Effect of Asperity Interactions on Rough Surface Elastic Contact Behavior- Hard Film o

8/10/2019 Effect of Asperity Interactions on Rough Surface Elastic Contact Behavior- Hard Film o

http://slidepdf.com/reader/full/effect-of-asperity-interactions-on-rough-surface-elastic-contact-behavior- 1/11

Effect of asperity interactions on rough surface elastic contact behavior: Hard

film on soft substrate

Chang-Dong Yeo 1, Raja R. Katta, Jungkyu Lee, Andreas A. Polycarpou

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

a r t i c l e i n f o

Article history:

Received 3 December 2009

Received in revised form24 January 2010

Accepted 28 January 2010Available online 2 February 2010

Keywords:

Contact mechanics

Surface roughness

Asperity i nteractions

Coating

a b s t r a c t

An improved elastic micro-contact model of rough surfaces accounting for asperity interactions is

proposed. The contact behavior of a single asperity system is composed of a stiffer hemi-spherical

asperity deformation and bellowing softer substrate deformation, which is then extended to roughsurface contact including asperity interactions. Using the solution of substrate deformation, normal

positions of individual asperities are adjusted during quasi-static contact, from which surface

interactive forces are obtained. Analytical simulations are performed using the proposed rough surface

contact model, whose results are compared to Greenwood–Williamson-based models and with

experimental measurements.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The Greenwood–Williamson contact model (GW model) [1]

has been used as the basis of rough surface contact models. In theGW model, only elastic asperity deformation is considered

neglecting substrate deformation and asperity interactions that

could occur during contact. Moreover GW assumes that all surface

asperities are spherical with the same radius of curvature, whose

heights follow a normal (Gaussian) distribution. Chang et al.

(CEB model) extended the GW model to include elastic–plastic

deformation of the contacting asperities [2]. An improved version

of the basic adhesion model by Derjaguin et al. (improved DMT

model) [3] was employed in the CEB model to determine adhesive

forces during rough surface contacts. Kogut and Etsion (KE model)

proposed a further improvement in the CEB model based on finite

element method (FEM) simulations to describe elastic, elastic–

plastic, and fully plastic deformation of the contacting asperities

during rough surface contact [4]. Stanley et al. [5] and Polycarpouand Etsion [6] included the effects of molecularly thin lubricant

film on the adhesive forces during elastic–plastic rough surface

contacts.

In recent thin solid film applications, for improved reliability

and performance, very hard and extremely thin solid films are

typically applied onto softer substrate materials [7–9]. For

example, in magnetic recording disks of current hard-disk-drives

(HDD), an amorphous carbon film of few nanometers thickness is

deposited onto a softer magnetic material, and its root-mean-

square (RMS) roughness is sub-nanometer [9]. Considering the

small carbon film thickness and surface roughness values, it canbe said that surface asperities are composed of harder carbon

material, while the substrate is a softer magnetic material [10].

For the contact of such ultra-thin and hard film applications, the

contact behavior cannot be modeled by asperity deformation only

(GW-type model) due to the presence of considerable substrate

deformation. Shi and Polycarpou [11] experimentally measured

contact stiffness values between a recording slider and a typical

magnetic recording disk, and compared the measurements to a

GW-based model. It was found that the contact stiffness predicted

using GW-based asperity deformation alone was 107–108 N/m,

which was significantly higher than the experimental measure-

ments of 106 N/m. Therefore, to correctly predict the contact

behavior of such ultra-thin films, the contact model should

include the effects of substrate deformation and asperity interac-tions.

Researchers have proposed contact models to account for

substrate deformation and asperity interactions for rough con-

tacting surfaces [12–17]. Iida and Ono [12] used a stiffness model

for substrate deformation and an elastic point contact solution for

asperity interactions. Zhao and Chang [14] applied a uniform

mean contact pressure to calculate the mean of asperity heights

and asperity deformation caused by adjacent contacting aspe-

rities. Ciavarella et al. [15] used a discretized version of the GW

model which accounts for first order interaction effects. This was

done by assuming that for every asperity contact, a displacement

will be imposed on the half-space such that the magnitude

ARTICLE IN PRESS

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/triboint

Tribology International

0301-679X/$- see front matter & 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.triboint.2010.01.021

Corresponding author.

E-mail address: [email protected] (A.A. Polycarpou).1 Currently with Texas Tech University, Lubbock, TX.

Tribology International 43 (2010) 1438–1448

http://-/?-

http://www.elsevier.com/locate/triboint

http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.triboint.2010.01.021

mailto:[email protected]

mailto:[email protected]

http://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.triboint.2010.01.021

http://www.elsevier.com/locate/triboint

http://-/?-



ARTICLE IN PRESS

(but not shape) of the pressure distribution due to other

contacting asperities will also get affected accordingly. Subse-

quently, using a different approach, Ciavarella et al. [16] improved

upon the integral formulation of the original GW model to include

0th order asperity interactions by treating the contact pressure as

uniformly distributed over the apparent contact area and the

resulting deformation as uniform (in addition to the Hertzian

asperity deformation). The validity and applicability of these

contact models have not been investigated through experiments.

In this study, an elastic rough surface contact model of ultra-

thin harder film on a substrate accounting for asperity interac-

tions is proposed. Based on the Hertzian contact solution, the

normal positions of individual asperities are adjusted during

contact, which are then used to calculate the surface normal

interactive and lateral friction forces. Analytical contact simula-

tions were performed using the proposed model, and the results

were compared with GW-based models and experimental

measurements.

2. Modeling of elastic rough surface contact

2.1. Single asperity contact

A simplified single asperity contact model accounting for

substrate deformation was developed and verified by Yeo et al.

[10]. Fig. 1 shows a schematic of a single asperity contact, where R

is the asperity radius of curvature and r b is the semi-width at the

base of the asperity. Elastic asperity deformation (da) is expressed

as a function of applied displacement (d) and given by

da ¼ d

1þkx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid

1þkx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid=ð1þkÞp s " #

ð1Þ

where k ( =E a/E b) is a physical parameter of the elastic modulus

ratio between aseprity and substrate, and x ð ¼ 3 ffiffiffi

Rp

=2r bÞ is

a geometrical parameter determining the asperity size. The

parameter k shows the relative difference in the material

sitffness of the top layer and the substrate. For a rough surface

with a given R, the parameter x indicates the type of asperity,

whether tall or short, since for a tall asperity r b will be larger and

vice-versa. Note that to enable the derivation of Eq. (1), the

constant asperity base semi-width r b was used instead of the

variable contact half-width (contact radius) a. The validity of this

assumption is analyzed in this work and will be discussed later.

