Transcript Slide 1

Contact Mechanics
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SEM Image of Early Northeastern
University MEMS Microswitch
Asperity
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SEM of Current NU Microswitch
Asperities
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Two Scales of the Contact
• Contact Bump (larger, micro-scale)
• Asperities (smaller, nano-scale)
Nominal Surface
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Basics of Hertz Contact
The pressure distribution:
p ( r )  p0 1  ( r / a ) 2 , r  a
p(r)
p
produces a parabolic depression
on the surface of an elastic body.
(1  2 )
Depth at center  
p0 a
0
r
2E
Curvature in contact region
Pressure Profile
a
2
1 (1  ) p0

R
2 Ea
Resultant Force
P
a
0
2 2
p(r )2 rdr  a p0
3
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Basics of Hertz Contact
Elasticity problem of a very “large” initially flat body
indented by a rigid sphere.
P
R
rigid
r
δ
R
a
r
z
R  R2  r 2
We have an elastic half-space with a spherical
depression. But:
(r  R)
w(r )    ( R  R 2  r 2 )    R(1  1  r 2 / R 2 )    r 2 / 2R
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Basics of Hertz Contact
 So the pressure distribution given by:
p ( r )  p0 1  ( r / a ) 2 , r  a
gives a spherical depression and hence is the pressure
for Hertz contact, i.e. for the indentation of a flat elastic
body by a rigid sphere with
2
2
1 (1  ) p0

R
2 Ea
(1  )

p0 a
2E
 But wait – that’s not all !
 Same pressure on a small circular region of a locally
spherical body will produce same change in curvature.
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Basics of Hertz Contact
P
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Hertz Contact
Hertz Contact (1882)
 3P 
   * 1/ 2 
 4E R 
P
2/3
Interference
1/ 3
 3PR 
a
* 
4
E


1 1  12 1  22


*
E
E1
E2
1
1
1


R R1 R2
Contact Radius
Effective
Young’s modulus
Effective Radius
of Curvature
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E2,2
R2

2a
R1
E1,1
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Assumptions of Hertz
 Contacting bodies are locally spherical
 Contact radius << dimensions of the body
 Linear elastic and isotropic material properties
 Neglect friction
 Neglect adhesion
 Hertz developed this theory as a graduate student during
his 1881 Christmas vacation
 What will you do during your Christmas vacation ?????
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Onset of Yielding
 Yielding initiates below the surface when VM = Y.
Fully Plastic
(uncontained plastic flow)
Elasto-Plastic
(contained plastic flow)
 With continued loading the plastic zone grows and reaches
the surface
 Eventually the pressure distribution is uniform, i.e. p=P/A=H
(hardness) and the contact is called fully plastic (H  2.8Y).
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Round Bump Fabrication
Shipley 1818
Photo Resist Before Reflow
Shipley 1818
The shape of the photo
resist is transferred to the
silicon by using SF6/O2/Ar
ICP silicon etching
process.
Photo Resist After Reflow
Silicon Bump
Silicon Bump
• Critical issues for
profile transfer:
– Process
Pressure
– Biased Power
– Gas Ratio
O2:SF6:Ar=20:10:25
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O2:SF6:Ar=15:10:25
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Evolution of Contacts
After 10 cycles
After 102 cycles
After 103 cycles
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After 104 cycles
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Elasto-Plastic Contacts
(L. Kogut and I Etsion, Journal of Applied Mechanics, 2002, pp. 657-662)
4E aC
 KH 
 C   *  R, K  0.454 0.41 , H  2.8 Y , aC   C R , PC 
3R
 2E 
2
* 3
c, aC, PC are the critical interference, critical contact radius,
and critical force respectively. i.e. the values of , a, P for
the initiation of plastic yielding
Curve-Fits for Elastic-Plastic Region

