Quasi‐static modelling of stress histories during the earthquake cycle: precursory seismic and aseismic stress release

SUMMARY A functional form describing non‐linearity in the stress history during the quasistatic phase of anelasticity in the final stages of semi‐brittle failure is presented Semi‐brittle failure is known to occur in laboratory rock samples, and the existence of earthquake precursors indicates that the Earth's upper crust also exhibits an anelastic response in the final stages of the buildup to dynamic failure. the functional form of the stress history depends on the amount of localized aseism stress relaxation on a fault nucleation patch, and the amount of relatively more distributed seismic damage around the fault measured by a catalogue of earthquakes or acoustic emissions. Aseismic stress relaxation on the nucleation patch is modelled assuming slip occurs behind a crack front which is growing due to subcritical crack growth by the mechanism of stress corrosion. Under these conditions it is impossible to model correctly the observed duration of intermediate‐term earthquake precursors given realistic estimates of the size of the nucleation patch (0.1–1 km) and the stress corrosion index, n (where 10 < n < 50). Therefore, intermediate‐term earthquake precursors must result from more distributed seismic or aseismic damage, coupled with fluid transport, in a volume much larger than that of the nucleation patch. In terms of seismic moment or seismic energy release, the precursory seismic damage in the quasi‐static anelastic phase is dominated by the largest magnitudes for the usual case of b < c , where c is a scaling constant in the moment‐magnitude relation log M 0 = cm + d and b is the seismic b value. the precursory remote stress drop (i.e. the sum of local stress drops averaged over the eventual area of the main shock or the sample size in the laboratory) is also dominated by the largest magnitudes for b < 2 c /3. However when b = 2 c /3 the contribution to the remote stress drop is uniformly distributed through the magnitude range, with smaller magnitudes contributing equally to the total due to their greater numbers. In the case b > 2 c /3 the smaller events dominate the remote stress drop. In terms of the inferred power‐law exponent D = 3 b/c of the fault length distribution, D > 2 corresponds to distributed damage, and D < 2 corresponds to stress concentration on a few of the larger fractures, irrespective of the value of c. This prediction agrees with recent laboratory experiments where distributed damage by dilatant micro‐cracking and associated strain hardening is associated with D > 2, and strain softening and shear localization occurs when D < 2.