Citation: Lyashenko, I.A.; Popov,
V.L.; Borysiuk, V. Experimental
Verification of the Boundary Element
Method for Adhesive Contacts of a
Coated Elastic Half-Space. Lubricants
2023,11, 84. https://doi.org/
10.3390/lubricants11020084
Received: 30 January 2023
Revised: 13 February 2023
Accepted: 14 February 2023
Published: 15 February 2023
Copyright: © 2023 by the authors.
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/).
lubricants
Article
Experimental Verification of the Boundary Element Method for
Adhesive Contacts of a Coated Elastic Half-Space
Iakov A. Lyashenko 1,2,* , Valentin L. Popov 1,* and Vadym Borysiuk 1,3
1
Department of System Dynamics and Friction Physics, Institute of Mechanics, Technische Universität Berlin,
10623 Berlin, Germany
2
Department of Applied Mathematics and Complex Systems Modeling, Faculty of Electronics and Information
Technology, Sumy State University, 40007 Sumy, Ukraine
3
Department of Nanoelectronics and Surface Modification, Faculty of Electronics and Information Technology,
Sumy State University, 40007 Sumy, Ukraine
*Correspondence: i.liashenko@tu-berlin.de (I.A.L.); v.popov@tu-berlin.de (V.L.P.); Tel.: +49-(0)30-314-75917 (I.A.L.)
Abstract:
We consider analytical, numerical, and experimental approaches developed to describe the
mechanical contact between a rigid indenter and an elastic half-space coated with an elastic layer.
Numerical simulations of the indentation process were performed using the recently generalized
boundary element method (BEM). Analytical approximation of the dependence of contact stiffness on
the indenter diameter was used to verify the results of BEM simulations. Adhesive contacts of hard
indenters of different shapes with soft rubber layers have been experimentally studied using specially
designed laboratory equipment. The comparison of the results from all three implemented methods
shows good agreement of the obtained data, thus supporting the generalized BEM simulation
technique developed for the JKR limit of very small range of action of adhesive forces. It was shown
that the half-space approximation is asymptotical at high ratios of layer thickness hto cylindrical
indenter diameter D; however, it is very slowly. Thus, at the ratio h/D= 3.22, the half-space
approximation leads to 20% lower contact stiffness compared with that obtained for finite thickness
using both an experiment and simulation.
Keywords: indentation; elastomer; elastic layer; contact stiffness; BEM; experiment; adhesion
1. Introduction
Coated and layered materials are commonly used in numerous engineering appli-
cations. The main purpose of a coating technology is significant improvement of the
performance of certain devices or their parts, as changing the properties of the surface
can dramatically affect the behavior of the material. Coating technology is often applied
to enhance the properties of the materials that are involved in mechanical contacts, to
modify their tribological properties, and to improve wear resistance, adhesion, friction,
etc [
1
]. Furthermore, coatings are also used to achieve the desired level of biocompatibility
of implants [
2
,
3
], to obtain needed optical properties [
4
], to enable triboelectric energy
harvesting [
5
], and for many other technologies. Therefore, development of experimental,
numerical and analytical techniques for the modelling and studying of materials with a
layered structure is an important topic in various fields of science and technology.
Most of the analytical theories and numerical simulation methods used for describing
elastic contacts make use of “half-space approximation”, wherein an object with finite
sizes at certain conditions can be considered as a half-space. This approximation can be
applied to bodies coated with an elastic substrate provided the characterized size, D, of the
contact area is much smaller than the thickness of the coating [
6
]. A special case of a coated
body is an elastic layer placed on the rigid substrate. If an elastic layer with thickness
his indented by a cylindrical indenter of diameter D, this layer can be considered as a
half-space provided the thickness of the layer is much larger than the diameter of indenter.
Lubricants 2023,11, 84. https://doi.org/10.3390/lubricants11020084 https://www.mdpi.com/journal/lubricants
Lubricants 2023,11, 84 2 of 12
Classical theories of adhesion, such as JKR [
7
], DMT [
8
] or Maugis theory [
9
], as well as
the classic Hertz theory [
10
], all operate in the half-space approximation; therefore, much
of our intuitive understanding of contacts is based implicitly on the results of half-space
approximation. However, if the contact size is comparable with the layer thickness, the use
of the half-space approximation may lead to inadequate results so that the finite size has to
be taken into account. There is a number of theories considering these corrections [11–16].
In addition to purely analytical methodology, numerical simulations are also widely used
for solving the contact tasks involving layers of a given thickness. Commonly used numerical
methods are the finite element method (FEM) [
17
–
19
] and the Fourier-based residuals molecular
dynamics (RMD) [
20
], among others [
21
]. At present, the FFT-assisted boundary element
method (BEM) is considered the most powerful technique for simulation contacts. Recently, it
was generalized for the case of coated elastic half-space and also takes adhesion into account [
22
].
In the described generalization, adhesion is considered in the “JKR-limit”, meaning that the
range of action of adhesive forces is much smaller than any other characteristic length of the
problem (including gap and indentation depth) so that it can be considered to be zero. It is
not automatically guaranteed that this condition is fulfilled in real adhesive contacts. Thus, it
is important to “verify” the simulation method through comparison with experiments. This
comparison is the main purpose of the present paper.
