Micromechanics of the velocity and normal stress dependence of rock friction

Among the second-order effects on friction the most important are those of variable normal stress and of slip velocity. Velocity weakening, which is usually considered the source of the stick-slip instability in rock friction, has been observed in velocity stepping experiments with Westerly granite. The friction change, , was –0.01 to –0.008 for a tenfold velocity increase. Using normal closure measurements, we observed dilation upon each increase in sliding rate. We also observed, for the first time, time-dependent closure between surfaces during static loading. The dilation that occurred during the velocity stepping experiment was found to be that expected from the static time-dependent closure phenomenon. This change in closure was used to predict friction change with an elastic contact model. The calculated friction change which results from a change in contact area and asperity interlocking, is in good agreement with the observed velocity dependence of steady-state friction. Variable normal stress during sliding has two effects, first in creating new partial slip contacts and locking some existing fully sliding contacts and second in increasing interlocking, for instance when normal load is suddenly increased. As a result, a transient change in friction occurs upon a sudden change in normal load.     

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.     

Time-dependent friction and the mechanics of stick-slip

Time-dependent increase of static friction is characteristic of rock friction undera variety of experimental circumstances. Data presented here show an analogous velocity-dependent effect. A theor of friction is proposed that establishes a common basis for static and sliding friction. Creep at points of contact causes increases in friction that are proportional to the logarithm of the time that the population of points of contact exist. For static friction that time is the time of stationary contact. For sliding friction the time of contact is determined by the critical displacement required to change the population of contacts and the slip velocity. An analysis of a one-dimensional spring and slider system shows that experimental observations establishing the transition from stable sliding to stick-slip to be a function of normal stress, stiffness and surface finish are a consequence of time-dependent friction.     

An asperity model to simulate rupture along heterogeneous fault surfaces

A model has been developed to simulate the statistical and mechanical nature of rupture on a heterogeneous strike-slip fault. The model is based on the progressive failure of circular asperities of varying sizes and strengths along a fault plane subjected to a constant far-field shear displacement rate. The basis of the model is a deformation and stress intensity factory solution for a single circular asperity under a unidirectional shear stress. The individual asperities are unified through the fault stiffness and the far-field stress and displacement. During fault deformation asperities can fail and reheal, resulting in changes in the local stresses in the asperities, stress drops, and changes in the stiffness of the fault. Depending on how the stress is redistributed following asperity failure and on the strenghts of the neighboring asperities an earthquake event can be the failure of one or more asperities. Following an earthquake event seismic source parameters such as the stress drop, energy change, and moment magnitude are calculated. Results from the model show a very realistic pattern of earthquake rupture, with reasonable source parameters, the proper magnitude-frequency behavior, and the development of characteristic earthquakes. Also the progression ofb-values in the model gives some insight into the phenomenon of self-organized criticality.     

Stress Transfer During Pressure Solution Compression of Rigidly Coupled Axisymmetric Asperities Pressed Against a Flat Semi-Infinite Solid

In a previous work, we developed a numerical model of compression by pressure solution (PS) of a single axisymmetric asperity pressed against a flat semi-infinite solid. The dissolution rate at any point along the contact and at any time t was determined by (1) computing the normal stress distribution from the current shape of the asperity, and (2) solving the diffusion equation inside the fluid-saturated solid-solid interface, including local dissolution source terms corresponding to the stress field previously determined. The change in shape of the asperity during an infinitesimal time interval δt can then be calculated and the entire procedure repeated as many times as desired. Our results showed that, as the contact flattens and grows during PS, the initial elastic deformation is partially relaxed and the stress transferred from the contact center to the edge. Our goal in the present paper is to demonstrate that, among a population of asperities, stress can also be transferred from one contact to another and that the overall compaction rate can be significantly affected by this process. For this purpose we extended our previous numerical model to simulate PS of two rigidly coupled spherical asperities simultaneously pressed against a flat semi-infinite solid. We considered two end-member cases: 1) transfer of stress to a newly created, not initially present contact, 2) transfer of stress between asperities with different sizes. In both cases, stress was transferred from the most stressed asperity to the least, and, the overall PS displacement rate was reduced. Thus, formation of new contacts and heterogeneous distribution of asperity sizes, which are both expected to exist in rough fractures with self-affine aperture or in heterogeneous granular materials with variable grain-packing geometry, may significantly slow down PS creep compaction.     

