The rate- and state- dependent friction laws (RSF) are empirical relations based on laboratory experiments that have been used to model a variety of earthquake behaviors, including the mechanics of a seismic cycle, episodic aseismic slip, and triggered seismicity (Kame et. al, 2013). These laws describe variations in friction based on the loading rate and state of the sheared zone. There are several forms of the RSF laws. The paper summarized below is based on the RSF law proposed by Dieterich (1979) and a more recently revised version proposed by Nagata et al. (2012).
In 1994, Dieterich modeled aftershock seismicity after an imposed stress step using his RSF model. His model can predict the observed 1/t decay of aftershock rate but there are two major observational gaps: (1) The model under predicts the amount of aftershock productivity and (2) The model predicts too long a delay time before the onset of decay. In a recent paper, Kame et al. (2013) hoped to address these gaps by running similar models using the Nagata RSF law.
Dieterich’s model considered a fault of fixed size embedded in an elastic medium. He was able to solve for the aftershock rate analytically. Kame et al. (2013) applied the Nagata law, which contains a stress weakening effect, to a similar model but found that the problem required a numerical solution.
Main observations from Kame et al. (2013) study:
1) Although the revised model produced greater seismicity and shortened delayed times, these improvements were only by a small factor compared to the disparities with natural observations that span several orders of magnitude.
2) Unlike the Dieterich model , in which a stress step always advances the timing of an earthquake, the revised model showed two different types of behavior. In most cases, the timing of the earthquake was advanced. However, if the stress step occurred at a specific time in the loading history of the fault , oscillatory slow slip cycles began, effectively delaying the onset of the earthquake.
For more details on this study see:
Kame, Nobuki, et al. “Effects of a revised rate-and state-dependent friction law on aftershock triggering model.” Tectonophysics 600 (2013): 187-195.
http://www.sciencedirect.com/science/article/pii/S004019511200755X
Other sources:
K. Nagata, M. Nakatani, and S. Yoshida. A revised rate- and state-dependent friction law obtained by constraining constitutive and evolution laws separately with laboratory data, 2012.
http://onlinelibrary.wiley.com/doi/10.1029/2011JB008818/abstract
J.H. Dieterich. A constitutive law for rate of earthquake production and its application to earthquake clustering, 1994. http://onlinelibrary.wiley.com/doi/10.1029/93JB02581/abstract
J.H. Dieterich. Modeling of rock friction 1. Experimental results and constitutive equations, 1979.
http://onlinelibrary.wiley.com/doi/10.1029/JB084iB05p02161/abstract