Effect of Time-strengthening Static Friction on Earthquake Recurrence

he effect of time-strengthening static friction on earthquake recurrence is numerically studied based on the normalized equation of motion of a one-body spring-slider model with thermal-pressurized slip-weakening friction having the characteristic distance, Uc, which is in terms of static friction coefficient, 𝜇. Considering the time-strengthening static friction coefficient, 𝜇(𝑡) = 𝜇+𝐵×𝑙𝑜𝑔(𝑡), we assume Uc=Uco/(1+𝐵×𝑙𝑜𝑔(𝑡)). Simulation results exhibit that TR, 𝜏D, D, and Vm, which represent the recurrence time of two events, the duration time of slip of an event, the final slip of an event, and the peak value of particle velocity of an event, respectively, are all similar for five values of Uco when B<0.01 and clearly different when B≥0.01. In general, Vm, D, and TR increase with B; and 𝜏D slightly decreases with increasing B and increases with Uco. An increase in D is particularly remarkable when Uco>0.3. The earthquake recurrence is almost periodic for small Uco; while the degree of periodicity decreases when either Uco or B increases. Either the time-predictable model or the slip-predictable model can approximately interpret the simulated earthquake recurrences with small Uco and small B, yet not for those with large Uco and large B. Simulation results exhibit that time-strengthening static friction produce an opposite effect on earthquake recurrence from the time-widening slip zone.    

happens when the tectonic shear stress on a fault is higher than a critical level, which depends on the physical conditions of the fault and tectonic loading. In their studies earthquakes may happen regularly because of homogeneous physical properties on a fault and constant tectonic loading. Shimazaki and Nakata [1980] proposed three simple phenomenological models to interpret earthquake  However, the three models have been debated for a long time. Some dicussions are given below. Ando [1975] suggested that the second model worked for post-1707 events, yet not for pre-1707 ones in the Nankai trough, Japan. Wang [2005Wang [ , 2019 assumed that the second model could describe the earthquakes occurred on the Chelungpu fault, Taiwan in the past 1900 years. For the Parkfield earthquake sequence, Bakun and McEvilly [1984] took different models; while Murray and Segall [2002] considered the failure of the second model. From laboratory results, Rubinstein et al. [2012] assumed the failure of the time-and slip-predictable models for earthquakes.
Numerous physical models, including the kinetic, crack, and dynamical spring-sider models have been developed for approaching fault dynamics. Unfortunately, up to date there is not a comprehensive model which can completely describe fault dynamics. Two major factors in controlling fault dynamics and earthquake ruptures [Bizzarri, 2009;Wang, 2017bWang, , 2018 are friction [Nur, 1978;Belardinelli and Belardinelli, 1996] and viscosity [Jeffreys, 1942;Spray, 1983;Wang, 2007]. Earthquake recurrence has been simulated based on different models [Bizzarri 2012a, b;Franović et al., 2016;Wang, 2018]. A detailed description about the studies based on the spring-slider model [Burridge and Knopoff, 1967] and its simplified forms with various friction laws can be seen in Wang [2018]. Among the physical models to approach earthquake faults, the one-body spring-slider model is the simplest and useful one to represent a single fault. Wang [2018] numerically simulated earthquake recurrence by using the normalized equation of motion of a onebody spring-sider model with thermal-pressurized slip-weakening friction and viscosity. The main parameters are the normalized characteristic displacement, U c , of the friction law and the normalized damping coefficient, , which is used to represent viscosity. His simulation results show that T R increases with decreasing U c or increasing ; D and D decrease with increasing ; and D increases with U c . The time-and slip-predictable model can describe the temporal variation in cumulative slip. Considering the wear process, he assumed that the thickness of slip zone, h, is an important parameter influencing T R and D and also related h to the cumulated slip S=∑D in the form: h=CS (C=a dimensionless increasing rate of h with S from a cycle to next one). U c is a function of h and thus depends on cumulated slip, ∑U, with an increasing rate of C. His simulation results reveal that the wear process influences the recurrence of events and the effect increases with C when C>0.0001. Although viscosity can strengthen the effect due to wear process it cannot change the pattern of earthquake recurrence. Both T R and D decrease when the fault becomes more mature, thus suggesting that it is more difficult to produce large earthquakes along a fault when it is more mature with longer h. Neither the time-predictable model nor the slip-predictable one can describe the temporal variation in cumulative slip of earthquakes under the wear process with large C. Due to a lack of long-term variation in and viscosity can only strengthens the effect and cannot change the pattern of earthquake recurrence as mentioned in Wang [2018]. Hence, the viscous effect is not included in this study.
In addition to the wear process, static friction (represented by the static friction coefficient, f ) also affects U c , thus being able to change the pattern of earthquake recurrence. The value of f could be time-dependent. In the followings, we will investigate the effect on earthquake recurrence caused by time-strengthening static friction coefficient based on a one-body spring-slider model. From numerical simulations based on this model in the presence of thermal-pressurized friction and viscosity, we can obtain significant information of frictional and viscous effects on earthquake recurrence along a single fault. is usually a function of u or v. A driving force, Kv p t, caused by the moving plate through the leaf spring pulls the slider to move. The equation of motion is:

