Wear 412–413 (2018) 92–108

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Wear
journal homepage: www.elsevier.com/locate/wear

A dynamical FEA fretting wear modeling taking into account the evolution T
of debris layer
⁎
P. Arnaud, S. Fouvry
Ecole Centrale de Lyon, LTDS, Ecully, France

A R T I C LE I N FO A B S T R A C T

Keywords: A new Finite Element Analysis (FEA) strategy is developed to simulate fretting wear, taking into account the
Ti-6Al-4V evolution of the debris layer trapped in the interface, so called “third body”. To validate this approach, simu-
Fretting wear lations were compared to experimental results from gross slip Ti-6Al-4V cylinder/plane experiments. Adequate
FEM simulation worn surface analyses allow the estimation of both cylinder and plane friction energy wear rates and the debris
Third body
layer thickness evolution. A third body conversion factor (γ(x)), expressing the proportion of worn thickness
transferred to the third body layer (i.e. debris layer) at a given position in the fretted interface is introduced. A
coupled Matlab-Python-Abaqus algorithm is developed to simulate the surface wear on plane and cylinder
surfaces to formalize the continuous evolution of the debris layer trapped within interface. Quantitative com-
parisons with experimental results confirmed the interest of this FEA approach. The maximum wear depth,
which was underestimated by nearly 80% without considering the third body, is predicted with an error less
than 10%. A numerical investigation demonstrates that the elastic properties of the third body do not influence
the surface wear profile. Acting as a contact pressure concentrator, the third body effect appears more geo-
metrical than rheological. This third body FEA fretting wear modeling is extended in order to consider both test
duration and sliding amplitude effects. Rather good correlations with experiments confirm the interest of this
approach.

1. Introduction cracking induced by “notch” stress concentration can occur [6]. Hence,
there is a real interest to model the fretting scar profile and above all to
Fretting wear damage are observed in many industrial assemblies predict the maximum wear depth [7].
subjected to vibrations. Cracking or/and surface wear are occuring Various wear models may be considered for fretting wear damage
depending on the sliding amplitude [1,2]. Small displacement ampli- [8]. However two main strategies are usually adopted to predict the
tudes, by promoting a partial slip contact, induce cyclic stresses which fretting wear volume (V) extension. The first one, based on the Arch-
favor crack nucleation and propagation. However, this loading condi- ard's wear equation [9], expresses the total wear volume as a linear
tion is non dissipative (i.e. very closed tangential force Q – displace- function between the product of the normal force and the total sliding
ment δ* hysteresis) and the surface wear is very limited. Above gross distance. Due to the friction fluctuations recent developments suggest
slip transition, a full sliding condition is activated promoting a quad- that the friction work approach (i.e. friction energy wear concept)
ratic dissipative fretting loop. The surface wear is then significantly provides a more representative description of the surface damage so
increased, the contact area extended and the maximum contact pres- that [10]:
sure sharply reduced. By reducing the surface stresses and removing the
V = α× ∑ Ed (1)
top cracked surface, gross slip tends to decrease the cracking risk.
Competition between cracking and wear was extensively investigated Where α is the friction energy wear coefficient. Both Archard and re-
during the past decades [3–5]. This concept is currently considered in lated friction energy approaches are quantitative, require a limited
industrial components like dovetail blade-disk aeronautical turbines number of variables and are easily implemented in FEA contact mod-
where sacrificial high wear rate coatings are currently applied to reduce eling. However, these approaches do not explicitly consider the debris
the fretting cracking problem. However, when the wear volume and layer (third body) trapped in the interface which can drastically modify
maximum wear depth are too high, over clearance and potentially surface wear damage. Godet followed by Berthier and co-authors

⁎
Corresponding author.
E-mail address: siegfried.fouvry@ec-lyon.fr (S. Fouvry).

https://doi.org/10.1016/j.wear.2018.07.018
Received 3 February 2018; Received in revised form 19 June 2018; Accepted 19 July 2018
Available online 20 July 2018
0043-1648/ © 2018 Elsevier B.V. All rights reserved.
P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

