Investigating Dynamic Weakening in Laboratory Faults Using Multi‐Scale Flash Heating Coupled With mm‐Scale Contact Evolution

Flash‐weakening models typically show good agreement with the total magnitude of weakening in high‐speed rock friction experiments, however deviations during the acceleration and deceleration phases, and at low and intermediate sliding velocities, remain unresolved. Here, we incorporate inhomogeneous mm‐scale normal stress evolution into a model for flash heating and weakening to resolve outstanding transient and hysteretic friction observed in laboratory experiments and to identify unique solutions to constitutive parameters. We conduced 37 rock friction experiments on Westerly granite using a high‐speed biaxial apparatus outfitted with a high‐speed infrared camera. We initiated velocity steps from quasi‐static rates of 1 mm/s to sliding velocities ranging from 300 to 900 mm/s and conducted both constant‐ and decreasing‐velocity tests following the velocity step. Two sliding surfaces geometries were used to control mm‐scale life‐times and rest‐times. Constant‐strength sliding is achieved within 2–3 mm of initiating the velocity step in all constant‐velocity experiments. Macroscopic surface temperature is inhomogeneous and increases with slip distance, velocity, and decreasing rest‐time. Weakening increases with sliding velocity and decreasing rest‐time. We combine thermal models with measured surface temperatures to constrain the evolution of local normal stress at the mm‐scale and incorporate this evolution into a flash‐weakening model that considers weakening at both the µm‐ and mm‐scale. The flash‐weakening model improves when the effects of mm‐scale wear processes are incorporated and multi‐scale weakening is considered, however some transient friction remains undescribed. Models will be advanced by further incorporating wear processes and by considering processes at the mm‐scale and above.

Plain Language Summary Natural fault roughness results in a true contact area much smaller than the apparent area and concentrated loads at microscopic asperity contacts.Rapid slip during earthquakes dramatically increases the temperature at these asperities and reduces the frictional strength of asperity contacts, dramatically reducing the frictional strength of the surface.Theoretical models for this frictional weakening show good overall agreement with data from laboratory experiments, though models do not predict some hysteretic frictional behavior during the mechanically significant acceleration and deceleration phases of sliding, nor do they capture some transient frictional behaviors observed in experiments.Here, we conduct experiments to resolve this outstanding frictional behavior.We use an infrared camera to measure surface temperature on laboratory fault surfaces during earthquake experiments and combine measured temperature data with mechanical data and thermal models to resolve the spatial distribution of temperature and normal stress on fault surfaces during sliding.We incorporate this evolution into a frictional weakening model that considers contacts at the micrometer asperity scale and the millimeter scale and shows improved model predictions, though some frictional behavior is still not described.Further advancements can be made by considering wear processes and frictional processes occurring over mm or longer length scales.
Experimental investigations of frictional sliding in rocks at seismic slip rates document reductions in steady-state friction coefficients consistent with the conventional flash-weakening model that considers µm-scale contact junctions (e.g., Beeler et al., 2008;Faulkner et al., 2011;Goldsby & Tullis, 2011;Kohli et al., 2011;Passelègue et al., 2016;Proctor et al., 2014;Yao, Ma, Platt, et al., 2016;Yuan & Prakash, 2008); however, observations of transient and hysteretic friction during the acceleration and deceleration phases are often not well described (e.g., Beeler et al., 2008;Rice, 2006).Model predictions are improved when changes in the macroscopic, average surface temperature with slip are included (Passelègue et al., 2014;Proctor et al., 2014), but observations of transient and hysteretic friction remain unexplained.The conventional flash-weakening model considers only sliding motion between adhesive, micrometer-scale asperity contacts.Other types of motion, such as rolling, and wear processes involving fracture, fragmentation, and plowing, as well as thermo-chemical processes also likely contribute to macroscopic friction behavior during high-speed slip (e.g., Boneh et al., 2013;Boneh & Reches, 2018;Boulton et al., 2017;Hirose et al., 2012).In the case of faulting in rocks, the scale-dependent roughness of fault surfaces suggests variation in strength and wear processes as a function of length scale (Brodsky et al., 2016;Candela & Brodsky, 2016;Thom et al., 2017;Yamashita et al., 2015).Considerations of wear processes and of the multiscale roughness of faults may then improve models of dynamic weakening by flash heating.
In work by Barbery et al. (2021), inhomogeneous temperature and normal stress at the mm-scale is documented during high-velocity friction experiments.A transition from few contacts with high local normal stress at the beginning of sliding to more numerous and larger contacts with low local normal stress at the end of sliding is documented and attributed to wear processes and the formation of gouge on the sliding surface.Incorporating this inhomogeneous evolution of temperature and normal stress at the mm-scale into a flash-weakening model would indirectly incorporate the effects of wear processes into the model.This may help resolve some of the outstand ing transient and hysteretic friction observed in laboratory experiments by providing improved constraints on the mm-scale contacts that host microscopic asperities during sliding.Furthermore, key material parameters, including asperity dimension, weakening temperature, and fully weakening coefficient of friction, are less well constrained than other model parameters (e.g., density and thermal diffusivity).A model incorporating the evolution of mm-scale contacts may provide better constraints on these key material properties.
Here, we use high-speed rock friction experiments in combination with numerical modeling to explore the effects of multi-scale roughness and wear processes on flash-heating and weakening.We conduct double-direct shear velocity-step experiments using several velocity histories and two mm-scale contact geometries.We compare

