State-dependent Friction Research Articles

The Smoluchowski-Kramers Limit of Stochastic Differential Equations with Arbitrary State-Dependent Friction

We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein-Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient.

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 = kcr 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 ≥ kcr(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.

3D modeling of earthquake cycles of the Xianshuihe fault, southwestern China

We perform 3D modeling of earthquake generation of the Xianshuihe fault, southwestern China, which is a highly active strike-slip fault with a length of about 350km, in order to understand earthquake cycles and segmentations for a long-term forecasting and earthquake nucleation process for a short-term forecasting. Historical earthquake data over the last 300years indicates repeated periods of seismic activity, and migration of large earthquake along the fault during active seismic periods. To develop the 3D model of earthquake cycles along the Xianshuihe fault, we use a rate- and state-dependent friction law. After analyzing the result, we find that the earthquakes occur in the reoccurrence intervals of 400–500years. Simulation result of slip velocity distribution along the fault at the depth of 10km during 2694years along the Xianshuihe fault indicates that since the third earthquake cycle, the fault has been divided into 3 parts. Some earthquake ruptures terminate at the bending part of the fault line, which may means the shape of the fault line controls how earthquake ruptures. The change of slip velocity and displacement at 10km depth is more tremendous than the change of the shallow and deep part of the fault and the largest slip velocity occurs at the depth of 10km which is the exact depth of the seismic zone where fast rupture occurs.

Seismicity change revealed by ETAS, PI, and Z-value methods: A case study of the 2013 Nantou, Taiwan earthquake

On Mar. 27, 2013, a ML6.2 earthquake occurred in the Nantou area of central Taiwan, which caused one death and nearly 90 injured. Two months later, another ML6.3 earthquake struck the same region on June 2, 2013, the epicenter of which is close to the March ML6.2 earthquake. Seismicity is a sensitive indicator of stress rate and inelastic deformation process in crust. Therefore, examination of temporal changes in seismicity is important to understand the preparatory processes of damaging inland earthquakes. In this study, we applied the Epidemic-Type Aftershock-Sequences model (ETAS model) to the earthquake data covering broader Taiwan region, which is maintained by the Central Weather Bureau (CWB) of Taiwan, to investigate precursory temporal changes in seismicity for the ML6.2 Nantou earthquake. We regard the March ML6.2 and June ML6.3 earthquakes as an event sequence and especially focus on temporal changes in seismicity prior to the ML6.2 event. Application of more than one model to an earthquake catalog would be informative in elucidating the relationships between seismicity precursors and the preparatory processes of large earthquakes. Based on this motivation, we further applied two different approaches: the pattern informatics (PI) method and the ZMAP method, which is a gridding technique based on the standard deviate (Z-value) test to the same earthquake data of CWB. As a result, we found that the epicenter of the 2013 ML6.2 Nantou earthquake was surrounded by three main seismic quiescence regions prior to its occurrence. The assumption that this is due to precursory slip (stress drop) on fault plane or its deeper extent of the ML6.2 Nantou earthquake is supported by previous researches based on seismicity data, geodedic data, and numerical simulations using rate- and state-dependent friction laws.

A trial derivation of seismic plate coupling by focusing on the activity of shallow slow earthquakes

To understand the effect of plate coupling on very low-frequency event (VLFE) activity resulting from megathrust earthquakes, we performed long-term multiscale earthquake cycle simulations (including a megathrust earthquake and slow earthquakes) on a 3-D subduction plate boundary model, based on a rate- and state-dependent friction law. Our simulation suggests that quiescence of shallow VLFEs off Miyagi may be explained by the location in the shallow central part of the 2011 Tohoku earthquake because of the locally strong coupling, while observed activation of VLFEs off Iwate (northern part of Tohoku district), Fukushima (southern part of Tohoku district), and Ibaraki (northern part of Kanto district) is explained by the location on the outer rim. The area and duration of the quiescence off Miyagi may be a new clue to evaluate the potential for plate coupling strong enough to cause the next megathrust earthquake.

Nucleation process in the Burridge-Knopoff model of earthquakes

The nucleation process of the one-dimensional Burridge-Knopoff model of earthquakes obeying the rate- and state-dependent friction law is studied both analytically and numerically. The properties of the nucleation dynamics, the nucleation lengths and the duration times are examined together with their continuum limits.

