Localization of fast and slow slip in fault gouge and fracture energy scaling

Abstract

The localization of slow and fast slip in fault gouges may play a crucial role in understanding the mechanics of earthquakes and slow slip events. Here, we investigate the fracture energy accompanying this localization and the subsequent thermal weakening. We develop an analytical framework, complemented by numerical simulations, for a gouge governed by rate-and-state-dependent friction with flash-heating at high strain rate and thermal pressurization of pore fluids. The model captures the transition from initially distributed shearing to a co-seismic principal slip ``surface'' at slip $\delta_{\mathrm{loc}} \approx \gamma_c h$, and yields a decomposition of the fracture energy, $G = G_\mathrm{loc}(h) + \Delta G(\delta)$. The minimum, localization-related component $G_\mathrm{loc}$ scales with gouge thickness $h$, which in turn scales linearly with fault size. Flash heating is activated only upon localization for fast earthquake slip, producing an abrupt strength drop, and contributing to the magnitude of $G_\mathrm{loc}$. The post-localization term $\Delta G$ increases with co-seismic slip due to efficient thermal pressurization and is insensitive to $h$. Localization is predicted to occur for both rate-weakening and rate-strengthening gouges because transient state evolution drives apparent weakening after a slip-rate increase. These results unify field, laboratory, and seismological observations of shear band thickness, critical slip, and fracture-energy scaling, and they clarify why small events can be governed by scale-dependent $G_\mathrm{loc}$ whereas large ruptures become increasingly fault-invariant as $\Delta G$ dominates. Our framework provides testable predictions for the relation of gouge thickness to lower bounds of co-seismic fracture energy, and the mechanics of slow-slip transients and fast earthquakes.