Stick-slip behavior of Indian gabbro as studied using a NIED large-scale biaxial friction apparatus

This paper reports stick-slip behaviors of Indian gabbro as studied using a new large-scale biaxial friction apparatus, built in the National Research Institute for Earth Science and Disaster Prevention (NIED), Tsukuba, Japan. The apparatus consists of the existing shaking table as the shear-loading device up to 3, 600 kN, the main frame for holding two large rectangular prismatic specimens with a sliding area of 0.75 m² and for applying normal stresses σ_n up to 1.33 MPa, and a reaction force unit holding the stationary specimen to the ground. The shaking table can produce loading rates v up to 1.0 m/s, accelerations up to 9.4 m/s², and displacements d up to 0.44 m, using four servocontrolled actuators. We report results from eight preliminary experiments conducted with room humidity on the same gabbro specimens at v = 0.1-100 mm/s and σ_n = 0.66-1.33 MPa, and with d of about 0.39 m. The peak and steady-state friction coefficients were about 0.8 and 0.6, respectively, consistent with the Byerlee friction. The axial force drop or shear stress drop during an abrupt slip is linearly proportional to the amount of displacement, and the slope of this relationship determines the stiffness of the apparatus as 1.15×10⁸ N/m or 153 MPa/m for the specimens we used. This low stiffness makes fault motion very unstable and the overshooting of shear stress to a negative value was recognized in some violent stick-slip events. An abrupt slip occurred in a constant rise time of 16-18 ms despite wide variation of the stress drop, and an average velocity during an abrupt slip is linearly proportional to the stress drop. The use of a large-scale shaking table has a great potential in increasing the slip rate and total displacement in biaxial friction experiments with large specimens.     

Internal structures and high-velocity frictional properties of Longmenshan fault zone at Shenxigou activated during the 2008 Wenchuan earthquake

This paper reports internal structures of a wide fault zone at Shenxigou, Dujiangyan, Sichuan province, China, and high-velocity frictional properties of the fault gouge collected near the coseismic slip zone during the 2008 Wenchuan earthquake. Vertical offset and horizontal displacement at the trench site were 2.8 m (NW side up) and 4.8 m (right-lateral), respectively. The fault zone formed in Triassic sandstone, siltstone, and shale about 500 m away from the Yingxiu-Beichuan fault, a major fault in the Longmenshan fault system. A trench survey across the coseismic fault, and observations of outcrops and drill cores down to a depth of 57 m revealed that the fault zone consists of fault gouge and fault breccia of about 0.5 and 250–300 m in widths, respectively, and that the fault strikes N62°E and dips 68° to NW. Quaternary conglomerates were recovered beneath the fault in the drilling, so that the fault moved at least 55 m along the coseismic slip zone, experiencing about 18 events of similar sizes. The fault core is composed of grayish gouge (GG) and blackish gouge (BG) with very complex slip-zone structures. BG contains low-crystalline graphite of about 30 %. High-velocity friction experiments were conducted at normal stresses of 0.6–2.1 MPa and slip rates of 0.1–2.1 m/s. Both GG and BG exhibit dramatic slip weakening at constant high slip rates that can be described as an exponential decay from peak friction coefficient μ _p to steady-state friction coefficient μ _ss over a slip-weakening distance D _c. Deformation of GG and BG is characterized by overlapped slip-zone structures and development of sharp slickenside surfaces, respectively. Comparison of our data with those reported for other outcrops indicates that the high-velocity frictional properties of the Longmenshan fault zones are quite uniform and the high-velocity weakening must have promoted dynamic rupture propagation during the Wenchuan earthquake.     

The micromechanics of friction in a granular layer

A grain bridge model is used to provide a physical interpretation of the rate- and state-dependent friction parameters for the simple shear of a granular layer. This model differs from the simpler asperity model in that it recognizes the difference between the fracture of a grain and the fracture of an adhesion between grains, and it explicitly accounts for dilation in the granular layer. The model provides an explanation for the observed differences in the friction of granular layers deformed between rough surfaces and those deformed between smooth surfaces and for the evolution of the friction parameters with displacement. The observed evolution from velocity strengthening to velocity weakening with displacement is interpreted as being due to the change in the micromechanics of strain accommodation from grain crushing to slip between adjacent grains; this change is associated with the observed evolution of a fractal grain structure.     

