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Procedia Structural Integrity 41 (2022) 220–231
2nd Mediterranean Conference on Fracture and Structural Integrity
Modelling of delamination in rolling and sliding contacts
Irina Goryachevaa,b,*, Almira Meshcheryakovaa,b†
aa
Ishlinsky
Ishlinsky Institute for Problems in Mechanics RAS, Prospekt Vernadskogo 101-1, Moscow, 119526, Russia
b
bSirius University of Science and Technology, 1 Olympic Ave, 354340, Sochi, Russia
Abstract
The contact fatigue phenomenon in conditions of cyclic sliding or rolling is studied based on the model of contact fatigue damage
accumulation in the subsurface layers of the material. It is assumed that the rate of the damage accumulation depends on the
principal shear stress amplitudes. The results of numerical modelling show that the contact fatigue fracture process in the conditions
of a constant load, acting on the sliding or rolling body, consists of the following stages: the incubation period, the running-in
period with the alternating detachment of material’s fragments of certain thickness (the mechanism of delamination) and the surface
wear, and then the steady-state stage, characterizing by the surface wear with a constant rate. The influence of the sliding friction
coefficient, the relative slippage and the strength properties of contacting bodies on the evolution of the accumulated damage in
the surface layers and on the fatigue wear kinetics in sliding and rolling contacts has been analyzed.
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Peer-review under responsibility of the MedFract2Guest Editors.
Peer-review under responsibility of the MedFract2Guest Editors.
Keywords: rolling contact, sliding contact, internal stresses, wear, delamination
1. Introduction
Fatigue wear is one of the most common mechanisms of fracture of machine components operating under sliding
or rolling friction and cyclic loading and can lead to the breakdown of tribocouplings. In fatigue wear the multiple
detachment of material particles from the friction surface occurs, and the corresponding change of contact geometry
during each fracture results in the redistribution of contact and internal stresses (Goryacheva, 1998). Modelling of this
* Corresponding author. Tel.: +7 495 434 3692.
E-mail address: goryache@ipmnet.ru
2452-3216 © 2022 The Authors. Published by ELSEVIER B.V.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the MedFract2Guest Editors.
2452-3216 © 2022 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the MedFract2Guest Editors.
10.1016/j.prostr.2022.05.025
2
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process consists of the following stages: the calculation of contact and internal stresses, analysis of the accumulated
damage in the surface layer of the material based on the appropriate dependence of the damage function on the internal
stress distribution, the calculation of the thickness of the detached fragment, taking into account the chosen material
failure criterion, the calculation of accumulated damage at the next step, etc. (Goryacheva, 1998; Goryacheva and
Chekina, 1999). At each step of wear modelling, the history of the process is taken into account, i.e. accumulated
damage in previous stages. As the main parameters affecting the wear rate, there are contact conditions, macro and
microgeometry of the surfaces of contacting bodies, strength properties of materials, residual stresses, etc.
Due to the fact that at the dominant number of cases, the contact fatigue is the main cause of the rail failure,
the accumulation of contact fatigue damage in wheel/rail contacts is considered in many studies. The review of rolling
contact fatigue defects in wheel/rail systems and the mechanisms of their formation is presented in (Magel, 2011).
The experimental studies of the influence of relative slippage on the contact fatigue damage accumulation and wear
in rolling contact of wheel and rail are described in (Guo et al., 2016; Pal et al., 2012; Zhang et al., 2022; Zhou et al.,
2016). The results of these studies show that the increase of the relative slippage leads to the evolution of the wear
mechanism from oxidation wear and local delamination to abrasive wear. In the tests carried out in (Hu et al., 2020),
in addition to the relative slippage, the influence of the ratio of the hardnesses of the wheel and rail materials on the
damage accumulation and wear was studied. A method of construction of the contact fatigue damage function
accumulated in the material for given loading conditions based on the two-disc testing results was developed in
(Hiensch and Burgelman, 2019). The rolling contact problem taking into account the relative slippage was solved
there using the FASTSIM (Kalker, 1982) and CONTACT (Kalker, 1990) algorithms. The twin disc tests were
conducted in (Santa et al., 2019) to study the fatigue wear in rails. As a result, the dependences of the wear rate on the
relative slippage for wheel and rail were obtained, and several wear regimes were identified depending on the contact
conditions. The method for calculation of the accumulated damage in a rail was developed in (Bernal et al., 2022),
taking into account the change in contact characteristics depending on the relative slippage and mechanical properties
of the materials of the wheel and rail, measured in the experiments. The results of experiments and field tests,
combined with the results of simulation of contact fatigue damage accumulation, can be used for the rail life
predictions.
The modelling of fatigue damage accumulation in material in rolling with friction of a cylinder over an elastic halfspace (contact problem in a plane formulation) was carried out in (Goryacheva and Torskaya, 2019), where the effect
of the relative slippage, the sliding friction coefficient, and residual stresses on the stress state of the subsurface layers
of the contacting bodies was studied. The results of the study show that the presence of residual stresses that occur,
for example, in the materials of a rail due to heat treatment of the rail surface, leads to an increase of the principal
shear stresses and a decrease of their amplitude values.
The results of wear modelling in wheel/rail system in rolling with slippage using the Archard model are presented
in (Sakalo et al., 2019), where the Dang Wang function (Dang Van et al., 1989), the equivalent Mises stresses and
principal shear stresses amplitudes were used in criteria of the material failure. Numerical calculations using the finite
element method make it possible to compare the results of wear simulation for different material failure criteria, taking
into account the shape of the contact region and the distributions of contact pressure and shear stress. The multiscale
finite element model is proposed in (Daves et al., 2016) to predict the initiation of contact fatigue cracks and the
formation of wear particles in the wheel/rail contact under sliding and rolling conditions, taking into account the
surface roughness. A review of empirical and analytical approaches for modelling the fatigue damage in rolling
bearings is given in (Sadeghi et al., 2009).
In this study, based on a single mechanism for the fatigue damage accumulation in the subsurface layers of materials
of interacting bodies, the surface wear and the detachment of material fragments of finite thickness (delamination) of
elastic bodies under the conditions of their cyclic interaction with a sliding or rolling ball are modelled. The influence
of the sliding friction coefficient (under conditions of full sliding), as well as the sliding friction coefficient and relative
slippage (under rolling conditions) on the stress state of subsurface layers of elastic bodies, the accumulation of fatigue
damage there, and the kinetics of surface (wear) and subsurface (delamination) fracture in the materials of contacting
bodies are studied.
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3
Nomenclature

