REVIEW
published: 03 July 2020
doi: 10.3389/fmech.2020.00051
Frontiers in Mechanical Engineering | www.frontiersin.org 1July 2020 | Volume 6 | Article 51
Edited by:
Marco Paggi,
IMT School for Advanced Studies
Lucca, Italy
Reviewed by:
Stanislaw Stupkiewicz,
Polish Academy of Sciences, Poland
Yoshitaka Nakanishi,
Kumamoto University, Japan
*Correspondence:
Ivan Argatov
Specialty section:
This article was submitted to
Tribology,
a section of the journal
Frontiers in Mechanical Engineering
Received: 13 March 2020
Accepted: 05 June 2020
Published: 03 July 2020
Citation:
Argatov I and Chai YS (2020) Contact
Geometry Adaptation in Fretting Wear:
A Constructive Review.
Front. Mech. Eng. 6:51.
doi: 10.3389/fmech.2020.00051
Contact Geometry Adaptation in
Fretting Wear: A Constructive Review
Ivan Argatov1*and Young Suck Chai2
1Institut für Mechanik, Technische Universität Berlin, Berlin, Germany, 2School of Mechanical Engineering, Yeungnam
University, Gyeongsan, South Korea
Fretting is a special type of wear, which appears at the contact interface between
two solids subjected to constant normal load and periodic tangential forces. Although
most studies on fretting have been executed experimentally, some approaches for
simulating fretting wear have also been introduced during the last decades. In
particular, fretting wear analysis is concerned with the evolution of the surface
profiles of the contacting bodies due to wear, and its modeling was executed using
numerical, finite-element, semi-analytical, and analytical methods, including the method
of dimensionality reduction. In the present review we discuss recent analytical results on
fretting wear contact geometry adaptation.
Keywords: fretting wear, elastic contact, limiting contact profile, steady state, wearing-in period,
wear accumulation
1. INTRODUCTION
Contact of two machine parts established under external compressive loads and subjected to
oscillating shear forces is often accompanied by friction and wear (Ciavarella and Demelio, 2001).
In the case of oscillatory tangential motion of small amplitude, the particular type of wear which
occurs at the sliding interface is called fretting (Vingsbo and Söderberg, 1988). Due to surface
wear and damage, the shapes of the contacting bodies change and this process is called the contact
geometry adaptation. Fretting phenomena, including fretting wear and accompanied variation of
contact geometry, are encountered in many industrial applications, where contact parts experience
oscillating small relative movements. For instance, the fretting wear characteristics of Inconel 690
U-tubes strongly influences the structural integrity of steam generators in nuclear power plants
(Chai et al., 2005; Lee et al., 2009).
The problem of contact geometry adaptation in fretting wear can be formulated as a
spatial-temporal contact problem with a variable contact geometry. In particular, fretting wear
analysis is concerned with the evaluation of the surface profiles of the contacting bodies due to wear.
Mathematical methods of solving elastic contact problems with wear were reviewed in a number of
review papers by Aleksandrov and Kovalenko (1984) and Kovalenko (2001). Numerical simulation
aspects of wear modeling were recently discussed in detail by Huajie and Hongzhao (2018), using
the integration size as a principal characteristic (which is absent in analytical models). Recent
studies on fretting wear damage in coated systems were reviewed by Ma et al. (2019). Fretting wear
mechanisms and modeling were considered by Yue and Wahab (2019) and Meng et al. (2020) with
a particular focus on models of debris and third-body fretting wear.
In the present review paper, we discuss different approaches to modeling fretting wear with
the emphasis on analytical and semi-analytical methods, including the method of dimensionality
reduction. It is to note here that though the present review is somewhat biased with the focus on
the recent work of the authors, an up-to-date account of relevant studies is given as well. The aim
of this review paper is to explore the theoretical ideas, analytical models and results relating to the
Argatov and Chai Contact Geometry Adaptation in Fretting Wear
concept of contact geometry adaptation in fretting wear to
facilitate their further development (for instance, by extending
the solutions of two-dimensional problems to the three-
dimensional case). For this reason, the term “constructive
review” is emphasized.
One of the things we shall concentrate on is the so-called
wearing-in period in gross-slip fretting wear, when the initial
contact state progresses into a kind of steady-state, in which the
applied contact load is redistributed along the contact area in
accordance with the wear equation. Under partial-slip fretting
wear conditions, no such steady state exists, and, theoretically
speaking, the initial contact state is expected to evolve into
a kind of steady state (called limiting state), characterized by
transferring the contact load primarily through the stick zone,
where no wear occurs. In both cases our particular interest is
focused on estimating the time needed to achieve the steady state
or the limiting state.
1.1. Archard Wear Equation and Its
Generalizations
The local wear is usually characterized by the linear wear rate,
˙w, where a dot denotes the derivative with respect to time.
According to Archard’s equation of wear (Archard, 1953), which
is adopted in the majority of studies on fretting wear published to
date, we have
˙w=kwpv, (1)
where pis the contact pressure, vis the absolute value of the
relative sliding velocity, and kwis the coefficient of wear.
A mathematically straightforward generalization of
Equation (1) leads to the Archard–Kragelsky wear equation
˙w=Kwpαvβ, (2)
which was considered in a number of studies (Kragelsky, 1965;
Meng and Ludema, 1995; Kragelsky et al., 2013). Recently,
Argatov and Chai (2019a) put the Archard–Kragelsky equation
into an ANN (artificial neural networks) framework, which
allows to account for the dependence of the wear coefficient Kw
on material parameters and operational conditions.
1.2. Reciprocal Sliding Wear
Let 1Tdenote the period of tangential oscillations. Then,
according to Equation (1), the linear wear resulting from one
cycle will be
1w(x,t)=kwZt+1T
t
p(x,¯
t)v(x,¯
t)d¯
t. (3)
One simplification of the wear relation (3) is that, under certain
conditions (e.g., under constant normal load and relatively small
wear coefficient), the contact pressure may be assumed to not
change appreciably during one cycle. In this way, Equation (3)
simplifies as
1w(x,t)=kwp(x,t)Zt+1T
t
v(x,¯
t)d¯
t. (4)
Another simplification is admissible in the gross slip regime,
when the relative sliding velocity is supposed to become
independent of the position of the point xon the contact
interface, and thus, Equation (4) simplifies further 1w(x,t)=
kwp(x,t)¯v1T, where ¯vis the average absolute value of sliding
velocity, that is
¯v=1
1TZt+1T
t
v(¯
t)d¯
t. (5)
Let now 1xdenote the stroke of tangential oscillations. Then,
Equation (5) can be rewritten as
¯v=21x
1T. (6)
In reciprocating sliding, it is convenient to operate both with the
number of cycles, N, and the effective time variable, t, such that
N= ⌊t/1T⌋, where ⌊x⌋denotes the floor function (that is the
largest integer less than or equal to x).
1.3. Energy Wear Equation
By introducing the sliding distance, s, such that ds =v dt,
Equation (1) can be represented in the differential form as dw =
kwp ds. Moreover, let µdenote the coefficient of friction. Then,
introducing the frictional shear stress is q=µp, the wear
increment can be further rewritten as
dw =αVq ds, (7)
where αVis the energy wear coefficient, such that
kw=µαV. (8)
In fretting wear, by combining Equations (3) and (7), (8), we
arrive at the following wear equation (Mróz and Stupkiewicz,
1994; Fouvry et al., 1996, 2003):
1w(x,t)=αV1Ed(x,t). (9)
Here, 1Ed(x,t) is the frictional dissipated energy during one
fretting cycle, that is
1Ed(x,t)=Zs+1s
s
q(x,t)ds. (10)
While, in view of (8), the wear equations (3) and (10) are
equivalent, the energy wear equation incorporates the friction
mechanism and allows to account for variable coefficient of
friction (Cheikh et al., 2007).