Feasibility of acoustic monitoring of strength drop precursory to earthquake occurrence

Rate- and state-dependent friction law (RSF), proposed on the basis of laboratory experiments, has been extensively applied to modeling of earthquake stick-slip cycles. A simple spring-slider model obeying RSF predicts a significant decrease of the frictional strength Φ (the state of contact) that is localized within a few years preceding the earthquake occurrence. On the other hand, recent laboratory experiments successfully monitored the history of the strength by simultaneously measuring the P-wave transmissivity |T| across the frictional interface using a 1-MHz transducer. This suggests a possibility of earthquake forecast by monitoring the strength of a natural fault by acoustic methods. The present paper explores the feasibility of such monitoring in the field on the basis of the physics of RSF combined with the linear slip model (LSM) employed in the classical acoustic methodology for monitoring an imperfectly welded interface. The characteristic frequency f c , around which |T| (or reflectivity |R|) has a good sensitivity to the interface strength, is shown to be proportional to the strength and inversely proportional to the representative scale of real contacts. For natural faults, f c is estimated to be 1 to 100 Hz, which is practicable in the field. The changes of |T| and |R| depend on the ratio of the strength drop to the absolute strength level, the latter of which is not constrained by RSF simulations. Expected changes in wave amplitude in the preslip period would be several percent for strong faults and several tens percent for weak faults, which may be detectable by acoustic methods such as seismic reflection surveys.

Energy versus information based estimations of dissipation using a pair of magnetic colloidal particles.

Using the framework of stochastic thermodynamics, we present an experimental study of a doublet of magnetic colloidal particles that is manipulated by a time-dependent magnetic field. Because of hydrodynamic interactions, each bead experiences a state-dependent friction, which we characterize using a hydrodynamic model. In this work, we compare two estimates of the dissipation in this system: the first one is energy based since it relies on the measured interaction potential, while the second one is information based since it uses only the information content of the trajectories. While the latter only offers a lower bound of the former, we find it to be simple to implement and of general applicability to more complex systems.

Thermodynamic aspects of rock friction

Rate- and state-dependent friction law for velocity-step tests is analyzed from a thermodynamic point of view. A simple macroscopic non-equilibrium thermodynamic model with a single internal variable reproduces instantaneous jump and relaxation. Velocity weakening appears as a consequence of a plasticity related nonlinear coefficient. Permanent part of displacement corresponds to plastic strain, and relaxation effects are analogous to creep in thermodynamic rheology.

The Effect on the Nucleation and Failure of MS7.0 Lushan Earthquake Induced by the MS8.0 Wenchuan Earthquake

AbstractBased on calculations of both the post‐seismic Coulomb failure stress change and coseismic dynamic Coulomb failure stress change caused by the 2008 Wenchuan Ms8.0 earthquake, we simulate the their effects on the time of the nucleation and failure of the 2013 Lushan earthquake fault. Our calculation uses the rate‐ and state‐dependent friction law and viscoelastic relaxation in the middle‐lower crust and upper mantle. Furthermore, the triggering relationship between the Wenchuan and Lushan earthquakes is studied. The results show that the peak value of dynamic stress change on the Lushan earthquake fault induced by Wenchuan event is 0.127 MPa. With the 5‐year accumulation of the viscoelastic relaxation in the middle‐lower crust and upper mantle, the post‐seismic stress change on the Lushan earthquake fault is 0.025 MPa. The results indicate that the process of the nucleation and failure of the Lushan earthquake fault is affected by the post‐seismic Coulomb failure stress change, dynamic stress change and static stress change induced by the Wenchuan earthquake, and the effect of dynamic stress change is more obvious than others. Finally, we investigate the field of stress changes in the study area after the Lushan earthquake.

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.

Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations

A simple model of earthquake nucleation that may account for the onset of chaotic dynamics is proposed and analyzed. It represents a generalization of the Burridge–Knopoff single-block model with Dieterich–Ruina's rate- and state-dependent friction law. It is demonstrated that deterministic chaos may emerge when some of the parameters are assumed to undergo small oscillations about their equilibrium values. Implementing the standard numerical methods from the theory of dynamical systems, the analysis is carried out for the cases having one or two periodically variable parameters, such that the appropriate bifurcation diagrams, phase portraits, power spectra, and the Lyapunov exponents are obtained. The results of analysis indicate two different scenarios to chaos. On one side, the Ruelle–Takens–Newhouse route to chaos is observed for the cases of limit amplitude perturbations. On the other side, when the angular frequency is assumed constant for the value near the periodic motion of the block in an unperturbed case, variation of oscillation amplitudes probably gives rise to global bifurcations, with immediate occurrence of chaotic behavior. Further analysis shows that chaotic behavior emerges only for small oscillation frequencies and higher perturbation amplitudes when two perturbed parameters are brought into play. If higher oscillation frequencies are assumed, no bifurcation occurs, and the system under study exhibits only the periodic motion. In contrast to the previous research, the onset of chaos is observed for much smaller values of the stress ratio parameter. In other words, even the relatively small perturbations of the control parameters could lead to deterministic chaos and, thus, to instabilities and earthquakes.