described this aspect, developing “third body theory” [11–14]. They Table 1
demonstrated that the wear rate can be expressed as a function of the Mechanical properties of the studied Ti-6Al-4V.
debris formation flow from the first bodies and the debris ejection flow Young's Poisson's Vickers Plastic yield
from the interface. This wear modeling states that the wear rate de- modulus, E ratio, υ hardness Hv0.3 σYO.2% (MPa)
pends on the thickness and rheological properties of the third body (GPa)
layer. This third body theory provides a more physical description of
Ti-6Al-4V 120 0.3 360 880
wear processes. However, the formulation provided by the authors re-
quires numerous variables, and appears complex to implement in FEA
simulations. 2.2. Plain fretting test
If the global wear volume prediction is still an open question, the
prediction of worn profiles is even less advanced. Indeed, the prediction This analysis is focused on a single cylinder/plane contact config-
of local wear depth requires a local description of surface wear pro- uration, with cylinder radius R = 80 mm, normal load P = 1066 N/mm
cesses. In addition to semi-numerical methods [15], a significant effort inducing a maximal Hertzian pressure of pmax = 525 MPa, and constant
was done during the past decades to implement surface wear modeling radius aH = 1.29 mm. The lateral width of the pad was fixed at L =
in finite element methods. A common approach consists in moving the 8 mm allowing a 2D plain strain hypothesis (i.e. aH/L < 0.16). As
surface nodes to simulate the worn profile [16]. However, a major shown in Fig. 1, the fretting test consists in a vertical contact where the
limitation of such approaches is the fact that they do not consider the normal force (P × L) is applied using a trolley system and the dis-
presence of a third body layer. Ding et al. [16] then Gosh et al. [17] and placement is monitored using a hydraulic actuator [24]. The resulting
Basseville et al. [18] were the first to investigate the presence of wear tangential force (Q × L) is recorded allowing the plotting of Q-δ fret-
debris in FEM fretting interfaces. They proposed to represent the third ting loops. From the fretting loop, we deduced the tangential force and
body by an individual partition in the FEM model of fretting interface. the displacement amplitude respectively Q* and δ*. Integrating the
Other models involving coupled FEM and DEM (discrete element hysteresis loop, we can also assess the friction energy Ed. The dis-
method) analysis were proposed by Haddad et al. [19] and Leonard placement amplitude includes the contact displacement but also the
et al. [20] to evaluate the influence of the third body on friction, con- tangential accommodation of the test apparatus and specimens.
tact pressure and stress fields. Therefore, to only consider the contact sliding, the experiments were
However, most of these developments consider a static description performed monitoring the residual displacement δ0, measured on the
of the third body and were not able to simulate simultaneously the fretting cycle when Q = 0. Indeed when Q = 0, no tangential de-
evolution of surface wear and third body layer. Besides these, numerical formations are generated in the test system and the corresponding δ0
investigations were barely related to experimental results. displacement corresponds well to the sliding amplitude imposed in the
Ding et al. [16,21] presented a debris evolution model, with con- interface.
sideration of micro-scale asperity-induced plasticity and included the A representative mean friction energy coefficient is considered to
high related effects of oxidation, for Ti-6Al-4V, to give a debris evolu- quantify the friction response of the interface [10].
tion model, including also reasonable comparisons with measured
debris layer thickness evolution across the contact width. Done et al. Ed
µe =
[22] recently proposed a 3D cylinder/plane surface wear modeling 4×δ*0 ×P × L (2)
including a third body description. Using semi analytical formulation,
For the studied conditions it was established at μe = 0.65 ± 0.05
the fretting wear profile of the plane is simulated assuming a flat dis-
Frequency was decreased to f = 0.11 Hz in order to limit adhesive
tribution of worn thickness converted to third body. The model also
wear phenomena and to ensure almost exclusively quasi-pure three
considers a unilateral wear process on plane. Despite such limitations,
bodies abrasion wear process, as previously described [26].
rather good correlations were observed with experiments. Simulta-
Two sets of experiments were applied to calibrate the surface profile
neously, a similar strategy using FE analysis was introduced [23]. This
modeling (Fig. 2). The first one consists in varying the test duration
2D model allows the bilateral wear of plane and cylinder surfaces.
from 2500 to 30,000 fretting cycles while keeping constant the sliding
Besides, the comparison with experiments suggest that better predic-
amplitude at δ0 = ± 75 μm. The second set of experiment consists in
tions are achieved if the γ conversion factor from worn surface to third
keeping constant the test duration at 10,000 fretting cycles while
body thickness is not constant but expressed as a parabolic function of
varying the sliding amplitude from δ0 = ± 62, ± 75, ± 100 and ±
the (x) distance from the center of the contact. Indeed, the probability
125 μm. Then various fretting test conditions outside the N-δ0 cali-
of wear debris to be included on the third body layer is larger in the
bration axis were applied to establish the stability of the model (Fig. 2)
center of the contact than on the lateral sides where it can more easily
be ejected from the interface.
The purpose of this research work is to deepen such a 2D FE analysis 2.3. Fretting scar analysis
by investigating the effect of the elastic properties of the debris layer
and by taking into account the effect of test duration and sliding am- 2.3.1. Fretting scar morphology
plitude regarding the debris layer extension and resulting fretting wear After the test, specimens were cleaned for 30 minutes in an ultra-
profile predictions. sonic ethanol bath to remove most of wear debris trapped in the fretting
scars. This was followed by 3D surface analysis of the worn profiles. The
cylinder shape on the pad was removed and wear volumes on plane (Vp)
2. Experiment and cylinder (Vc) are measured (Fig. 3).
The corresponding plane (αp) and cylinder (αc) energy wear rates
2.1. Materials [10] were estimated assuming a linear evolution of wear extension and
an equal distribution of friction energy toward plane and cylinder
The studied interface consists in a homogeneous Ti-6Al-4V/Ti-6Al- surfaces:
4V cylinder plane interface. Vp Vc
The Ti-6Al-4V titanium alloys consist in a 60% alpha and 40% beta αp = and α c =
∑ Ed/2 ∑ Ed/2 (3)
microstructure. Its mechanical properties are listed in Table 1.
The objective of this research work is to predict the 2D fretting scar
profiles. This implies to extract representative 2D experimental worn

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P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

Fig. 1. Illustration of the fretting test, definition of the loading variables extracted from the gross slip fretting cycle, illustration of plane and cylindrical Ti-6Al-4V
specimens.

J and αp = 1.6 × 10-4 mm3/J.

By summing both cylinder and plane worn profiles we can deduce
the total wear profile hw(x). This procedure is illustrated in Fig. 4 where
both worn and unworn cylinder and plane surfaces are compared to
extract respectively the cylinder hw,c(x) and plane hw,p(x) worn thick-
ness profiles so the total worn profile could be extrapolated:
hw (x) = hw,p (x) + hw,c (x) (4)

2.3.2. Third body layer analysis

One original aspect of this research work is the consideration of the
debris layer for the surface wear modeling. This implies to estimate the
experimental debris layer trapped between the fretting surfaces which
is quite more complex (Fig. 4).
The proposed methodology consists in measuring the void gap be-
Fig. 2. Illustration of an experimental test map used to establish the surface tween plane and cylinder worn profiles when they are just contacting
wear modeling (cylinder/plane, Ti-6Al-4V, R = 80 mm, L = 8 mm, P = on the lateral sides (Fig. 4(b)). A void gap is observed and it is assumed
1066 N/mm, f = 0.11 Hz); ⚫ test conditions used to calibrate the model, test that such a void gap hg(x) corresponds in fact to the third body layer
condition outside the calibration domain. htb(x) embedded between fretting surfaces before the surface cleaning
after the contact opening:
profiles for comparison. As illustrated in Fig. 3, such 2D equivalent hg (x) = htb (x) (5)
worn profiles were extracted from 3D profiles applying an averaging
procedure over the transverse width L. Former investigations suggest that a significant part of the debris
This 2Deq fretting scar was systematically applied for all the test layer is still adhering on the counterparts even after a surface cleaning,
conditions. For the reference test conditions (Nref = 10,000 cycle, and [25,26].
δ0,ref = ± 75 μm) it was found that Vc = 3.54 mm3, Vp = 2.67 mm3 If such adhering transfer structures are operating, the former third
and ΣEd = 16700 J which implies respectively αc = 2.12 × 10-4 mm3/ body interface gap correlation (i.e. Eq. 5) is indeed no more appro-
priated.

Fig. 3. Schematic illustration of the methodology used to quantify fretting wear of plane and cylinder fretting scar. The 3D surface profiles allow the wear volume (V)
estimation, then an averaging procedure over the lateral “L” width is applied to determine a representative 2Deq wear profile.

