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 experimental study of triggered stick-slip

Stick-slips have been studied in the laboratory on granite, labrodorite and sandstone samples of two different sizes. Different roughness was achieved on the sawcut surfaces by finishing them with different grinding compounds ranging from grit 40 to grit 1000. Stick-slips occurred as a result of 1) slowly increasing the shear and normal stresses, 2) superimposing a sinusoidal stress modulation (0.1 and 10 Hz) on the slowly increasing stresses, 3) triggering by a stress impulse when the shear stress was well below the levels where stick-slips occurred without the impulse, and 4) foreshocks.Stick-slips triggered by impulses or foreshocks occurred long after the beginning of the triggering events, i.e., long in comparison with elastic wave travel times through the sample. All triggered events were very rich in high frequencies (corner frequency of 100 kHz). The untriggered stick-slips did not contain much energy at the high frequencies (corner frequency of 10 kHz). The dynamic friction coefficients for the triggered stick-slips were smaller than for the untriggered events.The long delay between the onset of the trigger and the stick-slip, and the high frequencies may be a consequence of corrosion of asperities. The ultimate triggering and the rate of corrosion are likely related to the interplay of the normal and shear stresses as they load and unload the fault surface. The consistent shape of the high frequency spectra is probably due to sample resonances which are excited rather than being characteristic of the details of the stick-slips. If these laboratory observations are directly applicable to earthquake seismology, the spectra of earthquakes which were triggered by other earthquakes should be anomalously rich in high frequencies.     

黏滑实验的震级评估和应力降分析

本文通过三种结构模型的黏滑地震模拟实验,利用高频速度连续观测系统获得了地震失稳过程的速度特征,讨论了最大位移量的选取方法,估算了实验室黏滑型地震的矩震级,探讨了黏滑类型、应力降大小与震级的关系.结果表明,黏滑型地震的应力降过程可能包含一次到多次高频振荡,对应若干次黏滑子事件.高频振荡的摆动幅度很大,包含有多种频率成分,峰值速度0.003~0.008 m·s-1.初步估计黏滑型地震的震级范围为-4.4~-3级,断层构造面的差异对各种黏滑模型的地震震级分布有明显影响.总体来看应力降与地震震级没有明显相关性,决定地震震级的主要因素应当是震源尺度.     

断层黏滑动态变形过程的实验研究

实验研究断层黏滑过程的变形演化,尤其是失稳黏滑瞬间的断层位移演化特征,对于了解地震从孕育到发生的全过程具有重要意义.本文基于数字散斑相关方法(digital speckle correlation method, DSCM),用三套图像采集系统(两套低速和一套高速图像采集系统)搭建了断层黏滑过程的多观测区域、多时间尺度的变形场测量系统,并用此系统对一种花岗岩双剪滑动模型的黏滑过程进行了实验研究.对间黏滑期和黏滑期断层位移演化特征进行深入分析的结果表明:间黏滑期断层位移演化体现出空间上的非均匀性和时间上的"趋同化"特征,断层滑动趋同化也许是断层错动匀阻化的一种宏观表现形式;断层黏滑动态过程持续时间非常短(本文300 mm断层黏滑过程持续时间约在1 ms量级),黏滑失稳前会出现预滑,预滑出现到黏滑失稳发生所经历的时间与黏滑失稳过程所用时间相差一个量级;断层的一次黏滑由若干个滑动速度不同的、小的失稳滑动组成,黏滑失稳过程中断层的滑动速度呈现出波动性,整个滑动过程中断层经历了多个高速滑动和低速滑动的交错.     

Characteristics of the AE Distribution and Triggering Mechanism of a Simulated Meter-Scale Fault after Stick-Slip Events

With the more complete acoustic emission (AE) catalog improved by the multi-channel AE matched-filter technique (MFT), we study the spatiotemporal evolution of the AE activities after laboratory stick-slip events incorporate with the slip data recorded by displacement transducers on an ~1.5 m granite fault. The results show that the number of the AE events identified by MFT is about 9 times larger than that of the traditional method. A logarithmic expansion of early AE events along the fault strike is observed as a function of time, whereas the fault does not slip in the same manner. Thus, we related the expansion of the early AE events along the fault to the stress transfer caused by the adjacent AE events. Moreover, there is a good correlation between the cumulative number of the later AE events and the amount of fault slip. It suggests that the stress change caused by the continuous slip of the simulated fault after the stick-slip events response for the later AE events near or on the simulated fault.     

应力扰动对断层摩擦声发射活动影响的实验研究

利用中尺度岩石摩擦实验开展了应力扰动对断层摩擦滑动过程中声发射活动影响的实验研究,据此讨论同震应力变化引起的小震活动特征及其预测意义.实验结果表明方波状应力扰动对粘滑失稳前的声发射活动有明显影响,即触发了一些声发射事件,使得粘滑发生前声发射活动增强并使出现的时间提前.这种影响随平均正应力的提高和扰动振幅的增大而增强,其...     