Using this single asperity contact model, it was found that for

Nomenclature

An nominal contact area

Ar real contact area

a semi-contact width

ah asperity height

c 1–c 5 coefficients related to Poisson’s ratio (CEB friction

model)E a asperity effective elastic modulus

E b bulk substrate effective elastic modulus

F applied (external) force

F c contact force

F s adhesive force

f dimensionless function (CEB friction model)

H hardness

H a hardness of the asperity material

h separation based on surface heights (rough surface

contact)

K hardness coefficient

k combined asperity and bulk stiffness

ka asperity stiffness

kb bulk substrate stiffnessN total number of asperities on the nominal area

P c contact force (rough surface contact)

p0 maximum contact pressure

Q friction force

R asperity radius

r radial distance from the center of contact

r b radius of circular contact region on top of bulk

substrate (semi-width)

sp non-contact separation of an asperity

U z normal displacement of the contact surface

Z deformed spherical profile after contact

z 0 equilibrium spacing

d normal applied displacement (or surface approach)

da asperity deformationDg adhesion energy

Z areal density of asperities

k physical index parameter

m mean plane of asperity height

na asperity Poisson’s ratio

nb bulk substrate Poisson’s ratio

x geometrical index parameter

s RMS roughness (surface heights)

ss standard deviation of asperity heights

f distribution function of asperity heights

c plasticity index

oc critical interference at the inception of plastic defor-

mation

oT transition surface interference

Subscript

i represents a certain asperity (1 to N)

r b

F

b

a

R

ah

Softer substrate

Hemispherical

asperity

a

Softer substrate

Hemispherical

asperity

Fig. 1. Single asperity contact.

C.-D. Yeo et al. / Tribology International 43 (2010) 1438–1448 1439



ARTICLE IN PRESS

single asperity surfaces representative of micro and nanoscale

applications with harder films (coating) on a substrate

(x48.66 103 m1/2 and k41.25), the proposed model is more

accurate compared to the conventional Hertz contact solution by

20–70% [10].

2.2. Rough surface contact

As used in many GW-based contact models, the statistical(Gaussian) distribution function (f) is convenient to describe the

contact behavior of randomly distributed asperity heights.

However, it is not feasible to determine the contact behavior of

each individual asperity separately from the statistical distribu-

tion function, thus making it difficult to include the effects of

asperity interaction into rough surface contact. Hence in this

work, the approach of replacing the integrals with discrete sums

of each asperity contact, similar to the approaches taken by Iida

and Ono [12] and Ciavarella et al. [15] was adopted.

2.2.1. Surface asperity geometry and position

To account for asperity interaction as well as asperity and

substrate deformation, it is necessary to determine the geome-

trical dimensions (asperity height ah, semi-width of asperity r b,and asperity radius R) and spatial information ( x y z coordinate

data) for individual asperities on the rough surface. In this study it

was assumed that the asperity heights follow a normal (Gaussian)

distribution function. Note that other asperity height distribution

functions could also be used e.g. asymmetric distribution [18]

All asperities are positioned on the same base plane, whose

z -level (height) was determined from the distribution of asperity

heights as shown schematically in Fig. 2. Based on the value of

the nominal mean plane of asperity heights (m) and the

standard deviation of the asperity heights (ss) obtained from

experimentally measured roughness parameters, a plane which is

3 ss below the mean plane (m) was chosen as the base plane, on

which all asperities are positioned. In Fig. 2, spi is the non-contact

separation of an asperity and d is the surface interference (applieddisplacement) of an asperity. As described in Section 2.1,

the asperity deformation (da) is not the same as d but is given

by Eq. (1).

In accordance with the GW model, for a specified nominal

contact area ( An), the number of asperities (N ) can be obtained

from N = AnZ, where Z is the areal density of asperities. Since the

standard deviation (ss) and the mean plane (m) of asperity heights

are known roughness parameters, individual asperity heights (ahi)

for N number of asperities can be determined using the Gaussian

distribution function. Accordingly, the asperity semi-width (r bi)

for an individual asperity is given by

r bi ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2RahiÞ ahi

p ð2Þ

Thus, with the roughness parameters experimentally measuredusing for example an atomic force microscope or any other

topographical instrument and for a predetermined nominal

contact area, the geometrical dimensions and x y z coordinate

data for all asperities are mathematically determined. Once these

dimensions are determined we can account for asperity interac-

tions in the rough surface contact analysis.

2.2.2. Asperity interactions

Ciavarella et al. [15] used the solution of Hertzian pressure

applied on the circular region of the asperity contact (Eq. (3.42a)

in [19]) to obtain the displacement of the half-space surrounding

the asperity. Note that they treated the rough surface contact as a

rigid rough surface contacting an elastic flat half-space. Since thefocus of our work is to account for the coated rough surface

contact interactions, a rigid rough surface approximation is not

appropriate. Hence the contact was treated as a rigid flat

contacting a coated elastic rough surface. A simplified version of

Eq. (3.42a) from [19] is used to obtain the displacement of the

bulk substrate surrounding the contacting asperity and given by

Eq. (3) below. In Eq. (3) the asperity contact half-width a is

replaced with the constant asperity base half-width r b (Fig. 2).

This approximation works well for dimensions typical of

asperities pertaining to smooth surfaces used in micro/nanome-

chanical applications and does not need a nonlinear or a

linearized iterative scheme to solve a multi asperity contact

problem. In Eq. (3), the Hertzian pressure is applied on the bulk

through the base of the asperity when in contact. The appliedpressure causes bellowing displacement of the bulk outside the

contact region as shown in Fig. 3, which affect the normal position

and consequently the contact behavior of the neighboring

asperities. The magnitude of the normal displacement outside

the contact region (U z ) is dependent upon the distance (r ) and

given by

U z ¼ p0

2r bE b½ð2r 2b r 2Þsin

1ðr b=r Þþr 2ðr b=r Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1r 2

b=r 2Þ

q ; ðr Zr bÞ

ð3Þwhere p0 is the maximum pressure in a circular region expressed

as

p0 ¼

3

2

F c

pr 2b ð4Þ

From Fig. 3, it is seen that the contact center exhibits the largest

normal displacement and decreases away from the contact center.

