Implications of Coulomb plasticity for the velocity dependence of experimental faults

Simulated fault gouges often deform more stably than initially bare surfaces of the same composition. It is important to understand why the sliding stability is enhanced because the presence of gouge on natural faults may have the same effect as seen in experiments, and thus explain the absence of earthquakes at shallow depths. Gouge stabilization in experiments has been attributed to positive contributions to velocity dependence within gouge layers from either dilation (Marone et al., 1990) or grain fracture (Biegel et al., 1989). In this study we test the hypothesis that some aspects of gouge and initially bare surface velocity dependence are identical by measuring the time-dependent constitutive parameterb. An important result follows however from stress analysis: if both sample configurations are frictional in the Mohr-Coulomb sense, each configuration is required to deform on planes of distinctly different orientation. The measured strength and velocity dependence will reflect this geometric difference. Our observed values ofb for simulated granite and quartz gouge are two to two and a half times smaller thanb for initially bare surfaces. This difference is completely accounted for if gouge is represented as a cohesionless-Coulomb plastic material. The analysis demonstrates the following points: 1) gouge deformation is fully consistent with Coulomb plasticity, 2) observed gouge velocity dependence is a function of observed strength and 3) the constitutive parameterb is the same for both bare surfaces and gouge. Furthermore, the results suggest that there is no time-dependent strengthening associated with stabilizing effects in gouge. These observations provide a framework for understanding how slip on initially bare surfaces and gouge deformation are related.     

FRICTIONAL SLIDING OF PLAGIOCLASE GOUGE UNDER LOWER-CRUST TEMPERATURE AND RELATIVELY LOW EFFECTIVE NORMAL STRESS

The discovery of tremors on the lower crust portion of the San Andreas Fault has attracted more attention on the mechanical properties of the lower crust in recent years, and some experimental studies have been carried out to understand the mechanical behavior. Previous experiments under effective normal stresses of 200MPa have shown that pyroxene and plagioclase mineral separated from the gabbro and their mixtures all show velocity weakening in the lower-crust temperature range, which results in unstable slip when frictional sliding is the dominant deformation mechanism. This work is to examine whether the velocity-weakening behavior of plagioclase gouge also applies to relatively lower effective normal stress. Our experiments were performed under effective normal stress of about 100MPa, with a constant confining pressure control, with pore pressure of 30MPa and temperature of 100℃ to 600℃. We found that the frictional sliding of plagioclase are basically the same with the previous results obtained under effective normal stress of 200MPa, both of which show velocity weakening over the entire temperature range. The only difference is the out-of-trend drop of constitutive parameter a at 600℃ for the lower effective normal stress of 100MPa. It is thus concluded that reducing the effective normal stress has little effect on the sliding stability of plagioclase, and the previous conclusion made for mechanical behavior of the lower crust that unstable slips are possible therein also applies to the lower effective normal stress of 100MPa.     

热水条件下角闪石断层泥的摩擦滑动性质——与斜长石断层泥的对比

基于速率与状态依赖性摩擦本构关系理论框架,在热水条件下研究了角闪石断层泥的摩擦滑动性质并与闪长岩的另一种主要矿物斜长石的摩擦滑动性质进行了对比.摩擦实验是在三轴实验系统上完成,有效正应力200 MPa,孔隙压力30 MPa,并将加载速率在1.22 μm/s和 0.122 μm/s之间实施了切换.结果表明角闪石的摩擦系数均值为0.70±0.01,随着温度增加没有系统性的变化,整体低于斜长石的摩擦系数(0.75±0.01);计算与实验表明,角闪石和斜长石的摩擦系数的体积分数加权平均值与闪长岩的摩擦系数基本一致;角闪石在实验温度范围内(100～614 ℃)显示速率强化(a-b>0),与斜长石在整个温度范围内的速率弱化(a-b<0)正好相反;角闪石的速率依赖性在整个实验温度范围内无系统性的变化.     

