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.     

两种摩擦本构关系的对比研究

目前有两种最常用的岩石摩擦本构关系,其主要区别在于是否与静接触时间有关。对这两种本构关系的基本性质进行了对比研究,得出如下结论：（１）在匀速滑动的稳态附近,两种本构关系趋向一致;（２）在正应力恒定条件下,两种本构关系的主要差异在于克服摩擦力所需能量的大小。与静接触时间相关的本构关系在粘滑中需消耗较大的能量来克服摩擦阻力,在粘滑的减速段,这种本构关系可达到的最低速率比另一种本构关系低１０个数量级;（３）在正应力变化条件下,与静接触时间相关的本构关系基本继承了正应力恒定条件下的行为特征,而另一种本构关系在剪应力－速度相平面上却出现了一个长尾巴,使最低滑动速率降到比正应力恒定时的值低１０多个数量级     

Stability analyses and numerical simulations of the single degree of freedom spring-slider system obeying the revised rate- and state-dependent friction law

The linear and nonlinear stabilities of the single degree of freedom spring-slider system which accords to the revised rate- and state-dependent friction law (RSF) (Nagata et al. J Geophys Res 117 (B2):B2314, 2012) are analyzed. The revised ageing law obtained by Nagata et al. (J Geophys Res 117 (B2):B2314, 2012) incorporates the effects of changes in shear stress. Numerical simulations on the cyclic stick–slip motions of the system are developed and compared with the results of the systems according to the original ageing law or the slip law. From the insight of the stability analyses and numerical simulations, it is found that the revised ageing law integrates the “healing effect” feature of the original ageing law and the dynamic slip features of the slip law. In the stick–slip cycles, the velocity decreases with non-constant states during the dynamic overshoot for the revised ageing law, which is different from both the original ageing law and the slip law. Although the revised ageing law concluded from the low velocity friction experiments cannot account for the earthquake-like high velocity friction experiments, it can be used in earthquake nucleation with low velocity. The stability analyses and the results of numerical simulations are helpful to understanding the implications of the revised ageing law.     

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.     

Phase synchronization of slips by periodical (tangential and normal) mechanical forcing in the spring-slider model

In the present study, the character of slip regimes under weak external periodical (tangential and normal) mechanical forcing has been investigated in a laboratory spring-slider system. We report the experimental evidence of phase synchronization in a slip dynamics, induced by the external periodic mechanical impact. At certain conditions, we have a stick-slip effect in the spring-slider system. To describe this effect, we can use rate-and state-dependent friction law. In our experiments, the slip events are distinguished by acoustic emission bursts, which are generated by slider displacement. In addition to drag, the weak variable mechanical forcing was superimposed either tangential or normal to the slip plane. With increasing external forcing one can see increasing phase synchronization of the first arrivals (onsets) of stick-slip generated acoustic pulses. The grouping of the onsets in a certain phase of the external periodic forcing is considered as a hallmark of the phase synchronization. The onsets of stick-slip pulses in the case of normal mechanical forcing are shifted relative to onsets in the case of tangential forcing.     

基于速率-状态依从摩擦定律的前郭震群余震活动率及模型参数相关性研究

基于速率-状态依从摩擦定律的地震活动率时空预测模型,以同震库伦应力变化作为模型初始应力扰动,模拟了2013年吉林前郭MS5.8震群的余震活动率变化。考虑模型参数相关性,在模拟中采用2种不同的拟合方案,一是余震持续时间t_a不固定条件下的拟合,二是余震持续时间t_a固定条件下的拟合。结果显示,t_a不固定条件下的拟合方式可获得较好的AIC评价,适用于震后早期的趋势判定; t_a固定条件下的拟合计算耗时更短,拟合误差更小,理论模拟结果与前郭震群实际地震时序特征更为吻合。采用该方案对截至2016年10月24日的余震活动率变化进行了回溯性预测检验,结果显示模型预期的余震日频次与实际记录呈较好的正相关关系。研究还发现,主震破裂面附近的同震应力影区导致震后早期模型预测值相对于实际偏低,说明前郭序列余震活动可能还存在其他触发机制。     

