13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Figure 2. Plots of normalized stress as a function of normalized slip, for step velocity increases (solid lines) and decrease (dashed lines) of 1-4 orders of magnitude, for (a) the aging law and (b) the slip law. Stresses are relative to the future steady state value. For the aging law the curves for step decreases of 2-4 orders of magnitude appear indistinguishable, but they intersect the vertical axis at the same values of Δτ as the slip law. From figure 3 of Ampuero and Rubin [7] (in the present paper, the notation Dc is replaced by L). Recently Nagata et al. [1] proposed a revised RSF, using new rigorous methods of laboratory data analysis. Firstly, the direct effect coefficient a was constrained to be 0.05, about five times larger than previously believed. The difference came from their new method to constrain a without using any evolution laws, contrasting to conventional methods where the state change from imperfection of real-world ‘step’ tests was inferred by assuming ‘flawed’ evolution law. This large a immediately led to similarly large because was reliably constrained from velocity dependence of steady-state friction without any evolution laws. Secondly, a strong linear negative dependence of dΦ/dt on dτ/dt was newly found from the misprediction analysis of Φ between the observed Φ(= τ−aln V V* ( )) and the predicted Φ by using eq. (2). The shear-stress weakening effect was incorporated as dΦ dt = b σ L V* exp − Φ−Φ* b σ ⎛ ⎝⎜ ⎞ ⎠⎟ − b σ L V−c d τ dt , (4) where c is the stress weakening parameter. The term –cdτ/dt works to resolve the artifact of varying slip-weakening distance in the aging law as clearly seen in Fig. 3; when c=0, eq. (4) coincides with eq. (2) and the artifact remains unsolved, but with increasing c, the symmetric response in opposite sign of velocity-step tests was attained and the prediction curves became more symmetric like the slip law, keeping with the time-dependent healing term [14]. From the above misprediction analysis, the best-fit value of c was determined about 2.0. Nagata et al. [1] confirmed that the revised RSF could correctly reproduce both hold-slide and velocity-step tests with the same values of frictional parameters, which had never been attained in the existing RSFs. Very recently the revised RSF was employed to simulate earthquake cycle [15] and aftershock triggering [16]. Figure 3. Normalized change of shear stress in velocity step-up and step-down tests predicted for different c. Slip rate is increased or decreased tenfold. From figure 5.21 of Nagata [14].
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