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P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

(a) Z (mm) (b) Z (mm) hg(x): gap thickness

0.25 0.25 between worn
2Deq profile
0.20 0.20 profiles: assumed
inial cylinder (cylinder) equal to third body
surface 0.15 0.15
2Deq profile layer htb(x)=hg(x)
hw,c(x)
0.10 (cylinder) 0.10
0.05 0.05

0.00 0.00
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
-0.05 X (mm) -0.05 X (mm)
-0.10 hw,p(x) -0.10
2Deq profile
-0.15 2Deq profile
(plane) -0.15
hw(x)=hw,c(x)+hw,p(x) (plane)
-0.20 -0.20
worn material thickness
h th (x)
0.40 γ(x) = 1.1
thickness h w (x)
(mm) 0.35 htb(x) =hg(x) 1.0 Parabolic profile
0.30 hw(x) domain of interest γ x = γ 0 + K γ × x²
(c) (d) 0.9
0.25 γ 0 = 0.95
0.20 0.8 K γ = 0.05

0.15
0.7
0.10
0.6
0.05

0.00 0.5
-3 -1 1 X (mm) 3 -3 -1 1 X (mm) 3

Fig. 4. Illustration of the methodology used to quantify the fretting scar morphology from experiment fretting tests. (ref test condition: Nref = 10,000 cycle, and δ0,ref
= ± 75 μm); (a) identification of the worn profiles; (b) identification of the third body profile; (c) extrapolation of the third body and wear profile; (d) extrapolation
of third body conversion factor γ(x).

Fig. 5. Interface and debris SEM observation; a) cross section expertise after cleaning surface; b) third body particle collected at the contact opening; test conditions:
R = 80 mm, L = 8 mm, P = 1066 N/mm, f = 0.11 Hz, Nref = 10,000 cycle, and δ0,ref = ± 75 μm.

However, such adhering structures inducing typical “W-shape” debris are generated. Cross section observations of plane and cylinder
fretting scar are only observed above P.V friction power dissipation confirm this point (Fig. 5).
[26]. By imposing very slow sliding frequency (i.e. very low P.V va- For the studied condition only discontinuous and very thin debris,
lues), adhering transfers are prevented and only abrasive and oxidized layer less than 15 μm thickness are still observed after the surface

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P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

cleaning (Fig. 5a). This remaining layer can be neglected regarding the bodies abrasion wear process. This friction analysis suggests that the
300 μm gap measured between the worn surfaces which therefore jus- third body (i.e. debris layer) is stabilized above 10,000 fretting cycles.
tifies the third body-interface gap correlation (Eq. 5). Wear debris
collected at the contact opening is composed of thin oxide debris under
10 μm and bigger partially oxide plate-like particles of first bodies 3.1.2. Wear evolution
(Fig. 5b). Such plate-like agglomerated debris structure is assumed due Fig. 7 compares the maximum total wear depth hw,0 and maximum
to shearing solicitations. third body thickness htb,0 measured at the center of the fretting scar. A
Both worn and third body thickness profiles display similar “bell” fast increase is observed until 10,000 cycles followed by a smoother
distribution with a maximum value at the center of the fretting scar rising evolution. However, if the maximum wear depth keeps in-
(Fig. 4(c)). To quantify the proportion of worn material converted to creasing, the third body thickness stabilizes at a 0.35 mm maximum
third body, the conversion factor γ(x) is considered (Fig. 4(d)): thickness.
This stabilization of third body thickness is consistent with the third
γ(x) = htb (x)/hw (x) (6) body theory: it assumes that the global wear rate can be expressed
A systemic analysis of fretting scars confirms a parabolic distribu- through a balance between debris formation flow and the debris ejec-
tion so that: tion flow. When the third body thickness increases, a larger part of the
friction energy is consumed by the third body layer, so the debris for-
γ(x) = γ0 + Kγ × x2 (7) mation rate tends to decrease. Alternatively an increase of the third
body thickness increases the debris ejection flow. As illustrated in
The current investigation suggests that the fretting scar evolution Fig. 6(b), a steady state wear regimes is reached when both formation
can be formalized using a limited number of variables which are the and ejection flows are equal which corresponds to a constant third body
energy wear coefficients of plane (αp) and cylinder (αc) and the γ0 and thickness htb,ss.
Kγ third body conversion parameters. Such parameters evolve with the For the studied interface such a steady state wear regime seems to
test duration and sliding amplitude and it is necessary to formalize such be reached after 10,000 fretting cycles.
evolutions to predict the fretting scar profile whatever the loading Fig. 8 compares the evolution of αc and αp in function of fretting
conditions are. cycle.
For the reference test condition (Nref = 10,000 cycle, and δ0,ref As expected, according to the third body theory both αp and αc
= ± 75 μm), we obtained respectively γ0 = 0.89 and Kγ = display an asymptotic decreaseing. The thicker the third body layer, the
−0.05 mm−2. larger the proportion of friction energy comsumed by the third body
Such a parabolic evolution may be explained by the third body and the lower the energy wear rate. Constant energy wear rates are
theory which suggests that wear debris generated in the center of the however achieved when the third body thickness stabilizes which is
contact are more easily incorporated in the third body layer than debris observed on the contact center after 10,000 fretting cycles (Fig. 7a).
formed on the lateral side where they are more easily ejected from the Besides, before 10,000 fretting cycles both αp and αc display a very
interface. large discrepency fluctuation. This was explained by metal transfers
from cylinder to plane surface which minor the apparent plane wear
3. Influence of fretting loadings on αc, αp and γ0 and Kγ variables rate and exhibit the cylinder one. When the steady state is reached, the
three bodies observation wear process is dominating, transfers are
3.1. Influence of fretting cycles (δ0 = constant = ± 75 µm) progressively eliminated and both αp and αc converge asymptotically
toward a similar α∞ energy wear rate. It is interesting to note that wear
3.1.1. Friction behavior rate is still decreasing beyond 10 000 cycles whereas the maximum
Fig. 6 compares the friction evolution observed for reference test third body thickness measured at the center of contact was already
conditions up to 30,000 fretting cycles. stabilized (Fig. 7). To interpret such a tendency, it must be undelined
Starting from 0.45, μe rises nearly linearly up to 0.65 during the first that, in contrast to the local third body thickness, the wear rate con-
10,000 cycles before stabilizing. Such friction analysis suggest an initial cerns the global interface which also depends on the lateral extension of
transient period of 10,000 cycles where the initial metal and native the fretting scar and third body layer. Focusing on the steady state wear
oxides interactions are progressively replaced by an abrasive three regime (N = Nss = 10,000 fretting cycles), both αp and αc evolutions
were formalized only considering the test duration above 7500 fretting
cycles. Hence the following asymptotic functions are extrapolated:

α c (N) = α∞ + (α1,c − α∞) × exp (K α c,N × N) (8)

α p (N) = α∞ + (α1,p − α∞) × exp (K αp,N × N) (9)

For the studied conditions, we found K αp,N = −1.3 × 10−4, K α c,N

= 1.5 × 10−4, α1,p = 4.2 × 10−4 mm3/J and α1,c = 7 × 10−4 mm3/J
and α∞ = 9 × 10−5 mm3/J.
Where α∞ expresses the steady state wear rate for an infinite test
duration, α1, p and α1,c , the initial wear rate respectively for plane and
cylinder whereas K αp,N and K α c,N , express the decreasing evolution of
the wear rates with stabilization of the interface.
K αp,N and K α c,N exponents are quite similar but a significant fluc-
tuation is observed for α1 parameters. This can be explained by initial
transfer phenomena which are mainly occurring from cylinder to plane
surfaces. Deeper investigations are required to better interpret this
Fig. 6. Evolution of the Energetic coefficient of friction as a function of fretting tendency. However, it can be concluded that using such basic asymp-
cycles for the reference conditions with R = 80 mm, P = 1066 N/mm, N = totic formulations give a rather good description when the steady state
10,000 to 30,000 cycles, and δ0,ref = ± 75 μm. wear regime is achieved.

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Fig. 7. (a) Evolution of the maximum wear thickness and maximum third body thickness in function of fretting cycle; (b) third body theory applied to thickness
consideration; R = 80 mm, P = 1066 N/mm, and δ0,ref = ± 75 μm.

3.1.3. Third body evolution

For each test duration both total wear and third body profiles are
extrated and the corresponding γ(x) conversion profiles are defined
(Fig. 9(a)).
Like for the reference test condition, all these experimental dis-
tributions can be approximated using parabolic functions. The corre-
sponding γ0 and kγ values obtained for the different test durations are
plotted in Fig. 9(b) and compiled in Table 2.
The analysis suggests that Kγ factor remains nearly constant around
Kγ,N = −0.05 excluding the test done at 20,000 fretting cycles which
displays a larger contact extension than expected. The γ0 variable fol-
lows an asymptotic decrease which can be approximated using a power
law function:

γ0 (N) = K γ0,N × N n γ0,N (10)

For the studied conditions we found, Kγ,N = 1.4022 and nγ,N =

−0.049.
Fig. 8. Evolution of energy wear rate of plane and cylinder versus the number Unike the energy wear factors, the γ(x) third body conversion factor
of fretting cycle; R = 80 mm, P = 1066 N/mm, N = 2500 to 30,000 cycles, and displays a continuous evolution from the shortest to the longest test
δ0,ref = ± 75 μm. durations. Indeed, γ(x) parameter considers the total wear profile
(plane+cylinder) which indirectly eliminates the discrepancy induced
by initial transfer phenomena.
To conclude on the effect of test durations, the longer the test
duration, the lower the global conversion factor. In other words, the

Fig. 9. Evolution of the γ(x) conversion profile versus fretting cycle; (a) γ(x) profiles are extracted from the experimental wear and third body profiles measured at
the end of the N cycles fretting test; (b) plotting γ0 and Kγ versus fretting cycles; R = 80 mm, P = 1066 N/mm, N = 10,000 to 30,000 cycles, and δ0,ref = ± 75 μm.

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P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

Table 2 Table 3
γ0 and Kγ variables extracted from different test duration; R = 80 mm, P = γ0 and Kγ variables extracted from different sliding amplitudes; R = 80 mm, P
1066 N/mm, N = 10,000 to 30,000 cycles, and δ0,ref = ± 75 μm. = 1066 N/mm, N = 10,000 cycles, and δ0 = ± 62 μm to ± 125 μm.
N (cycles) γ0 Kγ (mm−2) δ0 ( ± μm)
γ0 Kγ (mm−2)

2,500 0.95 −0.05 62 0.92 − 0.055

5,000 0.95 −0.048 75 0.89 − 0.05
7,500 0.9 −0.051 100 0.86 − 0.015
10,000 0.89 −0.05 125 0.745 − 0.01
20,000 0.86 −0.029
30,000 0.855 −0.049
γ0 (δ0) = K γ0 × δ0 + γ0 (0) (12)

where Kγ (0) = Kγ (δ0 = 0µ m) = 0.1045, K Kγ = −0.0008, γ0 (0) =

γ0 (δ0 = 0µm) = 1.092 and K γ0 = −0.0026. Note that these values were
identified for N = Nss = 10,000 cycles.
To conclude concerning the effect of sliding amplitude, the larger
the sliding amplitude, the lower and flatter the global conversion factor.
In other words, an increase of sliding amplitude makes the debris
ejection easier, increases the contact size and consequently reduces and
flattens the conversion factor distribution.

