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Available online at www.sciencedirect.com

Wear 264 (2008) 349–358

Deterministic repeated contact of rough surfaces

J. Jamari ∗, D.J. SchipperLaboratory for Surface Technology and Tribology, Faculty of Engineering Technology, University of Twente,

Drienerloolaan 5, Postbus 217, 7500 AE Enschede, The Netherlands

Received 14 July 2006; received in revised form 16 February 2007; accepted 22 March 2007Available online 15 May 2007

bstract

A theoretical and experimental study to analyze the contact behavior of the repeated stationary deterministic contact of rough surfaces is

resented in this paper. Analysis focuses on the contact of rough surfaces on asperity level without bulk deformation. Contact area and deformationf isotropic and anisotropic contacting surfaces were simulated for each loading cycle and were plotted along with the experimental results. Veryood agreement is found between the theoretical prediction and the measurement. It was found that the contact behavior becomes elastic soon afterhe first loading has been applied for the same normal load.

2007 Elsevier B.V. All rights reserved.

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eywords: Contact mechanics; Elastic–plastic contact; Asperity; Loading–unlo

. Introduction

For many engineering applications knowledge about contactetween surfaces is of fundamental importance to understandriction, wear, lubrication, friction-induced vibrations and noise,hermal and electrical contact resistance, etc. Therefore, thenterest in studying the contact between rough surfaces is veryigh. This can be seen in the reviewing papers of Bhushan1], Liu et al. [2], Barber and Ciavarella [3] and Adams andosonovsky [4].Several approaches have been proposed to study the behav-

or of the contacting surfaces: statistical [5–11], fractal [12]nd direct or numerical simulation [13–16]. The statistical con-act model is pioneered by Greenwood and Williamson [5]here a nominal flat surface is assumed to be composed byemi-spherical asperities of the same radius and the height ofhe asperities is represented by a well-defined statistical dis-ribution function (i.e., Gaussian). In the fractal theory, thecale-dependency of the rough surface is diminished; however,he method can be applied only for fractal related surfaces.

or numerical contact modeling of surfaces, most of the pro-osed models consider pure elastic deformation; numericallastic–plastic contact models are rarely reported. Another

∗ Corresponding author. Tel.: +31 53 4892463; fax: +31 53 4894784.E-mail address: j.jamari@ctw.utwente.nl (J. Jamari).

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043-1648/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.wear.2007.03.024

mportant consideration to numerical contact modeling is theomputational cost. Karpenko and Akay [17] have introducedfast numerical method to calculate the friction force between

wo rough surfaces for the elastic contact situation. Contact forceistribution is computed, using the local contact geometry, untilhe sum of the local contact forces equals the normal load. Thisterating procedure is similar with the procedure as used in theresent paper. The method of Karpenko and Akay [17] has beenpplied to the elastic–plastic contact situation for the contactf rough wavy surfaces [18]. However, in their elastic–plasticontact analysis, a contact point deforms elasto-plastically oncets contact pressure reaches the hardness of the softer materialhile it has been widely accepted [1,2] that the elastic–plastic

ondition starts when the mean contact pressure reach about 0.4imes the hardness of the softer material.

There are many models available to analyze the contact ofough surfaces in the elastic, elastic–plastic and fully plasticontact regimes, however, the experimental investigations arearely published. A theoretically and experimentally study ofhe deterministic contact between rough surfaces in the fullylastic contact regime has been proposed by Jamari et al. [19].esults showed that the proposed model correlate very well with

he experimental investigation.

Most mechanical contact pairs carry their service load not

ust once but for a large number of (repeated) cycles. Thepplication ranges from micro/nano-systems, such as MEMSicroswitches [20], to normal systems, such as rail-wheel con-

3 / Wear 264 (2008) 349–358

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50 J. Jamari, D.J. Schipper

act [21]. Several studies have been performed to study thelastic–plastic contact behavior for the loading and unloadingase. By Finite Element Analysis calculation, Vu-Quoc et al.22] studied the relation between force and displacement of theoading–unloading of contacting spherical bodies and Li et al.23] analyzed the loading–unloading of the contact between aigid flat with an elastic–plastic sphere or a rigid sphere withn elastic–plastic half-space. Recently, Jones [24] modeled theoading and unloading of a rough surface based on the statisticalork of Greenwood and Williamson [5].This paper presents an analysis of the loading and unloading

f a stationary deterministic contact between rough surfaces.ough surfaces are modeled based on the measured real sur-

aces. Asperity or roughness deformation without any bulkeformation is considered. The model introduces a new methodo determine the initial asperity geometry and the remaining oreformed asperity geometry after unloading. In order to ver-fy the proposed model, an experimental investigation has beenerformed.

