Dynamic antiplane self-similar crack with distance-weakening friction: an analytical solution for source parameter estimation

Abstract:

In several analytical and numerical studies, the slip rate function and energy release rate for dynamic self-similar crack growth have been investigated, and the results obtained have contributed to a theoretical understanding and estimation of on-fault energetics. Analytical solutions to self-similar singular crack models have primarily provided unbounded slip-rate functions characterized by square-root singularity (Kostrov 1964 PMM; Nielsen & Madariaga 2003 BSSA). However, such solutions are unphysical, and we may have to introduce some weakening zone of friction behind crack tips to clarify some relationships among physical parameters, including stress state, process zone size, rupture velocity, peak slip rate, and energy release rate, etc.

Therefore, this study aims to derive an analytical solution of the slip rate distribution of a dynamic antiplane self-similar crack model under distance-weakening friction that mimics slip-weakening friction. We derive and solve a singular integral equation of the model. To guarantee the boundedness of the slip rate, we propose some trade-off relationships among peak slip rate, V_peak, normalized rupture velocity, k, energy release rate gradient, G/x, and breakdown stress-to-stress drop ratio, S. This process is similar to the strategy of Rice et al. (2005 BSSA) for a steady-state dynamic slip pulse model, but the result reveals the difference between the pulse and self-similar crack modes.

First, we quantify k as a function of G/x and S, which means that rupture velocity is determined when fracture energy gradient and initial stress are given. Second, we found that the rupture velocity is restricted to 80-90% of the shear wave speed if V_peak < 5.7 m/s holds, as previously implied by numerical work (Andrews 2005 JGR), meaning that our model can mimic off-fault inelastic energy dissipation. Third, we propose a convenient approximation to estimate fracture energy based on seismically observable parameters: k and V_peak. In conclusion, we discuss the applicabilities of our solution to forward and inverse problems of earthquake source mechanics.