Coulomb constitutive laws for friction: Contrasts in frictional behavior for distributed and localized shear

We describe slip-rate dependent friction laws based on the Coulomb failure criteria. Frictional rate dependence is attributed to a rate dependence of cohesionc and friction angle . We show that differences in the stress states developed during sliding result in different Coulomb friction laws for distributed shear within a thick gouge layer versus localized shear within a narrow shear band or between bare rock surfaces. For shear within gouge, shear strength is given by =c cos + _n sin, whereas for shear between bare rock surfaces the shear strength is =c cos + _n tan, where and _n are shear and normal stress, respectively. In the context of rate-dependent Coulomb friction laws, these differences mean that for a given material and rate dependence of the Coulomb parameters, pervasive shear may exhibit velocity strengthening frictional behavior while localized shear exhibits velocity weakening behavior. We derive from experimental data the slip-rate dependence and evolution ofc and for distributed and localized shear. The data show a positive rate dependence for distributed shear and a negative rate dependence for localized shear, indicating that the rate dependence ofc and are not the same for distributed and localized shear, even after accounting for differences in stress state. Our analysis is consistent with the well-known association of instability with shear localization in simulated fault gouge and the observation that bare rock surfaces exhibit predominantly velocity weakening frictional behavior whereas simulated fault gouge exhibits velocity strengthening followed by a transition to velocity weakening with increasing displacement. Natural faults also exhibit displacement dependent frictional behavior and thus the results may prove useful in understanding the seismic evolution of faulting.     

FRICTIONAL PROPERTIES OF A NATURAL FAULT GOUGE FROM DRILLED CORES IN THE LONGMENSHAN FAULT ZONE CUTTING GRANITIC ROCK

In this paper, we report friction experiments performed on natural fault gouge samples embedded in granitic rock from drilled core by a project entitled "the Longmenshan Fault Shallow Drilling(LMFD)". Compared with other natural fault gouge, this yellow-greenish gouge(YGG)is dominantly chlorite-rich. The maximum content of chlorite reaches 47%in the YGG. To understand the frictional properties of the YGG sample, experiments were performed at constant confining pressure of 130MPa, with constant pore pressure of 50MPa and at different temperatures from 25℃ to 150℃. The experiments aim to address the frictional behavior of the YGG under shallow, upper crustal pressure, and temperature conditions. Compared with previous studies of natural gouge, our results show that the YGG is stronger and shows a steady state friction coefficient of 0.47~0.51. Comparison with previous studies of natural gouge with similar content of clay minerals indicates a sequence of strengths of different clay minerals:chlorite > illite > smectite. At temperatures up to 150℃ hence depths up to~8km in the Longmenshan region, the YGG shows stable velocity-strengthening behavior at shallow crustal conditions. Combined with the fact of strong direct velocity effect, i.e., (a-b)/a>0.5, faults cutting the present clastic lithology up to~8km depth in the Longmenshan fault zone(LFZ)are likely to offer stable sliding resistance, damping co-seismic rupture propagating from below at not-too-high slip rates. However, as the fault gouge generally has low permeability, co-seismic weakening through thermal pressurization may occur at high slip rates(>0.05m/s), leading to additional hazards.     

Development of shear localization in simulated quartz gouge: Effect of cumulative slip and gouge particle size

Frictional sliding experiments were conducted on two types of simulated quartz gouge (with median particle diameters 5 m and 25 m, respectively) at confining pressures ranging from 50 MPa to 190 MPa in a conventional triaxial configuration. To investigate the operative micromechanical processes, deformation texture developed in the gouge layer was studied in samples which had accumulated different amounts of frictional slip and undergone different stability modes of sliding. The spatial patterning of shear localization was characterized by a quantitative measurement of the shear band density and orientation. Shear localization in the ultrafine quartz gouge initiated very early before the onset of frictional sliding. Various modes of shear localization were evident, but within the gouge zoneR ₁-shears were predominant. The density of shear localization increased with cumulative slip, whereas the angle subtended at the rock-gouge interface decreased. Destabilization of the sliding behavior in the ultrafine quartz gouge corresponded to the extension ofR ₁-shears and formation of boundaryY-shear segments, whereas stabilization with cumulative slip was related to the coalescence ofY-shear segments to form a throughgoing boundary shear. In the coarse quartz gouge, the sliding behavior was relatively stable, probably because shear localization was inhibited by distributed comminution. Two different models were formulated to analyze the stress field within the gouge zone, with fundamentally different predictions on the orientations of the principal stresses. If the rock-gouge interface is assumed to be bonded without any displacement discontinuity, then the maximum principal stress in the gouge zone is predicted to subtend an angle greater than 45° at the interface. If no assumption on displacement or strain continuity is made and if the gouge has yielded as a Coulomb material, then the maximum principal stress in the gouge zone is predicted to subtend an angle less than 45°. If the apparent friction coefficient increases with overall slip (i.e., slip-hardening), then the Riedel shear angle progressively decreases with increasing shear strain within the gouge layer, possibly attaining a zero value which corresponds to a boundaryY-shear. Our quantitative data on shear localization orientation are in reasonable agreement with this second model, which implies the coefficient of internal friction to be about 0.75 for the ultrafine quartz gouge and 0.8 for the coarse gouge. The wide range of orientations for Riedel shear localization observed in natural faults suggests that the orientations of principal stresses vary as much as in an experimental gouge zone.     