2. FRETTING WEAR IN GROSS SLIP
REGIME
2.1. Formulation of the Model Wear
Contact Problem
In the present review, we consider both two-dimensional and
three-dimensional settings, highlighting their similarities and
differences. The analysis of contact deformations is limited to
Frontiers in Mechanical Engineering | www.frontiersin.org 2July 2020 | Volume 6 | Article 51
Argatov and Chai Contact Geometry Adaptation in Fretting Wear
the framework of linear elasticity and small strain analysis.
While the Hertzian half-plane or half-space approximation can
be employed in modeling the local stress-displacement field, in
many cases, the developed analytical approach can be directly
generalized to the case of layered elastic bodies with planar
contact interface by utilizing the corresponding surface influence
function. However, the effect of finite geometry on the wear scar
profiles requires a special consideration. In what follows, the
effect of local contact geometry is accounted for by means of the
local gap function.
Let ϕ0(x) denote the undeformed gap between the contacting
surfaces as a function of Cartesian coordinate x. Then, according
to Equation (1), the worn gap under plane-deformation
conditions will be given by the time-continuous equation
ϕ(x,t)=ϕ0(x)+kwZt
0
p(x,¯
t)v(x,¯
t)d¯
t, (11)
or by the time-increment equation 1ϕ(x,ti+1)=
kwp(x,ti)v(x,ti)1t, where 1t=ti+1−t1is the time increment.
Note that, by the definition, the contact pressure p(x,t) is
positive, and, therefore, the value of the integral in (11)
monotonically increases with time, provided v(x,t)6= 0.
Another simplification which is implicitly or explicitly present
in many studies on contact with wear, is to replace the problem
for two contacting elastic bodies with the wear contact problem
for one equivalent elastic body (whose surface influence function
is composed from the surface influence functions of the given
bodies) and a rigid punch, whose shape function is determined
by the local gap function. Though, this is a usual approach in
studying contact problems (e.g., for elastic bodies with rough
surfaces), the formulation of the wear contact problem requires to
consider the partition of the linear wear between the two wearing
bodies, when the resulting wear scar profiles are determined.
A wear contact problem in gross slip fretting regime can be
formulated as follows (Goryacheva, 1998):
2
πE∗Za(t)
−a(t)
K(x− ¯x)p(¯x,t)d¯x=δ0(t)−ϕ(x,t). (12)
Here, K(x) is the normalized surface influence function, E∗is the
reduced elastic modulus, δ0(t) is the normal approach between
the contacting bodies, and a(t) is the variable half-width of the
contact interval.
Yet another simplifying assumption is incorporated into the
governing integral equation (12), which implicitly states that the
wear, while changing the contact geometry, does not significantly
affect the surface influence function. In other words, small
changes of the contact shape due to wear are assumed, and their
influence on the contact pressure distribution is neglected. The
geometry dependence of the Green functions was accounted for
by Peigney (2004) using a first-order perturbation approach.
In light of (11) and (6), we have
ϕ(x,t)=ϕ0(x)+2k1x
1TZt
0
p(x,¯
t)d¯
t. (13)
Finally, in unilateral contact, the extent of the contact zone is
determined by the condition of vanishing contact pressure at
x= ±a(t).
It is to note here that the gross-slip regime assumes that
the condition of sliding occurs at the entire contact interface.
It is clear that the simple equation of wear (13) is violated
at the turning points of the wear, since the sliding velocity
goes through zero. This aspect, as well as the relative value of
the displacement stroke, should be taken into account when
considering the application of the analytical model to the analysis
of experimental results.
The integral equation kernel K(x) depends on both the global
shapes of the contacting bodies and the boundary conditions.
Using the half-plane approximation, one obtains K(x− ¯x)=
−ln[|x− ¯x|/H]−d0, where His the characteristic length of
the contact pair, and d0is the asymptotic constant (Aleksandrov
et al., 1978; Argatov, 2001). It was Galin (1976) who first
considered a two-dimensional wear contact problem with a
constant area and applied the method of variables separation.
Galin’s method was further developed by Aleksandrov et al.
(1978) and Komogortsev (1985) and extended to three-
dimensional wear contact problems (Galin and Goriacheva, 1977;
Kovalenko, 1985).
Numerical methods for solving the fretting wear contact
problem of the type (12), (13) were developed in a number of
studies, of which we refer especially to the papers by McColl
et al. (2004),Chai et al. (2005),Mary and Fouvry (2007), and Bae
et al. (2009) on finite-element simulations and (Serre et al., 2001;
Sfantos and Aliabadi, 2006) on boundary-element simulations.
Alternatives to finite and boundary element methods were
proposed by Lee et al. (2009), using the influence function
method, and (Nowell, 2010), using a quadratic programming
technique. A general method for the analysis of plane contact
problems for layered elastic structures in the presence of
sliding wear was developed by Aleksandrov and Kovalenko
(1980). Recent modeling results for coated systems, including
functionally graded material (FGM) coatings, are discussed
elsewhere (Ma et al., 2019).
2.2. Force-Controlled Steady-State Regime
In gross slip fretting wear, the following natural assumption
makes sense (Galin, 1976; Komogortsev, 1985):
p(x,t)=p∞+q(x,t), p∞=P
2a, (14)
where q(x,t)→0 as t→ ∞. In other words, the
contact pressure is equalized during wear, i.e., p(x,t)→p∞
as t→ ∞. Of course, formula (14) holds true, provided
the following assumption is fulfilled: both the contact force
Pand the contact half-width aare kept constant during the
wear process.
While formula (14) has been proved to hold for a constant
area of contact (Aleksandrov et al., 1978; Komogortsev, 1985),
Argatov and Tato (2012) extended its applicability to the case
of constant contact load Pand variable contact half-width
Frontiers in Mechanical Engineering | www.frontiersin.org 3July 2020 | Volume 6 | Article 51
Argatov and Chai Contact Geometry Adaptation in Fretting Wear
a(t), when
P=Za(t)
−a(t)
p(¯x,t)d¯x, (15)
by including into q(x,t) the boundary layers appearing near the
ends of the contact interval. However, the problem of explicit
constructing these boundary layers still remains open.
In the steady-state regime, the variation of contact zone
depends on the contact geometry. For a paraboloidal gap ϕ0(x)=
x2/(2R), the following differential equation holds (Argatov et al.,
2011):
a2da
dt =1
2kw¯vRP. (16)
Observe that Equation (16) states that in quasi steady state the
contact zone evolves being governed by the wear coefficient
kw, the average sliding velocity ¯v, and the undeformed contact
geometry, which is characterized by the curvature radius R. This
fact was utilized by Lengiewicz and Stupkiewicz (2013) in their
model of evolution of contact zone and wear accumulation,
without referring to the underlying elasticity problem. The effect
of elastic deformations on the wear scar profiles was accounted
for by Argatov and Tato (2012). The model of fretting wear in
quasi-steady-state gross-slip regime was generalized by Argatov
et al. (2011) for the Archard–Kragelsky wear equation (2), which
assumes a power law dependence of the wear rate on the
contact pressure.
In contrast to the force-controlled loading, when the left-
hand side of Equation (15) is suggested to be known, in the
displacement-controlled loading, the variation of the contact
approach will be specified. In such a case, due to the dissipative
nature of wear accompanied by material removal, the total
contact load should gradually vanish to the end of the fretting
process. Such situation was analyzed by Peigney (2004), who
determined the asymptotically stabilized state reached by an
elastic body subjected to wear contact with a rigid indenter in the
displacement-controlled cyclic loading.