基于Chester-Higgs模型探讨摩擦生热对断层演化进程的影响

速率和状态相依赖的摩擦定律是本文采用的重要定律。结合Chester-Higgs摩擦模型和McKenzie-Brune摩擦生热模型,在一维弹簧-滑块-断层近似模型下,利用四阶变步长的Dormand-Prince算法,研究探讨了断层摩擦生热对断层演化的影响。结果表明:与忽略温度影响的情形相比,摩擦生热造成的温度上升可导致断层滑移时刻的略微提前,并伴随着摩擦系数和状态变量的下降,同时也使得断层的滑移量和应力降略有减小,而滑移速率有所增大;另外,在考虑温度影响时,有效正应力和临界滑移距离也会影响断层的演化过程,断层上的有效正应力越大,断层失稳时刻越提前,温度上升越明显;断层的临界滑移距离越大,断层失稳时刻则越迟,温度上升越显著,但当临界滑移距离超过5 cm时,具有不同临界滑移距离的断层,失稳时的温度则基本保持一致。      

Hydraulics of runoff and loess loam deposition

On many more or less loamy soils, rill erosion is reported to start on slopes that are equal to or steeper than 2–3°; critical Froude numbers for the start of rill wash on these slopes vary between 2·0 and 3·0. This explains why colluvial deposition often occurs on slopes below 2–3°, when water spreads out at the downslope extremities of the rills. The critical hydraulic conditions for loess loam deposition were tested in the laboratory for slopes of 0·5° and 2°, applying unit-discharges (q) up to 10 cm²/s. It appeared from these experiments that for afterflow, without raindrop impact, deposition starts for critical load concentrations (c_cr) varying between several g/1 and about 60 g/l. Under rain c_cr amounts to a minimum value of 100–125 g/l and it increases when the runoff film becomes thinner. Nevertheless, deposition in pluvial runoff is also possible, as was the case during the Weichselian, according to data from quarries in Belgium and in The Netherlands. A modified Kalinske equation is proposed for c_cr prediction, with the introduction of a typical empirical coefficient C_r and considering such factors as shear stress and mean particle size. Massive sedimentation may occur when it stops raining and afterflow starts, since c_cr values are then much lower. It is shown from the Shields' diagram that loamy suspensions are more sensitive to sedimentation than sands in clear water.     

Evaluation of combination rules for maximum response calculation in multicomponent seismic analysis

The existing rules for combining peak response to individual components of ground motion are evaluated. The response values r_e to two horizontal components of ground motion estimated by four multicomponent combination rules—SRSS‐, 30%‐, 40%‐ and simplified‐SRSS‐rules—are compared with the critical response, r_cr, obtained by the CQC3‐rule, which takes into account the direction of the principal ground components with respect to the structural axes and provides the largest response over all possible seismic incident angles. The following results are obtained in the first part of the paper and are valid for any elastic structure and any earthquake design response spectrum: For realistic values of the ratio γ of the design spectra for the two principal components of ground motion the SRSS‐rule estimate lies between 0.79r_cr and 1.00r_cr, the Simplified‐SRSS‐rule estimate lies between 1.00r_cr and 1.26r_cr, the 40%‐rule estimate lies between 0.99r_cr and 1.25r_cr, and the 30%‐rule estimate lies between 0.92r_cr and 1.16r_cr. None of the multicomponent combination rules account for the increase in response of systems if the vibration periods of the two modes that contribute most to the response to the x‐ and y‐components of ground motion are close to each other. Evaluated in the second part of the paper is the accuracy of the multicomponent combination rules in estimating the response of a range of one‐storey systems with (a) symmetrical plan and (b) unsymmetrical plan, and of two multistorey buildings. The SRSS‐rule underestimates the response by up to 16% and the other three rules overestimate it by up to 18%. Although these errors appear to be smaller than the many approximations inherent in structural design, they can be eliminated with very little additional computation by using an explicit formula for the critical response based on the CQC3 rule. Copyright © 2001 John Wiley & Sons, Ltd.     