2. Materials and Methods
As mentioned in introduction, in our study we used FFT-based BEM for coated half-
space [
22
]. This approach was recently implemented for a description of adhesive and
non-adhesive contacts between rigid indenter and elastic half-space coated with a layer
of different elastic properties. Within BEM, an elastic half-space with elastic modulus E
2
and Poisson ratio
ν2
coated with the elastic layer of thickness hand elastic parameters
E
1
and
ν1
is considered. To solve the contact problem between coated half-space and a
rigid indenter with arbitrary geometry numerically, we consider a square region on the
body surface with the size L
×
L, which has Ncells in each direction, while the size of each
of the N
2
square cells is
∆
x=
∆
y=
∆
. Pressure is assumed to be uniform in each cell. If
the pressure distribution
p
is given, the displacement
u
can be calculated according to a
numerical procedure [22],
u=IFFT[uz·FFT(p)], (1)
where
uz
is a Fourier-transformed fundamental solution (normal displacements at the
contact surface), and the analytical formula for
uz
is provided in [
22
]. The contact problem
is solved iteratively. In each step, the displacements
u
for a given pressure distribution
p
are determined through the evaluation of Equation (1). The inverse problem of finding
pressure
p
for producing given deformations
u
can be solved using the conjugate gradient
method [
23
]. For adhesive contacts, an additional detachment criterion is needed: a surface
element at the boundary of the contact area loses its contact as soon as tensile stress in this
element exceeds the critical value given by
σc=sE1∆γ
0.473201 ·∆·(1−ν2
1), (2)
where
∆γ
(J/m
2
) is the specific work of adhesion between the indenter and substrate. For
non-adhesive contacts, the detachment criterion is that normal pressure p> 0; for adhesive
contact the condition p> –
σc
must be used. Detailed information about numerical BEM
procedure can be found in [22].
Experiments concerning indenter–substrate contact are a well-known challenge in
many fields of tribology, contact mechanics and nanotechnology [
24
]. An experimental
study of adhesive contacts between rigid indenters and elastic layers of different thickness
was conducted on specially designed in-house laboratory equipment. Detailed description
of the designed facility and examples of its performance are given in our recent paper [
25
];
therefore, here, we provide only brief information about the experimental setup. General
Lubricants 2023,11, 84 3 of 12
view of the designed device together with an enlarged image of the indenter and sample
are shown in Figure 1.
Lubricants2023,11,xFORPEERREVIEW3of12
studyofadhesivecontactsbetweenrigidindentersandelasticlayersofdifferentthickness
wasconductedonspeciallydesignedin‐houselaboratoryequipment.Detaileddescrip‐
tionofthedesignedfacilityandexamplesofitsperformancearegiveninourrecentpaper
[25];therefore,here,weprovideonlybriefinformationabouttheexperimentalsetup.
Generalviewofthedesigneddevicetogetherwithanenlargedimageoftheindenterand
sampleareshowninFigure1.
Thefacilityoperatesasahigh‐precisiontribometer,whichiscapableofpreciseposi‐
tioningofthesampleinthreedimensionsandmeasuringallthreecomponentsofthein‐
teractionforce.Allfunctionalpartsofthedevicethataredenotedinthefigurearethe
sameforbothpanels:(1)and(2)arehigh‐precisionM‐403.2DGmotorizedlinearstages
(manufacturedbyPI),whicharehandledbyPIC‐863one‐axisservocontrollers;(3)isan
MEK3D40three‐axisforcesensor;(4)isanindenterthatismountedontheforcesensor;
(5)isthesamplebeingindentedplacedatthe8mmthicksilicateglassplate;(6)isatilt
mechanism;(7)isadigitalcameraXimea2.2MPMQ022CG‐CMwithFUJINONHF16SA‐
1,2/3”lens;and(8)isthe8MR190‐90‐4247‐MEn1motorizedrotationstage.Variousmod‐
ificationsofthedevelopeddevicehadalreadybeenusedtoperformseveralstudieson
contactmechanics[25].Below,wediscussexperimentswithindentationofhardsteelin‐
dentersinsoftelasticrubbersheets(elastomer)withgoodadhesiveproperties.Asanelas‐
tomer,TARNACCRGN3005rubbersheetswithlinearsizes100mm×100mm×5mm
wereused(seeFigure1b,position5).Intheexperiments,elastomerswiththicknessh=5,
10,15,20and25mmwereused;toobtaintheelastomerwithdifferentthickness,separate
rubbersheetswerestackedtogether.Duetothestrongadhesion,theserubbersheetsare
firmlyconcatenatedanddidnotslideovereachotherduringtheindentation.Forinden‐
tation,cylinderswithaflatbaseofdiameterD=4,7,10and15mm,aswellasspheres
withradiiR=30,50and100mm,wereused.Allexperimentswereperformedinlabora‐
toryunderroomtemperature(24˚C)andrelativehumidity(48%).
(a)(b)
Figure1.Photooftheexperimentalsetup:generalview(a)andenlargedimageoftheindenterand
thesample(b).Functionalpartsoftheequipmentaredenotedbylabels:(1)and(2)—M‐403.2DG
high‐precisionmotorizedlinearstages(manufacturedbyPI);(3)—three‐axisforcesensorMEK3D40
withmountedindenter(4);(5)—samplebeingindented;(6)—tiltmechanism;(7)—digitalcamera
Ximea2.2MPMQ022CG‐CMwithFUJINONHF16SA‐1,2/3”lens;and(8)—the8MR190‐90‐4247‐
MEn1motorizedrotationstage.
