Citation: Willert, E. Influence of
Profile Geometry on Frictional
Energy Dissipation in a Dry,
Compliant Steel-on-Steel Fretting
Contact: Macroscopic Modeling and
Experiment. Machines 2023,11, 484.
https://doi.org/10.3390/
machines11040484
Academic Editor: Sheng Li
Received: 24 March 2023
Revised: 13 April 2023
Accepted: 16 April 2023
Published: 18 April 2023
Copyright: © 2023 by the author.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
machines
Article
Influence of Profile Geometry on Frictional Energy Dissipation
in a Dry, Compliant Steel-on-Steel Fretting Contact:
Macroscopic Modeling and Experiment
Emanuel Willert
Institute of Mechanics, Technische Universität Berlin, Sekr. C8-4, Straße des 17. Juni 135, 10623 Berlin, Germany;
Abstract:
Dry, frictional steel-on-steel contacts under small-scale oscillations are considered experi-
mentally and theoretically. As indenting bodies, spheres, and truncated spheres are used to retrace
the transition from smooth to sharp contact profile geometries. The experimental apparatus is built
as a compliant setup, with the characteristic macroscopic values of stiffness being comparable to
or smaller than the contact stiffness of the fretting contact. A hybrid macroscopic–contact model is
formulated to predict the time development of the macroscopic contact quantities (forces and global
relative surface displacements), which are measured in the experiments. The model is well able to
predict the macroscopic behavior and, accordingly, the frictional hysteretic losses observed in the
experiment. The change of the indenter profile from spherical to truncated spherical “pushes” the
fretting contact towards the sliding regime if the nominal normal force and tangential displacement
oscillation amplitude are kept constant. The transition of the hysteretic behavior, depending on the
profile geometry from the perfectly spherical to the sharp flat-punch profile, occurs for the truncated
spherical indenter within a small margin of the radius of its flat face. Already for a flat face radius
which is roughly equal to the contact radius for the spherical case, the macroscopic hysteretic behav-
ior cannot be distinguished from a flat punch contact with the same radius. The compliance of the
apparatus (i.e., the macrosystem) can have a large influence on the energy dissipation and the fretting
regime. Below a critical value for the stiffness, the fretting contact exhibits a sharp transition to the
“sticking” regime. However, if the apparatus stiffness is large enough, the hysteretic behavior can be
controlled by changing the profile geometry.
Keywords: elastic contact; friction; fretting; fretting regime; hysteresis; profile geometry
1. Introduction
Fretting is a widespread source of surface damage and, possibly, structural failure
in many tribological contacts, which are subject to oscillating loads, e.g., in turbine blade
roots [
1
] or artificial joints [
2
]. The physical (or chemical) nature and mechanisms of the
tribological damage associated with fretting can be very diverse; in dry contacts of corrosion-
resistant materials, the main sources of damage are (fretting) wear and (fretting) fatigue [
3
].
Based on so-called “fretting loops”—i.e., hysteresis loops between contact forces and
macroscopic relative surface displacements—one can distinguish different fretting regimes:
the partial slip regime with narrow hysteresis, the sliding regime with a broad (round or
rectangular) hysteresis, and a mixed regime in between [4].
Although wear and fatigue in fretting contacts are separate (albeit interacting or even
competing [
5
]) damage phenomena, in many cases, the partial slip regime is dominated by
fatigue (due to the oscillating stress singularity at the edge of the permanent stick area),
while the dominant phenomenon in the sliding regime is wear [6].
From a contact mechanical perspective, it is clear that the fretting regime (and, thus,
the dominating damage mechanisms) can be heavily influenced by the contact profile, i.e.,
Machines 2023,11, 484. https://doi.org/10.3390/machines11040484 https://www.mdpi.com/journal/machines
Machines 2023,11, 484 2 of 13
the geometrical form of the gap between the contacting surfaces in the undeformed state of
first contact [
7
]. In that regard, the two main profile geometries that have been analyzed
in the context of fretting are the parabolic (or spherical) (see, e.g., [
8
–
10
]) and the rounded
flat punch profiles ([
11
,
12
])—which is due to the fact the these are the most common in
engineering contacts that suffer from small oscillations.
Nonetheless, contact profile geometry as an influencing factor and, possibly, control
variable in fretting has rarely been considered explicitly in the literature. Kim & Lee [
13
]
theoretically studied the influence of the geometrical parameters in a plane rounded flat
punch contact on the elastic edge crack propagation for the analysis of fretting fatigue
failure. Bartha et al. [
14
] considered the influence of profile geometry on fatigue crack
nucleation and propagation in an elastic fretting contact, validating their theoretical model
with experimental results. Gallego et al. [
15
] systematically analyzed the wear-contact
problem in fretting and evaluated the possibility of finding a corresponding optimal contact
geometry. Warmuth et al. [
16
] experimentally studied the influence of the parabolic radius
on the wear behavior in a steel-cylinder-on-steel-flat fretting contact. Zhang et al. [
17
]
numerically compared the parabolic and rounded flat punch geometries in terms of the
corresponding fretting behavior. Finally, Argatov & Chai [
18
] searched for the wear-optimal
version of a symmetrical punch with compound curvature, which creates an almost constant
contact pressure distribution in plane elastic contact.
Note that a related but slightly different question regards the influence of wear-induced
profile changes on especially fretting fatigue. This problem, as an aspect of the complex
interplay between fretting wear and fretting fatigue, has received a lot of attention in recent
years (see, e.g., [19–21]).
Theoretically speaking, the most important distinction with respect to the macroscopic
contact profile geometry, i.e., not considering roughness scales, is the one between (macro-
scopically) smooth profiles (e.g., paraboloids) and profiles with sharp edges (e.g., the flat
punch). As smooth profiles exhibit significant partial slip (propagating from the contact
boundary) but no stress singularities, they should be prone to wear rather than to fatigue,
while sharp edges can inhibit slip at the cost of introducing stress singularities, which
should make such profiles prone to fatigue rather than to wear.
To retrace the transition from smooth to sharp profile geometries—and, thus, to
possibly overcome the wear-fatigue dilemma in fretting—very recently [
22
], the truncated
parabolic geometry (which is also very easily realized experimentally) has been suggested
as an interesting object of study in the context of fretting contacts. Changing the radius (or
half-width) of the flat face of a truncated spherical (or cylindrical) profile geometry, one
might be able to tune the transition from smooth to sharp contact gaps to combine “the
best of both worlds”.
