Constitutive behavior and stability of frictional sliding of granite

An understanding of the frictional sliding on faults that can lead to earthquakes requires a knowledge of both constitutive behavior of the sliding surfaces and its mechanical interaction with the loading system. We have determined the constitutive parameters for frictional sliding of initially bare surfaces of Westerly granite, using a recently developed high pressure rotary shear apparatus that allows long distances of sliding and therefore a greater assurance of attaining steady state behavior. From experiments conducted at room temperature and normal stresses of 27–84 MPa several important results have been found. (1) A gouge layer 100 to 200 m thick was developed from the initially bare rock surfaces after 18 to 70 mm of sliding. (2) The steady state frictional resistance, attained after about 10 mm of sliding, is proportional to the negative of the logarithm of the sliding velocity. (3) Abrup changes in the velocity of sliding result in initial changes in the frictional resistance, which have the same sign as the velocity change, and are followed by a gradual decay to a new steady state value over a characteristic distance of sliding. This velocity weakening behavior is essentially identical with that found by several previous workers on the same material at lower normal stress. (4) Our results are well described by a two state variable constitutive law. The values of the constitutive parameters are quite similar to those found previously at low normal stress, but the characteristic distance is about an order of magnitude smaller than that found at 10 MPa normal stress with thicker layers of coarser gouge. (5) We have approximated our results with a one state variable constitutive law and compared the results with the predictions of existing nonlinear stability analysis; in addition, we have extended the stability analysis to systems possessing two state variables. With such formulations good agreement is found between the experimentally observed and theoretically predicted transitions between stable and unstable sliding. These results allow a better understanding of the instabilities that lead to earthquakes.     

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.     

Friction constitutive law with rate and state dependences

A general formula for the Dieterich-Ruina friction constitutive law with rate and state (n-state variables,n=1, 2,...) dependences has been obtained and discussed under the assumption that the slip acceleration a varies ion a linearly with the slip displacement , namelya = a ₀ + (-₀). Wherea ₀, ₀ are initial constants, is the acceleration rate and constant.a ₀ and may be arbitrary constants (positive, negative or zero).The extreme value of frictional resistance and the existence condition of the extreme value, which are very important and govern to some degree the motion process of a frictionally slipping mechanical system, have been analyzed. A critical value _c which is the measure of the velocity weakening and velocity strengthening of the mechanical system, and its properties and the relationship to the extreme problem have been studied. Again, according to the critical value _c, the concepts of light or strong velocity weakening (or strengthening) are introduced.A possibly new phenomenon that frictional resistance may vary in some kind of decayed oscillation is found. Finally, the condition for the smallest frictional resistance for a slipping mechanical system with nonuniform acceleration has been obtained.     

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., kc/ki max >~1), 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 (kc/ki max=0.4) between the two blocks. More complicated patterns are found with weak coupling stiffness (kc/ki max=0.2) and strong heterogeneity. With strong heterogeneity, very weak coupling leads to chaotic slip patterns. With coupling stiffness kc=5 ki 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.     

On a model of frictional sliding

A model of frictional sliding with anN-shaped curve for the sliding velocity dependence of the coefficient of friction is considered. This type of friction law is shown to be related to dynamic i.e., velocity dependent ageing of asperity junctions. Mechanisms of ageing for ductile (Bowden-Tabor) and brittle (Byerlee) materials, though different in nature, lead to qualitatively similarN-shaped velocity dependencies of the coefficient of friction. Estimates for the velocities limiting the range of negative velocity sensitivity of the coefficient of friction are obtained for the ductile case and—albeit with a lesser degree of reliability—for the brittle one. It is shown by linear stability analysis that discontinuous sliding (stick-slip) is associated with thedescending portion of theN-shaped curve. An instability criterion is obtained. An expression for the period of the attendant relaxation oscillations of the sliding velocity is given in terms of the calculated velocity dependence of the coefficient of friction. It is suggested that the micromechanically motivated friction law proposed should be used in models of earthquakes due to discontinuous frictional sliding on a crustal fault.     

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.     

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.     

Comprehensive study on stability of deep crust and unstable behavior of earthquake source by both failure mechanism and frictional sliding mechanism