Dimensionless critical shear stress evaluation from flume experiments using different gravel beds

Flume experiments were conducted using four different gravel beds (D₅₀ + 12–39 mm) and a range of marked particles (10–65 mm). The shear stresses were evaluated from friction velocities, when initial movement of marked particles occurred. Two kinds of equations were produced: first for the threshold of initial movement, and second for generalized movement. Equations of the type 0_c + a(D_i/D₅₀)^b, as proposed by Andrews (1983) are applicable even if the material is relatively well sorted. However, the values of a and b are lower (respectively 0·050 and -0·70) for initial movement. Generalized movement requires a higher shear stress (a + 0·068 and b + -0·80). D₉₀ of the bed material and y₀ (the bed roughness parameter) were also used as reference values in place of D₅₀. They produced lower values than in natural streams, mainly owing to the fact that the material used in the flume is better sorted: clusters are less well developed and the bed roughness is lower.     

Some properties of unstable slip on rough surfaces

In this paper we report results obtained from various friction experiments under direct and oblique shear loading conditions. We used four rock types of varying brittleness (quartzite, anhydrite, limestone, pyrophyllite) with different surface roughness. The observations concentrate on the time span several milliseconds before dynamic failure occurs. During this period a premonitory, unstable phase of slip (slip 2) occurs. This differs importantly from a premonitory, stable process (slip 1) with durations of hundreds of seconds. On smooth surfaces slip 2 is usually observed with ductile rocks and less reliably with brittle rocks. Slip 2 is mostly accompanied by acoustic emissions, which increase in rate of occurrence and in magnitude until the stick-slip event. Foreshocks are observed during approximately 50% of the slip 2 events on rough surfaces. Foreshocks far exceed the acoustic noise level, which is also prevalent before stick-slip events on rough surfaces. In the direct shear experiment, where two faults are being loaded simultaneously, in about 20% of the cases precursory slip 2 was observed on the opposite side on which the final stick-slip event occurred.     

Volumetric displacement of flow depth by obstacles,and the determination of friction factors in shallow overland flows

Large roughness elements such as stones or plant stems (obstacles) influence the depth of overland flows in two ways. The first effect is a dynamic one, involving frictional retardation of the flow and associated reduction in flow speeds. The second influence is static, and arises from the upward volumetric displacement of flow depth because of the submerged volume of the obstacles. Depending upon the distribution of submerged obstacle volume with height above the soil surface, the proportion of the flow volume occupied (and so, the perturbation of flow depth arising from volumetric displacement) can vary irregularly or systematically with flow stage. Furthermore, the amount of volumetric displacement of flow depth would vary among surfaces carrying different cover fractions of identical obstacles. Consequently, estimates of the change in friction factors arising from the drag on flow traversing varying obstacle cover fractions are confounded with the parallel shift volumetric displacement. To understand the true frictional drag arising from obstacles, a correction must be made for the volumetric displacement. A method for making this correction is outlined. New laboratory experiments provide precise observations of depths and friction coefficients in laminar flows passing fields of regular obstacles. After making the proposed correction for volumetric displacement, increases of 40 to 75 per cent in the derived value of the Darcy–Weisbach friction factor, f, are found for an obstacle cover of 20 per cent. Many published studies of friction coefficients in shallow overland flows, such as those on stone‐covered dryland soils, involve larger obstacle cover fractions, and evidently involve the significant confounding effect of volumetric displacement. Copyright © 2002 John Wiley & Sons, Ltd.     

Flow threads in surface run‐off: implications for the assessment of flow properties and friction coefficients in soil erosion and hydraulics investigations

Soil surface microtopography produces non‐uniform surface run‐off, in which narrow threads of relatively deep and fast ?ow move within broader, shallower, slower‐moving regions. This kind of ?ow is probably widespread, given that microtopography is itself common. Methods used to record the properties of surface run‐off include grid‐ or transect‐based depth observations, with a single mean ?ow speed derived by calculation from V = Q/WD, and the use of dye timing to estimate velocity, with an effective mean depth calculated from D = Q/WV. Because these methods allow only single, ?ow‐?eld mean values to be derived for V or D, neither is well suited to non‐uniform ?ows. The use of depth data to derive a ?ow‐?eld mean V furthermore implicitly applies area weighting to the depth data; likewise, the use of dye speeds for V inherently overestimates mean V because dye dominantly follows the faster ?ow threads. The associated errors in derived parameters such as friction coef?cients are not readily quanti?ed and appear not to have been addressed previously. New ?eld experiments made on untilled soil surfaces in arid western NSW, Australia, explore these circumstances and the implications for deriving meaningful measures of ?ow properties, including friction coef?cients. On surfaces deliberately chosen for their very subtle microtopography, average thread velocities are shown to be commonly 2·5 times greater than the ?ow‐?eld mean, and locally 6–7 times greater. On the other hand, non‐thread ?ow speeds lie below the ?ow‐?eld mean, on average reaching only 84 per cent of this value, and often considerably less. Flow‐?eld means conceal the existence of regions of the ?ow ?eld whose properties are statistically distinct. Results con?rm that a reliance on ?ow‐?eld average depths yields estimates of friction coef?cients that are biased toward the shallower, high‐roughness parts of the ?ow, while if dye speeds are relied upon the results are biased toward the deeper, smoother threads of ?ow. A new approach to the evaluation of friction coef?cients in non‐uniform ?ows is advanced, involving the determination of separate coef?cients for threads and non‐thread zones of the ?ow ?eld. In contrast, ?ow‐?eld friction coef?cients as they are customarily derived in run‐off plot experiments subsume these distinct coef?cients in proportions that are generally unknown. The value of such coef?cients is therefore questionable. Copyright © 2004 John Wiley & Sons, Ltd.     

