Strain-rate variance minimum as a mechanical precursor of shear localization across laboratory, induced, and tectonic earthquakes

Summary Earthquakes are preceded by a transition from diffuse to localized deformation, yet quantifying the mechanical conditions that define this onset and connecting them to routine seismic catalogue observables remains an open challenge. We derive, from the continuum damage constitutive law, a characteristic trajectory in the spatial mean and variance of the inelastic strain-rate field during progressive fault preparation: the mean $\langle \overset{.}{\gamma} ^{in}\rangle $ increases monotonically as damage-reduced viscosity declines, while the spatial variance $var( \overset{.}{\gamma} ^{in} )$ passes through a minimum as the production of new stress heterogeneity declines during the coalescence of damage into the forming shear band. This trajectory is a direct mechanical consequence of the damage law: under progressive localization, the variance minimum is expected to precede the mean inelastic strain-rate maximum, distinguishing coherent localization from distributed acceleration, and explaining why rate-only seismic precursors may fail to identify rupture nucleation. We show that two seismicity-derived metrics — seismic strain dynamics (SSD) and the seismic regularity coefficient (dc) — provide observational proxies for $\langle \overset{.}{\gamma} ^{in}\rangle $ and $\sqrt {var( \overset{.}{\gamma} ^{in} )} $, connecting routine catalog observables directly to the governing equations. Cross-scale validation across laboratory stick-slip experiments, physics-based dynamic rupture simulations, induced seismicity, and tectonic earthquakes spanning eight orders of magnitude in spatial scale confirms the predicted SSD–dc trajectory. These results provide a quantitative basis for assessing fault-zone evolution and for improving the physical interpretation of earthquake preparation processes.