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Sensitivity of the Stribeck curve to the pin geometry of
a pin-on-disc tribometerI
Erik Hansen∗
Institute of Fluid Mechanics, Karlsruhe Institute of Technology (KIT),
Kaiserstr. 10, 76131 Karlsruhe, Germany
Bettina Frohnapfel
Institute of Fluid Mechanics, Karlsruhe Institute of Technology (KIT),
Kaiserstr. 10, 76131 Karlsruhe, Germany
Andrea Codrignani
Department of Microsystems Engineering, Albert-Ludwigs-University Freiburg,
Georges-K¨ohler-Allee 103, 79110 Freiburg, Germany
Abstract
A considerable number of surface texture investigations is based on pin-on-
disc tribometers. This work shows that a crucial role in the reproducibilty of
the results, i.e. Stribeck curves, is played by the geometry of the pin surface.
The investigation is based on an elastohydrodynamic model of a pin-on-disc
tribometer which is validated with experimental data. Characteristic rough-
ness and pin shapes are introduced in this model to evaluate the sensitivity
of the Stribeck curve to these operating conditions. The obtained signifi-
cant variations in the friction coefficient indicate that studies which aim at
quantifying the influence of surface textures in the mixed lubrication regime
need to provide information about the pin geometry in order to enable a
meaningful comparison among literature data.
Keywords: mixed lubrication, modelling in tribology, pin-on-disc
IThis document is the result of a research project funded by the Friedrich und Elisabeth
Boysen-Foundation.
∗Corresponding author
Email address: erik.hansen@kit.edu (Erik Hansen)
URL: www.istm.kit.edu (Erik Hansen)
Preprint submitted to Tribology International October 21, 2019
tribometer, surface texturing
1. Introduction
Since 20% of the world’s total energy consumption goes to overcome fric-
tion [1], research on drag reduction bears great potential in energy saving.
One of these technologies is the introduction of surface textures in lubricated
contacts. Surface textures in the shape of dimples can cause an additional
pressure build-up within the lubrication flow, which leads to a thicker fluid
film and less friction [2]. This positive effect is highly sensitive to the con-
tact’s operating condition and the robust and versatile design is a subject of
intensive research [3, 4]. Due to the high sensitivity of the texture parameters,
the operating conditions during their investigations must be known and con-
trolled. A widely used way to investigate surface textures under such isolated
conditions is the pin-on-disc tribometer [5]. The knowledge of all operating
parameters in this experiment allows its numerical representation, while the
experimental results enable the meaningful validation of the created digital
twin. Among others, such studies were previously carried out in the context
of a thrust bearing [6] or a piston-ring-liner contact [7]. In the case of a pin-
on-disc tribometer, setting up a digital twin based on previously published
experimental data proves to be difficult because important information, such
as a measurement of the pin curvature, is often missing [8, 9, 10, 11, 12, 13].
For the present study additional data of the pin geometry employed in the
experiments of Braun et al. [8] is used in order to determine the sensitivity
of numerical predictions on this quantity. In consequence, a suggestion of the
surface parameters that should be published along with experimental results
can be made.
The paper is structured as follows. The reference pin-on-disc tribometer
is introduced in section 2 before the setup of the corresponding numerical
model is explained in section 3. This model is based on a two-scale approach
in which the roughness scale is considered through precalculated contact
pressure and homogenization factors [14] while the modelling of the elas-
tohydrodynamic lubrication (EHL) is performed through the homogenized
Reynolds equation including mass-conserving cavitation [5, 15, 16] and the
boundary element method [17]. In section 4, the implemented model is used
to quantify the influence of the pin geometry on the Stribeck curve.
2
2. Reference Tribometer set-up
A schematic setup of the Plint TE-92 HS tribometer from Phoenix Tri-
bology (Kingsclere, UK) that was used for the experiments of Braun et al.
