Friction: The rate-and-state constitutive law

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 http://www.servogrid.org/EarthPredict/

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

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 - 10-10 m/s • Axial shear testing - 10-7 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, 115-134, 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.