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Contact Modeling of Rough Surfaces

Robert L. JacksonMechanical Engineering Department

Auburn University

• The modeling of surface asperities on the micro-scale is of great interest to those interested in the mechanics of surface contact, friction and wear.

• When considering the area of contact between real objects, the roughness of their surfaces must be accounted for, in that it will determine the real area of contact between them.

Background

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Rough Surface Contact Models

• Statistical – Model surface as a statistical distribution of asperities with various heights and properties (Computationally inexpensive).

• Deterministic – Model the real features of the surface as with much detail as possible (Computationally expensive).

• FFT Methods: Problem solved in Frequency domain.

• Fractal: Multiple scale roughness is considered.

Statistical Contact Model (Greenwood & Williamson)

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Hertz Contact Solution (1882)• Closed-form expressions to the deformations

and stresses of two spheres in a purely elastic contact (Theory of Elasticity).

• The Hertz solution assumes that the interference is small enough such that the geometry does not change significantly.

• The solution also approximates the sphere surface as a parabolic curve with an equivalent radius of curvature at its tip.

• It is also assumed that the contact surfaces are frictionless.

Hertz Solution Results

ωπRAE = 2/3)(34 ωREPE ′=

2

22

1

21 111

EEEνν −

+−

=′ 21

111RRR

+=

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Fully Plastic Truncation Model

ωπRAP 2= HRPp ωπ2=H = 3⋅Sy

Hardness• The average contact pressure (P/A) when a

contact surface has fully yielded (the entire contact surface is plastically deforming).

• Usually assumed to be approximately 3⋅Sy as predicted by slip-line theory (Tabor, 1951).

• However, Williams (1994) suggests a hardness value of 2.83⋅ Sy.

• Hardness is not an independent material property and is dependant on the (deformed) contact geometry, as well as E, Sy, ν.

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Critical Interference (Initial Yielding)

• Using the von Mises Yield Criteria and the Hertz Contact solution the following numerically fit solution is obtained.

RE

SC yc

2

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅⋅

=π

ω

1.295exp(0.736 )C ν=

Normalization

* *2AFA ω=

cωωω /* = cPPP /* = cAAA /* =

** ω=EA

( ) 2/3** ω=EP * *3HAF

y

PCS

ω=

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CEB Model• CEB model (Chang et al., 1987) approximates elasto-

plastic contact by modeling a plastically deformed portion of a hemisphere using volume conservation.

• Assumes average contact pressure to be constant hardness once yielding occurs.

• Discontinuity at critical interference.

• For Elasto-Plastic Deformation:( )*/12 ωωπ −= RACEB

( )KHRPCEB*/12 ωωπ −=

ZMC Model• ZMC model (Zhao et.

al. 2000) interpolates between the elastic and fully plastic models.

• A template function satisfies continuity of the function and its slope at the two transitions.

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FEM Elasto-plastic Model

• Kogut and Etsion (2002) performed a FEM analysis of the same case of an elastic-perfectly plastic sphere in contact with a rigid flat.

• In this analysis, the value of H is set to be fixed at 2.8⋅Sy.

• Very similar to current model, although the finite element mesh used is much more course than the current mesh.

Spherical Contact Model

Just Before Contact Mostly Elastic Mostly Plastic Deformation Deformation

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Finite Element Model• Perfectly plastic material yields according to the von Mises

yield criterion.• 100 Contact Elements are used to model the contact at the

interface between the sphere and the rigid flat.• Iterative scheme used to relax problem to convergence.• Mesh convergence was satisfied.

Finite Element Mesh

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von Mises Equivalent stress at ω*=0.571

ω*=2.14

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ω*=5.72

ω*=31.4

11

ω*=62.9

ω*=114

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Empirical Formulation

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−−=

− 7.0

82.0exp184.2Ra

SH

y

G

.

Diagram of progression of change in hardness with geometry.