E
V
ω
P
Ω
p
pH
a
s
τ
µ
∆
σ1
σ3
Q
Q0
Q*
q
τmax
∆τmax
N
c
m
P
a
pH
Poisson’s ratio
Young's modulus
linear velocity
angular velocity
load
contact region
contact pressure
maximum value of contact pressure
radius of contact region
slip velocity
shear stress
sliding friction coefficient
relative slippage
maximum of principal value of stress tensor
minimum of principal value of stress tensor
accumulated damage
initial accumulated damage
critical value of accumulated damage
rate of damage accumulation
principal shear stress
principal shear stresses amplitude
number of cycles
strength parameter (coefficient)
strength parameter (exponent)
dimensionless load
dimensionless radius of contact region
dimensionless maximum value of contact pressure
2. Problem formulation
The contact of an elastic spherical body (ball) sliding/rolling over an elastic half-space made of the same material
in conditions of cyclic interaction is considered. The constant load P is applied to the ball. The movement of the ball
with the constant linear velocity V is described as steady relative to the moving coordinate system Oxyz, which origin
is located under the center of the ball. The contact scheme is presented in Fig. 1.
Fig. 1. Contact scheme of an elastic ball rolling over an elastic half-space
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223
Fixed coordinate system O’x’y’z’ is associated with the moving coordinate system Oxyz by the following relations:
x = x + Vt , y  = y , z  = z
(1)
In rolling contact the relative slip velocities within the contact region  are determined by the formula:
s ( x, y=
)

(x
2R
2
 u ( x, y ) u2 ( x, y )