[8] is depicted in Figure 1. It shows the rotating disc that is pressed with the
normal force FNagainst the pin. The pin is placed on a self alingning pin
holder and the contact of pin and disc is constantly fed with oil. The setup
is heated to keep the oil temperature at 100◦C. At this temperature, the ad-
ditive free poly-alpha-olefin (PAO), Kl¨uber Lubrication (Munich, Germany)
has a dynamic viscosity of approximately µl= 0.0031Pa ·s. The pin with
a diameter of 8mm consists of normalized steel C85 (Stahlbecker, Heusen-
stamm, Germany) with a Young’s modulus of approximately E= 206GPa
and a hardness of 400HV. The disc with a diameter of 70mm is made out
of hardened and tempered (190◦C) steel 100Cr6 (AISI 5210, Eisen Schmitt,
Karlsruhe, Germany) with a hardness of 800HV.
Figure 1: Schematic set-up of the pin-on-disc tribometer as presented in [8].
During the experiments, the friction force FTacting on the pin surface is
measured to determine the friction coefficient Cf:
Cf=FT
FN
. (1)
The Stribeck curve is measured for different disc velocites Uby approx-
imating them as U= Ω ·rs, where Ω is the angular velocity of the disc and
3
rs= 30mm is the distance from the pin center to the rotation center of the
disc. Consequently, this approximation neglects velocity gradient effects [18].
In order to numerically reproduce the experimental Stribeck curve, the
forces FNand FTon the pin surface Amust be computed. This is performed
by superposition of the hydrodynamic, ambient and contact pressures, phd,
paand pc, and the corresponding shear stresses, τhd and τc[5]:
FN=ZA
ptotdA=ZA
(phd −pa+pc) dA, (2)
FT=ZA
τtotdA=ZA
(τhd +τc) dA. (3)
Realistic values for these two quantities need to be determined in order to
study the influence of pin surface geometry and pin roughness in the digi-
tal twin of the experiment. Unfortunately, this information is typically not
provided in literature.
As reference for the present study we complement the results of Braun
et al. [8] with additional measurement data that are accessible. For the
macroscopic pin profile, a surface measurement along the centerline of an
unused pin is considered and extrapolated. The result is presented in Figure 2
where rounding of the pin surface is clearly visible. This macroscopic surface
geometry originates from the polishing step during manufacturing. It should
be noted that this pin surface has not yet been subjected to wear and running
in effects which are likely to generate geometrical changes. Therefore, the
available surface geometry can only serve as a rough estimate of the maximum
gap height variations across the pin.
In contrast to the macroscopic pin geometry, information about the pin
roughness is available for used pins. The roughness profile on a run-in pin
was measured with an optical 3D-profilometer of the type PLu from SensoFar
(Barcelona, Spain) in an interferometric measuring mode with a ten times
magnifying lens. The roughness profile is shown in Figure 3 with the mean
plane set to 0µm. The center-line average is Ra= 0.107µm, the standard
deviation σ= 0.143µm, the skewness Sk =−1.26, the kurtosis Ku = 4.50,
the maximum of Rp= 0.287µm and the minimum of Rv=−0.656µm, where
these values were calculated according to Bhushan [19].
4
Figure 2: Macroscopic pin profile with
curvature due to the polishing proce-
dure. Note that the x3-axis is more
than 1000 times magnified in order to
outline the curvature of the pin.
Figure 3: Representative roughness
patch with grooves due to manufactur-
ing and wear.
3. Numerical approach
3.1. Fluid mechanics
Based on the model of Wolozynski et al. [16] in the implementation of
Codrignani et al. [5], the hydrodynamic pressure distribution phd in the gap
of height hbetween pin and disc is described by the Reynolds equation con-
sidering a mass-conserving cavitation algorithm and the cavitation condition:
∇ · h3∇phd −6hµlU
0(1 −θ)= 0, (4)
(phd −pcav)θ= 0. (5)
The cavitation pressure is estimated to pcav = 0.08MPa following [20].
