Friction Why friction? Because slip on faults is resisted by frictional forces. We first describe the results of laboratory friction experiments, and then discuss the implications of the friction constitutive law for: Earthquake cycles, Earthquake depth distribution, Earthquake nucleation, The mechanics of aftershocks, and more...

Friction: From laboratory scale to crustal scale Figure from

Question: Given that all objects shown below are of equal mass and identical shape, in which case the frictional force is greater? Question: Who sketched this figure?

Leonardo Da Vinci ( ) showed that the friction force is independent of the geometrical area of contact. Friction: Da Vinci law and the paradox The paradox: Intuitively one would expect the friction force to scale proportionally to the contact area. Movie from:

Friction: Amontons’ laws Amontons' first law: The frictional force is independent of the geometrical contact area. Amontons' second law: Friction, F S, is proportional to the normal force, F N : Movie from:

A way out of Da Vinci’s paradox has been suggested by Bowden and Tabor, who distinguished between the real contact area and the geometric contact area. The real contact area is only a small fraction of the geometrical contact area. Friction: Bowden and Tabor (1950, 1964) Figure from: Scholz, 1990

where p is the penetration hardness. where s is the shear strength. Thus: Since both p and s are material constants, so is . The good news is that this explains Da Vinci and Amontons’ laws (but not the Byerlee law). Friction: Bowden and Tabor (1950, 1964)

Friction: Beyrlee law Byerlee, 1978

Friction: Modern experimental apparatus Animation and picture from Chris Marone’s site

Friction: Static versus kinetic friction The force required to start the motion of one object relative to another is greater than the force required to keep that object in motion. Ohnaka (2003)

Friction: Velocity stepping - Dieterich A sudden increase in the piston's velocity gives rise to a sudden increase in the friction, and vice versa. The return of friction to steady-state occurs over a characteristic sliding distance. Steady-state friction is velocity dependent. Dieterich and Kilgore, 1994

Friction: Slide-hold-slide - Dieterich Static (or peak) friction increases with hold time. Dieterich and Kilgore, 1994

Friction: Slide-hold-slide - Dieterich The increase in static friction is proportional to the logarithm of the hold duration. Dieterich, 1972

Friction: Monitoring the real contact area during slip - Dieterich and Kilgore

Friction: Change in true contact area with hold time - Dieterich and Kilgore The dimensions of existing contacts are increasing. New contacts are formed. Dieterich and Kilgore, 1994

Friction: Change in true contact area with hold time - Dieterich and Kilgore The real contact area, and thus also the static friction increase proportionally to the logarithm of hold time. Dieterich and Kilgore, 1994

Upon increasing the normal stress: The dimensions of existing contacts are increasing. New contacts are formed. Real contact area is proportional to the logarithm of normal stress. Friction: The effect of normal stress on the true contact area - Dieterich and Kilgore Dieterich and Kilgore, 1994

Friction: The effect of normal stress - Dieterich and Linker Linker and Dieterich, 1992 Instantaneous response linear response Changes in the normal stresses affect the coefficient of friction in two ways: Instantaneous response, whose trend on a shear stress versus shear strain curve is linear. Delayed response, some of which is linear and some not.

Static friction increases with the logarithm of hold time. True contact area increases with the logarithm of hold time. True contact area increases proportionally to the normal load. A sudden increase in the piston's velocity gives rise to a sudden increase in the friction, and vice versa. The return of friction to steady-state occurs over a characteristic sliding distance. Steady-state friction is velocity dependent. The coefficient of friction response to changes in the normal stresses is partly instantaneous (linear elastic), and partly delayed (linear followed by non-linear). Friction: Summary of experimental result

Friction: The constitutive law of Dieterich and Ruina where: V and  are sliding speed and contact state, respectively. A, B and  are non-dimensional empirical parameters. D c is a characteristic sliding distance. The * stands for a reference value.

