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

2. The elastic rebound theory (according to Raid, 1910)

3. The spring-slider analog

4. 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.

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

6. From laboratory scale to crustal scale
Figure from

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

8. Frictional instabilities governed by rate- and state-dependent friction
Dieterich-Ruina friction:
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.

9. Frictional instabilities governed by rate- and state-dependent friction
State [s]
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

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

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

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

13. The role of stress transfer
Stein et al., 1997
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

14. 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.

15. 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.

16. 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).

17. 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.

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

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

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

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

22. 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

23. System of two blocks
During static intervals:
During dynamic intervals:
To simplify matters we set:
We define:
Several situations:

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

25. 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???

26. 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.