# Fretting Wear of Coated Surfaces Under Gross Slip Conditions

Hana Jerbi, Daniel Nélias, Marie-Christine Baietto

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Hana Jerbi, Daniel Nélias, Marie-Christine Baietto. Fretting Wear of Coated Surfaces Under Gross Slip Conditions. ASME 2014 12th Biennial Conference on Engineering Systems Design and Analysis, Jul 2014, Copenhagen, Denmark. 10.1115/ESDA2014-20264. [hal-01951566](https://hal.science/hal-01951566)

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

Distributed under a Creative Commons Attribution 4.0 International License
FRETTING WEAR OF COATED SURFACES UNDER GROSS SLIP CONDITIONS

H. Jerbi

LaMCoS
INSA-Lyon
CNRS UMR5259
Villeurbanne F69621, France

D. Nelias*

LaMCoS
INSA-Lyon
CNRS UMR5259
Villeurbanne F69621, France
daniel.nelias@insa-lyon.fr

M-C. Baietto

LaMCoS
INSA-Lyon
CNRS UMR5259
Villeurbanne F69621, France
marie-christine.baietto@insa-lyon.fr

ABSTRACT

Blade-disk contact within high pressure compressor, is a
high temperature fretting contact zone submitted to high stress
levels. Therefore, it is important to estimate the various damages
encountered in this area in order to correctly predict the lifetime
of the parts. Coatings are used in order to improve the strength of
the parts. In the framework of Innolub project, this study aims at
developing a numerical model of coated contact accounting for
wear and temperature. The objective is to estimate the kinetics of
wear, to simulate worn surfaces and to allow the manufacturer
to produce analyses of design with worn surfaces using a fast,
efficient and reliable method of calculation.

Keyword: Coating, High temperatures, Fretting, Wear, Energy approach, Gross slip

INTRODUCTION

The aerospace is a sector of activities where many challeng-
ing problems arise. The area of contact between the blade and
disk in turbo-machinery has been identified as a critical area for
the nucleation of damage for the safety of the flying machine. To
control damage, coatings are applied. There is a great need for
innovative coatings given good lubricating properties under high
temperatures conditions and limited wear of the parts. The pre-
diction of wear is a challenging task. Many formalizations have
been proposed for quantitative approach to wear. More than 300

have been identified by Meng and Ludema [1], resting on dif-
ferent applications [2–5]. The energy theory of wear, inspired
by the Archard’s theory, is selected as it is very simple to im-
plement. This work aims at developing a numerical model of
coated contact accounting for wear and temperature. The objec-
tive is to perform wear prediction using semi analytical compu-
tational methods. This research is based on previous works [6–8]
developed at LaMCoS. Gallego et al. [9] proposed an elastic con-
tact model under stick-slip conditions. This model has been ex-
tended to the phenomenon of fretting wear [10] and under partial
slip conditions [11]. Analytical solutions to take into account
the plasticity in the resolution of the tangential contact have been
developed [12]. The presence of heterogeneities within the con-
tacting bodies has been later introduced and the model applied
to the elastic-plastic contact under fretting wear conditions [13].
Finally, Leroux [14] extended the formulation and proposed a
model to account for the presence of multiple spherical inclu-
sions based on the equivalent inclusion method originally pro-
posed by Eshelby. The first part of this paper focuses on coat-
ing. The coating is modeled as a cluster of inclusions based on
the Eshelby’s theory. Validation is obtained by comparison with
multigrid results. In the second part, wear simulation is carried
out illustrating the model robustness and efficiency. Results are
then presented and briefly discussed.

<footnote>*corresponding author.</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: 3D model of sphere/coated plane elastic contact.</figcaption></figure>

# 1 Coating: Modeling and Validation

## 1.1 Modeling

A rigid indenter is considered, with a spherical tip of radius R. It is pressed against an elastic flat of properties ($E_2 = 210\,\text{GPa}$, $v_2 = 0.3$) covered by a layer ($E_1, v_1$) - more or less rigid than the substrate - and of thickness 't', see Fig. 1. 'W' is the normal load and 'a' the contact radius. The equivalent inclusion method in the sense of Eshelby allows to describe accurately the effect of inhomogeneities. This method is modified and improved in order to take into account the mutual influence between neighboring heterogeneous inclusions. A few analytical solutions have been obtained for inclusions of simple geometry embedded in an infi-nite isotropic medium, such as ellipsoidal [15], cuboidal [16] and cylindrical [17, 18] inclusions. The aim of this part is to model an isotropic heterogeneous coated contact, based on [14, 19], using an arrangement of cuboidal inclusions to describe the layer. The Eshelby's equivalent inclusion method is considered to quantify the surrounding material stresses. An iterative process is imple-mented to determine the displacements and stress fields caused by the eigenstrains of all inclusions, accounting for interactions between close inclusions. The proposed method can be seen as an enrichment technique for which the effect of heterogeneous inclusions is superimposed on the homogeneous solution. 3D and 2D Fast Fourier Transforms are utilized to improve the com-putational efficiency. Displacements and pressure relationships account for this contribution. For more details on the method and algorithm the reader may refer to [14].