A way of estimating the characteristic slip displacement

During the ruptures of an earthquake,the strain energy.△E,.will be transferred into,at least,three parts,i.e..the seismic radiation energy(E_s),fracture energy(E_g),and frictional energy(E_f),that is,△E = E_s + E_g + E_f.Friction,which is represented by a velocity- and state-dependent friction law by some researchers,controls the three parts.One of the main parameters of the law is the characteristic slip displacement.D_c.It is significant and necessary to evaluate the reliable value of D_c from observed and inverted seismic data.Since D_c controls the radiation efficiency.η_R = E_s/(E_s+ E_g),the value of η_r is a good constraint of estimating D_c.Integrating observed data and inverted results of source parameters from recorded seismograms.the values of E_s and E_g of an earthquake can be measured,thus leading to the value of η_R.The constraint used to estimate the reliable value of D_c will be described in this work.An example of estimates of D_c.based on the observed and inverted values of source parameters of the September 20,1999 M_S 7.6 Chi-Chi(Ji-Ji).Taiwan region,earthquake will be presented.     

Interaction between two sliders in a system with rate- and state-dependent friction

This study examines slip recurrence patterns in a two-block spring-slider model with rate- and state-dependent friction. Both weak and strong heterogeneities are considered with different settings of coupling stiffness. The results show that the recurrence pattern of slips strongly depends on the degree of coupling between the two blocks. With strong coupling between the two blocks (e.g., k_c/k_{i max} τR1), the slip pattern of the system is simple and characterized by periodical stick-slips, with the two blocks slipping together. With strong heterogeneity in friction strength, period-2 motion is found for moderate coupling stiffness (k_c/k_{i mex}=0.4) between the two blocks. More complicated patterns are found with weak coupling stiffness (k_c/k_{i mex}=02) and strong heterogeneity. With strong heterogeneity, very weak coupling leads to chaotic slip patterns. With coupling stiffness k_c=5 k_{i max} and strong heterogeneity, chaotic slip patterns are not found, in contrast with the results by Huang and Turcotte who employed the classical static/kinetic friction law.     

A two degree-of-freedom earthquake model with static/dynamic friction

Can a simple multi-block-spring model with total symmetry make interesting predictions for fault behaviour? Our model consists of a symmetric, slowly driven, two degree-of-freedom block-spring system with static/dynamic friction. The simple friction law and slow driving rate allow the state of this fourth order system to be described between slip events by a single variable, the difference in the stretch of the driving springs. This stretch difference measures the locked-in stress and is closely related to fault stress inhomogeneity. In general,smoothing is not observed. A spatially homogeneous stress state is found to almost always be unstable, in that the system tends toward an inhomogeneous state after many slip events. The system evolves either to a cycle that alternates between two types of earthquakes, or to a cycle with repeating but identical asymmetric earthquakes. One type of alternating earthquake solution is structurally unstable, which implies a great sensitivity to model perturbations. For this simple model, spatial asymmetry necessarily occurs, despite the symmetry in the model, thus suggesting that spatial structure in seismicity patterns may be a consequence of earthquake dynamics, not just fault heterogeneity.     

