Modeling Dynamic-Weakening and Dynamic-Strengthening of Granite in High-Velocity Slip Experiments

Earthquakes are associated with slip along fault-zones in the crust, and the intensity of dynamic-weakening is one of the central questions of earthquake physics (Dieterich, 1979; Reches and Lockner, 2010). Since it is impossible to determine fault friction with seismological methods (Kanamori and Brodsky, 2004), the study of fault friction and earthquake weakening has been usually addressed with laboratory experiments (Dieterich, 1979) and theoretical models (Ohnaka and Yamashita, 1989).


Introduction
Earthquakes are associated with slip along fault-zones in the crust, and the intensity of dynamic-weakening is one of the central questions of earthquake physics (Dieterich, 1979;Reches and Lockner, 2010).Since it is impossible to determine fault friction with seismological methods (Kanamori and Brodsky, 2004), the study of fault friction and earthquake weakening has been usually addressed with laboratory experiments (Dieterich, 1979) and theoretical models (Ohnaka and Yamashita, 1989).
The experimental analyses of dynamic-weakening were conducted in several experimental configurations: bi-axial direct shear (Dieterich, 1979;Samuelson et al, 2009), tri-axial confined shear (Lockner and Beeler, 2002), and rotary shear apparatus (Tsutsumi and Shimamoto, 1997; Goldsby and Tullis, 2002;Di Toro et al., 2004;Reches and Lockner, 2010).The direct shear apparatus allows high normal stress and controlled pore water pressure (up to ~200 MPa) with limited slip velocity (up to 0.01 m/s) and limited slip distance (~10 mm) (Shimamoto and Logan, 1984).These slip velocities and displacements are significantly smaller than those of typical earthquakes (0.1-10 m/s and up to 5 m, respectively).In order to study high velocity and long slip distance, experiments have been conducted in rotary shear machines.
While many studies indicated a systematic weakening with increasing slip-velocity (Dieterich, 1979;Di Toro et al., 2011), recent experimental observations revealed an opposite trend of dynamic-strengthening particularly under high velocity (Reches and Lockner, 2010;Kuwano and Hatano, 2011).This strengthening was attributed to dehydration of the fault gouge due to frictional heating at elevated velocities (Reches and Lockner, 2010;Sammis et al., 2011).If similar dynamic-strengthening occurs during earthquakes, it should be incorporated in the analyses of earthquake slip.
We analyze here of the relations between friction coefficient and slip velocity for steady-state velocity the range of 0.001-1 m/s.These relations are derived for the Sierra White granite (SWG) experiments under normal stress up to 7 MPa.The results are presented by a model that is termed WEST (WEkening-STrengtheing).We first present the main experiment observations, for five rock types, and then we derive the numeric model for Sierra White granite (SWG).Finally, we apply the numerical model to tens of experiments with complex velocity history conducted on the same rock.

Experimental observations
The experiments were conducted with a rotary shear apparatus and cylindrical rock samples (Reches and Lockner, 2010;Chang et al., 2012).The experimental setup and sample configuration are described in Appendix A. The experimentally monitored parameters include slip distance, slip velocity, fault-normal displacement (dilation), shear stress and normal stress.Sample temperature was measured by thermocouples that are embdded ~3mm away from the slip surfaces.The normal load was maintained constant during a given experiment, and the experiments were performed at room conditions.Fault strength is represented by the friction coefficient, μ = τ/σ N (where τ is shear stress and σ N the normal stress).The slip velocity was either maintained constant, or increased or decreased in steps.We present results of tests with samples of Sierra White granite (after Reches and Lockner, 2010), and new results for samples of Blue quartzite, St. Cloud diorite, Fredricksburg syenite, and Karoo gabbro.The tests with SWG samples were run under the widest range of conditions and the numeric model was derived only for this rock.
Reches and Lockner (2010) determined the friction-velocity relations of SWG for a velocity range of.0003-1m/s and normal stress up to 7 MPa.Their results revealed three general regimes of friction-velocity relations (Fig. 1a): i.
Dynamic-weakening (drop of 20-60% of static strength) as slip velocity increased from ~0.0003 m/s to a critical velocity of V c ~ 0.03 m/s, during which the friction coefficient was 0.3-0.45.

