7

Chapter 2

Contact of rough surface: a literature survey

2.1. Basics of Contact Mechanics

2.1.1 Introduction

Contact phenomena are abundant in everyday life and play a very important role in

engineering structures and systems. They include friction, wear, adhesion and

lubrication, among other things; are inherently complex and time dependent; take place

on the outer surfaces of parts and components, and involve thermal, physical and

chemical processes. Contact Mechanics is the study of relative motion, interactive

forces and tribological behavior of two rigid or deformable solid bodies which touch or

rub on each other over parts of their boundaries during lapses of time. However, the

contact between deformable bodies is very complicated and it is not yet well

understood.

The contact theory was originally developed by Hertz [1] and it remains the

foundation for most contact problems encountered in engineering. It applies to normal

contact between two elastic solids that are smooth and can be described locally with

orthogonal radii of curvature such as a toroid. Furthermore, the size of the actual contact

area must be small compared to the dimensions of each body and to the radii of

curvature (non-conforming contact). Hertz made the assumption based on observations

that the contact area is elliptical in shape for such three dimensional bodies. The

equations simplify when the contact area is circular such as with spheres in contact. At

extremely elliptical contact, the contact area is assumed to have constant width over the

length of contact such as between parallel cylinders. The Hertz theory is restricted to

frictionless surfaces and perfectly elastic solids. It was not until nearly one hundred

years later that Johnson, Kendall, and Roberts [2] found a similar solution for the case

of adhesive contact. This theory was rejected by Boris Derjaguin and co-workers [3]

who proposed a different theory of adhesion in the 1970s. The Derjaguin model came to

be known as the DMT (after Derjaguin, Muller and Toporov) model, and the Johnson et

al. model came to be known as the JKR (after Johnson, Kendall and Roberts) model for

8

adhesive elastic contact. This rejection proved to be instrumental in the development of

the Tabor [4] and later Maugis [5] parameters that quantify which contact model (of the

JKR and DMT models) represent adhesive contact better for specific materials.

Further advancement in the field of contact mechanics in the mid-twentieth

century may be attributed to names such as Bowden and Tabor [58]. They were the first

to emphasize the importance of surface roughness for bodies in contact. Through

investigation of the surface roughness, the true contact area between friction partners is

found to be less than the apparent contact area. Such understanding also drastically

changed the direction of undertakings in tribology. The works of Bowden and Tabor

yielded several theories in contact mechanics of rough surfaces.

The contributions of Archard [6] must also be mentioned in discussion of

pioneering works in this field. Archard concluded that, even for rough elastic surfaces,

the contact area is approximately proportional to the normal force. Further important

insights along these lines were provided by Greenwood and Williamson [7], Bush

[8], and Persson [9]. The main findings of these works were that the true contact surface

in rough materials is generally proportional to the normal force, while the parameters of

individual micro-contacts (i.e. pressure, size of the micro-contact) are only weakly

dependent upon the load.

2.1.2 Elastic Contact

In the elastic contact surface area eA , contact force eP , maximum contact pressure mp ,

and average contact pressure p can be expressed in function of interference . Derived

from Hertz’s theory [1] equation contact surface area for elastic contact is given by eA

for elastic contact:

RAe ............................................................................................................. (2.1)

Contact force eP derived from equation

E

apm

2

, then:

9

31

2

2

16

9

RE

Pe

2

2

3

16

9

RE

Pe

322

9

16REPe

21

32

9

16

REPe

2321

3

4REPe

................................................................................................. (2.2)

Maximum contact pressure mp obtained from equation

E

Rpa m

2

, therefore:

21

2

2

R

E

a

Epm

21

2

R

Epm

................................................................................................. (2.3)

Average pressure contact elastic, p is given by:

21

3

4

3

2

R

Epp me

..................................................................................... (2.4)

In 1951 Tabor [13] stated that the maximum Hertz contact pressure reaches

Hpm 6.0 occurs on beginning of yield so that the average pressure Hp 4.0 . From

this relationship can be obtained an average contact pressure p with hardness H at the

time of the initial yield point is given by Chang, Etsion and Bogy (CEB model) [10] by

equation:

