Sine–Gordon Modulation Solutions: Application to Macroscopic Nonlubricant Friction

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The Frenkel–Kontorova (FK) model and its continuum approximation, the sine–Gordon (SG) equation, are widely used to model a variety of important nonlinear physical systems. Many practical applications require the wave-train solution, which includes many solitons. In such cases, an important and relevant extension of these models applies Whitham’s averaging procedure to the SG equation. The resulting SG modulation equations describe the behavior of important measurable system parameters that are the average of the small-scale solutions given by the SG equation.

A fundamental problem of modern physics that is the topic of this paper is the description of the transitional process from a static to a dynamic frictional regime. We have shown that the SG modulation equations are a suitable apparatus for describing this transition. The model provides relations between kinematic (rupture and slip velocities) and dynamic (shear and normal stresses) parameters of the transition process. A particular advantage of the model is its ability to describe frictional processes over a wide range of rupture and slip velocities covering seismic events ranging from regular earthquakes, with rupture velocities on the order of a few km/s, to slow slip events, with rupture velocities on the order of a few km/day.



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