Section: New Results

Travelling waves in a spring-block chain sliding down a slope

Participants : Guillaume James, Jose Eduardo Morales Morales, Arnaud Tonnelier.

Spatially discrete systems (lattice differential equations) have a wide range of applications in natural sciences, engineering and social sciences. They frequently occur in physics as mass-spring systems with nearest-neighbors coupling and they have been used extensively to describe the dynamics of microscopic structures such as crystals or micromechanical systems, or to model fragmentation phenomena.

In this work, we consider a spring-block system that slides down a slope due to gravity. Each block is subjected to a nonlinear friction force. This system differs from the Burridge-Knopoff model considered for the modeling of earthquakes, which incorporates local potentials. We perform numerical simulations of the coupled system and show that the bistability property induces traveling patterns, as fronts and pulses. For a piecewise-linear spinodal friction law, a closed-form expression of front waves is derived. Pulse waves are obtained as the matching of two travelling fronts with identical wave speeds. Explicit formulas are obtained for the wavespeed and the wave form in the anti-continuum limit. The link with propagating phenomena in the Burridge-Knopoff model is briefly discussed. These results have been published in [27].