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:
dF
K
du
• The area between B and C is equal to that between C and D.
Frictional instabilities
Frictional instabilities are commonly observed in lab experiments
and are referred to as stick-slip.
Brace and Byerlee, 1966
From laboratory scale to crustal scale
Figure from http://www.servogrid.org/EarthPredict/
Stress
Frictional instabilities governed by static-kinetic friction
The static-kinetic (or slipweakening) friction:
experiment
Constitutive law
static friction
kinetic friction
Lc
Ohnaka (2003)
slip
Time
Frictional instabilities governed by rate- and state-dependent
friction
Dieterich-Ruina friction:
* 




V

V
     A ln  *  B ln 

V 

 DC 
and
d
V  d /dt
 1

,
dt
DC
B 
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.
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 Parkfield example
A sequence of magnitude 6 quakes have
occurred in fairly regular intervals.
Magnitude
2004
Year
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.
Stein et al., 1997
• 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
The effect of a stress step
The effect of a stress
perturbation is to modify the
timing of the failure according
to:
time 
stress
.
dstress /dtime
That means that the amount
of time advance (or delay) is
independent of when in the
cycle the stress is applied.
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 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?
The static-kinetic friction:
static friction
kinetic friction
Lc
slope 
 N (static  kinetic)
Lc
Thus, the condition for instability is:

 N (static  kinetic)
Lc
K
slip
What are the conditions for instabilities in the spring-block system?
The rate- and state-dependent friction:
slope 
 N (b  a)
Dc
The condition for instability is:

 N (b  a)
Dc
K
Thus, a system is inherently unstable if b>a, and conditionally
stable if b<a.

How b-a changes with depth ?
• Note the
smallness of b-a.
Scholz (1998) and references therein
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
During static intervals:
k1 y1  kc (y1  y 2 )  FS1
k2 y 2  kc (y 2  y1)  FS 2
During dynamic intervals:
d 2 y1
m1 2  k1 y1  k c (y1  y 2 )  FD1
t
d 2 y2
m 2 2  k2 y 2  k c (y 2  y1 )  FD 2
t



Several situations:
To simplify matters we set:
• m1  m2  m
• k1  k2  k
• FS1 /FD1  FS2 /FD2  
We define: k
c
  0 versus   
and
  1 versus  1 .
FS1

and  
.
k
FS 2
System of two blocks
Next we show solutions for:
symmateric (  1)
asymmateric ( 1)

Turcotte, 1997

Were:
Yi  kyi F
Si
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 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, NewYork: Cambridge Univ. Press., 439 p., 1990.
• Turcotte, D. L., Fractals and chaos in geology and geophysics,
New-York: Cambridge Univ. Press., 398 p., 1997.