**2. Theoretical Description of Rubber Friction**

The friction properties of elastomers such as rubber have been extensively studied for decades [23–26]. The tribological properties of rubber depend on many parameters, e.g., surface roughness, speed, normal load, lubrication, temperature and material properties. A fundamental study in this field was conducted by Grosch in his pioneering work [25], where different types of rubber were experimentally analysed on diverse hard surfaces, mentioning two distinct processes taking part at the generation of the friction phenomena: the adhesion, akin to a molecular relaxation process, and a deformation process in which energy is lost due to the cyclic stress of the rubber due to the surface roughness [19], also known as the hysteretic component of friction. The friction coefficient evaluations obtained from experimental studies or mathematical models are usually plotted on a graph as a function of the sliding speed, this kind of representation is also called a "master curve". This curve can be also parametrized for other variables such as load and temperature. The master curve of rubber on a rigid rough surface exhibits, in general, two distinct peaks. The first, being attributed to the adhesive component of friction, occurs in general at low sliding velocity, whereas the second one, referring to the hysteretic contribution occurs at higher sliding velocity [27]. A typical example of a master curve is represented in Figure 2, where the friction coefficient for a styrene butadiene rubber (SBR) rubber sliding on three different surfaces is reported.

**Figure 2.** Master curve of the friction coefficient for SBR rubber sliding on three different surfaces [25].

Subsequently, other researchers such as Greenwood and Williamson (GW) [28] introduced a concept involving the contribution of the true contact area on friction. In particular, the formulation of the GW theory is based on Archard's [29] previous idea of multi-asperity contact. They approximated the roughness of the surface as an ensemble of spheres having the same radius randomly distributed over the mean plane to take into account of the surface statistics.

Over the years, some other studies such as the ones by Bush at al. [30,31], Heinrich, Kluppel and others [32–34] proposed further contributions based on the original GW theory. Multi-asperity contact models, based on Greenwood and Williamson theory, represent one of the two most used approaches to account for the true contact area on the friction mechanisms. Another widely adopted approach has been developed by Persson [35], whose theory, in contrast to the GW models, removes the assumption that the true contact area is smaller than the nominal contact area [36], considering the extreme case of full contact conditions between a rigid rough surface and an initially flat elastic half-space. Such theory takes partial contact into account requiring that, in case of adhesionless contacts the stress probability distribution vanishes when the local normal surface stress is fading. It also assumes that the power spectral density (PSD) of the deformed elastic surface is the same as the rough surface below [37]. The theory provides formulas, needing as inputs only the PSD surface and the elastic properties of the contacting bodies. A recent study by Carbone and Bottiglione [38] compared the two different approaches, stating that Persson's rubber contact and friction theory, and the subsequent theories based on it, are more accurate. Although theories on rubber friction have evolved over the years, they still present diverse limitations associated with the adhesive and the viscoelastic component of friction [39]. Under this light, appears evident the centrality of an experimental approach and how important the setup of the different testing fixtures is.
