Contact mechanics
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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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Contact mechanics

Contact Mechanics


Sem image of early northeastern university mems microswitch

Asperity

SEM Image of Early Northeastern University MEMS Microswitch


Sem of current nu microswitch

Asperities

SEM of Current NU Microswitch


Two scales of the contact

Nominal Surface

Two Scales of the Contact

  • Contact Bump (larger, micro-scale)

  • Asperities (smaller, nano-scale)


Basics of hertz contact

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.


Basics of hertz contact1

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 contact2

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.


Basics of hertz contact3

Basics of Hertz Contact

P


Hertz contact

P

E2,2

R2

Interference

2a

E1,1

R1

Contact Radius

Effective

Young’s modulus

Effective Radius

of Curvature

Hertz Contact

Hertz Contact (1882)


Assumptions of hertz

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 ?????


Onset of yielding

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).


Round bump fabrication

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

Evolution of Contacts

After 10 cycles

After 102 cycles

After 103 cycles

After 104 cycles


Contact mechanics

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

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

Nanoindenters

Hysitron Ubi®

Hysitron Triboindenter®


Nanoindentation test

Nanoindentation Test

Indent

Force vs. displacement


Depth dependent hardness

H0=0.58 GPa

h*=1.60m

Depth-Dependent Hardness

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


Surface topography

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

Contact of Surfaces

Flat and Rigid Surface

d

Reference Plane

Mean of Asperity

Summits

Typical Contact


Typical contact

2a

Typical Contact

P

Original shape

Contact area

R


Contact mechanics

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).


Contact mechanics

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 and williamson

Greenwood and Williamson


Greenwood williamson model

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

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


Forces of adhesion

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


Adhesion theories

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

Adhesion Theories

(A simple point-of-view)

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


Two rigid spheres bradley model

P

R2

R1

P

Two Rigid Spheres:Bradley Model

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

13, pp. 853-862.


Contact mechanics

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

Derivation of JKR Model

Stored Elastic Energy

Mechanical Potential Energy in the Applied Load

Surface Energy

Total Energy ET

Equilibrium when


Jkr model

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 model1

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.


Dmt model

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.


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)

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.


Maugis approximation

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


Elastic contact with adhesion

Elastic Contact With Adhesion


Elastic contact with adhesion1

Elastic Contact With Adhesion

w=


Elastic contact with adhesion2

Elastic Contact With Adhesion


Adhesion of spheres

1.5

JKR

Maugis

1

0.5

Lennard-Jones

TH

s

/

s

0

DMT

-0.5

-1

0

1

2

3

Z/Z

0

Adhesion of Spheres

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


Adhesion map

Adhesion Map

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


Multi asperity models with adhesion

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.


Surface tension

Surface Tension


Contact mechanics

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


Contact mechanics

 = 0.072 N/m for water at room temperature


Contact mechanics

p


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