1. Two Alternative Constitutive Theories *Rate-State Laws *Ginzburg-Landau

2. Rate-State Laws for Frictional Material Internal variable θ Yield τ/σ = + f(V, θ) Stress is no longer is rate independent i.e. rate dependent friction Recall Mohr-Coulomb has T ~ dev(V)/|dev(V)| Motivated by earthquake models

4. Aside:

5. Aside Analysis by Carlson and Langer, Phys Rev Lett (1989)

6. Evolution of Internal Variable Phenomenological Indentation, ploughing, slow fusing of contacts Typical ~.2-.8 Shear stress ~ 100 MPa - 4 GPa

15. A Different Take uAffine deformation is one for which all particle movement is related through a simple tensorial relation uObserve locally affine deformations with areas which are non-affine uFalk and Langer – rearrangement of particles in a 2 state system

16. Langer et al

17. Shearing

18. R +/- are pressure dependent probabilities, R+/- are number densities, w is the plastic work

21. Status of the Theory

24. And Now for Something Completely Different: Energetics Phenomenological Based on Cahn-Hilliard, Ginzburg-Landau theories Here, for partially fluidized granular material

25. Phys Rev E (2001)

26. ρ order parameter The base stress σ 0 is given as in a static stress; not specified in A-T

27. Several applications – thin and deep avalanche flows, shearing, rotating drum.