. Model

.1. Model condition

In the proposed deterministic contact of rough surfaces, theeformation of the contacting surfaces is studied only on asperityevel without any surface bulk deformation, therefore, a criterions was proposed in [25,26] is used to make sure that the contactystem is operating in the asperity deformation regime only. Aeformation criterion Cm is defined, according to [25], as:

m = p0

H

[1 − exp

(− 2

3α3/5

)]< 0.6 (1)

here H is the hardness of the softer material. p0 and α are theaximum Hertzian contact pressure and the roughness param-

ter, respectively, defined as [27]:

0 =(

16PE2

π3R2

)1/3

(2)

= σ

(16RE2

9P2

)1/3

(3)

here P is the load, R the radius of the curved body and σ ishe standard deviation of the height distribution. The effectivelastic modulus E is defined as:

1

E= 1 − v2

1

E1+ 1 − v2

2

E2(4)

here v1, v2, and E1, E2, are respectively the Poisson’s ratio andhe elasticity modulus of surfaces 1 and 2.

.2. Asperity determination

Schematic representation of the stationary repeated contactetween rough surfaces is presented in Fig. 1(a). Here, the con-act loading and unloading takes place at the same position.

twat

Fig. 1. Repeated contact of rough surfaces.

here are several ways to analyze the asperity-based determin-stic contact of rough surfaces. Greenwood [28], for instance,ntroduced a way to calculate the asperity properties using theummit method. In this method, a summit is defined as a localurface height higher than its neighboring points whether basedn five or nine points. Recently, Jamari et al. [19] proposednew method to determine the contacting asperity properties

ased on volume conservation. Contact areas (Fig. 1(b)) areetermined based on the measured rough surface loaded by aertain load P or separation given between the means of surfaceeights. These areas are converted into elliptical cross-sectionalreas (Fig. 1(c)) by conserving the total area of each asperityf Fig. 1(b). Other parameters such as asperity radii and asper-ty heights are determined based on the volume conservation

ethod, see Figs. 2 and 3. Details of the method can be foundn Appendix A.

Fig. 2 shows the discrepancy of the asperity propertiesetween the nine-points summit based asperity and the asperityolume conservation method. An example of surface protru-ion and its cut-off height is demonstrated in Fig. 2(a). Inig. 2(b)–(d), at the left-side are the properties obtained by

he volume conservation method and at the right-side are theroperties by using the nine-point summit method. Fig. 2(b)–(d)resenting the asperities property for the “indentation” by a flatf ω1, ω2 and ω3, respectively. As can be seen from Fig. 2(b)–(d),he number of asperity in contact (micro-contacts) changes as thendentation or separation changes. For the summit method, thisumber always increases as the indentation increases. For theolume conservation method, this number is changing depend-ng on the surface geometry and the location of the asperities inontact depending to the indentation. For d = ω3, for instance,

here are 40 asperities in contact according to the summit methodhilst for the volume conservation method there is only 1 large

sperity which represents a real surface in contact much betterhan the summit method. It can be concluded that the summit

J. Jamari, D.J. Schipper / Wear 264 (2008) 349–358 351

Fig. 2. 3D surface asperities and its properties. (a) Model “asperity” and its centre profile (b)–(d) location micro-contact points (not size) as a function of indentation�1 to �3, respectively. Left volume conservation method and right the nine-point summit method.

