# Contact Mechanics - PowerPoint PPT Presentation

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Contact Mechanics. Asperity. SEM Image of Early Northeastern University MEMS Microswitch. Asperities. SEM of Current NU Microswitch. Nominal Surface. Two Scales of the Contact. Contact Bump (larger, micro-scale) Asperities (smaller, nano-scale). Depth at center

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Contact Mechanics

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Asperity

Asperities

Nominal Surface

### Two Scales of the Contact

• Contact Bump (larger, micro-scale)

• Asperities (smaller, nano-scale)

Depth at center

Curvature in contact region

p(r)

p0

Pressure Profile

r

a

Resultant Force

### Basics of Hertz Contact

The pressure distribution:

produces a parabolic depression

on the surface of an elastic body.

rigid

R

r

### Basics of Hertz Contact

Elasticity problem of a very “large” initially flat body indented by a rigid sphere.

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

### 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 locallyspherical body will produce same change in curvature.

P

P

E2,2

R2

Interference

2a

E1,1

R1

Effective

Young’s modulus

of Curvature

### Hertz Contact

Hertz Contact (1882)

### 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 ?????

### 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  2.8Y).

Critical issues for profile transfer:

Process Pressure

Biased Power

Gas Ratio

### Round Bump Fabrication

Shipley 1818

Shipley 1818

The shape of the photo resist is transferred to the silicon by using SF6/O2/Ar ICP silicon etching process.

Photo Resist Before Reflow

Photo Resist After Reflow

Silicon Bump

Silicon Bump

O2:SF6:Ar=20:10:25

O2:SF6:Ar=15:10:25

### Evolution of Contacts

After 10 cycles

After 102 cycles

After 103 cycles

After 104 cycles

Elasto-Plastic Contacts

(L. Kogut and I Etsion, Journal of Applied Mechanics, 2002, pp. 657-662)

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=2.8Y

### 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

### Nanoindenters

Hysitron Ubi®

Hysitron Triboindenter®

### Nanoindentation Test

Indent

Force vs. displacement

H0=0.58 GPa

h*=1.60m

### Depth-Dependent Hardness

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

Mean of Asperity Summits

Mean of Surface

### Surface Topography

Standard Deviation of Surface Roughness

Standard Deviation of Asperity Summits

Scaling Issues – 2D, Multiscale, Fractals

### Contact of Surfaces

Flat and Rigid Surface

d

Reference Plane

Mean of Asperity

Summits

Typical Contact

2a

P

Original shape

Contact area

R

z

(z)

### 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)

### 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.

### 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.

### 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

• Important in MEMS Due to Scaling

• Characterized by the Surface Energy () and

• For identical materials

• Also characterized by an inter-atomic potential

1.5

Some inter-atomic potential, e.g. Lennard-Jones

1

Z0

0.5

TH

Z

s

/

s

0

-0.5

-1

0

1

2

3

Z/Z

0

(A simple point-of-view)

For ultra-clean metals, the potential is more sharply peaked.

P

R2

R1

P

13, pp. 853-862.

P

P1

a

a

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.

### Derivation of JKR Model

Stored Elastic Energy

Mechanical Potential Energy in the Applied Load

Surface Energy

Total Energy ET

Equilibrium when

Deformed Profile of Contact Bodies

P

• JKR model

• Stresses only remain compressive in the center.

• Stresses aretensile at the edge of the contact area.

• Stresses tend to infinityaround the contact area.

P

p(r)

JKR

a

r

a

a

### JKR Model

Pressure Profile

p(r)

• Hertz model

Only compressive stresses can exist in the contact area.

Hertz

a

r

### JKR Model

• When  = 0, JKR equations revert to the Hertz equations.

• Even under zero load (P = 0), there still exists a contact radius.

• F has a minimum value to meet the equilibrium equation

• i.e. the pull-off force.

p(r)

Applied Force, Contact Radius & Vertical Approach

a

r

### 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

• Tensile stresses exist outside the contact area.

• Stress profile remains Hertzian inside the contact area.

DMT theory applies

JKR theory applies

### JKR-DMT Transition

Tabor Parameter:

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.

1.5

Maugis approximation

1

0.5

TH

s

/

s

0

h0

-0.5

-1

0

1

2

3

Z/Z

0

### Maugis Approximation

where

w=

1.5

JKR

Maugis

1

0.5

Lennard-Jones

TH

s

/

s

0

DMT

-0.5

-1

0

1

2

3

Z/Z

0

Tabor Parameter

JKR valid for large 

DMT valid for small 

 and TH are most important

E. Barthel, 1998, J. Colloid Interface Sci., 200, pp. 7-18

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

• 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.

### Surface Tension

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

 = 0.072 N/m for water at room temperature

p