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

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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

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Aside:

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

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

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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

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Langer et al

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Shearing

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

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Status of the Theory

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

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Phys Rev E (2001)

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

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

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