Figure 1.
Photo of the experimental setup: general view (
a
) and enlarged image of the indenter and
the sample (
b
). Functional parts of the equipment are denoted by labels: (1) and (2)—M-403.2DG high-
precision motorized linear stages (manufactured by PI); (3)—three-axis force sensor ME K3D40 with
mounted indenter (4); (5)—sample being indented; (6)—tilt mechanism; (7)—digital camera Ximea
2.2MP MQ022CG-CM with FUJINON HF16SA-1, 2/3” lens; and (8)—the 8MR190-90-4247-MEn1
motorized rotation stage.
The facility operates as a high-precision tribometer, which is capable of precise po-
sitioning of the sample in three dimensions and measuring all three components of the
interaction force. All functional parts of the device that are denoted in the figure are the
same for both panels: (1) and (2) are high-precision M-403.2DG motorized linear stages
(manufactured by PI), which are handled by PI C-863 one-axis servo controllers; (3) is an
ME K3D40 three-axis force sensor; (4) is an indenter that is mounted on the force sensor;
(5) is the sample being indented placed at the 8 mm thick silicate glass plate; (6) is a tilt
mechanism; (7) is a digital camera Ximea 2.2MP MQ022CG-CM with FUJINON HF16SA-1,
2/3” lens; and (8) is the 8MR190-90-4247-MEn1 motorized rotation stage. Various modifica-
tions of the developed device had already been used to perform several studies on contact
mechanics [
25
]. Below, we discuss experiments with indentation of hard steel indenters
in soft elastic rubber sheets (elastomer) with good adhesive properties. As an elastomer,
TARNAC CRG N3005 rubber sheets with linear sizes 100 mm
×
100 mm
×
5 mm were
used (see Figure 1b, position 5). In the experiments, elastomers with thickness h= 5, 10, 15,
20 and 25 mm were used; to obtain the elastomer with different thickness, separate rubber
sheets were stacked together. Due to the strong adhesion, these rubber sheets are firmly
concatenated and did not slide over each other during the indentation. For indentation,
cylinders with a flat base of diameter D= 4, 7, 10 and 15 mm, as well as spheres with radii
R= 30, 50 and 100 mm, were used. All experiments were performed in laboratory under
room temperature (24 ◦C) and relative humidity (48%).
Lubricants 2023,11, 84 4 of 12
3. Comparison of Computer Simulations and Analytical Solutions
In the case of indentation of the rigid cylindrical stamp with a flat base of radius ainto
an elastic half-space, dependence of the normal force Fon indentation depth dis defined
by a classical expression:
F=2aE∗d,E∗=E
1−ν2, (3)
where Eand
ν
—elastic modulus and Poisson ratio of the elastic half-space. According to
Equation (3), the contact stiffness can be expressed as:
Khalf space =2aE∗. (4)
Within half-space approximation, it is assumed that the contact radius ais significantly
smaller than the thickness of the indented elastic layer. In the case of a layer with thickness
hand elastic parameters Eand
ν
placed onto rigid substrate, the approximate contact
stiffness can be estimated using the expression [11,12]:
Kah≃2aE∗n1+2εa0
π1+2εa0
π+8ε3
πa3
0
π2+a1
3+
+16ε4a0
π2a3
0
π2+2a1
3,
(5)
where small parameter εis introduced as
ε=a
h1, (6)
while the coefficients aiare defined as
am=(−1)m
22m(m!)2
∞
Z
0
Λ(u)u2mdu, (7)
Λ(u) = 2Le−4u−(L2+1+4u+4u2)e−2u
Le−4u−(L2+1+4u2)e−2u+L,L=4ν−3. (8)
Approximation (5) is valid only in the case when the radius of contact ais smaller than
the elastic layer thickness hand thus ε< 1 (6).
It is worth noting that in the opposite limit
ε
>> 1, analytical approximation is also
possible [
16
,
26
,
27
]; however, this case will not be discussed here as we consider inden-
tation into an elastomer, which is almost incompressible, and for such a material, the
abovementioned analytics are in bad agreement with both simulation and experiments.
In Figure 2, solid lines show the dependence of the contact stiffness K
a<<h
(5), normal-
ized on half-space stiffness K
half space
(4), while the thickness of elastic layer hvaries from 5
to 25 mm, with increments of 5 mm. Symbols in the figure show the results of the computer
simulations within the BEM method. The horizontal dashed line shows the threshold a=h,
i.e., when the layer thickness equals the radius of the indenter. Thus, the area of the plot
located above threshold line relates to the values
ε
=a/h> 1, and therefore, according to (5),
(6) analytical approximation is expected not to be valid and may lead to incorrect results.
However, as it can be seen from the figure, approximation (5) shows good agreement with
simulations in a certain region of ε> 1.
Lubricants 2023,11, 84 5 of 12
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0 1020304050
0
2
4
6
8
10
D,
m
m
K
a<<h
/
(2
aE
*
)
a=h
h =
5mm
h =
10mm
h =
15mm
h =
20mm
h =
25mm
Figure2.Solidlines—dependenciesofthecontactstiffnessKa<<h(5),normalizedonhalf‐spacecon‐
tactstiffnessKhalfspace(4),onthediameterofcylindricalindenterD=2a.Figureshowsfivecurvesfor
differentmagnitudesoftheelasticlayerthicknesshvaryingfrom5to25mmwithincrements5mm.