352 J. Jamari, D.J. Schipper / Wear 264 (2008) 349–358

asper

msmoc

v

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wafha

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z

wt

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ac

In the proposed model also an elastic–perfectly plastic mate-rial property without any strain hardening effect is considered.Therefore, the elastic–plastic elliptical asperity contact modelof Jamari and Schipper [29] is employed.

Fig. 3. Asperity height determination (a) surface, (b) original

ethod represents the micro-contacts well only for the verymall interference whilst the volume conservation method theicro-contacts are represented very well for all the interferences

r cut-off heights. Therefore, in the present model the volumeonservation method is used.

The curvatures of the asperity based on the volume conser-ation method, see Appendix A, are defined as:

x = 4πV

A2

Ly

Lx

(5)

y = κx

L2x

L2y

(6)

here V is the volume and A is the contact area. V and Are determined from the micro-geometry measurement, there-ore, the elliptic paraboloid representing the actual asperityas the same volume and contact area as the as measuredsperity.

After determining the asperity curvatures, by Eqs. (5) and6), another important parameter in the present deterministicontact model is the asperity height. Fig. 3 shows the methodow the asperity height is determined. The asperity height, z,s determined based on the calculated radius in x direction Rx

nd the diameter of the cross-sectional elliptical contact area indirection Lx as:

= d + (Lx/2)2

2Rx

(7)

here d is the contact separation measured from the mean con-act of the surface height of the contacting surfaces.

.3. Elastic–plastic elliptical asperity contact model

The elastic–plastic to fully plastic contact condition is consid-red in the present analysis, however, for subsequence loadinghe asperity changes due to plastic deformation, resulting into F

ity represented by paraboloid and (c) height z of paraboloid.

new asperity geometry, and as a result shifts the operatingontact regime to the elastic contact situation.

ig. 4. Flow diagram for the contact calculation of repeated stationary contacts.

J. Jamari, D.J. Schipper / Wear 264 (2008) 349–358 353

st (a),

2

cffon

ltoa

Fig. 5. Matching procedure: surface before te

.4. Calculation procedure

Principally, the procedure to simulate the repeated stationaryontact situation is similar to the calculation of single contact asor instance was presented in [19]. The only difference is that

or the repeated stationary contacts the new surfaces, as a resultf the previous loading condition, are used as the input for theext loading cycle.

Fig. 6. Setup of the experiment.

nfstctsccddtt2lattε

ctHft

surface after test (b) and their difference (c).

Fig. 4 shows schematically the iterative procedure for ana-yzing the repeated stationary contact of rough surfaces usinghe volume conservation method. The 3D surface height dataf the contacting surfaces, z1(x, y) and z2(x, y) are taken fromroughness measurement and are used as inputs. The hard-

ess properties of the material H of the surfaces are obtainedrom indentation tests. A normal load is applied to bring theseurfaces into contact. In the present study, analysis of the con-act is based on the contact interference; therefore, the load isalculated based on the separation between the surfaces. Forhis purpose, a separation is given as an initial guess. At thiseparation, there will be a number of asperities in contact (micro-ontacts). Subsequently, all the contact input parameters such asurvatures, height and location of each asperity in contact areetermined based on the volume conservation method as wasiscussed early. Given all the contact input parameters, the con-act load, the contact area, etc. can be calculated readily fromhe analysis of a single asperity as was discussed in Section.3 and formulated in [29]. Summation of all the micro-contactoads (Pi) gives the macro-contact load (input load), therefore,n iterative procedure is applied by changing the separation untilhe difference between summation of all the micro-contacts andhe applied load reach a certain criterion, ε. When the criterion

(=0.01) is satisfied the iteration loop is terminated and theontact parameters such as load, contact area, plastic deforma-

ion, etc. for each asperity are taken from the last separation.ere, the surface topography z′

1(x, y) and z′2(x, y) are the output

rom the contact model simulation. These outputs are used ashe input for the next loading cycle. The condition where there

354 J. Jamari, D.J. Schipper / Wear 264 (2008) 349–358

d (a)

itpr

3

3

cfEaoycEcsra

3

atdetm

Ftctcshti

Fig. 7. Isotropic aluminium surface before contact is applie

s no plastic deformation anymore, i.e., the difference betweenhe subsequent input and output values of the surface topogra-hy satisfies a certain running-in surface criterion εr (=0.01), iseferred as the run-in condition.