HG /Sy ≈3 3> HG /Sy >1 HG/Sy ≈1

a/R=0+ 0<a/R<1 a/R=1

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Statistical Equations

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛= −

22/1 5.0exp)2(

ss

zσσ

σπφ

∫∞

−=d

n dzzdzAAdA )()()( φη

∫∞

−=d

n dzzdzPAdP )()()( φη

c

s

ωσ

ψ =Plasticity Index

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Surface AdhesionFuller and Tabor (1975) proposed an Adhesion Parameter to predict if the effect of adhesion will be significant between the contact of two rough surfaces. They find that for the rubber surfaces they investigated, if θ>10 then adhesion is negligible.

2/1

⎟⎠⎞

⎜⎝⎛

∆′

=R

E ss σγσ

θ

∆γ is the change in surface energy (work of adhesion) and σs is the standard deviation of asperity heights (McCool, 1986).

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FFT Methods (Stanley and Kato, 1997)

1( ( ))FFT w FFT−= ⋅u(p) p

FFT Methods (cont)

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Fractal Methods

http://mathforum.org/alejandre/applet.mandlebrot.html

Fractal Methods for Contact

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Deterministic Methods

• Some claim to have assembled accurate deterministic models.

• This seems questionable since over 10,000 elements were used in the FEM analysis shown here to model a single asperity and results in long computation times! While these deterministic models model entire surfaces containing many asperities.

Unloading

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Other Factors

• Strain Hardening• Material Scale Effects• Effect of Asperity Shape• Thermal Effects• Sliding Contact• Lubrication

Conclusions

• Friction and Contact between real surfaces is a complicated issue which requires the use of simplified models.

• Care must be taken when using these simplified models as they may be very inaccurate for certain cases.

• Specifically, the use of hardness and hardness tests to model contact between rough surfaces may provide misleading results.

• Hardness as defined here is not an independent material property and it depends on the elastic properties and contact geometry.

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Literature• Greenwood, J. A. and Williamson, J. B. P., “Contact of Nominally Flat Surfaces,” Proc. R.

Soc. Lond. A 1966; 295, pp. 300-319.• Majumdar, A., Bhushan., B., “Fractal model of elastic-plastic contact between rough

surfaces.” ASME J. of Tribol., 1991. 113(1): p. pp. 1-11. • Jackson, R. L., Green, I., “A Finite Element Study of Elasto-plastic Hemispherical Contact,”

In press for ASME J. of Tribol.• Kogut, L., & Etsion, I., “Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat,” J. of

Applied Mechanics, Trans. ASME 2002; 69(5), pp. 657-662.• McCool, J. I., “Comparison of Models for the Contact of Rough Surfaces,” Wear 1986; 107,

pp. 37-60.• Chang, W. R., Etsion, I., and Bogy, D. B., “An Elastic-Plastic Model for the Contact of

Rough Surfaces,” ASME J. Tribol. 1987; 109, pp.257-263.• Zhao, Y., Maletta, D. M., Chang, L., “An Asperity Microcontact Model Incorporating the

Transition From Elastic Deformation to Fully Plastic Flow,” ASME J. Tribol. 2000; 122, pp.86-93.

• Timoshenko, S., and Goodier, J. N., Theory of Elasticity, New York, McGraw-Hill, 1951.• Greenwood, J. A., Tripp, J. H., “The Contact of Two Nominally Flat Rough Surfaces,” Proc.

Instn. Mech. Engrs. 1971; 185, pp. 625-633.• Kogut, L., and Etsion, I., "A Finite Element Based Elastic-Plastic Model for the Contact of

Rough Surfaces,” Tribology Transactions. 2003; 46, pp. 383-390.• Mesarovic, S. D. and Fleck, N. A., “Frictionless Indentation of Dissimilar Elastic-plastic

Spheres,” Int. J. Solids and Structures 2000; 37, pp.7071-7091.• Fuller, K.N.G. and D. Tabor, 1975, The effect of surface roughness on the adhesion of

elastic solids. Proc. R. Soc. Lond. A, Math. Phys. Sci., 345: p. 327-342. • Tabor, D., The Hardness of Materials, Clarendon Press, Oxford, 1951. • Williams, J. A. Engineering Tribology, New York, Oxford, 2000.