ResultsofcomputersimulationswithinBEMmethodareshowninsymbols.
AsfollowsfromFigure2,themagnitudeofKa<<h/2aE*decreaseswhenthelayerthick‐
nesshgrows.Inthelimitcaseofinfinitethicknessh¥,thesolutionisreducedtothe
half‐spaceapproximationKa<<h/2aE*=1whenthecontactstiffnessdoesnotdependonthe
indenterdiameter.AsitwasmentionedintheIntroduction,ifthethicknessofthein‐
dentedsubstrateexceedsthediameteroftheindenter,thesubstratecanbeconsidereda
half‐space.However,Figure2showsthatinrealcontact,thehalf‐spaceapproximationis
validonlyforverylargeratiosh/D.Forh=D=5mm,thestiffnessratioKa<<h/2aE*≈1.9is
almost2,meaningthatthecontactstiffnessisalmosttwicethehalf‐spaceapproximation.
Curveh=5mmreachesmagnitudeKa<<h/2aE*=1.2atD≈1.55mm.Therefore,half‐space
approximationleadstoanerrorof20%whenh/D=5/1.55=3.22,i.e.,whenthethickness
ofanelastomerexceedsthediameterofanindenterbymorethanthreetimes.There‐
strictionsofthehalf‐spaceapproximationwerediscussedindetailinourrecentstudy
[28].
4.ExperimentalVerificationoftheComputerSimulationsandTheoreticalModel
Figure3showsdependenciesofthenormalforceFNontheindentationdepthd,ob‐
tainedthroughtheindentationofthecylinderswithaflatbaseofdiameterD=4,7,10
and15mmintolayersofrubberTARNACCRGN3005ofdifferentthicknessh.Eachpanel
ofthefigureshowsfivedependenciescorrespondingtothedifferentmagnitudesofthe
layerthickness:h=5,10,15,20and25mm.Solidlinesareexperimentalresults,whilethe
resultsofthecomputersimulationsareshownwithsymbols.Everydependencemeas‐
uredinanexperimentatconstantDandhconsistsofthreecurvesobtainedinthreecon‐
secutivecyclesofindentation.Forallmeasurements,thesecurvesvisuallyoverlap.Note
thatinallexperiments,thevelocityofindentermotionwasequalto1μm/sinbothdirec‐
tions.Withthisindentervelocity,thecontactcanbeconsideredquasi‐staticandthevis‐
coelasticitycanbeneglected[29].
Figure 2.
Solid lines—dependencies of the contact stiffness K
a<<h
(5), normalized on half-space
contact stiffness K
half space
(4), on the diameter of cylindrical indenter D= 2a. Figure shows five curves
for different magnitudes of the elastic layer thickness hvarying from 5 to 25 mm with increments
5 mm. Results of computer simulations within BEM method are shown in symbols.
As follows from Figure 2, the magnitude of K
a<<h
/2aE
*
decreases when the layer
thickness hgrows. In the limit case of infinite thickness
h→∞
, the solution is reduced to
the half-space approximation K
a<<h
/2aE
*
= 1 when the contact stiffness does not depend
on the indenter diameter. As it was mentioned in the Introduction, if the thickness of the
indented substrate exceeds the diameter of the indenter, the substrate can be considered a
half-space. However, Figure 2shows that in real contact, the half-space approximation is
valid only for very large ratios h/D. For h=D= 5 mm, the stiffness ratio K
a<<h
/2aE
*≈
1.9
is almost 2, meaning that the contact stiffness is almost twice the half-space approximation.
Curve h= 5 mm reaches magnitude K
a<<h
/2aE
*
= 1.2 at D
≈
1.55 mm. Therefore, half-space
approximation leads to an error of 20% when h/D= 5/1.55 = 3.22, i.e., when the thickness of
an elastomer exceeds the diameter of an indenter by more than three times. The restrictions
of the half-space approximation were discussed in detail in our recent study [28].
4. Experimental Verification of the Computer Simulations and Theoretical Model
Figure 3shows dependencies of the normal force F
N
on the indentation depth d,
obtained through the indentation of the cylinders with a flat base of diameter D= 4, 7, 10
and 15 mm into layers of rubber TARNAC CRG N3005 of different thickness h. Each panel
of the figure shows five dependencies corresponding to the different magnitudes of the
layer thickness: h= 5, 10, 15, 20 and 25 mm. Solid lines are experimental results, while the
results of the computer simulations are shown with symbols. Every dependence measured
in an experiment at constant Dand hconsists of three curves obtained in three consecutive
cycles of indentation. For all measurements, these curves visually overlap. Note that in all
experiments, the velocity of indenter motion was equal to 1
µ
m/s in both directions. With
this indenter velocity, the contact can be considered quasi-static and the viscoelasticity can
be neglected [29].