The Role of Adsorbed Water on the Friction of a Layer of Submicron Particles

Anomalously low values of friction observed in layers of submicron particles deformed in simple shear at high slip velocities are explained as the consequence of a one nanometer thick layer of water adsorbed on the particles. The observed transition from normal friction with an apparent coefficient near ???=?0.6 at low slip speeds to a coefficient near ???=?0.3 at higher slip speeds is attributed to competition between the time required to extrude the water layer from between neighboring particles in a force chain and the average lifetime of the chain. At low slip speeds the time required for extrusion is less than the average lifetime of a chain so the particles make contact and lock. As slip speed increases, the average lifetime of a chain decreases until it is less than the extrusion time and the particles in a force chain never come into direct contact. If the adsorbed water layer enables the otherwise rough particles to rotate, the coefficient of friction will drop to ???=?0.3, appropriate for rotating spheres. At the highest slip speeds particle temperatures rise above 100°C, the water layer vaporizes, the particles contact and lock, and the coefficient of friction rises to ???=?0.6. The observed onset of weakening at slip speeds near 0.001?m/s is consistent with the measured viscosity of a 1?nm thick layer of adsorbed water, with a minimum particle radius of approximately 20?nm, and with reasonable assumptions about the distribution of force chains guided by experimental observation. The reduction of friction and the range of velocities over which it occurs decrease with increasing normal stress, as predicted by the model. Moreover, the analysis predicts that this high-speed weakening mechanism should operate only for particles with radii smaller than approximately 1???m. For larger particles the slip speed required for weakening is so large that frictional heating will evaporate the adsorbed water and weakening will not occur.     

G: Fracture energy,friction and dissipation in earthquakes

Recent estimates of fracture energy G ^′ in earthquakes show a power-law dependence with slip u which can be summarized as G ^′ ∝ u ^a where a is a positive real slightly larger than one. For cracks with sliding friction, fracture energy can be equated to G _f : the post-failure integral of the dynamic weakening curve. If the dominant dissipative process in earthquakes is friction, G ^′ and G _f should be comparable and show a similar scaling with slip. We test this hypothesis by analyzing experiments performed on various cohesive and non-cohesive rock types, under wet and dry conditions, with imposed deformation typical of seismic slip (normal stress of tens of MPa, target slip velocity > 1 m/s and fast accelerations ≈ 6.5 m/s²). The resulting fracture energy G _f is similar to the seismological estimates, with G _f and G ^′ being comparable over most of the slip range. However, G _f appears to saturate after several meters of slip, while in most of the reported earthquake sequences, G ^′ appears to increase further and surpasses G _f at large magnitudes. We analyze several possible causes of such discrepancy, in particular, additional off-fault damage in large natural earthquakes.     

The characteristic displacement in rate and state-dependent friction from a micromechanical point of view

The physical meaning of the characteristic displacement that has been observed in velocity-stepping friction experiments was investigated based on the micromechanics of asperity contact. It has been empirically found for bare rock surfaces that the magnitude of the characteristic displacement is dependent only on surface roughness and insensitive to both slip velocity and normal stress. Thus the characteristic displacement has been interpreted as the displacement required to change the population of contact points completely. Here arises a question about the physical mechanism by which the contact population changes. Because individual asperity contacts form, grow and are eliminated with displacement, there are at least two possible interpretations for the characteristic displacement: (1) it is the distance over which the contacts existing at the moment of the velocity change all fade away, being replaced by new asperity contacts, or (2) it is the distance required for a complete replacement in the real contact area that existed at the moment of the velocity change. In order to test these possibilities, theoretical models were developed based on the statistics of distributed asperity summits. A computer simulation was also performed to check the validity of the theoretical models using three-dimensional surface topography data with various surface roughnesses. The deformation was assumed to be elastic at each asperity contact. The results of both the simulation and the theoretical models show that the characteristic displacement in (1) is about three times longer than that in (2). Comparison of the results with the experimental observations obtained by others indicates that the possibility (2) is the correct interpretation. This means that the state in the rate and state variable friction law is memorized in a very confined area of real contact. Further, our results explain why the characteristic displacement is insensitive to normal stress: this comes from the fact that the microscopic properties such as the mean contact diameter are insensitive to normal stress. The approach based on the micromechanics of asperity contact is useful to investigate the underlying mechanism of various phenomena in rock friction.     

Effect of varying normal stress on stability and dynamic motion of a spring-slider system with rate- and state-dependent friction