10

kHp ................................................................................................................. (2.5)

Relationship between maximum contact pressures with hardness at the time of the

initial yield point is given by Chang, Etsion and Bogy. It expressed by the equation:

kHpm .............................................................................................................. (2.6)

By subtituting Equation (2.6) into Equation (2.3), critical interference CEB1 is given

by:

RE

kHCEB

2

12

............................................................................................... (2.7)

Kogut dan Etsion (KE model) [12] use value k from [10], so critical interference KE1

at the begining of yield:

RE

HkKEKE

2

12

............................................................................................ (2.8)

where 41.0454.0 KEk and is Poisson’s ratio.

Zhao, Maietta and Chang (ZMC model) [11] obtain critical interference based

from average contact pressure, by subtituting Equation (2.5) in Equation (2.4) obtained

critical interference ZMC1 :

RE

Hk ZMCZMC

2

1

14

3

.................................................................................... (2.9)

where 4.0ZMCk

when 1 contact that occurs is contact elastic. In other hand, when 1

contact that occurs is contact elastic-plastic or fully plastic.

11

2.1.3 Fully Plastic Contact

Fully plastic contact occurs when interference increased to reach 2 with average

contact pressure p reach value H (Fig 2.1). In fully plastic contact, Zhao, Maietta and

Chang using contact plastic model [11].

Figure 2.1: Deformation on asperity [8].

During deformation fully plastic ZMC 2 average contact pressure remains

constant at a value H or:

Hp ZMCp ........................................................................................................ (2.10)

The contact surface area for the fully plastic contact ZMC model using modeling of

plastic contacts [11] is given by:

RA ZMCp 2 .................................................................................................. (2.11)

Contact forces pP equal with the contact surface area multiplied by the

average contact pressure.

HRP ZMCp 2 ............................................................................................... (2.12)

Zhao, Maietta and Chang estimated minimum value 2 based on the results

of Johnson [14], the fully plastic condition occurs when the contact force on

12

the fully plastic pP ( = 2) approximately equal to four hundred times the contact

force at the point of initial yieldyP ( =1) or:

400yp PP ...................................................................................................... (2.13)

By using equation (2.2) is obtained as follows:

23

1

21

3

4REPy

.............................................................................................. (2.14)

and

23

2

21

3

4REPp

............................................................................................. (2.15)

By dividing Equation (2.15) with Equation (2.14) is obtained:

40023

12 yp PP ................................................................................. (2.16)

or

12 54 ........................................................................................................... (2.17)

From the equation above, Zhao, Maietta and Chang (ZMC model) [11] defines the value

of interference in the fully plastic ZMC2 limit to the value of critical interference at the

yield point ZMC1 by the equation:

ZMCZMC _12 54 ............................................................................................. (2.18)

While on Kogut and Etsion (KE model) [12] the value of interference in the

fully plastic KE2 limit to the value of critical interference at the yield point KE1 is

defined by the equation:

13

KEKE _12 110 .................................................................................... .......... (2.19)

2.1.4 Elastic-plastic Contact

Contact of elastic-plastic contact is a transition from elastic to fully plastic contact.

Elastic-plastic contact occurs when the interference is between 1 and ( 21 ).

At the contact of elastic-plastic, deformation that occurs consists of elastic and plastic

deformation. Relations between contact surface area and average contact pressure as a

function of the interference is a very complex relationship. Zhao, Maietta and Chang

(ZMC model) [11] gives the relationship between the average contact pressure and

interference in elastic-plastic contact is given by:

ZMCZMC

ZMC

ZMCep kHHp

12

2

lnln

lnln1

.................................................... (2.20)

Whereas the surface area contact on the elastic-plastic expressed as:

2

12

1

3

12

1 321ZMCZMC

ZMC

ZMCZMC

ZMC

ZMCep RA

................... (2.21)

By using equation (3.20) and (3.21) the elastic-plastic contact force is the product

between the average contact pressures with the contact surface area which yields:

R

ZMCkHHP

ZMCZMC

ZMC

ZMCZMC

ZMC

ZMCZMC

ZMCZMCep

2

12

1

3

12

1

12

2 321lnln

lnln1

........................................(2.22)

Kogut and Etsion [12] give the relationship an average contact pressure against

interference in elastic-plastic contact with the equation:

289.0

1

19.1

KE

KEep

Y

p

for 61

1

KE

........................................(2.23)

14

117.0

1

61.1

KE

KEep

Y

p

for 1106

1

KE

where Y is the yield strength of the material.

