Numerical simulations

This protocol is extracted from research article:

The equation of motion for supershear frictional rupture fronts

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Sci Adv**,
Jul 18, 2018;
DOI:
10.1126/sciadv.aat5622

The equation of motion for supershear frictional rupture fronts

Procedure

Our numerical results were generated by solving elastodynamic equations with the spectral boundary integral method (*32*, *42*). The propagation of a dynamic crack between two half spaces was modeled with a cohesive-type approach to describe the tractions along the weak interface. An explicit time integration was applied. The spectral formulation of the tractions and displacements at the interface results in a periodic setup. In our simulations, we used a replication length of 1.2 m with a discretization of 8192 nodes. The half-space material is linear elastic. To compare with the experiments, we applied a dynamic value of the elastic modulus *E* = 5.65 GPa, Poisson’s ratios *v* = 0.35, and density ρ = 1170 kg/m^{3} and used a plane strain assumption (*C*_{S}/*C*_{L} ≈ 0.48). The interface tractions were governed by a linear slip-weakening cohesive law, τ(δ) = τ_{p}(1 – δ/*d*_{c}) for 0 < δ < *d*_{c}, which imposes a strength that decreases linearly with slip δ from a peak value τ_{p} to zero over a characteristic slip weakening distance *d*_{c}. Reference values applied in the uniform setup were τ_{p} = 1.0 MPa and *d*_{c} = 2 μm, which led to a fracture energy of Г = 1.0 J/m^{2}. Rupture nucleation was triggered via slowly propagating a seed crack of imposed velocity 0.1*C*_{R} [following (*9*)] through the nucleation zone. In the nucleation zone, the value of τ_{p} was gradually reduced to zero (over a length of ≈6 mm). Once the seed crack reached a critical distance *l*_{c} (the Griffith length), rupture acceleration initiated and the ruptures propagated dynamically.

While *v* = 0.35 occurs in serpentinized mantle material (*43*) as, for instance, along the Denali fault (*44*) that hosted a supershear earthquake (*12*, *13*, *45*), *v* = 0.25 is a common value for granite. Additional numerical results were obtained with *v* = 0.25 and plane strain boundary conditions (*C*_{S}/*C*_{L} = 0.577). Comparisons with the theoretical predictions are shown in fig. S1. The results are qualitatively identical to Fig. 2. The main quantitative difference is that the ruptures accelerate over shorter distances to considerably higher speeds, that is, *C*_{f}/*C*_{L} > 0.9. Therefore, the differences in rupture speed for very different prestress levels are already relatively small shortly after transition.

Here, we used spatially nonuniform τ_{0}/τ_{p} profiles. Direct supershear transition was triggered by using high τ_{0}/τ_{p} levels for very short crack lengths. Supershear propagation was then studied as cracks entered into regions with lower values of τ_{0}/τ_{p} (for example, brown curve in Fig. 5). We have also tested the quality of the theoretical prediction for supershear ruptures after transition through the well-known Burridge-Andrews mechanisms (orange curve in Fig. 5). The rupture first propagates in the sub-Rayleigh regime until a radiated shear wave ahead of the crack tip nucleates a secondary crack that then propagates at supershear speeds. A third simulated rupture was initiated by a seed crack that propagated at supershear speeds already during the nucleation procedure. All three tested supershear transition mechanisms lead to supershear ruptures with propagation speeds that are, after brief transient differences, quantitatively well described by our theoretical model. The observed discrepancies within this transient period, we believe, are related to the history dependence, discussed by Huang and Gao (*24*), which was neglected in the derivation of Eq. 1.

Three different transition mechanisms are considered. Top: The first setup (brown curve) has a spatially nonuniform τ_{0}/τ_{p} profile with reduced local τ_{p} for *l*/*l*_{0} < 50 (see main text for details). Two additional examples have spatially uniform τ_{0}/τ_{p} profiles (orange and red curves). Bottom: Colors represent the crack velocities *C*_{f}(*l*) corresponding to the stress profiles in the top panel. Brown curve indicates continuous crack acceleration to supershear speeds (direct transition) within a weakened nucleation (nucl.) patch (high τ_{0}/τ_{p} level). Orange curve indicates sub-Rayleigh rupture transitions at *l*/*l*_{0} ≈ 65 to supershear speed through the Burridge-Andrews (BA) mechanism (*4*, *5*). Red curve indicates an imposed supershear seed crack leads to a self-sustained supershear crack propagation. The black dashed line denotes theoretical prediction for a spatially uniform prestress level (τ_{0}/τ_{p} = 0.45).

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