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: The mechanics of aftershocks Earthquake depth distribution, Earthquake nucleation, Earthquake cycles, and more...
Friction: From laboratory scale to crustal scale Figure from http://www.servogrid.org/earthpredict/
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?
Friction: Da Vinci law and the paradox Leonardo Da Vinci (1452-1519) showed that the friction force is independent of the geometrical area of contact. Movie from: http://movies.nano-world.org The paradox: Intuitively one would expect the friction force to scale proportionally to the contact area.
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 : F S = µf N Movie from: http://movies.nano-world.org
Friction: Bowden and Tabor (1950, 1964) 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. Figure from: Scholz, 1990
Friction: Bowden and Tabor (1950, 1964) F N = pa r, where p is the penetration hardness. where s is the shear strength. Thus: F S = sa r, µ F S F N = s p. Since both p and s are material constants, so is µ. The good news is that this explains Da Vinci and Amontons laws (but not the Byerlee law).
Friction: Beyrlee law For σ N < 200MPa : µ = 0.85 For σ N > 200MPa : µ = 0.60 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. µ static µ dynamic Ohnaka (2003) µ static > µ dynamic
Friction: Velocity stepping - Dieterich Dieterich and Kilgore, 1994 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.
Friction: Slide-hold-slide - Dieterich Dieterich and Kilgore, 1994 Static (or peak) friction increases with hold time.
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 Dieterich and Kilgore, 1994 The dimensions of existing contacts are increasing. New contacts are formed.
Friction: Change in true contact area with hold time - Dieterich and Kilgore Dieterich and Kilgore, 1994 The real contact area, and thus also the static friction increase proportionally to the logarithm of hold time.
Friction: The effect of normal stress on the true contact area - Dieterich and Kilgore 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 - Dieterich and Linker Changes in the normal stresses affect the coefficient of friction in two ways: Linker and Dieterich, 1992 Instantaneous response Instantaneous response, whose trend on a shear stress versus shear strain curve is linear. Delayed response, some of which is linear and some not. linear response
Friction: Summary of experimental result 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).
Implications for aftershock mechanics The elastic rebound theory (according to Raid, 1910)
Implications for aftershock mechanics
Implications for aftershock mechanics The static-kinetic (or slipweakening) friction: Stress experiment Constitutive law static friction kinetic friction Lc slip Ohnaka (2003) Time
Implications for aftershock mechanics The effect of a stress perturbation is to modify the timing of the failure according to: Δstress Δtime = dstress/dtime. This means that the amount of time advance (or delay) is independent of when in the cycle the stress is applied.
Implications for aftershock mechanics Dieterich-Ruina friction: τ σ = µ = " µ + Aln V % " $ '+ Bln θv * $ # V * & # and dθ θv =1, dt D C D C % ' & were: V and q 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.
Implications for aftershock mechanics 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. loading point (I.e., plate) velocity
The effect of a stress step is to increase the sliding speed, and consequently to advance the failure time. Implications for aftershock mechanics
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). Implications for aftershock mechanics
Implications for aftershock mechanics
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, static-kinetic model cannot explain the Omori-like spatiotemporal clustering, but rate-and-state friction can.
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, DV, due to a stress step of Dt: V = V 0 exp( Δτ Aσ ). Steady-state friction: # µ ss = µ * + (A B)ln V & # ss % ( = µ * + (B A)ln θ ssv * % $ V * ' $ Static friction following hold-time, Dt hold : D c & (. ' µ static Bln( θ 0 + Δt ) 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): τ σ = µ = % µ + Aln V ( % ' * + Bln θv * ' & V * ) & dθ dt and =1 θv D C, or dθ dt = θv % ln' θv D C & D C ( * ) D C ( * )
Friction: Slip law fits velocity-stepping better than aging law Dieterich law Ruina 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 Rotary shear at Kochi Core Center Di Toro et al., 2006
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: 643-696, 1998. 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 Changes in static friction are due to changes in the true contact area. Dieterich and Kilgore, 1994
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?
Friction is rate- and state-dependent Recall that: " µ ss = µ * + (a b)ln V % ss $ ' # & V * Thus, a-b may be inferred from the slope of µ ss versus ln(v ss ).
Friction is rate- and state-dependent Additionally: V = V 0 exp( Δτ aσ ). Thus, a may be inferred from the slope of ln(v/v 0 ) versus Dt/s.
Friction is rate- and state-dependent Finally: µ static bln( θ 0 + Δt ) hold. Thus, the b parameter may be inferred from the slope of µ static versus Dt hold.
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: df du > K The area between B and C is equal to that between C and D.
The seismic cycle: Frictional instabilities Frictional instabilities are commonly observed in lab experiments and are referred to as stick-slip. Brace and Byerlee, 1966
The seismic cycle: Frictional instabilities governed by static-kinetic friction The static-kinetic (or slipweakening) friction: Stress experiment Constitutive law static friction kinetic friction Lc slip Ohnaka (2003) Time
The seismic cycle: Frictional instabilities governed by rate- and state-dependent friction Dieterich-Ruina friction: τ σ = µ = " µ + Aln V % " $ '+ Bln θv * % $ ' # V * & # & and dθ θv =1, dt D C were: V and q 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. D C
The seismic cycle: Frictional instabilities governed by rate- and state-dependent friction 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. loading point (I.e., plate) velocity
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 A sequence of magnitude 6 quakes have occurred at fairly regular intervals. Magnitude Year The next magnitude 6 quake was anticipated to take place within the time frame 1988 to 1993.