The cavity fraction θ= 1 −ρ
ρldescribes the amount of cavitated lubricant,
where ρlis the density of the liquid lubricant and ρis the density of the
mixture phase. This equation system is discretized with the finite-volume-
method and solved using the Dirichlet boundary condition of ambient pres-
sure pa= 101325Pa and no cavitation at the domain boundaries. The corre-
sponding hydrodynamic shear stress τhd on the pin surface is determined as
5
[7]:
τhd =−h
2
∂phd
∂x1
+µlU
h(1 −θ). (6)
In order to consider roughness effects on the pin, the gap coordinate h0
is introduced. It descibes the distance of the disc to the mean plane of the
undeformed roughness profile on the pin. The disc is assumed to be perfectly
flat. For a value of h0≤Rp= 0.287µm, surface contact between the disc
and the roughness profile occurs. Since this causes a deformation of the
roughness profile, h0is not an adequate description of the mean gap height
during surface contact. Therefore, in analogy to Forooghi et al. [21], the
meltdown gap height hmis defined as the distance of the disc to the mean
plane of the deformed roughness profile. It is equal to h0for h0> Rpbut
differs during surface contact. Additionally, it is assumed that the surface
contact of disc and roughness profile cuts off roughness asperities but leaves
a remaining gap height of = 10−8m to improve numerical stability. This is
schematically displayed in Figure 4 and the resulting meltdown gap height
as a function of the gap coordinate h0is shown in Figure 5. Using this
dependency, the gap height hin equation (4) can be approximated by the
roughness meltdown gap height hmas a function of the gap coordinate h0:
h=hm(h0). (7)
6
h0Rp
h0hm
h0> Rp:
h0< Rp:
hm
Figure 4: Schematic comparison of gap
coordinate and meltdown gap height
with and without surface contact.
h0=Rp
Figure 5: Meltdown gap height as a
function of the gap coordinate during
surface contact for the specific rough-
ness shown in Figure 3.
While using equation (7) as a description of the gap height in the Reynolds
equation (4) only allows to take the mean plane of a roughness profile into
account, the homogenization of the Reynolds equation enables the considera-
tion of the roughness’s general shape and orientation [15]. This approach de-
composes the gap height hinto the meltdown gap height hmand a roughness
gap height hr, which depend on macroscopic coordinates ~x and roughness
coordinates ~
ξ:
h=h(~x, ~
ξ) = hm(~x) + hr(~x, ~
ξ). (8)
It is assumed that hris periodic over the representative roughness domain
A~
ξwith the lengths Y1and Y2and that these lengths are significantly smaller
than any characteristic length in the macroscopic coordinates ~x. In this
case, certain terms can be neglected when equation (8) is substituted into
the Reynolds equation (4), which is after an asymptotic expansion of the
hydrodynamic pressure phd averaged over the periodic domain A~
ξ. The result
is the homogenized Reynolds equation and the analogously derived expression
for the homogenized shear stress in dependence of the homogenization factors
A,~
b,~c and d[15, 22]:
∇ · A∇phd +~
b(1 −θ)= 0, (9)
τhd =~c · ∇phd +d(1 −θ) . (10)
7
Summarizing, the homogenization approach for the representation of the
microscale roughness is as follows: during the preprocessing, the local prob-
lems summarized in Appendix A are solved for various imposed gap co-
ordinates h0and the employed roughness profile under consideration of a
minimum remaining gap height of = 10−8m in between roughness and disc.
From their solutions, the homogenization factors are computed as a function
of the gap coordinate. This dependency of the normalized homogenization
factors is displayed in Figure 6. Since the homogenization factors are normal-
ized with the meltdown gap height hm, the homogenization method coincides
with the concept of a hydrodynamically smooth surface along the roughness
mean plane as long as the normalizations of A11,A22,b1are equal to 1 and
A12,A21,b2are equal to 0. Otherwise, the flow factors of both methods differ
from each other, which becomes visible for values below h0= 1µm.
