Between microscopic and continuum length scales, meso-scale phenomena are fundamental in understanding shear between material interfaces, and can be considered as the bridge between the two length scales for studying material response. A huge amount of experimental and theoretical work has been devoted to modeling and understanding the properties of microscopic defects but the question of how microstructural properties link to the macroscopic constitutive equations of continuum mechanics is crucial and poorly understood. Often the transition from discrete defects to continuum mechanics is accomplished by simple homogenization procedures neglecting the complex features of the process, employing tools of continuum mechanics to model deformations and slip, e.g., in the crust during earthquakes, as well as the rheology of snow avalanches and landslides. However the presence at the meso-scale of many structural elements like surface roughness, grains, fluid films, different material mixtures, and of their combined effects in determining the system response, make the problem of bridging the associated length scales very complex.
Computer simulations are powerful theoretical tools to study frictional phenomena at so many different length scales; they allow controlled numerical “experiments” where the geometry, sliding conditions and interactions between the constitutive parts (whether atoms or macroscopic grains) can be varied at will, and where the full dynamics of the system can be followed, unlike in real laboratory experiments or in nature. Thanks to the computational resources available nowadays, it is often possible to carry out numerical simulations of system sizes approaching those of real physical interest, albeit not always on the appropriate time scales.
Different models and computational techniques are employed to mimic and finally understand interfacial processes from nano up to the macroscale:
1) We can consider simple, “minimalistic” models, which are based on simplified interaction potentials and focus only on the most relevant degrees of freedom of the system, trying to retain the most important features. In spite of their simplicity these models can explain phenomena of high complexity and have contributed to unravel some important concepts of the mechanisms of friction (e.g., the role of commensurability of the contacting surfaces, the transition from intermittent stick-slip to smooth sliding, the onset of sliding motion, the mechanisms of detachment fronts preceding overall macroscopic sliding, the scaling laws describing statistics of earthquakes, some aspects of the creeping-to-sliding transition in fault like mechanics).
2) A different kind of simplified approach which allows to connect microscopic and meso-macroscopic scales consists in the inclusion of suitable noise terms in otherwise homogeneous equations. Recent results show that this approach is successful in effectively describing intermittency and fluctuations often observed in experiments.
3) Another possible approach is via extensive molecular dynamics simulations with more realistic interaction potentials and geometries. This route is usually taken to understand intricate processes such as, for example, wear and plastic deformations in the contact of sliding solid surfaces. The time and length scales for a truly realistic approach are still beyond reach and the computer simulations still rather heavy. An important issue, therefore, is how to reduce the large-scale, many-parameter MD simulations to simpler descriptions with only a few equations of motion.
4) An additional understanding of friction phenomena come from phenomenological Rate-State (RS) models that have been used to describe a wide range of observed frictional behaviors, such as the dilation of a liquid under shear and the transition between stick–slip (regular or chaotic) and smooth sliding friction. However, most “state variables” in RS models cannot yet be quantitatively related to physical system properties, and the main challenge is to define these variables starting from a microscopic description.
Each of these methods has its strengths and weaknesses. Large-scale simulations allow to reproduce quite accurately some experimental features, but are not really well suited to extract general information at a fundamental level; Rate-State models capture many experimental features quantitatively, but the physical nature of the state variable is unspecified, not allowing to fully rationalize the results of observations; minimalistic models have the advantage of being computationally cheap and simple enough to enable us to work out the general mechanisms at play of the problem, but, obviously, they are not system specific.