4. FEM surface wear modeling

4.1. Finite element model

The main aspects of the applied FEM wear modeling were pre-
viously exposed in [5] and [21]. Using an Abaqus-Python-Matlab code,
bilateral surface wear simulations were obtained by considering a
Fig. 10. Evolution of the energetic friction coefficient in function of sliding symmetrical “wear-box” part on each side of the cylinder/plane coun-
amplitude and fretting cycle; R = 80 mm, P = 1066 N/mm, N = 10,000 cycles, terparts. Two-dimensional linear plane strain elements with four nodes
and δ0 = ± 62 μm and ± 125 μm. (CPE4R) and fine elements: 40 μm width and 40 μm depth are con-
sidered [5]. A coarser meshing was applied to the outer domains using
thickness of third body converges to a constant maximum steady state triangular elements (CPE3). Contact integrations were monitored using
admissible thickness (Fig. 7b). When this thickness is reaching this a penalty contact algorithm which is faster than the Lagrange multiplier
value, the debris ejection flow increases and the conversion factor de- method and provides a very good description of the studied gross slip
creases. conditions. The plane surface was defined as the master surface and the
cylinder surface as slave. The penetration depth between slave and
master nodes was allowed up to a 1 μm. Interfacial shear was computed
3.2. Influence of sliding amplitude
using a Coulomb law with friction coefficient equal to the experimental
value μFEM = μe,mean = 0.65. All the studied test conditions induce
3.2.1. Friction behavior and wear analysis
macroscopic elastic stressing so the given simulation was restricted to
A similar procedure is applied keeping the reference test conditions
elastic conditions. Boundary conditions were ensured on the bottom
as a key loading point (N = 10,000 fretting cycles) but varying the
and lateral sides of the plane part through the reference point RF2. The
sliding amplitude (Fig. 2).
contact loading was applied to the cylinder counterpart through the
Fig. 10 compares the friction evolution obtained for the two ex-
reference point RF1. Firstly, rotations were blocked and an indentation
tremum sliding amplitudes, δ0 = ± 62 μm and δ0 = ± 125 μm. The
step was adjusted to reach the required normal load. Then, periodic
two curves are superimposed, which suggests that the sliding amplitude
horizontal displacement amplitude was imposed to achieve the re-
do not influence the friction response. Initial and steady state values at
quired δ0 sliding amplitude (Fig. 12(a)).
10,000 cycles are similar whatever the sliding amplitude applied, re-
spectively at 0.45 and 0.65.
Independence of the energetic friction coefficient in function of the 4.2. Numerical implementation of the energy wear modeling
sliding amplitude is so assumed. Like steady state coefficient of friction,
the energy wear coefficients appear independent of the sliding ampli- Wear volumes of plane and cylinder are expressed as a linear
tude and tend to constant values. For the studied condition, we found:
function of the accumulated friction energy in the interface.
α c,δ0 = 2.05 × 10−4 mm3/J and α p,δ0 = 1.7 × 10−4 mm3/J.
∑ Ed
V = α×
3.2.2. Third body analysis 2 (13)
Third body and γ(x) profiles are extracted using the proposed
methodology. Again a parabolic approximation can be considered to where α is the energy wear coefficient related to plane and cylinder
express the γ(x) distribution (Table 3). counterparts respectively.
The best fitted γ0 and Kγ coefficients are identified, compiled and The FEA surface wear modeling consists in transposing the global
plotted versus the sliding amplitude in Fig. 11. wear volume approach to a local wear depth simulation [27]. After
Both γ0 and Kγ show a linear decreasing which can be approximated each ith fretting cycle the φ(i) (x) friction energy density profile is com-
by the following expressions: puted. The related wear depth increment ∆h (i) (x) was established with a
β “numerical jump” which allows to simulate β experimental cycles
Kγ (δ0) = K Kγ × δ0 + Kγ (0) (11) during a single numerical cycle [18].

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Fig. 11. Influence of sliding amplitude on γ(x) distribution: (a) Experimental γ(x) distribution; (b) plotting of identified γ0 and Kγ coefficients extracted from the
parabolic approximation; R = 80 mm, P = 1066 N/mm, N = 10,000 cycles, and δ0 = ± 62 μm to ± 125 μm.

φ(i) (x) later was however updated to include a simultaneous smoothing of

∆h (i) (x) = β × α×
2 (14) pressure and worn surface profiles. Then, surface and subsurface mul-
tilayer node translations were activated to reset contact geometry for
This numerical jump was set to β = 50. This rather small value successive iteration (Fig. 11). New contact pressure and shear profiles
provided stable wear profile evolution without requiring excessive si- and also new stress fields are therefore identified [28]. This procedure
mulation time. The wear depth increment is then introduced in the was repeated until the complete simulation of the experimental test was
mesh structure of cylinder and plane surfaces. To limit numerical dis- reached.
continuities at the lateral side of the contact, a Gaussian smoothing Such surface wear modeling is commonly applied to simulate wear
procedure equivalent to the Basseville [18] proposal was adopted. This profile or to quantify the competition between wear and cracking

Fig. 12. (a) Illustration of the FE cylinder/plane contact model and applied loading conditions [27]; (b) Illustration of the numerical strategy used to simulate surface
wear extension (cylinder/plane contact) [18] and plane surface wear profiles computed for different test durations (without third body consideration).

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Fig. 13. Comparison between experimental and simulated

wear profiles of the reference test condition (R = 80 mm,
P = 1066 N/mm, δ0 = ± 75 μm, N = 10,000 cycles, αc =
2.12 × 10-4 mm3/J, αp = 1.6 × 10-4 mm3/J, μe = 0.65);
(a) without third body consideration (γ = 0); (b) with
third body consideration (γ(x) = γ0+Kγ.x² with γ0 = 0.89
and Kγ = −0.05).

processes for fretting fatigue application. Bell shape fretting scar dis- 4.3. FEM Third body consideration
playing significant wear lateral and depth extension are usually ob-
served. This approach provides relevant predictions when the debris 4.3.1. FE Third body modeling
layer is limited from the interface like observe in liquid solutions or The main idea of third body fretting wear modeling consists in in-
small contact. cluding between the fretted surfaces a debris layer “FEM part” which
However, when third body layer is maintained in the interface like evolves with the surface wear extensions. As illustrated in Fig. 14 for a
for the current dry contact, the model tends to underestimate the simplified unilateral flat wear element, a γ proportion of the wear depth
maximum wear depth and overestimates the lateral extension. generated on the upper body is converted to third body and added to
Investigations of Ti-6Al-4V fretting interface show that the maximum the former debris layer for the successive simulation.
wear depth can be underestimated by a factor 3 (Fig. 13(a)). Hence Considering the friction energy density dissipated φi between the
from dry Ti-6Al-4V interface, the presence of the third body layer needs third body surface and the upper body during the ith fretting cycle and
to be considered in the model. considering an equipartition of the friction energy density φi, dissipated
in the interface, the increment of wear generated on the upper body is

Fig. 14. Illustration of the conversion process of incremented worn volume to third body layer during the ith fretting cycle (unilateral wear description).

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P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

Fig. 15. Illustration of the algorithm used to simulate the surface wear of plane and cylinder counterpart taking into account a dynamical third body layer.

expressed by: where αc is the energy wear rate of cylinder.

