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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.