High-Speed Velocity-Step Experiments With High-Speed Thermography
Using a high-speed biaxial apparatus (HSB) outfitted with a high-speed infra-red (IR) camera, we conducted 37 rock friction experiments in a double-direct shear configuration on samples of Westerly granite.The HSB employs a pneumatic cylinder combined with a hydraulic damper to drive the piston rod and control the slip rate (Saber et al., 2016).A hydraulic cylinder equipped with a gas accumulator generates and maintains a constant normal stress over the sliding surfaces during experiments.The HSB performs velocity-steps from quasi-static sliding rates of 1 mm/s to target, high-velocity rates up to 1 m/s.After completing the velocity step and reaching a target velocity (V t ), either constant-velocity sliding or decreasing-velocity sliding is maintained for up to 3.5 cm of displacement (d).
In all experiments, sliding commenced at a velocity (V) of 1 mm/s.After approximately 5 mm of slip, the velocity step was initiated and a V t was achieved within 1-3 mm of displacement.High-velocity sliding was then maintained for 10-30 mm of high-velocity displacement (d hv ).Five target velocity and velocity path combinations were employed: constant velocity tests with V t of 900, 700, 600, and 300 mm/s in addition to decreasing velocity tests with initial V t of 700 mm/s (Table 1).Experiments were conducted at room temperature and room humidity.Forces and displacements were measured at a sampling rate of 25 kHz using a custom data acquisition software developed in LabVIEW™.Shear force was measured via a load cell in contact with the moving block, normal force was measured via a load cell in contact with a stationary block, and the sliding velocity was measured via a displacement transducer attached to the piston rod.All measurements are accurate to within 0.03% of the span.Post-processing employing a bypass filter was performed to remove high-frequency noise and extract the coefficient of friction (µ), normal stress (σ n ), velocity and displacement.
Thermographs of the moving block were captured during experiments via a high-speed IR camera.The camera images the moving block directly below the terminal contact with the stationary block via a window in the supporting steel block following procedures detailed in Barbery et al. (2021).Thermographs were captured at a frame rate of 300 Hz and a resolution of ∼75 μm to image an area approximately 14 mm parallel to the sliding direction and 18 mm perpendicular to the sliding direction.During a single experiment, consecutive thermographs image a total area 35 and 18 mm parallel and perpendicular to slip, respectively.This is equivalent to ∼16% of the moving block and stationary block contact area with ∼6% of the sliding surface imaged per individual thermograph.The IR camera is capable of measurements within pre-set, calibrated temperature ranges, several of which utilize a neutral density filter.Temperatures were measured using six different ranges, though most experiments utilize one of four primary ranges: 250-600, 100-300, 80-200, and 10-90°C (Table 1).All temperature ranges except 10-90°C utilize the neutral density filter.

Sample Preparation
Three rectangular blocks, two stationary blocks and one moving block, constitute the double-direct shear configuration that forms two sliding surfaces measuring 38.76 cm 2 each.Block faces were initially ground flat and square using a surface grinder equipped with a #60 grit precision grinding wheel.The bottom of the stationary blocks were ground to generate a 0.3° tilt of the stationary block away from the moving block to reduce torque generated on the sliding surfaces inherent to double-direct shear configurations and to achieve more uniform wear on the sliding surfaces during experiments (Dieterich & Linker, 1992).
To control the history of contacts at the mm-scale, we ground a series of regularly spaced, equally sized grooves oriented 22.5° counterclockwise from the sliding direction and ∼80 microns deep following procedures introduced by Barbery et al. (2021).Formation of the grooves generates flat-topped ridges that host all mm-scale contact during sliding and form uniformly distributed, diamond shaped contacts when the moving block adjoins the stationary blocks (Figure 1).Contact lifetimes (LT) occur when a ridge on the moving block opposes a ridge on the stationary block.Contact rest times (RT) occur when a ridge on the moving block opposes a groove on the stationary block.During sliding, mm-scale contacts migrate such that all ridge locations experience periods 10.1029/2023JB027110 of LT and RT, out of phase and cycle with adjacent locations on the same ridge.Contacts in the same position on adjacent ridges experience the same mm-scale contact history, and thus each ridge experiences the same mm-scale contact history.
We utilized two groove geometries in our experiments to generate two mm-scale contact histories.In the first geometry, 3.44 mm wide grooves are spaced 3.44 mm apart resulting in a surface geometry where LT = RT.In this geometry, the total maximum contact area is equal to 25% of the total sliding interface area, or approximately 9.69 cm 2 .In the second geometry, 1.72 mm wide grooves are spaced 3.44 mm apart resulting in a geometry where LT = 2RT, or LT > RT.For this geometry, the total maximum contact area is equal to 44% of the total sliding area, or approximately 17.22 cm 2 .To ensure the maximum mm-scale contact area is equal for both geometries, two grooves oriented parallel to the sliding direction were ground on the outer edges of stationary blocks with the LT > RT surface geometry (Figure 1).As flat-topped ridges are 3.44 mm wide in both geometries, LT are consistent between the two geometries and equal to 8.3 mm of displacement parallel to the sliding direction.
Given the maximum high-velocity displacement of 30 mm, points on ridges of the moving block in the LT = RT geometry will undergo a maximum of 1.8 LT-RT cycles during high-velocity sliding.In contrast, points on the ridges of the moving block in the LT > RT geometry will undergo a maximum of 2.4 LT-RT cycles during high-velocity sliding.Ridge locations that do not undergo the maximum number of cycles during sliding develop as (a) upper sections of the moving block initially out of contact with the stationary block enter contact during sliding and (b) lower sections of the moving block initially in contact with the stationary block emerge from under the base of the stationary block during sliding.The grooved geometry of the moving and stationary blocks allows for precise, detailed determination of the mm-scale LT-RT history for all points on the moving block during sliding.
Prior to conducting experiments, newly prepared sample blocks were run in for ∼70 mm of displacement, accomplished over the course of two consecutive experiments, to remove grinding features and to generate wear and a self-organized roughness on the sliding surface.The sliding surfaces were brushed lightly with a camel hair brush between successive experiments to remove any loose, disaggregated gouge while leaving behind any compacted gouge.For each velocity condition, a set of blocks were re-ground completely (both surface and grooves) for each geometry and run-in before re-commencing experimentation.Four block sets were used in total for all experiments.).Contact occurs only when ridges on the moving block juxtapose ridges on the stationary block, illustrated here by white diamonds.The interface area, illustrated by black dashed lines, is reduced in the LT > RT geometry to ensure a consistent mm-scale contact area across both geometries.The total imaged area captured by the high-speed infrared camera is illustrated with blue boxes in both geometries.Modified from Barbery et al. (2021). 10.1029/2023JB027110 6 of 22