. Experiment

.1. Specimens

Experiments are performed by loading a hard and smoothurved surface onto a deformable nominally flat rough sur-ace. A hard Silicon Carbide ceramic SiC ball (H = 28 GPa,= 430 GPa and v = 0.17) with a diameter of 6.35 mm was used

s hard smooth curved indenter. To comply with the assumptionf perfectly smooth surface as was used in the present anal-sis, the r.m.s. roughness of the ceramic ball of 0.01 �m washosen. An elastic–perfectly plastic aluminium (H = 0.24 GPa,= 75.2 GPa and v = 0.34), as was used for the single asperity

ontact experiment in [30], was used for the rough flat surfacepecimen. Isotropic and anisotropic flat surfaces were used. The.m.s. roughness is respectively, 1.3 and 1.4 �m for the isotropicnd the anisotropic flat surfaces.

bttm

Fig. 8. Contact area of the isotropic aluminium surface aft

and location (not size) of the corresponding asperities (b).

.2. Matching and stitching tool

The plastic deformation of the flat surface by a spherical bodyfter being loaded against each other, for instance, is distinc-ive but the plastic deformation of the asperities (without bulkeformation) cannot be observed with the naked eye. Sloetjest al. [31] have developed a new robust technique to determinehe changes of the surface topography locally by matching the

easurement images before and after an “indentation” test.Procedure of the final matching process is described in Fig. 5.

ig. 5(a) shows the 3D measurement result of the surface beforehe test. This surface is brought into a contact with a hard spheri-al indenter. This surface is measured again after the indentationest and is presented in Fig. 5(b). In this case, to determine thehange of the surface cannot be done by simply subtracting theurface before and after the test since the surface after the testas translated and rotated relatively to the surface taken beforehe test, therefore, the matching process is needed. In the match-ng process the mutual translations and rotations are determined

y aligning or repositioning both surfaces. Once these mutualranslations and rotations are found the surfaces are matched andhe difference image is determined readily by subtracting the

atched surfaces before and after test which results in Fig. 5(c).

er the first load cycle: (a) model and (b) experiment.

/ Wear 264 (2008) 349–358 355

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J. Jamari, D.J. Schipper

.3. Experiment details

Fig. 6 shows the experimental setup used in the present exper-ments. The setup can bear a maximum load of about 300 N. Thepherical and the flat specimens were cleaned with acetone andried in air prior to any test. An optical interference microscopeas used for measuring the three-dimensional surface roughnessf the surface before and after a test.

As is shown in Fig. 6, an X–Y table which is controlled bytepper motors is used to positioning the flat specimen fromhe loading position (position A) to the surface measuring posi-ion (position A′) and the other way around. The flat surfaceas measured at the statically optical interference microscope

position A′) before the loading test. After finishing the sur-ace measurement in position A′, the flat surface was moved tohe loading position (position A). In this loading position thetatically mounted spherical specimen was moved down by aoading screw and subsequently loaded by a dead weight loadystem. The contact region was lubricated in order to reduceossible effects of friction on the contact condition. The loadas applied to the contact system for 30 s and then removedy lifting up the loading arm manually. Next, the flat surface isrought to the surface measuring position (position A′) by usinghe X–Y table. Before measuring the surface after loading withhe optical interference microscope, again the flat specimen wasleaned and dried using the same procedure as was describedarlier. Subsequently, the surface is acquisitioned by the inter-erence microscope for surface topography data. Until this step,he first loading cycle is finished.

To continue the test for the next loading cycle the same pro-edure is applied. When position A is reached, the load is gentlyeapplied. All the surface topography images from every load-ng cycle are matched with the initial surface topography image.his was done separately by a personal computer.