While the relationship between surface areas in contact in the interference

functions of elastic-plastic contact is given in the equation:

136.1

1

93.0

KEc

KEep

A

A for 61

1

KE

........................................(2.24)

146.1

1

94.0

KEc

KEep

A

A for 1106

1

KE

Ac-KE is critical contact area when KE 1 .

2.2 Surface Topography: Surface Texture, Roughness, Waviness

Surface topography is the three-dimensional representation of geometric surface

irregularities. A surface can be curvy, rough or smooth depending upon the magnitude

and spacing of the peaks and valleys and also depending upon how the surface is

produced. Surface texture refers to the locally limited deviation of a surface from the

ideal intended geometry of the part. A realistic characterization of surface roughness is

required to analyze the effect of surface roughness on different tribological parameters.

Roughness relates to the closely-spaced irregularities left on a surface from a production

process. It is a measure of the fine, closely spaced, random irregularities or surface

texture. Waviness is the component of texture upon which roughness is superimposed. It

relates to the more widely-spaced irregularities than roughness caused by vibration,

chatter, heat treatment, or warping strains [16]. Surface roughness cannot be easily

defined by a single parameter. In fact, there are several ways to represent roughness. All

rough surfaces having height and wavelength, with the former measured at right angle

15

to the surface and the latter in the plane of the surface. The distribution of height is

measured from a reference plane (say, mean plane).

The characterization of a surface may be one dimensional (1-D) or two

dimensional (2-D) depending upon the machining and finishing process. For 1-D case,

height (z) varies with one of the coordinates, whereas in other coordinate there exists a

lay where variation of z is comparatively small. But for surfaces made by conventional

manufacturing processes, when 1-D characterization is not proper, 2-D surface

roughnesses description is required. Atomic Force Microscope (AFM) and 3D surface

profilometer are used to improve the resolution and accuracy of the roughness

measurement. Typical 1-D and 2-D representation of nominally flat surfaces (polished)

are shown in Figure 2.2. (a) and 2.2. (b). Scale used for the height is much larger than

that for the wavelength because the height of the asperities from the mean plane is very

less compared to their wavelengths [16].

Figure 2.2: Typical representation of a surface: (a) one-dimensional (b) two-

dimensional [16].

2.3 Contact Problem of Smooth Surfaces

The study of the contact behavior of deformable bodies has divided into two

approaches. The first, regard the bodies are smooth and can be adequately described by

16

their nominal geometry, the second admits that all surfaces are comprised of multitude

of peaks and valleys which have multiple asperities which is regarded as a rough

surface. Hertz [1] was first to analyze the elastic contact between two non-conforming

spheres. He gave the analytical solution for the normal contact between two curved

bodies for contact pressure and subsurface stress field.

In Hertz analysis, following assumptions were made.

a) Radii of curvature of the contacting bodies are large compared with radius of

circle of contact.

b) Contact is frictionless.

c) Surfaces are continuous and non-conforming.

d) Strain is small.

e) Each solid can be considered as half space.