## 1.2 Validation

The contact solution and internal stress field obtained for the 3D coated model are compared with results obtained using a 3D multigrid solver [20]. A sphere is pressed against a coated half space. The coating elastic constants are: $E_1 = 420\,\text{GPa} = 2 * E_2$, $v_1 = 0.3$ and $t=0.75 \, \mu\text{m}=0.25*a$. The normal load is kept con-stant $W=0.04\,\text{N}$. Figure 2 presents the distribution of pressure ob-tained using a 3D multigrid solver and the semi-analytical solver.

<figure><img src="image_3.png" /><figcaption>Figure 2: Distribution of pressure: fine mesh (dx=0.15 µm)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: Distribution of stress $\sigma_{xx}$ -for a thickness $t= 0.25a$- at different depths $z$: fine mesh (dx=0.05 µm=0.05a)</figcaption></figure>

$\sigma_{xx}$ within the layer and the substrate are plotted in Fig. 3 for a mesh size of 0.15 µm. The depth at which the stress is calculated is different, which explains the difference here.

# 2 Wear: Methodology and simulation

## 2.1 Methodology

The tribological analysis of a contact can be understood through the energetic balance. Friction is a dissipated phe-nomenon, and a part of this frictional dissipated energy is con-sumed by different processes that conduct to wear [21]. To quan-tify the wear of a sliding contact, Archard [22] published in 1953 his famous equation: $V=\alpha_{ARCHARD} \cdot S \cdot \frac{P}{K}$, where $\alpha_{ARCHARD}$ is the wear coefficient. The worn volume V is a function of the sliding distance s, the normal applied load P, and the hardness of the softer material K. The friction coefficient does not appear in this equation. However it is often observed that the greater the friction coefficient the larger the wear. In addition, the friction dissipated energy is equal to the work of the tangential load. For these reasons Fouvry and co-workers [2–4, 23, 24] de-veloped a modified Archard’s wear law: $V=\alpha \cdot \sum_N E_d$, with $E_d$ the frictional dissipated energy density. A method to simulate wear of contacting surfaces is proposed. A wear law based on the en-ergy dissipated by friction, derived from the work of Fouvry, is used. The local sliding amplitude is assumed constant along the
<figure><img src="image_5.png" /><figcaption>Figure 4: Profile of the punch [mm] in (x,z) plane</figcaption></figure>

wear cycles. Wear can concern only one of the contacting bodies, or both. In addition, the worn material is assumed in the numerical procedure to be removed at the end of each cyclic movement only. The input data of the problem are:

$P^k$, k=1, ..., N: the normal load history of each cycle

$s^k$, k=1, ..., N: the corresponding local sliding amplitude

$\delta_0$: the sliding amplitude

$\Delta N$: the number of cycles to update the worn geometry

$N_{max}$: the maximum number of cycles

$H_{max}$: the maximum value for the wear depth to end the computation

For each cycle, $E_d$ is computed at each point. It corresponds to the product of the tangential stress with the sliding distance; thus, in gross slip regime: $E_d = \sum_{k=1}^{N} \mu x^k x_s^k$.

The wear depth $\Delta h$ for one cycle is calculated at each point with an appropriate wear law $\Delta h= f(E_d)= \alpha \cdot S \cdot \sum_N \delta_0(N) E_d(N)$. The value is then multiplied by $\Delta N$, and the geometry is updated before next contact solution and wear calculation. The corresponding wear of each surface is taken into account according to a wear partitioning coefficient $WP$, equal to 0.5 in the case of an equal wear repartition for each body. The simulation can be stopped through two ways, one when the number of cycles $N$ exceeds the maximum number of cycles $N_{max}$, the other when a maximum value of a wear depth is reached. Finally the computation gives the final geometry and the wear distribution.