Lubricants 2023,11, 84 6 of 12
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-0.1 0 0.1 0.2
-0.2
0
0.2
0.4
0.6
d
, mm
F
N
, N
(a)
h = 20 mm
h = 15 mm
h = 10 mm
h = 5 mm
h = 25 mm
-0.1 0 0.1 0.2
-0.4
0
0.4
0.8
1.2
d
, mm
F
N
, N
(b)
h = 20 mm
h = 15 mm
h = 10 mm
h = 5 mm
h = 25 mm
-0.2 -0.1 0 0.1 0.2
-1
0
1
2
d
, mm
F
N
, N
(c)
h = 20 mm
h = 15 mm
h = 10 mm
h = 5 mm
h = 25 mm
-0.2 -0.1 0 0.1 0.2
0
2
4
6
d
,
m
m
F
N
, N
(d)
h = 20 mm
h = 15 mm
h = 10 mm
h = 5 mm
h = 25 mm
D = 4 mm D = 7 mm
D = 10 mm D = 15 mm
Figure3.DependenceofthenormalforceFNonindentationdepthd,obtainedthroughtheindenta‐
tionofcylinderswithaflatbaseofdiameterD=4mm(a),7mm(b),10mm(c)and15mm(d)into
layersofrubberTARNACCRGN3005ofdifferentthicknessh=5,10,15,20and25mm.
Consideredcontactbetweenrubberandsteelindenterischaracterizedbyastrong
adhesion;thus,intheregionsrelatedtothedetachmentd<0mm,themagnitudeofthe
normalforceisnegativeF<0N.However,whilecomparingtheresultsofexperiments
withtheoryandsimulations,wewillconsideronlymagnitudesofindentationdepthd>
0mmwhereindenterswithflatbaseexhibitexactlythesamebehaviorinbothadhesive
andnon‐adhesivecontacts.Thiscanbeexplainedbythefactthatdetachmentofadhesive
contactstronglydependsonadhesionspecificwork,beingafunctionofthesurfaceener‐
giesofcontactingbodies.Surfaceenergy,meanwhile,isaffectedbytheoxidationanddirt
onthesurfaceduringthetimeofexperiment.Thus,thoroughcleaningofthesurfacesis
neededwhentheaimofthestudyistodetecttheeffectoftheindenterradius,surface
roughness,etc.Surfacesmustbecleanedbeforeeachcycleofindentationstrictlyaccord‐
ingtothepredeterminedprocedure.Inthepresentedexperiments,suchcleaningproce‐
dureswerenotperformedastheadhesionphenomenawerenottheaimofthecurrent
study.Asanexampleoftheexperimentsonadhesioninvolvingsurfacecleaning,wecan
refertoourpreviouswork[30].
ExperimentaldependenciesFN(d)showninFigure3exhibitonedistinguishedfea‐
ture:allobtainedcurvesdonotcrossthecoordinateorigin,evenwhenthezeroindenta‐
tiondepthd=0mmistheoreticallycorrespondingtozeronormalforceFN=0N.This
featureiscausedbycertainpeculiaritiesoftheexperiment,namelythepresenceofthe
asperitiesofvarioustypeonthesurfacesofelastomerandindenter,andtheimpossibility
Figure 3.
Dependence of the normal force F
N
on indentation depth d, obtained through the indenta-
tion of cylinders with a flat base of diameter D=4mm(
a
), 7 mm (
b
), 10 mm (
c
) and 15 mm (
d
) into
layers of rubber TARNAC CRG N3005 of different thickness h= 5, 10, 15, 20 and 25 mm.
Considered contact between rubber and steel indenter is characterized by a strong
adhesion; thus, in the regions related to the detachment d< 0 mm, the magnitude of the
normal force is negative F< 0 N. However, while comparing the results of experiments
with theory and simulations, we will consider only magnitudes of indentation depth
d> 0 mm where indenters with flat base exhibit exactly the same behavior in both adhesive
and non-adhesive contacts. This can be explained by the fact that detachment of adhesive
contact strongly depends on adhesion specific work, being a function of the surface energies
of contacting bodies. Surface energy, meanwhile, is affected by the oxidation and dirt on the
surface during the time of experiment. Thus, thorough cleaning of the surfaces is needed
when the aim of the study is to detect the effect of the indenter radius, surface roughness,
etc. Surfaces must be cleaned before each cycle of indentation strictly according to the
predetermined procedure. In the presented experiments, such cleaning procedures were
not performed as the adhesion phenomena were not the aim of the current study. As an
example of the experiments on adhesion involving surface cleaning, we can refer to our
previous work [30].
Experimental dependencies F
N
(d) shown in Figure 3exhibit one distinguished feature:
all obtained curves do not cross the coordinate origin, even when the zero indentation
depth d= 0 mm is theoretically corresponding to zero normal force F
N
= 0 N. This feature
is caused by certain peculiarities of the experiment, namely the presence of the asperities of
various type on the surfaces of elastomer and indenter, and the impossibility of positioning
substrate and indenter exactly parallel to each other. Thus, after the appearance of the
Lubricants 2023,11, 84 7 of 12
first contact point, the contact area is spreading due to the adhesion, resulting in negative
normal force FN.
