The Elastic Energy Balance within Periodic, Chaotic and Localized Slip Pulse Solutions with Dieterich-Ruina Friction

We investigate the emergent dynamics when the nonlinear Dieterich-Ruina rate and state friction law is attached to a Burridge-Knopoff spring-block model. We derive both the discrete and continuous equations governing the system in this framework. The discrete system (ODEs) exhibits both periodic and chaotic motion, where the system's transition to chaos is size-dependent, i.e. how many blocks are considered. When the system of blocks is in the chaotic regime, we are interested in studying how the elastic energy of the spring connecting the moving plate to the block is transferred into the elastic energy stored in the springs interconnecting the blocks. The elastic energy travels from one spring to the others via the kinetic energy minus energy lost because of the friction law. The question is how the nonlinear friction law affects this transfer of energy as a function of time. From the discrete model we derive the nonlinear elastic wave equation by taking the continuum limit. This results in a nonlinear partial differential equation (PDE) and we find that both temporal and spatial chaos ensues when the same parameter is increased. This critical parameter value needed for the onset of chaos in the continuous model is much smaller than the value needed in the case of a single block and we discuss the implications this has on dynamic modeling with this specific friction law. Most importantly, these results suggest that the friction law is scale-dependent, thus caution should be taken when attaching a friction law derived at laboratory scales to full-scale earthquake rupture models. Furthermore, we find solutions where the initial slip pulse propagates like a traveling wave, or remains localized in space, suggesting the presence of soliton and breather solutions. In the case of a traveling wave we see evidence of a soliton, a wave with permanent form, that is localized in space while it travels at a constant speed through the medium. The breather solution is a time-periodic, exponentially decaying (in space) solution of a nonlinear wave equation. We discuss the significance of these pulse-like solutions and how they can be understood as a proxy for the propagation of the rupture front across the fault surface during an earthquake. We compute analytically the conditions for soliton solutions and by exploring the resulting parameter space, we may determine a range for suitable parameter values to be used in dynamic earthquake modeling.