Mechanical Results
The mechanical results from eight representative experiments demonstrate the typical evolution of V, µ, and σ n as a function of displacement (Figure 2).Experiment conditions and average mechanical results from all experiments are reported in Table 1.The target velocity is achieved within 1-3 mm of displacement following the velocity step.Normal stress is maintained within ±10% of 9 MPa during sliding.In constant velocity experiments, velocity is maintained within ±10% of V t .For experiments with sliding velocities of 900 mm/s, slip is arrested by a mechanical stop resulting in an almost instantaneous drop in friction unrelated to sample processes.This drop is removed from the data in our later modeling and the restrengthening phase is not available for these experiments.Plots for the mechanical data from additional experiments can be found in Supporting Information S1.
The µ weakens by as much as 50% from the quasi-static value to a dynamic, steady-state friction coefficient (µ d ) within 3 mm of displacement following the velocity step.After this initial weakening, µ d increases with increasing displacement in decreasing velocity experiments.In constant velocity experiments, a constant-strength µ d is maintained following the initial weakening, reproducible to within 4% of the average for each RT and target velocity combination.The magnitude of µ d is dependent on both target velocity and RT; µ d decreases as velocity increases and as RT decreases.From experiments with the same velocity condition and RT history, we calculated the mean and the standard deviation from the mean for the coefficient of friction during sliding and present representative results in Figure 3.For experiments with a constant velocity of 900 mm/s, the average µ d when LT = RT is 0.36 whereas this is reduced to 0.29 when LT > RT (Figure 3a).As velocity reduces to 600 mm/s, the average µ d decreases from 0.38 to 0.35 for LT = RT and LT > RT, respectively.At the lowest velocity of 300 mm/s, the maximum average µ d of 0.46 is observed when LT = RT, which is reduced to 0.42 when LT > RT (Figure 3b).
Audible stick slip events occur in all experiments during quasi-static sliding prior to initiating the velocity step.These events are represented by small drops in µ accompanied by small increases in the sliding velocity.Due to dynamics of the HSB, the maximum amount of high-velocity sliding decreases as velocity decreases.In experiments with target velocities less than 900 mm/s, stick slip events reoccur following the end of high-velocity sliding.In constant velocity experiments, these stick slip events show lower quasi-static coefficient of friction, higher frequency, and lower stress drop per event when compared with stick-slip events preceding high-velocity sliding.As stick-slip events continue for up to 20 mm of displacement, the quasi-static µ increases approaching values preceding high-velocity sliding.Similar behavior is observed in decreasing velocity experiments, however the first two stick-slip events following the end of high-velocity sliding typically show larger stress drops over longer durations than events preceding high-velocity sliding.

Thermography
Experiments can be assigned to five groups based on the velocity conditions (Table 1).For both RT geometries examined, experiments were repeated across 3-5 IR temperature sensing ranges to ensure the full spectrum of elevated surface temperature during all stages of sliding and cooling were measured for each group.A total of 2-16 thermographs were captured per experiment, with 361 thermographs captured in total across 37 experiments.
Following procedures detailed by Barbery et al. (2021), thermograph dimensions were used in combination with the total displacement to determine the amount of displacement and high-velocity displacement at the time of capture for all thermographs.Composite thermographs were constructed by stitching together the maximum surface temperatures recorded for each ridge pixel in an experiment.Consistent with results from Barbery et al. (2021), temperatures across the sliding surfaces are inhomogeneous and the spatial distribution of highs and lows in surface temperature varies from ridge to ridge (Figure 4).Surface temperature distributions in repeat experiments with different IR temperature ranges show good overall agreement (Figure 5).Surface temperature increases with cumulative high-velocity displacement.Average surface temperature and peak surface temperature increase as RT decreases and velocity increases.In constant velocity experiments with velocities of 900 mm/s, average surface temperature after 30 mm of high-velocity displacement is 126.5°C when LT > RT and 99.3°C when LT = RT.As velocity decreases to 700 mm/s, average surface temperature at 20 mm of high-velocity displacement is 90.0°C when LT > RT and 52.5°C when LT = RT (Figure 5).At the lowest velocity of 300 mm/s, average surface temperature following 15 mm of high-velocity displacement is 38.9°C when LT > RT and 31.3°Cwhen LT = RT.High temperature hotspots (∼200°C) are typically elongate in the slip direction and are approximately 2-4 mm in length parallel to slip and 1-2 mm in width perpendicular to slip.In the early stages of sliding, after one LT-RT cycle or less, high temperature hotspots have aspect ratios of ∼2-3.As sliding continues, the aspect ratio decreases approaching 1 at the end of sliding (Figure 4).These observations are consistent with findings by Barbery et al. (2021) who suggest this transition reflects wear processes and gouge accumulation on the sliding surfaces.Narrow contacts during the early stages of sliding are subject to locally high normal stresses, leading to locally high wear rates and a transition to wider contacts supporting lower local normal stresses by the end of sliding.Very high temperature hotspots (>300°C) are typically equidimensional and ∼1 mm in diameter (Figure 5), though high-temperature contacts with diameters as low 0.225 mm do occur (contacts are The lines represent the statistical mean calculated using all experiments at the given velocity and RT combination (i.e., velocity group denoted in Table 1) and the shaded region is one standard deviation from the mean for the same set of experiments.As RT decreases and sliding velocity increases, average µ decreases.
distinguishable down to diameters of ∼0.15 mm or the equivalent of 2 pixels).The highest surface temperatures are typically recorded in the last 5-10 mm of d hv , representing contacts that experience more than one LT-RT cycle during high-velocity sliding.The development ∼1 mm diameter hotspots is consistent with hotspots documented by Saber (2017) in experiments on flat sliding surfaces with similar velocities, normal stresses, and total displacements suggesting hotspot characteristics are not a function of the sliding surface geometry.

Flash-Weakening Coupled With Evolving Contact-Scale Normal Stress
Our experiments document a dependence of surface temperature and of frictional response on sliding velocity in addition to mm-scale contact history governed by the ridge and groove surface geometry (i.e., RT history).High temperature, mm-scale hot spots develop in all experiments indicative of localized normal stress at the mm-scale.
Here, we examine distributions of local normal stress (σ l ) informed by measured temperature distributions to evaluate any variations with cumulative slip due to velocity history or mm-scale contact history.We then incorporate the distribution and evolution of σ l into a model for flash heating and weakening and solve for constitutive parameters for Westerly granite.