In the experiment, during the indentation phase, adhesive interaction between the
surfaces is weaker compared to that in the pull-off phase. This well-known fact leads
to secondary adhesive hysteresis and corresponding differences in F
N
(d) dependencies
measured during indentation and pull-off [
30
–
32
]. Such behavior can be described by
introducing two different magnitudes of the adhesion specific work
∆γ
for both phases,
respectively, where
∆γ1
related to pull-off is significantly larger than
∆γ0
related to indenta-
tion. We found that the contact of the steel indenters with elastomer TARNAC CRG N3005
is characterized by the empirically estimated value of
∆γ0
= 0.0175 J/m
2
for indentation
phase and a range of values
∆γ1
from about 0.3 to 1 J/m
2
for pull-off [
33
]. It is worth noting
that larger-value
∆γ1
can be reached by chemical treatment of the indenter. For instance, in
ref. [
33
], after short-time treatment of the surface of steel indenter with 40% water solution
of FeCl
3
, magnitudes of
∆γ1
up to 13 J/m
2
were observed. Notably, even though chemical
treatment significantly increases
∆γ1
, it has almost no effect on
∆γ0
(which is related to
contact propagation). Dependencies plotted using symbols in Figure 3show the results of
BEM computer simulations related to the pull off of the indenter starting from maximal
indentation depth d= 0.2 mm. All simulations were performed with the same values of
elastic and adhesive parameters: E= 0.324 MPa,
ν
= 0.48,
∆γ1
= 0.326 J/m
2
. Magnitudes of
elastomer layer thickness hin simulations were chosen to be the same as in the experiments.
In the simulations, an elastic layer was located at the half-space substrate with elastic
modulus equal to E
2
= 10
100
Pa. Such an extremely large value ensures absolute rigidity
of the substrate in simulations. At the same time, in the experiment, rubber layers were
placed on the 8 mm thick silicate glass substrate with an elastic modulus exceeding that of
rubber by five orders of magnitude. This glass substrate was fixed on the aluminum table
as it is shown in Figure 1b.
On the other hand, within the BEM simulations, contacting surfaces are ideally flat
and parallel to each other; therefore, all dependencies F
N
(d) obtained from simulations start
from the coordinate origin, which makes it difficult to compare them to experimental data.
With this purpose, theoretical curves were shifted to the right along the abscissa axis by
∆
d
so both groups of data (theory and experiment) would overlap in the starting point F
N
(
∆
d)
= 0. Such type of data processing is applicable in our case, as it did not change the slope
of F
N
(d) dependencies and corresponding contact stiffness K= dF
N
(d)/dd, which is the
main subject of the current study. It is worth noting that another option for data correction
is to shift experimental curves to the left as in ref. [
29
]. Therein, both experimental and
theoretical dependencies cross the coordinate origin.
Another important detail of the experimental setup is the preparation of the substrates
of different thickness h. To obtain the elastomer with a certain hvalue, separate rubber
sheets, each with h= 5 mm, were stacked together. Due to the strong adhesion, these rubber
sheets are firmly concatenated and did not slide over each other during the indentation.
However, separate rubber sheets may have slightly different elastic properties and may
thus cause an extra disagreement between experiments and simulations. Nevertheless,
dependencies plotted in Figure 3show good agreement between the experimental results
and simulations.
In all four panels of Figure 3, dependencies F
N
(d) obtained for rubber layer with thickness h
= 5 mm show a distinctly high value of contact rigidity (highest slope of the F
N
(d) curve), while
other F
N
(d) dependencies are characterized by close values of related contact rigidity, especially
for higher h. This situation is caused by the fact that with the growth of elastomer thickness h,
the conditions of the experiment become closer to the half-space approximation limit
h>> D
;
thus, as the stiffness of the half-space K
half space
=2aE
*
is constant (at fixed indenter radius
a=D/2), all curves behave similarly. For instance, in the experiment with indenter of
a diameter D= 4 mm (see Figure 3a), the dependencies F
N
(d) measured for elastomers
with thickness h= 20 mm and 25 mm almost overlap as the ratio h/Dequals 5 and 6.25,
respectively, which practically satisfies the condition h>> D. The largest difference between
Lubricants 2023,11, 84 8 of 12
measured F
N
(d) curves was observed in the experiment with indenter of largest diameter D
= 15 mm (see Figure 3d) for elastomers of different thickness h= 20 mm and 25 mm h/D
≈
1.33 and 1.67, respectively.
As mentioned above (see description of Figure 2), even at magnitude h/D= 3.22,
application of half-space approximation leads to a contact stiffness reduction of 20%. Thus,
all dependencies shown in Figure 3d are not within the range of application of half-space
approximation. For more detailed analysis, magnitudes of contact stiffness K= dF
N
/dd
were estimated from the experimental dependencies F
N
(d) shown in Figure 3. Estimated
values are plotted as symbols in Figure 4. Figure 4also shows the contact stiffness calculated
through the BEM simulations (solid lines) and theoretical approximation (5) (dashed lines).
Lubricants2023,11,xFORPEERREVIEW8of12
equals5and6.25,respectively,whichpracticallysatisfiestheconditionh>>D.Thelargest
differencebetweenmeasuredFN(d)curveswasobservedintheexperimentwithindenter
oflargestdiameterD=15mm(seeFigure3d)forelastomersofdifferentthicknessh=20
mmand25mmh/D≈1.33and1.67,respectively.
Asmentionedabove(seedescriptionofFigure2),evenatmagnitudeh/D=3.22,ap‐
plicationofhalf‐spaceapproximationleadstoacontactstiffnessreductionof20%.Thus,
alldependenciesshowninFigure3darenotwithintherangeofapplicationofhalf‐space
approximation.Formoredetailedanalysis,magnitudesofcontactstiffnessK=dFN/dd
wereestimatedfromtheexperimentaldependenciesFN(d)showninFigure3.Estimated
valuesareplottedassymbolsinFigure4.Figure4alsoshowsthecontactstiffnesscalcu‐
latedthroughtheBEMsimulations(solidlines)andtheoreticalapproximation(5)(dashed
lines).
