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

The linear slip–weakening (SW) law, predicting that the traction decreases
for increasing fault slip, is one of the most widely adopted governing
models to describe the traction evolution and the stress release
processes occurring during coseismic slip failures. We will show that, contrary
to other constitutive models, the SW law inherently poses the problem
of considering the Euclidean norm of the slip vector or its cumulative
value along its path. In other words, it has the intrinsic problem of its
analytical formulation, which does not have a solution a priori. By considering
a fully dynamic, spontaneous, 3–D rupture problem, with rake
rotation allowed, in this paper we explore whether these two formulations
can lead to different results. We prove that, for homogeneous configurations,
the two formulations give the same results, with a normalized
difference less than 1%, which is comparable to the numerical error due
to grid dispersion. In particular, we show that the total slip, the resulting
seismic moment, the fracture energy density, the slip–weakening curve
and the energy flux at the rupture front are practically identical in the
two formulations. These findings contribute to reconcile the results presented
in previous papers, where the two formulations have been differently
employed. However, this coincidence is not the rule. Indeed, by
considering models with a highly heterogeneous initial shear stress distribution,
where the rake variation is significant, we have also demonstrated
that the overall rupture history is quite different by assuming the
two formulations, as well as the fault striations, the traction evolution
and the scalar seismic moment. In this case the choice of the analytical formulation
of the governing law does really matter.