While the microscale is represented through the homogenization approach
described above, the macroscopic pin geometry is taken into account through
the assignment of a gap coordinate h0to each coordinate ~x. This allows
solving the homogenized Reynolds equation on the macroscopic scale during
the main processing.
Figure 6: Normalized homogenization factors as function of the gap coordinate for the
specific roughness shown in Figure 3.
8
3.2. Contact mechanics
In analogy to the homogenization factors, the contact pressures and shear
stresses are determined during the preprocessing as a function of the gap co-
ordinate h0and the employed roughness profile. For each imposed h0, the
real area of contact Ac(h0) between the roughness profile and the smooth disc
is determined. Note that at this point, the earlier considered minimum re-
maining gap height of = 10−8m is not necessary for any numerical stability
and therefore not employed. Tabor [23] states that the contact of two rough
bodies does not actually occur on their whole macroscopic surface, but only
on a small fraction of it called the real area of contact. This area is described
by the contact of the surface asperities which are almost immediately plasti-
cally deformed until the real area of contact is large enough to support the
whole contact load. Since almost the entire real area of contact is plastically
deformed, the occuring contact pressures can be linked to the material’s yield
strength σY. If it is further assumed that the asperites are of a small height
and there is no relative movement between the surfaces, the contact pressure
pcin a macroscopic discretization cell of area Atot is described as:
pc= 3σY·Ac(h0)
Atot
. (11)
The precalculated contact pressure in depence of the gap coordinate is
visualized in Figure 7. Bowden and Tabor [24] also give an expression for
the shear stress τcin the contact surface of two metals without a normal
load. They explain that the metals form welded bridges which have to be
sheared off by relative motion. By applying the Mises criterion of equivalent
stress, the shear stress can be calculated as a function of the material’s yield
strength:
τc=σY
√3·Ac(h0)
Atot
(12)
Assuming ideal elastic-plastic material properties, the yield strength can
be approximated from the Vickers hardness as described in Appendix B.
It is important to mention that equation (11) assumes no shear load while
equation (12) assumes no normal load. If both are present, contact and shear
stresses are actually lower than those given by the equations. However, these
equations are used in the employed model since they excel in computation
efficiency and the resulting friction coefficent in the boundary lubrication
regime reflects experimental data reasonably well.
9
Figure 7: Contact pressure as a function of the gap coordinate. The value of h0= 0
corresponds to the roughness meltdown plane for the specific roughness shown in Figure
3.
3.3. Elastohydrodynamic deformation
The sum ptot of the hydrodynamic and contact pressure fields elastically
deforms the pin on the macroscopic scale, thereby changing the initial gap
height distribution. To take this into account, the elastic deformation hel of
the pin surface Ais described by the elastic half-space model [17]:
hel(x1, x2) = (1 −ν2)
πE ZZ
A
ptot(x0
1, x0
2)
q(x1−x0
1)2+ (x2−x0
2)2dx0
1dx0
2, (13)
where νis Poisson’s ratio and Eis Young’s modulus. When the surface A
is discretized into rectangles, equation (13) can be expressed in dependency
of a Kernel function K[17, 25]:
hel(x1, x2) = X
x0
1X
x0
2
K(x1−x0
1, x2−x0
2)·ptot(x0
1, x0
2). (14)
The computation of the elastic deformation due to a pressure field is
accelerated by using the Fourier transformation F[17]:
hel =F−1(F(K)· F (ptot )) . (15)
10
On the downside, at this point the discretized domain has to be increased
and padded with zeros to perform a linear convolution instead of a circular
one, which increases the computational costs of the convolution. After its
calculation, the domain is resized to its old range. The resulting gap coordi-
nate h0is then computed as the superposition of the gap coordinate above
the initially rigid pin h0,ri and its elastic deformation hel:
h0(ptot) = h0,ri +hel (ptot ). (16)
Since h0depends upon the total pressure profile ptot , which also depends
upon the gap coordinate h0, finding the equilibrium requires an iterative
procedure. At first, for an initial pressure profile p(i)
tot, the elastic displace-
ment and its resulting pressure distribution ptot,II are computed. Then, the
residuum rel of the two pressure fields is calculated and the pressure field for
the next iteration step p(i+1)
tot is determined by underrelaxation as long as rel
is higher than a threshold of tol = 10−5:
rel =1
NpZNp
|ptot,II −p(i)
tot|
pa
dn, (17)
p(i+1)
tot =p(i)
tot +α(ptot,II −p(i)
tot). (18)
where Npis the total number of the discretization points nabove the
pin, pais the ambient pressure and αis the underrelaxation factor. It is set
to α= 0.5 in the hydrodynamic and α= 0.05 in the mixed and boundary
lubrication regime to achieve a good tradeoff between convergence speed and
stability.
