On the relations between fracture energy and physical observables in dynamic earthquake models

We explore the relationships between the fracture energy density (E-G) and the key parameters characterizing earthquake sources, such as the rupture velocity (v(r)), the total fault slip (u(tot)), and the dynamic stress drop (Delta tau(d)). We perform several numerical experiments of three-dimensional, spontaneous, fully dynamic ruptures developing on planar faults of finite width, obeying different governing laws and accounting for both homogeneous and heterogeneous friction. Our results indicate that E-G behaves differently, depending on the adopted governing law and mainly on the rupture mode (pulselike or cracklike, sub-or supershear regime). Subshear, homogeneous ruptures show a general agreement with the theoretical prediction of EG proportional to root 1 - (v(r)(2)/v(s)(2)), but for ruptures that accelerate up to supershear speeds it is difficult to infer a clear dependence of fracture energy density on rupture speed, especially in heterogeneous configurations. We see that slip pulses noticeably agree with the theoretical prediction of E-G proportional to u(tot)(2), contrarily to cracklike solutions, both sub-and supershear and both homogeneous and heterogeneous, which is in agreement with seismological inferences, showing a scaling exponent roughly equal to 1. We also found that the proportionality between E-G and Delta tau(2)(d), expected from theoretical predictions, is somehow verified only in the case of subshear, homogeneous ruptures with RD law. Our spontaneous rupture models confirm that the total fracture energy (the integral of E-G over the whole fault surface) has a power law dependence on the seismic moment, with an exponent nearly equal to 1.13, in general agreement with kinematic inferences of previous studies. Overall, our results support the idea that E-G should not be regarded as an intrinsic material property.