+ y2 ) −V  1
−
−  ,
x
x


(2)
where  is the angular velocity of the ball rotation in respect to the y -axis (Fig. 1), ui ( x, y ) are the elastic
displacements of the contacting bodies (i = 1, 2) in the direction of x -axis due to deformation, R is the radius of the
ball, Δ is the relative slippage:
V − R
.
=
V
(3)
In rolling the contact region consists of the slip and stick subregions. In the stick subregion the shear stress  ( x, y ) is
related to the contact pressure p( x, y ) by the following inequality:
 ( x, y )   p ( x, y ) .
(4)
and the slip velocity s ( x, y ) (Eq. 2) is equal to zero:
s ( x, y ) = 0 .
(5)
In the slip subregion in rolling and in sliding contacts the Coulomb-Amonton law is satisfied:
 ( x, y ) =  p ( x, y ) .
(6)
Outside the contact region, the normal and shear stresses are equal to zero. In cyclic interaction of the system of
spherical indenters and the elastic half-space the mutual effect of indenters deforming the elastic half-space is
neglected.
3. Method of solution
The fatigue wear model is developed here based on a macroscopic approach, in which damage at some point of the
material is described by a positive nondecreasing function that depends on the stress state at this point, the number of
passed cycles, and parameters describing the strength properties of the material.
3.1. Calculation of contact and internal stresses
The stress state of an elastic half-space is determined by its elastic characteristics, contact pressure and shear
stresses. Since the materials of the contacting bodies are considered the same, shear stresses do not influence the
distribution of a contact pressure. Therefore, the problem of the contact normal and shear stress calculation is solved
in two stages: first, the contact pressure is calculated, and then the shear stress is identified under sliding and rolling
conditions.
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The contact pressure p( x, y ) is calculated from the Hertz theory:
1
 x2 + y 2  2
p(=
x, y ) p H  1 −

a2 


3PE 2
pH = 
 2 3 R 2 1 − 2 2
( )

(7)
1
(
3
 3PR 1 − 2
 , a=


2E


) 
1
3


(8)
Here a is the contact radius, E ,  are the elastic moduli, pH is the maximum value of contact pressure, R is the
ball radius.
The shear stress  ( x, y ) in the contact region is calculated using a numerical-analytical approach based on
the variational method (Goldstein et al., 1982; Meshcheryakova and Goryacheva, 2021) which leads to minimization
of the following functional:
=
F  , s ( ) 
 (  p ( x, y ) s ( ( x, y ) ) −  ( x, y ) s ( ( x, y )) )dxdy ,
(9)

where the slip velocity function is expressed as a function of shear stresses (Goldstein et al., 1982). The internal
stresses within the elastic half-space are calculated using the distributions of contact normal and shear stresses and the
Boussinesq and Cerruti potentials (Johnson, 1985).
3.2. Modelling of damage accumulation and fatigue wear
We assume that the cyclic loading of the elastic half-space is provided by identical rollers located at a sufficiently
large distance from each other (which allows us to neglect their mutual influence on the distribution of contact and
internal stresses), and the rate q ( z ) of damage accumulation at the depth z depends on the principal shear stress
amplitudes  max ( z ) and is calculated by the formula (Collins, 1981):
m
Q ( z, N )
  ( z ) 
=
q ( z ) = c  max
 .
N
pH


(10)
Here Q ( z , N ) is the damage accumulated over N cycles at the depth z, с and m are the parameters which describe the
strength properties of the material, and they are determined experimentally.
The following relation is used to calculate the principal shear stress amplitude:
1

 max ( z ) = max  ( 1 ( x, y, z ) −  3 ( x, y, z ) )  ,
x, y
2


(11)
where  1 ( x, y, z ) and  3 ( x, y, z ) are the maximum and minimum values of the principal stresses at the point with
coordinates ( x, y, z ) . Since the rollers are located at a large distance from each other, the principal stresses amplitudes
coincide with their maximum values at a fixed depth.
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The damage Q ( z , N ) accumulated over the first N cycles at the depth z is calculated by the formula:
m
  ( z ) 
Q ( z, N ) Nc  max
=
 + Q0 ( z ) ,
pH


