Prof. George G. Adams

THE MECHANICS OF ADHESION – A TUTORIAL

George G. AdamsMechanical Engineering Department

Northeastern University, Boston, MA 02115Email: adams@coe.neu.edu

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