2.3. Steady-State Contact Profile
Apparently, the problem of determining the worn shape in the
operating state, that is the contact shape function ϕ∞(x) that
produces a uniform contact pressure under a constant load was
first solved by Dundurs and Comninou (1980) in the case of an
elastic half-space. In a more general case (12), the steady-state
profile is given by
ϕ∞(x)=P
πE∗aZa
−a
K(¯x)d¯x−Za
−a
K(x− ¯x)d¯x. (17)
We note that ϕ∞(0) =0, and this explains the first term
in the curly braces in (17). The steady-state solution in the
case of Archard–Kragelsky model of wear (2) was obtained by
Goryacheva (1998), who also studied its asymptotic stability.
Interestingly, the steady-state profile (17) is shown to be
optimal, if the wear coefficient is assumed to be constant
(Banichuk et al., 2010). The optimal shapes generated by wear
process were evaluated by exploiting the dissipative nature of the
wear process, which can be characterized, e.g., by minimization
of friction dissipation power (Páczelt and Mróz, 2007). Based
on the Hertzian half-space approximation, a three-dimensional
computational method for determining the optimum contact
geometry in fretting under the gross slip regime was developed
by Gallego et al. (2006). Recently, Argatov and Chai (2019c)
considered a practically important question of approximating the
ideal profile (17) with a symmetric smooth profile composed of
three parabolic arcs, which was introduced by Vázquez et al.
(2010), exploiting the idea of compound curvature. The effect
of friction is shown to result in the profile asymmetry, which
depends on the direction of sliding (Argatov and Chai, 2020b).
2.4. Wearing-In Period
Any fretting wear test starts from the initial contact state, which
is fully characterized by the initial contact geometry and loading
conditions. The initial time interval, during which the contact
pressure evolves from the initial one to the steady state pattern,
is called the wearing-in period. Based on the Galin type analysis
of the wear contact problem with a fixed contact zone, Argatov
and Fadin (2011) have estimated the duration of the wearing-in
period, Tin, as follows:
Tin ∼a
λminkw¯vE∗. (18)
Here, λmin is the minimum characteristic value of the
corresponding integral eigenvalue problem.
It is to note that the right-hand side of the relation (18)
does not depend on the loading level. It is interesting that the
wearing-in period in the displacement-controlled regime is about
five times greater than that under the force-controlled loading
(Argatov and Fadin, 2011). It is also shown (Argatov and Chai,
2020b) that the effect of friction extends the wearing-in period.
2.5. Wear of Functionally-Graded
Wear-Resisting Materials
A range of wear contact profiles with variable wear resistance
of a sliding punch was considered by Goryacheva (1998) in the
case of the Archard–Kragelsky wear model (2) with a particular
focus on the steady-state solutions. The transient wear contact
problems for composite materials were considered recently using
different approaches, including the method of dimensionality
reduction with application to an axisymmetric heterogeneous
annular cylindrical punch (Li et al., 2018) and a level-set based
shape and topology optimization method with application to
a Pasternak elastic foundation model (Feppon et al., 2017). By
using an appropriate symmetrization of the integral equation
kernel, Argatov and Chai (2019b) extended the Galin method for
analyzing the transient contact pressure distribution and derived
an upper estimate for the wearing-in period. Also, the effective
wear coefficient was represented as
Keff(t)=K∞
eff +1
PZa
−a
kw(x)p(x,t)−p∞(x)dx, (19)
where kw(x) is a variable wear coefficient, p∞(x) is the steady-
state contact pressure distribution, and K∞
eff is the steady-state
Frontiers in Mechanical Engineering | www.frontiersin.org 4July 2020 | Volume 6 | Article 51
Argatov and Chai Contact Geometry Adaptation in Fretting Wear
value of the effective wear coefficient given by
K∞
eff =2a Za
−a
dx
kw(x)!−1
. (20)
Moreover, it was shown that the second term on the right-hand
side of Equation (19) decreases exponentially with time during
the wearing-in period, for which the following estimate was
established (Argatov and Chai, 2019b):
Tin ≤2a
πkmax
w¯vE∗sZ1
−1Z1
−1
|L(ξ,¯
ξ)|2d¯
ξdξ. (21)
Here, kmax
wis the maximum wear coefficient, and L(ξ,¯
ξ) is the
normalized integral kernel.
Interestingly, Equation (20) represents a generalization of the
mixture rule of Khruschov (1974) for functionally graded wear-
resisting composite materials (Friedrich, 1993; Yen and Dharan,
1996). Observe also (Argatov and Chai, 2019b) that while K∞
eff
is independent of the distribution of the wear resistance, that
is independent of the distribution of the phases in a wearable
multiphase material the duration of the wearing-in period Tin,
in contrast, is sensitive to the phase fraction distribution.
2.6. Three-Dimensional Fretting Wear
Contact Problems
The main difference between the 2D and 3D cases is a higher
variability of the sliding velocity orientation that can occur
in practice, so that the concept of anisotropic wear (Mróz
and Stupkiewicz, 1994; Zmitrowicz, 2006) can be introduced
in the spatial case. Otherwise, many features of the gross slip
fretting wear are similar in the two cases. In particular, in the
axisymmetric case with a constant contact radius a, Equation (14)
still applies with p∞=P/(πa2). Also, formula (17) for the
steady-state profile can be simply generalized by the appropriate
choice of the integral equation kernel (which is determined by
the corresponding surface influence function) and, of course,
by extending the integration to the whole contact area. For the
Hertzian type contact geometry with the initial gap ϕ0(x)=
(x2
1+x2
2)/(2R) and a variable contact area of radius a(t), the
analog of Equation (16) reads as follows (Argatov, 2011):
a3da
dt =1
πkw¯vRP. (22)
The axisymmetric model of fretting wear based on the Archard–
Kragelsky wear model (2) was developed by Argatov et al. (2011),
who have observed the phenomenon of the decrease of the
wear coefficient due to the increase of the contact area followed
by decrease of the contact pressure. Apparently, such an effect
depends on the exponent αin Equation (2). In the mentioned
study, it was evaluated to be greater than one.
The general Hertzian type contact with a variable elliptical
contact area was considered by Argatov (2011) in application
to local interwire contact under reciprocal sliding. A special
consideration is required in the case of torsional fretting wear
with an annular contact area, when the relative sliding velocity
varies proportionally to the distance from the axis of symmetry.
The analogous rotational contact problem with a sliding wear was
analyzed by Galin and Goriacheva (1977), and Kovalenko (1985).
The finite element method study of torsional fretting for a ball-
on-flat configuration was conducted by Liu et al. (2014) under the
assumption of variable coefficient of friction, whose variation due
to the abrasive wear degradation is governed by the local contact
history and the accumulated slip distance.
3. FRETTING WEAR IN PARTIAL SLIP
REGIME
3.1. Stick Zone
Recall that the local contact of two elastically similar bodies
during cyclic loading-unloading by a normal force only (i.e.,
at zero tangential force) is not accompanied by the relative
tangential displacements at the contact interface, whereas some
fretting can occur at the cyclic normal contact of dissimilar
bodies. However, the tangential displacements mismatch appears
for elastically similar bodies even in the case of cyclic tangential
loading with constant normal load.
In a certain regime of fretting, called partial slip regime,
the contact area, ω(t), which may vary in time, contains inside
a stick zone, ω∗, where the contacting surfaces stick one to
another, so that the contact geometry inside the stick zone
remains untouched by wear, whereas wear occurs in a slip
zone, where the contacting surfaces experience relative tangential
movement. The theory of tangential contact with partial slip
for the Hertzian geometry was developed by Cattaneo (1938)
and Mindlin (1949). The 2D theory of tangential contact for
elastically similar semi-infinite solids was developed by Jäger
(1998) and Ciavarella (1998). The 3D Cattaneo–Mindlin model
was extended by Jäger (1996) for the case of stepwise oblique
loading. Further progress in its development is associated with
devising the method of memory diagrams (Aleshin and Van
Den Abeele, 2013; Aleshin et al., 2015). Recently, the 3D
Cattaneo–Mindlin model was outlined for transversely isotropic
materials by Argatov et al. (2018).