Evaluation of combination rules for peak response calculation in three‐component seismic analysis

The responses, r_e, given by several multicomponent combination rules used in seismic codes for determining peak responses to three ground motion components are evaluated for elastic systems and compared with the critical response r_cr; this is defined as the largest response for all possible incident angles of the seismic components and obtained by means of the CQC3‐rule when a principal seismic component is vertical, or the GCQC3‐rule when it departs from the vertical direction. The combination rules examined are the SRSS‐, 30%‐, 40%‐ and IBC‐rules, considering different alternatives for the design horizontal spectrum. Assuming that a principal seismic component is along the vertical direction, the upper and lower bounds of the ratio r_e/r_cr for each combination rule are determined as a function of the spectral intensity ratio of the horizontal seismic components and of the responses to one seismic component acting alternately along each structural axis. Underestimations and overestimations of the critical response are identified for each combination rule and each design spectrum. When a component departs from the vertical direction, the envelopes of the bounds of the ratio r_e/r_cr for each combination rule are calculated, considering all possible values of the spectral intensity ratios. It is shown that the inclination of a principal component with respect to the vertical axis can significantly reduce the values of r_e/r_cr with respect to the case when the component is vertical. Copyright © 2003 John Wiley & Sons, Ltd.     

Using Grain‐Size Distribution Methods for Estimation of Air Permeability

Knowledge of air permeability (k_a) at dry conditions is critical for the use of air flow models in porous media; however, it is usually difficult and time consuming to measure k_a at dry conditions. It is thus desirable to estimate k_a at dry conditions from other readily obtainable properties. In this study, the feasibility of using information derived from grain‐size distributions (GSDs) for estimating k_a at dry conditions was examined. Fourteen GSD‐based equations originally developed for estimating saturated hydraulic conductivity were tested using k_a measured at dry conditions in both undisturbed and disturbed river sediment samples. On average, the estimated k_a from all the equations, except for the method of Slichter, differed by less than ± 4 times from the measured k_a for both undisturbed and disturbed groups. In particular, for the two sediment groups, the results given by the methods of Terzaghi and Hazen‐modified were comparable to the measured k_a. In addition, two methods (e.g., Barr and Beyer) for the undisturbed samples and one method (e.g., Hazen‐original) for the undisturbed samples were also able to produce comparable k_a estimates. Moreover, after adjusting the values of the coefficient C in the GSD‐based equations, the estimation of k_a was significantly improved with the differences between the measured and estimated k_a less than ±4% on average (except for the method of Barr). As demonstrated by this study, GSD‐based equations may provide a promising and efficient way to estimate k_a at dry conditions.     

Sediment transport in high‐speed flows over a fixed bed: 2. Particle impacts and abrasion prediction

Single bed load particle impacts were experimentally investigated in supercritical open channel flow over a fixed planar bed of low relative roughness height simulating high‐gradient non‐alluvial mountain streams as well as hydraulic structures. Particle impact characteristics (impact velocity, impact angle, Stokes number, restitution and dynamic friction coefficients) were determined for a wide range of hydraulic parameters and particle properties. Particle impact velocity scaled with the particle velocity, and the vertical particle impact velocity increased with excess transport stage. Particle impact and rebound angles were low and decreased with transport stage. Analysis of the particle impacts with the bed revealed almost no viscous damping effects with high normal restitution coefficients exceeding unity. The normal and resultant Stokes numbers were high and above critical thresholds for viscous damping. These results are attributed to the coherent turbulent structures near the wall region, i.e. bursting motion with ejection and sweep events responsible for turbulence generation and particle transport. The tangential restitution coefficients were slightly below unity and the dynamic friction coefficients were lower than for alluvial bed data, revealing that only a small amount of horizontal energy was transferred to the bed. The abrasion prediction model formed by Sklar and Dietrich in 2004 was revised based on the new equations on vertical impact velocity and hop length covering various bed configurations. The abrasion coefficient k_v was found to be vary around k_v ~ 10⁵ for hard materials (tensile strength f_t > 1 MPa), one order of magnitude lower than the value assumed so far for Sklar and Dietrich's model. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

