# a, P - Northeastern University

Mechanics

Oct 29, 2013 (4 years and 8 months ago)

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ME6260/EECE7244

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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

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

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)

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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(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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Important in MEMS Due to Scaling

Characterized by the Surface Energy (

)
and


)

For identical materials

Also characterized by an inter
-
atomic potential

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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:

P

P

R
2

R
1

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

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

(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

JKR theory applies

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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w=


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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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K.L. Johnson and J.A. Greenwood, J. of Colloid Interface Sci., 192, pp. 326
-
333, 1997

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Multi
-
Asperity Models

Replace Hertz Contacts of GW Model with
JKR

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

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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