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The seismic cycle. The elastic rebound theory. The spring-slider analogy. Frictional instabilities. Static-kinetic versus rate-state friction. Earthquake depth distribution. The elastic rebound theory (according to Raid, 1910). The spring-slider analog. Frictional instabilities. - PowerPoint PPT Presentation
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The seismic cycle
• The elastic rebound theory.
• The spring-slider analogy.
• Frictional instabilities.
• Static-kinetic versus rate-state friction.
• Earthquake depth distribution.
The elastic rebound theory (according to Raid, 1910)
The spring-slider analog
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.
€
dF
du>K
Frictional instabilities are commonly observed in lab experiments and are referred to as stick-slip.
Frictional instabilities
Brace and Byerlee, 1966
From laboratory scale to crustal scale
Figure from http://www.servogrid.org/EarthPredict/
Frictional instabilities governed by static-kinetic friction
Str
ess
Slip
Time
The static-kinetic (or slip-weakening) friction:
stre
ss
slipLc
static friction
kinetic friction
experiment Constitutive law
Ohnaka (2003)
Frictional instabilities governed by rate- and state-dependent friction
€
τσ=μ =μ∗+ A ln
V
V *
⎛
⎝ ⎜
⎞
⎠ ⎟+ B ln
θV *
DC
⎛
⎝ ⎜
⎞
⎠ ⎟
and
dθ
dt=1−
θV
DC−αθ
B
dσ /dt
σ ,
were:• V and are sliding speed and contact state, respectively.• A, B and are non-dimensional empirical parameters.• Dc is a characteristic sliding distance.• The * stands for a reference value.
Dieterich-Ruina friction:
Frictional instabilities governed by rate- and state-dependent friction
Sta
t e [
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 Parkfield example
Mag
nitu
de
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 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 effect of a stress step
The effect of a stress perturbation is to modify the timing of the failure according to:
That means that the amount of time advance (or delay) is independent of when in the cycle the stress is applied.
€
Δtime =Δstress
dstress /dtime .
The effect of a stress step
stat
e [t
]
The effect of a stress step is to increase the sliding speed, and consequently to advance the failure time.
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).
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.
What are the conditions for instabilities in the spring-slider system?
stre
ss
slipLc
static friction
kinetic friction
€
slope =σ N (μ static −μkinetic)
Lc
Thus, the condition for instability is:
€
σN (μ static −μkinetic)
Lc>K
The static-kinetic friction:
€
slope =σ N (b− a)
Dc
The condition for instability is:
Thus, a system is inherently unstable if b>a, and conditionally stable if b<a.
€
σN (b− a)
Dc>K
The rate- and state-dependent friction:
What are the conditions for instabilities in the spring-block system?
How b-a changes with depth ?
Scholz (1998) and references therein
• Note the smallness of b-a.
The depth dependence of b-a may explain the seismicity depth distribution
Scholz (1998) and references therein
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
System of two blocks
€
k1y1 + kc (y1 − y2) = FS1k2y2 + kc (y2 − y1) = FS2
€
m1
d2y1
t 2+ k1y1 + kc (y1 − y2) = FD1
m2
d2y2
t 2+ k2y2 + kc (y2 − y1) = FD2
During static intervals:
During dynamic intervals:
Several situations:
To simplify matters we set:• • •
We define:
€
=kck
and β =FS1FS2
.
€
=0 versus α → ∞
and
β =1 versus β ≠1 .
€
m1 = m2 = m
€
k1 = k2 = k
€
FS1 /FD1 = FS2 /FD2 = φ
System of two blocks
Turcotte, 1997
Next we show solutions for:asymmateric ( )
€
β ≠1symmateric ( )
€
β =1
Were:
€
Yi = ky i FSi
Breaking the symmetry of the system gives rise to chaotic behavior.
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, 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.