Stick-slip and strain waves in the physics of earthquake rupture: experiments and models

The study of stick-slip is directly connected with the formulation of the consistent concept of seismicity migration and explanation of the nature of strain waves observed at the contacts of blocks and plates. We propose a phenomenological model to describe the initiation of stick-slip at the rough contact of blocks of rocks. The model contains the leading factors of stick-slip process (friction, roughness of contact surfaces, and asperity). The model reproduces the universal profile of slip velocity and displacement, velocity of dynamic rupture observed in experiments. The stick-slip motion at a rough surface of a fault is shown to be a nonlinear process and can be described by the generalized sine-Gordon equation. The results of calculations and a comparison with experimental data testify to the existence of waves of a new type — solitary waves of sliding. The analogies are found between the strain waves generated due to stick-slip at the contact of the blocks of rocks, strain waves in the crustal faults, and those within the lithosphere.     

由宽频带辐射能量目录和地震矩目录给出的视应力及其地震学意义

对NEIC宽频带地震辐射能量目录和哈佛矩心矩张量（CMT）目录的比较，给出了关于视应力的一些可能是有意义的结果，尽管目前的结果误差仍很大，可靠性也是有限的，但这种比较所提供的线索却颇值得注意，视应力的计算给出关于地震断层面上非线性动摩擦函数的线索，能量／地震矩之比随地震大小的变化表明，在BK模型框架下，对于走滑型地城,依赖于滑动速度的摩擦似乎占主要地位，而对于非走滑型地震，依赖于位移的摩擦似乎占主要地位，主震和余震的能量／地震矩之比的比较表明，对于走滑型地震，余震的视应力平均地说来低于主震的视应力，而对于非直滑型地震，余震的视应力即有高于主震的也有低于主震的，这对于障碍体和凹凸模式的讨论及模型中地震破裂停止条件的设置可能具有一定的参考意义。     

Effect of the time-weakening friction law during the nucleation process

Based on dynamic rupture simulations on a planar fault in a homogeneous half-space, we investigated the nucleation processes using the time-weakening friction law. Both the characteristic time and the rupture speed in the nucleation asperity play an important role in determining rupture behaviors on a fault plane following the time-weakening friction law, with which rupture starts from a single point in the nucleation asperity and propagates at a given speed toward the boundary of the nucleation area. Rupture with a small characteristic time or a large rupture speed in the nucleation asperity propagates earlier from the hypocenter. Rupture following the slip-weakening friction law requires a smaller radius of nucleation patch to have similar rupture front contours of the time-weakening friction law. Even if the rupture velocity in the nucleation patch of the time-weakening friction law increases to infinity, the peak slip rate in the nucleation asperity is smaller than that of the slip-weakening law. The peak ground velocity distributions of ruptures following the two friction laws are also compared.     

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.     

障碍体和凹凸体对地震矩的影响

本文利用具有非均匀剪切应力分布的圆盘状裂缝问题的解,定量计算了障碍体和凹凸体对地震矩的影响.我们发现,除非障碍体的尺寸很小或很接近于整个断层的尺寸,含有障碍体的断层的地震矩,大致为具有均匀应力降的等面积断层的地震矩的40%.由此可以推知,在障碍体的这种尺寸范围内,视应力降大致也是△Af=SA-f的40%.这里SA是远场应力,f是残余摩擦应力.另外,已存在滑动带对于凹凸体破坏的地震矩有明显的放大作用,这种作用在凹凸体尺寸较小时特别显著.例如,当凹凸体半径是整个断层半径的2/10时,凹凸体上的矩被放大7倍多;当其半径为整个断层半径的1/10时,凹凸体的矩将被放大约30倍. 文中还比较了三维效应和二维效应的差异.我们发现,Mx0/M0c在三维情形对障碍体尺寸的变化并不敏感,这是与二维情形很不相同的.这里,Mx0是具有障碍体的断层破裂所产生的地震矩,M0c是相应的均匀断层破裂的矩.所以,二维分析不适用于具有障碍体的三维断层.然而,在分析已存在滑动带对凹凸体破裂的矩的影响时,平面应变结果与三维结果差异并不很大.      