One-body model and numerical methodology
When Kv p t≥F o , F drops from static to dynamic frictional force and then pushes the slider to move.
Numerous models have been proposed to describe F(u,v) [Wang, 2016]. One of them is proposed based on thermal pressurization (abbreviated as TP below) which is caused by the combined effect from temperature and fluids in a fault zone and thus yields a shear stress (resistance) on the fault plane [Rice, 2006;Wang, 2009Wang, , 2011Wang, , 2016Wang, , 2017aWang, , b, 2018. The 1-D TP model proposed by Rice [2006] has two end-member models: one is the adiabatic-undrained-deformation (AUD) model and the other the slip-on-a-plane (SOP) model. Since the characteristic distance of the SOP model is not associated with the wear process, the model is not used in this study. The AUD model is related to a homogeneous simple strain at a constant normal stress n on a spatial scale of the sheared layer. Its shear stress-slip function, (u), is [Rice, 2006]: C v , h, f , and are, respectively, the fluid density, heat capacity, the thickness, frictional strength, and the undrained pressurization factor of the fault zone. The detailed description about the model parameters can see Rice [2006].
Based on the AUD model, Wang [2009] first took a simplified slip-weakening friction law (denoted by the TP law hereafter) in the following form: which is only dependent on a slip and independent on v. Clearly, F(u) at u=0 is F o . The plot of F(u) versus u when u c =0.1, 0.3, 0.5, 0.7, and 0.9 m and F o =1 N/m 2 is displayed in Figure 4. F(u) decreases with increasing u and its decreasing rate, , decreases with increasing u c . The force drop decreases with increasing u c for the same final slip.
Substituting Equation (2) into Equation (1) leads to The easily conduct numerical computations, Equation (3)  Note that all normalization parameters are dimensionless. Define = / o to be the dimensionless angular frequency. This makes the phase t be . Substituting all normalization parameters into Equation (3) leads to To re-write Equation (4) as two first-order differential equations by using two new parameters, i.e., y 1 =U and Equation (5) can be numerically solved by using the fourth-order Runge-Kutta method [Press et al., 1986].
The general values of these model parameters are evaluated by Wang [2018] and also briefly explained below.
The values of D o are usually several meters and o ranges from 0.1 Hz to few Hz [Wang, 2016]. This leads to that D o o has an order of magnitude of 1 m/s. The value of V p is much smaller than 1 because of v p ≈10 -10 m/s. Wang [2018] assumed that since the value of V p mainly influences the recurrence time, T R , between two events and can only make a very small influence on the pattern of time variations in velocities and displacements of events.
In order to study long-term earthquake recurrence, there must be numerous modeled events with clear and visualized time functions of displacements and velocities for an event in the computational time period. If V p =10 -10 is considered, T R is very long and thus D is much shorter than T R . This makes the time function of an event displayed in the long-term temporal variation in slip looks like just a step function for the displacements and an impulse for the velocities. Hence, in order to get fine visualization a larger value of V p is necessary. The value of V p is usually very small during an event and cannot influence the rupture because of a very tiny value of V p . From numerical tests Wang [2018] stressed that when V p >10 -2 , the value of V p is not small during an event and can influence the rupture. Hence, he took V p =10 -2 for numerical simulations. This value is also used in this study. Note that backward slip is not allowed in the simulations, because of common behavior of forward faulting.
Some researchers [e.g., Bizzarri, 2012a, b;and Franović et al., 2016]  He assumed that U c varies with cumulative slip in the following way: U c =U co +C∑U where U co is the initial value of U c and ∑U is the sum of final displacements of cycles. His results remarkably reveal the effect of timewidening h on earthquake recurrence.
In addition to h, the time-varying static friction coefficient, f , of the slipping zone which can also influences earthquake recurrence is considered in this study. The f is influenced by several factors including humidity, temperature, sliding velocity, strain rate, normal stress, thermally activated rheology etc. [Marone, 1998;Rice, 2006], and thus can change with time [Sibson, 1992;Rice, 2006]. Hirose and Bystricky [2007] observed that serpentine dehydration and subsequent fluid pressurization due to co-seismic frictional heating may reduce f and thus promote further weakening in a fault zone. The pore fluid pressure exists in wet rocks, yet not in dry rocks. From the laboratory experiments for Tennessee sandstone, Shimamoto and Logan [1981] observed a decrease in f with increasing clay content in the fault-zone gouge and the decreasing rate depends on the types of minerals in the gouge. From the laboratory experiments for Westerly granite, Marone et al. [1990] found an increase in f with gouge thickness. From the laboratory experiments, several researchers [e.g., Ohnaka, 1996;Kamer and Marone, 1998;  Here, we assume that U c varies with time in the following manner: A phase portrait, which is a plot of a physical quantity, Y, versus another, X, i.e., Y=f(X), is commonly used to represent nonlinear behavior of a dynamical system [Thompson and Stewart, 1986]. X, f(X), f 2 (X), f 3 (X), …, converges to X f . Chaos can also be generated at some attractors. The details can be seen in Thompson and Stewart [1986]. In this study, there are three physical quantities, i.e., the acceleration, velocity, and displacement of the slider. The phase portrait can be made from a pair of any two of the three quantities.
Here, I take the velocity and displacement to form a phase portrait. Hence, "Y" represents V/V max and "X" denotes U/U max in Figures 5−9.

