Contact Mechanics

MTMG260/ECEG244

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

ararprp ,)/(1)( 20

The pressure distribution:

produces a parabolic depression on the surface of an elastic body.

Depth at center

Curvature in contact region

apE 0

2

2

)1(

Ea

p

R 2

)1(1 02

Resultant Force

02

0 3

22)( pardrrpP

a

Pressure Profile

p(r)

r

a

p0

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Basics of Hertz ContactElasticity problem of a very “large” initially flat body indented by a rigid sphere.

P

r

z

a

δ

R rigid

We have an elastic half-space with a spherical depression. But:

R

r22 rRR

RrRrRrRRrw 2/)/11()()( 22222

)( Rr

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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.

ararprp ,)/(1)( 20

apE 0

2

2

)1(

Ea

p

R 2

)1(1 02

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Basics of Hertz Contact

P

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P

Hertz ContactHertz Contact (1882)

2a

R1

R2

E1,1

E2,2

Interference3/2

2/1*4

3

RE

P

3/1

*4

3

E

PRa Contact Radius

21

111

RRR Effective Radius

of Curvature

EffectiveYoung’s modulus2

22

1

21

*

111

EEE

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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 2.8Y).

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 SF6/O2/Ar ICP silicon etching process.

Shipley 1818

O2:SF6:Ar=20:10:25 O2:SF6:Ar=15:10:25

Silicon Bump Silicon Bump

Shipley 1818

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Evolution of Contacts

After 10 cycles After 102 cycles After 103 cycles After 104 cycles

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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

Note when /c=110, then P/A=2.8Y

R

aEPRaHKR

E

KH CCCCYC 3

4,,8.2,41.0454.0,

2

3*2

*

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

662)

1106,94.0,40.1

61,93.0,03.1

146.1263.1

136.1425.1

CCCCC

CCCCC

A

A

P

P

A

A

P

P

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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 Triboindenter®Hysitron Ubi®

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Nanoindentation Test

Force vs. displacementIndent

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Depth-Dependent Hardness

0 1 2 3 4 5 6 70

2

4

6

8

10

12

1/h (1/m)

(H/H

0)2

Depth Dependence of Hardness of Cu

h

h

H

H *1

0

H0=0.58 GPa

h*=1.60m

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

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Microscale Testing – Scale Effect

John W. Hutchinson, “Plasticity at the Micron Scale,”International Journal of Solids and Structures, Vol. 37, 2000, pp. 225-238.

Tension Test Torsion Test

Uniform StressNonuniform Stress

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Microscale Testing – Scale Effect

Bending Test Hardness Test

Nonuniform Stress Nonuniform Stress

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Geometrically Necessary and Statistically Stored Dislocations

B. Bhushan and M. Nosonovsky, Acta Materialia, 2003, Vol. 51, pp. 4331-4345.

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Stress is related to strain and also to the strain-gradient

Strain gradient plays a primary role in yielding and in

post-yield behavior

Smaller is stronger

Theory applicable down to length scales of 100’s of nm

Competition between geometrically necessary and statistically

stored dislocations

Semi-empirical relation for scale-dependent hardness

h

h

H

H *1

0

Strain Gradient Plasticity Theory

(Hutchinson & Fleck; Gao & Nix)

, H0 is the macro-hardness, h* is a characteristic length

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Surface Topography

N

iSS zz

N iS

1

22 )(1

Standard Deviation of Surface Roughness

Standard Deviation of Asperity Summits

Scaling Issues – 2D, Multiscale, Fractals

Mean of Surface

Mean of Asperity Summits

L

dxmzL 0

22 )(1

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Contact of Surfaces

d

Reference PlaneMean of AsperitySummits

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 ~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 ()

For identical materials

Also characterized by an inter-atomic potential

1221

2

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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

Z0

(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

R2

R1

21

111

RRR

RP OffPull 2

Bradley, R.S., 1932, Philosophical Magazine, 13, pp. 853-862.

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JKR ModelJohnson, 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

aa

P1

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Derivation of JKR ModelDerivation of JKR Model

Total Energy Total Energy EETT

Stored Stored Elastic Elastic Energy Energy

Mechanical Potential Mechanical Potential Energy in the Applied Energy in the Applied

LoadLoad

Surface Surface EnergyEnergy

Equilibrium when 0da

dET

*23

3

4,)3(63 EKRRPRP

R

Ka

K

a

R

a

3

82 RP OffPull 5.1

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JKR ModelJKR Model

• Hertz modelHertz model Only compressive stresses can exist in the contact area.

Pressure Profile

HertHertzz

a r

p(r)

Deformed Profile of Contact Bodies

JKR modelJKR 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.

JKRJKRp(r)

a r

P

a

a

P

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JKR ModelJKR Model1. 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.

3

12

0

6

K

Ra

3

1

2

2220

0 3

4

3

K

R

R

a

RP 2

3min

3/12

min 22

3

K

R

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DMT ModelDMT Model

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.

,23

RPR

Ka R

a2

Applied Force, Contact Radius & Vertical Approach

RP OffPull 2

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Tabor Parameter:

JKR-DMT TransitionJKR-DMT Transition

1

3/1

30

2

2

ZE

R

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. 203–211W. Jiunn-Jong, J. Phys. D: Appl. Phys. 41 (2008), 185301.

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Maugis Approximation

00

00

,0

,

hZZ

hZZTH

where

0 1 2 3-1

-0.5

0

0.5

1

1.5

Z/Z 0

/

TH

Maugis approximation

h0

00

0

Zh

h TH

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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

3/1

30

2*

2

ZE

R

JKR valid for large

DMT valid for small

Tabor Parameter

0 1 2 3-1

-0.5

0

0.5

1

1.5

Z/Z0

/

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 MapK.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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JKR

DMT

Morrow, Lovell, Ning

???( )

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Microscale Friction

Amontons-Coulomb FrictionPF

Friction at the Microscale ?

PAF

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Two Separation Modes

Brittle Separation: Little if any plastic deformation during separation (Au-Au).

Ductile Separation: varying degrees of plastic deformation during separation(Au-Au).

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Contact Radius vs. Interference

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External Force vs. InterferenceExternal Force vs. Interference

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Sphere Profile Before Separation

Neck

Ru Au

No Neck