In this paper the relation between fault movement and stress state in deep crust is discussed, based on synthetic analysis of the crustal stresses measured over the world and the concerned data of focal mechanism. Using Coulomb criterion for shear failure and frictional slip, analytical expressions for estimating stabilities of intact rock and existing fault in the crust and for identifying the type of faulting (normal, strike-slip or thrust fault) are derived. By defining the Failure FunctionF _m and the Fraction FunctionF _f, which may describe steadiness of crustal rock and existing fault, respectively, a synthetic model is set up to consider both fracturing mechanism and the sliding mechanism. By this model, a method to study stability and unstable behavior of crustal rock and fault at different depths is given. According to the above model, quantitative study on the crustal stability in the North China plain is made in terms of the measured data of hydraulic fracturing stress, pore-fluid pressure, terrestrical heat flow in this region. The functionsF _m andF _f and the shear stresses on faults with different strike angle and dip angle at various depths in this region are calculated. In the calculation the constraint condition of fault movement obeys Byerlee’s Law, and the depth-dependent nonlinear change in the vertical stress due to inhomogeneity of crustal density and the high anomalous pore-fluid pressure in deep crust of this region are considered. The conclusions are: the unstable behavior of the crust in the North China plain is not failure of crustal rock but slip on existing fault; the depth range where stick-slip of fault may happen is about from 8 to 20 km or more; stability of steep fault is lower than that of gentle sloping fault; the shear stresses in the range where may occur stick-slip are nearly horizontal; the steep faults trending from NNE to NE in this region are liable to produce strong earthquakes, whose co-seismic faultings are, for the most part, right lateral slip; the change in pore-fluid pressure in depth remarkably affects the stability of the crust and the increase in pore-fluid pressure, therefore, would be an important factor exciting strong earthquake in this region. The above theoretical inferences are consistent with the data measured in this region. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologia Sinica,13, 450–461, 1991. This work is supported by Chinese Joint Seismological Science Foundation.     

Controls on the stability and inclinations of hillslopes formed on hard rock

In identifying controls on rock slope form a distinction is made between: (1) rock slopes with joints which dip steeply out of a cliff and hence are subject to mass failure of the rock mass above a critical joint; and (2) rock slopes with inclinations which are either in equilibrium with the mass strength of their rocks, or have profiles which will develop towards strength equilibrium as cross joints open. In the first class of slope, stability results not just from the basic frictional resistance of the rock but also from the frictional roughness along the critical joint and from the normal stress acting across that joint. Stability may be reduced by weathering and loss of strength of the joint wall rock. As a result of normal stress variations with depth, induced by overburdens, high cliffs which are not undercut have a concave profile. The second group of slopes includes those with inclinations controlled at the scale of individual joint blocks, buttressed slopes and those on unjointed rock masses. Buttressed and unjointed rock masses develop towards a condition of mass strength equilibrium as cross joints open. Strength equilibrium slopes may be recognized by application of a rock mass strength classification proposed for geomorphic purposes. Eleven propositions are formulated which identify controls on rock slope development and some consequences of these controls.     

Lyapunov exponent and dimension of the strange attractor of elastic frictional system

ＬｙａｐｕｎｏｖｅｘｐｏｎｅｎｔａｎｄｄｉｍｅｎｓｉｏｎｏｆｔｈｅｓｔｒａｎｅａｔｔｒａｃｔｏｒｏｆｅｌａｓｔｉｃｆｒｉｃｔｉｏｎａｌｓｙｓｔｅｍＺｈｉ－ＲｅｎＮＩＵ（牛志仁）ａｎｄＤａｎｇ－ＭｉｎＣＨＥＮ（陈党民）（ＳｅｉｓｍｏｌｏｇｉｃａｌＢｕｒ...     

2003年6、7月西昌二次4.8级地震的预测与分析

2003年6月17日和7月10日,先后在四川省西昌市和西昌-昭觉间发生2次ML4 8级强有感地震.对于这2次地震,我们曾作了较好的中期和短期预测.但是对于6月17日发生的第一次ML4 8级地震,未能做出临震预测;对于第二次地震,则在7月1日向上级填报的周会商表中作了明确的临震预测,7月10日,在西昌-昭觉间发生了ML4.8级地震.根据西昌地震遥测台网各子台的P波初动求解震源机制,结合地震所处的地质构造以及现场烈度考察圈定出的结果分析,认为这二次地震是由安宁河断裂带、则木河断裂、凉山断裂所围成的凉山小菱形块体活动的结果.     

The optimum design of a base isolation system with frictional elements

A linear multistorey structure with a seismic base isolation system consisting of rubber bearings and frictional elements is considered. The non-linear equations of motion are derived for the first mode vibration and the stochastic response to a white noise ground acceleration is determined. Based on this response, suitable objective functions are defined and the optimum design of the isolation system is performed. It is shown that a small amount offriction increases the effectiveness of the system compared with the same system but without frictional elements.     

滑移隔震结构的抗风稳定性分析

滑移隔震结构要求在风荷载作用下保持安定。应用多种方法研究了不同摩擦系数、不同基本风压作用下滑移隔震体系的抗风稳定性：（1）考虑一阶振型的等效静力法;（2）采用谐波合成法模拟近似的具有给定功率谱密度函数的脉动风,作用于结构上,进行动力时程分析;（3）根据风压的功率谱密度函数计算脉动风压的概率法。研究结果表明时程分析法与概率法得到的风荷载相近,大于等效静力法得到的风荷载。摩擦系数为0.1可以满足基本风压小于0.90kN／m^2地区滑移隔震结构的抗风稳定性。     

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.     

