1. Aftershocks tend to fall preferentially in area of static Coulomb stress increase but there are also earthquakes in area of decrease Coulomb stress
Aftershocks follow the Omori Law (seismicity rate decays as 1/t)
>> the stress history at the location of each aftershocks is a step function and a static Coulomb failure criterion is rejected. Failure needs being a time depend process. OR
>> the postseismic stress is time dependent in a way that controls the seismicity rate.

2. What about dynamic triggering:
Required to explain distant ‘aftershocks’ (hill et al, 1993)
Probably essential to explain the rupture dynamics during the main shock
…then, why would it be effective at large distance and not be the dominant factor a small distances?
(e.g., Kilb et al, 2003)

3. Ge277-Experimental Rock Friction implication for seismic faulting
Review articles by
- Scholz (1998): ‘Earthquakes and Friction Laws’
- Chris Marone (1998): ‘ Laboratory derived friction laws and their application to seismic faulting
Some other references:
Byerlee, 1978; Dieterich, 1979; Ruina, 1983; Tse and Rice, 1986; Blanpied et al, 1991; Chester, 1995; Lockner et Beeler, 1999; Cocco and Bizarri, 2002; Beeler et al, 2003.

4. Constant stress experiment (creep experiment).
In creep experiments three stages are generally observed, primary, secondary and tertiary creep leading to failure.
The sample ultimately fails by Static Fatigue following a phase of sub-critical crack growth. Rock strength is time dependent.
Failure of a rock sample is not instantaneous. It is preceded by a phase of ‘nucleation’
(Lockner, 1998)

5. Premonitory slip is also observed in friction epxeriments.
(Lockner and Beeler, 1999)

6. Rate Weakening Dc
Static friction depends on hold time and dynamic friction decreases with slip rate (‘rate weakening’). These phenomena contribute to an (apparent) slip-weakening friction law. Friction can also be ‘rate strengthening’.
(Marone, 1998)

7. Previous figure (a) Measurements of the relative variation in static friction with hold time for initially bare rock surfaces (solid symbols) and granular fault gouge (open symbols). The data have been offset to ms = 0.6 at 1 s and thus represent relative changes in static friction. (b) Friction data versus displacement, showing measurements of static friction and s in slide-hold-slide experiments. Hold times are given below. In this case the loading velocity before and after holding Vs/r was 3 m/s (data from Marone 1998). (c) The relative dynamic coefficient of friction is shown versus slip velocity for initially bare rock surfaces (solid symbols) and granular fault gouge (open symbols). The data have been offset to d = 0.6 at 1 m/s. (d) Data showing the transient and steady-state effect on friction (see Figure 2 for identification of friction parameters) of a change in loading velocity for a 3-mm-thick layer of quartz gouge sheared under nominally dry conditions at 25-MPa normal stress (data from J Johnson & C Marone).
(Marone, 1998)

8. Rate-and-State Friction
(Scholz, 1998)
t/s=m=m*(T)+a ln(V/V*)+b ln(q/q*)
dq/dt=1-Vq /Dc
(Dieterich 1981)
Generally a and b are of the order of ~
and Dc is of the order of mm
NB: There are other possible formalisms (e.g., Ruina, 1983 or Perrin et al, 1995)

9. Rate-and-State Friction
(Scholz, 1998)
t/s=m=m*(T)+a ln(V/V*)+b ln(q/q*)
dq/dt=1-Vq /Dc
(Dieterich 1981)
mss= m*+(a-b) ln(V/V*)
At Steady State

10. Rate-and-State Friction
(Scholz, 1998)
mss= m*+(a-b) ln(V/V*)
At Steady State
If a-b> 0, friction is rate-strengthening, only stable sliding is possible.
If a-b< 0, friction is rate-weakening, sliding can be stable or unstable.