含有单-粗糙面的走滑断层的滑动弱化不稳定性h

本文研究了有限长度含有单一粗糙面的垂直走滑断层的滑动弱化不稳定性.我们假定断层强度在其深度方向是均匀的,而在走滑方向是不均匀的,即存在一个高强度段.并假定断层上的粗糙段及断层上的其它部分具有相同类型的本构律,只是峰值应力不同.断层周围的地壳材料用上、下表面应力自由的弹性平板来模拟.我们用有限元方法研究了理论位移场、应力场和应变场随远场位移的演变情况.此时,断层位移和摩擦应力并不是预先给定的,而是在求解过程中同时确定的.根据计算结果,我们比较了理论位移场及断层上摩擦应力分布在稳定和不稳定滑动情况时的差别.并与内部含有低强度段的走滑断层的场合做了比较.      

Source parameters of mining-induced seismic events: An evaluation of homogeneous and inhomogeneous faulting models for assessing damage potential

Source parameter estimates based on the homogeneous and inhomogeneous source models have been examined for an anomalous sequence of seven mine-induced events located between 640 and 825 m depth at Strathcona mine, Ontario, and having magnitudes ranging betweenm _N 0.8 and 2.7. The derived Brune static stress drops were found to be similar to those observed for natural earthquakes (30 bars), whereas dynamic stress drops were found to range up to 250–300 bars. Source radii derived from Madariaga's model better fit documented evidence of underground damage. These values of source radii were similar to those observed for the inhomogeneous model. The displacement at the source, based on the observed attenuation relationship, was about 60 mm for three magnitude 2.7 events. This is in agreement with slip values calculated using peak velocities and assuming the asperity as a Brune source within itself (72 mm). By using Madariaga's model for the asperity, the slip was over 3 times larger than observed. Peak velocity and acceleration scaling relations with magnitude were investigated by incorporating available South African data, appropriately reduced to Canadian geophysical conditions. The dynamic stress drop scaled as the square root of the seismic moment, similar to reported results in the literature for crustal earthquakes. This behavior suggests that the size of the asperities responsible for the peak ground motion, with respect to the overall source size, follow distributions that may be similar over a wide range of magnitudes. Measurements of source rupture complexity (ranging from 2 to 4) were found to agree with estimates of overall source to asperity radii, suggesting, together with the observed low rupture velocities (0.3 to 0.6 ), that the sources were somewhat complex. Validation of source model appropriateness was achieved by direct comparison of the predicted ground motion level to observed underground damage in Creighton mine, located within the same regional stress and geological regime as Strathcona mine. Close to the source (<100 m), corresponding to relatively higher damage levels, a good agreement was found between the predicted peak particle velocities for the inhomogeneous model and velocities derived based on established geomechanical relationships. The similarity between asperity radii and the regions of the highest observed damage provided additional support for the use of the inhomogeneous source model in the assessment of damage potential.     

Slip-weakening instability on a strike-slip fault with asperity

In this paper, we studied the slip-weakening instability on a finite and vertical strike-slip fault with asperity. We assumed that the fault strength is uniform in depth, but it is non-uniform in the strike direction, i.e., there is an asperity on the fault. The segments of the fault with and without asperity follow the same type of constitutive law, but the peak stresses are different. The material surrounding the fault is represented by elastic plates whose upper-surface and bottom are stress-free. We use finite element method to study the evolution of theoretical displacement, stress and strain fields that vary with a remote displacement. Here the displacement and frictional stress are not given in advance, but are determined in the solving process. According to the results, we compared the theoretical displacemeut and the distribution of frictional stress on the fault between the stable and unstable slips. In addition, we compared the results in this paper with known results for the strike-slip fault with a segment of lower strength.     