Based on the fundamental scientific principle of separating different aspects or parts
of a system or complex phenomenon, with the purpose of studying them individually, the
“academic” analysis of fretting is usually concerned with laboratory contacts, which are
not part of a larger macrosystem anymore (or the influence of the macrosystem is kept
negligible). Accordingly, apparatuses for the experimental analysis of fretting contacts
are usually constructed to be much stiffer than the contact itself, although it is known
that system inertia and compliance can significantly influence tribological properties, e.g.,
wear rates [
23
]. While this approach is, of course, and undoubtedly, necessary for the
understanding of the complex processes and mechanisms active in fretting contacts, it also
discards the properties of the macrosystem as a possible control variable of the damage
resulting from the tribological interaction. In contrast, for the analysis documented in
the present manuscript, the experimental apparatus was built as a compliant setup, with
the characteristic macroscopic values of stiffness being comparable to or smaller than the
contact stiffness of the fretting contact that is to be studied during the analysis.
The aim of the analysis is, therefore, to study the influence of the profile geometry
in fretting for a generic steel-on-steel oscillating laboratory contact, accounting for the
mechanical properties (stiffness) of the macrosystem. Before studying local phenomena,
Machines 2023,11, 484 3 of 13
like wear or fatigue, the fretting regime shall be characterized based on the hysteretic (i.e.,
macroscopic) behavior, which is due to the frictional energy dissipation. A macroscopic
model shall be built and validated, which can later be used as a component of a multiscale
model for the analysis of the local phenomena.
The remainder of this manuscript is organized as follows: In Section 2, the experi-
mental setup for the analysis is described in detail. Section 3gives a hybrid macroscopic-
contact-model that was used to predict the time development of the macroscopic contact
quantities (forces and global relative surface displacements), which are measured in the ex-
periments. In Section 4, the theoretical predictions are compared to experimental results to
substantiate the model. After that, the model is used to analyze the influence of the profile
geometry on the frictional energy dissipation, especially considering the high compliance
of the experimental apparatus (which can have a significant qualitative impact, as will be
demonstrated) in Section 5. A discussion of the methodology used and some conclusive
remarks finish the manuscript.
2. Experimental Setup
A photograph of the experimental setup is shown in Figure 1. For a clearer under-
standing, a reduced scheme of the setup is given in Figure 2.
Machines 2023, 11, x FOR PEER REVIEW 3 of 13
The aim of the analysis is, therefore, to study the influence of the profile geometry in
fretting for a generic steel-on-steel oscillating laboratory contact, accounting for the me-
chanical properties (stiffness) of the macrosystem. Before studying local phenomena, like
wear or fatigue, the fretting regime shall be characterized based on the hysteretic (i.e.,
macroscopic) behavior, which is due to the frictional energy dissipation. A macroscopic
model shall be built and validated, which can later be used as a component of a multiscale
model for the analysis of the local phenomena.
The remainder of this manuscript is organized as follows: In Section 2, the experi-
mental setup for the analysis is described in detail. Section 3 gives a hybrid macroscopic-
contact-model that was used to predict the time development of the macroscopic contact
quantities (forces and global relative surface displacements), which are measured in the
experiments. In Section 4, the theoretical predictions are compared to experimental results
to substantiate the model. After that, the model is used to analyze the influence of the
profile geometry on the frictional energy dissipation, especially considering the high com-
pliance of the experimental apparatus (which can have a significant qualitative impact, as
will be demonstrated) in Section 5. A discussion of the methodology used and some con-
clusive remarks finish the manuscript.
2. Experimental Setup
A photograph of the experimental setup is shown in Figure 1. For a clearer under-
standing, a reduced scheme of the setup is given in Figure 2.
Figure 1. Photograph of the experimental setup.
Figure 2. Scheme of the experimental setup with linear stage (LS), force sensor (FS), piezo actuator
(PA), and laser vibrometers (LVMs).
Figure 1. Photograph of the experimental setup.
Machines 2023, 11, x FOR PEER REVIEW 3 of 13
The aim of the analysis is, therefore, to study the influence of the profile geometry in
fretting for a generic steel-on-steel oscillating laboratory contact, accounting for the me-
chanical properties (stiffness) of the macrosystem. Before studying local phenomena, like
wear or fatigue, the fretting regime shall be characterized based on the hysteretic (i.e.,
macroscopic) behavior, which is due to the frictional energy dissipation. A macroscopic
model shall be built and validated, which can later be used as a component of a multiscale
model for the analysis of the local phenomena.
The remainder of this manuscript is organized as follows: In Section 2, the experi-
mental setup for the analysis is described in detail. Section 3 gives a hybrid macroscopic-
contact-model that was used to predict the time development of the macroscopic contact
quantities (forces and global relative surface displacements), which are measured in the
experiments. In Section 4, the theoretical predictions are compared to experimental results
to substantiate the model. After that, the model is used to analyze the influence of the
profile geometry on the frictional energy dissipation, especially considering the high com-
pliance of the experimental apparatus (which can have a significant qualitative impact, as
will be demonstrated) in Section 5. A discussion of the methodology used and some con-
clusive remarks finish the manuscript.
2. Experimental Setup
A photograph of the experimental setup is shown in Figure 1. For a clearer under-
standing, a reduced scheme of the setup is given in Figure 2.
Figure 1. Photograph of the experimental setup.
Figure 2. Scheme of the experimental setup with linear stage (LS), force sensor (FS), piezo actuator
(PA), and laser vibrometers (LVMs).
Figure 2.
Scheme of the experimental setup with linear stage (LS), force sensor (FS), piezo actuator
(PA), and laser vibrometers (LVMs).
A flat steel block (stainless steel X5CrNi18-10) rests on a three-point bedding of silicon
nitride half-spheres, arranged as a regular triangle with the center of gravity below the
center of gravity of the block. The point of first contact with the steel indenter (also stainless
steel X5CrNi18-10, to ensure elastic similarity with the flat) lies above the center of gravity of
the block. The normal axis of the apparatus consists of a 3D force sensor (Kistler 9317C, with
Machines 2023,11, 484 4 of 13
charge amplifier Kistler 5167A) and a linear stage (PI M-403.2DG, with controller PI C-863
Mercury). The force sensor is oriented in such a way that the sensor’s normal axis—along
which the sensor sensitivity is worse than along the sensor’s tangential
axes—coincides
with the lateral axis of the contact so that the most relevant contact forces (in the contact’s
normal and tangential directions) are measured with the highest possible sensitivity. This
setup also leads to a relatively high tangential compliance of this axis of the apparatus.
