Dependent Friction Law Research Articles

Rate and state dependent friction and the stability of sliding between elastically deformable solids

We study the stability of steady sliding between elastically deformable continua using rate and state dependent friction laws. That is done for both elastically identical and elastically dissimilar solids. The focus is on linearized response to perturbations of steady-state sliding, and on studying how the positive direct effect (instantaneous increase or decrease of shear strength in response to a respective instantaneous increase or decrease of slip rate) of those laws allows the existence of a quasi-static range of response to perturbations at sufficiently low slip rate. We discuss the physical basis of rate and state laws, including the likely basis for the direct effect in thermally activated processes allowing creep slippage at asperity contacts, and estimate activation parameters for quartzite and granite. Also, a class of rate and state laws suitable for variable normal stress is presented. As part of the work, we show that compromises from the rate and state framework for describing velocity-weakening friction lead to paradoxical results, like supersonic propagation of slip perturbations, or to ill-posedness, when applied to sliding between elastically deformable solids. The case of sliding between elastically dissimilar solids has the inherently destabilizing feature that spatially inhomogeneous slip leads to an alteration of normal stress, hence of frictional resistance. We show that the rate and state friction laws nevertheless lead to stability of response to sufficiently short wavelength perturbations, at very slow slip rates. Further, for slow sliding between dissimilar solids, we show that there is a critical amplitude of velocity-strengthening above which there is stability to perturbations of all wavelengths.

State, rate and temperature–dependent sliding friction of elastomers

We present an experimental investigation of the non–stationary frictional properties of multicontact interfaces between rough elastomers and rough hard glass at low velocities (not more than 200 μm s⊃-1). These systems, for which the deformation contribution to friction is negligible, are shown to exhibit a phenomenology that is similar to what is observed for non–elastomeric materials in the same multi–contact configuration, and which are quantitatively described by the state– and rate–dependent friction laws. This permits us to identify clearly the two contributions to adhesive friction which are mixed in steady sliding: the interfacial shear stress, which appears as thermally activated formation and breaking of molecular bonds, and the real area of contact which evolves through viscoelastic creep of the load bearing asperities.

Upgrading a synthetic seismicity model for more realistic fault ruptures

A quasi‐static synthetic seismicity model for multiple interacting faults of any orientation is upgraded to include a more realistic rupture model. In the original model entire ruptures, covering many fault patches, occur instantaneously and rupture propagation is ignored. Now time is introduced into the rupture process via a patch interaction delay in proportion to distance/shear wave velocity. Also, to try to mimic observed rise times for real earthquakes, a simple time dependent friction law is used. Comparison of results for the old and new models for a simple case of two strike‐slip faults show that clustering of large events is somewhat reduced in the new model. It remains to be seen if this conclusion will be verified in more complex fault networks. The new model takes about twice as much computation time, but the results may now be used to calculate synthetic seismograms.

Periodic loading on a creeping fault: Implications for tides

We study the effect of time varying normal and shear stress perturbations on a creeping fault and in which inertia is neglected. When interpreted in terms of earthquake triggering by earth tides, our results suggest that tidal triggering may exist but be very difficult to detect. We use a spring‐block model with a rate‐and state dependent friction law, loaded at constant velocity, and in which inertia is neglected. When a periodic stress is applied, a resonance exists which can destabilize sliding (i.e. earthquakes). However, when factional parameters slightly vary along the fault, the observed phase lag between the response of the fault and the perturbating stresses close to resonance shows a broad and uncorrelated scattering.

Binary Galaxy Spin Correlations in Small Galaxy Groups

AbstractStatistics of spin orientations in binary galaxies may be used as a tool to study the formation process of galaxies. Helou (1984) found that the spins of spiral galaxies in binary systems are anticorrelated, and he suggested that it may have to do with the inclination dependence of the galaxy merger process. We study this dependence in small groups of galaxies. An N-body code is developed which uses an inclination dependent dynamical friction law to affect mergers of galaxies. It is found that some correlation results in this way in groups, which have initially random spin orientations.