In the axisymmetric case, Jäger (1995) generalized the model
of local tangential contact for two elastically similar bodies
with arbitrary gap function. In the general non-axisymmetric
three-dimensional case, Ciavarella (1998) introduced a simplified
version of the Cattaneo–Mindlin theory (without Poisson’s
effect), which was recently extended for the case of transversely
isotropic materials by Chai and Argatov (2018), who also applied
the self-similarity solutions by Borodich (1983, 1989) to derive
explicit tangential force-displacement relations in the case of
self-similar gap between the contacting surfaces.
Further, Hills and Sosa (1999) reviewed analytical solutions
for general elastic contact problems with partial slip, which can
be used, for instance, in estimating frictional energy and the
local wear rate in the initial stage of fretting. Analytical aspects
of fretting fatigue damage were considered by Ciavarella and
Demelio (2001) with application to dovetail joints. Numerical
method for partial-slip frictional contact problems have been
Frontiers in Mechanical Engineering | www.frontiersin.org 5July 2020 | Volume 6 | Article 51
Argatov and Chai Contact Geometry Adaptation in Fretting Wear
developed in a number of studies (Chen and Wang, 2008; Wang
et al., 2013).
Let ϕ(x1,x2) be the gap between the surfaces of two elastically
similar transversely isotropic semi-infinite bodies that do not
exhibit the effect of coupling between the distributions of
shearing contact tractions (in the case of identical materials this
assumption implies the zero Poisson’s ratio, see, e.g., Ciavarella,
1998). In many cases it may be assumed that the gap is described
by a homogeneous function of degree d, such that
ϕ(cx1,cx2)=cdϕ(x1,x2), (23)
where cis an arbitrary positive constant.
Let also land l∗be the characteristic sizes of the contact
area ωand the stick zone ω∗, respectively. Then, the following
relations hold between the relative size of the stick zone, the ratio
of the tangential to the normal contact force, and the ratio of the
tangential to the normal displacement (Chai and Argatov, 2018):
l∗
l=1−F1
µF31/(d+1)
,l∗
l=1−M1
M3
δ1
µδ31/d
. (24)
Here, M1and M3are effective elastic moduli. The Cattaneo–
Mindlin model is recovered from Equation (24) for d=2.
3.2. MDR-Based Approach
It is well-known (De Mul et al., 1986) that for a wide class
of contact geometries, the Hertzian half-space analysis of local
contact gives reliable results. The method of dimensionality
reduction (MDR) developed by Popov and Heß (2015), and
their collaborators, combines the axisymmetric Hertz theory of
normal contact and the axisymmetric Cattaneo–Mindlin theory
of tangential contact into a unified modeling framework by
transforming a given 3D contact problem for two elastically
similar solids into one-dimensional contact problem for an
equivalent rigid punch and the Popov foundation, which is a
linearly-elastic spring-like foundation that possesses both normal
and tangential stiffnesses. The relation between the gap function
ϕ(r,t) and the equivalent 1D profile g(x,t) represents the direct
mapping rule from the original 3D contact problem into the
1D equivalent contact problem. Since the Popov foundation is
spring-like, its normal reaction is given by qz(x,t)=E∗δ(t)−
g(x,t), where |x| ≤ a(t) and a(t) is a root of the equation
g(a,t)=δ(t). The integration of the 1D normal contact reaction
over the contact interval −a(t), a(t)yields the total normal
force, F3(Note that in the displacement controlled mode, the
contact approach δ(t) is assumed to be known).
The Popov foundation can be discretized by introducing
a discretization step 1x, so that the normal and tangential
stiffnesses of every individual spring element will be 1kz=E∗1x
and 1kx=G∗1x, respectively, where G∗is the effective shear
modulus. Correspondingly, if an individual elastic spring element
with a coordinate xreceives normal, uz(x,t), and tangential,
ux(x,t), displacements, the values of normal and tangential
reaction forces, respectively, will be 1fN(x,t)=uz(x,t)1kz
and 1fT(x,t)=ux(x,t)1kx(with compression-positive sign
convention taken into account).
In a stick zone, the tangential displacement ux(x,t) is
determined by the punch’s tangential displacement, u(0)
x(t),
whereas in a slip zone 1fT(x,t)= ±µ1fN(x,t), as it is prescribed
by Coulomb’s law, where µis the coefficient of friction. It is
suggested (Dimaki et al., 2014, 2016) to consider the fretting
wear as an incremental process, such that 1u(0)
x(t)=u(0)
x(t+
1t)−u(0)
x(t) is the tangential displacement increment of the 1D
equivalent punch, which exactly corresponds to the increment
of the relative tangential displacement of the contacting solids.
According to the Archard wear equation (1), the linear change of
the 3D profile is given as follows (Dimaki et al., 2016):
1ϕ(r,t)=kwp(r,t)1u(0)
x(t)−1u(3D)
x(r,t). (25)
Here, u(3D)
x(r,t) and 1u(3D)
x(r,t) are the relative tangential
displacement at the contact interface and its increment.
Thus, the numerical implementation of the MDR-based
approach using Equation (25) will require the application of
the inverse mapping (from the 1D contact problem to the
3D contact problem) for evaluating u(3D)
x(r,t) and the normal
contact pressure p(r,t). The corresponding numerical procedures
have been developed for both the gross-slip (Dimaki et al., 2016)
and partial-slip (Dimaki et al., 2014) regimes.
3.3. Limiting Profile
Evidently, in the partial slip fretting wear, by the definition,
there is a stick zone which remains untouched by wear for
the entire periodic loading process. For instance, in the force-
controlled mode with a constant normal contact load and a
constant amplitude sinusoidal tangential force, the effect of wear
on the contacting surfaces will exhibit itself in an increase of the
contact approach as well as in an increase of the contact area.
However, as it was observed by Ciavarella and Hills (1999), this
process eventually leads to some limiting contact geometry that is
characterized by the absence of wear outside the stick zone. In the
case of the Archard wear model, the latter means that the contact
pressure vanishes in the final slip zone.
In the axisymmetric case under displacement-controlled
loading (when δ=const), the solution for the limiting profile
can be easily obtained in terms of the limiting profile for the
equivalent punch as follows (Popov, 2014):
g∞(x)=g0(x), |x| ≤ c,
δ,c<|x| ≤ a∞.(26)
Here, g0(x) is the initial equivalent profile, cis the radius of
the stick zone, which depends on the tangential displacement
amplitude u(0)
x,δis the contact approach, a∞is the limiting
radius of the contact area, which is determined by the equation
ϕ∞(a)=ϕ0(a). Finally, the function ϕ∞(r) itself is determined
by the inverse transform applied to g∞(x).
Of course, formula (26) can be easily extended to the case
of force-controlled loading by replacing δwith δ∞, where δ∞
is the limiting contact approach, and expressing cin terms of
the normal and tangential contact loads F3and F1, e.g., using
Equation (24). However, it is instructive to distinguish the two
cases, especially since it makes sense in the 3D case.
Frontiers in Mechanical Engineering | www.frontiersin.org 6July 2020 | Volume 6 | Article 51
Argatov and Chai Contact Geometry Adaptation in Fretting Wear
Table 1 shows the state of the art of the analysis of the final
(limiting or asymptotic) contact geometry. We note that the
solutions of Heß (2019) and Willert et al. (2019) were obtained
for the case of power-law graded materials with Young’s modulus
varying with depth as E(z)=E0(z/z0)k, where k∈(−1, 1).
