at the critical distance for each elastic-plastic law is presented in the Figure 20(b). As expected, the elastic model overestimates the stress level compared to the monotonic or the cyclic plastic laws. Indeed, the plastic models induce more irreversible deformation and then reduce the local stress. In this example, regardless of the law used to describe the material, the Crossland criteria was superior to 1 during the first loading blocks, suggesting that a crack will be nucleated. Figure 20(c) shows the evolution of the accumulated damage as a function of the loading cycles. The condition of crack nucleation (D=1) is therefore reached faster using the elastic hypothesis at $N_{CN,pred}(E) = 12 117$ then followed by MI $N_{CN,pred}(MI) = 33 309$ and CK $N_{CN,pred}(CK) = 46 821$ simulations. Hence, the choice of the material law clearly influences the evolution of the crack nucleation prediction. As seen in Figure 20(d) the crack propagation in the elastic model is faster and more brutal which leads to a failure prediction of around 135 015 compared to the CK law for example that is characterized by a much slower and a softer crack propagation mechanism so that failure was estimated above 968 272. Compared to the experimental result, the average failure number of cycles among the three experimental tests was 636 916 with one of the tests breaking at 1 002 001 cycles. From this comparison, we can conclude that for both plastic laws MI and CK provide acceptable estimations comparatively to the experimental dispersion whereas elastic simulations are too pessimistic.
<figure><img src="image_28.png" /><figcaption>Figure 20: comparison of (a) Crossland criterion on the surface, (b) Crossland criterion at the critical distance, (c) damage evolution as a function of number of cycles, (d) crack evolution as a function of number of cycles between the elastic model E: —, Monotonic law MI: -, and cyclic law CK: -, experimental failures : ↓, for Test L.</figcaption></figure>

## 5.2 Comparison between experiments and simulations

The simulations were divided into two separate groups. The first group consisted of tests in which crack propagation occurred without reaching failure, while the second group involved tests under finite endurance conditions (i.e. failure occurs during the experiment). The loading conditions and the experimental results are summarized in Table 4.
### 5.2.1 Quantitative comparison of the prediction

The initial series of tests featured a low $Q^*/P$ ratio (<0.7) and a low fatigue loading. These tests were simulated using each of the material law hypothesis detailed in the previous section and the resulting crack length at the end of the simulation were compared with the experimental value. Since three tests for each loading condition were experimentally performed, an average value of the 3 three crack lengths obtained from the experimental test was used to illustrate the experimental results.

Additionally, a new approach was introduced to estimate the crack length for tests that did not reach the crack nucleation conditions ($b_{max} < b_0 = 50 \ \mu m$). The crack length at the end of the simulation was calculated using the following equation:

$$b_{max} = b_0 * D \qquad (22)$$

Where D is the cumulated damage at the end of the simulation

Using this approximation, cracks shorter than $b_0$ can be compared through a semi quantitative description.

Figure 21 compares the experimental crack lengths and the simulated ones, as well as to $b_0 = 50 \ \mu m$ the short-long crack transition.

The elastic model greatly overestimated the crack length for all tests conditions, and even predicted failure for tests such as C, E and F where the experimental crack length did not exceed 100µm. This result was expected since an elastic hypothesis is not able to capture the plastic deformations occurring within the spiral interface and therefore the decrease in pressure and shear stress fields on the surface. Hence elastic simulations lead to very conservative predictions. The error in prediction is too substantial to be considered reliable. In contrast, monotonic and cyclic elastic-plastic laws seem to accurately predict the crack nucleation, whereas crack arrest conditions are reached immediately after the propagation. Hence MI hypothesis tends to dangerously underestimate the crack propagation extension as illustrated in Figure 21 whereas the CK hypothesis seems to predict well the crack length compared to experimental values.

<figure><img src="image_29.png" /><figcaption>Figure 21: Analysis of unfailed test conditions: Comparison between predicted crack lengths ($b_{\{pred\}}$) and experimental crack lengths ($b$).</figcaption></figure>
The outcome of the second series of tests leading to failure is illustrated in Figure 22(a). The elastic model exhibits a behavior identical to that of the crack arrest results; it predicts failure at a shortened number of loading cycles comparatively to the experiments providing a conservative approximation of the fretting fatigue lifespan. By contrast, elastic-plastic simulations provide predicted endurances closer to the experiment values. However, in many situations, MI and CK simulations are too optimistic with predicting endurances slightly longer than observed experimentally. The mean square logarithmic error (MSLE) between the experiment results and the predicted results was introduced to better evaluate the prediction (Eq 23).