ii.
Transition to dynamic-strengthening regime in the velocity range of V = 0.06-0.2m/s, during which the fault strength almost regained its static strength; and iii.Quasi-constant strength for V > 0.2 m/s, with possible further drops as velocity approaches ~1 m/s.Only few experiments were conducted in this range due to sample failure by thermal fracturing.
Similar pattern of weakening-strengthening of Westerly granite samples was recently observed by Kuwano and Hatano (2011) (Fig. 1b).They showed that the friction coefficient dropped in the velocity range of 0.001-0.06m/s, and rose in the velocity range of 0.06-0.2m/s (Fig. 1b).Earlier work of Tsutsumi and Shimamoto (1997) also indicated temporal periods of strengthening, in which the friction sharply increased, and visible melting was observed at the strength peak when the slip rate was increased from 0.55 to 0.73 m/s.To further explore the occurrence of dynamic-strengthening, we tested four other rocks under conditions similar to Reches and Lockner (2010) tests.The new experiments were conducted on samples made of Blue quartzite (Fig. 2a), St. Cloud diorite (Fig. 2b), Fredricksburg syenite (Fig. 2c), and Karoo gabbro (Fig. 2d).The diorite (Fig. 2b) and syenite (Fig. 2c) samples displayed a distinct transition into dynamic-strengthening regime (red arrows).The critical transition velocity, V C , depends on the sample lithology; it is ~0.02 m/s and ~0.01 m/s for the diorite and syenite, respectively.The Blue quartzite tests displayed only negligible weakening and strengthening (Fig. 2a).

Approach
The most striking feature of the relations between the friction coefficient and slip-velocity for the above experiments is the systematic transition from weakening under low velocity to strengthening at higher velocities (Figs. 1, 2).We refer to this feature of WEakening-STrengthening as the WEST mode, and developed a numerical model to describe its character.
The model is based on two central assumptions.First, the friction coefficients, in both weakening and strengthening regimes, can be presented as dependent of the slip distance, and slip velocity.This assumption was employed in many previous models (Dieterich, 1979

Model formulation
Following the above assumptions, we model the relations of the steady-state friction coefficient, μ, and the slip distance, D, and slip-velocity, V (Fig. 3).The steady-state is defined for a given slip velocity during which the weakening (or strengthening) intensifies with increasing slip distance.The model addresses the three major features of the experimental observation: (I) the slip-weakening relation, μ(V, D), represents the drop from static friction coefficient, μ S , to the kinematic friction coefficient, μ K during slip to the critical distance, D C , at a constant velocity; (II) the dynamic-friction coefficient, μ K (V), at V < V C , under steady-state; and (III) the dynamic-strengthening at high velocity regime of V > V C .We regard the friction coefficient decrease from (1) to (2) (Fig. 3) as the dynamic-weakening, and the friction coefficient increase from ( 2) to (3) as dynamic-strengthening.The dependence of the experimentally monitored friction coefficient, μ, on the two experimentally controlling factors of D and V has the general form of ( ) The friction coefficient, μ, has two end members, the static friction coefficient, μ S , which is the friction coefficient during slip initiation, and the kinematic friction coefficient, μ K , the steady-state coefficient after large slip distance under a constant velocity.The transition between the weakening regime and the strengthening regime occurs at the critical slip-velocity, V C , that is determined from the general plot of friction coefficient as function of slip-velocity (Fig. 1, 2, 3).
We assume that during the weakening stage and under constant velocity, the drop of μ from μ S to μ K is controlled only by the slip distance D, ( ), where, where D C is the critical slip distance.
It is also assumed that μ K is a function only of slip velocity V, ( ), Similarly, we assume that during the strengthening stage, μ is a function of both slip-velocity and slip distance, ( ) , , , This approach for the empirical relations is in the spirit of the weakening relations developed by Dieterich (1979) who introduced the rate-and state-friction law.

General
Solutions for the model functional relations were searched in the following steps.
1. Prepare a table of the experimental velocity-friction relation;

2.
Select experiments with representative weakening stage; 3. Search for a suitable functional fit between slip velocities, friction coefficient, and slip distance.For this search, we chose the simplest functional relation that provided good fit.
The search revealed the relations between the μ K and velocity.

4.
Integrate the above relations for the effect of distance and velocity.
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5.
Apply the above functional relations to set up the specific WEST model (Equations 2, 4, 5).

Dynamic-weakening regime
The experimental data set has large data scatter at low velocities (Fig. 1; Reches and Lockner, 2010).For the range of V < 0.03 m/s (Fig. 4), we found the following relations of μ K and slip velocity, μ K (V) = 0.742 -0.375V 0.00183 + V , for V ≤ 0.03 m / s (6) The RMS (root mean square) of this solution is 0.83 while the correlation coefficient is 0.73.This simple solution provides reasonable fit to the scattered friction data (Fig. 4).

Dynamic-strengthening regime
In the strengthening regime of 0.03 m/s < V < 1.0 m/s (Fig. 5), the selected solution for velocitycontrolled friction coefficient is μ K (V) = 0.824 exp ( -0.0275 V ) for V > 0.03m / s (7) The RMS of this solution is 0.91 and the correlation coefficient is 0.74.The trend of this exponential relation provide good fit for SWG (Reches and Lockner, 2010) in the strengthening trend (Fig. 5), and this trend generally fit the experimental results of Kuwano and Hatano (2011).Note: Friction during 0.305-0.4m has been locally adjusted to comply with Fig. 4.

