1. Friction:
The rate-and-state constitutive law
Connecting the dots between Bowden-Tabor and Dieterich
Nucleation of slip
Insight from high speed friction experiments

2. The rate-and-state constitutive law
The following constitutive law provides reasonable fit to the experimental data (Dieterich, 1979; Ruina, 1983):
where:
“delta dot” is the slip rate.
theta is a state variable.
a and b are constitutive dimensionless numbers of the order of 10-2.
Dc is a characteristic slip distance.
Question: What are the units of the state variable.

3. The rate-and-state constitutive law
It is useful to examine the properties of this law under three simple situations:
1. Steady state.
2. Hold time.
3. Velocity stepping.

4. The rate-and-state constitutive law
1. When at steady-state:
Thus:
and:
Note that steady-state friction is velocity weakening if b>a and is velocity strengthening if b<a.

5. The rate-and-state constitutive law
Question: Which materials exhibit velocity-weakening and which velocity-strengthening

6. The rate-and-state constitutive law
2. During hold time:
The solution of which is:
Experimental data shows that:
In terms of the state variable, the real contact area is thus:

7. The rate-and-state constitutive law
Recall that according to Bowden and Tabor:
where p is the contact hardness, previously considered as time-independent. The results presented above clearly show that p can be related to the hold time and the state variable as follows:

8. The rate-and-state constitutive law
3. To examine consequences of velocity stepping, we write the stress just before and just after a shear stress perturbation of  are - and +, respectively:
and thus the stress change is:
Since +=-, we get:
Note that a modest change in /a results in a big change in the sliding velocity.

9. The rate-and-state constitutive law

10. The rate-and-state constitutive law
Similarly, we can write an expression for the shear stress after an instantaneous stress change due to a velocity jump from the reference velocity to “delta dot” as:
Because instantaneous change in shear stress causes no change in the contact area, we can safely write.
Comparison with Bowden-Tabor 2nd equation suggests that the shear strength, c, is velocity dependent:

11. Slip nucleation in the lab
Figure from

12. Slip nucleation in the lab
Okubo and Dieterich, 1984

13. Slip nucleation in the lab
Okubo and Dieterich, 1984

14. Slip nucleation in the lab
Ohnaka’s (1990) stick-slip experiment
Figures from Shibazaki and Matsu’ura, 1998

15. Slip 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

16. Slip 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).

17. The critical stiffness:
Reminder: The notion of critical stiffness and the condition for slip acceleration in a spring-slider system
Slope=k
Slope=kcrit
The critical stiffness:
For the slip-weakening law:
The condition for slip acceleration:

18. From a spring-slider to a crack embedded within elastic medium
The elastic stiffness is:
where:
 is a geometrical constant
G is the shear modulus
The critical stiffness:
Dieterich (1992) identified the constant with: L
Rice and Ruina (1983) identified the constant with:

19. From critical stiffness to the notion of a critical crack length
Ziv, 2007

20. From critical stiffness to the notion of a critical crack length
Ziv, 2007

21. High speed friction experiments
Neither triaxial nor biaxial (double-shear) testing apparatus can be used for high speed (and large slip) experiments.
Tectonic m/s
Axial shear testing to 10-5 m/s
Seismic slip - 1 m/s

22. High speed friction experiments
Tonalites under 1.28 m/s
Di Toro et al., 2006.
Question: Does melt act as a lubricant or a viscous brake?

23. High speed friction experiments
Di Toro et al., 2006.

24. High speed friction experiments
Di Toro et al., 2006.

25. Further reading
Dieterich, J. H., Earthquake nucleation on faults with rate- and state-dependent strength, Tectonophysics, 211, , 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 high-speed rupture propagation: scaling from stick-slip experiments to natural earthquakes, Geophys. J. Int., 132, 14-30, 1998.
Di Toro et al., Natural and experimental evidence of melt lubrication of faults during earthquakes, Science, 311, 647, 2006.