$$E_N f(\%) = \ln \left( \frac{1 + N_{f,pred}}{1 + N_{f,exp}} \right) \qquad (23)$$

Table 8 presents the experimental and numerical results for the different elastic-plastic laws and the corresponding error for each configuration. A decrease of the error can be seen when using either a MI or CK plastic hypotheses, as the error goes from -62% with pure elastic law to 22% and 18% respectively for MI and CK. Figure 22(b) shows that even though the error between the experimental results and the elastic simulation is significant, the elastic simulations provide systematic conservative results and therefore could be preferred for a safe fretting fatigue designing strategy.

<figure><img src="image_30.png" /><figcaption>Figure 22: (a) Analysis of test conditions leading to failure: experimental (NF,exp) and predicted (NF,pred : Eq. 21) fretting fatigue endurances, (b) the mean error between the predicted (NF,pred) and the experimental result.</figcaption></figure>

<table><caption>Table 8: Numerical life estimation using the Numerical damage accumulation rule for the different plastic laws hypothesis compared with experimental results.</caption><thead><tr><th rowspan="2">Name</th><th rowspan="2">N<sub>F,exp</sub></th><th colspan="3">N<sub>F,pred</sub></th><th colspan="3">Error</th></tr><tr><th>E</th><th>MI</th><th>CK</th><th>E</th><th>MI</th><th>CK</th></tr></thead><tbody><tr><td>I</td><td>876 640</td><td>219 610</td><td>402 736</td><td>385 710</td><td>-60%</td><td>-34%</td><td>-36%</td></tr><tr><td>J</td><td>140 200</td><td>62 875</td><td>247 884</td><td>235 184</td><td>-35%</td><td>25%</td><td>22%</td></tr><tr><td>K</td><td>181 807</td><td>54 780</td><td>256 942</td><td>190 419</td><td>-52%</td><td>15%</td><td>2%</td></tr><tr><td>L</td><td>636 916</td><td>135 015</td><td>906 173</td><td>968 272</td><td>-67%</td><td>15%</td><td>18%</td></tr><tr><td>M</td><td>171 201</td><td>46 449</td><td>272 033</td><td>179 939</td><td>-57%</td><td>20%</td><td>2%</td></tr><tr><td>N</td><td>124 934</td><td>43 828</td><td>320 616</td><td>252 270</td><td>-45%</td><td>41%</td><td>31%</td></tr><tr><td>O</td><td>1 005 471</td><td>70 330</td><td>922 245</td><td>709 445</td><td>-116%</td><td>-4%</td><td>-15%</td></tr></tbody><tfoot><tr><td></td><td></td><td></td><td></td><td>mean</td><td>-62%</td><td>22%</td><td>18%</td></tr></tfoot></table>
<figure><img src="image_31.png" /><figcaption>Figure 23: Comparison experimental (NF,exp) and predicted (NF,pred; Eq. 21) fretting fatigue endurances. A perfect correlation is achieved when the endurance points are aligned along the correlation curve (slope = 1; NF,pred = NF,exp).</figcaption></figure>

Another chart to evaluate the stability of the modelling consists in plotting the evolution of the predicted endurances (NF, pred) against the experiments (NF, exp) (Figure 23). By providing a better description of the contact area extension and consequently surface pressure and shear fields, both MI and CK plastic simulations display a better correlation with experiments and a less scattering than elastic analysis (Figure 23). It is surprising to note that the very basic MI law leads to equivalent results than CK simulation. This indirectly suggests that most of the contact accommodation may occur during the initial contact loading. Therefore, a monotonic plastic law is sufficient to describe the plastic accommodation of the interface even if fretting is a cyclic load. This good correlation of the MI approach in terms of prediction of the lifetime is to be compared with the good prediction of the extension of the contact area (Figure 20) and therefore indirectly of the pressure profile imposed in the interface. In other words, it can be assumed that if the applied constitutive law allows a good estimation of the plastically deformed contact area, then it is possible to approximate the fretting fatigue lifetime in a representative way. This result is very important since a cyclic plastic law is much more complicated and fastidious to identify than a monotonic law established from a simple tensile test. In comparison, the predictions from elastic simulations lie well below the correlation curve and present strong dispersions. However, although more dispersed, the elastic hypothesis has two advantages: on the one hand, it allows very fast predictions, with computation times less than 30 minutes compared to 2 hours for an elastoplastic calculation that can sometimes go up to 4 hours for high loading tests. On the other hand, we show that the predicted endurances are systematically lower than those observed experimentally. This conservative behavior is very important in the field of mechanical design and can justify the use of an elastic hypothesis to avoid any risk of fretting fatigue failure under variable loading conditions.
## Conclusion