Simulation results
Here    Meanwhile, the number of events increases with decreasing B and slightly increases with U co . For the interseismic time T R in general increases with B especially for U co >0.3. This means that a higher increasing rate of f with time results in a longer interseismic time especially for U co >0.3. The earthquake recurrence is almost periodic when U co is small; while the degree of aperiodicity increases when either U co or B increases. The three simple phenomenological models of earthquake recurrence proposed by Shimazaki and Nakata [1980] can approximately interpret the simulation results with smaller U co and smaller B, yet not for those with larger U co and larger B. For the present model, the critical friction force (related to the stress level, c , for failure) is not constant and increases with time; while the base friction force (associated with the base stress level, b ) also increases with time. The two phenomena are particularly remarkable when U co >0.3. Since non-constant c and non-constant o cannot meet the stress conditions proposed by Shimazaki and Nakata [1980], their critical friction force (related to the stress level, c , for failure) is constant because of constant f and the base friction force (associated with the base stress level, b ) also decreases with time. Hence, the time-predictable model proposed by Shimazaki and Nakata [1980] seems able to interpret the simulation results in Wang [2018] when the viscous effect is not included.
Figures 5-9 also reveal that V m slightly increases with B when U co <0.3 and clearly increases with B when U co ≥0.3.
The D increases with B and decreases with increasing U co . The value of D is an integral of velocity function from =0 to = D as displayed in Figure 1. Figures 5-9 reveal that D slightly decreases with increasing B and increases with U co . In other words, smaller B and larger U co produce longer duration time of an earthquake. This produces the results that D increases with B and decreases with increasing U co . Larger V m and D are associated with larger events.
Hence, a higher increasing rate of f with time together with a smaller value of U co can produce a larger-sized earthquake. This phenomenon is particularly remarkable when U co >0.3. Figure 4 displays that and smaller U c due to smaller B result in a higher force drop, F, and lower static friction. Hence, higher F or lower f is more easily to produce a larger event and to lead to a perfectly periodic earthquake recurrence than lower F or higher f . This phenomenon is particularly clear for smaller U co .
The right-handed-side panels of Figures 5-9 exhibit that the phase portraits are similar when B<0.01 and different when B≥0.01 for various values of U co even though the patterns of their variations in V and U are similar.
The size of a phase portrait increases with B. This reflects increases in both T R and D of events with B. The absolute values of slope at non-zero fixed point are higher than 1 and slightly increase with time when B≥0.010. This suggests that the non-zero fixed points for all cases in study are not always an attractor. In addition, the zero fixed points are not an attractor, because the absolute values of slope at them are always higher than 1.
An interesting question arises: Can the time-widening slip zone change the static friction coefficient on a fault?
This might be possible because the wear process causing the time-widening slip zone seems able to increase the content of gouge inside the fault zone, thus changing the static friction coefficient. Of course, further studies should be conducted to answer this question.

Summary
Considering the effect due to time-strengthening static frictional coefficient, we assume f (t)=1+B×log(t) and thus U c =U co /(1+B×log(t)). Simulation results exhibit that V m , D, D , and T R are all similar for five values of U co when B<0.01 and different when B≥0.01. In general V m , D, and T R increase with B; and D slightly decreases with increasing B. An increase in D is particularly remarkable when U co >0.3. The earthquake recurrence is periodic when U co or B is small; while the degree of periodicity decreases when either U co or B increases. The three simple models of earthquake recurrence proposed by Shimazaki and Nakata [1980] can approximately interpret the simulation results with small U co and small B, yet not for those with large U co and large B. Simulation results exhibit that the time-strengthening static friction coefficient and the time-widening slip zone produce opposite effects on earthquake recurrence.