0246810121416
1
2
3
4
5
6
D,
m
m
K
/
(2
a
E
*
)
simulation
experiment
theory
h
Figure4.Dependenciesofthecontactstiffnessnormalizedbythehalf‐spacestiffnessKhalfspace(4)on
thediameterofcylindricalindenterD.Figureshowsfivecurves,obtainedformagnitudesofelasto‐
merthicknesshfrom5to25mmwithincrement5mm,arrowshowstheincreasingofh.Solidlines
denoteBEMsimulations,whiledashedlinesandsymbolsdenotetheoreticalapproximationand
experimentaldata,respectively.
Comparingdependenciesobtainedfromthethreedifferentmethods(experiment,
simulationsandtheory),wecanconfirmtherangedeterminedearlierwheretheanalytical
solutioncanbeapplied,andalsoconcludethattheexperimentaldataareingoodagree‐
mentwiththecomputersimulations.
5.Discussion
Intheclosingpartofourstudy,wediscussanadditionalseriesofexperimentscon‐
cerningindentationofthesphericalindenterswithdifferentradii.Intheseexperiments,
steelsphereswithradiiR=30,50and100mmwereindentedintotherubbersheetswith
thicknessh=5,10,15,20and25mm.TheobtainedresultsareshowninFigure5.Asin
thepreviouscase,solidlinesrepresenttheexperimentaldataobtainedduringthreecycles
ofindentationineveryexperiment(measuredcurvesareoverlap),whilesymbolsshow
theresultsofBEMsimulations.
Figure 4.
Dependencies of the contact stiffness normalized by the half-space stiffness K
half space
(4) on the diameter of cylindrical indenter D. Figure shows five curves, obtained for magnitudes of
elastomer thickness hfrom 5 to 25 mm with increment 5 mm, arrow shows the increasing of h. Solid
lines denote BEM simulations, while dashed lines and symbols denote theoretical approximation and
experimental data, respectively.
Comparing dependencies obtained from the three different methods (experiment,
simulations and theory), we can confirm the range determined earlier where the analytical
solution can be applied, and also conclude that the experimental data are in good agreement
with the computer simulations.
5. Discussion
In the closing part of our study, we discuss an additional series of experiments con-
cerning indentation of the spherical indenters with different radii. In these experiments,
steel spheres with radii R= 30, 50 and 100 mm were indented into the rubber sheets with
thickness h= 5, 10, 15, 20 and 25 mm. The obtained results are shown in Figure 5. As in the
previous case, solid lines represent the experimental data obtained during three cycles of
indentation in every experiment (measured curves are overlap), while symbols show the
results of BEM simulations.
Lubricants 2023,11, 84 9 of 12
Lubricants2023,11,xFORPEERREVIEW9of12
-0.1 0 0.1 0.2
0
0.2
0.4
d
, mm
F
N
, N
(a)
h = 20 mm
h = 15 mm
h = 10 mm
h = 5 mm
h = 25 mm
R = 30 mm
-0.1 0 0.1 0.2
-0.2
0
0.2
0.4
0.6
0.8
d
, mm
F
N
, N
(b)
h = 20 mm
h = 15 mm
h = 10 mm
h = 5 mm
h = 25 mm
R = 50 mm
-0.1 0 0.1 0.2
0
0.5
1
1.5
2
d
,
m
m
F
N
, N
(c)
h = 20 mm
h = 15 mm
h = 10 mm
h = 5 mm
h = 25 mm
R = 100 mm
Figure5.DependenciesofnormalforceFNonindentationdepthd,obtainedinexperimentsonin‐
dentationofsteelsphereswithradiiR=30mm(a),50mm(b)and100mm(c)intolayersofrubber
TARNACCRGN3005.Eachpanelshowsfivedependencies,relatedtodifferentthicknessesofthe
rubbersubstratebeingindented:h=5,10,15,20and25mm(shownindifferentcolors).Experi‐
mentaldataareplottedinsolidlines,whileresultsofBEMsimulationsareshownwithsymbols.
Theconditionsoftheperformedexperimentsarethesameasintheexperiment,the
resultsofwhicharepresentedinFigure3,withtheonlydifferencebeingindentationwith
sphericalindentersinsteadofcylindrical.Theelasticparametersusedforsimulationsare
alsothesame:E=0.324MPa,ν=0.48.However,inthecomputerexperiment,bothinden‐
tationandpull‐offwerealsosimulated.Fortheindentationphase,adhesionspecificwork
wassetequaltoΔγ0=0.0175J/m2,whileforthepull‐offphase,itwaschosenfromthe
experimentaldata.DependenciesshowninFigure5wereobtainedwiththemagnitudes
ofΔγ1varyingintherangefrom0.27J/m2to0.722J/m2.SucharangeoftheΔγ1valuesis
causedbythespecificfeatureoftheexperimentalprocedure,wheresurfacesofelastomer
andindenterwerenotclearedaftereverycycleofindentation,whichsignificantlyaffects
theΔγ1.
Figure5showsgoodagreementbetweenthesimulationandtheexperiment,which
confirmstheaccuracyoftheperformedcalculations.Itisimportanttonotethatelastic
parametersoftheelastomerwerethesameineachnumericalexperimentandwerenot
adjustedtoreproducetheexperimentaldata.Theonlyparameterthatwasadjustedinthe
simulationshowninFigure5isthespecificadhesionworkinthepull‐offphaseΔγ1,which
naturallychangesaftereachcycleoftheexperiment.
