a, P - Northeastern University

baconossifiedΜηχανική

29 Οκτ 2013 (πριν από 3 χρόνια και 11 μήνες)

138 εμφανίσεις

ME6260/EECE7244

1

Contact Mechanics

ME6260/EECE7244

2

SEM Image of Early Northeastern
University MEMS Microswitch

Asperity

ME6260/EECE7244

3

SEM of Current NU Microswitch

Asperities

ME6260/EECE7244

4

Two Scales of the Contact

Nominal Surface


Contact Bump (larger, micro
-
scale)




Asperities (smaller, nano
-
scale)

ME6260/EECE7244

5

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



ME6260/EECE7244

6

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

ME6260/EECE7244

7

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.


ME6260/EECE7244

8

Basics of Hertz Contact

P

ME6260/EECE7244

9

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

ME6260/EECE7244

10

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 ?????

ME6260/EECE7244

11

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)

ME6260/EECE7244

12

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

ME6260/EECE7244

13

Evolution of Contacts

After 10 cycles

After 10
2

cycles

After 10
3

cycles

After 10
4

cycles

ME6260/EECE7244

14


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)

ME6260/EECE7244

15

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

ME6260/EECE7244

16

Nanoindenters

Hysitron Triboindenter®

Hysitron Ubi®

ME6260/EECE7244

17

Nanoindentation Test

Force vs. displacement

Indent

ME6260/EECE7244

18

Depth
-
Dependent Hardness

H
0
=0.58 GPa


h*=1.60

m

Data from Nix & Gao, JMPS, Vol. 46, pp. 411
-
425, 1998.

ME6260/EECE7244

19

Surface Topography

Standard Deviation of Surface Roughness



Standard Deviation of Asperity Summits



Scaling Issues


2D, Multiscale, Fractals

Mean of Surface

Mean of Asperity Summits

ME6260/EECE7244

20

Contact of Surfaces

d

Reference Plane

Mean of Asperity

Summits

Typical Contact

Flat and Rigid Surface

ME6260/EECE7244

21

Typical Contact

Original shape

2a



P

R

Contact area

ME6260/EECE7244

22

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

ME6260/EECE7244

23

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

ME6260/EECE7244

24

Greenwood and Williamson

ME6260/EECE7244

25

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.

ME6260/EECE7244

26

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


ME6260/EECE7244

27

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

ME6260/EECE7244

28

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.

ME6260/EECE7244

29

Two Rigid Spheres:

Bradley Model

P

P

R
2

R
1

Bradley, R.S., 1932, Philosophical Magazine,

13
, pp. 853
-
862.

ME6260/EECE7244

30

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

ME6260/EECE7244

31

Derivation of JKR Model

Total Energy E
T


Stored Elastic
Energy

Mechanical Potential
Energy in the Applied Load

Surface
Energy

Equilibrium when

ME6260/EECE7244

32

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

ME6260/EECE7244

33

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.

ME6260/EECE7244

34

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

ME6260/EECE7244

35

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.

ME6260/EECE7244

36

Maugis Approximation

where

0

1

2

3

-
1

-
0.5

0

0.5

1

1.5

Z/Z

0



/



TH

Maugis approximation

h
0

ME6260/EECE7244

37

Elastic Contact With Adhesion

ME6260/EECE7244

38

Elastic Contact With Adhesion

w=


ME6260/EECE7244

39

Elastic Contact With Adhesion

ME6260/EECE7244

40

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

ME6260/EECE7244

41

Adhesion Map

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

ME6260/EECE7244

42

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.

ME6260/EECE7244

43

Surface Tension

ME6260/EECE7244

44

http://www.unitconversion.org/unit_converter/surface
-
tension
-
ex.html

ME6260/EECE7244

45



= 0.072 N/m for water at room temperature

ME6260/EECE7244

46

ME6260/EECE7244

47


p

ME6260/EECE7244

48

ME6260/EECE7244

49

ME6260/EECE7244

50

ME6260/EECE7244

51