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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