Based on these assumptions, the stress fields generated by an indenter contacting

an elastic solid can be analyzed [17]. For the case of elastic contact with a spherical

indenter of radius R, the radius of the circle of contact a between the indenter and the

specimen surface increases with the load. The contact pressure distribution p proposed

by Hertz is given by

(

)

⁄

The total contact load P can be obtained from the above pressure distribution as,

∫ ( )

Where maximum pressure = 3/2 , with denoting the mean pressure. The contact

circle radius a is given by,

17

is the effective modulus of elasticity and is the effective radius of curvature

defined as,

, , are the Young’s modulus, Poisson’s ratio and radius of curvature of the

indenter material and , , are the corresponding parameters of the specimen. The

depth of indentation h is related to indenter radius by,

When two nominally flat surfaces are brought into contact under load, contact

occurs only at discrete spots. The real area of contact is the summation of all individual

spots. Real area is only fraction of the nominal contact area. The real area of contact

between two solid surfaces has a profound influence on friction and wear of machine

parts, thermal and electrical conduction, contact stresses and joint stiffness. Greenwood

and Williamson [7] explains how Gaussian distribution can be applied to surface

parameters in practice and the reasons for regular occurrence of such a distribution

governing each individual effect. For a rough surface in contact with a rigid plane, GW

model which assumes constant tip radius for the asperities gives good order of

magnitude estimates the number of contacts, real area of contact and nominal pressure

at any given separation. Some other important models are available in literature like

Greenwood and Tripp model (GT Model) [18], Whitehouse and Archard Model (WA

Model) [19], Onions and Archard Model (OA Model) [20], Nayak Model [21], The

elliptic Model [22], Hisakado Model [23], Francis Model [24], McCool model [25]. In

most of these models, the asperities are assumed to be either spherical, paraboloidal or

elliptic paraboloidal in shape and the corresponding Hertzian solution for single contact

is used in the analysis.

18

2.4 Contact Problem of Rough Surface

Traditionally, surfaces were modeled analytically using assumption and simplifications.

Surface produced by any conventional machining/manufacturing processes are never

smooth. The surface irregularities are termed as asperities. Asperities were modeled as a

variety of geometric shapes. In the past, a number of authors study rough surface

contact problem using analytical method (Whitehouse & Archad [19]; Onions & Archad

[20]; Bush, Gibson, et al, [8]; Hisokado [23]). Their result was very useful but their

application is limited to a relatively small range of loads.

Surface asperity height and contact pattern were treated as probability

distributions. Behavior of a single pair of interacting asperities was often extrapolated to

describe the behavior of a pair of interacting surfaces covered in asperities [25].

Investigation of the contact itself classically follows two types of approach, either

stochastically or deterministically. One of the first models has been proposed by

Greenwood and Williamson [7], who assumed that the asperity summits are spherical

with a constant radius, the asperities deform elastically and their height follows a

Gaussian distribution. Statistical models have had a considerable impact on contact

analysis and have been considered by many authors [28-31]. Nevertheless, these models

do not take into account the real geometry of the surface and the interactions between

the asperities. Deterministic approaches were then developed to introduce a more

precise geometric description [41].

Surface roughness can affect the performance of components and system in a

wide variety of fields including tribology, fluid sealing, heat transfer, electronic

packaging, dentistry, and medicine. Although it is possible to measure the topography

of a real surface and incorporate that data into a finite element model [32-34], this

practice is still relatively uncommon [35].

In fact, most analysts create probabilistic surfaces based on assumed, known, or

desired surface geometry [7, 36-39], in part because the real geometry cannot always be

measured. In recent time, researchers work on real surfaces analysis by either

experimentally [42] or developed rough surface model in finite element software. Finite

element modeling permits contact simulation with complex geometry, boundary

condition, material properties, and material models. The finite elements method has

19

been used to solve the contact problem for artificial fractal surfaces [46]. Starting from

roughness measurements, synthesized fractal surfaces were also used in the studies of

Vallet et al. [44-45] where they used a numerical procedure to solve the contact

problem.

2.5 Modeling Rough Surface

The classical analysis of rough surface contact problems were based upon statistical

models. Their asperities were assumed to have a certain shape and their physical

dimensions such as the widths and the heights were assumed to have a certain statistical

distribution. The rough surface is represented by a collection of asperities of prescribed

shape scattered over a reference plane. The height of the summits has a statistical

distribution and is assumed that the contacting asperities deform elastically according to

Hertz theory.