In the tangential direction (from left to right in Figures 1and 2), the steel block is
connected to a custom linear piezo actuator via a bridge of two thin brass plates—that
is supposed to act as a parallel motion. The tangential displacements of the block and
the indenter foundation are measured by two laser vibrometers (OFV-505 and OFV-503,
with controllers OFV-5000). The data from the charge amplifier and the laser vibrometers
are collected as analog input from a measuring board (NI USB-6353). The whole setup is
controlled with LabVIEW scripts.
Before every experiment, the contact between the indenter and the flat steel block,
as well as the contacts between the block and the silicon nitride half-spheres, are cleaned
(the same bodies are used again for all experiments, as only a few dozen oscillations are
performed in each experiment). Moreover, the contacts between the silicon nitride half-
spheres and the flat block are lubricated with a drop of standard lubrication oil to reduce
the corresponding coefficient of friction. Note that the research interest solely lies in the
(dry) steel-on-steel contact between the indenter and the flat block.
For the experiments, two types of indenter profiles are used. First, the experiments
are performed with full spheres. After that, the indenter profile is truncated, and a flat
planform is introduced mechanically, in parallel to the contact plane.
The rms roughness of the original spherical steel indenter is approximately 1
µ
m, and
the rms roughness of the flat steel block is approximately 3
µ
m. The rms roughness of the
flat face of the truncated spherical indenter after the common turning process to generate
the flat face is approximately 2
µ
m, so it is decided that no surface finish of the flat face
is necessary. The tilting angle of the flat face with respect to the contact plane is less than
π
/3600. The width (in theradial direction) of the transition region from the flat part to the
spherical part of the truncated indenter is less than the rms roughness of the surfaces.
3. Hybrid Macroscopic-Contact Model
As the energy dissipation in the fretting contact (as well as the fretting regime) can
be determined from the hysteresis loops between contact forces and global displacements
(i.e., macroscopic quantities), a hybrid macroscopic model was constructed that considers
the equilibrium of the flat block and the indenter, and which is shown in Figure 3. As the
frequency of the piezo actuator oscillation (f= 20 Hz) is much smaller than the smallest
eigenfrequencies of the block or the normal axis, inertial forces are neglected. In other
words, the motion of the apparatus is considered in the quasi-static limit. Moreover, elastic
deformations of the block as a whole are neglected. The brass plates are modeled as bent
beams within the framework of the Euler–Bernoulli theory. All deformations are assumed
to be small and within the elastic range.
The hybrid character of the model stems from the fact that all contact interactions
(between the indenter and the block, as well as between the block and the silicon nitride
bedding) are modeled within subroutines based on the method of dimensionality reduction
(MDR) [
24
]. The MDR description of elastic tangential contact with friction operates within
the framework of the Hertz–Mindlin-approximation ([25,26]), specifically:
•
The characteristic length of the contact area is small compared to all linear dimen-
sions of the contacting bodies; the gradients of the deformed surfaces in the vicinity
of the contact are small (together, these two assumptions constitute the so-called
“half-space-approximation”).
•
The frictional interaction can be described by a local-global Amontons–Coulomb
friction with a constant coefficient of friction.
Machines 2023,11, 484 5 of 13
•
All deformations are elastic. Elastic coupling between the normal and tangential
contact problem and local lateral displacements are neglected.
•Effects of roughness or adhesion are neglected.
Machines 2023, 11, x FOR PEER REVIEW 5 of 13
Figure 3. Macroscopic model of the experimental setup with relevant geometrical and mechanical
quantities. The considered global displacements (i.e., degrees of freedom) are shown in red.
The hybrid character of the model stems from the fact that all contact interactions
(between the indenter and the block, as well as between the block and the silicon nitride
bedding) are modeled within subroutines based on the method of dimensionality reduc-
tion (MDR) [24]. The MDR description of elastic tangential contact with friction operates
within the framework of the Hertz–Mindlin-approximation ([25,26]), specifically:
• The characteristic length of the contact area is small compared to all linear dimen-
sions of the contacting bodies; the gradients of the deformed surfaces in the vicinity
of the contact are small (together, these two assumptions constitute the so-called
“half-space-approximation”)
• The frictional interaction can be described by a local-global Amontons–Coulomb fric-
tion with a constant coefficient of friction.
• All deformations are elastic. Elastic coupling between the normal and tangential con-
tact problem and local lateral displacements are neglected.
• Effects of roughness or adhesion are neglected.
Within the MDR framework, the axisymmetric contact of elastic continua is exactly
mapped onto the contact between a rigid plane profile with an elastic foundation of inde-
pendent linear springs [27]. It has been proven that the MDR formalism, within the re-
strictions of the Hertz–Mindlin approximation, will provide the correct contact solution
for arbitrary 2D oblique loading [28].
Within the model, the measured normal force (in the contact between the indenter
and the flat) and the tangential displacement of the flat block (which is enforced by the
piezo actuator) are taken as external excitations. For the determination of the fretting hys-
teresis loops, this leaves three degrees of freedom (the normal displacement w3 of the
block, the rotation φ of the block, and the tangential displacement u1 of the indenter) to be
calculated from the vertical and angular equilibrium of the block, and the tangential equi-
librium of the indenter. In that regard, Euler–Bernoulli theory will provide force laws for
the vertical force and bending moment in the brass plates, and the MDR formalism pro-
vides (incremental) force laws for the contact interaction.
Hence, the macroscopic equation system is closed. Note that the real tangential in-
denter displacement in the contact is assumed to be u2 = u1(h1 + h2)/h1 from the intercept
theorem.
4. Comparison between Experimental and Numerical Results
As was stated above, the measured normal force in the contact between the indenter
and flat, as well as the experimentally determined tangential displacement of the block,
are taken as external loads/excitations for the model. That leaves the frictional force in the
Figure 3.
Macroscopic model of the experimental setup with relevant geometrical and mechanical
quantities. The considered global displacements (i.e., degrees of freedom) are shown in red.
Within the MDR framework, the axisymmetric contact of elastic continua is exactly
mapped onto the contact between a rigid plane profile with an elastic foundation of
independent linear springs [
27
]. It has been proven that the MDR formalism, within the
restrictions of the Hertz–Mindlin approximation, will provide the correct contact solution
for arbitrary 2D oblique loading [28].
