ME6260/EECE72441

Contact Mechanics

ME6260/EECE72442

SEM Image of Early Northeastern University MEMS

Microswitch

Asperity

ME6260/EECE72443

SEM of Current NU Microswitch

Asperities

ME6260/EECE72444

Two Scales of the Contact

Nominal Surface

• Contact Bump (larger, micro-scale)

• Asperities (smaller, nano-scale)

ME6260/EECE72445

Basics of Hertz Contact

ararprp ,)/(1)( 20

The pressure distribution:

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

Resultant Force

02

0 3

22)( pardrrpP

a

Pressure Profile

p(r)

r

a

p0

Depth at center

Curvature in contact region

apE 0

2

2

)1(

Ea

p

R 2

)1(1 02

ME6260/EECE72446

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

ME6260/EECE72447

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

ME6260/EECE72448

Basics of Hertz Contact

P

ME6260/EECE72449

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 modulus

2

22

1

21

*

111

EEE

ME6260/EECE724410

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/EECE724411

Onset of Yielding Yielding initiates below the surface when VM = Y which for Poisson’s

ratio of 0.3 occurs when the average contact pressure is 1.07*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)

ME6260/EECE724412

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

ME6260/EECE724413

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

ME6260/EECE724414

Nanoindenters

Hysitron Triboindenter®Hysitron Ubi®

ME6260/EECE724415

Nanoindentation Test

Force vs. displacementIndent

ME6260/EECE724416

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.

ME6260/EECE724417

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

ME6260/EECE724418

Contact of Surfaces

d

Reference PlaneMean of AsperitySummits

Typical Contact

Flat and Rigid Surface

ME6260/EECE724419

Typical Contact

Original shape

2a

P

R

Contact area

ME6260/EECE724420

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

ME6260/EECE724421

Greenwood and Williamson

ME6260/EECE724422

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 many are not important.

Especially important is plastic deformation and adhesion.

ME6260/EECE724423

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

ME6260/EECE724424

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

ME6260/EECE724425

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.

ME6260/EECE724426

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.

ME6260/EECE724427

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

ME6260/EECE724428

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

ME6260/EECE724429

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

ME6260/EECE724430

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)

RP OffPull 2

RP OffPull 5.1

Intermediate values of µ

ME6260/EECE724431

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

ME6260/EECE724432

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/EECE724433

Surface Tension

ME6260/EECE724434

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

ME6260/EECE724435

= 0.072 N/m for water at room temperature

ME6260/EECE724436

ME6260/EECE724437

pp

ME6260/EECE724438

ME6260/EECE724439

ME6260/EECE724440

ME6260/EECE724441