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We present a nonlinear stability analysis of quasi-static slip in a spring-block model. The sliding interface is governed by rate-and-state friction, with an intermediate state evolution law spanning ageing and slip laws. We examine the robustness of prior results to changes in the evolution law, including the unconditional stability of the ageing law for spring stiffnesses above a critical value. We investigate two scenarios: a spring-block model with stationary and non-stationary point loading rate. In the former scenario, deviations from the ageing law lead to only conditional stability for spring stiffnesses above a critical value: finite perturbations can trigger instability, consistent with prior results for the slip law. For a given supercritical stiffness, the perturbation size required to induce instability grows as the state evolution law approaches the ageing law. By contrast, for a stationary point loading rate, there exists a maximum critical stiffness above which instability can never develop, for any perturbation size. This critical stiffness vanishes as the slip law is approached. For a limited extension of the results found for a spring-slider to continuum faults, we derive relations for an effective spring stiffness as a function of elastic moduli and a characteristic fault dimension or perturbation wavelength.
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This content will become publicly available on June 6, 2026
Stability and minimality of the ball for attractive-repulsive energies with perimeter penalization
We consider perimeter perturbations of a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. We prove that there exist curves in the perturbation-volume parameter space that separate stability/instability and global minimality/nonminimality regions of the ball, and provide a precise description of these curves for certain interaction kernels. In particular, we show that in small perturbation regimes there are (at least) two disconnected regions for the mass parameter in which the ball is stable, separated by an instability region.
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- Award ID(s):
- 2306962
- PAR ID:
- 10598922
- Publisher / Repository:
- European Mathematical Society Press
- Date Published:
- Journal Name:
- Interfaces and Free Boundaries, Mathematical Analysis, Computation and Applications
- ISSN:
- 1463-9963
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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