Many models describing rough surface have followed the pioneering work of

Abbot & Firestone [43]. They attempt to characterize roughness by a series of

indicators, such as the arithmetic average of vertical deviation Ra and the mean line m,

the root mean squared Rs or the standard deviation σ. However, the surface can’t be

fully described using only a surface profile in the vertical direction. These were done in

order to simplify the problem. With the rapid advances of faster computers within the

last decade, the development of more realistic models for contact simulation becomes

more feasible.

Webster and Sayles [50] made a computer model for the dry, frictionless contact

of real rough surface which uses data recorded directly from stylus measuring

instrument. It presented a numerical method of studying the elastic contact of real

roughness and topography in order to investigate a wide range of frictionless contact

problem. Moreover, they investigated numerical elastic-plastic contact model of a

smooth ball on a directionally structured anisotropic rough surface. It’s obtained the

contact information such as the extent of plastic deformation and the states of contact

over a range of rough surface and result from the analysis have been used to correlate

results from friction test. The result is used in Poon and Sayles [50] studied about the

effect of surface roughness and waviness upon real contact areas, gasp between contact

20

spots, and asperity contact along with their distribution for different surface roughness

and effect of σ (height distribution/surface roughness) and β* (spatial structure of a

surface). The result is compared with Bush’s model [8] using stochastic approach.

Lee and Cheng [52] also using a computer simulation to made a model for the

contact between longitudinally oriented rough surface for simulating contact between

purely longitudinal surface. During that time, simulating a model in computer was really

taking a lot of time. Several methods were investigated to increase time efficiency and

reduce the requirement of the computer memory size. Lee & Ren [53] developed

simulating dry contact of three-dimensional rough surface based upon Moving Grid

Method (MGM). Its method was able to reduce required RAM (320 Kb from previous

12.8 GB). It method reduce required storage space for deformation matrix to the order

of N (Fig 2.3). The computing time to construct the matrix is also proportional to N.

Figure 2.3: A schematic of representation of the three dimensional moving grid

method [52].

Chang and Gao [57] determined a new method for contact problem to optimize

the efficiency of computer. Two surfaces brought into a contact, a pressure is developed

at every surface point inside, and only inside, the true contact area is modeled in

algorithm principal. The method is used to calculate the pressures and surface

displacements in contacts of rough surfaces. Karpenko & Akav [54] also worked on a

computational method for analysis of contact between two rough wavy surfaces for

21

which the nominal contact area may be arbitrarily large that both wavy and rough (Fig

2.4).

Figure 2.4: A schematic three dimensional model of Karpenko [54].

Another method to generate rough surface such as successive random midpoint

algorithm is used by Pei, et al. [55] when they presented a finite element calculation of

frictionless, non-adhesive, contact between a rigid plane and an elasto-plastic solid with

a self-affine fractal surface (Fig. 2.5.). All of the rough surface research mentioned

above are generated by digitized measured profile of contacting surface and used them

for computer simulation.

Figure 2.5: A self-affine fractal surface for L = 256 generated by the successive random

midpoint algorithm. Heights are magnified by a factor of 10 to make the roughness

visible, and the color varies from dark (blue) to light (red) with increasing height [54].

22

Bryan et,al. [47] lately analyze elastic-plastic finite element of line contact

between cylinder and rigid plane using ABAQUS. However, they still generated rough

surface from measured real surface which imported to ABAQUS using a Phyton script

(Fig 2.6). Another finite element research lately from Yastrebov & Durand [48]

presented normal frictionless mechanical contact between an elastoplastic material and

rigid plane using finite element analysis (FEA) and representative surface element

(RSE) approach. Their research also introduced a new reduced model for the analysis of

rough surface. Their new model can solve problem in a few second instead of FEA that

need a few days. The new model is a series of basic curves obtained by means of

elementary finite element computation on a single asperities and phenomenological

relations to take into account the interaction between neighboring asperities of the rough

surface.

Figure 2.6: Measured rough surface model by Bryant & Evans [46].