Stability of quasi-static slip in a single degree of freedom elastic system with rate and state dependent friction

The stability of quasi-static frictional slip of a single degree of freedom elastic system is studied for a Dieterich–Ruina rate and state dependent friction law, showing steady-state velocity weakening, and following the ageing (or slowness) version of the state evolution law. Previous studies have been done for the slip version. Analytically determined phase plane trajectories and Liapunov function methods are used in this work. The stability results have an extremely simple form: (1) When a constant velocity is imposed at the load point, slip motion is always periodic when the elastic stiffness, K, has a critical value, K cr. Slip is always stable when K > K cr > 0, with rate approaching the load-point velocity, and unstable (slip rates within the quasi-static model become unbounded) when K < K cr. This is unlike results based on the slip version of the state evolution law, in which instability occurs in response to sufficiently large perturbations from steady sliding when K > K cr. An implication of this result for slip instabilities in continuum systems is that a critical nucleation size of coherent slip has to be attained before unstable slip can ensue. (2) When the load point is stationary, the system stably evolves towards slip at a monotonically decreasing rate whenever K ⩾ K cr > 0. However, when K < K cr, initial conditions leading to stable and unstable slip motion exist. Hence self-driven creep modes of instability exist, but only in the latter case.

Modelling of orthogonal cutting with a temperature dependent friction law

In this paper the process of orthogonal cutting is studied by analytical means. A thermomechanical model of the primary shear zone is combined with a modelling of the contact problem at the tool-chip interface. A friction law is introduced that accounts for temperature effects. The effects of cutting conditions and material behaviour on the temperature distribution along the contact zone, on the mean friction and on the global cutting forces are evaluated. The experimental trends are shown to be well described by the proposed model. © 1998 Elsevier Science Ltd. All rights reserved.Keywords : A. cutting and forming, thermomechanical processes, B. viscoplastic material, friction.

Reduction of Dietrich-Ruina attractors to unimodal maps

Abstract. We present a geometric analysis of a quasi-static single degree of freedom elastic slider with a state and rate dependent friction law. In particular, we examine and characterize the regime of chaotic motions displayed by the Dieterich-Ruina model. We do so by numerically reducing the chaotic attractors to a family of unimodal maps and discuss why this suggests complex behaviour in the dynamical system.

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 ϕ. 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 τ=c cosφ + σ n sinφ, whereas for shear between bare rock surfaces the shear strength is τ=c cosφ + σ n tanφ, where τ and σ 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 ϕ 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 ϕ 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.

The mechanics of earthquakes and faulting

This essential reference for graduate students and researchers provides a unified treatment of earthquakes and faulting as two aspects of brittle tectonics at different timescales. The intimate connection between the two is manifested in their scaling laws and populations, which evolve from fracture growth and interactions between fractures. The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws - producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events. The third edition of this classic treatise presents a wealth of new topics and new observations. These include slow earthquake phenomena; friction of phyllosilicates, and at high sliding velocities; fault structures; relative roles of strong and seismogenic versus weak and creeping faults; dynamic triggering of earthquakes; oceanic earthquakes; megathrust earthquakes in subduction zones; deep earthquakes; and new observations of earthquake precursory phenomena.

Dynamic motion of a single degree of freedom system following a rate and state dependent friction law : J Geophys ResV91, BB1, Jan 1986, P521–530

Sequences of dynamic instabilities are analyzed for a single degree of freedom elastic system which slides along a surface having frictional resistance depending on slip rate and slip rate history, in the manner of Dieterich, Ruina and others. The system is represented as a rigid block in contact with a fixed surface and having a spring attached to it whose opposite end is forced to move at a uniform slow speed. The resulting stick-slip motions are well understood in the classical case for which there is an abrupt drop from static to sliding frictional resistance. We analyze them here on the basis of more accurate frictional constitutive models. The problem has two time scales, an inertial scale set by the natural oscillation period T of the analogous frictionless system as T/2:rr and a relaxation scale L/V occurring in evolution, over a characteristic slip distance L, of frictional stress u towards a state value uss(v) associated with slip speed V. We show that u _ uss(v) during motions for which acceleration a satisfies aL/V 2 > T/2n during much of the essentially quasi-static stick part of the cycle when there is a sufficiently small imposed velocity at the load point. Thus the physically irrelevant time scale (L/V during inertial controlled motion, T/2n during quasi-static motion) is much shorter than the relevant scale, which is troublesome from a numerical point of view as it is the shorter time scale which constrains allowable step size. We propose efficient numerical procedures to deal with such response, in which the full equations with inertia and relaxation are solved only in a transition regime when both time scales are significant. We show results for several friction laws, all having history dependence based on a single evolving variable and all having properties that &/V > 0 for instantaneous changes in V, that u evolves towards uss(v) as exp (--6/L) with ongoing slip 6 when V- const, and that duSS(V)/dV < 0 except possibly at high V. During the dynamic instabilities we find that motion continues at a nearly steady condition, u _ 'rss(v), until dynamic overshoot becomes so significant that begins. In the arrest stage, V drops rapidly to very much lower values (never zero in our models) under nearly fixed conditions, and then the long quasi-static stick phase of the motion begins again.