P
 1.03
PC
 C

P
 1.40
PC
 C
1.425



,
1.263



,

A
 0.93
AC
 C

A
 0.94
AC
 C
1.136



,
1

6
C
,
6

 110
C
1.146



Note when /c=110,ME6260/EECE7244
then P/A=2.8Y
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Fully Plastic Single Asperity Contacts
(Hardness Indentation)
Contact pressure is uniform and equal to
the hardness (H)
Area varies linearly with force A=P/H
Area is linear in the interference  = a2/2R
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Nanoindenters
Hysitron Ubi®
Hysitron Triboindenter®
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Nanoindentation Test
Indent
Force vs. displacement
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Depth-Dependent Hardness
Depth Dependence of Hardness of Cu
12
10
H
h*
 1
H0
h
0
(H/H )2
8
6
H0=0.58 GPa
4
h*=1.60m
2
0
0
1
2
3
4
1/h (1/ m)
5
6
7
Data from Nix & Gao, JMPS, Vol. 46, pp. 411-425, 1998.
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Surface Topography
Mean of Asperity Summits
Mean of Surface
1 L
2
Standard Deviation of Surface Roughness   0 ( z  m) dx
L
2
Standard Deviation of Asperity Summits
1 N
 S   ( z Si  z S ) 2
N i 1
2
Scaling Issues – 2D, Multiscale, Fractals
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Contact of Surfaces
Flat and Rigid Surface
d
Reference Plane
Mean of Asperity
Summits
Typical Contact
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Typical Contact
P
Original shape
Contact area

2a
R
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Multi-Asperity Models
(Greenwood and Williamson, 1966, Proceedings of the Royal Society
of London, A295, pp. 300-319.)
Assumptions
 All asperities are spherical and have the same summit
curvature.
 The asperities have a statistical distribution of heights
(Gaussian).
z
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(z)
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Multi-Asperity Models
(Greenwood and Williamson, 1966, Proceedings of the Royal Society
of London, A295, pp. 300-319.)
Assumptions (cont’d)
 Deformation is linear elastic and isotropic.
 Asperities are uncoupled from each other.
 Ignore bulk deformation.
z
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(z)
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Greenwood and Williamson
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Greenwood & Williamson Model
 For a Gaussian distribution of asperity heights the
contact area is almost linear in the normal force.
 Elastic deformation is consistent with Coulomb friction
i.e. A  P, F  A, hence F  P, i.e. F = N
 Many modifications have been made to the GW theory to
include more effects  for many effects not important.
 Especially important is plastic deformation and adhesion.
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Contacts With Adhesion
(van der Waals Forces)
 Surface forces important in MEMS due to scaling
 Surface forces ~L2 or L; weight as L3
 Surface Forces/Weight ~ 1/L or 1/L2
 Consider going from cm to m
 MEMS Switches can stick shut
 Friction can cause “moving” parts to stick, i.e. “stiction”
 Dry adhesion only at this point; meniscus forces later
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Forces of Adhesion
 Important in MEMS Due to Scaling
 Characterized by the Surface Energy () and
the Work of Adhesion ()    1   2   12
 For identical materials
  2
 Also characterized by an inter-atomic potential
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Adhesion Theories
(A simple point-of-view)
1.5
Some inter-atomic
potential, e.g.
Lennard-Jones
1
Z0
 /
TH
0.5
0
Z
-0.5
-1
0
1
2
3
Z/Z 0
For ultra-clean metals, the potential is more sharply peaked.
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Two Rigid Spheres:
Bradley Model
P
R2
1
1
1


R R1 R2
PPullOff  2 R
Bradley, R.S., 1932, Philosophical Magazine,
13, pp. 853-862.
R1
P
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JKR Model
Johnson, K.L., Kendall, K., and Roberts, A.D., 1971, “Surface Energy and the Contact
of Elastic Solids,” Proceedings of the Royal Society of London, A324, pp. 301-313.
•
•
•
•
Includes the effect of elastic deformation.
Treats the effect of adhesion as surface energy only.
Tensile (adhesive) stresses only in the contact area.
Neglects adhesive stresses in the separation zone.
P
P1
a
a
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Derivation of JKR Model
Stored Elastic
Energy
Mechanical Potential
Energy in the Applied Load
Surface
Energy
Total Energy ET
Equilibrium when
3
a K
 P  3R  6RP  (3R) 2 ,
R
a2
8a
 
