Coulomb failure criterion, fault types, stress drop, and Omori aftershock law
τ = μσₙ + c
τ = shear stress on fault, σₙ = normal stress, μ = friction coefficient (~0.6–0.85, Byerlee's law), c = cohesion (~0 for pre-existing faults). Failure when resolved shear exceeds this threshold. Pore pressure reduces effective σₙ: σₙ_eff = σₙ − P_fluid
Normal: hanging wall drops; extensional regime (σ₁ vertical). Thrust/Reverse: hanging wall rises; compressional (σ₁ horizontal). Strike-slip: horizontal motion; σ₁ and σ₃ both horizontal. Anderson's theory: principal stress axes align with Earth's surface.
Rate of aftershocks decays as n(t) = K/(t+c)^p, where p ≈ 1 (Omori 1894), refined as p ≈ 0.9–1.5 (Utsu). Modified Omori law widely used. Aftershock sequences follow GR relation: log N = a − bM (b ≈ 1). Epidemic-type aftershock sequence (ETAS) model includes secondary triggering.
Seismic moment M₀ = μ × A × D (rigidity × area × slip). Magnitude: Mw = (log M₀ − 9.1)/1.5. Stress drop Δσ = 7M₀/(16r³) for circular rupture. Typical Δσ = 1–10 MPa, independent of earthquake size. Larger earthquakes rupture larger areas, not higher stress drops.