4. Results
A digital twin of a pin-on-disc tribometer should eventually allow the
prediction of Stribeck curves. While exact agreement with experimental
data is challenging [5], the present model can be used to indicate poten-
tial sensitivities of the Stribeck curve to the macroscopic pin geometry and
the microscopic surface roughness. Therefore, different roughness represen-
tations on the measured macroscopic pin geometry as well as variations of
the macroscopic geometry are considered in the following.
The parameters used for the simulations are summarized in Table 1. In
order to have a stationary numerical problem, the disc is assumed to be
11
perfectly flat. Since it is of a harder material than the pin, the disc is also
assumed to be rigid. Consequently, elastic deformations are only modelled
for the pin surface.
Parameter Value Unit Description
E206 ·109Pa Young’s modulus
FN150 N tribometer load force
Np1012−number of pin discretization cells
Nr3712−number of roughness discretization cells
pa101325 Pa ambient pressure
pcav 80000 Pa cavitation pressure
U0.01...5 m/s disc velocity
tol 10−5−relative error tolerance
µl0.0030758 Pa ·s dynamic viscosity of uncavitated lubricant
ν0.321 −Poisson’s ratio
σY1200 ·106Pa yield strength
Table 1: Numerical parameters.
4.1. Influence of the roughness on the Stribeck curve
In section 3.1, two approaches of considering the surface roughness in the
Reynolds equation were presented and will be compared in the following.
The first approach is simply using the meltdown gap height in the Reynolds
equation. The second method consists of using the homogenized Reynolds
equation. While the homogenization method allows the consideration of aver-
aged roughness effects, it also increases the computational costs compared to
just using the meltdown height gap in the Reynolds equation. The reason is a
less sparse system matrix because of the off-diagonal homogenization factors
in Matrix Aand the additional interpolation of the homogenization factors.
Based on the measured pin and roughness profiles shown in the Figures 2
and 3, the Stribeck curves are computed with both roughness methods and
the obtained results are displayed Figure 8. Note that both methods use the
roughness profile for the computation of the contact mechanics as described
in section 3.2. Their difference is only in the consideration of roughness ef-
fects on the hydrodynamics and it can be seen that almost identical results
are obtained. However, the computation using the homogenized Reynolds
equation took about 3 times longer than the other one. Since roughness
12
effects on the hydrodynamics are apparently negligible for the presently con-
sidered roughness profile, only the more efficient concept of the meltdown
gap height is used in the following.
Nonetheless, it should be noted that the roughness profile is important
for determining the onset of the contact pressure contribution which in turn
strongly influences the transition from the purely EHL to the mixed lubrica-
tion regime. If the gap coordinate h0at any point ~x above the pin becomes
less than the value of the highest roughness asperity Rp, surface contact oc-
curs through the contribution of the contact mechanics. Thus, the transition
point of the purely EHL to the mixed lubrication regime can be defined as
the critical disc velocity Ucat which the minimum gap coordinate above the
pin min (h0(~x)) is equal to Rp. This is visualized in Figure 9 with the results
of the meltdown gap height method.