φ(i) Then the third body increment induced by the cylinder wear is es-
∆hw,i = α× tablished:
2 (15)
∆htb,c(i) (x) = γ(i) (x)× ∆hw,c(i) (x) (19)
with α: the energy wear factor of the upper body
Then, it is assumed that the proportion γ of worn thickness is con- th
where γ(i) (x) is the conversion profile at the i fretting cycle approxi-
verted to third body whereas the complement proportion (1−γ) is mated by a simple parabolic function (Eq. 7).
ejected from the interface. Hence, a worn layer Δhw,i is removed from A similar strategy is applied for the bottom plane/third body in-
the upper specimen. The ith increment of third body thickness is terface. The ith third body is stuck to the worn cylinder surface and the
therefore expressed by: friction energy profile φtb/p(i) (x) between third body and bottom plane is
computed. The increment of worn thickness removed from the plane is
∆htb,i = γ×∆hw,i (16)
given by:
The third body thickness at the successive i+1 numerical fretting is φtb/p(i) (x)
finally given by: ∆hw,p(i) (x) = β × α p ×
(20)
2
htb,i + 1 = htb,i + ∆htb,i = htb,i + γ×∆hw,i (17) A similar conversion profile is assumed on both friction interfaces
which allows expressing the increment of third body thickness induced
This approach is extended to successive stages up to the end of the
by the plane surface wear:
test simulation. The application of such concept to a real interface is
more complex. Pressure, friction energy density and therefore wear ∆htb,p(i) (x) = γ(i) (x)× ∆hw,p(i) (x) (21)
profiles are not flat but evolve along the axial fretted interfaces.
Besides, a bilateral wear needs to be considered. Then plane and cylinder surface profiles are updated:
Fig. 15 illustrates the extension of this FEM third body surface wear Zc(i + 1) (x) = Zc(i) (x)+∆hw,c(i) (x) (22)
modeling for more generate case of 2D cylinder/plane interface.
To adress the bilateral wear, each numerical cycle requires two FEM Z p(i + 1) (x) = Z p(i) (x)+∆hw,p(i) (x) (23)
cylinder plane simulations. Considering the ith numerical fretting cycle
the following sequence is applied: As well as the third body profile (Fig. 15):
The third body layer is stuck to the bottom plane and the friction htb(i + 1) (x) = htb(i) (x)+∆htb,p(i) (x)+∆htb,c(i) (x) (24)
energy profile dissipated φtb/c(i) (x) between third body and upper cy-
linder is computed. Succesive iterations are then applied up to the end of simulation.
The increment of worn thickness removed from the cylinder surface It is interesting to note that the model requires a limited number of
is defined by: variables to quantify surface wear (αp, αc) and third body layer (Kγ, γ0).
In a first approximation the elastic properties of the third body layer
φtb/c(i) (x) were assumed to be similar to the Ti-6Al-4V bulk material (ie Etb = ETi-
∆hw,c(i) (x) = β × α c ×
2 (18) 6Al-4V). Despite such abusive approximations the simulation of the wear

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profiles at the reference test conditions show a very good correlation test condition was simulated (R = 80 mm, P = 1066 N/mm; δ0
with experiments (Fig. 13(b)). = ± 75 μm, Nss = 10,000 cycles applying αc = 2.12 × 10−4 mm3/J,
The wear depth which was underestimated by 300% without the αp = 1.6 × 10-4 mm3/J, γ0 = 0.89 and Kγ = −0.05) varying the
third body consideration was predicted with an error less than 10% elastic modulus of the third body from Etb = 0.5 × ETi-6Al-4V to 2 × ETi-
using this third body modeling. 6Al-4V. Fig. 16(a) compares the cumulated wear profile obtained for the
two extremum elastic approximations.
4.3.2. Influence of the third body elastic properties Very similar profiles are obtained wich suggests a very small in-
As illustrated previoulsy, all these variables required by the model fluence of the elastic properties of third body. Whereas the elastic
can easily be extrated from an adequate expertise of a restricted number modulus of the third body varies by more than 400%, the fluctuation of
of fretting experiments. the predicted maximum wear depth was lower than 2.8% (Fig. 16(b)).
A last aspect however concerns the effect of the mechanical prop- Similar conclusions are derived regarding the contact radius which
erties of the third body and its potential influence on the wear profile remains unchanged.
simulations. This, confirms the interest of the model which can be applied even if
This is a key issue according that it is extremly difficult to establish the third body elastic modulus remain unknown. Futher investigations
the mechanical properties of the third body layer [29]. Hence, if its are still required to investigate other aspect like plasticity or porous
influence is significant, this aspect must be considered for the simula- description of the debris layer.
tions. However at this stage of our analysis, we can conclude that the
Yue et al. [30] applying a constant third body thickness showed that elastic properties of the third body layer do not influence the worn
a fluctuation of the third body elastic properties do not influence sig- profile simulations.
nificantly the pressure profile and therefore they conclude that the Therefore to simplify the following simulations will be performed
elatic properties of third body plays a minor influence on the surface assuming that the third body elastic modulus is similar to the Ti-6Al-4V
wear modeling. However, such analysis do not consider the dynamical bulk material, Etb = ETi-6Al-4V.
evolution of the third body layer. To clarify this aspect, the reference

Fig. 16. Analysis of the reference test condition (R = 80 mm, P = 1066 N/mm, δ0 = ± 75 μm, N = 10,000 cycles, αc = 2.12 × 10−4 mm3/J, αp = 1.6 × 10−4
mm3/J, μe = 0.65, γ(x) = γ0+Kγ.x² with γ0 = 0.89 and Kγ = −0.05); (a) comparison of acumulated worn profiles for different third body elastic modulus Etb; (b)
evolution of computed wear depth versus Etb; (c) evolution of worn lateral extention versus Etb.

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P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

Fig. 17. Evolution of pressure profiles, third body debris layer and final interface morphology (N = 10,000 cycles) for the reference test condition; (a) without
consideration of third body; (b) with consideration of third body.

4.3.3. Comparison of FEM worn profile with and without third body increase compared to γ = 0 condition whereas a fast increase of
Fig. 17 clearly shows the effect of debris layer on the surface profile. maximum wear depth is observed.
Whitout third body, the pressure profile fastly evolves from an el-
liptical to a flat distribution characterized by a large lateral extention.
This very fast reduction of the maximum reduces the maximum friction 4.3.4. Implementation of test duration and sliding amplitude effects in the
density in the center part of the contact and consequently slows down model
the wear depth extension. In contrast the presence of a debris layer The following analysis consists now in implementing the test
tends to localize the pressure in the center of the contact. duration and sliding amplitude effects in the model. The study focuses
Acting as a pressure concentrator, the presence of the debris layer on the steady state wear regime which corresponds to test duration
maintains a very high pressure level in the contact center equivalent to longer than 10,000 fretting cycles and sliding amplitude larger
the initial hertzian one. This exibits the friction dissipation in the than ± 75 μm.
contact center favoring the wear depth increase and reducing the lateral As illustrated during the experimental investigation, the interface
contact extension. parameter (i.e. αc, αp, γ0 and Kγ) varies with the loading conditions.
This tendency is confirmed in Fig. 18 where maximum contact The global formulation of surface wear modeling implies to explicit
pressure, worn contact radius and maximum wear depth are compared. αc, αp, γ0 and Kγ variables as a function of N and δ0 using continuous
The simulation without third body displays an asymptotic reduction polynomial functions. Combining the different formulations developed
of the maximum pressure which decreased by a factor 3 after 10,000 in Section 3, the following expressions are derived:
cycles. The contact radius is multiplied by 3 whereas the depth displays The energy wear coefficients seem not influenced by the sliding
a smooth asymptotic rising. amplitude which implies:
In contrast, the simulation with third body displays a quasi-constant
α c (N,δ0) = α c (N) = α c (N) = α∞ + (α1,c − α∞)× exp (K α c,N × N) (25)
high maximum pressure value. The contact radius shows a small

