In this chapter, we proposed a model for friction between macroscopic surfaces con- sidering asperities at the microscale to be viscoelastic and modeling rough surfaces
as a stochastic process. Our main contribution is a framework to link properties of single asperities and surface features to the macroscopic static and sliding frictional behavior. We showed that, because of the collective response of contacts, the behavior at the macroscale can be very different from that at the microscale.
Even in our relatively simple model, rate and state effects appear naturally as a consequence of the collective asperity response. The model reproduces the strength- ening of the static friction with hold time. This strengthening is approximately log- arithmic for 3-4 decades in time. For sliding friction, the model reproduces many experimental observations. To capture the direct effect of these experiments, we need to endow the microscopic asperity-level friction law with a velocity-strengthening de- pendence. This does not, however, imply that the macroscopic response is velocity- strengthening. The velocity strengthening of individual contacts and the collective be- havior together determine whether the macroscopic response is velocity-strengthening or weakening.
The power n in the local friction law plays a crucial role in the evolution of the friction coefficient. If n > 1, the static friction coefficient decreases with hold time and the transient evolution of sliding friction in velocity jump experiments happens in the same sense as the instantaneous change. Since we expect the power n to be less than or equal to 1, this explains why the above two features are not observed in experiments.
At the same time, the model appears to be too simple to reproduce all experimen- tal observations. First, the strengthening of static friction with hold time saturates in 3-4 decades in our model. This is consistent with experiments on some materials but the strengthening persists for longer times in others. A complete understanding of this issue appears to be beyond the scope of this model. Second, our model results in velocity dependence of the characteristic evolution length scale Dc and parame- ter b that quantifies the transient change in friction in the rate and state equations, whereas these quantities are largely velocity-independent in experimental studies.
There are a number of ways to improve the model. In the present model, individual asperities are independent, the only interaction between them being through a mean
field (dilatation). Long-range elastic interactions through the bulk may, however, play an important role. We have also assumed one of the surfaces to be rigid. While this is reasonable when one of the surfaces is much more deformable than the other, when the two surfaces are similar (with respect to deformability), it will be important to incorporate the non-rigidity, especially during sliding. We have neglected the spatial distribution of asperities and contacts; this will, however, be important for reproducing realistic frictional behavior, especially when the long-range interactions are incorporated. Depending on the material of the sliding surfaces, the model for a single asperity can be modified to incorporate effects such as plasticity, adhesion, etc.
We explore some of these issues in the next chapters.
Chapter 3
Static and sliding contact of rough surfaces: effect of surface
roughness, material properties, and long-range elastic interactions
Here, we study the static and sliding contact of three-dimensional rough surfaces with viscoelastic and viscoplastic material models and long-range elastic interactions between contacts. We present, as far as we know, the first sliding simulations of three-dimensional rough surfaces with long-range interactions. Simulations show that the qualitative features of static and sliding friction are determined by the material and surface properties and the long-range elastic interactions only change the results quantitatively.
3.1 Introduction
Most surfaces have roughness features at many length scales. To bridge the micro and macro scales, the smallest relevant length scales must be resolved while at the same time, the system must be large enough to be representative of a macroscopic body. A numerical method like the Finite Element Method, though useful to study the stress distribution at contacts, plasticity, and such, results in a large number of degrees of freedom [68, 69], and this can be especially intractable during sliding of surfaces. To overcome this, a boundary-element like method has been proposed in
the literature, and many aspects of rough surface contact have been studied using this method [70, 71,72, 73, 74, 75,76]. However, as far as we know, time dependent behavior and sliding of rough surfaces has not been studied, and this is the main focus here.
As mentioned earlier, high stresses at the contacts result in time-dependent be- havior of the materials. As a starting point, the model developed here considers viscoelastic material behavior.
Elastic fields are long-range, the displacement field due to a point load on the surface of an elastic half-space decays only inversely with the distance from the point of application. This means that when two rough surfaces are pushed against each other, the interactions between contacts can be important.
Here, we develop a model that considers all of the above mentioned effects. An important question to answer is: what role does each of the factors: material prop- erties, surface roughness and elastic interactions, play in determining macroscopic friction?