The MDR-based approach (Popov, 2014) was generalized by
Chai and Popov (2016) for fretting wear in an adhesive contact in
the Dugdale approximation when the adhesive (attractive) stress
outside the contact area is assumed to be constant up to some
critical distance and vanishing beyond this range. Recently, the
analytical approach developed by Popov (2014) was extended by
Dmitriev et al. (2016) and Mao et al. (2016) (see also Li, 2016)
to a dual-mode fretting under the influence of superimposed
normal oscillations (with amplitude 1u0
zand frequency ωz)
and tangential oscillations (with amplitude 1u0
xand frequency
ωx). It is interesting that Dmitriev et al. (2016) also provided
experimental evidence for the correctness of the theoretically
predicted limiting shape. We note also that recent experimental
aspects of fretting wear were discussed in an extensive review by
Meng et al. (2020).
3.4. Wear Accumulation
A number of numerical methods have been devised for
simulating the contact geometry adaptation during fretting wear
in the partial slip regime (Gallego and Nelias, 2007; Dimaki et al.,
2014; Wang et al., 2015; Cardoso et al., 2019), when both the
contact profiles (outside the stick zone) and the contact pressure
distribution evolve in relation to each other.
Interestingly, a special focus on modeling of the evolution of
the worn volume, V, was not shown until recently (Kasarekar
et al., 2007). A simple mathematical model of wear accumulation
in the case of initial Hertzian contact was developed by Chai
and Argatov (2019) based on the dissipation energy model for
the volume wear rate (Fouvry et al., 2003). The following one
fitting parameter formula was suggested for the non-monotonic
variation of the volume wear rate:
dV
dN =w01+β1
N
N1exp−N
N1. (27)
TABLE 1 | Limiting shapes of profiles in fretting wear.
Initial
profile
Displacement-
controlled
regime
Froce-controlled
regime
2D case Hertzian Goryacheva et al.,
2001; Hills et al.,
2009
Arbitrary Goryacheva and
Goryachev, 2006
3D axisymmetric Hertzian Popov, 2014 Dini et al., 2008
Arbitrary Popov, 2014 Argatov and Chai,
2018
3D non-axisymmetric Hertzian Argatov et al.,
2018
3D axisymmetric FGM Hertzian Willert et al., 2019 Heß, 2019
Arbitrary Willert et al., 2019
Here, w0is the initial volume wear rate, which can be evaluated
using the Cattaneo–Mindlin theory (Johnson, 1955), Nis the
number of cycles, N1, is an auxiliary parameter, which is related
to the total worn volume V∞=w0(1 +β1)N1, and β1is the only
fitting parameter.
The analytical model (27) implies that the wearing-in period
is proportional to N1, which, in turn, is estimated as
N1∼G∗
(E∗)5/3
R2/3
µkwP1/3n1(χ), (28)
where χ=c/a0is the relative stick-zone radius, and a0is the
initial (Hertzian) contact radius. In contrast to Equation (18),
formula (28) shows that the duration of the wearing-in period
(measured in number of cycles) in partial-slip fretting wear
depends on the load level.
4. DISCUSSION AND CONCLUSIONS
The obtained results for wearing-in period (18), (21), and (28)
merit comment. First, as we might intuitively expect, the larger
the initial contact zone, the larger the wearing-in period. Second,
as it could be foreseen from the physical dimension of the wear
coefficient, the duration of the wearing-in period is inversely
proportional to the characteristic value of the coefficient of wear.
Third, in the gross-slip and partial-slip regimes, the wearing-in
period is inversely proportional to E∗and (E∗)2/3, respectively,
so that the elasticity effect weakens in the second case.
4.1. Limitations of the Analytical Approach
It goes without saying that analytical methods are not so
flexible as numerical ones, especially in applications to specific
engineering problems. In contact mechanics, the success of
analytical approach is critically dependent on the possibility
to approximate the surface influence function, which, in turn,
strongly depends on the contact geometry. The phenomenon of
wear manifests itself in the variation of the latter, and, generally
speaking, the main limitation of the current state-of-the-art
analytical techniques is in their inability to effectively deal with
varying geometry due to wear loss.
Perhaps, another reason for the slow progress in solving
transient wear contact problems is the difficulty which may
arise with the introduction of non-linear equations of wear.
However, a further rapid advance can be achieved, for instance,
in estimating the duration of wearing-in period, provided the
problem formulation admits the existence a steady-state regime.
4.2. Open Problems
First, observe that Table 1 has a few empty cells, which indicate a
number of still unsolved problems on limiting shapes of profiles
in fretting wear. The limiting profile problem formulation
assumes that in the limiting state the contact pressure vanishes
outside of the stick zone. It makes sense to investigate whether
a threshold model of wear, which assumes no wear below
certain level of contact pressures, is suitable for describing the
limiting state in practical fretting problems. Second, we point
out that the analytical solutions (16) and (22) were obtained
Frontiers in Mechanical Engineering | www.frontiersin.org 7July 2020 | Volume 6 | Article 51
Argatov and Chai Contact Geometry Adaptation in Fretting Wear
for the Hertzian local gap, though the method with which
they were derived allows such generalizations. By the way, it
is still interesting to analyze the boundary-layer problem in
the wear contact problem with variable contact zone. Third,
it is to note here that the effective wear coefficient (20) for
functionally-graded wear-resisting materials was evaluated under
the simplifying assumption of homogeneous elastic properties,
which is implicitly or explicitly present in a majority of studies
of wear contact problems for composite materials. Further, as it
was already mentioned in section 1.3, the energy wear equation
allows to account for variable coefficient of friction, which can be
done in a straightforward manner in the gross slip regime.
It should be noted that the wearing-in period in gross-
slip fretting wear is associated with the redistribution of the
macroscopic contact pressure from the initial pattern, which
is caused by the intact contact geometry (e.g., the Hertzian
contact pattern in ball-on-plate contact), to an approximately
uniform distribution, which reproduces the steady-state shape
profile. In the case of non-conforming contact, there is still
uncertainty regarding the experimental relation between the
wearing-in period and the running-in period, which is primarily
associated with the evolution of surface topography and shows
the exponential evolution of the wear rate as well (Zhang et al.,
2018). It is to note here that the recently developed artificial
neural network modeling framework (Argatov and Chai, 2020a)
can be utilized in analysis of experimental studies of wearing-in
period and the so-called true wear coefficient. It is also apparently
an open question as to whether the duration of the wearing-in
period in periodic contact is longer than that in the wear contact
problem for a single contact.
It was shown (Argatov and Chai, 2020b) that the eigenvalue
problem associated with the wear contact problem with friction,
generally speaking, may possess the complex-valued spectrum,
and therefore, the contact pressure is predicted to approach
the steady state exponentially decaying and oscillating. Such an
oscillating behavior of the contact pressure during the wearing-
in period, if exists, would be experimentally observed for large
values of the friction coefficient.
4.3. Directions for Future Research
Based on the above review, one can suggest that further
progress is expected in studying transient wear contact problems
for functionally-graded wear-resisting materials. Also, further
attention needs to be paid to the optimization problems in
fretting wear, while combining different strategies for optimizing
contacting parts, including geometrical (shape optimization) and
material grading.
Observe that Archard’s equation of wear or the work rate
model are usually adopted in the majority of analytical studies
to date. Further progress is needed in extending the results
obtained for these linear wear models to other models, including
non-linear, like the Archard–Kragelsky equation, or models
which account for the effect of debris formation and third-body
fretting wear.