## 2.2 Simulation of a Fretting Wear Test

The numerical simulations presented here correspond to experimental fretting tests [25]. It consists of fretting tests with a Ti6242 punch/Ti17 plan. The indenter (Ti6242 punch) is considered as a plan with crown radius. The geometry of this punch is shown in Fig. 4. Materials properties are ($E_0$ = 114GPa, $v_0$ = 0.36) and ($E_2$ = 110GPa, $v_2$ = 0.34). A slip amplitude $\delta_0$ = 100µm is considered. The normal load $W$ and the friction coefficient $\mu$ are assumed constant all over the cycles and equal to 1150N and 0.68, respectively.

<figure><img src="image_6.png" /><figcaption>Figure 5: Evolution of wear: pressure and depth</figcaption></figure>

# 3 Results and discussions

For industrial application, the most important element is the lifetime of the disk, which can not be repaired, and whose failure is the most critical. In contrast, the blade root can be repaired, and breaking of the blade can be contained within the engine. Thus, the main objective is to develop a sacrificial deposit which wears out preferentially but which will avoid disk wear (Ti17). Before studying the response of deposit, we aim to characterize wear for uncoated contact to quantify the gain provided by the deposition.

## 3.1 Uncoated contact

A first simulation was carried out to analyze the wear response of uncoated contact Ti6242g/Ti17g. The simulation were conducted by imposing the sliding amplitude $\delta_0$ = 100µm. The normal load was chosen equal to 1150N, the friction coefficient to 0.68 and the wear coefficient $\alpha$ to $3e^{-5} mm^3/J$. 10 000 cycles were realized. Wear is assumed to affect only the plan (WP=0). The evolution of wear is detailed in the graphs of Fig. 5 in terms of contact pressure history and wear depth history. It can be observed that the pressure distribution has a profile similar to the surface of the punch for the first cycles and after it evolves naturally through a flat profile, i.e., with a uniform increment of wear inside the contact area as the wear law used in our case is proportional to the contact pressure. The area where the wear is maximum is located at the edge of the punch because of the peaks of pressure produced by the geometrical discontinuity. The resulting worn geometry of the plane after $N_{max}$ (=10 000) wear cycles is given in Fig. 6. These data characterize the uncoated interface, they will allow us to quantify the gain provided by the coating studied in the next section.

## 3.2 Coated contact

The coating was supposed two times harder than the matrix (the plan): $E_1$ = 2 * $E_2$ = 220GPa with a Poisson's ratio $v_1$=0.3.
<figure><img src="image_7.png" /><figcaption>Figure 6: Worn geometry after $N_{MAX} = 10000$ cycles</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Comparison of coated and uncoated contacts after $N_{MAX} = 20000$ cycles</figcaption></figure>

It is characterized by a thickness $t = 400\mu m$. The coefficient of wear is assumed identical. Simulation loading conditions considered are $W=1150N$ and $\mu = 0.32$. Wear is assumed to affect the plan and the punch ($WP=0.5$). The simulation is carried out for 20,000 cycles. Comparison between coated and uncoated results are shown in Fig. 7. It is observed that the wear depth for the coated case is reduced by more than half compared to the uncoated case. This result tends to confirm the role of sacrificial coating. The distribution of the shear stress component $\sigma_{xz}$ is plotted in the plane (x,z) with consideration of wear. It is shown in Fig. 8. The asymmetry is due to the effect of the tangential loading.

## CONCLUSION AND PERSPECTIVES

A semi-analytical contact method has been employed for the simulation of fretting wear. The contact solver is based on the CG method and uses optimized FFT techniques to reduce

<figure><img src="image_9.png" /><figcaption>Figure 8: Distribution of the shear stress component $\sigma_{xz}$ [MPa] in (x,z) plane</figcaption></figure>

CPU time for contact resolution. A contact model permitting us to simulate coating, based on the equivalent inclusion method in the sense of Eshelby, has been first presented and validated through comparison with Multigrid solutions. The contact solver has been coupled to a wear law proposed by Fouvry and coworkers. The model then enables one to account for surface geometry changes due to wear. Several academic simulations have been performed thanks to the efficiency of the contact solver. Results obtained through this analysis show that wear has a significant impact on the contact pressure distribution and on the geometry of the contacting surfaces, leading to changes even for limited wear. First, the simulations for uncoated contact show that the pressure distribution tends to be flattened, erasing the peaks of pressure at the contact edges. Secondly, for coated surfaces, the numerical method permits to obtain a contact solution for a coating of non-uniform thickness, with is not possible with other analytical techniques such as integral transforms.

## ACKNOWLEDGMENT

This work was supported by the French FUI Project Innolub. The authors gratefully acknowledge all partners of this consortium.

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