13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- On the contrary, following the slip-weakening curves of the slip law (Fig. 2b), the fracture energy becomes Gc ∝(Δ τp−r ) 1 ∝(lnV)1 due to the constant slip-weakening distance, keeping G∝(lnV)2 remain the same extent. The energy balance suggests that the accelerating V is incompatible with the crack-like expansion of the nucleation zone and numerical simulation demonstrated that the nucleation instead takes the form of unidirectional slip pulse (Fig. 1b). It must be noted that the nucleation was, however, wrongly predicted at the very beginning where the slip rate was very low and healing should have dominated. A genuine regime following laboratory experiments should be investigated by using the revised RSF. 2. Simulation method and choice of parameters Following Dieterich [5], quasi-static nucleation is modeled here by adopting the revised RSF. A fault is divided by n equally spaced segment with a length Δs and loaded by a constant stressing rate τ r : τ i = τ i 0 + τ rt +Δ τ i (i =1,2, ,n), (5) where τi is the shear stress, τi 0 is the initial stress, Δ τ i (=ΣSij δj ) is the change of stress resulting from the slip δj over the fault, and Sij is stress kernel obtained from elastic dislocation solution [5]. By equating eq. (5) with eq. (1) and by substituting the revised law of eq. (4) to eliminate Φ, we obtain a couple of nondimensional differential equations: d ′τi d ′t = ′τ r + ′Sij ′Vj (1.1) d ln ′Vi ( ) d ′t = a/ b ( ) −1 (1+c)d ′τ i / d ′t −exp[−( ′τi −(a/ b)ln ′Vi )]+ ′Vi { } (1.2) where ′τi =( τi − τ*) / (b σ), ′Vi =Vi /V*, ′t =t /(L/V*), ′τ r = τ r /(b σ/ (L/V*)) , ′Sij =Sij /(b σ) and frictional parameters (a,b,c,L) are assumed the same at all points on the fault. A factor (1+c) is newly appeared by the shear-stress dependent term. Nagata et al. [1] obtained a best fit parameter set (a,b,c,L)=(0.051, 0.0565, 2.0, 0.33micron) and this is the only set ever constrained by laboratory experiments. It leads to a/b=0.90 and (1+c)=3.0 in simulating nucleation. In our controlled numerical experiments, c is chosen for a tuning parameter: no stress weakening c=0.0 is for the aging law, the optimum c=2.0 for the revised law, and an excessed c=4.0 for comparison. a/b is chosen as 0.95, 0.90, 0.85, 0.80, 0.75, implying that smaller a/b corresponds to stronger velocity weakening. Except for a/b and c, all simulations are identical in all respects. Fault discretization is done with n=2400 and Δs = Lb /10, small enough to resolve nucleation length. Initial velocity is assumed to be uniformly distributed ′Vi 0 =1.0(= ′V*) and initial stress ′τ i 0 is assumed to be randomly distributed between [-1,0] in order to start nucleation from below-steady state where the time-healing term dominates the slip-weakening term. ′τ r =0.1 and ′μ* =11.56×10 3 are chosen the same as in Ampuero and Rubin [7]. If V* =10 −9m/s is taken, our simulation results are directly comparable with Fig. 1. Time integration is numerically done by using the Runge-Kutta method until ′Vmax =10 9.
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