Di32687692

Biswajit Bera / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 2, March -April 2013, pp.687-692

Coefficient of Static Friction of Elastic-Plastic Micromechanical
Surface Contact

Biswajit Bera
(Department of Mechanical Engineering, National Institute of Technology Durgapur, India)

ABSTRACT
has developed classical adhesion theory of friction
In this study, the effect of elastic and
for plastically deformed spherical contact. These
plastic adhesion index on deformation force and
theories are revisited for present study of
friction force of MEMS surface contact has
multiasperity contact of adhesive MEMS surfaces.
investigated in term of surface roughness. It is
However, Fuller and Tabor [6], and K L Johnson [7]
found that at low value of θ, i.e. for smooth
have developed elastic adhesion index and plastic
surfaces, the contact is mostly elastic in nature
adhesion index respectively for multiasperity contact.
and at high value of θ, i.e. for rough surfaces,
Elastic and plastic adhesion index is basically a
mostly plastic contact occurs. This supports the
number which is developed combination of
elastic-plastic concept of Greenwood and
mechanical and tribological parameters of a material.
Williamson for rough surface contact. Total
Here, effect of both the adhesion index on
deformation force and friction force are mainly
deformation force and friction force of MEMS
supported by plastically deformed asperities.
surface contact has investigated in term of surface
From the study of coefficient of static friction of
roughness. Value of both the indices has considered
elastic-plastic MEMS surface contact, it is found
in small range so that study is limited to soft MEMS
that COF is almost constant of the order of value
surface contact. It should be mentioned that adhesive
of 0.4
force effect within contact zone of asperity with
respect to deformation force has neglected to avoid
Keywords – Adhesion index, Deformation force,
complexity of the study.
Friction force, Coefficient of friction
2. Theoretical Formulation
1. Introduction 2.1 Single Asperity Contact
Combined study of load and friction of
2.1.1Single asperity deformation force
rough surface contact is an important study in the
2.1.1a Single asperity elastic deformation force
field of Tribology. Empirical correlation in between
Hertzian Elastic loading force for a single asperity
load and friction was developed by Leonardo da
contact is given by
Vanci as well as Amontos [1] which are stated as 3
Kae
laws of friction. When two rough surfaces comes in Pae   KR 0.5 1.5
contact due to application of normal force, contact R
occurs at the tip of asperities and finite tangential 2.1.1b Single Asperity Plastic Deformation
force is required for sliding motion which is measure Force
of friction force. Under normal loading condition, CEB Plastic loading force for a single asperity
asperities of rough surface contact would deform contact on the basis of volume conservation principle
elastically and also, plastically after critical value of is given by
  
deformation. Correspondingly, adhesive bond would Pap  Aap Y  R  2  c 0.6H
be developed at the contact zone of elastically and   
plastically deformed asperity due to cold welding by 2.1.2 Single asperity friction force
interatomic adhesion at contact zone [2]. So, finite 2.1.2a Single Asperity Elastic Friction Force
force is required for shearing of adhesive asperity CEB elastic friction force for a single asperity
junction at real area of contact which is measure of contact on the basis of Hamilton stress field is given
friction force. First, Hertz developed the expression by
of loading force for elastically deformed spherical
contact and thereafter, Chang, Etsion, and Bogy [3]
Tae 
1
27 0.5 c1
 
4 2Y 2 a e4  9c 2 Pae
2 2 0.5

developed expression of loading force for plastically
deformed spherical contact considering volume 
1
5.2c1
4 2 0.6 H  ae4  9c 2 Pae
2 2 2

0. 5

conservation. Also, Chang, Etsion, and Bogy [4] where
developed expression of friction force for elastically
2
c1  1  tan 1   1  
3
 
deformed spherical contact considering Hamilton
stress field. On the other hand, Bowden and Tabor [5] 2   2 1  2  

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d  c 
(Take |C1| value) 
   ( z )dz    ( z )dz
c2  1    tan 1  1   1 
  