Further, it is well-known and practically important that the
Archard equation (1) as well as the Archard–Kragelsky equation
(2) treat wear as a local process. In this respect, it would of interest
to investigate what role the non-local nature of wear damage
plays in the partial lip regime of fretting, especially near the
boundary of the stick zone.
Lastly, caution is urged when the simple analytical models are
applied for analyzing practical problems. For instance, one may
argue that the analytical solutions for limiting profiles outlined in
Table 1 do not predict the final state that is practically achieved in
partial-slip fretting, since they assume complete removal of worn
material. Nevertheless, the limiting shape profiles are useful, as
they allow to upper estimate the differences in contact behavior
that can be observed both in practice and experiment (Dini et al.,
2008). An important asset of the developed analytical solutions is
their explicit dependence on the model parameters, which can be
effectively used for solving design and optimization problems.
AUTHOR CONTRIBUTIONS
IA: methodology, formal analysis, and writing—original draft
preparation. YC: conceptualization, methodology, and writing—
reviewing and editing. All authors contributed to the article and
approved the submitted version.
FUNDING
This work was supported by the National Research Foundation
of Korea (NRF) grant funded by the Korean government (MSIT)
(No. NRF-2017M2B2A9072449).
REFERENCES
Aleksandrov, V. M., Galin, L. A., and Piriev, N. P. (1978). A plane contact problem
for an elastic layer of considerable thickness in the presence of wear [in
Russian]. Mech. Solids 4, 60–67.
Aleksandrov, V. M., and Kovalenko, E. V. (1980). Plane contact problems of the
theory of elasticity for nonclassical regions in the presence of wear. J. Appl.
Mech. Tech. Phys. 21, 421–427. doi: 10.1007/BF00920786
Aleksandrov, V. M., and Kovalenko, E. V. (1984). “Mathematical methods in
contact problems with wear,” in Nonlinear Models and Problems of Mechanics
of Deformable Solids, ed K. V. Frolov (Moscow: Nauka), 77–89.
Aleshin, V., Matar, O. B., and Van Den Abeele, K. (2015). Method of memory
diagrams for mechanical frictional contacts subject to arbitrary 2D loading. Int.
J. Solids Struct. 60, 84–95. doi: 10.1016/j.ijsolstr.2015.02.016
Aleshin, V., and Van Den Abeele, K. (2013). General solution to the Hertz-
Mindlin problem via Preisach formalism. Int. J. Nonlinear Mech. 49, 15–30.
doi: 10.1016/j.ijnonlinmec.2012.09.003
Archard, J. F. (1953). Contact and rubbing of flat surfaces. J. Appl. Phys. 24,
981–988. doi: 10.1063/1.1721448
Argatov, I., and Tato, W. (2012). Asymptotic modeling of reciprocating sliding
wear-comparison with finite-element simulations. Eur. J. Mech. A Solids 34,
1–11. doi: 10.1016/j.euromechsol.2011.11.008
Argatov, I. I. (2001). Solution of the plane Hertz problem. J.
Appl. Mech. Tech. Phys. 42, 1064–1072. doi: 10.1023/A:10125344
32055
Argatov, I. I. (2011). Asymptotic modeling of reciprocating sliding wear
with application to local interwire contact. Wear 271, 1147–1155.
doi: 10.1016/j.wear.2011.05.028
Frontiers in Mechanical Engineering | www.frontiersin.org 8July 2020 | Volume 6 | Article 51
Argatov and Chai Contact Geometry Adaptation in Fretting Wear
Argatov, I. I., Bae, J. W., and Chai, Y. S. (2018). The limiting shape of the
transversely isotropic elastically similar solids in fretting. Int. J. Appl. Mech.
10:1850089. doi: 10.1142/S1758825118500898
Argatov, I. I., and Chai, Y. S. (2018). Limiting shape of profiles in fretting wear.
Tribol. Int. 125, 95–99. doi: 10.1016/j.triboint.2018.04.026
Argatov, I. I., and Chai, Y. S. (2019a). An artificial neural network
supported regression model for wear rate. Tribol. Int. 138, 211–214.
doi: 10.1016/j.triboint.2019.05.040
Argatov, I. I., and Chai, Y. S. (2019b). Effective wear coefficient and wearing-
in period for a functionally graded wear-resisting punch. Acta Mech. 230,
2295–2307. doi: 10.1007/s00707-019-2366-9
Argatov, I. I., and Chai, Y. S. (2019c). A note on optimal design of contact
geometry in fretting wear. Int. J. Mech. Mater. Design 16, 415–422.
doi: 10.1007/s10999-019-09467-9
Argatov, I. I., and Chai, Y. S. (2020a). Artificial neural network modeling of sliding
wear. J. Proc. Instit. Mech. Eng. J. doi: 10.1177/1350650120925582. [Epub ahead
of print].
Argatov, I. I., and Chai, Y. S. (2020b). Wear contact problem with friction: Steady-
state regime and wearing-in period. Int. J. Solids Struct. 193–194:213–221.
doi: 10.1016/j.ijsolstr.2020.02.019
Argatov, I. I., and Fadin, Y. A. (2011). A macro-scale approximation for the
running-in period. Tribol. Lett. 42, 311–317. doi: 10.1007/s11249-011-9775-9
Argatov, I. I., Gómez, X., Tato, W., and Urchegui, M. A. (2011). Wear evolution
in a stranded rope under cyclic bending: Implications to fatigue life estimation.
Wear 271, 2857–2867. doi: 10.1016/j.wear.2011.05.045
Bae, J. W., Lee, C. Y., and Chai, Y. S. (2009). Three dimensional fretting wear
analysis by finite element substructure method. Int. J. Precis. Eng. Manufact.
10, 63–69. doi: 10.1007/s12541-009-0072-6
Banichuk, N. V., Ragnedda, F., and Serra, M. (2010). Some optimization problems
for bodies in quasi-steady state wear. Mech. Based Design Struct. Mach. 38,
430–439. doi: 10.1080/15397734.2010.483574
Borodich, F. M. (1983). Similarity in the problem of contact between elastic bodies.
J. Appl. Math. Mech. 47, 440–442. doi: 10.1016/0021-8928(83)90077-1
Borodich, F. M. (1989). Hertz contact problems for an anisotropic
physically nonlinear elastic medium. Strength Mater. 21, 1668–1676.
doi: 10.1007/BF01533408
Cardoso, R. A., Doca, T., Néron, D., Pommier, S., and Araújo, J. A. (2019). Wear
numerical assessment for partial slip fretting fatigue conditions. Tribol. Int. 136,
508–523. doi: 10.1016/j.triboint.2019.03.074
Cattaneo, C. (1938). Sul contatto di due corpi elastici: distribuzione locale degli
sforzi. Rendiconti dell’Accademia nazionale dei Lincei 27, 342–348, 434-436,
474-478.
Chai, Y., Lee, C., Bae, J., Lee, S., and Hwang, J. (2005). Finite element analysis of
fretting wear problems in consideration of frictional contact. Key Eng. Mater.
297–300:1406–1411. doi: 10.4028/www.scientific.net/KEM.297-300.1406
Chai, Y. S., and Argatov, I. I. (2018). Local tangential contact of elastically
similar, transversely isotropic elastic bodies. Meccanica 53, 3137–3143.
doi: 10.1007/s11012-018-0870-y
Chai, Y. S., and Argatov, I. I. (2019). Fretting wear accumulation in
partial-slip circular Hertzian contact. Mech. Res. Commun. 96, 45–48.
doi: 10.1016/j.mechrescom.2019.02.005
Chai, Y. S., and Popov, V. L. (2016). Limiting shape due to fretting wear in an
adhesive contact in Dugdale approximation. Phys. Mesomech. 19, 378–381.
doi: 10.1134/S1029959916040044
Cheikh, M., Quilici, S., and Cailletaud, G. (2007). Presentation of KI-COF, a
phenomenological model of variable friction in fretting contact. Wear 262,
914–924. doi: 10.1016/j.wear.2006.10.001
Chen, W. W., and Wang, Q. J. (2008). A numerical model for the point contact of
dissimilar materials considering tangential tractions. Mech. Mater. 40, 936–948.
doi: 10.1016/j.mechmat.2008.06.002
Ciavarella, M. (1998). Tangential loading of general 3D contacts. ASME J. Appl.