Evolution of Local Normal Stress With Displacement
Our sliding surface geometries ensure that points on ridges of the moving blocks undergo repeating cycles of LT and RT, out of phase and total number of cycles with neighboring ridge locations.Points in the same location on adjacent ridges undergo the same mm-scale contact history, and each ridge experiences the same mm-scale LT-RT history for a given RT geometry.For modeling purposes then, we identify a minimum model area that covers the total displacement in experiments for a single ridge.The model area comprises 544 discrete 1 mm 2 elements.The unique LT-RT history for each element is precisely calculated for both sliding surface geometries to model local surface temperature (T l ) coupled with a local coefficient of friction (µ l ) following procedures detailed in Barbery et al. (2021).To model T l , we employ a 1-D heat conduction equation (e.g., Carslaw & Jaeger, 1959;Proctor et al., 2014) incorporating thermal properties of Westerly granite at room temperature and known LT-RT history for each discrete model element.Local surface temperature is calculated by: where T 0 is the ambient, initial surface temperature (20°C), t is time, Q l is a local heat generation rate, ρ is the density, c p is the specific heat capacity, and α th is the thermal diffusivity.The material properties ρ, c p , and α th , are fairly well constrained for most rock types (e.g., Eppelbaum et al., 2014).A density of 2,700 kg m −3 is assumed, and the specific heat capacity and thermal diffusivity used for Westerly granite are 800 J kg −1 K −1 and 1.25 × 10 −6 m 2 s −1 , respectively (Goldsby & Tullis, 2011).The local heat generation rate is determined by: where V is taken directly from the measured data and σ l is assumed to be constant.The local, temperature dependent coefficient of friction is defined according to the flash-weakening relation: where µ qs is the quasi-static friction coefficient and μ w is the fully weakened friction coefficient.We use a µ qs of 0.76, obtained by averaging the mechanical results from all experiments during the initial quasi-static sliding, and a value for μ w of 0.215 from Barbery et al. (2021).In low velocity and decreasing velocity experiments, the  437, 436, 434. (c) 452, 451, 450, 449. (d) 455, 454, 453, 447.coefficient of friction is slightly reduced after the end of high-velocity sliding in some experiments (Figure 2), however in many experiments there is no significant change in friction after high-velocity sliding ends.Here, we idealize µ qs and assume it is constant.V w is the critical weakening velocity above which flash-weakening will occur, defined by: where C is a single constant representing all micrometer-scale contact junctions and material properties.For analysis of local normal stress distributions with slip, we assume C is equal to 2.58 × 10 −7 m s −1 °C −2 and T w is the breakdown temperature of 900°C, both of which are taken from Rice (2006) and largely consistent with values for Westerly Granite used by others (e.g., Goldsby & Tullis, 2011).
Local (mm-scale) surface temperature is assumed to remain at ambient temperature prior to initiating the velocity step.This assumption likely underestimates surface temperature in some experiments with large stick slip events during quasi-static sliding (i.e., events with displacements greater than ∼2 mm), however there is sufficient time following stick slip events for elevated surface temperature to return to ambient temperature making this assumption reasonable.During high-velocity sliding, T l is calculated over 300 discrete time steps for each element.
During each time step, the weakening velocity and a local friction coefficient are calculated using Equations 3 and 4 for the current T l .The heat generation rate is then calculated using Equation 2, and then T l for the next time step is calculated using Equation 1.A logic diagram for the model is provided in Supporting Information S1.
Surface temperatures were modeled for 14 σ l magnitudes ranging from 1 to 40 times the applied σ n (9 MPa) and used to produce 14 synthetic thermographs for each experiment.We modeled σ l of: 1, 2, 4,6,8,10,12,16,20,24,28,32,36, and 40 times σ n .Synthetic thermographs were compared to captured thermographs and used as a basis to contour σ l during sliding.Representative contoured thermographs from two constant velocity experiments with V t of 900 mm/s show contours for local normal stress factors of 4, 8, and 16, representing local normal stress magnitudes of 4, 8, and 16 times the applied σ n (Figure 6).Consistent with observations by Barbery et al. (2021), high-temperature hotspots correlate with high σ l and there is a transition from highly stressed contacts in the early stages of sliding to low stressed contacts in the final stages of sliding.Likewise, gradients for temperature and local normal stress at mm-scale hot spots decrease with progressive slip.
To evaluate any variation in σ l distribution with slip due to velocity history or RT history, thermographs from 14 experiments encompassing each unique RT and V combination (denoted by a in Table 1) were contoured for σ l .The fractional area for each σ l range was calculated by counting the total number of pixels within a set range and dividing by the total number of pixels within the known ridge area in each thermograph.While calculating fractional areas, any incompletely mapped σ l ranges were disregarded to prevent underestimations of area percent due to local element temperatures above or below the IR measurable temperature limits.Subsequently, only fully imaged σ l ranges were included in calculations, that is, ranges bounded everywhere in a thermograph by closed contours or open contours terminating at either ridge boundaries or image boundaries.
For each σ l range, changes in area percent were examined as a function of high-velocity displacement (Figure 7).The σ l distributions evolve approximately linearly with displacement though there is significant scatter and we idealize this relationship as linear.This scatter likely reflects limitations in our thermography as each thermograph represents only 6% of the sliding surface but is assumed to represent the full sliding surface.There may also be real, subtle effects related to either RT history or velocity history, though there are no clear trends related to either.At σ l less than 6 times σ n , area percent increases as a function of displacement (Figure 7a).At σ l greater than or equal to 6 times σ n , area percent decreases as a function of displacement (Figure 7b).These findings are consistent with results by Barbery et al. (2021).We performed simple linear regressions analysis for each range of σ l contingent on the condition that the initial fractional area is greater than 0.0 for all ranges of σ l .The linear regressions for each σ l range were combined to model σ n for comparison with measured σ n .There is excellent agreement between the modeled and measured macroscopic normal stress (Figure 7c).Plots similar to Figures 7a and 7b showing different ranges of σ l are available in Supporting Information S1, as are more plots for modeled σ n similar to plot 7c.
Model results for T l for each σ l range were combined with the linear regression results to generate a synthetic thermograph (Figure 8).When generating synthetic distributions, pixels in each row are randomly assigned a σ l value from a distribution that evolves with displacement based on our measurements for changes in σ l during sliding.Synthetic temperatures are assigned based on σ l values and modeled T l for each σ l range.As each pixel is modeled independently from neighboring pixels, we do not reproduce temperature clusters and hotspots observed in the measured data, however there is good overall agreement between measured and modeled temperature distributions (Figure 8c).Within the IR measured temperature range of 100-300°C, the average synthetic surface temperature is 150.6°C and the average measured surface temperature is 149.9°C.There is good overall agreement between these measured and modeled data, suggesting the evolution of local normal stress at the mm-scale is reasonably well characterized by our thermal model.