Figure 8: Stribeck curves using melt-
down gap height and homogenization.
Uc
mixed
lubrication
purely
EHL
Figure 9: Minimum gap coordinate
above the tribometer pin compared to
Rpas a function of the disc velocity.
4.2. Influence of the pin shape on the Stribeck curve
To investigate the influence of a different macroscopic gap height, the
pin profile is approximated by two parabolas. One parabola is designed to
closely fit the measured pin profile in the center while the other one is chosen
(within a parameter study) such that it captures the experimentally deter-
mined transition point from EHL to mixed lubrication in the Stribeck curve.
The corresponding pin profiles and computed Stribeck curves are depicted in
13
Figures 10 and 11. The predicted friction coefficient is in very good agree-
ment with the experimental data [8] in the boundary regime while there is a
difference by about one order magnitude in the EHL regime. This difference
might be related to the neglected temperature and pressure dependence of
the fluid viscosity on the numerical side or the low signal-to-noise-ratio of
the experiment in this regime. While the numerical predictions almost coin-
cide in EHL regime, the transition points from the purely EHL to the mixed
lubrication regime and the corresponding friction coefficients show a clear
dependence on the different macroscopic pin geometries. This dependence is
further investigated in the following. Therefore, the parabola center height
lcis defined as the difference of the parabola profile height in the center of
the pin to the profile height at the pin’s rim. The pin parabola with a center
height of lc= 2µm is taken as a reference profile because it fits the experi-
mental data closely. Further simulations are carried out with pin parabolas
of the center heights lc= 1.9µm, lc= 1.5µm and lc= 1µm. They correspond
to a relative decrease of 5%, 25% and 50% of the reference center height. The
resulting Stribeck curves are displayed in Figure 12. Afterwards, the change
in the friction coefficient relative to the reference profile is computed for each
velocity as displayed in Figure 13. It shows that a measurement deviation of
25% or 0.5µm in the characteristic length of the reference pin causes a max-
imum difference in the friction coefficient of more than 80%. Therefore, the
observed difference between the experimentally determined Stribeck curve
and the model prediction for the measured pin geometry in the mixed lubri-
cation regime could be caused by the fact that the macroscopic pin geometry
differs for a run-in pin.
14
lc= 2µm
Figure 10: Gap coordinates for different
pin profiles.
Figure 11: Stribeck curves for different
pin profiles and the experimentally de-
termined Stribeck curve of Braun et al.
[8].
Figure 12: Stribeck curves obtained
by the variation of the reference pin
parabola.
Figure 13: Relative difference in the
friction coefficient compared to the ref-
erence pin parabola.
5. Conclusions
In order to investigate the sensitivity of a predicted Stribeck curve we
studied the macroscopic pin shape and its microscopic roughness in a simpli-
15
fied digital twin of a pin-on-disc tribometer. The numerical model consists
of the homogenized Reynolds equation with mass-conserving cavitation, the
boundary element method and the contact mechanics model of Bowden and
Tabor [23, 24]. The model is capable of simulating the tribometer in EHL,
mixed and boundary lubrication conditions.
The experimental results of Braun et al. [8], complemented by informa-
tion about the macroscopic pin geometry and surface roughness, serve as
reference data for the numerically predicted Stribeck curve.
The main findings of the work can be summarized as follows:
•The measured surface roughness that is employed in this work has neg-
ligible influence on the tribometer’s Stribeck curve in the EHL regime
but significantly affects the relative velocity at which the transition
from the purely EHL to the mixed lubrication regime occurs.
•The mixed lubrication regime of the Stribeck curve is highly sensitive
to the macroscopic pin geometry. An approximation of the pin profile
through a parabola allows a quantitative estimation of the pin geometry
influence on the Stribeck curve. It indicates that a 0.5µm variation in
the characteristic length of the pin profile can cause a deviation in the
predicted friction coefficient of more than 80%.