For the contact of rough surfaces, since one asperity contact

affects the normal positions ( z -level) of the adjacent asperities,

the contact behavior should account for the effect of all contacting

asperities. In this study, when two surfaces are in contact, the

normal positions of the surface asperities are adjusted for each

asperity contact in sequence. Since the contact is elastic, the

displacement arising from each asperity contact can be super-

imposed linearly to obtain the total displacement. Fig. 4 shows

schematically the normal position change of individual asperities

on the surface during rough surface contact. When the tallest

Mean planeof asperity heights

- 3 s

spi

R

Baseline (Baseplane) of asperities

Flat Surface Asperity height

distribution

h- 3 s

i

R

Baseline (Baseplane) of asperities

Flat Surface Asperity height

distribution

h

Fig. 2. Surface asperities positioned on a base plane with the relevant parameters

R, d, m, ss, h, spi) indicated.

Applied Surface Displacement

U z (r)

r b

r

Fig. 3. Single asperity normal displacement outside the contact region.

C.-D. Yeo et al. / Tribology International 43 (2010) 1438–14481440



ARTICLE IN PRESS

asperity (denoted a in the figure) initiates contact with the

counteracting surface, the neighboring asperities also move

upward with a magnitude that depends on their distance from

the contacting asperity determined by Eq. (3). Thus, the normal

positions of all asperities are adjusted accordingly. Since the

force on one contacting asperity results on neighboring asperities

adjust their positions, this process is termed as asperity

interaction. In the new configuration, subsequent contact on the

next asperity (denoted b in the figure) enables the neighboring

asperities to move further, thus re-adjusting the normal positions

of the surface asperities. In this manner, the normal positions of

all asperities are adjusted based on the deformation induced by

the pressure acting on the substrate through the contacting

asperities. Once the equilibrated normal positions of the surfaceasperities are determined from all the asperity interactions, the

adjusted interference on each contacting asperity is determined

from which the interfacial forces, namely contact, adhesion and

friction could be calculated.

2.2.3. Interfacial forces

Fig. 5 shows the flowchart to obtain the normal position

of all asperities and to obtain the surface interactive forces at

a prescribed surface separation (interference). As described in

Section 2.2.1, once the roughness parameters and nominal contact

area are known (from measurements), the geometrical

dimensions (ahi, r bi, and R) and x y z coordinate data for all

asperities are determined mathematically. At a given surface

separation value (hi), the existence of contacting asperities is

examined first. If there are no contacting asperities, the surface

interactive forces are calculated from this non-contact condition.

However, if one or more asperities are contacting, the normal

positions of all asperities are adjusted for every asperity contact in

sequence (using Eq. (3)). With the adjusted normal position data,

the interactive forces for individual asperities are calculated, thus

the surface interactive forces for the given normal contact area are

determined by summing up the values from individual asperities.In this work, the contact force (F ci) for an individual asperity is

based on the work of Yeo et al. [10] and given by

F ci ¼ k d ð5Þwhere

ka ¼ 2 ffiffiffi

Rp

E a ffiffiffiffiffida

p ð6Þ

kb ¼ 43r bE b ð7Þ

k ¼ 1

kaðdaÞþ 1

kb

1

ð8Þ

Subscript i (=1–N ), indicates an individual asperity. The contact

force obtained from Eq. (5) includes the effects of both the harderasperity and the softer bulk substrate. Note that this solution is

accurate for cases where the asperity is stiffer than the substrate.

For a homogeneous asperity-substrate system, the simple contact

force calculated from Hertz solution will be sufficient [10]. Thus

the contact force given by Eq. (5) accounts for both asperity and

substrate deformation. This single asperity model is implemented

into a ‘‘discrete’’ GW rough surface normal contact model

framework, also accounting for the effect of asperity interactions

using a simplified method as described in Section 2.2.2. Conse-

quently the interfacial forces i.e., adhesive and friction forces are

calculated based on the proposed normal contact model.

The adhesive force (F s) was based on the improved DMT

adhesion theory [3], where the contact and non-contact adhesive

forces are determined separately by

F si ¼ 8pRDg z 20

3

1

ð z 0spiÞ2 0:25 z 60

ð z 0spiÞ8

" #; ðnon-contactÞ ð9Þ

F si ¼ 16pDg

3 z 0

Z 1

a

z 0 Z i

3

z 0 Z i

9" #

r dr ; ðcontactÞ ð10Þ

where spi is the non-contacting separation for an individual

asperity (Fig. 2), z 0 is the equilibrium spacing and for carbon is

0.1nm [20], and Dg is the adhesion energy of the contacting

surfaces. In Eq. (11), Z i is the deformed sphere profile based on

Hertz contact and is expressed as

Z iðr ; aiÞ ¼ 1

pR ai ffiffiffiffiffiffiffiffiffiffiffiffiffi

r 2a

2i

q ð2a2i r

2

Þtan1 ffiffiffiffiffiffiffiffiffiffiffiffiffir 2

a2i1

s " #þ z 0 ð11Þ

where a i is the semi-contact width of an asperity and calculated

by ai ¼ ffiffiffiffiffiffiffiffiffi

Rdaip

. As shown in Eqs. (9)–(11), the adhesive force is

determined from the geometry of a spherical asperity in contact

or non-contact condition. The implementation of the above

adhesion model into GW-type statistical contact models is

described for example in Chang et al. (CEB) [21] and Polycarpou

and Etsion [6]. Note however that in this work we use a ‘‘discrete’’

approach, so that the total force is the summation of individual

forces on each asperity. An important assumption made is that

the adhesive force F s does not affect the deformed profile of the

asperity (caused by the contact force), but the deformed profile

due to contact influences the adhesion force as seen in through

Eq. (10) in the improved DMT model.

r i

z r

U z ( r i )

U z (r ) = 0

Applied Surface Displacement

a b spi

r i

z r

z ( r i )

Fig. 4. Asperity interactions during rough surface contact.

Surface Separation (Interference): h = hi

Max. interference

< tolerance

Determine forces of individual asperities ( F ci, F si, Qi)

Determine tallest asperity

Adjust normal position

of all N asperities

Final separation?

End

no

yes

Determine forces on nominal contact area ( F c, F s, Q): Store

h = hi -

yes

no

Surface Separation (Interference): h = hi

Max. interference

< tolerance

Determine forces of individual asperities ( F ci, F si, Qi)

Determine tallest asperity

Adjust normal position

of all N asperities

Final separation?