Predicting the displacement of triple pendulum™ bearings in a full‐scale shaking experiment using a three‐dimensional element

The accuracy of a series spring model to predict the peak displacement and displacement history of Triple Pendulum? (TP) bearings in a strongly shaken, full‐scale building is evaluated in this paper. The series spring model was implemented as a self‐contained three‐dimensional TP bearing element in OpenSees and is now available for general use. The TP bearing element contains the option for constant friction or a generalized friction model that accounts for the effect of instantaneous velocity and compression load on the friction coefficient. Comparison between numerical simulation and experimental data of a five‐story steel moment frame building shows that the peak displacement of isolation system can generally be predicted with confidence using a constant friction coefficient model. The friction coefficient model accounting for the effect of axial load and velocity leads to minor improvement over the constant friction coefficient models in some cases. Copyright © 2013 John Wiley & Sons, Ltd.     

2011年M_W9.0东日本大地震动力学破裂过程的数值模拟

力学上,地震可以看作在应力场作用下由于断层带介质的突然损伤或软化导致的断层带失稳事件.本文基于这个地震动力学模型,利用一种可以模拟断层大位错的有限元方法,研究了2011年MW9.0东日本大地震(Tohoku-Oki)的动力学破裂过程.比较了无障碍体和具有不同刚度障碍体的断层带模型产生的断层位移、位错和应力降.主要结果表明,障碍体的存在并不明显地改变障碍体区域的初始构造应力场.对有障碍体情形,准静态结果显示断层上盘最大逆冲位移和最大剪切位错分别为51m和58m,均发生在海底表面海沟处,与无障碍体的结果(最大剪切位错约55m)相比差别不大;下盘最大倾向位移(-10m)并不与上盘最大值出现在同一位置,而是在障碍体处.障碍体处剪应力降(约11 MPa)大于周围非障碍体区域.障碍体处正应力降的最大值约为3 MPa.模拟结果似乎不支持海山是导致本次地震异乎寻常大位错的原因,而倾向于断层带剪切刚度在地震过程中极度损伤或软化.     

The effect of normal stress on the real area of contact during frictional sliding in rocks

The real area of contact during frictional sliding has been determined as a function of changing normal stress in triaxial experiments through the use of thermodyes. Utilizing the technique, described by Teufel and Logan in 1978, with saw-cut surfaces inclined 35° to the load axis, determinations were made for monolithologic sliding of Tennessee sandstone and Indiana limestone and dilithologic sliding of the same rocks. Confining pressures to 200 MPa were investigated at a constant shortening rate of 10^–2 mm/sec and at room temperature. Direct measurements were made of single-asperity areas and the asperity density. The product of these measurements gives the percent area of real contact across the sliding surface. Single-asperity area and density are found to remain relatively constant during the displacement. Single-asperity areas are in the ranges of 0.4 to 6×10^–2 mm² for sandstone, 0.8 to 2×10^–2 mm² for limestone, and 0.2 to 24×10^–2 mm² for sandstone sliding against limestone. These values are smaller than the grain size of either rock. The values increase with increasing normal stress for both monolithologic and dilithologic sliding. In sandstone the asperity density increases from about 0.8 to 2.75 contacts per square millimeter in a logarithmic fashion. Monolithologic limestone has values of about 0.9 contacts per square millimeter and does not show significant change with increasing normal stress. The percent area of real contact increases in all cases, with average maximum values of 16% of the apparent area at a normal stress of 374 MPa in sandstone, 18% at 25 MPa in limestone, and 22% at 123 MPa in the dilithologic specimens. The normal stress recalculated for the real area of contact approaches the unconfined compressive strength for sandstone and limestone.     

Direct observation of frictional contacts: New insights for state-dependent properties

Rocks and many other materials display a rather complicated, but characteristic, dependence of friction on sliding history. These effects are well-described by empirical rate- and state-dependent constitutive formulations which have been utilized for analysis of fault slip and earthquake processes. We present a procedure for direct quantitative microscopic observation of frictional contacts during slip. The observations reveal that frictional state dependence represents an increase of contact area with contact age. Transient changes of sliding resistance correlate with changes in contact area and arise from shifts of contact population age. Displacement-dependent replacement of contact populations is shown to cause the diagnostic evolution of friction over a characteristic sliding distance that occurs whenever slip begins or sliding conditions change.