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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.

19

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