Huihui Weng, Jean Paul Ampuero 

The largest earthquakes propagate laterally after saturating the fault’s seismogenic width and reach large length-to-width ratios L/W. Smaller earthquakes can also develop elongated ruptures due to confinement by heterogeneities of initial stresses or material properties. The energetics of such elongated ruptures is radically different from that of conventional circular crack models: they feature width-limited rather than length-dependent energy release rate. However, a synoptic understanding of their dynamics is still missing. Here we combine computational and analytical modeling of long ruptures in 3D and 2.5D (width-averaged) to develop a theoretical relation between the evolution of rupture speed and the along-strike distribution of fault stress, fracture energy and rupture width. We find that the evolution of elongated ruptures in our simulations is well described by the following rupture-tip-equation-of-motion:
G_c=G_0 (1-(v ̇_r W)/(v_s^2 ) 1/(〖Aα〗_s^P ))
where G_c is the fracture energy, G_0 the steady-state energy release rate, v_s the S wave speed, v_r the rupture speed, v ̇_r=dv_r/dt the rupture acceleration, α_s=√(1-(v_r/v_s )^2 ), A = π and P = 3 for rupture acceleration and A = 1.2π and P = 2.6 for rupture deceleration. The steady energy release rate is limited by rupture width as G_0=〖∆τ〗^2 W/πμ, where ∆τ is the stress drop (spatially smoothed over a length scale smaller than W) and μ the shear modulus. If G_c is a constant and exactly balanced by G_0, the rupture can in principle propagate steadily at any speed. If G_c increases with rupture speed, steady ruptures have a well-defined speed and are stable. When G_c≠G_0, the rupture acquires an inertial effect: the rupture-tip-equation-of-motion depends explicitly on rupture acceleration. This inertial effect does not exist in the classical theory of dynamic rupture in 2D unbounded media and unbounded fault in 3D, but emerges in 2D bounded media or, as shown here, as a consequence of the finite rupture width in 3D. These findings highlight the essential role of the seismogenic width on rupture dynamics. Based on the rupture-tip-equation-of-motion we define the rupture potential, a function that determines the size of next earthquake, and we propose a conceptual model that helps rationalize one type of “super-cycles” observed on segmented faults. More generally, the theory developed here can yield relations between earthquake source properties (final magnitude, moment rate function, radiated energy) and the heterogeneities of stress and strength along the fault, which can then be used to extract statistical information on fault heterogeneity from source time functions of past earthquakes or as physics-based constraints on finite-fault source inversion and on seismic hazard assessment.