Linking the Recurrence Time of Earthquakes to Source Parameters: A Dream or a Real Possibility?

By using a single-degree-of-freedom spring-slider analog fault model, we generate a synthetic catalog of nearly 500 different seismic sequences. We explore the parameter space by assuming different values of constitutive parameters and tectonic environment. We also consider three different versions of the rate-dependent and state-dependent friction laws [the Dieterich-Ruina (DR), the Ruina-Dieterich (RD) and the Chester-Higgs (CH) models], and different approximations of the behavior of the friction at high sliding speeds, as well as the radiation damping effects. Our results indicate that for all the considered models, the recurrence time (T cycle) exhibits an inverse proportionality on the loading rate; a linear, positive dependence on the effective normal stress; and a linear, negative dependence on the characteristic distance controlling the state variable evolution. These results confirm and generalize previous studies. Remarkably, we found here that the coefficients of proportionality strongly depend on the adopted friction model, on the high speed behavior and on the reference set of parameters. Notably, we also found that the positive proportionality between T cycle and the difference b – a, confirmed for DR and RD laws, does not hold in general for the CH law. Overall, we conclude that even in the simplest (and idealized) case of characteristic earthquakes considered here, in which the limiting cycle is reached by the system, and even in the framework of a very simplified fault model, the possibility to a priori predict, through an universal analytical relation, the inter-event time of an impending earthquake still remains only a dream. On the other hand, a numerical prediction of T cycle would require the exact knowledge of the rheological model (and its parameters at all times over the entire life of the fault) and the actual state of the fault, which indeed are often unknown.

Earthquake Nucleation on Faults with a Revised Rate- and State-Dependent Friction Law

Recently Nagata et al. (J Geophys Res 117:B02314, 2012) have proposed a new version of rate- and state-dependent friction law (RSF) that seems to have eventually resolved all the previously known discrepancies in the existing RSFs from laboratory observations. The values of a and b, empirical RSF parameters determined by fitting the same laboratory experiments, have been revised to be five times greater and a newly noticed weakening effect by shear stress with a coefficient c has been introduced. By using this revised RSF, we reinvestigated a problem of 2D quasi-static nucleation on faults. A crack-like nucleation-zone expansion known for the ‘aging’ version of RSF is not sustainable with the ‘Nagata’ law, which is understandable as the Nagata law does not produce a slip-weakening distance proportional to the involved strength reduction, an aging law’s feature that contradicts laboratory observations. The later stage of Nagata-law nucleation shows localization of quasi-static slip within a limited spatial extent, but the localization is much milder than that predicted by the ‘slip’ version of RSF. With an appropriate c parameter of the Nagata law, the nucleation size seems to be reduced only by a factor from that of the aging law.

Variational formulation of rate‐ and state‐dependent friction problems

We propose a variational formulation of rate‐ and state‐dependent models for the dynamic sliding of a linearly elastic block on a rigid surface in terms of two coupled variational inequalities. Classical Dieterich–Ruina models are covered as special cases. We show existence and uniqueness of solutions for the two spatial subproblems arising from time discretisation. Existence of solutions to the coupled spatial problems is established for Dieterich's state equation through a fixed point argument. We conclude with some numerical experiments that suggest mesh independent convergence of the underlying fixed point iteration, and illustrate quasiperiodic occurrence of stick/slip events.

Comparison of deterministic and stochastic earthquake simulators for fault interactions in the Lower Rhine Embayment, Germany

Time-dependent probabilistic seismic hazard assessment requires a stochastic description of earthquake occurrences. While short-term seismicity models are well-constrained by observations, the recurrences of characteristic on-fault earthquakes are only derived from theoretical considerations, uncertain palaeo-events or proxy data. Despite the involved uncertainties and complexity, simple statistical models for a quasi-period recurrence of on-fault events are implemented in seismic hazard assessments. To test the applicability of statistical models, such as the Brownian relaxation oscillator or the stress release model, we perform a systematic comparison with deterministic simulations based on rate- and state-dependent friction, high-resolution representations of fault systems and quasi-dynamic rupture propagation. For the specific fault network of the Lower Rhine Embayment, Germany, we run both stochastic and deterministic model simulations based on the same fault geometries and stress interactions. Our results indicate that the stochastic simulators are able to reproduce the first-order characteristics of the major earthquakes on isolated faults as well as for coupled faults with moderate stress interactions. However, we find that all tested statistical models fail to reproduce the characteristics of strongly coupled faults, because multisegment rupturing resulting from a spatiotemporally correlated stress field is underestimated in the stochastic simulators. Our results suggest that stochastic models have to be extended by multirupture probability distributions to provide more reliable results.

