Linear and nonlinear stability of rate-and-state faults

TL;DR

The paper advances the analytic understanding of rate-and-state faults by performing linear stability analyses for finite patches and non-uniform loading, identifying a critical size $L_c$ above which linear instability arises and showing how it depends on the frictional ratio $a/b$ and loading. It then revisits non-linear nucleation, deriving free-boundary and pinned nucleation lengths $L_f$ and $L_p$, and extends these results to aging-slips laws with a universal blow-up profile $V(x,t)=\frac{D_c}{t_f-t}\mathcal{W}(x)$ that is largely independent of loading conditions. The authors provide accurate algebraic fits for the transition boundaries, highlight the role of the state-evolution law through a scaling parameter $\epsilon$, and present 3D extensions (circular patches) with explicit radii $R_c$, $R_f$, and $R_p$. Together, these results yield practical phase diagrams and design rules to delimit parameter spaces in seismic-cycle models and to anticipate conditions under which small asperities can nucleate dynamic rupture. The findings illuminate the interplay between elastic transfer, frictional weakening, and boundary conditions, offering a framework to interpret observations and constrain model parameterizations with analytic, not just numerical, guidance.

Abstract

Models of faults incorporating slip rate- and state-dependent friction have reproduced phenomena from spontaneous slow, aseismic slip to earthquake-generating dynamic rupture. Exhaustive explorations of model parameter space regularly show sudden transitions in behavior. However these boundaries are poorly constrained analytically, with commonly used scalings derived assuming conditions that do not resemble those of the models. In this work, we demonstrate that an analysis of linear stability can reflect model conditions. We examine two scenarios that move beyond the classical case of the uniform sliding of an unbounded fault: an asperity driven by the steady creep of its surroundings, and a finite fault experiencing a constant rate of shear loading. We identify the critical fault dimension $L_c$ at which point linear stability is lost. Beyond this linear regime, the non-linear nature of friction law implies the loss of memory of loading conditions as instability progresses and the existence of universal solutions describing this process. We refine prior analyses of this non-linear instability and find the minimum fault size that can support self-sustaining, unstable acceleration towards dynamic rupture. We examine the role of the state evolution law and delineate conditions under which faults may be linearly stable but non-linearly unstable, requiring finitely large perturbations to trigger instability. On the basis of numerical solutions, approximate but accurate algebraic expressions for the transition boundaries are presented. These results provide a means for careful model design and to easily delimit plausible regions of parameter space when considering physical observations.

Figures (7)

• Figure 1: (a) Schematic of finite-sized rate-weakening patch of length $2L$ embedded within a much longer fault that creeps at a constant rate $V_{pl}$. (b) Schematic profile of a fault of length $2L$ that is loaded by a constant rate of shear stress $\dot\tau_b$ (not illustrated). The fault may terminate at $x=\pm L$ or the fault may continue along $|x|>L$ but is locked there. (c,d) Plot of steady-state distribution of slip rate, $V_o(x)$ for scenarios (a) and (b), respectively, scaled by a characteristic slip rate from loading, $V_{pl}$. (e,f) Plot of the rate of "background" shear stress, $\partial \tau_b/\partial t$, defined as the rate of shear stress on the fault in the absence of slip for the scenarios (a) and (b), respectively.
• Figure 2: (a,b) Plot of the critical length $L_c$ (solid back) and its approximation (red-dashed) for the loss of stability of a finite fault, locked at its edges and loaded by a constant rate of shear stress. (c,d) Plot of the imaginary part of the eigenvalue $s_c$ at stability loss (solid black) as well as its approximation (red dashed).
• Figure 3: (a,b) Plot of the unpinned $L_f$ (solid back) and pinned $L_p$ (solid blue) nucleation lengths length and approximations (\ref{['eq:Ln']}), (\ref{['eq:Lp']}) (red-dashed) for an aging-law fault, determined by solution for the compact support of the function $\mathcal{W}(x)$ in the non-linear instability solutions (\ref{['eq:bu']}), under singularity-free (unpinned) and singularity-present (pinned) conditions.
• Figure 4: (a) Plot of the the critical fault length $L_c$ at which point linear stability is lost for two loading conditions: steady creep at boundaries (dary grey), loading at a constant rate of shear stress (black). (b) Comparison of $L_c$ for the these two conditions (dark grey and black) with the nucleation lengths $L_f$ (red) and $L_p$ (blue). For these cases $L_c<L_p<L_f$ for all $0<a/b<1$.
• Figure 5: Phase diagrams for the expected response of a rate-weakening fault within $|x|<L$, driven by a constant rate of creep $V_{pl}$ along $|x|>L$. The response may be either steady, uniform creep at $V=V_{pl}$ for $L<L_c$, aseismic transients for $L_c<L<L_p$, or the emergence of dynamic rupture for $L_p<L$. The phase boundaries are determined by the linear stability critical fault size $L_c$, (\ref{['eq:Lc1']}), and minimum pinned nucleation length $L_p$, (\ref{['eq:Lp']}). While $L_c$ is independent of the chose state evolution law (aging, slip or an intermediate law), $L_p$ is dependent on that choice; here $L_p$ is drawn for aging-law state evolution.