Fig. 1

Schematic illustration of the numerical setting of the dynamic rupture simulations. a Slip-weakening friction law used in this study. \({\sigma }_{p},{ \sigma }_{0},\) and \({D}_{C}\) are the peak stress, initial stress, and slip-weakening distance, respectively. This study treats the residual friction as zero owing to the symmetry of the problem. The shaded region represents the fracture energy, \({G}_{C}\). b Spatial distribution of \({D}_{C}\) (Eq. 1). The model space is divided into an inner nucleation zone and outer target region, with the two separated at \(\left|x\right|={R}^{dyn}\). The step in \({G}_{C}\) at \(\left|x\right|={R}^{dyn}\) is quantified by the proportionality constant, \(\gamma\). c, d Time evolution of slip (black line) for an anti-plane case, where the seismic nucleation initiated at \(x=0\) as a crack with a half-width of \(20\mathrm{\Delta x}\), which is negligible, under the \({D}_{C}\) distribution indicated by the blue line. The following parameters are used in the simulations: \({\sigma }_{0}=4 \mathrm{MPa}\), \({\sigma }_{p}=12 \mathrm{MPa}\), \({D}_{C}^{\prime}=6\times {10}^{-4}\), and \({D}_{C}^{\infty }=0.06\Delta x\). The time interval between each adjacent pair of solid lines is \(160\Delta t\), and each gray dashed line is spaced 80 \(\Delta t\) from the next solid line to clarify the self-similarity. c Example of a failed nucleation, whereby the rupture decelerates and terminates. d Example of a successful nucleation, whereby rupture propagation initially decelerates until the crack reaches the static critical crack size \({R}_{C}^{sta}\), and rupture propagation then accelerates again. The difference between \({D}_{C}^{\infty }\) in c and d is 0.6%