Instabilities at Frictional Interfaces: Creep Patches, Nucleation and Rupture Fronts

Abstract: The strength and stability of frictional interfaces, ranging from tribological systems to earthquake faults, are intimately related to the underlying spatially-extended dynamics. Here we provide a comprehensive theoretical account, both analytic and numeric, of spatiotemporal interfacial dynamics in a realistic rate-and-state friction model, featuring both velocity-weakening and strengthening behaviors. Slowly extending, loading-rate dependent, creep patches undergo a linear instability at a critical nucleation size, which is nearly independent of interfacial history, initial stress conditions and velocity-strengthening friction. Nonlinear propagating rupture fronts -- the outcome of instability -- depend sensitively on the stress state and velocity-strengthening friction. Rupture fronts span a wide range of propagation velocities and are related to steady state fronts solutions.

• The paper presents a comprehensive theoretical framework for analyzing frictional instabilities using a quasi-one-dimensional rate‐and‐state friction model.
• The model reveals that creep patches become unstable at a critical size, leading to nucleation of rupture fronts with speeds that vary based on stress and friction properties.
• Numerical simulations show that velocity-strengthening friction slows rupture propagation, offering practical insights for understanding earthquake dynamics and tribological failures.

Instabilities at Frictional Interfaces: Creep Patches, Nucleation and Rupture Fronts

This paper presents a comprehensive theoretical exploration of the spatiotemporal dynamics at frictional interfaces, focusing on instabilities such as creep patches, nucleation, and rupture fronts. The analysis is conducted within the framework of the rate-and-state friction model, incorporating both velocity-weakening and velocity-strengthening behaviors. Detailed analytic and numerical investigations are carried out to elucidate the nature of these instabilities and their implications for the strength and stability of frictional interfaces across various scales, from tribological systems to earthquake faults.

Model Overview

The study employs a quasi-one-dimensional rate-and-state friction model. Friction within the model is defined by the interaction between contact asperities at the interface, with frictional resistance decomposing into elastic and viscous contributions. A characteristic feature of this model is the crossover from velocity-weakening at low slip velocities to velocity-strengthening at higher velocities. This behavior is crucial in capturing the complex dynamics leading to different forms of instabilities at the interface.

Dynamics of Creep Patches and Their Instabilities

Creep patches emerge as slowly extending regions of slip driven by external loading. These patches propagate quasi-statically until they reach a critical size, $L_c$, at which point they become linearly unstable. The derived expression for $L_c$ indicates that it is primarily independent of the stress state and insensitive to the presence of velocity-strengthening friction. The propagation speed of these patches decreases as they grow in size, inversely proportional to their length.

Nucleation and Rupture Fronts

The instabilities that arise from oversized creep patches give birth to rupture fronts. These fronts represent nonlinear propagating solutions with velocities sensitive to the initial stress state and the velocity-strengthening properties of the friction law. Numerical simulations reveal that rupture speeds can vary from much slower than the shear wave speed to near-elastic wave speeds, depending significantly on the frictional model employed.

Sensitivity to Stress and Frictional Properties

Through parametric studies, the paper shows that rupture front propagation speeds are highly dependent on the pre-existing stress field and the presence of a velocity-strengthening regime in the friction law. Notably, systems with pure velocity-weakening behavior tend to result in fast rupture propagation, whereas the inclusion of velocity-strengthening characteristics can lead to reduced slip velocities and more localized failure, aligning with observations of slow ruptures in certain natural earthquakes.

Implications and Future Directions

The insights from this study provide substantial implications for predicting and understanding failure at frictional interfaces, particularly in seismically active regions. The critical role of the velocity-strengthening friction law suggests that future models should incorporate such properties to realistically simulate observed rupture behaviors. The potential extension to two-dimensional analyses could further refine our understanding of real-world earthquake dynamics and contribute to more accurate predictions of fault line behaviors under stress.

In summary, this paper advances the theoretical understanding of frictional instabilities, offering a detailed exploration of the mechanisms underpinning creep patches, rupture nucleation, and front propagation in frictional interfaces. The implications are significant for fields ranging from materials science to earthquake physics, highlighting the complex interplay between frictional properties and failure mechanisms.