End

no

yes

Determine forces on nominal contact area ( F c, F s, Q): Store

h = hi -

yes

no

Fig. 5. Flowchart of normal asperity position adjustment and surface interactive

force calculation during rough surface contact.




ARTICLE IN PRESS

The friction force (Q ) was based on the CEB friction model [22],

where the shear strength is influenced by the normal contact load

and uses the von Mises yield criterion to calculate the tangential

(friction) force that will cause failure of the contacting asperities.

The friction force is given by

Q i ¼ F ci f di

oc ; n

ð12Þ

where n is Poisson’s ratio and the dimensionless function, f , isgiven by

f ¼ c 4 þðc 244c 3c 5Þ1=2

2c 3ðdiooT Þ ð13Þ

f ¼ 0:2045

K jc 1joc

di

1

1=2

ðdiZoT Þ ð14Þ

where

oc ¼ pKH

2E

2

R ð15Þ

oT

oc ¼ 0:9860:111n ð16Þ

oc is the critical interference to initiate plastic deformation

(material failure) and oT is the transition interference, where the

location of material yield changes: When the surface interference

(di) is smaller than oT , material failure occurs on the surface

and the dimensionless function, f , is expressed by Eq. (13).

When di4oT , material failure occurs below the surface and the

dimensionless function, f , is expressed by Eq. (14). The hardness

coefficient (K ) is given by K =0.454+0.41n, and the coefficients

c 1–c 5 in Eqs. (13) and (14) are related to the Poisson’s ratio and

can be found in [22]. Note that more advanced friction models

than the CEB friction model have been proposed; see for example

[23]. However the CEB model was chosen due to its seminal

nature and accuracy despite its simplicity (note that we only use

the elastic part of the CEB model).

The rough surface adhesive and friction models are imple-

mented in this work to demonstrate the effect/significance of

the rough surface with asperity interaction model in practical

predictions of interface parameters (e.g., friction and friction

coefficient). As has been shown in the literature, e.g., Refs. [2,4],

once the normal contact model is established, at a specified

external normal load, one calculates the mean normal separation

between the two contacting bodies and at this mean separation,

one can then calculate the interfacial contact parameters, such as

real contact area, normal, adhesion and friction forces. This is

what is also done in this work, namely the proposed improved

normal contact model that accounts for asperity interactions

and the presence of a hard coating, is first derived and then

implemented to show its effects in the contact parameters. At a

given surface separation, the interactive forces for individualasperities could be calculated from Eqs. (9)–(16), and then the

total interactive forces and real contact area Ar for the given

nominal contact area could be determined by summing up the

values of N number of asperities expressed as

F c ¼XN

i

F ci ð17Þ

F s ¼XN

i

F si ð18Þ

Q ¼ XN

i

Q i ð19Þ

Ar ¼XN

i

pRoi ð20Þ

3. Simulation results and discussion

Using the rough surface contact model described in Section 2,

analytical simulations were performed for a typical head-disk-

interface (HDI) contact and compared with experimental data

reported in [11]. The measured combined roughness parameters

are as follows: average radius of asperity curvature R=3.73mm,

RMS roughness (standard deviation of surface heights)

s=1.03 nm, standard deviation of asperity heights ss=0.94nm,

areal density of asperities Z=12.48mm2, asperity elastic mod-

ulus E a=139.6 GPa, asperity Poisson’s ratio na=0.24, substrate

elastic modulus E b=80 GPa, and substrate Poisson’s ratio nb=0.30.

Since the above roughness measurements depend on scan size

and sampling interval [24], a digital filtering methodology to

reduce scale-of-measurement effects was implemented to obtain

the above roughness parameters [25]. Note that these parameters

are representative of extremely smooth surfaces used in nano-

mechanical applications. Based on contact experiments per-

formed on a typical HDI [11], the nominal contact area wasestimated to be around 250mm2 (though in some cases it could be

as large as 1000mm2). Once the nominal contact area is known,

the number of asperities (N ) is calculated by N = AnZ. For N

number of asperities, the individual asperity height is mathema-

tically determined using the one standard deviation (ss) and

normal distribution function of the asperity heights.

Fig. 6 shows a histogram for the asperity height distribution

and a nominal contact area of 250 mm2 used in this study, which

follows a Gaussian distribution. Within the given nominal contact

area, N number of surface asperities are randomly distributed,

whose x y z coordinate data are stored and used to calculate

the interfacial forces during the simulation. For comparison, in

addition to the proposed model, simulations were also performed

using the ‘‘discrete’’ version GW contact model assuming that thewhole surface is made of the harder coating material [1], Iida and

Ono (IO) asperity interaction model [12] and Ciavarella et al.

(CGP) asperity interaction model [14]. Brief descriptions of the IO

and CGP models are provided in Appendix A.

The proposed model was also modified to obtain simplified

versions of the model to separately investigate the effects of

asperity interactions versus hard coating. In one case asperity

-2 0 2 4 6 8

x 10-9

0

50

100

150

200

250

300

350

400

450

Asperity Height (m)

A s p e r i t y C o u n

t

Histogram

Gaussian function

Fig. 6. Histogramand probability distribution functionof asperity heights (Table1,

Interface 1), An=250mm2.




ARTICLE IN PRESS

interactions are not accounted for and the model is referred as

‘‘w/o interaction.’’ This model is similar to the discrete GW model

but accounting for the hard asperity over soft substrate contact

behavior. The asperity interaction model is also implemented

assuming the whole surface is made of the hard asperity coating

material along with the discrete GW model and is referred to as

‘‘GW w/interaction.’’ ‘‘Proposed’’ model is referred to the complete

model that captures both the asperity interactions and the hard

asperity over soft substrate behavior. Note that another casewhere the discrete GW model with asperity interaction included

and with the rough surface assumed to be made of the softer bulk

material could also be examined. For the cases considered in this

work, the results are similar to the proposed model since asperity

heights are extremely small in nanoscale applications, thus

showing that the softer bulk dominates the contact. For conven-

tional hard coated surfaces where asperities tend to be taller, the

coating will also influence the contact. In these cases it will

require a fair amount of trial-and-error approach to determine an

appropriate composite modulus or use only the softer substrate

properties to treat the contact. Hence the proposed model is a

unified approach where it can capture the effects of both hard

asperities and softer bulk without the need of a trial-and-error

approach to obtain an appropriate composite reduced modulus

for the contact.