3 c     ( z )dz  

  
  2 1  2  d
d d  c
Elasticasperities Plasticasperities
where η is number of total asperity per unit cross-
2.1.2b Single Asperity Plastic Friction Force sectional area and φ(z) is the Gaussian asperity height
Bowden and Tabor plastic friction force for a distribution function.
single asperity contact is given by
Tap  Aap 2.2.1 Multiasperity Deformation Force
   2.2.1a Multiasperity Elastic Deformation Force
  R  2  c 
   Elastic deformation force due to multiasperity
contact is given by
d  c
2.2 Multiasperity Contact Pe    KR
0.5
 1.5 z dz
First of all, Greenwood and Williamson [8] d

developed statistical multiasperity contact theory of Dividing both side byR , we have
rough surface under very low loading condition and it C
 h   2 
1
was assumed that asperities are deformed elastically P e   1.5 exp  d
according to Hertz theory. Same theory is extended 0 2  2 
here in elastic and plastic rough surface contact and it
is based on following assumptions: 2.2.1b Multiasperity Plastic Deformation Force
a) The rough surface is isotropic. Plastic deformation force due to multi asperity
b) Asperities are spherical near their contact is given by

summits.   
Pp    R  2  c H ( z )dz
c) All asperity summits have the same d     c
radius R but their heights very randomly. Dividing both side by ηRγ, we have
d) Asperities are far apart and there is no 
   1  h   2 
interaction between them. Pp   2.19 0.5 0.25  2  c  exp  d
e) Asperities are deformed elastically as c    2  2 
well as plastically. Total deformation force due to all asperities is
f) There is no bulk deformation and only, given by
d  c 
the asperities deform during contact.  c 
P   KR 0.5 1.5 z dz    0.6R  2   H z dz
d d  
Flat plane c

Dividing both side by ηRγ, we have

δ C
1  h   2  
0.5 0.25  c  1  h   2 
P    1.5 exp  d   2.19   2   exp  d
Z 0 2  2  c    2  2 
d

Mean 2.2.2 Multiasperity Friction Force
2.2.2a Multiasperity Elastic Friction Force
Rough surface
Elastic friction force for multiasprity is given by
d  c

Te   
1
5.2c1

4 2 0.6 H  R 2  2  9c 2 K 2 R 3
2 2

0.5
 z dz
d
Fig. 1 Rough surfaces contact

Multiasperity contact of elastic and plastic rough Dividing both side byR , we have
c

 5.2c 19.2 2  9c 2  2 3    d
surface has shown in Fig.1. Two rough surface 1 0.5
Te  0.5 2

contact could be considered equivalently, contact 0 1
between rough surface and smooth rigid surface. Let z
and d represents the asperity height and separation of 2.2.2b Multiasperity Plastic Friction Force
the surfaces respectively, measured from the reference Plastic friction force for multiasprity is given by
plane defined by the mean of the asperity height. δ 
 

Tp     R  2    z dz
c
denotes deformation of asperity by flat surface.
Number of contacting asperity per unit cross-sectional d c
 
area is Dividing both side by ηRγ, we have

    
T p   3.65 0.25 0.5  2  c   d
c  H   

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.
Total friction force due to all asperities is given by
d  c λ=5,һ= 1 λ=5,һ= 0 λ=5,һ= -1
  c 


4 2 0.6 H  R 2  2  9c 2 K 2 R 3  z dz    z dz
1
 
0.5
T   R  2 
2 2

d
5.2c1 d   

Elastic deformation force (P e*)
c

Dividing both side by ηRγ, we have 0.40
c 
  c 
T 
 1
19.2   9c      d   3.65 H 
0.5 2 2
2
2 3 0.5
 
0.25
 0.5  2 
 
  d 0.35
5.2c0 1 c   
where, 0.30
z d  d  0.25
 ,  h  , h 
   , 0.20
2
δ H R λ 0.5 0.15
Δ c  c  1.256     1.7
σ K  σ   , 0.10
τ 0.05
 0.2
  0.3 , H 0.00
5 10 15 20 25
Elastic adhesion index (θ)
3. Result and Discussion
In this combined analysis of deformation force
and friction force of adhesive MEMS surface contact, Fig.2a Elastic deformation force Vs Elastic adhesion index
elastic adhesion index ( θ ), plastic adhesion index ( λ
) and dimensionless mean separation ( h ) have used λ=5,һ= 1 λ=5,һ= 0 λ=5,һ= -1
as input parameters. Before analyzing the results on
the basis of surface roughness point of view, the 0.6
Elastic friction force(Te*)

definition and physical significance of these indices 0.5
should be mentioned. First, Fuller and Tabor
introduced an elastic adhesion index for elastic solid 0.4
which is defined as ratio of elastic deformation force
to adhesive force experienced by elastically deformed 0.3
sphere and given in following form; 0.2
  1 .5

R 0.5 0.1
Similarly, Johnson developed a plastic adhesion 0
index for plastic solid which is the ratio of plastic 5 10 15 20 25
deformation force to adhesive force experienced by Elastic adhesion index (θ)
plastically deformed sphere and given in following
form;
 2 RH 4 Fig.2b Elastic friction force Vs Elastic adhesion index