The planetary system and solar-terrestrial phenomena

Summary The Sun moves at different distances round the barycentre of the Solar System with different velocities depending on the distribution of the planets. The mean period of the Sun's motion round the barycentre of 11.8627 years and its time variability were observed. A relation between the basic period of 178.4 years and other periods p_i in the Sun's motion (p_i=178.4/i; i=1, 2, ...) was found. The consistency of the periods in the Sun's motion round the barycentre with the periods of different solar-terrestrial phenomena was studied. It seems that the planetary system governs these phenomena.

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Analysis of stick-slip and earthquake mechanism

A mathematical model of a sliding system that contains a frictional surface embedded in an elastic specimen and machine is proposed to analyze the frictional behavior observed in laboratory and field. It is shown that stick-slip occurs if the slope of the velocity-stress relation on the sliding surface exceeds a critical value at a certain point between static and kinetic frictions. This condition, coupled with Amontons' law and other subsidiary relations, predicts the effects of normal stress, gouge thickness, temperature, and loading rate etc. on the stick-slip instability, consistent with known experimental evidence. The elastic-wave radiation associated with stick-slip is governed by Brune's source time function, in which rise time and effective stress are proportional to fault dimension and stress drop, respectively.     

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.     

Kinematic response functions and dynamic stiffnesses of bridge embankments

Recognizing that soil–structure interaction affects appreciably the earthquake response of highway overcrossings, this paper compares approximate analytical solutions and finite element results to conclude on a simple procedure that allows for the estimation of the kinematic response functions and dynamic stiffnesses of approach embankments. It is shown that the shear‐wedge model yields realistic estimates for the amplification functions of typical embankments and reveals the appropriate levels of dynamic strains which are subsequently used to estimate the stiffness and damping coefficients of embankments. The shear‐wedge model is extended to a two‐dimensional model in order to calculate the transverse static stiffness of an approach embankment loaded at one end. The formulation leads to a sound closed‐form expression for the critical length, L_c, that is the ratio of the transverse static stiffness of an approach embankment and the transverse static stiffness of a unit‐width wedge. It is shown through two case studies that the transverse dynamic stiffness (‘spring’ and ‘dashpot’) of the approach embankment can be estimated with confidence by multiplying the dynamic stiffness of the unit‐width wedge with the critical length, L_c. The paper concludes that the values obtained for the transverse kinematic response function and dynamic stiffness can also be used with confidence to represent the longitudinal kinematic response function and dynamic stiffness, respectively. Copyright © 2002 John Wiley & Sons, Ltd.     

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.

     

Velocity anomalies: An alternative explanation based on data from laboratory experiments

Locations and velocities were calculated for microseisms occurring in samples of rock subjected to triaxial loading and injection of pore fluid. This was accomplished by analyzing arrival times of acoustic emission using an automatic first arrival picker. Apparent velocity anomalies were observed prior to both failure of intact samples and violent slip in samples containing saw cuts. Further analysis revealed that these fluctuations in calculated velocity were not due to changes in the true seismie velocity. Instead, variations in calculated velocity are shown to be related to sampling errors in picking first arrivals. The systematic picking of late first arrivals for small magnitude events was found to be a persistent bias resulting in low calculated velocities. This has encouraged the reexamination of earthquake records to determine how important sampling biases are in contributing to reported velocity anomalies.     

Some comparisons between mining-induced and laboratory earthquakes

Although laboratory stick-slip friction experiments have long been regarded as analogs to natural crustal earthquakes, the potential use of laboratory results for understanding the earthquake source mechanism has not been fully exploited because of essential difficulties in relating seismographic data to measurements made in the controlled laboratory environment. Mining-induced earthquakes, however, provide a means of calibrating the seismic data in terms of laboratory results because, in contrast to natural earthquakes, the causative forces as well as the hypocentral conditions are known. A comparison of stick-slip friction events in a large granite sample with mining-induced earthquakes in South Africa and Canada indicates both similarities and differences between the two phenomena. The physics of unstable fault slip appears to be largely the same for both types of events. For example, both laboratory and mining-induced earthquakes have very low seismic efficiencies where _a is the apparent stress and is the average stress acting on the fault plane to cause slip; nearly all of the energy released by faulting is consumed in overcoming friction. In more detail, the mining-induced earthquakes differ from the laboratory events in the behavior of as a function of seismic momentM ₀. Whereas for the laboratory events 0.06 independent ofM ₀, depends quite strongly onM ₀ for each set of induced earthquakes, with 0.06 serving, apparently, as an upper bound. It seems most likely that this observed scaling difference is due to variations in slip distribution over the fault plane. In the laboratory, a stick-slip event entails homogeneous slip over a fault of fixed area. For each set of induced earthquakes, the fault area appears to be approximately fixed but the slip is inhomogeneous due presumably to barriers (zones of no slip) distributed over the fault plane; at constant , larger events correspond to larger _a as a consequence of fewer barriers to slip. If the inequality _a / 0.06 has general validity, then measurements of _a =µE _a /M ₀, where is the modulus of rigidity andE _a is the seismically-radiated energy, can be used to infer the absolute level of deviatoric stress at the hypocenter.     