Within the model, the measured normal force (in the contact between the indenter and
the flat) and the tangential displacement of the flat block (which is enforced by the piezo
actuator) are taken as external excitations. For the determination of the fretting hysteresis
loops, this leaves three degrees of freedom (the normal displacement w
3
of the block, the
rotation
ϕ
of the block, and the tangential displacement u
1
of the indenter) to be calculated
from the vertical and angular equilibrium of the block, and the tangential equilibrium of the
indenter. In that regard, Euler–Bernoulli theory will provide force laws for the vertical force
and bending moment in the brass plates, and the MDR formalism provides (incremental)
force laws for the contact interaction.
Hence, the macroscopic equation system is closed. Note that the real tangential
indenter displacement in the contact is assumed to be u
2
=u
1
(h
1
+h
2
)/h
1
from the
intercept theorem.
4. Comparison between Experimental and Numerical Results
As was stated above, the measured normal force in the contact between the indenter
and flat, as well as the experimentally determined tangential displacement of the block,
are taken as external loads/excitations for the model. That leaves the frictional force in the
indenter–flat contact, as well as the tangential indenter displacement u
1
, as variables that
allow for a comparison between the model prediction and the experimental results.
The following parameters, shown in Table 1, are known a priori and were used for
the model (the tangential stiffness k
x
of the normal axis of the apparatus was determined
before the experiments):
As b(i.e., the radius of the flat face of the truncated spherical indenter) is much larger
than the characteristic contact radius for the full spherical (i.e., parabolic) profile under the
considered nominal normal loads (below 1 N), the model results for the truncated sphere
Machines 2023,11, 484 6 of 13
can, actually, not be distinguished from the respective results, if a flat punch with the radius
bis used as the indenting body.
Table 1.
List of a priori known parameters for the numerical simulation of the existing experimental
apparatus (for notations, see Figure 3and the text below).
l= 34 mm E1= 200 GPa B= 42 mm h1= 50 mm m1= 335 g ν1= 0.29
l1= 8 mm E2= 300 GPa b= 0.6 mm h2= 20 mm m2= 47 g ν2= 0.29
l2= 13 mm E= 100 GPa R= 6 mm h= 19 mm kx= 1.9 N/µmt= 0.2 mm
In Figure 4, a comparison between experimental results and model predictions for the
fretting loops of frictional force over normal force (left) and frictional force over relative
tangential displacement (right) are shown for a spherical indenter, with a nominal normal
force of F
N,0
= 0.35 and a piezo oscillation amplitude of u
A
= 90 nm. The red line is the
model prediction for a stationary fretting cycle with
µ1
= 0.24 and
µ2
= 0.06, and the black
lines are the experimental results for seven fretting cycles in the initial stationary state
(after 60 oscillation cycles to exclude the initial acceleration state, but before running-
in). The choice of
µ1
was based on the ratio of tangential force oscillation amplitude and
nominal normal force (which varies about 0.01–0.02 between the three different experiments
performed for the same parameter combination);
µ2
was chosen arbitrarily but had only a
very weak influence on the theoretical prediction for the hysteresis loop. Three experiments
have been performed for each parameter combination. However, as the apparent coefficient
of friction varies slightly between the experiments (even if they are done consecutively and
with the same parameter combinations), the experimental hysteresis curves of only one
experiment are shown to avoid confusion. The agreement between the model prediction
and the experimental fretting loops is very good. The fretting contact is in the partial slip
regime, with a narrow, slightly irregular hysteretic behavior.
Machines 2023, 11, x FOR PEER REVIEW 6 of 13
indenter–flat contact, as well as the tangential indenter displacement u1, as variables that
allow for a comparison between the model prediction and the experimental results.
The following parameters, shown in Table 1, are known a priori and were used for
the model (the tangential stiffness kx of the normal axis of the apparatus was determined
before the experiments):
Table 1. List of a priori known parameters for the numerical simulation of the existing experimental
apparatus (for notations, see Figure 3 and the text below).
l = 34 mm
E1 = 200 GPa
B = 42 mm
h1 = 50 mm
m1 = 335 g
ν1 = 0.29
l1 = 8 mm
E2 = 300 GPa
b = 0.6 mm
h2 = 20 mm
m2 = 47 g
ν2 = 0.29
l2 = 13 mm
E = 100 GPa
R = 6 mm
h = 19 mm
kx = 1.9 N/μm
t = 0.2 mm
As b (i.e., the radius of the flat face of the truncated spherical indenter) is much larger
than the characteristic contact radius for the full spherical (i.e., parabolic) profile under
the considered nominal normal loads (below 1 N), the model results for the truncated
sphere can, actually, not be distinguished from the respective results, if a flat punch with
the radius b is used as the indenting body.
In Figure 4, a comparison between experimental results and model predictions for
the fretting loops of frictional force over normal force (left) and frictional force over rela-
tive tangential displacement (right) are shown for a spherical indenter, with a nominal
normal force of FN,0 = 0.35 and a piezo oscillation amplitude of uA = 90 nm. The red line is
the model prediction for a stationary fretting cycle with μ1 = 0.24 and μ2 = 0.06, and the
black lines are the experimental results for seven fretting cycles in the initial stationary
state (after 60 oscillation cycles to exclude the initial acceleration state, but before running-
in). The choice of μ1 was based on the ratio of tangential force oscillation amplitude and
nominal normal force (which varies about 0.01–0.02 between the three different experi-
ments performed for the same parameter combination); μ2 was chosen arbitrarily but had
only a very weak influence on the theoretical prediction for the hysteresis loop. Three ex-
periments have been performed for each parameter combination. However, as the appar-
ent coefficient of friction varies slightly between the experiments (even if they are done
consecutively and with the same parameter combinations), the experimental hysteresis
curves of only one experiment are shown to avoid confusion. The agreement between the
model prediction and the experimental fretting loops is very good. The fretting contact is
in the partial slip regime, with a narrow, slightly irregular hysteretic behavior.
Figure 4. Comparison between experimental results and model predictions for the fretting loops of
frictional force over normal force (left) and frictional force over relative tangential displacement
(right) for a spherical indenter, with a nominal normal force of FN,0 = 0.35 and an oscillation ampli-
tude of uA = 90 nm. Red: model prediction for stationary fretting cycle with μ1 = 0.24 and μ2 = 0.06.