R
3K
dET
0
da
K
4 *
E
3
PPullOff  1.5R
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JKR Model
Deformed Profile of
Contact Bodies
Pressure Profile
P
p(r)
• Hertz model
Hertz
Only compressive stresses
can exist in the contact area.
a
a




JKR model
Stresses only remain
compressive in the center.
Stresses are tensile at the
edge of the contact area.
Stresses tend to infinity
around the contact area.
r
P
p(r)
JKR
a
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a
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JKR Model
1. When  = 0, JKR equations revert to the Hertz equations.
2. Even under zero load (P = 0), there still exists a contact radius.
 6R
a 0  
 K
2



1
3
0 
 4  R 
a0

 
2
3R  3K

2
2
2
1
3
3. F has a minimum value to meet the equilibrium equation
Pmin
3
  R
2
1/ 3
 min
 3 R 2 

 
 2 2K 
i.e. the pull-off force.
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DMT Model
Derjaguin, B.V., Muller, V.M., Toporov, Y.P., 1975, J. Coll. Interf. Sci., 53, pp. 314-326.
Muller, V.M., Derjaguin, B.V., Toporov, Y.P., 1983, Coll. and Surf., 7, pp. 251-259.
DMT model


p(r)
Tensile stresses exist outside
the contact area.
Stress profile remains Hertzian
inside the contact area.
a
r
Applied Force, Contact Radius & Vertical Approach
a3 K
 P  2R,
R
a2

R
PPullOff  2  R
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JKR-DMT Transition
Tabor Parameter:
1/ 3
  R 
   *2 3 
 E Z0 
2
  1 DMT theory applies
(stiff solids, small radius of curvature, weak energy of adhesion)
  1 JKR theory applies
(compliant solids, large radius of curvature, large adhesion energy)
Recent papers suggest another model for DMT & large loads.
J. A. Greenwood 2007, Tribol. Lett., 26 pp. 203–211
W. Jiunn-Jong, J. Phys. D: Appl. Phys. 41 (2008), 185301.
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Maugis Approximation
1.5
 TH , Z  Z 0  h0
 
 0, Z  Z 0  h0
Maugis approximation
1
0.5
 /
TH
where
h0 TH  
0
h0
-0.5
 h0  Z 0
-1
0
1
2
3
Z/Z 0
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Elastic Contact With Adhesion
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Elastic Contact With Adhesion
w=
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Elastic Contact With Adhesion
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Adhesion of Spheres
1.5
JKR
Tabor Parameter
Maugis
  
 /
TH
0.5
1/ 3
 R 

*2 3 
 E Z0 
1
Lennard-Jones
0
2
JKR valid for large 
DMT valid for small 
DMT
-0.5
-1
0
1
2
Z/Z0
3
 and TH are most important
E. Barthel, 1998, J. Colloid Interface Sci., 200, pp. 7-18
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Adhesion Map
K.L. Johnson and J.A. Greenwood, J. of Colloid Interface Sci., 192, pp. 326-333, 1997
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Multi-Asperity Models
With Adhesion
• Replace Hertz Contacts of GW Model with JKR Adhesive
Contacts: Fuller, K.N.G., and Tabor, D., 1975, Proc.
Royal Society of London, A345, pp. 327-342.
• Replace Hertz Contacts of GW Model with DMT Adhesive
Contacts: Maugis, D., 1996, J. Adhesion Science and
Technology, 10, pp. 161-175.
• Replace Hertz Contacts of GW Model with Maugis
Adhesive Contacts: Morrow, C., Lovell, M., and Ning, X.,
2003, J. of Physics D: Applied Physics, 36, pp. 534-540.
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Surface Tension
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http://www.unitconversion.org/unit_converter/surface-tension-ex.html
44
 = 0.072 N/m for water at room temperature
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
p
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