A double actuator fretting fatigue rig was applied to investigate the fretting fatigue performance of a cylinder/plane contact. This rig allows independent monitoring of the normal force, the fatigue stress, and the fretting tangential force amplitude. The following results were obtained:

* A predictive algorithm (combining finite elements analysis on Abaqus and a numerical analysis on matlab) was developed to predict fretting fatigue crack nucleation and propagation for loading blocks under plastic regime.

* An experimental fretting fatigue analysis of a cylinder/plane configuration and loads that reached plasticity was performed with decreasing loading blocks and was used to validate the model. The loadings and contact geometry are representative of the pressure armor of flexible risers. Depending on the loading sequences, no crack nucleation, crack arrest or failure after $N_{F,exp}$ loading cycles were observed.

* An incremental damage analysis was implemented to simulate the crack nucleation ($N_{CN}$), then the crack propagation ($N_{CP}$) endurances in order to estimate the predicted fretting fatigue lifespan (i.e. $N_{F,pred}=N_{CN} + N_{CP}$). This model, transposed to the loading block analysis, captures the stress gradient effect using a critical distance method. The parameters of the crack nucleation law were identified from the inverse analysis of a limited number of plain fretting tests while the propagation law and the crack arrest threshold were extracted from literature. Two Monotonic Isotropic (MI) and Kinematic Cyclic (KC) plastic laws were considered to describe the plastic response of the materials.

* The simulations demonstrated that both MI and KC plastic hypotheses, by better describing the plastic accommodation of the interface, provide a good correlation with experimental results either regarding crack arrest extension or the total lifespan in the finite endurance domain.

* The simple MI approach directly extracted from a plain tensile test yields predictions equivalent to those provided by the complex cyclic kinematic formalism (KC). This typical behavior can be explained by the very fast plastic accommodation of the interface and the dominating influence of the normal force regarding the plastic indentation of the contact. Hence, even with fretting cycle, the plastic response of a fretting fatigue experiment can be adequately addressed using a simple monotonic plastic law.

* The proposed approach demonstrates significant advantages, particularly in the reduction of lead time for risk analysis by eliminating the need for dedicated test for each configuration to evaluate the risk of fretting fatigue failure. In addition to being very fast, elastic simulations mostly predict longer crack extensions than observed and systematically conservative fretting fatigue endurances. This conservative behavior is crucial in the field of mechanical design and can justify the use of an elastic hypothesis to mitigate any risk of fretting fatigue failure.

Future developments will focus on studying more complex fatigue fretting loading cycles by imposing the loading blocks randomly and no longer in a decreasing manner. Furthermore, various studies have demonstrated that wear can significantly modify the fretting fatigue response. It therefore seems interesting to integrate into these studies the effect of alternating loading blocks in partial slip but also in gross slip.

## Acknowledgments

The authors would like to express their sincere gratitude to The Ecole Centrale de Lyon (ECL, France) and TechnipFMC for financially supporting this work and enabling the publication of this paper.
References

[1] Specification for unbonded flexible pipe API 17 J 5th edition - 2024.

[2] Liu JP, Vaz MA, Chen RQ, Duan ML, Hernandez I. Axial mechanical experiments of unbonded flexible pipes. Pet Sci. oct 2020;17(5):1400-10.

[3] Szolwinski MP, Farris TN. Observation, analysis and prediction of fretting fatigue in 2024-T351 aluminum alloy. Wear. oct 1998;221(1):24-36.

[4] Araújo JA, Nowell D. Mixed high low fretting fatigue of Ti6Al4V: Tests and modelling. Tribol Int. sept 2009;42(9):1276-85.

[5] Gandiolle C, Fouvry S. Stability of critical distance approach to predict fretting fatigue cracking: a "ℓ opt – b opt" concept. Int J Fatigue. janv 2016;82:199-210.

[6] Cortez R, Mall S, Calcaterra JR. Investigation of variable amplitude loading on fretting fatigue behavior of Ti-6Al-4V. Int J Fatigue. 1999;

[7] Kantimathi A, Alic JA. The Effects of Periodic High Loads on Fretting Fatigue. J Eng Mater Technol. 1 juill 1981;103(3):223-8.