Mech. 65, 998–1003. doi: 10.1115/1.2791944
Ciavarella, M., and Demelio, G. (2001). A review of analytical aspects of fretting
fatigue, with extension to damage parameters, and application to dovetail joints.
Int. J. Solids Struct. 38, 1791–1811. doi: 10.1016/S0020-7683(00)00136-0
Ciavarella, M., and Hills, D. A. (1999). Brief note: Some observations on oscillating
tangential forces and wear in general plane contacts. Eur. J. Mech. A Solids 18,
491–497. doi: 10.1016/S0997-7538(99)00117-5
De Mul, J. M., Kalker, J. J., and Fredriksson, B. (1986). The contact between
arbitrarily curved bodies of finite dimensions. J. Tribol. 108, 140–148.
doi: 10.1115/1.3261134
Dimaki, A. V., Dmitriev, A. I., Chai, Y. S., and Popov, V. L. (2014).
Rapid simulation procedure for fretting wear on the basis of the
method of dimensionality reduction. Int. J. Solids Struct. 51, 4215–4220.
doi: 10.1016/j.ijsolstr.2014.08.003
Dimaki, A. V., Dmitriev, A. I., Menga, N., Papangelo, A., Ciavarella, M.,
and Popov, V. L. (2016). Fast high-resolution simulation of the gross
slip wear of axially symmetric contacts. Tribol. Trans. 59, 189–194.
doi: 10.1080/10402004.2015.1065529
Dini, D., Sackfield, A., and Hills, D. A. (2008). An axi-symmetric Hertzian
contact subject to cyclic shear and severe wear. Wear 265, 1918–1922.
doi: 10.1016/j.wear.2008.04.031
Dmitriev, A. I., Voll, L. B., Psakhie, S. G., and Popov, V. L. (2016). Universal
limiting shape of worn profile under multiple-mode fretting conditions: theory
and experimental evidence. Sci. Rep. 6:23231. doi: 10.1038/srep23231
Dundurs, J., and Comninou, M. (1980). Shape of a worn slider. Wear 62, 419–424.
doi: 10.1016/0043-1648(80)90183-0
Feppon, F., Michailidis, G., Sidebottom, M. A., Allaire, G., Krick, B. A.,
and Vermaak, N. (2017). Introducing a level-set based shape and
topology optimization method for the wear of composite materials
with geometric constraints. Struct. Multidiscipl. Optimizat. 55, 547–568.
doi: 10.1007/s00158-016-1512-4
Fouvry, S., Kapsa, P., and Vincent, L. (1996). Quantification of fretting damage.
Wear 200, 186–205. doi: 10.1016/S0043-1648(96)07306-1
Fouvry, S., Liskiewicz, T., Kapsa, P., Hannel, S., and Sauger, E. (2003).
An energy description of wear mechanisms and its applications to
oscillating sliding contacts. Wear 255, 287–298. doi: 10.1016/S0043-1648(03)0
0117-0
Friedrich, K. (1993). “Wear models for multiphase materials and synergistic effects
in polymeric hybrid composites,” in in Advances in Composite Tribology, Vol. 8
of Composite Materials Series, ed K. Friedrich (Amsterdam: Elsevier), 209–273.
doi: 10.1016/B978-0-444-89079-5.50010-6
Galin, L. A. (1976). Contact problems of the theory of elasticity in the presence of
wear. J. Appl. Math. Mech. 40, 931–936. doi: 10.1016/0021-8928(76)90132-5
Galin, L. A., and Goriacheva, I. G. (1977). Axisymmetric contact problem of the
theory of elasticity in the presence of wear. J. Appl. Math. Mech. 41, 826–831.
doi: 10.1016/0021-8928(77)90164-2
Gallego, L., and Nelias, D. (2007). Modeling of fretting wear under gross slip and
partial slip conditions. J. Tribol. 129, 528–535. doi: 10.1115/1.2736436
Gallego, L., Nélias, D., and Jacq, C. (2006). A comprehensive method to predict
wear and to define the optimum geometry of fretting surfaces. J. Tribol. 128,
476–485. doi: 10.1115/1.2194917
Goryacheva, I. G. (1998). Contact Mechanics in Tribology. Dordrecht: Kluwer.
doi: 10.1007/978-94-015-9048-8
Goryacheva, I. G., and Goryachev, A. P. (2006). The wear contact
problem with partial slippage. J. Appl. Math. Mech. 70, 934–944.
doi: 10.1016/j.jappmathmech.2007.01.010
Goryacheva, I. G., Rajeev, R. T., and Farris, T. N. (2001). Wear in partial slip
contact. J. Tribol. 123, 848–856. doi: 10.1115/1.1338476
Heß, M. (2019). A study on gross slip and fretting wear of contacts involving a
power-law graded elastic half-space. Fact. Univers. Ser. Mech. Eng. 17, 47–64.
doi: 10.22190/FUME190121010H
Hills, D. A., Sackfield, A., and Paynter, R. J. H. (2009). Simulation of fretting
wear in halfplane geometries: part I-the solution for long term wear. J. Tribol.
131:031401. doi: 10.1115/1.3118785
Hills, D. A., and Sosa, G. U. (1999). Origins of partial slip in fretting-a review
of known and potential solutions. J. Strain Anal. Eng. Design 34, 175–181.
doi: 10.1243/0309324991513731
Huajie, L., and Hongzhao, L. (2018). “An overview of the numerical simulation
wear models in mechanical systems,” in 2018 IEEE 9th International Conference
on Mechanical and Intelligent Manufacturing Technologies (ICMIMT) (IEEE),
105–109. doi: 10.1109/ICMIMT.2018.8340430
Jäger, J. (1995). Axi-symmetric bodies of equal material in contact under torsion or
shift. Arch. Appl. Mech. 65, 478–487. doi: 10.1007/BF00835661
Jäger, J. (1996). Stepwise loading of half-spaces in elliptical contact. ASME J. Appl.
Mech. 63, 766–773. doi: 10.1115/1.2823361
Frontiers in Mechanical Engineering | www.frontiersin.org 9July 2020 | Volume 6 | Article 51
Argatov and Chai Contact Geometry Adaptation in Fretting Wear
Jäger, J. (1998). A new principle in contact mechanics. J. Tribol. 120, 677–684.
doi: 10.1115/1.2833765
Johnson, K. (1955). Surface interaction between elastically loaded bodies
under tangential forces. Proc. R. Soc. Lond. Ser. A 230, 531–548.
doi: 10.1098/rspa.1955.0149
Kasarekar, A. T., Bolander, N. W., Sadeghi, F., and Tseregounis, S. (2007). Modeling
of fretting wear evolution in rough circular contacts in partial slip. Int. J. Mech.
Sci. 49, 690–703. doi: 10.1016/j.ijmecsci.2006.08.021
Khruschov, M. M. (1974). Principles of abrasive wear. Wear 28, 69–88.
doi: 10.1016/0043-1648(74)90102-1
Komogortsev, V. F. (1985). Contact between a moving stamp and an
elastic half-plane when there is wear. J. Appl. Math. Mech. 49, 243–246.
doi: 10.1016/0021-8928(85)90110-8
Kovalenko, E. V. (1985). Study of the axisymmetric contact problem of the wear
of a pair consisting of an annular stamp and a rough half-space. J. Appl. Math.