(12)
where Q0 ( z ) is the initial damage at the depth z, which is assumed to depend only on the coordinate z.
When the damage function reaches a critical value Q* at some depth z * , the fracture of the material occurs, which
involves the separation from the surface layer thickness z * :
Q ( z*, N * ) = Q* .
(13)
Here N * is the number of cycles until the first delamination occurs. The remaining part of the half-space z  z * with
Q( z , N *) for z  z * comes into contact and becomes the initial damage
the accumulated damage Q1 ( z − z*) =
function for the next stage calculation of the accumulated damage ( ( N − N *)  0 ) following the procedure, described
above.
Thus, the damage function significantly depends on the principal shear stress distribution under the ball.
4. Analysis of principal shear stress amplitude values in sliding and rolling contacts
The stress state of elastic bodies in sliding contact is determined by the following parameters: the elastic
characteristics of the material (the elastic modulus E , the Poisson ratio  ), the geometry of the contacting surfaces
(the radius of the indenter R ) and the acting load P , linear sliding velocity V , as well as the sliding friction
coefficient  . In addition in rolling contact, an important parameter that affects the stress state of the contacting
bodies and the fatigue damage accumulation is the relative slippage (3), which is confirmed by the results of
experimental studies (Guo et al., 2016; Pal et al., 2012; Zhang et al., 2022; Zhou et al., 2016).
In this study, the elastic constants of materials, the geometry of the contacting bodies and the total load are
considered fixed, so the following dimensionless parameters do not change in calculations:
=
P
P
= 0.00001 ,  = 0.3 ,
ER 2
(14)
where P is the dimensionless load.
For these parameters the dimensionless radius a of the contact region and the dimensionless maximum value
of contact pressure pH have the following values:
=
a
a
= 0.024 , p=
H
R
pH
= 0.008 .
E
(15)
The dependences of the principal shear stresses amplitudes on depth for a sphere sliding over an elastic half-space
with different sliding friction coefficients are presented in Fig. 2.
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Fig. 2. Dependences of the principal shear stress amplitude on depth in sliding contact for different values of sliding friction coefficient
The results indicate that the distribution of the principal shear stress amplitudes on the depth in the elastic halfspace under the sliding body has two maxima: at the surface and below the surface. For the selected values of
the dimensionless parameters (14) and the sliding friction coefficient not exceeding 0.2, the surface maximum of
the principal shear stresses amplitude function is less than the subsurface maximum. With an increase in the sliding
friction coefficient (for   0.2 ), the surface maximum of the function  ( z ) becomes greater than the subsurface
maximum of this function.
In rolling contact there are two main parameters which influence essentially on the dependence of the principal
shear stress amplitude on the depth: they are the relative slippage and the sliding friction coefficient. Fig. 3 illustrates
the influence of the relative slippage and the sliding friction coefficient on the distribution of the principal shear
stresses amplitudes.
Fig. 3. Dependencies of the principal shear stress amplitude on depth in rolling contact for different values of relative slippage and sliding friction
coefficient
The results illustrate that in the rolling contact a change in the relative slippage significantly affects the surface
maximum value of principal shear stresses amplitudes, while the subsurface maximum value practically does not
change. At low values of the relative slippage, the surface maximum of principal shear stress amplitudes is less than
the subsurface one. With an increase in the relative slippage up to a value corresponding to the total slip in the contact
region (  =0.0045) , the surface and subsurface maxima values become comparable. At the same value of relative
slippage, the increase in the sliding friction coefficient corresponds to the increase in the surface and subsurface
maxima of principal shear stresses amplitudes, and the surface maximum changes significantly.
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5. Analysis of damage accumulation and wear kinetics
The accumulated damage function and its evolution during the process of the fracture of subsurface layers of the
material were calculated using the relations (12) and (13). In addition to the parameters that affect the stress state of
the contacting bodies under the conditions of sliding and rolling friction mentioned in Section 4, the strength
characteristics of the material (coefficient c and exponent m in (10)) influence the fatigue damage accumulation. In
calculations the dimensionless damage value Q / Q * , which depends only on the parameter m, is used:
m
Q ( z, N )
Q( z , N ) N   max ( z ) 
 + 0 *
= *
(16)
Q*
N   max ( z * ) 
Q