The seismic cycle: The Parkfield example 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. Stein et al., 1997 Animation from the USGS site
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: Δstress Δtime = dstress/dtime. 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 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: k 1 y 1 + k c (y 1 y 2 ) = F S1 k 2 y 2 + k c (y 2 y 1 ) = F S 2 During dynamic intervals: m 1 d 2 y 1 t 2 m 2 d 2 y 2 t 2 Several situations: + k 1 y 1 + k c (y 1 y 2 ) = F D1 + k 2 y 2 + k c (y 2 y 1 ) = F D2 α = 0 versus α and β =1 versus β 1. To simplify matters we set: m 1 = m 2 = m k 1 = k 2 = k F S1 /F D1 = F S2 /F D 2 = φ We define: α = k c k and β = F S1. F S2
The seismic cycle: System of two blocks Next we show solutions for: symmateric ( β =1) asymmateric ( ) β 1 Turcotte, 1997 Were: Y i = ky i FSi 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 earthquakelike 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, 1998. Scholz, C. H., The mechanics of earthquakes and faulting, New- York: Cambridge Univ. Press., 439 p., 1990. 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: df du > K The area between B and C is equal to that between C and D.
Nucleation: What are the conditions for instabilities in the springslider system? The static-kinetic friction: static friction kinetic friction Lc slip slope = σ N (µ static µ kinetic ) L c Thus, the condition for instability is: σ N (µ static µ kinetic ) L c > K
Nucleation: What are the conditions for instabilities in the springblock system? The rate- and state-dependent friction: slope σ N (b a) D c The condition for instability is: σ N (b a) D c > K Thus, a system is inherently unstable if b>a, and conditionally stable if b<a.
Nucleation: How b-a changes with depth? Note the smallness of b-a. Scholz (1998) and references therein
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, 1986. Fialko et al., 2005.
Nucleation: Consequences of depth-dependent b-a Lapusta and Rice, 2003.
Nucleation: In the lab Hydraulic flatjacks & Teflon bearings 2.0 m 1.5 m Thickness = 0.42 m Strain gage Displacement transducer Okubo and Dieterich, 1984
Okubo and Dieterich, 1984 Nucleation: In the lab
Nucleation: In the lab Ohnaka s (1990) stick-slip experiment Figures from Shibazaki and Matsu ura, 1998
Nucleation: In the lab The hatched area indicates the breakdown zone, in which the shear stress decrease from a peak stress to a constant friction stress. 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 The condition for acceleration: k crit > k. Slope=k
Nucleation: The critical stiffness and the self-accelerating approximation Recall that: F S # = µ = µ + aln V & # % ( + bln θv * & % (. F N $ V * ' $ ' It is convenient to write this equation as follows: µ C where now contains all the constant parameters. The state evolution law (the aging law) is: D C F S F N = µ = µ C + aln V ( ) + bln θ ( ), dθ dt =1 θv D C.
Nucleation: The critical stiffness and the self-accelerating approximation For large sliding speeds, the following approximation holds: the solution of which is: dθ dt = θv D C, θ = θ 0 exp( δ D C ), and the quasi-static (i.e., slider mass=0) spring-slider force balance equation may be written as: k(δ lp δ) F N = µ C + aln(v ) + bln(θ 0 ) bδ D C. δ δ lp k
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 dv /dδ = 0. This approach leads to: The block will accelerate if: k c = F Nb D C. k > F Nb D C. should be < k = k c.
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, e, as: τ = Gε = ηgδ L. Were h is a geometrical constant with a value close to 1. Writing the stress balance equation, and taking the slip derivative as before leads to: ηg L = σ b N. D c
Nucleation: From a spring-slider to a crack embedded within elastic medium The elastic stiffness is: k = ηg L, where: h is a geometrical constant G is the shear modulus L The critical stiffness: k crit = ξσ. D c Dieterich (1992) identified the x constant with: ξ = b.
Nucleation: Slip instability on a crack embedded within an elastic medium So now, the equivalent for critical stiffness in the spring-slider system is the critical crack length: In conclusion: L crit = ηgd c σ N b. 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.
Dieterich, 1992 Nucleation: Numerically simulated nucleation
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? 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. patches of a>b? Rubin et al., 1999
Nucleation: What controls the size of the nucleation patch? L crit = ηgd c ξσ 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. Ziv, 2007
Nucleation: What controls the size of the nucleation patch? L D = ηgd c bσ The size of the nucleation patch depends not only on the constitutive parameters, but also on the stressing history.. Ziv, 2007
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 steadystate 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? Near source recording of the 1994 Northridge earthquake Ellsworth and Beroza, 1995 Note that real seismograms do not show the linear increase of velocity versus time that is predictable by the self-similar model predicts.
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: seismogram = source path instrument. 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 statedependent strength, Tectonophysics, 211, 115-134, 1992. Iio, Y., Observations of slow initial phase generated by microearthquakes: Implications for earthquake nucleation and propagation, J.G.R., 100, 15,333-15,349, 1995. Shibazaki, B., and M. Matsu ura, Transition process from nucleation to highspeed rupture propagation: scaling from stick-slip experiments to natural earthquakes, Geophys. J. Int., 132, 14-30, 1998. Ellsworth, W. L., and G. C. Beroza, Seismic evidence fo an earthquake nucleation phase, Science, 268, 851-855, 1995. 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