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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
Nominal Surface
•
Contact Bump (larger, micro

scale)
•
Asperities (smaller, nano

scale)
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Basics of Hertz Contact
The
pressure distribution
:
produces a parabolic depression
on the surface of an elastic body.
Resultant Force
Pressure Profile
p(r)
r
a
p
0
Depth at center
Curvature
in contact region
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Basics of Hertz Contact
Elasticity problem of a very “large” initially flat body
indented by a rigid sphere.
rigid
We have an elastic half

space with a spherical
depression. But:
R
r
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Basics of Hertz Contact
So the pressure distribution given by:
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
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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P
Hertz Contact
Hertz Contact (1882)
2a
R
1
R
2
E
1
,
1
E
2
,
2
Interference
Contact Radius
Effective Radius
of Curvature
Effective
Young’s modulus
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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
.
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
)
.
Fully Plastic
(uncontained plastic flow)
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Round Bump Fabrication
•
Critical issues for
profile transfer:
–
Process
Pressure
–
Biased Power
–
Gas Ratio
Photo Resist Before Reflow
Photo Resist After Reflow
The shape of the photo
resist is transferred to the
silicon by using SF
6
/O
2
/Ar
ICP silicon etching
process.
Shipley 1818
O
2
:SF
6
:Ar=20:10:25
O
2
:SF
6
:Ar=15:10:25
Silicon Bump
Silicon Bump
Shipley 1818
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Evolution of Contacts
After 10 cycles
After 10
2
cycles
After 10
3
cycles
After 10
4
cycles
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c
, a
C
, P
C
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
Note when
/
c
=110, then P/A=2.8
Y
Elasto

Plastic Contacts
(L. Kogut and I Etsion, Journal of Applied Mechanics, 2002, pp. 657

662)
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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
=
a
2
/2R
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Nanoindenters
Hysitron Triboindenter®
Hysitron Ubi®
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Nanoindentation Test
Force vs. displacement
Indent
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Depth

Dependent Hardness
H
0
=0.58 GPa
h*=1.60
m
Data from Nix & Gao, JMPS, Vol. 46, pp. 411

425, 1998.
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Surface Topography
Standard Deviation of Surface Roughness
Standard Deviation of Asperity Summits
Scaling Issues
–
2D, Multiscale, Fractals
Mean of Surface
Mean of Asperity Summits
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Contact of Surfaces
d
Reference Plane
Mean of Asperity
Summits
Typical Contact
Flat and Rigid Surface
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Typical Contact
Original shape
2a
P
R
Contact area
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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)
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)
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 ~L
2
or L; weight as L
3
Surface Forces/Weight
~ 1/L or 1/L
2
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 (
)
For identical materials
Also characterized by an inter

atomic potential
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Adhesion Theories
Z
0
1
2
3

1

0.5
0
0.5
1
1.5
Z/Z
0
/
TH
Some inter

atomic
potential, e.g.
Lennard

Jones
Z
0
(A simple point

of

view)
For ultra

clean metals, the potential is more sharply peaked.
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Two Rigid Spheres:
Bradley Model
P
P
R
2
R
1
Bradley, R.S., 1932, Philosophical Magazine,
13
, pp. 853

862.
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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
a
a
P
1
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Derivation of JKR Model
Total Energy E
T
Stored Elastic
Energy
Mechanical Potential
Energy in the Applied Load
Surface
Energy
Equilibrium when
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JKR Model
•
Hertz model
Only
compressive
stresses
can exist in the contact area.
Pressure Profile
Hertz
a
r
p(r)
Deformed Profile of
Contact Bodies
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.
JKR
p(r)
a
r
P
a
a
P
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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.
3.
F has a
minimum value
to meet the equilibrium equation
i.e. the pull

off force.
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DMT Model
DMT model
Tensile stresses exist
outside
the contact area.
Stress profile remains Hertzian
inside
the contact area.
p(r)
a
r
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.
Applied Force, Contact Radius & Vertical Approach
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Tabor Parameter:
JKR

DMT Transition
DMT theory applies
(stiff solids, small radius of curvature, weak energy of adhesion)
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
where
0
1
2
3

1

0.5
0
0.5
1
1.5
Z/Z
0
/
TH
Maugis approximation
h
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
JKR valid for large
DMT valid for small
Tabor Parameter
0
1
2
3

1

0.5
0
0.5
1
1.5
Z/Z
0
/
TH
Maugis
JKR
DMT
Lennard

Jones
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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