Black: Experimental results for seven fretting cycles in the initial stationary state.
Figure 4.
Comparison between experimental results and model predictions for the fretting loops
of frictional force over normal force (
left
) and frictional force over relative tangential displacement
(
right
) for a spherical indenter, with a nominal normal force of F
N,0
= 0.35 and an oscillation amplitude
of u
A
= 90 nm. Red: model prediction for stationary fretting cycle with
µ1
= 0.24 and
µ2
= 0.06. Black:
Experimental results for seven fretting cycles in the initial stationary state.
Figure 5gives the comparison between experimental results and model predictions
for the fretting loops for a spherical indenter, with a nominal normal force of F
N,0
= 0.61
and a piezo oscillation amplitude of u
A
= 220 nm. The model prediction uses
µ1
= 0.26 and
µ2
= 0.1. The agreement between theory and experiment is still very good, although slightly
worse than in Figure 4. Note that the slight offset in the hysteresis curve (right) is not
due to an offset of the displacement measurement (which, of course, is arbitrary) but
Machines 2023,11, 484 7 of 13
due to a misprediction of the phase difference between F
x
and u
1
(which is zero in the
model—because
the corresponding force law is simply the one of a linear
spring—and
nonzero in the experiments). However, the offset, naturally, has no influence on the
prediction quality for the energy dissipation or the fretting regime.
Machines 2023, 11, x FOR PEER REVIEW 7 of 13
Figure 5 gives the comparison between experimental results and model predictions
for the fretting loops for a spherical indenter, with a nominal normal force of FN,0 = 0.61
and a piezo oscillation amplitude of uA = 220 nm. The model prediction uses μ1 = 0.26 and
μ2 = 0.1. The agreement between theory and experiment is still very good, although
slightly worse than in Figure 4. Note that the slight offset in the hysteresis curve (right) is
not due to an offset of the displacement measurement (which, of course, is arbitrary) but
due to a misprediction of the phase difference between Fx and u1 (which is zero in the
model—because the corresponding force law is simply the one of a linear spring—and
nonzero in the experiments). However, the offset, naturally, has no influence on the pre-
diction quality for the energy dissipation or the fretting regime.
Figure 5. Comparison between experimental results and model predictions for the fretting loops of
frictional force over normal force (left) and frictional force over relative tangential displacement
(right) for a spherical indenter, with a nominal normal force of FN,0 = 0.61 and an oscillation ampli-
tude of uA = 220 nm. Red: model prediction for stationary fretting cycle with μ1 = 0.26 and μ2 = 0.1.
Black: Experimental results for seven fretting cycles in the initial stationary state.
In Figure 6, a comparison is shown between experimental results and model predic-
tions for the fretting loops of frictional force over normal force (left) and frictional force
over relative tangential displacement (right) for a truncated spherical indenter, with a
nominal normal force of FN,0 = 0.31 and a piezo oscillation amplitude of uA = 90 nm. The
model prediction uses μ1 = 0.2 and μ2 = 0.1. The agreement between theory and experiment
is good; the fretting contact is in the gross slip regime, which is expected, as the contact
stiffness is much higher, and the indentation depth d, therefore, much lower. Hence, the
tangential displacement necessary to cause gross slip—which is of the order of μd—is also
much lower.
Figure 5.
Comparison between experimental results and model predictions for the fretting loops
of frictional force over normal force (
left
) and frictional force over relative tangential displacement
(
right
) for a spherical indenter, with a nominal normal force of F
N,0
= 0.61 and an oscillation amplitude
of u
A
= 220 nm. Red: model prediction for stationary fretting cycle with
µ1
= 0.26 and
µ2
= 0.1. Black:
Experimental results for seven fretting cycles in the initial stationary state.
In Figure 6, a comparison is shown between experimental results and model predic-
tions for the fretting loops of frictional force over normal force (left) and frictional force over
relative tangential displacement (right) for a truncated spherical indenter, with a nominal
normal force of F
N,0
= 0.31 and a piezo oscillation amplitude of u
A
= 90 nm. The model
prediction uses
µ1
= 0.2 and
µ2
= 0.1. The agreement between theory and experiment is
good; the fretting contact is in the gross slip regime, which is expected, as the contact
stiffness is much higher, and the indentation depth d, therefore, much lower. Hence, the
tangential displacement necessary to cause gross slip—which is of the order of
µ
d—is also
much lower.
Machines 2023, 11, x FOR PEER REVIEW 7 of 13
Figure 5 gives the comparison between experimental results and model predictions
for the fretting loops for a spherical indenter, with a nominal normal force of FN,0 = 0.61
and a piezo oscillation amplitude of uA = 220 nm. The model prediction uses μ1 = 0.26 and
μ2 = 0.1. The agreement between theory and experiment is still very good, although
slightly worse than in Figure 4. Note that the slight offset in the hysteresis curve (right) is
not due to an offset of the displacement measurement (which, of course, is arbitrary) but
due to a misprediction of the phase difference between Fx and u1 (which is zero in the
model—because the corresponding force law is simply the one of a linear spring—and
nonzero in the experiments). However, the offset, naturally, has no influence on the pre-
diction quality for the energy dissipation or the fretting regime.
Figure 5. Comparison between experimental results and model predictions for the fretting loops of
frictional force over normal force (left) and frictional force over relative tangential displacement
(right) for a spherical indenter, with a nominal normal force of FN,0 = 0.61 and an oscillation ampli-
tude of uA = 220 nm. Red: model prediction for stationary fretting cycle with μ1 = 0.26 and μ2 = 0.1.
Black: Experimental results for seven fretting cycles in the initial stationary state.
In Figure 6, a comparison is shown between experimental results and model predic-
tions for the fretting loops of frictional force over normal force (left) and frictional force
over relative tangential displacement (right) for a truncated spherical indenter, with a
nominal normal force of FN,0 = 0.31 and a piezo oscillation amplitude of uA = 90 nm. The
model prediction uses μ1 = 0.2 and μ2 = 0.1. The agreement between theory and experiment
is good; the fretting contact is in the gross slip regime, which is expected, as the contact
stiffness is much higher, and the indentation depth d, therefore, much lower. Hence, the
tangential displacement necessary to cause gross slip—which is of the order of μd—is also
much lower.
Figure 6.
