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Contact Mechanics

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Asperity

Asperities

Nominal Surface

- 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

The pressure distribution:

produces a parabolic depression

on the surface of an elastic body.

rigid

R

r

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:

- 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

Contact Radius

Effective

Young’s modulus

Effective Radius

of Curvature

Hertz Contact (1882)

- 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 ?????

- 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 2.8Y).

Critical issues for profile transfer:

Process Pressure

Biased Power

Gas Ratio

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

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=2.8Y

- 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

Hysitron Ubi®

Hysitron Triboindenter®

Indent

Force vs. displacement

H0=0.58 GPa

h*=1.60m

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

Mean of Asperity Summits

Mean of Surface

Standard Deviation of Surface Roughness

Standard Deviation of Asperity Summits

Scaling Issues – 2D, Multiscale, Fractals

Flat and Rigid Surface

d

Reference Plane

Mean of Asperity

Summits

Typical Contact

2a

P

Original shape

Contact area

R

z

(z)

Assumptions

- All asperities are spherical and have the same summit curvature.
- The asperities have a statistical distribution of heights (Gaussian).

z

(z)

Assumptions (cont’d)

- Deformation is linear elastic and isotropic.
- Asperities are uncoupled from each other.
- Ignore bulk deformation.

- 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.

- 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
the Work of Adhesion ()

- 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

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

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.

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

Pressure Profile

p(r)

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

Hertz

a

r

- 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

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

(stiff solids, small radius of curvature, weak energy of adhesion)

JKR theory applies

(compliant solids, large radius of curvature, large adhesion energy)

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

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.

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

= 0.072 N/m for water at room temperature

p