THE MECHANICS OF ADHESION – A TUTORIALfiles.asme.org/Divisions/Tribology/17449.pdf · THE...
Transcript of THE MECHANICS OF ADHESION – A TUTORIALfiles.asme.org/Divisions/Tribology/17449.pdf · THE...
Prof. George G. Adams
THE MECHANICS OF ADHESION – A TUTORIAL
George G. AdamsMechanical Engineering Department
Northeastern University, Boston, MA 02115Email: [email protected]
Presented at the Nanotribology Tutorial/Panel SessionSTLE/ASME International Joint Tribology Conference
October 20-22, 2008, Miami, Florida, USA
Prof. George G. Adams
Hertz Contact (No Adhesion)
Physical Basis of Adhesion
JKR Model
DMT Model
Maugis Model
Multi-Asperity Models
Greenwood-Williamson (No Adhesion)
Multi-Asperity Models With Adhesion
Tutorial Outline
Prof. George G. Adams
Basis of Hertz Contact
P
r
z
a
δ
R
Pressure Profile
p(r)
r
a
p0ararprp <−= ,)/(1)( 2
0
The pressure distribution:
produces a parabolic depression on the surface of an elastic body.
Depth at center
Curvature in contact region
Resultant Force
apE 0
2
2)1( πνδ −
=
Eap
R 2)1(1 0
2 πν−=
02
0 322)( pardrrpP
aππ == ∫
Prof. George G. Adams
P
Hertz ContactsHertz Contact (1882)
2aR1
R2δ
E1,ν1
E2,ν2
Applied Force2/32/1*
34 δREP =
3/1
*43
⎟⎠⎞
⎜⎝⎛=
EPRa Contact Radius
21
111RRR
+= Effective Radius of Curvature
EffectiveYoung’s modulus2
22
1
21
*
111EEEνν −
+−
=
Prof. George G. Adams
Assumptions of HertzContacting bodies are locally sphericalContact radius << dimensions of the bodyLinear elastic and isotropic material propertiesNeglect frictionNeglect adhesionHertz developed this theory as a graduate student during his 1881 Christmas vacationWhat did you do during your Christmas vacation ?????
Prof. George G. Adams
Onset of YieldingYielding initiates below the surface.
Elasto-Plastic
With continued loading the plastic zone grows and reaches the surfaceEventually the pressure distribution is uniform, i.e. p=P/A=H and the contact is called fully plastic.
Fully Plastic
Prof. George G. Adams
Contacts With Adhesion
Prof. George G. Adams
Forces of Adhesion
Important in MEMS Due to Scaling
Characterized by the Surface Energy (γ) and
the Work of Adhesion (Δγ)
For identical materials
Also characterized by an inter-atomic potential
1221 γγγγ −+=Δ
γγ 2=Δ
Prof. George G. Adams
Adhesion Theories
Z
0 1 2 3-1
-0.5
0
0.5
1
1.5
Z/Z 0
σ/σ
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.
Prof. George G. Adams
Two Rigid Spheres:Bradley Model*
P
P
R2
R1
21
111RRR
+=
RP OffPull γπΔ=− 2
*Bradley, R.S., 1932, Philosophical Magazine, 13, pp. 853-862.
Prof. George G. Adams
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
Prof. George G. Adams
Derivation of JKR ModelDerivation of JKR Model
Total Energy ETotal Energy ETT
Stored Elastic Stored Elastic Energy Energy
Mechanical Potential Mechanical Potential Energy in the Applied LoadEnergy in the Applied Load
Surface Surface EnergyEnergy
Equilibrium when 0=da
dET
*23
34,)3(63 EKRRPRP
RKa
=Δ+Δ+Δ+= γπγπγπ
Ka
Ra
382 γπδ Δ
−= RP OffPull γπΔ=− 5.1
Prof. George G. Adams
JKR ModelJKR Model
•• Hertz modelHertz modelOnly compressive stresses can exist in the contact area.
JKR modelJKR modelStresses only remain compressive in the center.Stresses are tensile at the edge of the contact area.Stresses tend to infinityaround the contact area.
Pressure Profile
JKRJKR
HertzHertz
a r
p(r)
Deformed Profile of Contact Bodies
p(r)
a r
P
a
a
P
Prof. George G. Adams
JKR ModelJKR Model1. When Δγ = 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.
31
2
06
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ=
KRa γπ 3
1
2
2220
0 34
3 ⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ==
KR
Ra γπδ
RP γπΔ−=23
min
3/12
min 223
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ=
KRγπδ
Prof. George G. Adams
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
,23
RPRKa γπΔ+=
Ra2
=δ
Applied Force, Contact Radius & Vertical Approach
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.
