Above we have described a simple mesoscopic model – the Burridge-Knopoff spring-block model, known also as the earthquake (EQ) model or the multi-contact model – which bridges the gap in scales and describes the main experimental observations, such as stick-slip and smooth sliding, in terms of the properties of local contacts (junctions) that break at a critical force. Typically for a dry contact of rough surfaces these contacts are associated with asperities, but they may otherwise represent molecular bonds or capillary bridges, or they may account for patches of solidified lubricant or its domains for the case of lubricated friction. The drawback of EQ simulations, however, is that heavy calculations with different parameter sets are required to determine the main features of the model, and it is hard to draw conclusions of general validity. Moreover, most of the studies based on the EQ model assume that all contacts have identical properties for simplicity. However, as we show below, this limit is singular and may lead to qualitatively incorrect conclusions.
In a general case, the contacts are characterized by a continuous distribution of the static threshold values. A contact itself behaves as follows from MD simulation and tip-based experiments – it operates as an elastic spring until the local shear force is below a threshold value, and breaks when the threshold is exceeded. When the upper block moves, the forces on the contacts increase, and at some moment they start to break in sequence, one after another, with weaker contacts breaking earlier and strongest contacts resisting to the last. Once a contact is broken, it slips and then is reformed again. Namely the EQ model with a continuous distribution of thresholds will be discussed in this Chapter.
The EQ model, being the cellular automaton model, allows no analytical treatment. Its kinetics, however, may be reduced to the so-called master equation (ME), also known as the kinetic equation, the Boltzmann equation, etc. Moreover, the ME approach may be generalized to incorporate thermal effects, and can also be supplemented by another equation describing the aging of contacts. The ME can often be solved analytically, thus allowing us to describe the dependence of friction on temperature and velocity, the stick-slip motion and the transitions between the stick-slip and smooth sliding regimes.
According to the second Amontons law (also known as the Coulomb law), the friction force does not depend on the sliding speed; however, this is not true in a general case. The friction force does depend on the speed – fk(v) increases with v at small velocities, reaches a maximum and then decreases. At low driving velocities the kinetic friction force increases linearly with speed – if the slider moves slowly, all contacts will break sooner or later, purely due to thermal fluctuations. The slower the slider moves, the longer time the contacts have to receive a fluctuation above the threshold, so the smaller is the friction force. The linear fk(v) dependence sometimes is treated as a (typically very high) "viscosity" of a thin lubricant film. At intermediate speed, the role of thermal fluctuations becomes more and more marginal, and friction is dominated by the so-called aging effects: when a contact breaks, soon it reforms and grows in size. This leads to a weak (logarithmic) fk(v) dependence. Eventually at high velocities the kinetic friction reaches a maximum and starts to decrease, when sliding is so fast that no time is left for contact reforming.
Next, one may incorporate the elasticity of the slider which leads to contact-contact interaction – when one of contacts breaks, the forces on surrounding contacts increase by some δf. Numerics shows that δf(r) decays with the distance r from the broken contact as δf(r) ∝ r-1 at short distances r << λc, and as δf(r) ∝ r-3 at long distances, where λc~ a2E/k is the elastic correlation length expressed in terms of the slider Young modulus E, the contact stiffness k and the average distance a between the contacts. The model may then be simplified by considering the slider as rigid over distances r < λc, and treating the contacts within each λc-area as one effective λ-contact with the parameters determined by a corresponding mean-field solution of the ME. Numerics also shows that most of the intercontact extra force arises in front and behind the broken contact, which means that the interface may be approximately considered as an effective 1D chain of λ-contacts.
If the λ-contacts do not undergo elastic instability, then a local perturbation spreads smoothly over the interface. Otherwise, if it is subject to the elastic instability, i.e. if it breaks and slides at a certain threshold stress, then the nearest neighboring λ-contacts may break too, and a sequence of breaks will propagate through the interface like in a domino effect. In the latter case the dynamics of the chain of λ-contacts can be addressed with the help of the FK-like model, where the sinusoidal substrate potential is replaced by a sawtooth-like potential of periodically repeated inclined pieces. With this approach one can find analytically the maximum and minimum shear stress for crack propagation (the latter corresponds to the Griffith threshold) as well as the crack velocity as function of the applied stress. When the shear stress is uniform and a λ-contact breaks somewhere along the chain, two self-healing cracks propagate from the initial break point in opposite directions as solitary waves similarly to the kink-antikink pair until they reach the boundary or meet with another crack created somewhere else.When an elastic slider is pushed from its trailing edge, the nonuniform shear stress is maximal at the trailing edge and falls off with distance inside the block. When the pushing force increases, the starting event is the breaking of the leftmost λ-contact. Due to interaction between the contacts, this will result in the increase of the stress on the second λ-contact which will break too, and so on until the self-consistent stress will occur below the threshold. Thus, a self-healing crack created at the trailing edge, propagates through the interface over some distance Λ (which can be found analytically), removing the stress at its tail but creating an extra stress in the region ahead. With a further increase of the pushing force, a second crack emerges and propagates, triggering the previously formed stressed state, and so on until the cracks will reach the leading edge of the system. When these cracks propagate through the interface, the whole slider undergoes slight slips, the so-called precursors, which may be detected and used to predict the large earthquake.
Finally note that the EQ-like model has been invented initially by Burridge and Knopoff to explain real earthquakes, not friction. The physics of these two problems – friction and earthquakes – is essentially similar and differs mainly by the spatio-temporal scale: nanometers and seconds to hours in tribology in comparison to kilometers and years to centuries in geology. Real earthquakes are characterized by two laws – the Gutenberg-Richter (GR) law and the Omori law. Both these laws are empirical, found through long-term statistical observations, and there are no more or less articulate explanations of these laws yet. The EQ-like models may be one of the approaches which would allow to explain both laws. In particular, the GR law may be explained as emerging due to contact aging, while the Omori law may be associated with a finite distance of crack propagation – after a large earthquake, not all the stress is released, but a part of it is stored at a distance Λ from the main shock.
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Master equation approach
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tribology pageLast updated on April 21, 2014 by O.Braun. Copyright © by O.Braun. Translated from LATEX by TTH