13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Figure 1. Snapshots of normalized slip (top panels) and slip rate (bottom) for two simulations identical in all respects except that the left panels use the aging law and the right panels the slip law. a/b=0.95, Dc =0.4×10 −3m,V0 =10 −9m/s, τ r =10 −2Pa/s, the normalizing length scaleL b ≡ μ*Dc (b σ) =4.6m. From figure 2 of Ampuero and Rubin [7] (in the present paper, the notation Dc is replaced by L). 1-1. Existing and revised RSFs RSF consists of two equations bearing logically separate roles [8]. One is the constitutive law, which describes the relationship between applied shear stress τ and slip rate V as V =Vexp τ−Φ a σ ⎛ ⎝⎜ ⎞ ⎠⎟ , or τ=Φ+aln V V* ⎛ ⎝ ⎜ ⎞ ⎠ ⎟, (1) where Φ is the state variable specifying the internal physical state of the interface, which may reflect the real contact area [9], a is called direct effect coefficient and play an important role in the constitutive law for which the physical mechanism has been attributed to thermally activated creep [8,10], σ is a normal stress and V* is a reference velocity. Another equation is the evolution law, which phenomenologically describes variations of the state Φ. Two empirical evolution equations for the evolution of Φ, first formalized by Ruina [11], are common use. These are dΦ dt = b σ L V* exp − Φ−Φ* b σ ⎛ ⎝⎜ ⎞ ⎠⎟ − b σ L V (Aging law), (2) dΦ dt =− V L Φ−Φ* −b σln V V* ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (Slip law) , (3) where b is a RSF parameter relating change in state, L is a length scale related with slip, Φ∗ is a reference state. The first and second terms of eq. (2) represent logarithmic time-dependent healing and linear slip weakening with a constant rate b/L per unit slip, while eq. (3) represents exponential slip weakening with a fixed distance L [8]. The aging law has trouble in reproducing a symmetric exponential change of friction over a fixed slip distance subsequent to stepwise velocity jumps to opposite signs observed in velocity-step tests (Fig. 2a) though it explains time-dependent healing in hold-slide tests very well as observed [8,11]. The slip law does explain the slip weakening (Fig. 2b), but has a well-known difficulty in reproducing time healing at low slip rate [12-13].
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