. Results and discussions

Fig. 7 to Fig. 9 show the deformation results for the isotropicurface. Elastic perfectly plastic aluminium as was used in [30]

hiic

Fig. 10. Anisotropic aluminium surface before load is applied (a)

ig. 9. Profile of the matched and stitched isotropic aluminium surface: (a)-profile at y = 120 �m and (b) y-profile at x = 129 �m; n = number of load cycles.

as a hardness constant for full plasticity ch of 0.71 and thenception of fully plastic deformation ω2 equals 80 times thenception of the deformation where the first yield occurs ω1 forircular contact, or cA = 160. These constants are used in the

and location (not size) of the corresponding asperities (b).

356 J. Jamari, D.J. Schipper / Wear 264 (2008) 349–358

face a

cf

italcfioctFoa

ioipctitrfdgcWsitaasstca

sis the only factor. Since the same loading condition is appliedfor cycle 1, 2, 3, 10 and 20, therefore, the same phenomena areobserved.

Fig. 11. Contact area of the anisotropic aluminium sur

ontact simulations. The explanation of these constants can beound in [30].

Fig. 7(a) presents the initial isotropic surface before broughtn contact with a hard smooth ball. A load of 0.2 N was appliedo the contact. The dashed line represents macroscopic contactrea. The position of the asperities in contact due to the appliedoad is presented in Fig. 7(b). The location (not size) of theorresponding asperities is represented by dot markers in thisgure. 0.2 N load together with the r.m.s. roughness value Rqf 1.3 �m and its material properties yields a Cm value for theriterion of Eq. (1) of 0.16 which means that deformation ofhe surface on asperity level is expected. As can be seen fromig. 9, there is no bulk deformation, all the plastic deformationccurs at asperity level so that the deformation criterion worksccurately.

The contact area prediction by the model for the cycle n = 1s depicted in Fig. 8(a). Impression of the size of the geometryf the asperities may be illustrated from this figure. The exper-mental results of the contact area from the contact system areresented in Fig. 8(b). As can be seen, the model predicts theontact area very well. More insight in the model prediction andhe measured results may be seen in the plot of the asperitiesn x and y direction as presented in Fig. 9. The model predictshe change of the surface topography accurately. Experimentalesults show that there is almost no difference between the profileor cycle 1, 2, 3, 10 and 20. In the first loading cycle the asperityeforms elastic, elastic–plastic or fully plastic depending on itseometry, location and height of the contact indentation. Theontact area for each asperity is developed to support the load.

hen the load is removed, plastic deformation will modify theurface geometry due to elastic–plastic and fully plastic deform-ng asperities which normally increase the degree of conformityo the counter-surface (indenter). If the same load is reappliedt exactly the same position (stationary), the same contact areas for the first loading will be developed. In this second loadingtep the asperities deform elastically as a result of the residual

tresses and geometrical changes induced by the first loading,herefore there is no change of the surface topography or of theontact area. Residual stresses will increase the yielding stressnd the change of the geometry will reduce the level of applied

Fx

fter the first load cycle: (a) model and (b) experiment.

tresses. In the present model the change of the surface geometry

ig. 12. Profile of the matched and stitched anisotropic aluminium surface: (a)-profile at y = 118 �m and (b) y-profile at x = 126 �m; n = number of load cycles.

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J. Jamari, D.J. Schipper

Results of the anisotropic aluminium surface are presentedn Figs. 10–12. The initial anisotropic aluminium surface andts asperity location are shown in Fig. 10. The same load asas applied to the isotropic aluminium surface is used for the

nisotropic surface. Fig. 11 presents the result of the contact areaf the model prediction and of the experimental investigation forhe first cycle. From this figure it can be seen that the surfaces dominantly represented by a single elliptic asperity. With an.m.s. roughness Rq value of 1.4 �m, it yields a Cm value of 0.15o that asperity deformation is expected. This is confirmed byhe experimental results of Fig. 12 where the plot of the x and yrofile are drawn. Results for cycle 1, 2, 3, 10 and 20 are similaro the isotropic aluminium surface, i.e., there are almost no dif-erences in the contact area and the surface topography. Smalleviations between the model prediction and the experimentalesults are observed for the y profile in Fig. 12(b). This may beaused by the assumption of the rigid indenter where, in fact,here is some elasticity of the ceramic ball indenter.