On strong ground motion synthesis with k −2 slip distributions

This study deals with the methodical aspects of k ^–2(Bernard et al., 1996) kinematic strong motions modelling: (1) it is shown how to incorporate the k-dependent rise time for 2D fault geometry in the strong motion synthesis according to the representation theorem, (2) it is suggested how to produce realistic k ^–2 slip models including asperity(ies), (3) modifications are introduced concerning the typeof used slip velocity function and the corner wave number in the slip distribution. High frequency effects of these generalized models are discussed.It is shown that, assuming the rise time proportional to the spatial slip wavelength at high wave numbers, the spectral decay of displacement at frequencies higher than the corner frequency is given just by the decay ofthe slip distribution spectrum, regardless of the type of slip velocity function. It is shown numerically that this model provides -squared source spectrum even in a vicinity of a 2D normal fault buried in 1D structure, which is an agreement with previous studies.     

Frictional resistance to accelerating slip

General analytical expressions for the friction stress and state variable, based on a rate and state-dependent constitutive friction law proposed by Dieterich and Ruina, have been obtained as an explicit function of slip rateV or slip timet or slip displacement δ under the assumption that slip accelerationa is constant or piecewise constant. Properties of the solutions have been discussed, and reviewed, for uniformly accelerating (or decelerating) slip, the following.

1. Frictional stress increases (or decreases) with increasing time, or slip rate, or slippage at the beginning of motion, until a maximum (or a minimum) value (when it exists) has been reached, then decreases (or increases), and finally approaches a special frictional state, namely a steady state, for which stress depends on instantaneous slip rate.
2. The maximal value of frictional stress is dependent on accelerationa; the larger thea, the larger the magnitude of the maximum.

The energy expense ε needed to overcome the frictional stress has been estimated. And therefore the optimal value of acceleration for saving energy for a slipping mechanical system has been obtained. The energy release rateG for an abrupt rupture process of a crack or fault has been estimated.     

The effect of long-term aerial exposure on intertidal mudflat erodibility

Intertidal zones by definition are exposed to air at low tide, and the exposure duration can be weeks (e.g. during neap tides) depending on water level and bed elevation. Here we investigated the effect of varying exposure duration (6 h to 10 days) on intertidal mudflat erosion (measured using the EROMES device), where the effects of water content and biofilm biomass (using chlorophyll-a content as a proxy, Chl-a μg g⁻¹) were taken into account. Sediments were collected between spring and summer (in October 2018, January 2019 and February 2019) from an intertidal site in the Firth of Thames, New Zealand. Longer exposure duration resulted in more stable sediments [higher erosion threshold (Ƭ_cr, N m⁻²) and lower erosion rate (ER, g m⁻² s⁻¹)]. After 10 days, exposure increased Ƭ_cr by 1.7 to 4.4 times and decreased ER by 11.6 to 21.5 times compared with 6 h of exposure. Chl-a and water content changed with exposure duration and were significantly correlated with changes in Ƭ_cr and ER. The stability of sediments after two re-submersion periods following exposure was also examined and showed that the stabilizing effect of exposure persisted even though water content had increased to non-exposure levels. Re-submersion was associated with an increase in Chl-a content, which likely counteracted the destabilizing influence of increased water content. A site-specific model, which included the interplay between evaporation and biofilm biomass, was developed to predict water content as a function of exposure duration. The modelled water content (W_Mod.) explained 98% of the observed variation in water content (W_Obs.). These results highlight how the exposure period can cause subtle changes to erosion regimes of sediments. An understanding of these effects (e.g. in sediment transport modelling) is critical to predicting the resilience of intertidal zones into the future, when sea-level rise is believed to exacerbate erosion in low-lying areas. © 2020 John Wiley & Sons, Ltd.     

Numerical Simulation of Earthquake Nucleation Process and Seismic Precursors on Faults

To understand precursory phenomena before seismic fault sliP,this work focuses onearthquake nucleation process on a fault plane through numerical simulation.Rate and statedependent friction law with variable normal stress is employed in the analysis.The resultsshow that in the late stage of nucleation process:(1)The maximum slip velocity ismonotonically accelerating;(2)The slipping hot spot(where the slip rate is maximum)migrates spontaneously from a certain instant,and such migration is spatially continuous;(3)When the maximum velocity reaches a detectable order of magnitude(at least one orderof magnitude greater than the loading rate),the remaining time is 20 hours or longer,andthe temporal variation of slip velocity beyond this point may be used as a precursoryindicator;(4)The average slip velocity is related to the remaining time by a log-log linearrelation,which means that a similar relation between rate of microseismicity and remainingtime may also exist;(5)when normal stress variation is taken int     

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.     