18 K 2 2
Fig.2a and Fig.2b shows that for particular mean
From roughness parameters data of rough surface
separation, elastic deformation force and friction force
contact, it is found that value of rms surface
is maximum at θ=5 and thereafter decreases and
roughness ( σ ) and asperity radius ( R ) are inversely
converges exponentially to a same value as the elastic
proportional and multiplication of both the two
adhesion index (θ) increases. So, it indicates rough
parameter is almost constant. Present study, low θ
surfaces are not able to support external load and
value is considered from 5 to 25 for soft polymeric
friction load by elastic deformation. Conversely,
MEMS material and λ value is considered almost smooth surface contact mainly occurs due to elastic
constant, 2.5 or 5. It is assumed that soft material
deformation of asperities. Fig.3a and Fig.3b shows
parameters, modulus of elasticity ( K ) and hardness ( that for particular mean separation, plastic
H ), and surface energy ( γ ) are almost constant and θ
deformation force and friction force is minimum at
values slightly increases from 5 to 25 due to only
θ=5 and thereafter increases and diverges linearly to a
variation of surface roughness keeping constant of λ
maximum value as the elastic adhesion index (θ)
value. So, keeping same λ value, as θ value increases,
increases. So, it indicates rough surfaces are much
soft surface becomes much more rougher.
more able to support external load and friction load by
plastic deformation. Conversely, smooth surface
contact is not supported by plastic deformation of
asperities.

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λ=5,һ= 1 λ=5,һ= 0 λ=5,һ= -1 λ=5,һ= 1 λ=5,һ= 0 λ=5,һ= -1
Plastic deformation force (P p *)

40
12

Total friction force (T*)
35
10
30
25 8
20 6
15 4
10
2
5
0
0
5 10 15 20 25 5 10 15 20 25
Elastic adhesion index (θ) Elastic adhesion index (θ)

Fig.3a Plastic deformation force Vs Elastic adhesion index Fig.4b Total friction force Vs Elastic adhesion index

λ=5,һ= 1 λ=5,һ= 0 λ=5,һ= -1 λ=5,һ= 1 λ=5,һ= 0 λ=5,һ= -1

12 Coefficient of static friction (μ) 0.7
Plastic friction force (T p *)

10 0.6
0.5
8
0.4
6
0.3
4 0.2

2 0.1
0
0
5 10 15 20 25
5 10 15 20 25

Fig.5 Coefficient of static friction Vs Elastic adhesion
Fig.3b Plastic friction force Vs Elastic adhesion index
index
λ=2.5,θ=5 λ=2.5,θ=10 λ=2.5,θ=20
λ=5,һ= 1 λ=5,һ= 0 λ=5,һ= -1
Elastic deformation force (P e*)

0.180
Total deformation force (P*)

40 0.160
35 0.140
30 0.120
25 0.100
20 0.080
15 0.060
10 0.040
5 0.020
0 0.000
5 10 15 20 25 4 3 2 1 0 -1
Mean separation (h)

Fig.6a Elastic deformation force Vs Mean separation
Fig.4a Total deformation force Vs Elastic adhesion index

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Fig.4a Fig.4b shows that nature of curves for
λ=2.5,θ=5 λ=2.5,θ=10 λ=2.5,θ=20
variation of total deformation and friction force with
adhesion index for a particular value of h is same with
9
Fig.3a and Fig.3b. Contribution of elastic deformation

Plastic friction force (T p *)
8
force and friction force is negligible with respect to
plastic deformation force and friction force and so, 7
total forces are mainly contributed by plastically 6
deformed asperity. Fig.5 depicts the variation 5
coefficient of friction with elastic adhesion index for a 4
particular value of h. It is found that COF gradually 3
decreases and converges to a constant value of 0.4. It 2
is noted that COF is constant after the value of θ=10 1
and independent with roughness effect. 0
λ=2.5,θ=5 λ=2.5,θ=10 λ=2.5,θ=20 4 3 2 1 0 -1
Mean separation (h)
0.30
Elastic friction force (Te*)

0.25 Fig.7b Plastic friction force Vs Mean separation

0.20
λ=2.5,θ=5 λ=2.5,θ=10 λ=2.5,θ=20
0.15
30
0.10
Total deformation force (P*)