Multi-Scale Flash-Weakening With Inhomogeneous Normal Stress
Here, we incorporate the evolution of local normal stress at the mm-scale detailed in the previous section into a model for flash-weakening.However, we deviate from previous model treatment regarding certain model parameters (μ qs , μ w , T w , D a , and τ c ) in addition to the scale at which flash-weakening occurs.
We observe stick slip events in all our experiments, both preceding and succeeding high-velocity sliding, with typical reductions of 0.1-0.2 in the coefficient of friction.Such rate weakening behavior suggests a lower μ qs during high-velocity sliding.This would reduce the local friction coefficient and heat generation rate during sliding.Without incorporating a rate-weakening μ qs , we may overestimate both the local surface temperature and the magnitude of weakening in our experiments.To address this, we incorporated the following rate-and-state friction formulation (Dieterich, 1979;Ruina, 1983) into our model: where μ 0 is a reference coefficient of friction measured at a sliding velocity V 0 and the parameters a and b are proportionality constants representing the direct and evolution effects, respectively.We use 0.76 for μ 0 based on the average, maximum friction coefficient during low velocity sliding and 1 mm/s for V 0 , corresponding to the average low velocity sliding rate.We use an empirically derived value of −0.017 for a-b.To solve for a-b, we  calculated the average coefficient of friction during stick slip events, μ ss , and the average stick slip velocity, V ss , for each experiment (Table 1).We averaged the values for V ss and μ ss , equal to 60 mm/s and 0.69 respectively, and inserted them into Equation 5 as V and μ qs along with the values of V 0 and μ 0 to calculate a-b.Two experiments, 438 and 440, with V ss exceeding 100 mm/s were excluded from these calculations to minimize any effects due to flash-weakening which has been documented in experiments with sliding velocities as low as 100 mm/s (Saber, 2017).
Conventional treatments for flash-weakening consider weakening at the microscopic asperity contact scale only.Beeler et al. (2008) explored the effects of varying asperity contact size on flash-weakening and suggest that weakening velocity decreases with asperity size.In the flash-weakening model, V w is calculated using: where D a is the asperity contact diameter and τ c is the asperity shear strength (Rice, 2006).If we consider typical estimates for D a of 5 μm and τ c of 3 GPa (Rice, 2006), the denominator in the flash-weakening calculation  At a sliding velocity of 500 mm/s and with a thermal diffusivity of 0.5 mm 2 /s, 5 μm and 1 mm sized quartz or feldspar contacts would have thermal diffusion lengths of 2 and 31 μm, respectively.For microscopic asperities, this rise occurs over a fraction of the host rock grain size and the weakening temperature is high.If we consider the longer thermal diffusion length of macroscopic contacts however, the temperature rise would occur over a significant portion of the gouge thickness if wear product is present.As gouge particles are typically microns to sub-micron in size with high surface energy, and likely with high intragranular crystalline disorder, macroscopic asperity contacts may weaken at lower temperatures than their microscopic counterparts due to gouge processes like structural superplasticity (De Paola et al., 2015;Green et al., 2015) and nanograin rolling (Chen et al., 2017a;Han et al., 2011).Weakening temperature may then decrease with contact size.This could allow for a lower coefficient of friction in highly stressed, mm-scale contacts.Combined with longer contact lifetimes of mm-scale contacts, reduced contact strength would lead to lower weakening temperatures and less weakening during the acceleration phase while continued weakening during the remainder of sliding may generate hysteretic behaviors observed in experiments.We do observe wear product on the sliding surfaces in all experiments and expect higher wear rates, favoring generation of wear product, at hot spots with high σ l in line with expectations from Archard's wear law (Archard, 1953) and from recent investigations by Acosta et al. (2020).Indeed, in experiments with similar sliding rates, macroscopic normal stress, and displacements conducted by Saber ( 2017) on initially flat samples, gouge layers 100-200 μm in thickness on the sliding surface were documented associated with hot spots identified in thermography.
To explore multi-scale flash-weakening, we model weakening at both the µm-and mm-scales and incorporate the evolution of σ l documented in the previous section into our model.To accomplish this, we discretized the sliding surface into four regions with each region representing 25% of the ridge sliding surface area, or equivalent to 6.25% of the total sliding surface area (Figure 9a).For regions 1-3, corresponding to the lower stress regions, σ l is less than ∼64 MPa and we consider flash-weakening only at the microscopic scale.For these regions we calculated a moving average for σ l as a function of d hv using the results from our linear regressions analysis described in the previous section (Figure 9b).This value represents the background mm-scale local normal stress with much higher normal stress expected at the asperity scale.
For region 4, our highest stress region, we consider flash-weakening at both the microscopic and macroscopic scales.For elements with σ l of ∼108 MPa or higher, equivalent to 12σ n or higher, we consider weakening at the macroscopic scale only.For elements with σ l less than ∼108 MPa, we consider flash-weakening at the microscopic scale only.For both scales, we calculated moving averages for local normal stress with displacement in addition to changes in area percent with displacement (Figures 9b and 9c).The sliding surface was then divided into five model areas: four areas where weakening occurs at the µm-scale only and one area where weakening occurs at the mm-scale only.
To model multi-scale flash-weakening, we follow modeling procedures similar to those described in the previous section, however we model each of the five areas independently.For each area, we calculate surface temperature for each model element over n discrete time steps, where n is equal to the total number of measured data points per experiment.For µm-scale weakening areas, we use Equation 1.For the mm-scale weakening area, local surface temperature at the sliding interface is calculated via: where W is the half-width of the gouge zone (Proctor et al., 2014).We use a half-width of 150 μm based on observations by Saber (2017) in experiments with the same macroscopic normal stress, displacement, and sliding velocities.During each time step, the local heat generation rate is calculated using Equation 2where the σ l is assigned based on the calculated moving averages for each model area.A μ l for each area is calculated using Equation 3, however μ w is no longer prescribed and instead becomes a fitting parameter.To calculate μ l , V w is computed, at both the mm-scale and um-scale, following: where D c is the contact diameter and τ c is the contact shear strength (Rice, 2006).We assume the thermal properties (c and α th ) do not change with scale or with temperature, though this is not necessarily a realistic view.
Changes in thermal properties with temperature have been explored by Nielsen et al. (2021) and they could play an important role in dynamic frictional processes.Additionally, large flaws and increased porosity of gouge layers at mm-scale contacts could result in density changes and subsequently changes in thermal diffusivity.Here though, we idealize the thermal properties and density and assume neither change with scale or temperature.