RP OffPull γπΔ=− 2
Prof. George G. Adams
Tabor Parameter:
JKRJKR--DMT TransitionDMT Transition
1<<μ
3/1
30
2
2
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ=
ZERγμ
DMT theory applies(stiff solids, small radius of curvature, weak energy of adhesion)
1>>μ 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.
Prof. George G. Adams
Maugis Approximation
0 1 2 3-1
-0.5
0
0.5
1
1.5
Z/Z 0
σ/σ
TH
Maugis approximation
⎩⎨⎧
>−≤−
=00
00
,0,
hZZhZZTHσ
σ
where
h0
00
0
Zh
h TH
≅⇒
Δ= γσ
Prof. George G. Adams
Elastic Contact With Adhesion
Prof. George G. Adams
Elastic Contact With Adhesion
P/πwR
a/(π
wR
2 /K)1/
3
-3 -2 -1 0 1 2 3 40.0
0.5
1.0
1.5
2.0
2.5
Hertz
JKR
λ=0.1
λ=0.5
λ=1λ=2
DMT
w=Δγ
Prof. George G. Adams
Elastic Contact With Adhesion
δ/(π2w2R/K2)1/ 3
P/πw
R
-1 0 1 2-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
Hertz
JKR
λ=0.1
λ=0.5
λ=1
λ=2
DMT
Prof. George G. Adams
Adhesion of Spheres
3/1
30
2*
2
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ=
ZER γμ
JKR valid for large μ
DMT valid for small μ
Tabor Parameter
0 1 2 3-1
-0.5
0
0.5
1
1.5
Z/Z0
σ/σ
TH
MaugisJKR
DMT
Lennard-Jones
Δγ and σTH are most important E. Barthel, 1998, J. Colloid Interface Sci., 200, pp. 7-18
Prof. George G. Adams
Adhesion MapK.L. Johnson and J.A. Greenwood, J. of Colloid Interface Sci., 192, pp. 326-333, 1997
Prof. George G. Adams
Multi-Asperity Contacts
Prof. George G. Adams
Surface Topography
∑=
−=N
iSS zz
N iS
1
22 )(1σ
Standard Deviation of Surface Roughness
Standard Deviation of Asperity Summits
Scaling Issues - Fractals
Mean of Surface
Mean of Asperity Summits
∫ −=L
dxmzL 0
22 )(1σ
Prof. George G. Adams
Contact of Surfaces
d
Reference PlaneMean of AsperitySummits
Typical Contact
Flat and Rigid Surface
Prof. George G. Adams
Typical Contact
Original shape
2a
δ
P
R
Contact area
Prof. George G. Adams
Multi-Asperity Models(Greenwood and Williamson, 1966, Proceedings of the Royal Society
of London, A295, pp. 300-319.)
AssumptionsAll asperities are spherical and have the same summit curvature.The asperities have a statistical distribution of heights (Gaussian).
φ(z)z
Prof. George G. Adams
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.
φ(z)z
Prof. George G. Adams
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.
Many modifications have been made to the GW theory to include more effects − especially important is plastic deformation and adhesion.
Prof. George G. Adams
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 MaugisAdhesive Contacts:– Adams, G.G., Müftü, S., and Mohd Azhar, N., 2003, J. of
Tribology, 125, pp. 700-708.– Morrow, C., Lovell, M., and Ning, X., 2003, J. of Physics D:
Applied Physics, 36, pp. 534-540.
Prof. George G. Adams
Pull-Off Force for Various α, β, and γ
α β γ 2/P NGb0.002 1000 0.001 2158.10.003 1000 0.001 320.10.004 1000 0.001 26.70.005 1000 0.001 1.0
0.01 100 0.001 2.90.01 300 0.001 0.0050.01 1000 0.005 9.50.01 1000 0.007 81.20.01 1000 0.010 2480.8
Adams, Muftu, Mohd-Azhar
1 2/
Rσα ⎛ ⎞= ⎜ ⎟⎝ ⎠
( )1 2/Rbσ
β =
wE b
γ =′
Prof. George G. Adams
Morrow, Lovell, Ning
???
Prof. George G. Adams
Summary of Topics Covered
• Hertz contact • Various theories of adhesion (Bradley,
JKR, DMT, Maugis) • Applicability of each (Tabor parameter,
adhesion map)• Rough surface models without adhesion• Rough surface models with adhesion• All the above pertain to elastic contacts