Fig. 7 shows the adjusted normal positions of the surface

asperities during contact obtained by implementing the proposed

model. In the case of the GW model, since the applied surface

displacement goes towards asperity deformation only, the

position of the plane on which the asperities lie does not

change (i.e., the baseline in Fig. 2 does not change) during

contact. For this reason, for a given applied load, more asperities

come into contact since the positions of their respective baselines

are not adjusted, thus the contact force and contact stiffness

values in the GW model dramatically increase even at small

displacements. In the proposed model, the applied surface

displacement causes significant substrate deformation as well as

asperity interactions, which leads to considerable normal

displacement of surface asperities as shown in Fig. 7. The non-

uniform normal positions of surface asperities in Fig. 7 were

caused by the differences of individual asperity height (ah) and

semi-width (r b) values. Therefore, in such a case, the applied load

is mainly supported by fewer asperities compared to the GW

model along with the increased effect of the softer substrate. Thus

the contact stiffness values would be decreased compared to the

GW model.

Fig. 8 shows the contact force F c as a function of surface

separation h for a nominal contact area An of 250mm2 obtained

using the proposed model. Fig. 8a plots F c at small separation

values where the different contact models are compared.

The proposed model (accounting for both asperity and bulkdeformations) clearly shows that the GW model (asperity

deformations only) over-predicts F c . The CGP model, which

includes asperity interactions without accounting for softer

substrate indicates that asperity interaction is also important.

This fact is reinforced by the IO model, where it also includes

asperity interaction. The proposed asperity interaction model

without the softer substrate referred to as ‘‘GW w/interaction’’

yields similar results as the IO model but with F c predictions

slightly lower. Both IO model and GW w/interaction model

predict lower F c values, indicating that asperity interaction is

higher than as seen in the CGP model. Lower F c values are

predicted using the proposed model which accounts for both bulk

substrate as well as asperity interaction effects. Fig. 8b shows F c for higher interference values, where the same relative difference

Baseline: Non-Contact

Baseline: h = 0.2 nm

Baseline: h = - 2.9 nm






1

0

-1

-2

-3

-4

-5

-6

-7

-8

X-position of one column of asperities, m




Baseline: h = - 6.0 nm N o r m a l D i s p l a c

e m e n t , n m

0 2 4 6 8 10 12 14 16





Fig. 7. Simulation results of adjusted normal positions of surface asperities during

contact (only one ‘‘column or row’’ of asperities is shown). Table 1, Interface 1.

-50 -40 -30 -20 -10 0 1010

-2

100

102

104

GW model

IO model

GW w/ interaction

YKP w/o interaction

YKP w/ interaction (Proposed)

0 0.5 1 1.5 20

5

10

15

Surface Separation (h), nm

GW

CGP model

IO model

GW w/ interaction

YKP w/o interaction


GW w/ only Substrate

Surface Separation (h), nmh

C o n t a c t F o r c e ( F c ) , m N

F

GW model

IO model

GW w/ interaction

YKP w/o interaction


C o

n t a c t F o r c e ( F c ) , m N

F

GW

CGP model

IO model

GW w/ interaction

YKP w/o interaction


GW w/ only Substrate

Fig. 8. Contact force vs. surface separation computed using different contact

models: (a) low separation values, (b) large separation values.




ARTICLE IN PRESS

between the contact models can be seen, however the increase in

F c with h is lower at higher h. The CGP model is not shown in

Fig. 8b since there were some complexities in implementing the

model for large interference values.

Since the proposed model was developed for elastic contact,

it is also critical to check if the ensuing contact will result in

most asperities deforming elastically. To determine if the contact

deformation is primarily elastic or plastic, a parameter called

plasticity index c proposed by GW [1] was used, where it is afunction of the material and geometric parameters of the

contacting interfaces:

c¼ 2E r

pKH

ffiffiffiffiffiss

R

r ð21Þ

If co0.6, as long as the applied nominal pressure is not

extremely high, asperities will undergo elastic deformation only.

However, when c41, even under light loads, most asperities

deform plastically. For the current case, conservatively estimating

the hardness H =10 GPa (asperity hardness H a is 30GPa, bulk

substrate hardness H b7.5 GPa), c =0.27, which clearly indicates

that the deformation is elastic, thus validating the applicability of

the proposed model for the HDI case. It also has to be noted that

the effect of asperity interactions is to reduce the interference of the asperities. This means that asperities which undergo plastic

deformation when asperity interaction is not accounted, might in

reality still be in elastic state due to asperity interactions.

4. Comparison with experiments

Fig. 9 depicts the contact stiffness (K c ) values with respect to

the applied contact force values. It is observed that the GW model

(asperity deformation only) shows contact stiffness values of the

order of 107 N/m at contact force values of 10–60 mN, while the

contact stiffness values from the proposed model are of the order

of 105 N/m. Comparing the model predictions to the experimental

measurements, it was found that the GW model significantlyoverestimates the contact stiffness values, whereas the model

predictions from the proposed model are close to the

experimental measurements. Comparing the contact stiffness

values between the simplified proposed model (substrate effect

only ‘‘w/o interaction’’) and complete proposed (substrate and

asperity interaction), it was indicative that even though the

substrate deformation itself contributes to reduce the contact

stiffness, its values were still higher than the experimental

measurements. In other words, for sub-nm surface roughness,

asperity interaction effect is more critical compared to the bulk

compliance effect, as far as affecting the contact stiffness in the

contact of rough surfaces.

Examining the trends of contact stiffness values from the

proposed model (Fig. 9), it was also observed that when the

applied force (or surface interference) is very low (less than10 mN in this case), contact stiffness values increased with

contact force. This trend is similar to the GW predictions because

asperity deformation (as compared to bulk) is dominant during

the contact behavior at very low applied forces. However, once

the applied force is beyond a certain value, the contact stiffness

dramatically decreased with the contact force, because the softer

substrate deformation and asperity interaction become dominant.

It was also observed that the effect of increased nominal area

An results in increased contact stiffness. Ono et al. [26] showed

that the contact stiffness K c increases as square-root of An for

smooth contact, i.e., when roughness is presumed not to affect the

contact compliance, K c ¼ 1:79E r

ffiffiffiffiffiffi An

p . The proposed contact model

was also used to estimate the contact stiffness when An=1000

mm2 (4 times larger). It was observed that the contact stiffness

initially was higher by a factor of 2.35 and with increased

interference it decreased to 1.4. This is because roughness

induced asperity interaction also affects the contact stiffness.