The set of constitutive equations is non-linear. Simultaneous solution of non-linear set of equations may be obtained numerically (but not analytically). Yet, analytical expressions may be derived for some special cases. The change in sliding speed,  V, due to a stress step of  : Steady-state friction: Static friction following hold-time,  t hold :

Friction: The constitutive law of Dieterich and Ruina Rate direct effect: State effect: Combined rate and state effect:

Friction: Aging-versus-slip evolution law Aging law (Dieterich law): Slip law (Ruina law):

Friction: Slip law fits velocity-stepping better than aging law Linker and Dieterich, 1992 Unpublished data by Marone and Rubin

Friction: Aging law fits slide-hold-slide better than slip law Beeler et al., 1994

Friction: Fast slip experiments Rotary shear apparatus Di Toro et al., 2006 Rotary shear at Kochi Core Center

Friction: Fast slip experiments Di Toro et al., 2006

Next we review the implications of the friction law to: Earthquake cycles and aftershocks, Earthquake nucleation, Earthquake depth distribution, Recommended reading: Marone, C., Laboratory-derived friction laws and their applications to seismic faulting, Annu. Rev. Earth Planet. Sci., 26: , Scholz, C. H., The mechanics of earthquakes and faulting, New- York: Cambridge Univ. Press., 439 p., 1990.

Friction is rate- and state-dependent Slide-hold-slide Velocity stepping

Friction is rate- and state-dependent Dieterich and Kilgore, 1994 Changes in static friction are due to changes in the true contact area.

Friction is rate- and state-dependent Experimental data may be fit with the following constitutive law: How can the a and b parameters be inferred?

Recall that: Thus, a-b may be inferred from the slope of  ss versus ln(V ss ). Friction is rate- and state-dependent

Additionally: Thus, a may be inferred from the slope of ln(V/V 0 ) versus . Friction is rate- and state-dependent

Finally: Thus, the b parameter may be inferred from the slope of  static versus  t hold. Friction is rate- and state-dependent

The seismic cycle The elastic rebound theory. The spring-slider analogy. Frictional instabilities. Static-kinetic versus rate-state friction. Earthquake depth distribution.

The seismic cycle: The elastic rebound theory (according to Raid, 1910)

The seismic cycle: The spring-slider analog

The seismic cycle: Frictional instabilities The common notion is that earthquakes are frictional instabilities. The condition for instability is simply: The area between B and C is equal to that between C and D.

Frictional instabilities are commonly observed in lab experiments and are referred to as stick-slip. The seismic cycle: Frictional instabilities Brace and Byerlee, 1966

The seismic cycle: Frictional instabilities governed by static-kinetic friction Stress Slip Time The static-kinetic (or slip- weakening) friction: stress slip Lc static friction kinetic friction experiment Constitutive law Ohnaka (2003)

The seismic cycle: Frictional instabilities governed by rate- and state-dependent friction were: V and  are sliding speed and contact state, respectively. A and B are non-dimensional empirical parameters. D c is a characteristic sliding distance. The * stands for a reference value. Dieterich-Ruina friction:

The seismic cycle: Frictional instabilities governed by rate- and state-dependent friction State [s] loading point (I.e., plate) velocity The evolution of sliding the speed and the state throughout the cycles. An earthquake occurs when the sliding speed reaches the seismic speed - say a meter per second.

According to the spring-slider model earthquake occurrence is periodic, and thus earthquake timing and size are predictable - is that so?

The seismic cycle: The Parkfield example Magnitude Year 2004 A sequence of magnitude 6 quakes have occurred in fairly regular intervals. The next magnitude 6 quake was anticipated to take place within the time frame 1988 to 1993, but ruptured only on 2004.

So the occurrence of major quakes is non-periodic - why?

The seismic cycle: The role of stress transfer Faults are often segmented, having jogs and steps. Every earthquake perturb the stress field at the site of future earthquakes. So it is instructive to examine the implications of stress changes on spring-slider systems. Animation from the USGS site Stein et al., 1997

The seismic cycle: The effect of a stress step The effect of a stress perturbation is to modify the timing of the failure according to: This means that the amount of time advance (or delay) is independent of when in the cycle the stress is applied.

The seismic cycle: The effect of a stress step state [t] The effect of a stress step is to increase the sliding speed, and consequently to advance the failure time.

The seismic cycle: The effect of a stress step The ‘clock advance’ of a fault that is in an early state of the seismic cycle (i.e., far from failure) is greater than the ‘clock advance’ of a fault that is late in the cycle (I.e., close to failure).

The seismic cycle: Implications for aftershock mechanics

In summary: The effect of positive and negative stress steps is to advance and delay the timing of the earthquake, respectively. While according to the static-kinetic model the time advance depends only on the magnitude of the stress step and the stressing rate, according to the rate-and-state model it depends not only on these parameters, but also on when in the cycle the stress has been perturbed. Thus, short-term earthquake prediction may be very difficult (if not impossible) if rate-and-state model applies to the Earth.