热水条件下黑云母断层泥的摩擦强度与稳定性

黑云母是自然界常见的层状硅酸盐矿物,其摩擦系数不高且化学稳定性好,对其摩擦性质的关注可能会对弱断层的研究有所帮助.本次工作选取的实验温度条件对应于典型地壳强度模型中脆塑性转化带的范围,为300 ℃和400 ℃.有效正应力为200 MPa,孔隙水压包括10 MPa和30 MPa,在此条件下对黑云母模拟断层泥进行摩擦实验研究.实验得出黑云母的摩擦系数平均在0.36左右.速度依赖性随温度升高速度弱化的程度增强,表现为300 ℃为十分微弱的速度弱化,而在400 ℃出现了黏滑行为,代表了更强的速度弱化.显微结构中同时出现了脆性剪切变形和塑性扭折变形,但决定宏观力学性质的显然是脆性剪切变形.在黑云母存在的情况下,本研究的实验结果有助于理解大陆地壳脆塑性转化带中地震的可能性和弱断层深部的变形机制、宏观力学行为以及地震活动.     

白云母对岩盐断层带摩擦速度依赖性影响的实验研究

为更好地理解层状硅酸盐对断层强度、滑动速度依赖性及地震活动特征的影响,利用双轴摩擦实验对含白云母岩盐断层带在干燥及含水条件下摩擦的速度依赖性进行了实验研究,并观测了摩擦滑动过程中的声发射,分析了断层带的微观结构.实验结果表明,干燥条件下含白云母岩盐断层带在0.1 ～ 100μm/s的速度范围内表现为黏滑和速度弱化,增大σ2会使断层带从速度弱化向速度强化转化,速度依赖性转换出现在0.1 μm/s,其中断层滑动表现为稳滑或应力释放时间较长的黏滑事件;含水条件下含白云母岩盐断层带在0.05 ～0.01μm/s的速度范围内表现为速度强化,0.1 ～10μm/s的速度范围内表现为速度弱化,50～100μm/s的速度范围内又转换为速度强化行为.含白云母岩盐断层带在干燥条件下一次黏滑伴随一个或一丛声发射事件,而在含水条件下与稳滑相对应,滑动过程中并未记录到声发射事件.显微结构观察表明,速度弱化域的主要变形机制是岩盐颗粒的脆性破裂和局部化的滑动;干燥条件下,速度强化域的主要变形机制是岩盐颗粒的均匀破裂;含水条件下2个速度强化域对应不同的微观机制,高速域的速度强化受控于岩盐颗粒在白云母相互连结形成的网状结构上的滑动及其均匀碎裂作用,而低速域的速度强化还受岩盐的压溶作用控制.通过与岩盐断层带摩擦实验结果对比可知,白云母的存在对于燥岩盐断层带摩擦滑动方式和速度依赖性没有显著影响,而在含水条件下白云母的存在使得岩盐断层带滑动趋于稳定.实验结果为分析含层状硅酸盐断层的强度和稳定性提供了依据.此外,在速度依赖性转换域上观察到的应力缓慢释放的现象进一步证实了在岩盐断层带摩擦滑动过程中观察到的现象,这对慢地震机制研究具有参考意义.     