Incorporating rate and state friction laws, stability of linearly stable (i.e., with stiffness greater than the critical value) spring-slider systems subjected to triggering perturbations was analyzed under variable normal stress condition, and comparison was made between our results and that of fixed normal stress cases revealed in previous studies. For systems associated with the slip law, the critical magnitude of rate steps for triggering unstable slips are found to have a similar pattern to the fixed normal stress case, and the critical velocity steps scale with a/(b - a) when k = k_cr for both cases. The rate-step boundaries for the variable normal stress cases are revealed to be lower than the fixed normal stress case by 7 %–16 % for a relatively large α = 0.56 with (b - a)/a ranging from 0.25 to 1, indicating easier triggering under the variable normal stress condition with rate steps. The difference between fixed and variable normal stress cases decreases when the α value is smaller. In the same slip-law-type systems, critical displacements to trigger instability are revealed to be little affected by the variable normal stress condition. When k ≥ k_cr(V_*), a spring-slider system with the slowness law is much more stable than with the slip law, suggesting that the slowness law fits experimental data better when a single state variable is adopted. In stick-slip motions, the variable normal stress case has larger stress drops than the constant normal stress case. The variable normal stress has little effect on the range of slip velocity in systems associated with the slowness law, whereas systems associated with the slip law have a slowest slip velocity immensely smaller than the fixed normal stress case, by ~10 orders of magnitude.     

Damage mechanics preceding the September–October 1989 flank eruption at Mount Etna volcano inferred by seismic scaling exponents

Seismicity preceding the September–October 1989 flank eruption at Mt. Etna volcano has been analysed by means of the fractal two-point correlation dimension D_c relative to epicentres distribution and the scaling exponent D inferred from the seismic b-value. Time evolution of D_c and D preceding the eruption onset revealed: a general decrease of D_c and increase of D (9–4 months prior the eruption onset), intermittent rates of D_c increase and D decrease (4 to 1 month), steady high values of D_c and low values of D (1 month) and sharp decreases of D_c and D marking the eruption onset. Results allowed to constrain the increment of seismic damage by recognising negative (mechanical hardening) and positive (mechanical softening) feedback prior the eruption onset.     

A numerical study of comparison of two one-state-variable,rate- and state-dependent friction evolution laws

The two one-state-variable, rate- and state-dependent friction laws, i.e., the slip and slowness laws, are compared on the basis of dynamical behavior of a one-degree-of-freedom spring-slider model through numerical simulations. Results show that two (normalized) model parameters, i.e., Δ (the normalized characteristic slip distance) and β?α (the difference in two normalized parameters of friction laws), control the solutions. From given values of Δ, β, and α, for the slowness laws, the solution exists and the unique non-zero fixed point is stable when Δ>(β?α), yet not when Δ < (β?α). For the slip law, the solution exists for large ranges of model parameters and the number and stability of the non-zero fixed points change from one case to another. Results suggest that the slip law is more appropriate for controlling earthquake dynamics than the slowness law.     

The Dependence of Constitutive Properties on Temperature and Effective Normal Stress in Seismogenic Environments

— We have evaluated how the parameters prescribing the slip-dependent constitutive law are affected by temperature and effective normal stress, by conducting the triaxial fracture experiments on Tsukuba-granite samples in seismogenic environments, which correspond to a depth range to 15 km. The normalized critical slip displacement D _c almost remains constant below 300^oC (insensitive to both temperature and effective normal stress σ _n ^eff); D _c increases with increasing temperature above 300 °C, and the rate of D _c increase with temperature tends to be largest at higher σ _n ^eff. The breakdown stress drop Δτ _b for the granite at constant σ _n ^eff is roughly 80 MPa below 300 °C, and does not depend on σ _n ^eff. Above 300 °C, Δτ _b decreases gradually with increasing temperature, and the rate of Δτ _b reduction with temperature increases at higher σ _n ^eff. The peak shear strength τ _p increases nearly linearly with increasing σ _n ^eff below 300 °C. However, τ _p becomes lower above 300 °C, deviating from the linear relation extrapolated from below 300 °C. This is consistent with the onset of crystal plastic deformation mechanisms of Tsukuba granite.     

Coseismic Slip from the 6 October 2008, M w6.3 Damxung Earthquake, Tibetan Plateau, Constrained by InSAR Observations

Coseismic deformation fields of the 6 October 2008 M _w6.3 Damxung earthquake were obtained from interferometric synthetic aperture radar by using three descending and two ascending Envisat images. Significant coseismic surface deformation occurred within 20?km?×?20?km of the epicenter with a maximum displacement of ~0.3?m along the satellite line of sight. We model a linear elastic dislocation in a homogeneous half space and use a nonlinear constraint optimized algorithm to estimate the fault location, geometry and slip distribution. The results indicate a moment magnitude M _w6.3, and the earthquake is dominated by oblique normal and right-lateral slip with a maximum slip of 2.86?m at depth of 8?km. The rupture plane is about 15?km?×?14?km with strike S190°W and dip 55° to NW, located at a secondary fault of the Southeastern Piedmont of the Nyainqentanglha Mountains. Slip on normal faults in the Tibetan Plateau contributes to the rift evolution.