General
We now apply the WEST model to simulate the friction-velocity-distance evolution in a set of experiments with Sierra White granite.In these simulations, we use the experimental slip distance, D, and slip velocity, V, which were controlled by the operator, as input parameters in Equations (6,7,9) and Table 1.This substitution predicts the evolution of the friction coefficient during the experiments.The predicted friction coefficient is then plotted with the experimentally observed equivalent.The simulations were done on four types of velocity histories; none of the simulated experiment was used for the derivation of the model.

Rising velocity steps
We start with three runs under same σ Ν = 5.0MPa, and each with three upward stepping velocities, 0.0025, 0.025 and 0.047 m/s.During these runs the cumulative slip was about 9 m (Fig. 7).These velocities are in the weakening regime, and they display intense friction drops that reflect both the rising velocity and increase of slip distance.The friction coefficient was 0.7-0.9 at the first step of V = 0.0025 m/s, and it decreased to μ ≅ 0.5 in the second velocity step of V = 0.025 m/s.A minimum of friction coefficient of μ = 0.4 developed under V = 0.05 m/s.This experimental evolution fits well the predictions by WEST model.

Rise and drop
We now examine two groups of experiments, each with five runs of the same loading conditions (σ N = 5.0MPa).Each group has three velocity stages: low, high, and back to low.In the first group, the three steps were 0.00075, 0.0075, and 0.00075 m/s.The initial friction coefficient ranged 0.4-0.7 for the different runs in the group, and during the first step, the friction coefficient was gradually decreased by ~0.1.During the second stage of V = 0.0075 m/s, which lasted 10 s, the friction coefficient reduced drastically, with larger reduction of the runs that started at higher friction coefficient.During the final, low velocity, the sample was slightly strengthened.
In Fig. 8, this group of experiments was performed on the same sample under σ N = 5.0MPa.The different initial friction coefficients (e.g., runs 1, 2 have higher friction coefficient than run 3, 4, 5) were attributed to the holding-times between experiments.The weakening was well noticed when slip velocity was raised and then the friction reached a relative steady value.The fit line shows the weakening immediately after the velocity has been raised.When the velocity drops, the friction coefficient is raised higher in the fit line than the data.In the second group (Fig. 9), the scenario was repeated but at velocities which were higher by an order of magnitude: 0.0075 m/s (first and last stages) and 0.075 m/s (second stage).This time, the friction coefficient in the second stage was increased from μ=~0.3 to the highest μ = 0.7.Then, μ gradually reduced to 0.3-0.4 during the final, low velocity stage.In Fig. 9, the friction coefficient of the fit results rises and drops earlier than the experimental data (running from 0.25 m to 0.75 m).The rest of the running, at the beginning low velocity and the final velocity, the friction coefficient between the model and experiments fit well each other.
Comparing the two groups, the different friction responses provide clear evidence for dynamic-strengthening at slip velocity ~ 0.07 m/s.The WEST simulations were also performed to investigate if the model can capture the dynamic-strengthening at the high velocity regime.Fig. 10 is an experiment with five alternating steps of velocity that again indicates the dynamic-strengthening at velocity 0.075 m/s (above the critical value V C ).The friction coefficient dropped from 0.7 to 0.5 at velocity of 0.0025 m/s and 0.025 m/s, then dynamically-strengthened to μ = 0.7 at V = 0.75 m/s, and finally slightly dropped to 0.55 with a slip distance of ~4.2 m (from 1.7 m to ~5.9 m) as the velocity decreased.
Similarly to the case of rise-drop experiment (Fig. 8), the observed friction coefficient changed only slightly in the end at dropping velocity (from 5.9 m to 7.6 m).The simulation illustrates the dynamic-strengthening during the three-step sliding.

Drop and rise during long distance slip experiments
In Fig. 11, the simulation presents the experiment behavior that involved three steps: an initial weakening at the higher slip velocity of 0.045 m/s, regaining strengthen at a lower speed of 0.0018 m/s, and a final weakening at high-speed step of 0.045 m/s after healing for about 2,000 seconds.The healing refers to the regaining of strength of the fault during hold time.
During the simulation of the second step, the friction coefficient was manually shifted to static value due to healing (indicated by blue arrow, Fig. 11).The increase of friction here is related to healing during hold time, which differs from dynamic-strengthening.The simulation results show smooth lines instead of noisy curves of the experiment.