In this paper author want to introduce a new way to generate rough surface using

numerical method and export its result into finite element commercial software

(ABAQUS). Furthermore, analysis to evaluate pressure and contact distribution will be

applied on contact problem of rough surface against hard smooth ball. Rough surface is

obtained from experimental data and the result is compared from the result from

simulation.

23

2.6 Three Dimensional Model of Rough Surface in Finite Element

Commercial Software

Traditionally, surfaces were modeled analytically using assumptions and

simplifications. Asperities were modeled as a variety of geometric shapes. Surface

asperities height and contact pattern were treated as probability distributions. The

behavior of single pair of interaction asperities was often extrapolated to describe the

behavior of a pair of interacting surfaces covered in asperities. These assumptions were

not made because they were shown to accurately represent the system of interest, but

because they made modeling possible. Therefore, people studied rough surface by mean

analyze one single asperity which assumed represent the others asperities of the surface.

The topography of surface was taken from measured real surface using optical micro

and macro scale surface features and record data digitally.

Schwarzer [56] geometrically construct all sort of rough surfaces by applying

mathematical functions. Figure 2.7 shown an example of two surface of equal roughness

in a mere mathematical contact situation, yet this model cannot represent the real

surface due to the asperities which are homogeny.

Figure 2.7: Model of rough surface from Schwarzer [56].

In recent time, people developed rough surface model in finite element software.

Finite element modeling permits contact simulation with complex geometry, boundary

condition, material properties, and material models. Bhowmik [16] modeled rough

surface with homogeny asperities as shown in Figure 2.8. In his work, the mechanics of

contact between a rigid, hard sphere and a surface with a well defined roughness profile

24

is studied through experiments and finite element simulation. The well defined

roughness profile is made up of a regular array of pyramidal asperities.

Figure 2.8: Rough surface modeled by Bhowmik [16].

David et, al.[49] demonstrated RF MEMS simulation. They used either an

optical profilometer (VEECO) or an Atomic Force Microscope (AFM) to capture three

dimensional data points of contact surfaces. Then, using Matlab functions they convert

the closed surface from a stereo-lithographic format to an ASCII file compatible with

ANSYS Parametric Design Language (APDL). In the final step, the rough surface was

obtained by creating key points from the imported file. Since the key points are not co-

planar, ANSYS uses Coons patches to generate the surface, and then we used a bottom

up solid modeling to create the block volume with the rough surface on the top. Figure

2.9 describes the full method developed on ANSYS platform. Meanwhile Figure 2.10.

shows rough surface interface in ANSYS.

Figure 2.9: Interface of rough surface on ANSYS [48].

25

Figure 2.10: Methodology for generating rough surface from David et,al [48].

M Kathrine Thompson [26] from mechanical department MIT, presents methods

for generating, using, and operating on nonuniform variates for the incorporation of

probabilistic rough surfaces in ANSYS (Fig. 2.11). Her work discusses how to decouple

the surface from the finite element model by transferring the surface information from

arrays to tables.

Figure 2.11: Block with normally distributed rough surface. Mesh created by moving

all nodes – actual scale (left), 100x displacement (right) [26].

26

The methods are presented for creating solid model geometry from the

metrology data. Three example surfaces are imported and used in a contact analysis. For

this work, surface metrology data is imported into the finite element program as a two

dimensional array. These techniques, combined with the ability to model real surfaces in

ANSYS, can be used to help researchers in material science, mechanical and electrical

engineering, and beyond to better understand micro scale surface phenomena. However,

Thompson’s model shows asperities with sharp peak instead of smooth. Meanwhile, in

the real rough surface, the geometry of the asperities is considered as either hemisphere

or ellipsoid as have been proved by previous works on modeling asperities from

measured real surface. Moreover, Thompson’s model is difficult to be meshed because

of its manipulated geometry. This paper will discuss a new way to generate surface in

ABAQUS with surface treatment in SolidWorks. Rough surface with smooth asperities

is expected as the result of this work. The model behavior from this method will be

compared with surface that created normally from ABAQUS.

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28

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