However, for simplicity the relationship provided by Ono et al.

[26] can be used to estimate the effect of An on the contact

stiffness.

The simulation results from the proposed model were

compared with a representative GW-based contact model (CEB

model) [11], assuming the whole surface properties are those of

the harder coating. Fig. 10 shows the comparisons (using a

nominal contact area of 250 mm2) between the proposed model

and the CEB model in regards to the net interface normal force

(F c F s), friction force (Q ), real area of contact ( Ar ) and static

friction coefficient (m). Since the proposed model is for elastic

contact, the implementation of the CEB adhesion [21] and CEB

friction [22] models was done only within the elastic contact

regime. In the case of the GW-based CEB model, since only

asperity interaction was not included, it showed dramatic

increase of the contact force and real contact area even at small

interference values as shown in Figs. 10(a) and (b), respectively.

As expected, the predicted increase is significantly smaller with

the proposed model. Examining the results of net normal force, it

was also observed that the maximum attractive force of the

proposed model was larger and occurred at a smaller surface

separation compared to the CEB model, as shown in Fig. 10a. The

increase in the friction force Q with respect to the contact force F c is also lower using the proposed model as seen in Fig. 10c, which

is due to considerable substrate deformation. Accordingly, the

friction coefficient using the proposed model was lower than thatof the CEB model at higher net forces where the effect of asperity

interaction is predominant, as shown in Fig. 10d.

5. Analysis for sub-10 nm and sub-100nm rough surfaces

The above analysis clearly showed that the effect of asperity

interaction and harder coating layer need to be considered to

accurately model the contact of sub-nm surface roughness. To

verify if the effect of asperity interaction will be critical for

rougher surfaces, two different rough surfaces indicative of higher

roughness values are examined for their contact behavior using

the proposed contact model. Table 1 shows the combined surface

parameters for the different rough surfaces used in this analysis.

0 10 20 30 40 50 60 7010

5

106

107

108

Contact Force (F c ), mN

C o n t a c t S t i f f n e s s ( K

c ) , N / m

GW YKP w/o interaction


Experiments

Fig. 9. Contact stiffness values with respect to the contact force values.




ARTICLE IN PRESS

Interface 1, indicative of modern magnetic storage HDI samples

has already been analyzed in Section 3. Interfaces 2 and 3

represent smooth and rough surfaces in typical microelectro-

mechanical systems [27]. For direct comparison, interfaces 2 and

3 are also assumed to have the same mechanical properties asused in interface 1. This means that the stiff and hard top layer

thickness on interface 2 is around 10nm (s=10.4), and on

interface 3 it is 40 nm. Note that the plasticity index c for

interface 2 is 0.63, which indicates that the contacting surface

asperities will be just above the elastic state. c for interface 3 is

higher at 2.33, which indicates most asperity deformation will be

in plastic state (since the stiffer top thin-films are thicker for

interfaces 2 and 3, film harness H a of 30 GPa was directly used to

calculate c for interfaces 2 and 3). Note that the effect of asperity

interaction is to reduce the interference on the contacting

asperities. Hence, though c is high, presumably most asperities

in contact might still be in elastic state. Nevertheless, caution

needs to be exercised in using the proposed elastic contact model

to analyze interface 3, especially for higher loads. However, to

examine the performance of the model for interfaces 2 and 3, we

will assume that elastic deformation is dominant and examine the

results.

Fig. 11 shows the evolution of contact force F c with decreasing

surface separation h (or increasing interference). The proposed

model results were compared to the GW model, GW with asperity

interaction included (GW w/interaction) and the hard asperity

on softer bulk model without the effect of asperity interaction

(w/o interaction). Fig. 11a shows F c vs. h for interface 2, which is

representative of a surface rougher than surface 1. We observe the

same trend as seen in Fig. 8 where the proposed model shows that

F c is significantly lower than the GW model. As h is decreasing, the

effect of asperity interaction becomes more prominent. The inset

in Fig. 11a shows the comparison of F c at lower interferences(h =25–30 nm), where most of the contact will be elastic: the

proposed model shows the lowest F c , but the contact is dominated

by the compliant substrate behavior since we observe that the

‘‘w/o interaction’’ model results are similar to the proposed model

results. This proves that at lower interference values (higher h),

the effect of asperity interaction might be negligible, but the effect

of compliant substrate is important. Fig. 11b shows F c vs. h for

interface 3, which is representative of a rougher interface. The

trends remain the same, where F c from the proposed model is

lower than the other cases. We observe that the same behavior for

larger separation h (lower interferences) in the inset of Fig. 11b,

but we can observe that F c from the GW model is still higher;

whereas the other models have lower F c . This indicates that both

asperity interactions as well as compliant substrate are important

CEB Model (Elastic)

Proposed Model

CEB Model (Elastic)

Proposed Model

8

CEB Model (Elastic)

Proposed Model

CEB Model (Elastic)

Proposed Model

CEB Model (Elastic)

Proposed Model

CEB Model (Elastic)

Proposed Model

CEB Model (Elastic)

Proposed Model

CEB Model (Elastic)

Proposed Model

CEB Model (Elastic)

Proposed Model

-2 -1 0 1 2 3 4 5 6 7

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


N e t F o r c e

( F - F ) , m N

CEB Model (Elastic)

Proposed Model

R e a l A r e a o f C

o n t a c t ( A ) , µ m 2

CEB Model (Elastic)

Proposed Model

-2 0 2 4 6

0

1

2

3

4

5

6

7


CEB Model (Elastic)

Proposed Model

CEB Model (Elastic)

Proposed Model

0 1 2 3 4 5 6 7 8 9

0

2

4

6

8

10

12

14

16

18

20

Contact Force (Fc), mN

F r i c t i o n F o r c e ( Q ) , m

N

CEB Model (Elastic)

Proposed Model

CEB Model (Elastic)

Proposed Model

1 2 3 4 5 6 7

0

1

2

3

4

5

6

7

8

External Force (F -F sc

), mN

F r i c t i o n c o e f f i c i e n t ,

CEB Model (Elastic)

Proposed Model

Fig. 10. Comparison between the proposed model and the CEB elastic model: (a) net interactive force, (b) real area of contact, (c) friction force, (d) friction coefficient.