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P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

Fig. 18. Comparison between wear modeling with (γ(x)) and without (γ = 0) third body consideration (reference condition, R = 80 mm, P = 1066 N/mm, δ0
= ± 75 μm, N = 10,000 cycles, αc = 2.12 × 10−4 mm3/J, αp = 1.6 × 10−4 mm3/J, μe = 0.65, γ(x) = γ0+Kγ.x² with γ0 = 0.89 and Kγ = −0.05) (a) maximum
contact pressure; (b) contact radius; (c) maximum wear depth and third body thickness.

Fig. 19. Comparison between experimental and third body simulated wear profiles using the global formulations detailed by Eqs. (25–30).

α p (N, δ0) = α p (N) = α p (N) = α∞ + (α1,p − α∞)× exp (K αp,N × N) (26) Kγ (N,δ0) = Kγ (δ0) = K Kγ × δ0 + Kγ (0) (27)

Regarding the third body conversion factor, we show that the Kγ In contrast, the γ0 offset depends on the test duration (Eq. 10) and
coefficient is constant versus test duration but proportional to the sliding amplitude (Eq. 12). The fretting cycle influence can be nor-
sliding amplitude (Eq. 11) which infers: malized by γ0,Nref reference value which leads to:

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P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

γ0 (N) = K γ0,N × N n γ0,N ⎫ γ0 (N) n γ0,N underestimated by a factor 3. In contrast, the wear modeling taking into
= ⎛ N ⎞
n
γ0,Nref = K γ0,N × Nref γ0,N ⎬ γ0,Nref
⎜ ⎟
account the third body layer (γ(x)), provides very good predictions
⎭ ⎝ Nref ⎠ (28)
although some discrepancies can be observed (Fig. 22(b)).
Assuming that Nref = 10 000 cycles and δ0,ref = ± 75 μm, we de- Such dispersion may be explained by the fact that the given wear
duce γ0 (Nref ) = γ0,ref = 0.89. simulations only considers the steady state wear regime. During the
To account for the sliding amplitude influence, γ0 (Nref ) is related transient period, transfers from one counter-face to the other scatter the
to Eq. (12) which implies: experimental wear profiles. Further analyses suggest that such transfer
artefacts are eliminated if rather than comparing the individual plane
γ0 (δ0 , Nref ) = K γ0 × δ0 + γ0 (0) (29) and cylinder profiles, the analysis considers the total wear profile (i.e.
And finally, we deduce: the sum of both plane and cylinder wear profiles). Fig. 23 compares the
experimental and simulated total 2Deq wear profiles.
N n γ0,N Better correlations and less dispersive results are systematically
γ0 (δ0 , N) = (K γ0 × δ0 + γ0 (0) ) × ⎛ ⎞
⎝ Nref ⎠ (30) observed. In fact the sum of plane and cylinder wear profiles eliminates
the dispersion induced by the initial transfer phenomena, and so less
These global formulations of α c (N,δ0) , α p (N,δ0) , γ0 (N,δ0) and
Kγ (N,δ0) were then implemented in the model to simulate the wear dispersive results are obtained. This result is confirmed in Fig. 24 where
profiles. both experimental and simulated maximum wear depth of total 2Deq
Fig. 19 compares the experimental and simulated wear profiles of wear profiles are compared. An excellent correlation is observed for
plane and cylinder counter faces for δ0 = ± 75 μm and varying the test studied conditions R = 80 mm and P = 1066 N/mm. The relative
duration, N = 10,000 to 30,000 fretting cycles. dispersion falls below 15% compared to the 45% when comparing re-
A very good correlation is again observed which confirms the sta- spectively plane and cylinder wear profiles (Fig. 22 (b)).
bility of the model. Some dispersion is however observed. At 20,000 From this analysis, we can nevertheless conclude that the transient
cycles, the model underestimates the wear depth of the plane and period and the initial activation of metal transfers alter the wear profile
overestimates the cylinder whereas the opposite is noticed at 30,000 prediction. This suggests that alternative strategies taking into account
cycles. metal transfer phenomena including statistical description need to be
Fig. 20 compares both experimental and simulated wear profiles of developed to better predict maximum wear depth.
plane and cylinder counter-faces. The test duration was set to N = Besides, other aspect like normal force or cylinder radius are not
10,000 cycles and different sliding amplitudes larger than δ0,ref included in the model. However, there is a critical interest to evaluate
= ± 75 μm were applied. how such incomplete formalism may simulate very distinct fretting test
Like for the fretting cycle analysis a rather good correlation is ob- conditions.
served. However some dispersion regarding the radial extension can be To illustrate this aspect, two test conditions were investigated im-
observed. An increase of the sliding amplitude tends to slightly over- posing constant frequency f = 0.11 Hz, sliding amplitude δ0
estimate the predicted contact radius and slightly underestimate the = ± 75 μm, test duration N = 10 000 cycles and maximum Hertzian
maximum wear depth. pressure phertz = 525 MPa but varying the cylinder radius and normal
All the parameters of the model (i.e. αc, αp, γ0 and Kγ) were in fact load at respectively test X (R = 40 mm, P = 533 N/mm), and text Y (R
extracted from the studied test condition, which explains the good = 20 mm, P = 266 N/mm).
correlation between experiments and simulations. Experimental and simulated total wear profiles are compared in
To better assess the stability of the model, tests given outside the Fig. 25.
calibration domain (Fig. 21 (test A, B, C)) have been investigated. As expected, a larger discrepancy is observed compared to the
Fig. 20 compares the experimental and simulated wear profiles. former analysis where R and P were kept constant, R = 80 mm and P =
Again a rather good correlation is observed which confirms the 1066 N/mm (Figs. 24 and 25). The smaller the cylinder radius, the
stability of the approach and suggests that the hypothesis of a linear larger the over estimation of the maximum wear depth. This indirectly
correlation of N and δ0 influences used to express αc, αp, γ0 and Kγ (Eqs. suggests that a reduction of the cylinder radius by favoring the debris
25-30) is suitable. ejection will reduce the conversion factor γ(x) and as a consequence the
The stability of the model is demonstrated by comparing experi- third body thickness. The friction dissipation in the center of the fret-
mental and predicted maximum wear depths of planes and cylinders in ting scar is decreased as well as the maximum wear depth extension.
Fig. 22. Hence, the reduction of the cylinder radius tends to reduce the effect of
A very poor correlation is observed if the third body layer is not the third body. This suggests that for very small contacts, simulations
considered (Fig. 22(a)). The predicted maximum wear depths are without third body could be considered. Fig. 26 illustrates the stability