So, for N  N * the surface or subsurface fracture (delamination) occurs if the value of the right-hand side of Eq. (16)
reaches 1.
For study of fatigue damage accumulation and wear, a numerical analysis of the damage function (16) was carried
out. In Fig. 4 the evolution of the damage function (curves 1-6) in sliding contact is presented, where each curve
corresponds to a number of cycles passed until the damage function reaches the critical value at the surface ( z = 0)
or under the surface ( z  0 ) of the elastic half-space.
Fig. 4. Accumulated damage for sliding contact, sliding friction coefficient μ = 0.2 and m = 4.8
After the first case of subsurface fracture (curve 1), the damage function is a monotonically decreasing function
with a maximum on the surface (curves 2 and 3). Then, with an increase in the number of cycles, there is an inflection
of the function curve and the next act of subsurface fracture occurs (curve 4). After that, the damage function again
takes the form of a monotonically decreasing (with the depth) function corresponding to surface wear (curves 5 and 6).
5.1. Sliding contact
In sliding contact the effect of the sliding friction coefficient  and the strength properties of the material,
describing by the parameter m in Eq. (16), on the damage accumulation in the elastic half-space, i.e. the function
Q( z, N ) / Q* (16), is studied.
Damage functions for different number of cycles (Fig. 5a) and the kinetics of wear (Fig. 5b) in the sliding contact
for sliding friction coefficient µ = 0.2 and different values of parameter m are shown in Fig. 5.
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Fig. 5. Accumulated damage and wear kinetics in sliding contact with sliding friction coefficient μ = 0.2 for different values of strength
parameter m
The analysis of the results shows that in sliding contact at the sliding friction coefficient µ = 0.2 and parameter
m = 3 after several acts of delamination the surface wear with a constant rate occurs. With the increase of the parameter
m periods of surface wear alternate with acts of subsurface fracture (delamination). The duration (number of cycles)
of the surface wear before each act of subsurface fraction (delamination) grows with increase of the parameter m.
The accumulated damage function and the kinetics of wear in sliding contact at the sliding friction coefficient
µ = 0.4 and different values of the parameter m are presented in Fig. 6.
Fig. 6. Accumulated damage and wear kinetics for sliding contact, sliding friction coefficient μ = 0.4 and different values of strength parameter m
The results indicate that in sliding with the sliding friction coefficient µ = 0.4 surface wear is the dominant wear
behavior. The results illustrate that after a certain number of cycles, the wear rate reaches a constant value. The number
of cycles up to the regime of steady surface wear increases with the growth of the parameter m, while the rate of
surface wear decreases.
The results presented in Fig. 7, allow us to compare the wear kinetics for different values of the parameter m and
the sliding friction coefficient µ under conditions of sliding friction.
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Fig. 7. Wear kinetics for sliding contact and different values of sliding friction coefficient μ and strength parameter m
The analysis of the wear kinetics for different values of the sliding friction coefficient and the strength properties
of the material shows that at a large value of the sliding friction coefficient (  = 0.4 ) , surface wear prevails. The
surface wear rate for a constant sliding friction coefficient depends on the strength properties of the material, given
by the parameter m. At low values of the sliding friction coefficient, the wear is accompanied by periodic separation
of fragments of the base material of finite thickness; the duration of the regime before the only surface wear takes
place, increases with the growth of the parameter m .
5.2. Rolling contact
Under conditions of rolling friction, the analysis of the relative slippage effect is of great interest. The study of
the principal shear stress distribution developed in Section 4 shows that a change in the relative slippage affects
the values of the surface and subsurface maxima of principal shear stress amplitudes and, accordingly, affects
the fatigue damage accumulation features within the contacting bodies.
Damage functions and the kinetics of wear in rolling contact for sliding friction coefficient  = 0.3 and different
values of the relative slippage are presented in Fig. 8.
Fig. 8. Accumulated damage and wear kinetics in rolling contact with sliding friction coefficient μ = 0.3, strength parameter m = 4 and different
relative slippages
The obtained results illustrate that in rolling contact two types of fatigue wear occur: the subsurface fracture
(delamination) and the surface wear. The rate and the duration of surface wear and the number of delamination acts,
as well as the thickness of separated fragments of the material depend on the relative slippage.
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The curves of the wear kinetics in conditions of rolling friction for different values of relative slippage are presented
in Fig. 9.
Fig. 9. Wear kinetics for rolling contact with μ = 0.3, m = 4 and different values of relative slippage
The results indicate that in rolling contact the surface wear alternates with acts of subsurface fracture
(delamination), and the rate of surface wear is constant and depends on the relative slippage. For sliding friction
coefficient equals to 0.3 and large value of relative slippage (Δ = 0.0045), which corresponds to the complete slip in
the contact region a single act of delamination occurs, after which, with an increase in the number of cycles, a regime
of surface wear with a constant rate is established.
6. Conclusion
The paper presents a model, developed to describe the fatigue damage accumulation and surface and subsurface
(delamination) wear under conditions of frictional interaction of elastic bodies. The sliding and rolling contacts of
elastic bodies were considered, and the sliding friction coefficient, the relative slippage and the strength properties of
the material of elastic bodies were used as variable parameters.
The obtained results illustrate the evolution of the accumulated damage function with an increasing number of
cycles, as well as the change of the wear behavior from acts of subsurface fracture (delamination) to continuous surface
wear with a constant rate. It is shown that the surface wear dominates in sliding contact for the sliding friction
coefficient equals to 0.4. At low values of the sliding friction coefficient, there is an alternation of subsurface fracture
(delamination) and surface wear. In rolling contact of elastic bodies, the wear rate increases with the growth of
the relative slippage, which is accompanied by a change of wear behavior from subsurface to surface.
The results of the study are used to analyze the influence of the mechanical and strength characteristics of materials
of interacting bodies, as well as the sliding friction coefficient under conditions of sliding friction and additionally
relative slippage in rolling contact on fatigue wear features.
Acknowledgements
The reported study was funded by RFBR, Sirius University of Science and Technology, JSC Russian Railways and
Educational Fund “Talent and success”, project number 20-38-51005.
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