Model studies of shear displacement along a pre-existing fault

The faiture of pre-existing faults was studied experimentally and theoretically. The stress concentration around open and healed slits in plexiglas under uniaxial stress was calculated and compared to observations by Schlieren photographs. The rupture velocities along these pre-existing faults were found to vary from 0.25 of the shear velocity up to velocities slightly exceeding the shear velocity. Sudden fault dislocation was often preceded by stable sliding. The amplitudes of elastic waves radiated in the forward quadrant of the rupture were larger than the amplitudes in the backward quadrant.     

Nonlinear dynamics of two degree of freedom systems with linear and nonlinear stiffnesses

In this study,a new analytical approach is developed to analyze the free nonlinear vibration of conservative two-degree-of-freedom(TDOF) systems.The mathematical models of these systems are governed by second–order nonlinear partial differential equations.Nonlinear differential equations were transferred into a single equation by using some intermediate variables.The single nonlinear differential equations are solved by using the first order of the Hamiltonian approach(HA).Different parameters,which have a significant impact on the response of the systems,are considered and discussed.Some comparisons are presented to verify the results between the Hamiltonian approach and the exact solution.The maximum relative error is less than 2.2124 % for large amplitudes of vibration.It has been established that the first iteration of the Hamiltonian approach achieves very accurate results,does not require any small perturbations,and can be used for a wide range of nonlinear problems.     

Simulation of triggered earthquakes in the laboratory

The experiments were conducted for the study of stick-slip at the contact between two granite blocks. Three cases were studied under the following conditions: 1) the increase of load at a constant rate; 2) the additional application of sinusoidal oscillations in the frequency range from 1 to 30 Hz; 3) subjection to the impulse in the kilohertz frequency range. The imposition of sinusoidal oscillations with the amplitude of 15% of the maximal load caused the reduction of time by 10% for the discussed in terms of durability. The high frequency impulse influence increased this effect and also caused essential changes in the amplitude of elastic oscillations generating during the stick-slip. The trigger phenomena should be integrated in prediction models of the time and magnitude of earthquakes.     

Instability on a weakening fault

Weakening of a prestressed sawcut in Westerly granite under laboratory condition is accomplished by injecting pressurized fluid into the sawcut. After injection a sequence of stick-slips is observed while the deviatoric stress decreases successively with each stick-slip. On the basis of the experimental observation we develop a model of fault instability due to inhomogeneous and progressive weakening of the fault. According to this model, the fault surface is divided into the slipped and the locked regions, depending on whether or not the local state of stress satisfies the friction criterion. The average shear stress in the slipped region decreases with time and, in order to maintain a quasi-static equilibrium, shear stress in the remaining locked region on the fault surface increases. This situation may last until a critical state of stress on the fault is met, at which a sudden instability (stick0slip) may occur. We suggest that this mechanism of stress transfer may be a viable mechanism of induced seismicity and aftershocks, in addition to the well-known mechanism of a local increase of pore pressure. By comparing the experimental data with model predictions we show that the critical condition for slip instability is when the average shear stress over the locked region becomes equal to the value given by the friction criterion. Thus the friction criterion established for slip on fractures on which the state of stress is macroscopically uniform may also be applicable to fractures on which the stress state is macroscopically heterogeneous.     

Boundary differential equation method: simplified dynamic soil stiffnesses for embedded rigid foundations

With a simplified model and Galerkin's weighted residual procedure, two simple differential equations of dynamic behavior of a bounded rectangular medium are established along the boundaries in the x- and y-direction in the medium. Solutions of these equations yield closed form expressions of soil stiffnesses for various cases of a partially embedded rigid foundation, including the stiffnesses per depth of foundation with rectangular base area and the stifnesses of strip foundation. The developed procedure provides the definition of the weight functions, which are used in Galerkin's method for weighted residual. In addition to these weight functions, their conjugators are also suitable for weight functions. When the soil depth is finite, the original weight functions fail to produce physically meaningful results in some frequency range but the conjugators do not fail at any frequencies. The developed equations to compute soil stiffnesses for embedded foundations are simple yet capable of calculating the responses close to those computed by the much more elaborated finite element method.     

集宁地电观测系统故障检查及处理

对集宁2017年4月1日-5月31日地电阻率原始观测数据曲线常规分析发现,NS、EW测道数值差异明显,采用电测深曲线进行对比分析,发现2测道电阻率数值基本一致。在同场范围内,地电阻率与电测深曲线应同步变化。对集宁地电阻率布极区、观测环境、观测系统、装置系统进行系列检查、测试,发现NS测道导线线芯与导线铠导通,导线铠与避雷箱箱体接触导致线路漏电,造成数值出现误差。通过对外线路以及电极引线等进行优化改造,集宁地电阻率观测数据恢复正常。