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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.
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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
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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
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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,
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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.
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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
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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
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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
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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.
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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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