Comparison between experimental results and model predictions for the fretting loops
of frictional force over normal force (
left
) and frictional force over relative tangential displacement
(
right
) for a truncated spherical indenter, with a nominal normal force of F
N,0
= 0.31 and a piezo
oscillation amplitude of u
A
= 90 nm. Red: model prediction for stationary fretting cycle with
µ1
= 0.2
and µ2= 0.1. Black: Experimental results for seven fretting cycles in the initial stationary state.
Machines 2023,11, 484 8 of 13
Figure 7gives a comparison between experimental results and model predictions for
the same fretting loops for a truncated spherical indenter, with a nominal normal force of
F
N,0
= 0.61 and a piezo oscillation amplitude of u
A
= 220 nm. The model prediction uses
the values
µ1
= 0.16 (which is quite low) and
µ2
= 0.1. The agreement is still good, and the
contact is in the sliding regime. It can be noted that the behavior of the friction force at
the “corners” of the hysteresis loop, i.e., at the beginning of the sliding phases, is much
smoother in the experiments than predicted by the Amontons law.
Machines 2023, 11, x FOR PEER REVIEW 8 of 13
Figure 6. Comparison between experimental results and model predictions for the fretting loops of
frictional force over normal force (left) and frictional force over relative tangential displacement
(right) for a truncated spherical indenter, with a nominal normal force of FN,0 = 0.31 and a piezo
oscillation amplitude of uA = 90 nm. Red: model prediction for stationary fretting cycle with μ1 = 0.2
and μ2 = 0.1. Black: Experimental results for seven fretting cycles in the initial stationary state.
Figure 7 gives a comparison between experimental results and model predictions for
the same fretting loops for a truncated spherical indenter, with a nominal normal force of
FN,0 = 0.61 and a piezo oscillation amplitude of uA = 220 nm. The model prediction uses the
values μ1 = 0.16 (which is quite low) and μ2 = 0.1. The agreement is still good, and the
contact is in the sliding regime. It can be noted that the behavior of the friction force at the
“corners” of the hysteresis loop, i.e., at the beginning of the sliding phases, is much
smoother in the experiments than predicted by the Amontons law.
Figure 7. Comparison between experimental results and model predictions for the fretting loops of
frictional force over normal force (left) and frictional force over relative tangential displacement
(right) for a truncated spherical indenter, with a nominal normal force of FN,0 = 0.61 and a piezo
oscillation amplitude of uA = 220 nm. Red: model prediction for stationary fretting cycle with μ1 =
0.16 and μ2 = 0.1. Black: Experimental results for seven fretting cycles in the initial stationary state.
5. Analysis of Profile Influence Based on the Numerical Model
Let us analyze the system and the important influencing variables a little deeper,
based on the macroscopic numerical model, which has been substantiated above.
The external excitations shall be harmonic, i.e., for the piezo-enforced tangential os-
cillation of the block,
( ) ( )
3sin 2 ,
A
u t u ft
=
(1)
and for the normal force,
( ) ( )
0sin 2 .
N
F t F F ft
= + +
(2)
Note that, in the experiments, the normal force oscillation is not externally controlled
but exhibited by the apparatus due to the piezo actuator oscillation. However, ΔF and the
phase angle α only have a weak influence on the frictional energy dissipation anyway.
To avoid showing the full fretting loops, let us introduce two quantities to character-
ize the frictional dissipation in the fretting contact between the steel indenter and the steel
flat: the energy dissipation per cycle,
( )
d,
x
W F u =
(3)
Figure 7.
Comparison between experimental results and model predictions for the fretting loops
of frictional force over normal force (
left
) and frictional force over relative tangential displacement
(
right
) for a truncated spherical indenter, with a nominal normal force of F
N,0
= 0.61 and a piezo
oscillation amplitude of u
A
= 220 nm. Red: model prediction for stationary fretting cycle with
µ1
= 0.16 and
µ2
= 0.1. Black: Experimental results for seven fretting cycles in the initial stationary state.
5. Analysis of Profile Influence Based on the Numerical Model
Let us analyze the system and the important influencing variables a little deeper, based
on the macroscopic numerical model, which has been substantiated above.
The external excitations shall be harmonic, i.e., for the piezo-enforced tangential
oscillation of the block,
u3(t)=uAsin(2πf t), (1)
and for the normal force,
FN(t)=F0+∆Fsin(2πf t +α). (2)
Note that, in the experiments, the normal force oscillation is not externally controlled
but exhibited by the apparatus due to the piezo actuator oscillation. However,
∆
Fand the
phase angle αonly have a weak influence on the frictional energy dissipation anyway.
To avoid showing the full fretting loops, let us introduce two quantities to characterize
the frictional dissipation in the fretting contact between the steel indenter and the steel flat:
the energy dissipation per cycle,
∆W=IFxd(∆u)
, (3)
with the frictional force F
R
and the macroscopic relative surface displacement
∆
u, as well
as a slip index [29]
δ=∆W
4µ1uAF0
. (4)
Machines 2023,11, 484 9 of 13
A small slip index corresponds to the partial slip regime, while a slip index of approxi-
mately 1 corresponds to the gross sliding regime. Note that definition (4) slightly differs
from the slip index proposed in the original paper by Varenberg et al. [
29
], as the dissipated
energy is used directly, instead of slopes of the hysteresis curve, to account for the bimodal
character of the fretting oscillation.
Out of the several dimensional variables, which will influence the frictional dissipation,
F
0
defines the characteristic scales of the contact problem (indentation depth, contact radius,
and, hence, contact stiffness) and shall therefore be kept constant. The normal force
oscillation amplitude and phase angle, as well as the coefficient of friction between the
steel flat and the silicon nitride bedding (if it is small enough), only have little influence
on the dissipation in the fretting contact between the indenter and the flat. That leaves the
actuator amplitude, u
A
, the coefficient of friction between the indenter and the flat,
µ1
, the
linear tangential stiffness of the normal axis of the apparatus, k
x
, and the indenter profile
as the main influencing properties. In that regard, the influence of u
A
is trivial (except for
a very interesting stiffness effect, which will be discussed below): increasing the actuator
amplitude will increase the energy dissipation and the slip index. Similarly, the influence
of the friction coefficient
µ1
is relatively simple, except for the aforementioned stiffness
effect: increasing
µ1
will generally reduce the slip index and (non-trivially) increase the
energy dissipation. However, as for dry contacts of a given material pairing, the coefficient
of friction can only be somewhat controlled within a very small margin; we shall not bother
ourselves with the intricacies of these dependencies.