Figure 5.
Dependencies of normal force F
N
on indentation depth d, obtained in experiments on
indentation of steel spheres with radii R= 30 mm (
a
), 50 mm (
b
) and 100 mm (
c
) into layers of rubber
TARNAC CRG N3005. Each panel shows five dependencies, related to different thicknesses of the
rubber substrate being indented: h= 5, 10, 15, 20 and 25 mm (shown in different colors). Experimental
data are plotted in solid lines, while results of BEM simulations are shown with symbols.
The conditions of the performed experiments are the same as in the experiment, the
results of which are presented in Figure 3, with the only difference being indentation with
spherical indenters instead of cylindrical. The elastic parameters used for simulations
are also the same: E= 0.324 MPa,
ν
= 0.48. However, in the computer experiment, both
indentation and pull-off were also simulated. For the indentation phase, adhesion specific
work was set equal to
∆γ0
= 0.0175 J/m
2
, while for the pull-off phase, it was chosen from
the experimental data. Dependencies shown in Figure 5were obtained with the magnitudes
of
∆γ1
varying in the range from 0.27 J/m
2
to 0.722 J/m
2
. Such a range of the
∆γ1
values is
caused by the specific feature of the experimental procedure, where surfaces of elastomer
and indenter were not cleared after every cycle of indentation, which significantly affects
the ∆γ1.
Figure 5shows good agreement between the simulation and the experiment, which
confirms the accuracy of the performed calculations. It is important to note that elastic
parameters of the elastomer were the same in each numerical experiment and were not
adjusted to reproduce the experimental data. The only parameter that was adjusted in the
simulation shown in Figure 5is the specific adhesion work in the pull-off phase
∆γ1
, which
naturally changes after each cycle of the experiment.
It is worth noting that the theory of the adhesive contact of the parabolic indenter
and elastic layer fixed on the elastic half-space is also presented in [
11
]. The developed
Lubricants 2023,11, 84 10 of 12
solution therein allows for the attainment of the dependence of the normal force Fon
indentation depth d, similar to the data shown in Figure 5. Here, we are not comparing the
results of simulations and experiments with the abovementioned work, as the accuracy of
analytical approximation from [
11
] has already been confirmed for cylindrical stamps. We
expect similar agreement of the results obtained through the experiment and theory for
any other shape of the indenter, as the theoretical solution obtained in ref. [
11
] is based on
the analytical formalism that defines elastic properties of the elastomer and is valid for any
indenter shape [
11
,
12
]. Nevertheless, in our previous study [
34
], we compared the results
of BEM simulation with the theoretical solution from ref. [
11
]. The comparison showed
good agreement of all the data (theory, experiment and computer simulations) obtained in
the experiment on indentation of the steel sphere with a radius R= 33 mm into the rubber
layer with a thickness h= 25 mm.
6. Conclusions
We performed a theoretical, numerical and experimental study of the normal contact
between the elastic elastomer layer of a finite thickness placed on the hard substrate and
rigid indenters of different geometrical shapes (cylindrical stamps with a flat base and
spheres of different radii). The main focus of the performed research was to investigate how
both indenter radius and elastomer thickness affect the contact stiffness. The study shows
that the magnitudes of contact stiffness measured in the series of performed experiments
on indentation of the elastomer layers are in a good agreement with the ones that were
calculated through the computer simulations with the boundary elements method (BEM).
In addition, the results obtained from both simulation and experiments were compared
with the existing analytical solution. Such comparison showed partial agreement between
the theoretical and experimental data, namely when the radius of the indenter is smaller or
slightly larger than the thickness of the elastomer layer. Thus, it seems that the assumptions
behind the numerical procedure based on the BEM as formulated in ref. [
22
] are confirmed
experimentally. It is worth noting that adopted BEM simulations can be preferred over all
existing analytical solutions, as these are valid for any values of elastic layer thickness and
any geometrical shape of an indenter.
In addition to a verification of the numerical method experimentally, the present study
reveals the magnitude of error that occurs when the half-space approximation is used
to describe the indentation of a plate with finite thickness. In particular, it was shown
that when the ratio of substrate thickness to indenter diameter equals 3.22, the half-space
approximation gives a value of contact stiffness reduced by 20%.
Furthermore, the verification of the developed BEM opens up the possibility of its
application for scientific and engineering purposes. As an example of possible applications,
we can refer to experiments concerning nanoindentation of materials coated with thin films.
In this case, BEM can be used for additional analysis of the data, obtained from experiments
performed with a relatively low ratio of substrate thickness to indenter diameter. Such a
type of analysis may help to save experimental time and does not require the usage of more
expensive nanosized indenters.
Author Contributions:
Conceptualization, supervision, project administration, writing—review and
editing, V.L.P.; methodology, hardware, software, validation, experiments, simulations, experimental
data analysis, visualization, writing—original draft preparation, I.A.L.; data analysis, writing—
original draft preparation, V.B. All authors have read and agreed to the published version of the
manuscript.
Funding:
This research was funded by the Deutsche Forschungsgemeinschaft (Project DFG PO
810-55-3). V.B. is grateful to Technische Universität Berlin for support.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Lubricants 2023,11, 84 11 of 12
Data Availability Statement:
The datasets generated for this study are available on request to the
corresponding authors.
Conflicts of Interest: The authors declare no conflict of interest.
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