13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 1-2. Nucleation regimes and lengths We here briefly summarize nucleation regimes and their lengths in the previous studies [5-7] examined both numerically and analytically. For strongly velocity-weakening faults characterized by a/b<0.5, nucleation occurs in a ‘fixed-length patch’ regime, where regions of quasi-static slip remains relatively small. In the patch regime, the state along patch kept well above steady state with insignificant healing. With a no-healing approximation, Rubin and Ampuero [6] analytically estimated the patch half-length as Lν =1.3774Lb (Lb ≡ μ*L b σ ), (5) where μ* = μ/(1− ν) is the stiffness of a medium for edge dislocation, μ is the rigidity, and ν is the Poisson's ratio. For the slip law, the patch length is slightly smaller than Lν [7]. For weakly velocity-weakening faults (0.5<a/b<1), nucleation must propagate spatially. Propagation occurs in a form of an expanding crack (‘crack-like expansion’ regime) in the aging-law case (Fig. 1a) and in a form of a migrating slip pulse (‘unidirectional slip-pulse’ regime) in the slip-law case (Fig. 1b). In the crack regime, the state along expanding portion instead kept steady state for increasing slip, that is, healing effect was not negligible. The nucleation half-length was estimated with a steady-state approximation as L∞ = b b−a ⎛ ⎝⎜ ⎞ ⎠⎟ 2 L b π (b−a>0). (6) Ampuero and Rubin [7] attributed the crack-like expansion to the slip-weakening curves predicted with the aging law in response to velocity step ΔV (Fig. 2a), which would be experienced at the expanding nucleation front where the concentrated stress yields with a sudden jump in the slip rate. In their analysis, nucleation zone was proven to be approximated by a crack; stress uniformly drops by Δτ on the crack and locally concentrates at the tips by peak-to-residual stress drop Δτp-r. Fracture mechanics tells us the energy release rate G in terms of the crack length l and fracture energy Gc at the crack tip, which were then connected to slip rate V on RSF faults as (7) (8) where V' and V'' are nearly constant slip rates, δc is the effective slip-weakening distance following the linear slip-weakening rate (b/L), and . Setting G=Gc and solving for the instantaneous nucleation length l lead to l = Lb π Δ τp−r Δ τ ⎛ ⎝⎜ ⎞ ⎠⎟ 2 =l(V). (9) At every moment in accelerating V, the energy balance G=Gc is satisfied; this is why the nucleation can be the crack-like expansion. By considering the ratio Δτp-r/ Δτ approaching b/(b-a) in the limit of large slip rate V, the nucleation size L∞ in eq. (6) was derived. However, the prediction curves with increasing slip-weakening distances are an artifact due to the linear slip-weakening rate and contradict with laboratory experiments [8].
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