Two Alternative Constitutive Theories

*Rate-State Laws*Ginzburg-Landau

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

Aside:

Aside

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

Evolution of Internal Variable

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

A Different Take

Affine deformation is one for which all particle movement is related through a simple tensorial relation

Observe locally affine deformations with areas which are non-affine

Falk and Langer – rearrangement of particles in a 2 state system

Langer et al

Shearing

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

Status of the Theory

And Now for Something Completely Different: Energetics

Phenomenological Based on Cahn-Hilliard, Ginzburg-Landau theories Here, for partially fluidized granular material

Phys Rev E (2001)

ρ order parameter

The base stress σ0 is given as in a static stress; not specified in A-T

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