Table 1

Roughness properties.

r (nm) g (lm2) R (lm) w

Interface 1 1.03 12.48 3.73 0.27

Interface 2 10.4 11.34 0.81 0.64

Interface 3 38.8 17.68 0.23 2.33




ARTICLE IN PRESS

for smaller interference values and as interference increases

asperity interactions dominate. Based on these additional

simulations performed with rougher surfaces, the effects of

asperity interaction and bulk substrate also need to be

considered. However, care must be taken to ensure that the

deformation regime of the contact is also verified, since for such

rougher surfaces, contact deformation might be plastic.

Fig. 12 shows the contact stiffness K c plotted against F c . For

interface 2, we can clearly observe that K c is influenced by theasperity interaction effect at higher F c , though the effect of

compliant substrate is also observed as seen in Fig. 12a. The inset

in Fig. 12a shows the results at lower F c , where the effect of

compliant substrate as well as the asperity interaction will result

in a lower K c than as computed using the GW model. The contact

stiffness behavior for interface 3 is shown in Fig. 12b. Here both

the GW model as well as ‘‘w/o interaction’’ model exhibit similar

behaviors, whereas the when asperity interaction is also included

(proposed model), K c is significantly lower. This is because the

stiffer top film is fairly thick (40 nm) and most of the contact is

dominated by the top film. But the effect of asperity interaction is

still important at larger F c values. At lower F c values as seen in the

inset of Fig. 12b, K c computed by all three models is similar,

indicating that for these situations, neither the effect of compliant

substrate nor asperity interaction will be important and a simple

GW model calculation will suffice to model the contact behavior.

The goal of this section was to elucidate the importance of the

need for an advanced micro-contact model, like the proposed

model, for modern micro and nanomechanical contacting inter-

faces where the presence of stiffer thin film coatings is routine.

It has been shown that for conventional rougher surfaces, GW-based models will still capture the contact behavior as long as the

applied nominal pressures are within the elastic limit. However,

for smaller scales and smoother surfaces the effect of compliant

bulk as well asperity interaction needs to be considered.

As a final note, in the current work each asperity contact

is treated independently and the total interface quantities are

obtained using discrete summation of the individual quantities.

This indicates that the computational cost associated with the

solution, directly depend on the number of asperities present on

the surface. As a first approximation, for light loads, the classical

GW model may still be a good tool to use since the solution is

obtained using direct integration and thus requiring less compu-

tational time. However due to the development of extremely

smooth surfaces, the asperity interaction effects become more

Fig. 11. Contact force vs. surface separation for rougher surfaces with An=1000

mm2: (a) Interface 2, Table 1, (b) Interface 3, Table 1.

Fig. 12. Contact stiffness vs. contact force computed for rougher surfaces with

An=1000mm2: (a) interface 2, Table 1, (b) interface 3, Table 1.




ARTICLE IN PRESS

dominant and need to be considered. The proposed model is one

such model to capture this effect without dealing with computa-

tional efficiency. In the future greater emphasis also needs to

place in developing contact solutions that include asperity

interaction and are computationally more efficient.

6. Conclusions

An elastic rough surface contact model accounting for asperity

interactions and substrate deformation was proposed. Using the

proposed model, simulations were performed to investigate the

contact behavior of sub-nanometer roughness and microscale

contact areas. Compared to a representative GW-based contact

model where asperity interaction is not considered, the proposed

model showed significantly lower contact stiffness values, which is

in good agreement with experimental measurements. The applic-

ability of the proposed model to rougher surfaces under elastic

contact was also performed. It was found that for smooth as well as

slightly rough surfaces (sub-10nm RMS roughness), consideration of

the bulk deformation and asperity interaction, along with asperity

deformation are important. For RMS roughness in the range of 10 s

of nanometer and meso-scale rough surfaces, the GW model will

suffice to estimate the contact behavior under light loads [28], but

asperity interaction becomes important at heavier loads. However,

caution needs to be exercised in using the proposed model for

situations where significant plastic deformation may occur. Due to

the influence of asperity interaction and compliant bulk substrate,

the resulting real contact area and contact force are lower than

estimated using a GW-based model. Also, the maximum attractive

force is higher and occurs at a smaller surface separation. At higher

contact forces, where the effects of softer substrate and asperity

interactions become significant, friction force is lower in magnitude

compared to elastic CEB model-based results.

Acknowledgments

The initial motivation of this work was through a sponsored

research program from Seagate Technology LLC through Grant no.

SRA-64828. Support by Grants from the National Science

Foundation under Grant number CAREER CMS-0239232 and

Information Storage Industry Consortium (INSIC) Extremely High

Density Recording (EHDR) program are also acknowledged.

Appendix A. Summary of asperity interaction models

A.1. Iida and Ono (IO) model [12]

The contact force F ci and contact area Ai of each asperity are

calculated from Hertzian theory as follows:

F ci ¼4

3E r R

1=2d3=2i

ðdi40Þ0 ðdir0Þ

8<: ðA:1Þ

Aci ¼pRdi ðdi40Þ0 ðdir0Þ

( ðA:2Þ

When the rigid plane comes in contact with an asperity i, the

bases of the other asperities are deformed by the contact force F ciof the contacting asperity i. If the ratio of the real contact area Ar is

o1% of the nominal area An, then the contact on the asperity

summit can be regarded as a concentrated force. Thus, the normal

deformation of asperity i by contact force F cj on another asperity j

is obtained using a Green function for a semi-infinite elastic body

as follows:

u zij ¼F cj

pE r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið xi x jÞ2ð yi y jÞ2

q ðA:3Þ

where ( xi, yi) and ( x j, y j) are positions of asperities i and j on the

surface, respectively. The total deformation depth of asperity i due

to contact force on other asperities is given by

u zi ¼XN

j ¼ 1

u zij ðia jÞ ðA:4Þ

and correspondingly the asperity height from the original base

plane is adjusted to z iu zi. In the contact calculation, asperities

are generated randomly using a Gaussian distribution with

heights z i and positions ( xi, yi) calculated similar to the method

used in Section 2.2. During the calculation, F ci and Ai are

iteratively calculated after updating the normal positions of the

asperities until the change in total contact force F c converges to a

value with o0.01% error.

