The equation of motion for rate-and-state supershear earthquake rupture

Zekang Gong, Federico Ciardo, & Dmitry I. Garagash

Submitted August 30, 2026, SCEC Contribution #15483, 2026 SCEC Annual Meeting Poster #TBD

Supershear earthquakes propagate faster than the shear-wave speed V_S and generate shear Mach fronts that can produce intense ground motions. Although the mechanics of supershear rupture has been extensively investigated with rate-independent cohesive zone laws (e.g., linear distance / slip-weakening laws), how rupture speed evolves during propagation, how the energy balance and the effective fracture energy changes, and how these processes control rupture growth remains poorly understood for faults governed by rate-and-state friction. Here, we address this problem using a semi-analytical approach that combines a steady-state near-tip solution with an energy-based equation of motion for evolving mode-II supershear ruptures. The near-tip fields are characterized by a rupture-speed-dependent singularity exponent -q, rather than the classical fixed exponent of -1/2 associated, for example, with subshear rupture. The equation of motion is formulated through the energy balance G=G_c, evaluated from singular boundary integral equations, and coupled to a stress-closure condition that matches the stress predicted by the near-tip solution to the prescribed background shear stress. Together, these two equations determine the two unknowns, the rupture speed v_r and rupture length L. For the parameter regime considered, v_r converges to the Eshelby speed, V_E=√2 V_S, as L increases, rather than the compressional-wave speed V_P predicted by the corresponding linear slip-weakening model. This behavior reflects the rupture-speed dependence of the fracture energy under rate-and-state friction and the progressive contraction of cohesive zone during rupture propagation. Because q also depends strongly on the elastic wave-speed ratio V_P/V_S, the framework further reveals how elastic properties influence the potential for supershear propagation and the near-tip rupture process. These results connect near-tip frictional physics to source-scale rupture dynamics and provide a basis for assessing supershear rupture potential and the resulting strong ground motion.

Key Words
supershear, equation of motion

Citation
Gong, Z., Ciardo, F., & Garagash, D. I. (2026, 08). The equation of motion for rate-and-state supershear earthquake rupture. Poster Presentation at 2026 SCEC Annual Meeting.

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