The seismic cycle: But a spring-slider system is too simple… Fault networks are extremely complex. More complex models are needed. In terms of spring-slider system, we need to add many more springs and sliders. Figure from Ward, 1996

The seismic cycle: System of two blocks During static intervals: During dynamic intervals: Several situations: To simplify matters we set: We define:

The seismic cycle: System of two blocks Turcotte, 1997 Next we show solutions for: asymmateric ( )symmateric ( ) Were: Breaking the symmetry of the system gives rise to chaotic behavior.

The seismic cycle: Summary Single spring-slider systems governed by either static-kinetic, or rate- and state-dependent friction give rise to periodic earthquake- like episodes. The effect of stress change on the system is to modify the timing of the instability. While according to the static-kinetic model the time advance depends only on the magnitude of the stress step and the stressing rate, according to the rate-and-state model it depends not only on these parameters, but also on when in the cycle the stress has been perturbed. Breaking the symmetry of two spring-slider system results in a chaotic behavior. If such a simple configuration gives rise to a chaotic behavior - what are the chances that natural fault networks are predictable???

Recommended reading Scholz, C., Earthquakes and friction laws, Nature, 391/1, Scholz, C. H., The mechanics of earthquakes and faulting, New- York: Cambridge Univ. Press., 439 p., Turcotte, D. L., Fractals and chaos in geology and geophysics, New-York: Cambridge Univ. Press., 398 p., 1997.

Earthquake nucleation Stability analysis of the spring-slider system How do earthquakes begin? Are large and small ones begin similarly? Are the initial phases geodetically or seismically detectable?

Nucleation: Frictional instabilities The common notion is that earthquakes are frictional instabilities. The condition for instability is simply: The area between B and C is equal to that between C and D.

Nucleation: What are the conditions for instabilities in the spring- slider system? stress slip Lc static friction kinetic friction Thus, the condition for instability is: The static-kinetic friction:

The condition for instability is: Thus, a system is inherently unstable if b>a, and conditionally stable if b<a. The rate- and state-dependent friction: Nucleation: What are the conditions for instabilities in the spring- block system?

Nucleation: How b-a changes with depth ? Scholz (1998) and references therein Note the smallness of b-a.

Nucleation: The depth dependence of b-a may explain the seismicity depth distribution Scholz (1998) and references therein

Nucleation: Consequences of depth-dependent b-a Figure from Scholz (1998) After Tse and Rice, Fialko et al., 2005.

Nucleation: Consequences of depth-dependent b-a Lapusta and Rice, 2003.

Nucleation: In the lab Okubo and Dieterich, 1984

Nucleation: In the lab Okubo and Dieterich, 1984

Nucleation: In the lab Ohnaka’s (1990) stick-slip experiment Figures from Shibazaki and Matsu’ura, 1998

The hatched area indicates the breakdown zone, in which the shear stress decrease from a peak stress to a constant friction stress. Nucleation: In the lab Ohnaka, 1990

Nucleation: In the lab The 3 phases according to Ohnaka are: Stable quasi-static nucleation phase (~1 cm/s). Unstable, accelerating nucleation phase (~10 m/s). Rupture propagation (~2 km/s).

Nucleation: The critical stiffness and the condition for slip acceleration Slope=k The condition for acceleration:

Nucleation: The critical stiffness and the self-accelerating approximation Recall that: It is convenient to write this equation as follows: where now contains all the constant parameters. The state evolution law (the aging law) is:

Nucleation: The critical stiffness and the self-accelerating approximation For large sliding speeds, the following approximation holds: the solution of which is: and the quasi-static (i.e., slider mass=0) spring-slider force balance equation may be written as:

Nucleation: The critical stiffness and the self-accelerating approximation The block neither accelerates nor decelerates if Thus, to obtain the critical stiffness, one needs to take the slip derivative of the force balance equation for This approach leads to: The block will accelerate if:

Nucleation: Slip instability on a crack embedded within an elastic medium So far we have examined spring-slider systems. We now consider a crack embedded within an elastic medium. In that case, Hook’s law is written in terms of the shear modulus, G, and the shear strain, , as: Were  is a geometrical constant with a value close to 1. Writing the stress balance equation, and taking the slip derivative as before leads to:

Nucleation: From a spring-slider to a crack embedded within elastic medium L The elastic stiffness is: where:  is a geometrical constant G is the shear modulus The critical stiffness: Dieterich (1992) identified the  constant with:

So now, the equivalent for critical stiffness in the spring-slider system is the critical crack length: In conclusion: The condition for unstable slip is that the crack length be larger than the critical crack length. The dimensions of the critical crack scale with b. Nucleation: Slip instability on a crack embedded within an elastic medium

Nucleation: Numerically simulated nucleation Dieterich, 1992

Nucleation: What controls the size of the nucleation patch? L crit provides only a minimum estimate of the nucleation patch size. The actual size of the nucleation patch is asymptotic to L crit for small a/b, but increases with decreasing a/b. Rubin and Ampuero, 2005

Nucleation: What controls the size of the nucleation patch? Rubin et al., 1999 patches of a>b ? Precise location of seismicity on the Calaveras fault (CA) suggest that a/b~1. Slip episodes on patches of a/b>1 may trigger slip on patches of a/b<1, and vice versa. It is, therefore, instructive to examine slip localization around a/b~1.

Nucleation: What controls the size of the nucleation patch? Ziv, 2007 Positive stress changes applied on a>b interfaces can trigger quasi-static slip episodes. Similar to the onset of ruptures on a<b, the creep on intrinsically stable fractures too are preceded by intervals, during which the slip is highly localized.

Nucleation: What controls the size of the nucleation patch? Ziv, 2007 The size of the nucleation patch depends not only on the constitutive parameters, but also on the stressing history.

Nucleation: The effect of negative stress perturbations

Nucleation: The effect of negative stress perturbation Following a negative stress step, sliding velocity drops below the load point velocity, and the system evolves towards restoring the steady-state. The path along which the system evolves overshoots the steady- state curve, and the amount of overshoot is proportional to the magnitude of the stress perturbation. Consequently, a crack subjected to a larger negative stress perturbation intersects the steady state at a higher sliding speed, undergoes more weakening and more slip during the nucleation stage. Similar to the results for positive stress changes, the size of the localization patch depends on the magnitude of the stress perturbation. The greater the stress change is, the smaller is the localization patch.

Nucleation: Implications for prediction The bad news is that small and large quakes begin similarly. The good news is that, under certain circumstances, the nucleation phase may occupy large areas - therefore be detectable. Calculations employ: a/b=0.7. If a/b is closer to a unity and actual Dc is much larger than lab values, premonitory slip should be detectable. Dieterich, 1992

Nucleation: Seismically observed earthquake nucleation phase? Note that real seismograms do not show the linear increase of velocity versus time that is predictable by the self-similar model predicts. Near source recording of the 1994 Northridge earthquake Ellsworth and Beroza, 1995

Nucleation: Seismically observed earthquake nucleation phase? Ellsworth and Beroza, 1995 For a given event, it’s initial seismic phase is proportional to it’s final size; but this conclusion is inconsistent with the inference (a few slides back) that the dimensions of the nucleation patch depend on the constitutive parameters and the normal stress.

Nucleation: Seismically observed earthquake nucleation phase? Recall that: Some of the previously reported seismically observed initial phases are due to improper removal of the instrumental effect (e.g., Scherbaum and Bouin, 1997). Some claim that the slow initial phase observed in teleseismic records are distorted by anelastic attenuation or inhomogeneous medium.

Further reading Dieterich, J. H., Earthquake nucleation on faults with rate- and state- dependent strength, Tectonophysics, 211, , Iio, Y., Observations of slow initial phase generated by microearthquakes: Implications for earthquake nucleation and propagation, J.G.R., 100, 15, ,349, Shibazaki, B., and M. Matsu’ura, Transition process from nucleation to high- speed rupture propagation: scaling from stick-slip experiments to natural earthquakes, Geophys. J. Int., 132, 14-30, Ellsworth, W. L., and G. C. Beroza, Seismic evidence fo an earthquake nucleation phase, Science, 268, , Di Toro et al., Natural and experimental evidence of melt lubrication of faults during earthquakes, Science, 311, 647, 2006.

Seismological evidence for the dependence of static friction on the log of recurrence time Repeating quakes on the Parkfield segment in CA Nadeau and Johnson, 1998

Seismological evidence for the dependence of static friction on the log of recurrence time Nadeau and Johnson, 1998

Seismological evidence for the dependence of static friction on the log of recurrence time Chen, Nadeau and Rau, 2007