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TECHNICAL PAPERS

# A Greenwood-Williamson Model of Small-Scale Friction

[+] Author and Article Information
Reese E. Jones

Sandia National Laboratories, Livermore, CA 94551-0969

The term “pre-sliding tangential deflection” is used here to denote the frictional transition region of a pair of surfaces and to distinguish this phenomenon from the “micro-slip” of an individual microscopic protuberance on a counterface.

The solution to the fully sticking problem 4 includes singularities in the traction field. It is assumed that the total tangential force is only representative of the actual force on the asperity on average. A discussion of how normal adhesion is related to this tangential adherence is delayed to the last section.

The continuity of $q$ with respect to $v$ is not specifically required by the theoretical framework up to this point, as is demonstrated in (16). Furthermore, a Hertz-Mindlin solution (with a corresponding steady-state value of tangential traction) would merely soften the ramp-like transition of stick to large displacement slip, see Figure 7.8 of (10).

This is taken as a representative value for $c$ given that mechanical yield strengths are typically two to three orders of magnitude less than the corresponding elastic moduli.

The symbol $δ̂(h)$ should not to be confused with the symbol $δ$ used to denote the normal displacement of an asperity.

Since $v$, in fact, represents a relative displacement of the contacting surfaces, the material time derivative of $v$ is objective.

This is similar to crystal plasticity, where a distribution of states contributes to the stress response at a homogenized material “point.”

It is possible for the size of ${yi}$ to grow with the number of time steps; however, the number of points necessary to approximate $y$ can be limited by noting that the region $x∊(x¯,x1S)$ is of finite extent. Additionally, the highest portion of the asperity population may not contribute significantly to $y(x)ϕ(x)$ and therefore to $Q$ if the value of the population distribution $ϕ(x)$ decreases rapidly as the independent variable increases (which is certainly the case for the Gaussian).

J. Appl. Mech 74(1), 31-40 (Dec 07, 2005) (10 pages) doi:10.1115/1.2172269 History: Received May 09, 2005; Revised December 07, 2005

## Abstract

A Greenwood and Williamson based model for interfacial friction is presented that incorporates the presliding transition phenomenon that can significantly affect small devices. This work builds on previous similar models by developing: an analytical estimate of the transition length in terms of material and surface parameters, a general recursion formula for the case of slip in one direction with multiple reversals and constant normal loading, and a numerical method for the general three-dimensional loading case. In addition, the proposed model is developed within a plasticity-like framework and is shown to have qualitative similarities with published experimental observations. A number of model problems illustrate the response of the proposed model to various loading conditions.

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## Figures

Figure 1

Tangential force vs. tangential displacement for c=0.01 and x¯=1

Figure 2

Tangential force vs. tangential displacement for various population sizes and a fixed normal force

Figure 3

Tangential force vs. tangential displacement for constant normal displacement

Figure 4

Intersection of the yield surface and the negative slip branches

Figure 5

Prescribed tangential displacement history and slip distribution at t=40,100,140,170

Figure 6

Tangential force vs. tangential displacement for reverse slip for x¯=1

Figure 16

The circular path and the yield surface for a typical asperity

Figure 7

Tangential force and work per cycle for cyclic loading in a fixed tangential direction

Figure 8

Return map for a specific step

Figure 9

Comparison of tangential force vs. normal force reverse slip for x¯=1

Figure 10

Normal and tangential force for varying approach and increasing applied slip

Figure 11

Elastic slip distribution at y¯=1.5 and y¯=0.028

Figure 12

Evolution of slip distribution for circular displacement of radius 0.004

Figure 13

Tangential force history for circular displacement of radius 0.004

Figure 14

Tangential force history for circular displacement of radius 0.04

Figure 15

Angular difference between displacement and force for circular displacement of radius 0.04

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