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 2.8Y).
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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=2.8Y
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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*=1.60m
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
PPullOff 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 3R 6RP (3R) 2 ,
R
a2
8a
R
3K
dET
0
da
K
4 *
E
3
PPullOff 1.5R
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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.
6R
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 2R,
R
a2
R
PPullOff 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
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= 0.072 N/m for water at room temperature
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p
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