水热作用条件下蛇纹石断层泥的摩擦强度和速度依赖性及其地震地质意义

在恒定的正应力和孔隙压力、不同温度下测量了三种蛇纹石断层泥摩擦强度的速度依赖性。在室温下，纤蛇纹石具有很低的摩擦系数（０．２～０．２５），而利蛇纹石和叶蛇纹石的摩擦系数较高，分别为０．３９和０．４５左右。在２５～２００℃范围内，温度变化对纤蛇纹石和叶蛇纹石强度的影响不明显，而利蛇纹石的摩擦系数随温度升高明显增加。室温下，三种蛇纹石的摩擦在较快的滑动速率下表现为微弱的速度弱化，而在较慢的滑动速度下为明显的速度强化，温度的提高有促使速度弱化向速度强化转变的趋势。研究表明，蛇纹石的存在有利于断层的无震蠕动，但这种影响可能主要限于浅部；尽管纤蛇纹石是最软弱的造岩矿物之一，但仍不足以解释圣安德烈斯断层的软弱性     

Effects of pore fluid chemistry on stable sliding of Berea sandstone

Single-cycle and multiple-cycle frictional-sliding experiments were employed to evaluate the effects of pore fluid environments on yield strength, frictional-sliding dynamics, and gouge production and morphology. Circular right cylinders cored from Berea sandstone sawcut at 35° to the axes were saturated in water, an inorganic brine, and various anionic, cationic, and nonionic aqueous surface-active agents. Samples were deformed under an effective confining pressure of 50 MPa and an axial strain rate of 6×10^–5 sec^–1 until a 2% axial strain beyond yield (defined as the onset of sliding) was achieved. All samples were displaced by stable sliding. In the single-cycle tests the unsaturated and water-saturated samples displayed small stress peaks at yield. During stable sliding samples saturated with DTAB and SDS displayed slight increases in differential stress and statistically significant higher frictional coefficients than other environments (including water) but were very similar in behavior to dry, unsaturated samples. In the multiple-cycle tests, samples were loaded to 2% strain beyond yield and unloaded to a differential stress of approximately 5–10 MPa a total of four times. These results further suggest that DTAB exerts a strengthening effect on the sandstone relative to water which, to a limiting value, increased with displacement. The DTAB and SDS environments also produced a coarser grain-size distribution in the gouge relative to gouge produced in the other environments. Investigation of the gouge by scanning electron microscope revealed that these larger grains were composed of dense, apparently cemented aggregates of ultrafine, platy quartz particles.     

The frictional behavior of lizardite and antigorite serpentinites: Experiments,constitutive models,and implications for natural faults

Laboratory studies of the frictional behavior of rocks can provide important information about the strength and sliding stability of natural faults. We have conducted friction experiments on antigorite and lizardite serpentinites, rocks common to both continental and oceanic crustal faults. We conducted both velocity-step tests and timed-hold tests on bare surfaces and gouge layers of serpentinite at room temperature. We find that the coefficient of friction of lizardite serpentinite is quite low (0.15–0.35) and could explain the apparent low stresses observed on crustal transform faults, while that of antigorite serpentinite is comparable to other crustal rocks (0.50–0.85). The frictional behavior of both types of serpentinite is well described by a two-mechanism model combining state-variable-dominated behavior at high slip velocities and flow-dominated behavior at low velocities. The two-mechanism model is supported by data from velocity-step tests and timed-hold tests. The low velocity behavior of serpentinite is strongly rate strengthening and should result in stable fault creep on natural faults containing either antigorite or lizardite serpentinite.     

Simulation of Frictional Strength and Steady Relaxation Using the Rate and State Dependent Friction Model

In this paper, frictional strength of hard solids, such as rock–rock sliding surfaces, is studied as a function of waiting time and shearing velocity. A one dimensional spring–mass sliding system is numerically simulated under the quasistatic condition using the rate and state dependent friction model. It is established that frictional strength varies linearly with the logarithm of waiting time (also known as time of stationary contact or relaxation time, etc.) as well as logarithm of shearing velocity. Analytical expression developed for frictional strength is found to be valid only in the case of high stiffness of the connecting spring. In the steady relaxation simulation, a steadily sliding mass is suddenly brought to zero velocity and relaxation of the interfacial stress and corresponding velocity at the sliding interface is studied as a function of relaxation time in the velocity strengthening regime of friction. A mathematical relation is derived between state variable and waiting time using the concept of steady relaxation. The relaxation model is also compared with the experimental data from the literature. Finally, the present study enables one to unify the slide–hold–slide friction experiments.     