As was mentioned, there is an interesting stiffness effect, which, in fact, includes all
important influencing factors, and which is due to the compliance of the experimental
apparatus (or, generally speaking, the macrosystem). One can imagine the normal axis of
the apparatus being very soft, i.e., kxbeing very small—but small compared to what? If
kxuAµ1F0, (5)
the axis is able to elastically follow the tangential excitation of the block without significant
phases of sliding in the contact, and, therefore, the fretting contact will always be in the
partial slip regime. In fact, the resulting slip index might suggest the existence of a “sticking
regime” (without any frictional hysteresis), which, however, is only due to the apparatus
compliance, as was pointed out before [30].
In other words, there is a critical apparatus stiffness of the order
kc≈µ1F0
uA
, (6)
below which the indenter profile (via changing the contact stiffness) or other influencing
variables cannot “pull” the fretting contact out of the partial slip (or “sticking”) regime.
Let us illustrate these considerations with some small numerical parameter studies.
Only k
x
and the indenter profile geometry were varied for the study; all other parameters
and known quantities can be taken from Tables 1and 2.
Table 2. List of chosen values for the fixed variables in the numerical parameter studies.
F0= 0.5 N ∆F= 0.1 N µ1= 0.2 µ2= 0.1 uA= 150 nm
As F
0
,
µ1
, and u
A
are kept constant, the dissipated energy
∆
Wand the slip index
δ
incorporate the same information because the latter, in this case, is just a non-dimensional
measure of the frictional hysteretic loss. Therefore, only the results of the macroscopic
model for the slip index will be given.
In Figure 8, the numerical results for the slip index
δ
as a function of the radius
bof the flat face of the truncated spherical indenter (in logarithmic scaling) are shown
for different values of the tangential apparatus stiffness. The thin lines of the respective
Machines 2023,11, 484 10 of 13
colors correspond to the numerical solution if a flat punch with radius bis used as the
indenting body. It can be seen that already for values b> 30
µ
m (log
10
(30)
≈
1.5) (which is
approximately equal to the characteristic contact radius in the parabolic contact, under a
normal load F0), the solution for the hysteretic behavior cannot be distinguished from the
respective solution for a flat punch indenter with radius bof the flat face.
Machines 2023, 11, x FOR PEER REVIEW 10 of 13
measure of the frictional hysteretic loss. Therefore, only the results of the macroscopic
model for the slip index will be given.
In Figure 8, the numerical results for the slip index δ as a function of the radius b of
the flat face of the truncated spherical indenter (in logarithmic scaling) are shown for dif-
ferent values of the tangential apparatus stiffness. The thin lines of the respective colors
correspond to the numerical solution if a flat punch with radius b is used as the indenting
body. It can be seen that already for values b > 30 μm (log10(30) ≈ 1.5) (which is approxi-
mately equal to the characteristic contact radius in the parabolic contact, under a normal
load F0), the solution for the hysteretic behavior cannot be distinguished from the respec-
tive solution for a flat punch indenter with radius b of the flat face.
Figure 8. Slip index δ as a function of radius b of the flat face of the truncated spherical indenter (in
logarithmic scaling), for different values of the tangential apparatus stiffness kx, according to the
numerical model. The thin lines of the respective colors correspond to the numerical solution if a
flat punch with radius b is used as the indenting body.
Moreover, for b < 10 μm, the solution has already converged to the fully spherical (or
parabolic) case, b = 0. In other words, the “window” for the geometrical transition in the
hysteretic behavior from the smooth parabolic profile to a sharp flat-punch indenter is
very small, independently of the apparatus stiffness. Changes in the hysteretic behavior
for b > 30 μm are only due to the change in the contact stiffness of the flat-punch contact
(which is proportional to its radius).
Also, it is apparent that—while the curves in Figure 8 for kx = 2 N/μm, 10 N/μm and
20 N/μm are all qualitatively very similar—the curve for kx = 1 N/μm is very different, and
the slip index is close to (or even equal to) zero. This sharp transition is shown again in
Figure 9, giving the slip index as a function of the tangential apparatus stiffness, for dif-
ferent values of the radius of the flat face of the truncated spherical indenter, according to
the numerical model. For kx > 2 N/μm, all three curves are in the mixed or sliding regime
of the fretting contact, while between 1 N/μm and 2 N/μm, there is a steep transition to
the “sticking” regime, where the dependence of the slip index on the apparatus stiffness
somewhat resembles the behavior of the order parameter in a second-order phase transi-
tion. Notably, that resemblance is especially good in the case of the curves for b = 50 μm
and b = 100 μm—for which the contact macroscopically already behaves like a flat-punch
contact, as was discussed above—while the transition behavior for b = 10 μm (which basi-
cally corresponds to the spherical limit) is slightly smoother.
Figure 8.
Slip index
δ
as a function of radius bof the flat face of the truncated spherical indenter (in
logarithmic scaling), for different values of the tangential apparatus stiffness k
x
, according to the
numerical model. The thin lines of the respective colors correspond to the numerical solution if a flat
punch with radius bis used as the indenting body.
Moreover, for b< 10
µ
m, the solution has already converged to the fully spherical (or
parabolic) case, b= 0. In other words, the “window” for the geometrical transition in the
hysteretic behavior from the smooth parabolic profile to a sharp flat-punch indenter is very
small, independently of the apparatus stiffness. Changes in the hysteretic behavior for
b> 30
µ
m are only due to the change in the contact stiffness of the flat-punch contact (which
is proportional to its radius).
Also, it is apparent that—while the curves in Figure 8for k
x
= 2 N/
µ
m, 10 N/
µ
m and
20 N/
µ
m are all qualitatively very similar—the curve for k
x
= 1 N/
µ
m is very different,
and the slip index is close to (or even equal to) zero. This sharp transition is shown again
in Figure 9, giving the slip index as a function of the tangential apparatus stiffness, for
different values of the radius of the flat face of the truncated spherical indenter, according to
the numerical model. For k
x
> 2 N/
µ
m, all three curves are in the mixed or sliding regime
of the fretting contact, while between 1 N/
µ
m and 2 N/
µ
m, there is a steep transition to
the “sticking” regime, where the dependence of the slip index on the apparatus stiffness
somewhat resembles the behavior of the order parameter in a second-order phase transition.
Notably, that resemblance is especially good in the case of the curves for b= 50
µ
m and
b= 100
µ
m—for which the contact macroscopically already behaves like a flat-punch
contact, as was discussed above—while the transition behavior for b= 10
µ
m (which
basically corresponds to the spherical limit) is slightly smoother.