Mech. 49, 641–647. doi: 10.1016/0021-8928(85)90085-1
Kovalenko, E. V. (2001). “Contact problems for coated solids [in Russian],” in
Mechanics of Contact Interactions, eds I. I. Vorovich and V. M. Aleksandrov
(Moscow: Fizmatli), 459–475.
Kragelsky, I. V. (1965). Friction and Wear. London: Butterworths.
Kragelsky, I. V., Dobychin, M. N., and Kombalov, V. S. (2013). Friction and Wear:
Calculation Methods. Oxford: Pergamon Press.
Lee, C. Y., Tian, L. S., Bae, J. W., and Chai, Y. S. (2009). Application of influence
function method on the fretting wear of tube-to-plate contact. Tribol. Int. 42,
951–957. doi: 10.1016/j.triboint.2009.01.005
Lengiewicz, J., and Stupkiewicz, S. (2013). Efficient model of evolution
of wear in quasi-steady-state sliding contacts. Wear 303, 611–621.
doi: 10.1016/j.wear.2013.03.051
Li, Q. (2016). Limiting profile of axisymmetric indenter due to the initially
displaced dual-motion fretting wear. Fact. Univers. Ser. Mech. Eng. 14, 55–61.
doi: 10.22190/FUME1601055L
Li, Q., Forsbach, F., Schuster, M., Pielsticker, D., and Popov, V. L. (2018).
Wear analysis of a heterogeneous annular cylinder. Lubricants 6:28.
doi: 10.3390/lubricants6010028
Liu, J., Shen, H. M., and Yang, Y. R. (2014). Finite element implementation of a
varied friction model applied to torsional fretting wear. Wear 314, 220–227.
doi: 10.1016/j.wear.2014.01.006
Ma, L., Eom, K., Geringer, J., Jun, T.-S., and Kim, K. (2019). Literature review on
fretting wear and contact mechanics of tribological coatings. Coatings 9:501.
doi: 10.3390/coatings9080501
Mao, X., Liu, W., Ni, Y., and Popov, V. L. (2016). Limiting shape of profile due to
dual-mode fretting wear in contact with an elastomer. Proc. Instit. Mech. Eng.
C J. Mech. Eng. Sci. 230, 1417–1423. doi: 10.1177/0954406215619450
Mary, C., and Fouvry, S. (2007). Numerical prediction of fretting contact durability
using energy wear approach: optimisation of finite-element model. Wear 263,
444–450. doi: 10.1016/j.wear.2007.01.116
McColl, I. R., Ding, J., and Leen, S. B. (2004). Finite element simulation
and experimental validation of fretting wear. Wear 256, 1114–1127.
doi: 10.1016/j.wear.2003.07.001
Meng, H. C., and Ludema, K. C. (1995). Wear models and
predictive equations: their form and content. Wear 181, 443–457.
doi: 10.1016/0043-1648(95)90158-2
Meng, Y., Xu, J., Jin, Z., Prakash, B., and Hu, Y. (2020). A review of recent advances
in tribology. Friction 8, 221–300. doi: 10.1007/s40544-020-0367-2
Mindlin, R. D. (1949). Compliance of elastic bodies in contact. ASME J. Appl. Mech.
16, 259–268.
Mróz, Z., and Stupkiewicz, S. (1994). An anisotropic friction and wear model. Int.
J. Solids Struct. 31, 1113–1131. doi: 10.1016/0020-7683(94)90167-8
Nowell, D. (2010). Simulation of fretting wear in half-plane geometries-Part II:
analysis of the transient wear problem using quadratic programming. J. Tribol.
132:021402. doi: 10.1115/1.4000733
Páczelt, I., and Mróz, Z. (2007). Optimal shapes of contact interfaces due to
sliding wear in the steady relative motion. Int. J. Solids Struct. 44, 895–925.
doi: 10.1016/j.ijsolstr.2006.05.027
Peigney, M. (2004). Simulating wear under cyclic loading by a minimization
approach. Int. J. Solids Struct. 41, 6783–6799. doi: 10.1016/j.ijsolstr.2004.05.022
Popov, V. L. (2014). Analytic solution for the limiting shape of
profiles due to fretting wear. Sci.Rep. 4:3749. doi: 10.1038/srep
03749
Popov, V. L., and Heß, M. (2015). Method of Dimensionality Reduction in
Contact Mechanics and Friction. Heidelberg: Springer. doi: 10.1007/978-3-642-
53876-6
Serre, I., Bonnet, M., and Pradeilles-Duval, R.-M. (2001). Modelling an
abrasive wear experiment by the boundary element method. Comptes
Rendus de l’Académie des Sciences Series IIB Mechanics 329, 803–808.
doi: 10.1016/S1620-7742(01)01402-7
Sfantos, G. K., and Aliabadi, M. H. (2006). Application of BEM and
optimization technique to wear problems. Int. J. Solids Struct. 43, 3626–3642.
doi: 10.1016/j.ijsolstr.2005.09.004
Vázquez, J., Sackfield, A., Hills, D., and Domínguez, J. (2010). The mechanical
behaviour of a symmetrical punch with compound curvature. J. Strain Anal.
Eng. Design 45, 209–222. doi: 10.1243/03093247JSA633
Vingsbo, O., and Söderberg, S. (1988). On fretting maps. Wear 126, 131–147.
doi: 10.1016/0043-1648(88)90134-2
Wang, Z., Jin, X., Keer, L. M., and Wang, Q. (2013). Novel model for partial-
slip contact involving a material with inhomogeneity. J. Tribol. 135:041401.
doi: 10.1115/1.4024548
Wang, Z., Yu, C., and Wang, Q. (2015). An efficient method for solving
three-dimensional fretting contact problems involving multilayered
or functionally graded materials. Int. J. Solids Struct. 66, 46–61.
doi: 10.1016/j.ijsolstr.2015.04.010
Willert, E., Dmitriev, A. I., Psakhie, S. G., and Popov, V. L. (2019). Effect of elastic
grading on fretting wear. Sci. Rep. 9:7791. doi: 10.1038/s41598-019-44269-1
Yen, B. K., and Dharan, C. K. H. (1996). A model for the abrasive
wear of fiber-reinforced polymer composites. Wear 195, 123–127.
doi: 10.1016/0043-1648(95)06804-X
Yue, T., and Wahab, M. A. (2019). A review on fretting wear mechanisms,
models and numerical analyses. Comput. Mater. Contin. 59, 405–432.
doi: 10.32604/cmc.2019.04253
Zhang, Y., Kovalev, A., and Meng, Y. (2018). Combined effect of boundary layer
formation and surface smoothing on friction and wear rate of lubricated
point contacts during normal running-in processes. Friction 6, 274–288.
doi: 10.1007/s40544-018-0228-4
Zmitrowicz, A. (2006). Wear patterns and laws of wear-a review. J. Theor. Appl.
Mech. 44, 219–253.
Conflict of Interest: The authors declare that the research was conducted in the
absence of any commercial or financial relationships that could be construed as a
potential conflict of interest.
Copyright © 2020 Argatov and Chai. This is an open-access article distributed
under the terms of the Creative Commons Attribution License (CC BY). The use,
distribution or reproduction in other forums is permitted, provided the original
author(s) and the copyright owner(s) are credited and that the original publication
in this journal is cited, in accordance with accepted academic practice. No use,
distribution or reproduction is permitted which does not comply with these terms.
Frontiers in Mechanical Engineering | www.frontiersin.org 10 July 2020 | Volume 6 | Article 51