[8] Troshchenko V. Fatigue damage accumulation in aluminium and titanium alloys subjected to block program loading under conditions of stress concentration and fretting. Int J Fatigue. mars 1999;21(3):271-9.

[9] Mutoh Y, Tanaka K, Kondoh M. Fretting fatigue in JIS S45C steel under two-step block loading. JSME Int J. 1987;30(261):386-93.

[10] Aid A, Amrouche A, Bouiadjra BB, Benguediab M, Mesmacque G. Fatigue life prediction under variable loading based on a new damage model. Mater Des. janv 2011;32(1):183-91.

[11] Gandiolle C, Fouvry S, Charkaluk E. Lifetime prediction methodology for variable fretting fatigue loading: Plasticity effect. Int J Fatigue. nov 2016;92:531-47.

[12] Gandiolle C, Fouvry S. FEM modeling of crack nucleation and crack propagation fretting fatigue maps: Plasticity effect. Wear. mai 2015;330-331:136-44.

[13] Bosch C, Delafosse D, Longaygue X. Effects of strain and trapping on hydrogen-induced cracking in high strength low alloy steels. In: European Corrosion Congress 2010 (Eurocorr 2010) [Internet]. Moscou, Russia; 2010. p. 1558-70.Disponible sur: https://hal-emse.ccsd.cnrs.fr/emse-01063599

[14] Chaboche JL. Viscoplastic constitutive equations for the description of cyclic and anisotropic behaviour of metals. 1977;Disponible sur: https://api.semanticscholar.org/CorpusID:117808407

[15] Montalvo S, Fouvry S, Martinez M. A hybrid analytical-FEM 3D approach including wear effects to simulate fretting fatigue endurance: Application to steel wires in crossed contact. Tribol Int. sept 2023;187:108713.
[16] Said J, Garcin S, Fouvry S, Cailletaud G, Yang C, Hafid F. A multi-scale strategy to predict fretting-fatigue endurance of overhead conductors. Tribol Int. mars 2020;143:106053.

[17] Mindlin RD, Deresiewicz H. Elastic Spheres in Contact Under Varying Oblique Forces. J Appl Mech. 1953;

[18] Nowell D, Hills DA. Mechanics of fretting fatigue tests. Int J Mech Sci. janv 1987;29(5):355-65.

[19] Crossland B. Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. In: Proceedings of the International Conference on Fatigue of Metals. 1956. p. 138-49.

[20] Fouvry S, Kapsa Ph, Sidoroff F, Vincent L. Identification of the characteristic length scale for fatigue cracking in fretting contacts. J Phys IV. nov 1998;08(PR8):Pr8-159-Pr8-166.

[21] Taylor D. Geometrical effects in fatigue: a unifying theoretical model. Int J Fatigue. mai 1999;21(5):413-20.

[22] Araújo JA, Nowell D, Vivacqua RC. The use of multiaxial fatigue models to predict fretting fatigue life of components subjected to different contact stress fields. Fatigue Fract Eng Mater Struct. oct 2004;27(10):967-78.

[23] Fouvry S, Gallien H, Berthel B. From uni- to multi-axial fretting-fatigue crack nucleation: Development of a stress-gradient-dependent critical distance approach. Int J Fatigue. mai 2014;62:194-209.

[24] Miner MA. Cumulative Damage in Fatigue. J Appl Mech. 1 sept 1945;12(3):A159-64.

[25] Gandiolle C, Christine BM, Gilbert H, Henri-Paul L, Vincent M, Siegfried F, et al. ETUDE ET MODELISATION DE L'ENDURANCE EN FRETTING FATIGUE : EFFET DE LA PLASTICITE ET DES SOLLICITATIONS VARIABLES.

[26] Fouvry S. Étude quantitative des dégradations en fretting.

[27] Bueckner HF. Weight functions and fundamental fields for the penny-shaped and the half-plane crack in three-space. Int J Solids Struct. 1987;23(1):57-93.

[28] Fouvry S, Nowell D, Kubiak K, Hills DA. Prediction of fretting crack propagation based on a short crack methodology. Eng Fract Mech. avr 2008;75(6):1605-22.

[29] Kujawski D. A fatigue crack driving force parameter with load ratio effects. Int J Fatigue. 2001;23:239-46.

[30] El Haddad MH, Smith KN, Topper TH. Fatigue Crack Propagation of Short Cracks. J Eng Mater Technol. 1 janv 1979;101(1):42-6.