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Non-linear stability analysis of slip in a single-degree-of-freedom elastic system with different frictional evolution laws

Federico Ciardo, & Robert C. Viesca

Submitted September 10, 2023, SCEC Contribution #13116, 2023 SCEC Annual Meeting Poster #136

Pioneer work on stability of steady sliding to infinitesimal perturbations in a spring-block model with rate- and state-dependent friction showed there exists a critical stiffness below which the block would be unstable to perturbations [Rice and Ruina (1983),JGR]. The result of this linear stability analysis is the same for either slip or aging laws, owing to the identical linearization. However, when non-linearity of the friction law is non-negligible, the model response strongly depends on the state evolution law. Notably, for a spring at or above critical stiffness, the model is unconditionally stable if the interface follows the aging law [Ranjith and Rice (1999),JMPS]. In contrast, for the slip law, a block whose spring is at or above a critical stiffness is only conditionally stable: a finite perturbation (such as a sudden increase in loading) can trigger instability, implying that the block can slide dynamically at any slipping distances [Gu et al. (1984),JMPS]. Because stiffness for continuum faults is roughly inversely proportional to size, these results imply that a minimum possible earthquake size exists with the aging law, while it may not be the case with the slip law.

Here, we use an intermediate state evolution law bridging aging and slip laws using a dimensionless parameter [Ruina (1983),JGR; Viesca (2023),JMPS] and perform a non-linear stability analysis to investigate whether deviations from either evolution law would impact conditional or unconditional slip stability. Our results show that a well-defined critical stiffness only exists when the interface strictly follows the aging law. A slight deviation leads to conditional stability: sufficiently large perturbations can trigger instability. We show the critical perturbation required to trigger dynamic slip is at a minimum with the slip law, and only increases slowly (logarithmically) as the frictional evolution law tends towards the form of the aging law. Our results also indicate that for a stationary load point, critical perturbation follows a similar trend. However, when the spring stiffness is above the critical value, critical perturbation only exists for a subset range of frictional evolution laws. Finally, when dynamic slip is not triggered, the interface is more likely to host transient slip acceleration with the slip law.

Our results have important implications in a geophysical context, specifically regarding the dynamics of a single fault or interacting fault networks.

Key Words
Slip stability, spring-block model, rate-and-state friction

Ciardo, F., & Viesca, R. C. (2023, 09). Non-linear stability analysis of slip in a single-degree-of-freedom elastic system with different frictional evolution laws. Poster Presentation at 2023 SCEC Annual Meeting.

Related Projects & Working Groups
Fault and Rupture Mechanics (FARM)