## 1. Introduction

[2] Time-dependent healing of frictional strength [e.g., *Dieterich*, 1972] is one of the key mechanisms that govern earthquake cycles [e.g., *Beeler et al.*, 2001]. It is also a cornerstone of rate and state friction laws [e.g., *Ruina*, 1983], which have been successfully applied to explain a wide spectrum of earthquake phenomena [e.g., *Scholz*, 1998].

[3] When the frictional interface is in (quasi-) stationary contact, frictional strength often increases logarithmically with time *t* [e.g., *Dokos*, 1946; *Dieterich*, 1972].

Here, Θ is the frictional strength normalized by the applied effective normal stress . Since it is known that more shear stress (*τ*) is necessary to cause faster slip on the interface at the same state (strength), we need to define frictional strength Θ as the shear stress required to cause a slip of an arbitrarily chosen reference velocity *V*_{*}, using the following constitutive equation:

where *a* is an empirical constant called the coefficient of direct effect [e.g., *Dieterich*, 1979; *Ruina*, 1983]. Thus defined, Θ, taken distinct from the shear stress, is a natural extension of the classical frictional strength as a threshold for slip to occur [*Nakatani*, 2001]. Since this defined strength is determined by the internal physical states of the frictional interface, Θ can be regarded as a state variable in rate and state friction laws [*Ruina*, 1983]. In the present paper, Θ will be referred to as “strength” or “state” interchangeably.

[4] The logarithmic healing law (1) has two parameters. The first parameter *b* is the magnitude of strengthening per an *e*-fold increase of contact time (Figure 1a). Although *b* is often called the “healing rate,” we avoid this wording. As seen from the plot of Θ against linear time (Figure 1b), the real healing rate is always decreasing with time. We reserve the term healing rate for , not *b*.

[5] The present paper focuses on the other parameter *t*_{c}. This parameter is necessary to avoid the negative divergence of logarithm as *t* → 0 and is often called the “cutoff time” because the increase of Θ per an *e*-fold increase of time is much less than *b* for 0 ≤ *t* ≪ *t*_{c} (Figure 1a). Again, however, “cutoff” is a misleading word because healing is more active at a smaller *t* even for *t* ≪ *t*_{c}, as seen from Figure 1b. It is not that the process of healing is cut off for *t* ≪ *t*_{c}. Rather, it is just that 0 ≤ *t* ≪ *t*_{c} is too short a time interval to see a significant increase of Θ. Nonetheless, we keep the familiar terminology of cutoff time to refer to *t*_{c}.

[6] The cutoff time *t*_{c} is important not only for *t* ≪ *t*_{c} but also for *t* ≫ *t*_{c}. For *t* ≫ *t*_{c}, ΔΘ(*t*) ≡ Θ(*t*) − Θ(*t* = 0) has a dependency of −*b*ln(*t*_{c}) on the cutoff time. So, without knowing the value of *t*_{c}, one cannot tell the amount of strength recovery even for *t* ≫ *t*_{c}. In many laboratory healing tests at room temperature [e.g., *Dieterich*, 1972], *t*_{c} seems to be order 0.1–1 s, while *t*_{c} up to half a day (5 × 10^{4} s) has been registered in laboratory tests under hydrothermal conditions [*Nakatani and Scholz*, 2004a]. A cutoff time as long as 100 days (9 × 10^{6} s) has been observed for healing of natural faults [*Marone et al.*, 1995]. Assuming a typical value of *b* = 0.01 from laboratory experiments, a difference of *t*_{c} by 8 orders of magnitude, for example, results in a difference of ΔΘ by 0.18 in terms of frictional coefficient. Predicting the amount of frictional healing ΔΘ in terms of frictional coefficient is important. For example, we can estimate the effective normal stress of earthquake faults, an enigmatic problem [e.g., *Wang et al.*, 1995; *Scholz*, 2000], by = Δ*τ*/ΔΘ, where Δ*τ* is the interseismic strength recovery in terms of stress, which can be independently estimated from seismological and geodetic observations.

[7] From an experimental point of view, we should note that the −*b*ln(*t*_{c}) dependence of ΔΘ for *t* ≫ *t*_{c} means that the value of *t*_{c} can be determined from the data for *t* ≫ *t*_{c} only. This is good news because conducting healing experiments for very short *t* is often technically difficult.

[8] Furthermore, recent process-based theories of logarithmic healing [e.g., *Brechet and Estrin*, 1994; *Nakatani and Scholz*, 2004b] suggest that *t*_{c} is inversely proportional to the reaction rate constant of the underlying process. For example, experiments by *Nakatani and Scholz* [2004a, 2004b] have found an Arrhenius-type temperature dependence of 1/*t*_{c}. On the other hand, there is a laboratory data set [*Marone*, 1998a] suggesting that *t*_{c} depends on the slip velocity employed in the experiment. This implies that *t*_{c} is not necessarily an intrinsic constant of the healing process.

[9] In this paper, we will theoretically show in section 2 that the observed *t*_{c} is the sum of (1) the process rate-related characteristic time that appears in the original Brechet and Estrin (BE) model and (2) the effective contact time of asperities at the beginning of (quasi-) stationary contact. The latter factor was not considered in the BE model, but it will be shown that this is a natural consequence of general physical systems that result in logarithmic growth, including the BE model. Further, thus developed cutoff time theory is incorporated into the existing friction laws. This leads to a quantitative prediction of the high-velocity cutoff for the velocity dependence of strength in steady state sliding [e.g., *Okubo and Dieterich*, 1986], which is important in fault dynamics [e.g., *Ruina*, 1983]. In section 3, we will show that these theoretical predictions are consistent with the two experimental data sets mentioned above.