For the mm-scale flash-weakening area, we use a   mm of 1 mm based on typical hotspot size observed in thermography.For the µm-scale weakening areas,   m becomes a fitting parameter.Likewise, T w is a fitting parameter at both the micrometer and millimeter scale represented by   m and   mm , respectfully.We assume single, constant values for both D c and T w at both modeled scales.While variations in contact sizes at both scales are expected, these constant, scale-dependent values represent the average contact diameter and average weakening temperature for each set of contact populations.We assume that changes in contact dimension will engender changes to the contact shear strength, so we couple τ c with D c using: thereby keeping the product of τ c and D c proportional to the product of the same parameters used by Rice (2006): 3 GPa and 5 μm, respectively.As a result, τ c will decrease with increasing D c to incorporate a scale dependence of rock strength (Adey & Pusch, 1999;Yamashita et al., 2015).As with D c and T w , τ c will be a single, constant value at each modeled scale representing the average contact shear strength for each set of contact populations.Finally, the macroscopic coefficient of friction is calculated using: where i represents the ith area, a i is the fractional area percent of the ith area, µ li is the local friction coefficient for the ith area, and σ li is the local normal stress for the ith area.A logic diagram for this model is provided in Supporting Information S1.Following this modeling procedure, we utilize four total fitting parameters: μ w ,   m ,   m , and   mm .We consider a μ w of 0.15-0.45,with divisions of 0.05, constant during sliding and with scale.For µm-scale weakening, we explore   m of 2-7 μm with divisions of 1 μm and   m of 750-100°C with divisions of 50°C.For mm-scale weakening, we explore   mm of 200-300°C with divisions of 50°C.To identify the best fitting parameter set, we modeled eight experiments (denoted by b in Table 1) using over 500 unique parameter combinations.Each modeled experiment represents a unique combination of sliding surface geometry and velocity history.The parameter spaces for   m and   m were iteratively expanded until the global, best fitting values to all eight of the modeled experiments were bound twice over by the upper and lower limits of the range.For μ w , the lower limit was expanded following the same procedures as   m and   m , but the upper limit was set to 0.45, which is more than one standard deviation above the mean friction coefficient from all 37 experiments during high-velocity sliding calculated from the mechanical data reported in Table 1.For   mm , the lower limit of 200°C was set based on the typical temperature of 1 mm 2 hotspots observed in thermography and the upper limit was again iteratively expanded until the best fitting global parameter value was bounded twice over by the upper limit.
During experiments, the largest transient and hysteretic friction occurs during the acceleration and deceleration phases.Correctly modeling and describing these periods of slip in addition to the steady-state sliding phase is essential to accurately model weakening and strengthening due to flash-weakening, however the flash-weakening model is a steady-state model.We therefore evaluated model fits to three phases (the acceleration, deceleration, and steady-state sliding) in addition to the entire data set independently.Here, we focus on the steady-state sliding phase as this is the most appropriate phase to evaluate model fits.Fits for the two other phases and to the entire data set are presented in Supporting Information S1.
To evaluate the goodness of fit for model results, we calculated residuals between the modeled and measured data for each time step (i.e., the difference between the measured data and the modeled data).We then calculated the mean squared error of the model residuals, equal to the average of the squared residuals.The best fitting model for each experiment is the model with the lowest mean squared error.To identify a single, global parameter set common to all eight modeled experiments, we iteratively removed the worst fitting parameter set until only one parameter set was common to all experiments and this parameter set was identified as the best fitting, global parameter set.This process was performed for (a) a conventional flash-weakening model with homogenous σ n stress (hereafter the conventional model), (b) a single-scale, inhomogeneous σ n flash-weakening model with weakening only at the asperity scale (hereafter the single-scale model), and (c) a multi-scale, inhomogeneous σ n flash-weakening model (hereafter the multi-scale model).To compare the performance of each model, we calculated Akaike information criterion (AIC) values using the global, best fitting parameter sets to each model type following Yang (2019): where k is the number of model parameters, n is the number of data points, and RSS is the sum of the squared model residuals.The best fitting model is the model with the lowest AIC value and this metric includes a penalty for additional model parameters by incorporating the 2k term.
Comparisons between the measured data and model data for the multi-scale model, single-scale model, and conventional model are shown in Figure 10.AIC values for the best fitting global parameter sets to each model Notably, the fully weakened friction coefficient predicted by the conventional and single-scale model is 0.45, higher than the average dynamic friction coefficient for all constant-velocity experiments, 0.38, with two experiments having average dynamic friction coefficients as low as 0.28 (Table 1).In contrast, the best fitting fully weakened friction coefficient to the multi-scale model is 0.25, below the average dynamic friction coefficient for all experiments.This is quite reasonable when you consider the forced cooling related to our mm-scale RT geometries.The fully weakened friction coefficient can only be reached when the weakening velocity goes to zero, which requires a local temperature equal to the weakening temperature.As we have cooled ridge sections constantly coming into contact during sliding, this is unlikely to be achieved in our experiments.The predicted values of   m and   m between the three model types do not vary significantly.
Though the multi-scale model performs better, none of the model fits are ideal using a global parameter set.All models overestimate the rate of weakening during the acceleration phase and the rate of strengthening during the deceleration phase, though the fits are improved in the multi-scale model compared to a single-scale model (Figures 10b and 10c) or a conventional model (Figures 10d and 10e).Notably, the multi-scale model generates large, spontaneous stick slip events that occur in the decreasing velocity experiments after the end of high-velocity sliding (Figures 10b and 10c).Though we overestimate the magnitude of weakening in these events, they are not predicted beyond rate-and-state weakening by the single-scale model (Figures 10b and 10c) nor by a conventional model.At sliding velocities of 300 mm/s, the magnitude of weakening is underestimated (Figures 10d and 10e).At velocities of 600 mm/s or greater, the magnitude of weakening is overestimated for LT = RT experiments and the rate of weakening is increasingly overestimated with larger V t (Figures 10f and 10g).There is much better overall agreement between model results and the mechanical data using non-global, best fitting parameter sets to individual experiments (Figures 10f and 10g).