AN EXPERIMENTAL STUDY ON THE PROCESS OF INTERGRANULAR PRESSURE SOLUTION OF PLAGIOCLASE GOUGE UNDER HIGH TEMPERATURE AND PRESSURE: METHOD AND PRELIMINARY RESULTS

To understand the mechanism of lower-crust earthquake and slow slips, it is necessary to study the frictional properties of mafic rocks and their major rock-forming minerals. Previous studies have performed a series of experimental researches on gabbro, basalt and their major constituents. According to the results of previous experiments, frictional sliding of plagioclase under hydrothermal conditions(100~600℃)shows a property of velocity weakening, and the experimental results show that both the direct rate effect parameter(a)and the healing effect parameter(b)increase with temperature, a typical feature for thermally-activated processes. Velocity weakening means property of a shear band that has a stronger friction healing effect than the direct rate effect in the rate and state friction constitutive framework, and the healing effect(b value)in constitutive relation mainly reflects the increase in contact area with time under hydrothermal conditions, with some minor effect of structural changes. Since the microphysical mechanism of feldspar minerals at the contacts is mainly brittle cataclasis for temperatures below 600℃, the significant frictional healing effect in this case can only be explained by the mechanism of pressure solution. In order to determine if the dissolution process of plagioclase actually occurs on the laboratory time scale, we conducted hydrostatic experiments on plagioclase powder samples under hydrothermal conditions whereby frequent contact switch between particles seen in frictional sliding experiments can be avoided, making the observation on the dissolution sites possible. Experimental temperatures were 400℃ and 500℃, with confining pressure of 90~150MPa, pore pressure of 30MPa, with 2mm initial thickness of fault gouge. The mechanical data show that a creep process occurred in the plagioclase fault gouge in the experimental temperature and pressure range; and the microstructures of the experiment show that precipitation of new grains is prevalent as the product of pressure solution process between plagioclase particles. At the same time, it is observed that the contact points have an appearance similar to fused, fuzzy structure as signatures of dissolution. The results of our experiments provide a definite experimental evidence for the healing mechanism in friction of plagioclase and for the theoretical relation between unstable slip and the pressure solution process. The results of the experiments are summarized as follows: (1)Drainage rate of pore water in plagioclase gouge was high in the first few hours of experiment, but gradually decreases over time for both temperature and pressure series of experiments slowing down to a steady state. This feature indicates that there is a creep process that evolves inside the plagioclase gouge. In the temperature-series experiments, the drainage rate of the pore water in the plagioclase gouge at 400℃ is relatively low than the cases for higher temperatures. Thus, the applied temperature is positively correlated with the creep of plagioclase gouge. (2)Scanning electron microscopy(SEM)observations of the experimentally deformed samples were performed on thin sections cut along the sample axis. Firstly, from the images of microstructure, it was found that the degree of particle fracture became more significant at a higher effective pressure, with smaller pore volume between particles. In the temperature-series experiments it was found that the degree of compaction of plagioclase gouge increased with increasing temperature. Precipitation of plagioclase grains in layered structures was generally observed in high-magnification images, indicating the presence of pressure solution processes. Contact points were also found to be in a state of ambiguity that seems to be a fused morphology, but the details of the structure remain to be determined by further observations. The above results indicate that the pressure solution process of plagioclase particles can occur on a typical laboratory time scale, and the results of this study provide robust experimental evidences for the theory that links between pressure solution and the mechanism of frictional healing and unstable slips for plagioclase.     

岩盐断层带摩擦滑动的速度依赖性转换及其地震学意义

为了深入理解断层带摩擦滑动速度依赖性转换及其机制,利用双轴摩擦实验对干燥及含水条件下岩盐断层带摩擦的速度依赖性进行了实验研究,并观测了摩擦滑动过程中的声发射,分析了断层带的微观结构.实验结果表明,干燥岩盐断层带在0.1~100 μm/s的速度范围内表现为速度弱化,增大σ₂会使断层带向速度强化转变;含水条件下岩盐断层带在1~100 μm/s的速度范围内表现为速度弱化,而在0.1~0.01 μm/s的速度范围内表现为速度强化,速度依赖性转换出现在0.1~1 μm/s,其中断层表现为振荡或应力释放时间较长的黏滑事件;岩盐断层带在干燥条件下表现出很强的声发射活动,每个黏滑均对应一丛声发射事件,而在含水条件下一次黏滑只对应一个声发射事件.显微观察表明,局部化的脆性破裂是速度弱化域的主要变形机制,分布式的碎裂流动是干燥岩盐断层带在速度强化域的变形机制,颗粒边界迁移以及压溶作用的塑性变形是含水条件下岩盐断层带在速度强化域的主要变形机制,而脆性破裂和塑性变形共同控制着速度依赖性转换域断层带的变形.水的存在促进岩盐发生塑性变形,进而导致断层带从速度弱化向速度强化转换.上述结果有助于理解断层带上地震活动的特征和慢地震的机制.     

