Unsteady slip pulses under spatially-varying prestress

It was recently established that self-healing slip pulses under uniform prestress $\tau_b$ are unstable frictional rupture modes, i.e., they either slowly expand/decay with time t. Furthermore, their dynamics were shown to follow a reduced-dimensionality description corresponding to a special $L(c)$ line in a plane defined by the pulse propagation velocity $c(t)$ and size $L(t)$. Yet, uniform prestress is rather the exception than the rule in natural faults. We study the effects of a spatially-varying prestress $\tau_b(x)$ on 2D slip pulses, initially generated under a uniform $\tau_b$ along a rate-and-state friction fault. We consider periodic and constant-gradient prestress $\tau_b(x)$ around the reference uniform $\tau_b$. For a periodic $\tau_b(x)$, pulses either sustain and form quasi-limit cycles in the $L-c$ plane or decay predominantly monotonically along the $L(c)$ line, depending on the instability index of the initial pulse and the properties of the periodic $\tau_b(x)$. For a constant-gradient $\tau_b(x)$, expanding/decaying pulses closely follow the $L(c)$ line, with systematic shifts determined by the sign and magnitude of the gradient. We also find that a spatially-varying $\tau_b(x)$ can revert the expanding/decaying nature of the initial reference pulse. Finally, we show that a constant-gradient $\tau_b(x)$, of sufficient magnitude and specific sign, can lead to the nucleation of a back-propagating rupture at the healing tail of the initial pulse, generating a bilateral crack-like rupture. This pulse-to-crack transition, along with the above-described effects, demonstrate that rich rupture dynamics merge from a simple, nonuniform prestress. Furthermore, we show that as long as pulses exist, their dynamics are related to the special $L(c)$ line, providing an effective, reduced-dimensionality description of unsteady slip pulses under spatially-varying prestress.