Discussion
Experimental investigations of flash-weakening generally utilize rock samples with surfaces ground flat or lined with fine-grained natural or synthetic gouge, so contact sizes are likely similar to those assumed in theoretical models of flash-weakening and suggested by our model results (i.e., several micrometers in length); however, natural faults surfaces display direction dependent, anisotropic roughness that increases with decreasing length scale (Brodsky et al., 2016;Candela & Brodsky, 2016;Power & Tullis, 1991;Scholz & Aviles, 1986).Microstructural investigations of natural fault surfaces suggest that these roughness characteristics reflect a scale dependence of the yield strength of contacts and that different yielding processes operate at different length scales during dynamic slip (Brodsky et al., 2016;Candela & Brodsky, 2016;Thom et al., 2017).Here, we present a model for flash heating and weakening that considers multi-scale flash-weakening and indirectly incorporates wear processes via the evolution of local normal stress at the mm-scale.Though our model shows significant improvements relative to a conventional µm-scale only flash-weakening model, transient and hysteretic friction observed in our experiments is not fully described and the rate of weakening is consistently overestimated.Additionally, when comparing model and measured data using a single, unique parameter set, the magnitude of weakening for experiments with shorter RT is consistently overestimated.When using a single set of constitutive parameters to describe and model a wide range of experiment conditions, some outstanding transient and hysteretic weakening is to be expected, however the model fits are not presently sufficient to definitively establish these parameters as representative for Westerly granite.Incorporating healing processes related to chemical bonding at asperity contacts (Bedford et al., 2023) or water related healing (Violay et al., 2019) into the restrengthening phases of sliding may improve fits during the deceleration phase.Additionally, descriptions of frictional behavior during all phases of sliding may be improved by (a) further incorporating wear processes, or the effects of wear processes, into constitutive equations for dynamic weakening including flash heating and weakening and (b) considering dynamic weakening mechanisms occurring at the mm-scale or larger.
Steady-state sliding is observed in all constant velocity experiments, and the coefficient of friction decreases as mm-scale RT decreases.During the early phases of our experiments, prior to steady-state, the wear rate is expected to be highest (Boneh & Reches, 2018) with fresh gouge and wear product developing on the sliding surfaces due to moderate to severe predicted wear rates (Reches & Lockner, 2010).The reduction in friction due to decreasing RT, holding all other conditions equal, suggests that wear rate is dependent on local, contact scale surface temperature in addition to previously documented dependencies on sliding velocity and macroscopic normal stress (Boneh et al., 2013).In other words, decreasing RT results in increased surface temperature and more weakening when all other experiment conditions are equal.This indicates that wear rate changes with surface temperature.Incorporating the effects of wear processes at the mm-scale into flash-weakening constitutive equations, in our case via evolving local normal stress at the mm-scale, improves model results.Further constraints on wear processes across all roughness scales may help resolve the outstanding transient and hysteretic friction observed in experiments.Weakening at velocities below the critical weakening velocity for flash-weakening are consistently observed, in combination with slower weakening during the acceleration phase than predicted by the flash-weakening model (Figure 10).Deviations from model predictions may relate to an energy sink associated with wear product surface energy and extreme comminution.Assuming a gouge layer 150 μm thick as documented by Saber (2017) in experiments with similar normal stress, total displacement, and sliding velocities, with an average gouge grain size of 0.1 μm, measured in similar experiments on granite (e.g., Chen et al., 2017aChen et al., , 2017b)), the work done to produce gouge can be estimated following Scholz (1987) where the gouge surface energy, E s , is function of the surface area, a, the specific surface energy, γ, (assumed from Brace and Walsh (1962) to be ∼1 J m −2 ), the thickness of the gouge layer, t g , and the gouge diameter, D g , by E s = 6aγt g / D g .The resulting gouge surface energy is ∼35 J.This value is on par with the frictional work during the initial acceleration phase (i.e., the first ∼3-4 mm of d hv ) where the work done by friction is the product of the force of friction and displacement, ∼44 J in our experiments.Microstructural investigations of deformed samples at different stages of sliding will help to characterize deformation features, identify active wear processes, and provide constraints on the amount of work done to generate gouge on the sliding surface and is a focus of our ongoing work.
Considering the effects of mm-scale processes on flash-weakening, and the well-documented scaling roughness of faults attributed to scaling yield strength, increased focus on processes at the mm-scale and longer may also help resolve frictional behavior in experiments.In our experiments with sliding rates of 300 mm/s, we underestimate the magnitude of weakening for both RT geometries when a single, unique parameter set is used.This could be the result of a secondary weakening mechanism, concurrent with flash-weakening, acting at the mm-or longer scale.At sliding rates of 300 mm/s, the velocity is close to the critical weakening velocity and the reduction in friction due to flash heating is diminished relative to faster sliding rates.Consequently, a secondary weakening process may contribute more to the total reduction in friction at lower velocities that is masked at higher velocities where flash heating greatly reduces the coefficient of friction.If this secondary mechanism is a thermal weakening mechanism, it would also contribute more as RT decreases.This could explain our model results at higher sliding rates as well.For the unique, best-fitting parameter set, weakening at larger RT (LT = RT) is typically overestimated.If a secondary weakening mechanism is active, it will contribute more in shorter RT experiments (LT > RT).To fit both RT conditions with a single, unique parameter set, weakening will be either overestimated for longer RT or underestimated for shorter RT, with the former matching our observations.While we do identify unique parameter values for asperity dimension, fully weakened coefficient of friction, and weakening temperatures at both the µm and mm scales, some uncertainty remains without further resolving the transient and hysteretic friction observed in experiments due to either wear processes or secondary weakening processes, or a combination of both.
In high-speed rock friction experiments by Saber (2017) a gradual, temperature dependent mm-scale weakening process was coupled with the flash-weakening model to generate good fits between mechanical data and modeled data, however model solutions were non-unique without well constrained mm-scale contact history and heat generation rates, and no weakening mechanism was identified.Considering potential secondary weakening mechanisms in our experiments, several proposed dynamic weakening mechanisms are unlikely to be activated here.Our experiments are conducted on room dry samples meaning water content is negligible, so thermal pressurization and other pore fluid related weakening processes are doubtful.Other proposed weakening mechanisms including melt lubrication and silica gel formation are also unlikely to contribute meaningfully to weakening in our experiments as surface temperatures do not exceed 400°C and experiments are performed at room pressure.
Considering the significance of surface temperature and wear rate on observed weakening, wear-related weakening processes could contribute to a longer length-scale weakening mechanism.Dynamic weakening related to nanoparticle formation has been suggested in several high-speed rock friction experiments over millimeters-long and meters-long length scales (Chen et al., 2017a;De Paola et al., 2011;Han et al., 2010Han et al., , 2011;;Hirose & Bystricky, 2007;Reches & Lockner, 2010;Rowe et al., 2019).Theoretical mechanisms proposed to account for nanoparticle driven weakening include powder lubrication (Wornyoh et al., 2007), rapid superplastic deformation of fine-grained materials (De Paola et al., 2011;Han et al., 2010Han et al., , 2011;;Hirose & Bystricky, 2007;Reches & Lockner, 2010), and structural superplasticity (Green et al., 2015).Detailed characterization of deformed samples, including the presence of nanoparticles and the chemical composition of wear products, will be necessary to evaluate any contribution to weakening due to nanoparticles.