Seismicity simulation with a mass-spring model and a displacement hardening-softening friction law

Dieterich simulated aftershocks numerically, using a one-dimensional mass-spring model with a time-dependent friction law. But an important precursory phenomenon called quiescence cannot be produced by this model unless, as Mikumo and Miyatake showed with a three-dimensional continuum model, a somewhat arbitrary bimodal distribution of frictional strength is assumed. Here we used the friction law proposed by Stuart, which is a displacement hardening-softening model, and simulated the quiescence. By varying the parameters of the friction law in our mass-spring model, we found a variety of seismicity patterns. When we choose extremely large critical displacement we get a recurrent sequence of creep followed by mainshock without small earthquakes. But when we choose a critical displacement in the same order of magnitude as the slip-weakening critical displacement estimated by Papageorgiou and Aki from strong motion data, we get a normal seismicity pattern, including quiescence before large events. This simple model points to a promising approach for the interpretation of the rupture process during an earthquake by the same physical model.     

断层系统摩擦动力学行为的有限元模拟分析

绝大多数大地震发生在先存断层构造上,被广泛认为是一种断层摩擦失稳行为.研究断层上的摩擦动力学行为对地震数值预报具有重要意义.本文围绕地震断层动力学行为,首先回顾了断层摩擦动力学有限元模拟相关的进展,尤其是有限元模拟过程中断层摩擦行为的处理方法,重点分析比较了不同时间积分方案在有限元模拟计算中的差异,探讨了传统有限元方法在模拟断层摩擦失稳非线性的收敛性等难题.考虑到相关差异及问题,本文主要简明介绍了基于R-minimum策略自适应地控制时间步长方案及其有限元计算算法和相关软件,其兼顾了静态隐式与动态显式的优点,将不稳定的隐式迭代计算转变为显式计算;同时利用库仑摩擦定律及速率和状态相关的摩擦本构方程,通过节点-点任意接触单元模拟变形体之间的三维非线性接触摩擦动力学行为,并小结了相关的成功应用实例.此外,为了构建刻画复杂断层形态的有限元网格模型,本文在总结了相关有限元网格生成算法基础上,举例说明了我们最近发展的基于图像的复杂断层约束下非连续四边形和六面体网格的生成方法.最后以此为例,对生成的复杂断层系统模型进行了有限元数值模拟分析.

     

The frictional resistance characteristics of a bedrock‐influenced river channel

Much of the published literature relating to reach‐scale total ?ow resistance concentrates on alluvial rivers and little mention is made of the more extreme resistance found in bedrock‐in?uenced river systems. This paper presents the results of a detailed investigation of total ?ow resistance variation for ?ve channel types on the Sabie River, South Africa (cohesive mixed anastomosed, uncohesive mixed anastomosed, mixed pool–rapid, alluvial braided and alluvial single thread) over a range of discharges. Manning's ‘n’ and Darcy‐Weisbach f resistance coef?cients are calculated using Barnes' methodology (H. H. Barnes, US Geological Survey Water Supply Paper 1849, 1967). These data are analysed at the scales of channel type and morphologic unit and the resultant resistance values are compared with the literature. Generally the low‐?ow friction values estimated for the bedrock‐in?uenced channel types are higher than any reported. Flow resistance quanti?cation of the alluvial channel type study sites on the Sabie River produced values slightly higher than earlier studies of similar morphologies. In situations where signi?cant areas of irregular bedrock are exposed (cohesive mixed anastomosed and mixed pool–rapid channel types) the high ?ow resistance coef?cients at low discharges probably re?ect energy dissipation due to hydraulic jumps and internal distortions; however, these values fall as discharge increases to values comparable to those recorded for step–pool rivers. It is suggested that the slightly elevated resistance coef?cients recorded in the alluvial channel types are the result of the vegetation roughness component, operating when the macro‐channel bar features are inundated. Copyright © 2004 John Wiley & Sons, Ltd.     

Reduction of structural response to near fault earthquakes by seismic isolation columns and variable friction dampers

<正>This paper focuses on the investigation of a hybrid seismic isolation system with passive variable friction dampers for protection of structures against near fault earthquakes.The seismic isolation can be implemented by replacing the conventional columns fixed to the foundations by seismic isolating ones.These columns allow horizontal displacement between the superstructure and the foundations and decouple the building from the damaging earthquake motion.As a result, the forces in the structural elements decrease and damage that may be caused to the building by the earthquake significantly decreases.However,this positive effect is achieved on account of displacements occurring in the isolating columns.These displacements become very large when the structure is subjected to a strong earthquake.In this case,impact may occur between the parts of the isolating column yielding their damage or collapse.In order to limit the displacements in the isolating columns,it is proposed to add variable friction dampers.A method for selecting the dampers' properties is proposed.It is carried out using an artificial ground motion record and optimal active control algorithm.Numerical simulation of a seven-story structure shows that the proposed method allows efficient reduction in structural response and limits the displacements at the seismic isolating columns.