Reconciling aging and slip state evolutions from laboratory-derived canons of rate-and-state friction

The aging law and the slip law are two representative evolution laws of the rate- and state-dependent friction (RSF) law, based on canonical behaviors in three types of laboratory experiments: slide-hold-slide (SHS), velocity-step (VS), and steady-state (SS) tests. The aging law explains the SHS canon but contradicts the VS canon, and vice versa for the slip law. The later proposed composite law, which switches these two laws according to the slip rate $V$, explains both canons but contradicts the SS canon. The present study constructs evolution laws satisfying all three canons throughout the range of variables where experiments have confirmed the canons. By recompiling the three canons, we have derived constraints on the evolution law and found that the evolution rates in the strengthening phases of the SHS and VS canons are so different that complete reconciliation throughout the entire range of variables is mathematically impossible. However, for the limited range of variables probed by experiments so far, we have found that the SHS and VS canons can be reconciled without violating the SS canon by switching the evolution function according to $\Omega$, the ratio of the state $\theta$ to its steady-state value $\theta_{\rm SS}$ for the instantaneous slip rate. We could generally show that, as long as the state evolution rate $\dot \theta$ depends only on the instantaneous values of $V$ and $\theta$, simultaneous reproduction of the three canons, throughout the experimentally confirmed range, requires the aging-law-like evolution for $\Omega$ sufficiently below a threshold $\beta$ and the slip-law-like evolution for $\Omega$ sufficiently above $\beta$. The validity of the canons in existing experiments suggests $\beta \lesssim 0.01$.