Conclusions
Using results from 37 double-direct shear, high-velocity friction experiments, we characterized the spatiotemporal evolution of local normal stress at the mm-scale during sliding.We then incorporated this evolution into a model for flash heating and weakening that considers weakening at both the µm-and mm-scale.We tested over 500 unique constitutive parameter combinations and identified the best fitting, non-global parameter sets to individual experiments in addition to a global, best fitting parameter set for Westerly granite.While our multi-scale flash-weakening model incorporating inhomogeneous contact-scale normal stress resolves some outstanding transient and hysteretic friction observed in laboratory experiments, weakening is still not fully described and uncertainty regarding Westerly granite constitutive parameter values remains.Dynamic weakening models will be advanced by further incorporating wear processes and by considering processes acting over the mm-scale and above.

Figure 1 .
Figure1.Initial contact distributions for the two geometries examined: LT = RT and LT > RT (LT = 2RT).Contact occurs only when ridges on the moving block juxtapose ridges on the stationary block, illustrated here by white diamonds.The interface area, illustrated by black dashed lines, is reduced in the LT > RT geometry to ensure a consistent mm-scale contact area across both geometries.The total imaged area captured by the high-speed infrared camera is illustrated with blue boxes in both geometries.Modified fromBarbery et al. (2021).

Figure 2 .
Figure 2. Representative mechanical data from eight experiments.(a-c) Velocity, coefficient of friction, and normal stress versus displacement for six representative constant velocity experiments with target velocities of 300 mm/s (yellow lines), 600 mm/s (brown lines), and 900 mm/s (blue lines).(d-f) Velocity, coefficient of friction, and normal stress versus displacement for two representative decreasing velocity experiments with target velocities of 700 mm/s.LT = RT experiments are plotted with solid lines and LT > RT experiments are plotted with dashed lines.

Figure 3 .
Figure3.Comparisons between the averaged frictional behavior in LT = RT experiments (solid lines) and LT > RT experiments (dashed lines) at two different velocity conditions: (a) 900 mm/s constant velocity and (b) 300 mm/s constant velocity.The lines represent the statistical mean calculated using all experiments at the given velocity and RT combination (i.e., velocity group denoted in Table1) and the shaded region is one standard deviation from the mean for the same set of experiments.As RT decreases and sliding velocity increases, average µ decreases.

Figure 4 .
Figure 4. Composite thermographs captured in constant velocity experiments with a target velocity of 900 mm/s.Composite thermographs are constructed using the maximum measured temperature at each pixel during sliding.(a) Experiment 412 with a LT = RT geometry.(b) Experiment 418 with a LT > RT geometry.Average surface temperature increases as RT decreases.High temperature hotspots elongated in the slip direction form during all phases of sliding.Very high temperature hotspots are typically equant and ∼1 mm in diameter.

Figure 6 .
Figure 6.Composite thermographs captured in constant velocity experiments with a target velocity of 900 mm/s.Thermographs are contoured for local normal stress factors of 4, 8, and 16, representing local normal stresses of 4, 8, and 16 times the applied macroscopic normal stress of 9 MPa.For regions with local temperature exceeding the measurable range, contours are mapped after contacts cool to within the measurable range.These experiments followed the experiments plotted in Figure 4 and show some repeating high-temperature hotspots.(a) Experiment 413 with a LT = RT geometry.(b) Experiment 419 with a LT > RT geometry.For a given local normal stress factor, local temperature increases as RT decreases for contacts with equivalent mm-scale contact history and high-velocity displacement.

Figure 7 .
Figure7.(a, b)  Changes in area percent with displacement following the velocity step for two ranges of local normal stress.The lower limits of each range are contoured in Figure6.Individual data points show measured area percent from contoured thermographs from experiments presented herein in addition to experiments fromBarbery et al. (2021) (blue circles).Linear regression fits fromBarbery et al. (2021) (dashed lines) have been updated using all data points (solid lines) and with the added condition that the area percent is not permitted to drop to or below 0% within 35 mm of high-velocity displacement.(a) Local normal stress range of ≥4 and <6 times the applied macroscopic normal stress of 9 MPa.(b) Local normal stress range of ≥8 and <10 times the applied macroscopic normal stress.(c) Modeled changes in macroscopic normal stress with displacement (dashed blue line) for Experiment 412 where LT = RT based on linear fits for all 15σ l ranges in comparison with measured macroscopic normal stress (red line) for the same experiment.

Figure 8 .
Figure 8. Measured and synthetic temperature distributions for Experiment 418 where LT > RT.(a) Measured temperatures within a temperature range of 100-300°C after 32.7 mm of high-velocity displacement.(b) Synthetic temperatures within a temperature range of 20-400°C for the same thermograph, representing temperatures both inside and outside of the infrared (IR) range.When generating synthetic distributions, pixels in each row are randomly assigned a σ l value from a distribution that evolves with displacement based on our measurements for changes in σ l during sliding.Synthetic temperatures are then assigned based on σ l values and modeled T l for each σ l range.As each pixel is modeled independently from neighboring pixels, we do not reproduce temperature clusters and hotspots observed in the measured data.(c) Comparisons of temperature distributions in the measured thermograph (yellow) and the synthetic thermograph both within the IR temperature range (blue) and outside of the IR temperature range (pink).We show good overall agreement between the measured and synthetic temperature distributions, though we do slightly underestimate temperatures suggesting the model may slightly underestimate the magnitude and rate of weakening.

Figure 9 .
Figure9.Changes in mm-scale local normal stress distributions with high-velocity displacement.(a) Changes in area percent with displacement for all of the modeled ranges of local normal stress.Each range is represented by a different color, labeled in the legend to the right of the plot, with cold to hot colors representing low to high local normal stress magnitudes, respectively.For readability, the six highest ranges were combined into a single range.The four discretized regions identified for modeling flash-weakening are identified and separated by black horizontal lines.(b) Moving averages for local normal stress as a function of high-velocity displacement corresponding to regions 1-3 in addition to the µm-and mm-scale areas of region 4. (c) Changes in area percent with high-velocity displacement for the µm-and mm-scale areas of region 4, and constant area percent value of 6.25 for regions 1-3 where weakening is considered at the µm-scale only.

Figure 10 .
Figure 10.Comparisons between measured data and model results.(a) Akaike information criterion (AIC) values for the global, best fitting parameter sets to the steady-state sliding phase of each model explored: a homogeneous local normal stress model (red), a single-scale inhomogeneous local normal stress model (pink), and a multi-scale inhomogeneous local normal stress model (blue).LT = RT experiments are potted as circles and LT > RT experiments are plotted as diamonds.Lower AIC values indicate better fitting models.(b, c) Comparisons between modeled friction (dashed lines) and measured friction (solid lines) with sliding velocity (a) and displacement (b) for Experiment 436.Multi-scale model results for the non-global, best fitting parameter set (brown dashed line) are plotted alongside single-scale model results (pink dotted-dashed line).Acceleration and deceleration phases of slip are marked with green and yellow arrows, respectively.(d, e) Comparisons between measured friction (green line) and modeled friction for Experiment 474 with sliding velocity (d) and displacement (e).Multi-scale model results for the non-global, best fitting parameter set (red dashed line) are plotted alongside a homogeneous stress model (blue dotted-dashed line).(f, g) Multi-scale model results for experiments 412 (purple) and 463 (green).The non-global, best fitting parameter set to each experiment (dashed lines) are plotted with the global best fitting parameter set (dotted-dashed lines).

Table 1
Experiment Conditions and Mechanical Results for All Experiments a Denotes experiments contoured for each modeled σ l .b Denotes experiments modeled to identify flash-weakening model parameters.