Attenuation and dispersion ofSH waves due to scattering by randomly distributed cracks

The effect of randomly distributed cracks on the attenuation and dispersion ofSH waves is theoretically studied. If earthquake ruptures are caused by sudden coalescence of preexisting cracks, it will be crucial for earthquake prediction to monitor the temporal variation of the crack distribution. Our aim is to investigate how the property of crack distribution is reflected in the attenuation and dispersion of elastic waves.We introduce the stochastic property, in the mathematical analysis, for the distributions of crack location, crack size and crack orientation. The crack size distribution is assumed to be described by a power law probability density (p(a) a ^– fora _minaa _max according to recent seismological and experimental knowledge, wherea is a half crack length and the range 13 is assumed. The distribution of crack location is assumed to be homogeneous for the sake of mathematical simplicity, and a low crack density is assumed. The stochastic property of each crack is assumed to be independent of that of the other cracks. We assume two models, that is, the aligned crack model and the randomly oriented crack model, for the distribution of crack orientation. All cracks are assumed to be aligned in the former model. The orientation of each crack is assumed to be random in the latter model, and the homogeneous distribution is assumed for the crack orientation. The idea of the mean wave formalism is employed in the analysis, and Foldy's approximation is assumed.We observe the following features common to both the aligned crack model and the randomly oriented crack model. The attenuation coefficientQ ^–1 decays in proportion tok ^–1 in the high frequency range and its growth is proportional tok ² in the low frequency range, wherek is the intrinsic wave number. This asymptotic behavior is parameter-independent, too. The attenuation coefficientQ ^–1 has a broader peak as increases and/ora _min/a _max decreases. The nondimensional peak wave numberk _p a _max at whichQ ^–1 takes the peak value is almost independent ofa _min/a _max for =1 and 2 while it considerably depends ona _min/a _max for =3. The phase velocity is almost independent ofk in the rangeka _max<1 and increases monotonically ask increases in the rangeka _max>1. While the magnitude ofQ ^–1 and the phase velocity considerably depend on the orientation of the crack in the aligned crack model, the above feature does not depend on the crack orientation.The accumulation of seismological measurements suggests thatQ ^–1 ofS waves has a peak at around 0.5 Hz. If this observation is combined with our theoretical results onk _p a _max, the probable range ofa _max of the crack distribution in the earth can be estimated for =1 or 2. If we assume 4 km/sec as theS wave velocity of the matrix medium,a _max is estimated to range from 2 to 5 km. We cannot estimatea _max in a narrow range for =3.     

Evaluation of various relationships of reaeration rate coefficient for modeling dissolved oxygen in a river with extreme flow variations in Pakistan

Mathematical relationships have been developed for reaeration rate coefficient (K_a) by various researchers. These relationships have a number of variables such as depth, velocity, width, slope, Froud number, molecular diffusion coefficient, kinematic viscosity and the gas‐transfer Reynolds number. From these variables, 29 relations have been developed and divided into four groups. To evaluate their predictive capability for highly variable flow rivers receiving high pollution loads form large cities, these relationships have been used to model dissolved oxygen (DO) in the River Ravi. Such rivers are either saturated with DO during high flows or anaerobic during critical low‐flow conditions. The evaluation is based on the agreement between model DO values calculated using K_a obtained from the available equations and the measured DO concentrations in the river samples in terms of sum of square of residuals (SSR) and coefficient of determination (R²). It has been found that in general, the group of equations containing depth and velocity as the only two variables affecting K_a performed better than the equations in other groups as reflected by lower SSR and higher R² values. The study results also reveal that the turbulence‐based reaeration rate coefficient equation containing additional variables also resulted in close agreement between DO model results and the measured values. The study results identify the most important parameters affecting the reaeration rate coefficient and the suitability of various K_a relationships as well for rivers with highly variable flows. Copyright © 2012 John Wiley & Sons, Ltd.