Machines 2023,11, 484 11 of 13
Machines 2023, 11, x FOR PEER REVIEW 11 of 13
Figure 9. Slip index δ as a function of the tangential apparatus stiffness kx, for different values of the
radius b of the flat face of the truncated spherical indenter, according to the numerical model.
6. Discussion and Conclusions
Being able to predict the macroscopic contact behavior in terms of forces, global rel-
ative displacements, and, as a result, energies is, of course, only the first step in the anal-
ysis of the fretting contact. The study of damage mechanisms inevitably requires
knowledge of the behavior on (several) smaller scales and over much more oscillation
cycles—in the present study, only the initial stationary states, before running-in, i.e., only
several hundred cycles, were considered. It is, therefore, self-evident that the present
study can only be the first step in a larger project, with the final aim of retracing the com-
petition between wear and fatigue in steel-on-steel fretting contacts through the “lenses”
of the contact profile geometry.
It is interesting that, while “fretting” is the result of many integrated, complex, tribo-
logical processes, a simple mechanical model with extremely few degrees of freedom, ap-
parently, can successfully describe the contact’s macro-behavior. This, at first glance,
might be especially surprising, considering that the contact interaction is captured within
the framework of a very basic contact mechanical description—the Hertz–Mindlin formal-
ism. It must be noted, however, that the experimental setup was designed to comply with
most of the restrictions of that formalism. For example, the normal load was kept very
small to ensure the absence of relevant plastic deformations, equal materials were chosen
for the contact pairing, and the contacts were cleaned before each experiment to avoid the
influence of the tiniest amounts of wear debris in the contact.
On the other hand, the simplicity of the model also constitutes its strength, as all
parameters of the model can be determined a priori, except for the global coefficients of
friction (which are extremely difficult to predict precisely, anyway, for the type of contact
considered here). In this regard, the model represents the “absolute minimum” of com-
plexity—in the spirit of the principle to explain things “as simple as possible, but not any
simpler”—which, in future work, can be extended step-by-step to improve its predictive
power.
The next phase of research will be concerned with the influence of the profile geom-
etry on the running-in process and the long-term wear behavior, studying the surface to-
pography evolution and modeling the contact interaction on a smaller scale, based on the
boundary element method—which has already been used to model adhesive wear within
the framework of an asperity-free Rabinowicz criterion [31].
The main findings of the analysis documented in the present manuscript can be sum-
marized as follows:
Figure 9.
Slip index
δ
as a function of the tangential apparatus stiffness k
x
, for different values of the
radius bof the flat face of the truncated spherical indenter, according to the numerical model.
6. Discussion and Conclusions
Being able to predict the macroscopic contact behavior in terms of forces, global
relative displacements, and, as a result, energies is, of course, only the first step in the
analysis of the fretting contact. The study of damage mechanisms inevitably requires
knowledge of the behavior on (several) smaller scales and over much more oscillation
cycles—in the present study, only the initial stationary states, before running-in, i.e., only
several hundred cycles, were considered. It is, therefore, self-evident that the present study
can only be the first step in a larger project, with the final aim of retracing the competition
between wear and fatigue in steel-on-steel fretting contacts through the “lenses” of the
contact profile geometry.
It is interesting that, while “fretting” is the result of many integrated, complex, tri-
bological processes, a simple mechanical model with extremely few degrees of freedom,
apparently, can successfully describe the contact’s macro-behavior. This, at first glance,
might be especially surprising, considering that the contact interaction is captured within
the framework of a very basic contact mechanical description—the Hertz–Mindlin formal-
ism. It must be noted, however, that the experimental setup was designed to comply with
most of the restrictions of that formalism. For example, the normal load was kept very
small to ensure the absence of relevant plastic deformations, equal materials were chosen
for the contact pairing, and the contacts were cleaned before each experiment to avoid the
influence of the tiniest amounts of wear debris in the contact.
On the other hand, the simplicity of the model also constitutes its strength, as all
parameters of the model can be determined a priori, except for the global coefficients
of friction (which are extremely difficult to predict precisely, anyway, for the type of
contact considered here). In this regard, the model represents the “absolute minimum”
of complexity—in the spirit of the principle to explain things “as simple as possible, but
not any simpler”—which, in future work, can be extended step-by-step to improve its
predictive power.
The next phase of research will be concerned with the influence of the profile geometry
on the running-in process and the long-term wear behavior, studying the surface topog-
raphy evolution and modeling the contact interaction on a smaller scale, based on the
boundary element method—which has already been used to model adhesive wear within
the framework of an asperity-free Rabinowicz criterion [31].
The main findings of the analysis documented in the present manuscript can be
summarized as follows:
Machines 2023,11, 484 12 of 13
•
The hybrid macroscopic–contact model is well able to predict the time behavior of
the macroscopic contact quantities (forces and displacements) and, accordingly, the
frictional hysteretic losses observed in the experiment.
•
In the experiments and the simulation, the change of the indenter profile from spherical
to truncated spherical “pushes” the fretting contact towards the sliding regime if the
nominal normal force and tangential displacement oscillation amplitude are kept
constant; this is mainly due to the higher contact stiffness (and, accordingly, smaller
indentation depth) for the truncated profile.
•
The transition of the hysteretic behavior, depending on the profile geometry from the
perfectly spherical to the sharp flat-punch profile, occurs for the truncated spherical
indenter within a small margin of the radius of its flat face. Already for a flat face
radius which is roughly equal to the contact radius for the spherical case, the macro-
scopic hysteretic behavior cannot be distinguished from a flat punch contact with the
same radius.
•
The compliance of the apparatus (i.e., the macrosystem) can have a large influence
on the energy dissipation and the fretting regime. Below a critical value for the
stiffness, which is of the order of
µ
F
0
/u
A
, the fretting contact exhibits a sharp transition
to the “sticking” regime (with almost no dissipation). However, if the apparatus
stiffness is large enough, the hysteretic behavior can be controlled by changing the
profile geometry.
Funding:
We acknowledge support by the German Research Foundation and the Open Access
Publication Fund of TU Berlin. This research was funded by the German Research Foundation under
project number PO 810/66-1.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments:
The author is grateful to Valentin L. Popov for valuable discussions on the topic.
Conflicts of Interest: The author declares no conflict of interest.
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