Dynamic motion of a single degree of freedom system following a rate and state dependent friction law

Sequences of dynamic instabilities are analyzed for a single degree of freedom elastic system which slides along a surface having frictional resistance depending on slip rate and slip rate history, in the manner of Dieterich, Ruina and others. The system is represented as a rigid block in contact with a fixed surface and having a spring attached to it whose opposite end is forced to move at a uniform slow speed. The resulting “stick‐slip” motions are well understood in the classical case for which there is an abrupt drop from “static” to “sliding” frictional resistance. We analyze them here on the basis of more accurate frictional constitutive models. The problem has two time scales, an inertial scale set by the natural oscillation period T of the analogous frictionless system as T/2π and a state relaxation scale L/V occurring in evolution, over a characteristic slip distance L, of frictional stress τ towards a “steady state” value τss(V) associated with slip speed V. We show that τ ≃ τss(V) during motions for which acceleration a satisfies aL/V2 ≪ 1, and that this condition is met during an inertia controlled instability in typical circumstances for which the unstable slip is much greater than L. Since V/a is of order T/2π during inertia controlled motion, one has L/V ≪ T/2π, whereas L/V ≫ T/2π during much of the essentially quasi‐static “stick” part of the cycle when there is a sufficiently small imposed velocity at the load point. Thus the physically irrelevant time scale (L/V during inertial controlled motion, T/2π during quasi‐static motion) is much shorter than the relevant scale, which is troublesome from a numerical point of view as it is the shorter time scale which constrains allowable step size. We propose efficient numerical procedures to deal with such response, in which the full equations with inertia and state relaxation are solved only in a transition regime when both time scales are significant. We show results for several friction laws, all having history dependence based on a single evolving state variable and all having properties that ∂τ/∂V &gt; 0 for instantaneous changes in V, that τ evolves towards τss(V) as exp (−δ/L) with ongoing slip δ when V = const, and that dτss(V)/dV &lt; 0 except possibly at high V. During the dynamic instabilities we find that motion continues at a nearly steady state condition, τ≃τss(V), until dynamic overshoot becomes so significant that “arrest” begins. In the arrest stage, V drops rapidly to very much lower values (never zero in our models) under nearly fixed state conditions, and then the long quasi‐static “stick” phase of the motion begins again.

Effect of slip rate on stress drop

Recent observations made by Kanamori and Allen about earthquake recurrence time and average stress drop revealed a very interesting relation: earthquakes with longer recurrence times have higher average stress drops. They attributed the difference in stress drop to the difference in long-term average slip rate. To interpret their result in terms of the healing effect, we simulated earthquake recurrence with a one-dimensional mass-spring model, incorporating a recently developed rate-and-state dependent friction law for different loading rates and heterogeneous strength distributions. We first calculated the stress drop and recurrence time as functions of loading rate for a homogenous fault model. We found that the stress drop increases up to 30% when the loading rate decreases from 10 cm/yr to 0.01 mm/yr. Thus, the observed great variability of stress drop, from a few bars to a few hundred bars, which is obtained by replotting the data of Kanamori and Allen in the form of stress drop versus long-term slip rate, may not be attributable to the healing effect alone. Our numerical simulation shows that the variability may be due primarily to the spatial heterogeneity of strength on the fault. Our simulation also suggests that of the two empirical laws that were inferred from the same laboratory friction data, called the power law and the logarithmic law by Shimamoto and Logan, the former can explain the observed relation between stress drop and slip rate better than can the latter, at least for strike-slip fault. The logarithmic law is an earlier and simpler version of the rate-state-dependent friction law.