Friction memory effect in complex dynamics of earthquake model

In present paper, an effect of delayed frictional healing on complex dynamics of simple model of earthquake nucleation is analyzed, following the commonly accepted assumption that frictional healing represents the main mechanism for fault restrengthening. The studied model represents a generalization of Burridge–Knopoff single-block model with Dieterich–Ruina’s rate and state dependent friction law. The time-dependent character of the frictional healing process is modeled by introducing time delay τ in the friction term. Standard local bifurcation analysis of the obtained delay-differential equations demonstrates that the observed model exhibits Ruelle–Takens–Newhouse route to chaos. Domain in parameters space where the solutions are stable for all values of time delay is determined by applying the Rouche theorem. The obtained results are corroborated by Fourier power spectra and largest Lyapunov exponents techniques. In contrast to previous research, the performed analysis reveals that even the small perturbations of the control parameters could lead to deterministic chaos, and, thus, to instabilities and earthquakes. The obtained results further imply the necessity of taking into account this delayed character of frictional healing, which renders complex behavior of the model, already captured in the case of more than one block.

The role of radiation damping in the modeling of repeated earthquake events

<p>We have investigated the role of the radiation damping term (RDT) on repeated earthquake ruptures by modeling the faulting process through a single one-dimensional analog fault system governed by different constitutive laws. The RDT expresses the energy lost by the seismic waves. The RDT is inherently accounted for in more elaborated, fully dynamic models of extended fault, whereas it is neglected in one-dimensional fault models. In this study, we adopt various formulations of the laboratory-derived rate-dependent and state-dependent friction constitutive laws: the Dieterich-Ruina law, the Ruina-Dieterich law and the Chester and Higgs law. Our numerical results clearly indicate that the RDT significantly affects the system dynamics. More specifically, the more the RDT is effective, the more frequent the slip failures are (with a cycle-time reduction of ca. 30%). We also show that inclusion of the RDT tends to promote smaller but more frequent earthquake instabilities, irrespective of the choice of the governing law. Our data shed light on the limitations implied by the conventional formulation of the equation of motion for the spring system, in which the energy radiation is ignored.</p>

Re-examination of possible great interplate earthquake scenarios in the Nankai Trough, southwest Japan, based on recent findings and numerical simulations

Abstract Great earthquakes have occurred repeatedly along the Nankai Trough in southwestern Japan with recurrence intervals of 100–200 years. The magnitudes of Nankai earthquakes have historically varied, with many studies to date asserting that an event's magnitude is controlled only by the number of broken segments arranged along the trough. Recent evidence suggests that large tsunami with recurrence intervals of several hundred years have occurred along the Nankai Trough. Therefore, it is now essential to reconsider the scenario of the Nankai earthquakes, including the possibility of seismic slip on the shallow subduction interface. Recently, model calculations of interplate seismic cycles have been conducted using rate- and state-dependent friction. In these calculations, assuming heterogeneous distribution of fracture energy in the seismogenic zone and its shallower extension, both massive earthquakes with slips up to the trough and ordinary earthquakes in the seismogenic zone can occur. Here, we apply a heterogeneous distribution of fracture energy in the shallow plate interface of the Nankai Trough. In the model, we represent the difference in fracture energy by the difference in the characteristic slip distance L . To model the conventional seismogenic zone as a zone of low fracture energy, we set L to be 5 cm at depths of 10–20 km. Conversely, the shallow plate boundary near the trough is modeled as a zone of high fracture energy by setting L to 5 m. As a result, two types of earthquake with different moment magnitude occur alternately. The recurrence interval of the larger type is about 370 years, which is comparable to the recurrence of larger tsunami deduced from geological findings. Large coseismic slip extends to the trough during the larger types. In contrast, the smaller one, whose seismic slip is distributed along the seismogenic zone, occurs after ~ 200 years of the larger one.

Precursory change in seismicity revealed by the Epidemic-Type Aftershock-Sequences model: A case study of the 1999 Chi-Chi, Taiwan earthquake

For gaining insight into the preparatory process of the 1999 Chi-Chi, Taiwan earthquake and its related statistical feature, the Epidemic-Type Aftershock-Sequences model (ETAS model) was applied to the earthquake data in Taiwan region. By means of the ETAS analysis for Taiwanese earthquakes larger than a magnitude of 2.4, we found seismic quiescence over broader regions of Taiwan. It was also found that inland areas near the epicenter of the Chi-Chi earthquake showed seismic activation in the period from Jan. 1, 1998 to Sep. 20, 1999, which is just before the occurrence of the Chi-Chi earthquake. The assumption that this is due to precursory slip (stress drop) on the fault plane of the Chi-Chi earthquake is supported by previous researches such as a numerical simulation using rate- and state-dependent friction laws and the observation of abnormal change in crustal displacement for a station of Taiwan GPS network near southern edge of the source area of the Chi-Chi earthquake.