•In order to enable the comparison of experimental results with numeri-
cal predictions, which is critical for the further development of a digital
twin of a pin-on-disc-tribometer, macro- and microscopic surface mea-
surements of the pin and disc profiles in worn conditions need to be
provided.
We note that further steps to create a digital twin do not only require the
knowledge of the geometric properties discussed in this work in addition to
the values listed in Table 1. Moreover, the self aligning pin holder may allow
variations of the pin inclination which induces changes in the gap height
distribution. Also geometrical imperfections and EHL effects on the disc
should be considered.
Acknowledgments
The authors want to express their gratitude to Dr. Paul Schreiber for
providing the measurements of the representative roughness used by Braun
16
et al. [8]. This document is the result of a research project funded by the
Friedrich und Elisabeth Boysen-Foundation.
17
Appendix A. Homogenization
The homogenization factors A,~
b,~c and dare determined as [15, 22]:
A(h0) =
1
Y1Y2R
A~
ξ
h31 + ∂χ1
∂ξ1dA~
ξ
1
Y1Y2R
A~
ξ
h3∂χ2
∂ξ1dA~
ξ
1
Y1Y2R
A~
ξ
h3∂χ1
∂ξ2dA~
ξ
1
Y1Y2R
A~
ξ
h31 + ∂χ2
∂ξ2dA~
ξ
, (A.1)
~
b(h0) =
1
Y1Y2R
A~
ξ
h3∂χ3
∂ξ1−6hµlUdA~
ξ
1
Y1Y2R
A~
ξ
h3∂χ3
∂ξ2dA~
ξ
, (A.2)
~c(h0) =
1
Y1Y2R
A~
ξ−h
21 + ∂χ1
∂ξ1dA~
ξ
1
Y1Y2R
A~
ξ−h
2
∂χ2
∂ξ1dA~
ξ
, (A.3)
d(h0) = 1
Y1Y2Z
A~
ξ
−h
2
∂χ3
∂ξ1
+µlU
hdA~
ξ. (A.4)
Bayada et al. [15] only employed longitudinal, transversal or oblique
roughness profiles and could therefore derive an analytical solution of the
homogenization factors. In order to consider arbitrary roughness profiles,
the follwoing local problems must be solved numerically for their solutions
χ1,χ2and χ3in [22]:
∇~
ξ·h3∇~
ξχ1=−∂h3
∂ξ1
, (A.5)
∇~
ξ·h3∇~
ξχ2=−∂h3
∂ξ2
, (A.6)
∇~
ξ·h3∇~
ξχ3= 6µlU·∂h
∂ξ1
. (A.7)
The indexed nabla operator ∇~
ξmeans using it with respect to the ~
ξ
coordinates.
Appendix B. Relation of Vickers hardness and yield strength
Depending on whether the Vickers hardness is calculated from the in-
dentation mass or force, there are two slightly different hardness definitions.
18
Tabor describes the Vickers hardness HVTas [23]:
HVT=W
AS
= 0.9272 W
AG
= 1.8544W
d2, (B.1)
where Wis the applied indentation mass in kg and ASis the area of
the sides of the indented volume. For a Vickers indentation pyramid, AS=
AG/0.927 holds, where AG=d2/2 is the projected surface of the indented
volume. The diagonal length of AGis given by din mm. The resulting hard-
ness HVThas the units kg/mm2. In this case, the ultimate tensile strength
in MPa roughly correlates to:
U T S ≈3HVT·m/s2. (B.2)
However, there exists another definition of the Vickers hardness HV in
dependency of the indentation foce FIin the unit N, which is commonly used
e.g. in Germany [26]:
HV = 0.189FI
d2. (B.3)
From experiments, an approximation for the ultimte tensile strength in
MPa was found [26]:
U T S ≈3HV . (B.4)
In both cases, the yield strength is equal to the ultimate tensile strength
σy=U T S if ideal elastic-plastic material properties are assumed.
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