Wide velocity range
Figs. 12 and 13 display the friction evolution in two experiments with wide velocity range.Fig. 12 displays a slide-hold-slide run under σ Ν = 1.1 MPa and hold times of 10 s between the multiple velocity steps.Under this short hold time, the friction coefficient curve displays quasicontinuous trend.There is a marked weakening as the velocity was increased to ~ 0.04 m/s, followed by a gentle strengthening at higher velocities.The modeling results were lower at beginning than expected and strengthening occurred earlier.

Field applications
The WEST model was developed for the experimental results of Sierra White granite for which we have multiple runs that well bound the solutions and simulations.The WEST relations (Eqn.6, 7, 9) represent the empirical frictional history of SWG in the slip-velocity range of 10 -4 -1m/s.The application of the model to the experimental results shows that friction history can be captured by a numerical combination of the slip velocity and slip distance for both the weakening and strengthening stages.Specifically, in the dynamic-strengthening regime has the following properties: 1) It is a high-velocity regime, V > V C , e.g.V ~ 0.03-0.6m/s for Sierra White granite; 2) The WEST model uses a kinematic friction coefficient for the strengthening stage, which is derived independently of the weakening stage; 3) The friction coefficient history is well simulated by combining the kinematic friction coefficient for strengthening and the slip distance.
The WEST model predicts the rate-dependence of friction over the full range of observed natural seismic slip-rates, and in this sense it is applicable to earthquake simulations, and modeling of earthquake ruptures.Accumulating evidence supports the presence of dynamicstrengthening. Kaneko et al (2008) stated that "the velocity-strengthening region suppresses supershear propagation at the free surface occurring in the absence of such region, which could explain the lack of universally observed supershear rupture near the free surface".Hence, it is important to understand how the dynamic friction at seismic rates affects earthquake rupture dynamics by adopting new friction model in the numerical simulations.Although this is beyond the scope of the present study, the abundance of efforts on experimental studies on dynamic friction provides a realistic velocity-friction relation to explore the ground motions during earthquake rupture.

Figure 2 .
Figure 2. Friction-velocity relations in shear experiements of Blue quartzite (a), St. Cloud diorite (b), Fredricksburg syenite (c), and Karoo gabbro (d).Connected lines indicated data from step-velocities experiments.Unconnected dots indicate single velocity experiments.Note velocity-weakening and velocity-strengthening stages (marked by red arrows) in (a)-(c) and no velocity-strengthening in (d).

Figure 3 .
Figure 3.A schematic presentation of WEST model, with velocity controlled kinematic friction coeefcient.

Figure 10 .Figure 11 .
Figure 10.WEST model simulation and experimental results for SWG 240, SWG, σ N = 1.1 MPa Figs. 12 and 13 display the friction evolution in two experiments with wide velocity range.Fig.12displays a slide-hold-slide run under σ Ν = 1.1 MPa and hold times of 10 s between the multiple velocity steps.Under this short hold time, the friction coefficient curve displays quasicontinuous trend.There is a marked weakening as the velocity was increased to ~ 0.04 m/s, followed by a gentle strengthening at higher velocities.The modeling results were lower at beginning than expected and strengthening occurred earlier.Fig.13displays two major features: an initial gradual weakening in the slip velocity range of ~0.0003 m/s to a critical velocity of ~0.03 m/s, and a fast strengthening at velocities from ~0.03 m/s to 0.2 m/s.In the final stage, the friction reaches ~0.8.The modeling simulates the weakening-strengthening pattern, but it fails to follow the experimental results in the region faster than the critical velocity.In the experiment, the friction coefficient remained relatively low from ~3 m to ~11 m, where the simulated results predicted earlier strengthening.Also, an abrupt friction rise was observed in the experiment (at ~ 11 m) whereas the model indicated smooth and continuous strengthening.

Figure 12 .Figure 13 .
Figure 12.WEST model simulation and experimental results for SWG 531, SWG, σ N = 1.1MPa.Friction coefficient is shown for a full-velocity by interval holding in sliding distance together with a best fit modeling result

Figure 14 .
Figure 14.The Rotary Shear Apparatus with builder Joel Young.The sample block assembled in the loading frame (after Reches and Lockner (2010) Sammis et al. (2011)2010)umi and Shimamoto, 1997;Reches and Lockner, 2010;Di Toro et al., 2011).Second, the weakening-strengthening mode reflects a transition between different frictional mechanisms.Reches and Lockner (2010)andSammis et al. (2011)suggested that the weakening is controlled by gouge powder lubrication due to coating of the powder grains by a thin layer of water that is 2-3 monolayers thick.The dehydration of this layer at elevated temperature under high slip velocity leads to strengthening.It is thus assumed that the weakening and strengthening regimes have different parametric relation between friction and velocity.