11. Condition for unstable slip of a spring slider model
mss= m*+(a-b) ln(V/V*)
At steady state:
rate-weakening, a-b<0,
- stick-slip slip occurs if the decrease of friction is more rapid than elastic unloading during slip:
i.e., K < Kc=-(a-b)s /Dc
F/u>K, i.e. (ms-md)s/Dc>K
(Scholz, 1990)

12. Condition for stable sliding
At steady state:
rate-weakening, a-b<0,
mss= m*+(a-b) ln(V/V*)
- stable frictional sliding occurs if
i.e., K > Kc=-(a-b)s /Dc
F/u>K, i.e. (ms-md)s/Dc<K

13. Conditionally Stable regime
K > Kc=-(a-b)s /Dc
The condition for stable sliding can be met if the system is ‘stiff’ or if normal stress is low (near the surface)
NB: Seismic rupture can propagate into the ‘conditionally stable zone’ due to dynamic loading
(Gu et al, 1984)

14. Implication for nucleation size
K > Kc=-(a-b)s /Dc
The effective stiffness, K, of a crack with length L embedded in an elastic medium with shear modulus G scales as G/L.
>> A critical length, the nucleation length, is needed for the rupture to run away. It scales with (G.Dc)/(a-b) s.
>>The nucleation time is approximatively ta= a s / dS/dt , where S is the Coulomb stress.
(Diterich, 1994;Dieterich and Kilgore, 1996; Beeler et al, 2003)

15. Slip-weakening behavior of rate-and state friction
(Cocco and Bizzarri, 2002) Dc
Dc is not the distance over which slip-weakening happens during seismic slip (Obuko, 1988; Cocco and Bizarru, 2002)

16. Some implications of Rate-and-State Friction
Earthquakes can only nucleate on a patch where frictional sliding is unstable, where a-b<0
Seismic rupture can propagate into the conditionally stable zones, where a-b>0
Rupture becomes unstable only for large enough slipping patches. There is a critical nucleation size, scaling as G.Dc/s (a-b) implying a physical cutoff of the Gutenberg-Richter distribution at low magnitudes (cf Heimpel and malin, 1998).
The duration of the nucleation phase scales with ta= a s / dS/dt (cf Beeler et al, 2003)

17. Friction depends on temperature
Example of the results and model simulations of slip velocity and temperature stepping experiments on wet quartz gouge [Chester, 1994]. Sequence of velocity and temperature steps imposed on the gouge layers is shown. The friction response of quartz gouge is indicated by the thin lines (observed) and thick lines (model). The model is based on the state variable friction constitutive relation described in the text. The plot illustrates that for an abrupt change in temperature or slip rate, friction shows an immediate change (direct effect) followed by a gradual change in the opposite sense (evolution effect) to a new steady state value
(Chester, 1995)

18. Effect of temperature and normal stress on friction
A, Dependence of (a - b) on temperature for granite . B, Dependence of (a - b) on pressure for granulated granite. This effect, due to lithification, should be augmented with temperature.
(Marone, 1998; Sholz, 1998)
Laboratory experiments show that stable frictional sliding is promoted at temperatures higher than about 300°C, and that non lithified gouge is dominantly rate-strengthening.

19. Stationary State Frictional Sliding
mss== t/s= m*+(a-b) ln(V/V*)
a-b < 0
slip is potentially unstable
a-b > 0
stable sliding
a-b
Correspond to
T~300 °C
For Quartzo-Feldspathic rocks
(Blanpied et al, 1991)

20. Modeling the Seismic Cycle from Rate-and-state friction
(e.g., Tse and Rice, 1986)
Slip as a function of depth over the seismic cycle of a strike–slip fault, using a frictional model containing a transition from unstable to stable friction at 11 km depth
(Sholz, 1998)

21. Implications for aftershicks will be discussed next week
Reading:
- Dieterich, 1994; Gomberg 2001
- Gross and Kisslinger, 1997.

22. Modeling the Seismic Cycle from Rate-and-state friction
Tinter= 96.2 ans
Modeling the Seismic Cycle
from Rate-and-state friction
(e.g., Tse and Rice, 1986)
Hugo Perfettini