25
0.05
20
0.00
15
4 3 2 1 0 -1
Mean separation (h)
10

5
Fig.6b Elastic friction force Vs Mean separation
0
λ=2.5,θ=5 λ=2.5,θ=10 λ=2.5,θ=20
4 3 2 1 0 -1
Mean separation (h)
Plastic deformation force (P p *)

30

25 Fig.8a Total deformation force Vs Mean separation
20
λ=2.5,θ=5 λ=2.5,θ=10 λ=2.5,θ=20
15
9
10
8
5
Total friction force (T*)

7
0 6
4 3 2 1 0 -1 5
Mean separation (h) 4
3
Fig.7a Plastic deformation force Vs Mean separation 2
1
In Fig.6a and Fig.6b the graphs are plotted 0
showing the variation of elastic deformation force and 4 3 2 1 0 -1
friction force with variation in dimensional mean Mean separation (h)
separation at λ=2.5 and θ=5, 10, 20. It shows that
deformation force and friction force for elastically Fig.8b Total friction force Vs Mean separation
deformed asperities increases exponentially with
decrease in dimensionless mean separation upto h=0
Fig.7a and Fig.7b depicts variation plastic
and thereafter it decreases to a finite value. For a deformation force and friction force with mean
particular mean separation value, elastic deformation separation. It shows that deformation force and
force and friction force decreases with increase in friction force for plastically deformed asperities
elastic adhesion index (θ) value as discussed earlier.

691 | P a g e


increases exponentially with decrease in References
dimensionless mean separation. For a particular mean 1. G Amontons, Histoire de l'Académie Royale
separation value, plastic deformation force and des Sciences avec les Mémoires de
friction increases with increase in elastic adhesion Mathématique et de Physique (1699)
index (θ) value as mentioned earlier. Like Fig.7a and 2. B Bera, Adhesional friction theory of
Fig.7b, similar nature of curve is obtained in Fig.8a micromechanical surface contact. IOSR
and Fig.8b when the graph is plotted between total Journal of Engineering, 3 (2), 2013, 38-44
deformation force and friction force with mean 3. W R Chang, I Etsion, D B Bogy, An elastic–
separation, because high plastic deformation force and plastic model for the contact of rough
friction force value diminishes the effect of low surfaces. ASME Journal of Tribology, 109,
elastic deformation force and friction force value. 1987, 257–63
4. W R Chang, I Etsion, D B Bogy, Static
λ=2.5,θ=5 λ=2.5,θ=10 λ=2.5,θ=20
friction coefficient model for metallic rough
Coefficient of static friction (μ)

surfaces, ASME Journal of Tribology, 110,
0.7
1988, 57-63
0.6 5. F P Bowden and D Tabor, The Friction and
0.5 Lubrication of Solids (Oxford University
Press, New York 1950)
0.4
6. K. N. G. Fuller and D. Tabor, The effect of
0.3 surface roughness on the adhesion of elastic
0.2 solid Proc. R. Soc. London. A 345, 1975,
327-345
0.1
7. K L Johnson, Adhesion at the contact of
0 solids, in theoretical and applied mechanics
3 2 1 0 -1 (Ed. W T Koiter), North Holland,
Mean separation (h) Amsterdam.
8. J A Greenwood, and J B P Williamson,
Contact of nominally flat surfaces. Proc. R.
Fig.9 Coefficient of static friction Vs Mean separation Soc. London., A 295, 1966, 300-319

Fig.9 represents variation of coefficient of static
friction with mean separation and it shows that COF
converges to the value of 0.4 with decrement of mean
separation. From the study of coefficient of static
friction of elastic-plastic soft surface contact, it is
found that COF is almost constant value of 0.4 which
is mostly independent of roughness effect ( θ ) and
mean separation (h ). So, COF is almost independent
of load and friction which supports Amonton’s law of
dry friction.

4. Conclusion
From the above study, it could be concluded that
at low value of θ, i.e. for smooth MEMS surfaces, the
contact is mostly elastic in nature and at high value of
θ, i.e. for rough MEMS surfaces, mostly plastic
contacts occur. This supports the elastic-plastic
concept of Greenwood and Williamson for rough
surface contact. Total deformation force and friction
force are mainly supported by plastically deformed
asperities of soft MEMS surfaces. From the study of
coefficient of static friction of elastic-plastic soft
MEMS surface contact, it is found that COF is almost
constant value of 0.4 which is mostly independent of
roughness effect ( θ ) and mean separation ( h ). The
evaluated COF=0.4 is almost equal to experimentally,
found COF=0.5.

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