. Conclusions

Theoretical and experimental studies have been performedo analyze the deterministic contact behavior of the repeatedtationary contact of rough surfaces. Bulk deformation of theurface is excluded in the analysis and concentrated to the con-act deformation on asperity level. Contact area and deformationf the isotropic and anisotropic contacting surfaces were sim-lated for every loading cycle and were plotted along with thexperimental findings. Results show that the theoretical modelredicts the measurement results very well.

In the repeated stationary contact of rough surfaces, it wasound that the contact behavior becomes elastic soon after therst loading has been applied for the same normal load. This isecause of the degree of conformity of the contacting surfaces iseached, so that the developed contact area supports the appliedoad elastically.

cknowledgements

The financial support of SKF ERC B.V. Nieuwegein, Theetherlands and The Dutch Technology Foundation (STW) areratefully acknowledged.

ppendix A. Volume conservation method foretermining asperity geometry

In this appendix, a method to model the micro-contacts ofeal rough surfaces which asperities are represented by ellipticalaraboloids will be described. The size of the asperities is basedn a volume conservation method.

An elliptical paraboloid is defined as a paraboloid having anlliptical cross-section in the xy-plane and paraboloids in the xz-nd the yz-plane respectively, see Fig. 2. In mathematical form,

his elliptical paraboloid is expressed as:

x2

a2 + y2

b2 − z

c= 0 (A.1)

V

t

r 264 (2008) 349–358 357

here x, y, and z are the coordinate system and a, b, c are con-tants. The volume displaced, V, due to contact of this ellipticalaraboloid is the same as the volume above a certain cut-offeight (volume conservation), hence:

=∫ xhigh

xlow

∫ yhigh

ylow

∫ zhigh

zlow

dx dy dz (A.2)

The limits of integral in Eq. (A.2) are determined as follows.or the z-coordinate the upper integration limit is the cut-offeight ω of the micro-contact as:

low = ω (A.3)

nd the lower limit is determined rearranging Eq. (A.1) into:

high = c

a2b2 (x2b2 + y2a2) (A.4)

For the y-coordinate the following equations are valid at thedge of the contact:

c

a2b2 (x2b2 + y2a2) = ω (A.5)

Solving y in Eq. (A.5) gives:

low = − b

ac

√c(a2ω − cx2) (A.6)

high = b

ac

√c(a2ω − cx2) (A.7)

Now, the integration limits for x are left to consider. Thesean be determined by substituting ylow = yhigh = 0 into Eqs. (A.6)nd (A.7) which results:

low = −a

√ω

c(A.8)

high = a

√ω

c(A.9)

By substituting Eqs. (A.3)–(A.9) into Eq. (A.2) prior to inte-ration and simplifying yields:

= π

2

ba

cω2 (A.10)

The length of the micro-contact area in x-direction Lx isalculated by subtracting Eq. (A.9) by Eq. (A.8) as:

x = 2a

√ω

c(A.11)

nd the length of the micro-contact area in y-direction Ly isalculated by subtracting Eq. (A.7) by Eq. (A.6) at x = 0 as:

y = 2b

ac

√ca2ω (A.12)

Substituting Eqs. (A.11) and (A.12) into Eq. (A.10) results aew expression for the volume as:

= 1

8πLxLyω (A.13)

The curvature is defined as the second derivative of the ellip-ical paraboloid, thus the curvature κx in x-direction and the

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κ

κ

√

tf

κ

κ

R

[

[

[

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[

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[

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58 J. Jamari, D.J. Schipper

urvature κy in y-direction are found by applying the seconderivative to Eq. (A.1) as:

x = 2

a2 c (A.14)

y = 2

b2 c (A.15)

A combination is made by rearranging Eqs. (A.10)–(A.15):

κxκy = 2c

ab= 8ω

LxLy

= 4πV

A2 (A.16)

κx

κy

= L2y

L2x

(A.17)

Substituting Eq. (A.17) into Eq. (A.16) and rearranging giveshe final expressions for the elliptic paraboloid that will be usedor ‘fitting’ the real micro-contact region as:

x = 4πV

A2

Ly

Lx

(A.18)

y = κx

L2x

L2y

(A.19)

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