Grain size and chemical controls on the ductile properties of mostly frictional faults at low-temperature hydrothermal conditions

A conceptually simple process which establishes a steady grain size distribution is envisioned to control the ductile creep properties of fault zones that mainly slip by frictional processes. Fracture during earthquakes and aseismic frictional creep tend to reduce grain size. However, sufficiently small grains tend to dissolve so that larger grains grow at their expense, a process called Ostwald ripening. A dynamic stedy state is reached where grain size reduction by fracture is balanced by grain growth from Ostwald ripening. The ductile creep mechanism within fault zones in hard rock is probably pressure solution where the rate is limited by diffusion along load-bearing grain-grain contacts. The diffusion paths that limit Ostwald repening are to a considerable extent the same as those for pressure solution. Active Ostwald ripening thus implies conditions suitable for ductile creep. An analytic theory allows estimation of the steady-state mean grain size and the viscosity for creep implied by this dynamic steady state from material properties and from the width, shear traction, and long-term slip velocity of the fault zone. Numerical models were formulated to compute the steady state grain size distribution. The results indicate that ductile creep, as suggested in the companion paper, is a plausible mechanism for transiently increasing fluid pressure within mostly sealed fault zones so that frictional failure occurs at relatively low shear tractions, 10 MPa. The relevant material properties are too poorly known, however, for the steady state theory (or its extension to a fault that slips in infrequent large earthquakes) to have much predictive value without additional laboratory experiments and studies of exhumed faults.     

The effects of sliding velocity on the frictional and physical properties of heated fault gouge

The frictional properties of a crushed granite gouge and of gouges rich in montmorillonite, illite, and serpentine minerals have been investigated at temperatures as high as 600°C, confining pressures as high as 2.5 kbar, a pore pressure of 30 bar, and sliding velocities of 4.8 and 4.8×10^–2 m/sec. The gouges showed nearly identical strength behaviors at the two sliding velocities; all four gouges, however, showed a greater tendency to stick-slip movement and somewhat higher stress drops in the experiments at 4.8×10^–2 m/sec. Varying the sliding velocity also had an effect on the mineral assemblages and deformation textures developed in the heated gouges. The principal mineralogical difference was that at 400°C and 1 kbar confining pressure a serpentine breakdown reaction occurred in the experiments at 4.8×10^–2 m/sec but not in those at 4.8 m/sec. The textures developed in the gouge layers were in part functions of the gouge type and the temperature, but changes in the sliding velocity affected, among other features, the degree of mineral deformation and the orientation of some fractures.     

Effects of temperature and sliding rate on frictional strength of granite

Layers of artificial granite gouge have been deformed on saw-cut granite surfaces inclined 30° to the sample axes. Samples were deformed at a constant confining pressure of 250 MPa and temperatures of 22 to 845°C. The velocity dependence of the steady-state coefficient of friction (^ss) was determined by comparing sliding strengths at different sliding rates. The results of these measurements are consistent with those reported bySolberg andByerlee (1984) at room temperature andStesky (1975) between 300 and 400°C. Stesky found that the slip-rate dependence of (^ss) increased above 400°C. In the present study, however, the velocity dependence of (^ss) was nearly independent of temperature.     

Application of the Pi theorem to the wear rate of gouge formation in frictional sliding of rocks

A simple law of wear rate is examined for the process of gouge generation during the frictional sliding of simulated faults in rocks, by use of the Pi theorem method (dimensional analysis) and existing experimental data. The relationship between wear rate (t/d) and the applied stress can be expressed by the power-law relations $$\frac{t}{d} = C_\sigma \sigma ^{m\sigma } ,\frac{t}{d} = C_\tau \tau ^{m\tau }$$ wheret is the thickness of the gouge generated on the frictional surfaces,d is the fault displacement, σ and τ are normal stress and shear stress, respectively, andC _σ,C _τ,m _σ andm _τ are constants. These results indicate that the exponent coefficientsm _σ andm _τ and the coefficientsC _σ andC _τ depend on the material hardness of the frictional surfaces. By using the wear rates of natural faults, these power-law relationships may prove to be an acceptable palaeopiezometer of natural faults and the lithosphere.