Rake rotation introduces ambiguity in the formulation of slip-dependent constitutive models: slip modulus or slip path?

摘要

The linear slip–weakening (SW) law, predicting that the traction decreasesfor increasing fault slip, is one of the most widely adopted governingmodels to describe the traction evolution and the stress releaseprocesses occurring during coseismic slip failures. We will show that, contraryto other constitutive models, the SW law inherently poses the problemof considering the Euclidean norm of the slip vector or its cumulativevalue along its path. In other words, it has the intrinsic problem of itsanalytical formulation, which does not have a solution a priori. By consideringa fully dynamic, spontaneous, 3–D rupture problem, with rakerotation allowed, in this paper we explore whether these two formulationscan lead to different results. We prove that, for homogeneous configurations,the two formulations give the same results, with a normalizeddifference less than 1%, which is comparable to the numerical error dueto grid dispersion. In particular, we show that the total slip, the resultingseismic moment, the fracture energy density, the slip–weakening curveand the energy flux at the rupture front are practically identical in thetwo formulations. These findings contribute to reconcile the results presentedin previous papers, where the two formulations have been differentlyemployed. However, this coincidence is not the rule. Indeed, byconsidering models with a highly heterogeneous initial shear stress distribution,where the rake variation is significant, we have also demonstratedthat the overall rupture history is quite different by assuming thetwo formulations, as well as the fault striations, the traction evolutionand the scalar seismic moment. In this case the choice of the analytical formulationof the governing law does really matter.