A.2. Ciavarella, Greenwood and Paggi (CGP) model [15]

In this model, the fundamental assumption was that forheavier loads, the asperity contacts can be treated as uniformly

distributed over the apparent area of contact and thus the

resulting deformation will be uniform. The average deformation

over a nominal area An due to a nominal pressure pnom is given

as mpnom


p =E r , where for a Hertzian pressure distribution

m ¼ ð 916Þ

ffiffiffiffip

p ¼ 0:997 1. Based on the GW model, the total

nominal pressure acting on the rough surface is

pnom ¼ F

An¼ 4

3 ffiffiffiffiffiffi

2pp ZE r R

1=2s3=2 f 3=2ðdÞ ðA:5Þ

where f 3=2ðdÞ ¼ R 1d ð z dÞ3=2expð0:5 z 2Þdz , and z is the height

of the asperity from the base plane of the asperity heights (which

is 3ss) below the mean plane of asperity heights and d is the

distance between the rigid plane and base plane of the asperity

heights. Therefore, the effect of the asperity interaction is to

modify Eq. (A.5) to

pnom ¼ 4

3 ffiffiffiffiffiffi

2pp ZE r R

1=2s3=2 f 3=2 dþ pnom


p E r

! ðA:6Þ

Hence, the total nominal pressure acting on all contacting

asperities is assumed to act on the bulk in an ‘‘average’’ sense

which would result in increasing the distance between the rigid

plane and base plane of the asperities. Equation (A.6) has to be

solved iteratively, but a good approximation can be found by the

first iteration only, i.e. using the original GW theory to estimate

pnom as shown in Eq. (A.5) and then substitute that pnom on the

right hand side of Eq. (A.6) to obtain the actual pnom.

References

[1] Greenwood JA, Williamson BP. Contact of nominally flat surfaces. Proc R SocLondon A 1966;295:300–19.

[2] Chang WR, Etsion I, Bogy DB. An elastic–plastic model for the contact of roughsurfaces. ASME J Tribol 1987;109:257–63.

[3] Derjaguin BV, Muller VM, Toporov YP. Effect of contact deformation on theadhesion of elastic solids. J Colloid Interf Sci 1975;53:314–26.

[4] Kogut L, Etsion I. Elastic–plastic contact analysis of a sphere and a rigid flat.ASME J Appl Mech 2002;69:657–62.

[5] Stanley HM, Etsion I, Bogy DB. Adhesion of contacting rough surfaces in thepresence of sub-boundary lubrication. ASME J Tribol 1990;112:98–104.

[6] Polycarpou AA, Etsion I. Static friction of contacting real surfaces in thepresence of sub-boundary lubrication. ASME J Tribol 1998;120:296–303.

[7] Tsui TY, Vlassak J, Nix WD. Indentation plastic deformation field: part II. Thecase of harder films on soft substrates. J Mater Res 1999;14:2204–9.

[8] Karimi A, Wang Y, Cselle T, Morstein M. Fracture mechanisms in nanoscale

layered hard films. Thin Solid Films 2002;420–421:275–80.




ARTICLE IN PRESS

[9] Yeo CD, Polycarpou AA, Kiely JD, Hsia YT. Nanomechanical properties of ultra-thin carbon film overcoats using the nanoindentation technique. J Mater Res2007;22:141–51.

[10] Yeo CD, Katta RR, Polycarpou AA. Improved elastic contact model accountingfor asperity and bulk substrate deformation. Tribol Lett 2009;35:191–203.

[11] Shi X, Polycarpou AA. Investigation of contact stiffness and contact dampingfor magnetic storage head-disk interfaces. ASME J Tribol 2008;130:021901–9.

[12] Iida K, Ono K. Design consideration of contact/near-contact sliders based on arough surface contact model. ASME J Tribol 2003;125:562–70.

[13] Bahrami M, Yovanovich MM, Chulham JR. A compact model for sphericalrough contacts. ASME J Tribol 2005;127:884–9.

[14] Zhao Y, Chang L. A model of asperity interactions in elastic–plastic contact of rough surfaces. ASME J Tribol 2002;123:857–64.

[15] Ciavarella M, Delfine V, Demelio G. A ‘‘revitalized’’ Greenwood and William-son model of elastic contact between fractal surfaces. J Mech Phys Solids2006;54:2569–91.

[16] Ciavarella M, Greenwood JA, Paggi M. Inclusion of ‘‘interaction’’ in theGreenwood and Williamson contact theory. Wear 2008;265:729–34.

[17] Sahoo P, Banerjee A. Asperity interaction in adhesive contact of metallicrough surfaces. J Phys D: Appl Phys 2005;38:4096–103.

[18] Yu N, Polycarpou AA. Contact of rough surfaces with asymmetric distributionof asperity heights. ASME J Tribol 2002;124:367–76.

[19] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press;1985.

[20] Yu N, Polycarpou AA. Adhesive contact based on Lennard-Jones potential: acorrection to the value of the equilibrium distance as used in the potential. JColl Int Sci 2004;278:428–35.

[21] Chang WR, Etsion I, Bogy DB. Adhesion model for metallic rough surfaces.ASME J Tribol 1988;110:50–6.

[22] Chang WR, Etsion I, Bogy DB. Static friction coefficient model for metallicrough surfaces. ASME J Tribol 1988;110:57–63.

[23] Cohen D, Kligerman Y, Etsion I. A model of contact and static friction of nominally flat surfaces under full stick contact condition. ASME J Tribol2008;130:031401.

[24] Thomas TR, Rosen B-G. Determination of the optimum sampling interval for

rough contact mechanics. Tribol Int 2000;33:601–10.[25] Suh A, Polycarpou AA. Digital filtering methodology used to reduce scale of

measurement effects in roughness parameters for magnetic storage super-smooth hard disks. Wear 2005;260:538–48.

[26] Ono K, Takahashi K, Iida K. Computer analysis of bouncing vibration andtracking characteristics of a point contact slider model over random disksurfaces. ASME J Tribol 1999;121:587–95.

[27] Tayebi N, Polycarpou AA. Reducing the effect of adhesion and frictionin microelectromechanical systems (MEMs) through surface roughening:comparison between theory and experiment. J Appl Phys 2005;78:0735281–13.

[28] Shi X, Polycarpou AA. Measurement and modeling of normal contact stiffnessand contact damping at the meso scale. ASME J Tribol 2008;127:52–60.


Documents

Effect of Asperity Interactions on Rough Surface Elastic Contact Behavior- Hard Film o