Fig. 20. Comparison between experimental and simulated wear profiles using global formulations developed Eqs. (25–30).

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Fig. 21. Comparison of experimental and simulated wear profiles for test conditions outside the N-δ0 calibration axis (Fig. 2).

of the model by comparing the relative error between experimental and

simulated maximum wear depths
N 2
1 hmax,exp − hmax,FEA ⎞
E hmax % =
N
∑ ⎜⎛ hmax,exp
⎟ × 100
i=1 ⎝ ⎠ (31)

As expected, as the third body wear modeling does not consider the
effect of cylinder radius and normal force, the relative error for X and Y
test conditions is increased. However, as previously underlined, the
smaller the contact radius, the lower the error given by the model
without third body consideration. Current researches are undertaken to
update the actual power law function model to describe both cylinder
radius and normal force effects. However, one limitation of the model is
the necessity to consider various fitting parameters to describe the third
body evolution. Hence, for each loading variable, up to 4 parameters
need to be identified. The given formulation can be simplified ne- Fig. 23. Comparison between experimental and simulated γ(x) total wear
glecting some aspects like for instance the effect of the fretting cycle profiles (R = 80 mm, P = 1066 N/mm, f = 0.11 Hz); ⚫ test conditions used to
regarding Kγ coefficient. However, it is clear that alternative strategies calibrate the model, test conditions outside the calibration domain.
considering a more physical description of third body and involving a
limited number of variables must be preferred in the future. In a sense, fretting interface.
this first approach appears as a first step that hopefully will stimulate This modeling was directly calibrated and compared using equiva-
more fundamental research in this area. lent fretting wear experiments, the following conclusion can be drawn
from this study:
5. Conclusion

A new FE friction energy wear modeling with consideration of the

• From 3D wear profiles, 2D wear profiles can be extracted and using
an adequate superposition, it appears possible to estimate the third
third body layer was introduced to simulate Ti-6Al-4V cylinder/plane

Fig. 22. Comparison between experimental and predicted maximum wear depths; (a) wear simulations without third body (γ = 0); (b) wear simulation with third
body, γ(x) (Eqs. 25–30); R = 80 mm, P = 1066 N/mm, f = 0.11 Hz.

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P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

Fig. 24. Comparison between experimental and simulated maximum wear Fig. 26. Comparison of the relative error between experiments and simulation
depths of total 2Deq wear profiles (simulation with third body layer γ(x)); R = versus the studied test condition (f = 0.11 Hz, phertz = 525 MPa).
80 mm, P = 1066 N/mm: ⚫ test conditions used to calibrate the model, test
condition outside the calibration domain. Various contact geometry: × test depth prediction. The effect of the debris layer on surface wear
condition with cylinder radius R = 20 mm and P = 266 N/mm, + test con- modeling is rather geometrical than rheological.
dition with cylinder radius R = 40 mm and P = 533 N/mm.
• The third body debris layer acts as a contact pressure concentrator
which, by accentuating the friction energy dissipation in the central
body layer profile trapped within the interface. part of the contact, favors the wear depth extension rather than the
• The third body layer profile can be extrapolated from the total lateral wear extension.
surface wear profile by applying a so called γ(x) third body con- • The general formulation of the conversion factor and energy wear
version factor which is a better approximation using a parabolic rates is implemented to simulate various N and δ0 fretting wear
function (γ(x): local proportion of wear thickness transferred to experiments. Rather good correlations are observed which confirms
third body layer). the interest of the proposal
• The experimental investigations of test duration and sliding ampli- • Lower scattering is even observed if the wear analysis considers the
tude show that an increase of sliding amplitude tends to decrease total wear profile which, by summing both plane and cylinder wears
the γ(x) parabola maximum and to widen its lateral distribution, profiles, eliminates the transfer perturbations.
whereas the increase of fretting cycles tends to only decrease the • The analysis of smaller cylinder radius (i.e. smaller contact size)
parabola maximum. Regarding the energy wear coefficients, we shows a higher discrepancy between experiments and simulations,
conclude that below 10,000 fretting cycles the wear response is as the actual model does not consider contact size effects. However,
unstable (transient wear response). Focusing above 10,000 fretting it suggests that the smaller the contact size, the smaller the third
cycle, the αc and αp energy wear coefficients asymptotically con- body effect and consequently for very small contacts, simulations
verge to a constant steady state wear rate response. without third body could be considered.
• The FE analysis of a reference test condition confirms that the model
that does not consider the third body layer (γ = 0) critically un- Further developments are still required to fully simulate the com-
derestimates the maximum wear depth (factor 3). Alternatively, the plex fretting wear processes. For instance, porous and/or elasto-plastic
FEA taking into account a dynamical constitution of the third body material properties could be considered to illustrate the rheological
layer using the γ(x) parameter provides reliable predictions. response of the third body. The model will also be improved taking into
• A parametric study demonstrates that the elastic properties of the account the adhesive metal transfers initially activated during the
third body layer do not influence the wear profile prediction. transient wear process.
Imposing a fluctuation of 400% of the elastic properties of debris Finally, the effect of debris layer regarding fretting cycle shape can
layer induces a fluctuation of less than 3% of the maximum wear also be examined to better illustrate how the presence of third body can

Fig. 25. Experimental and simulated wear profiles for various cylinder radius conditions: f = 0.11 Hz, δ0 = ± 75 μm, N = 10,000, phertz = 525 MPa; a) X condition
(R = 20 mm, P = 266 N/mm); b) Y condition (R = 40 mm, P = 533 N/mm).

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P. Arnaud, S. Fouvry Wear 412–413 (2018) 92–108

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