Date: Mon, 22 Aug 2005 15:56:34
From: Michael Falk
To: Earthquake Physics 2005
Subject: [earthq05] Suggestions for discussion Tuesday 8/23/05

Dear Earthq05 Participants:

The following are some discussion points that Peter Sollich and Anael
Lemaitre wrote up to help prompt discussion Tuesday.  Please feel free to
read them through if you have time before the discussion. 

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Constitutive modelling of slip and plasticity

What should it be able to predict?

(a) Bulk flows (mainly shear):
- linear response; broad relaxation timescale spectrum
- flow curves: yield stress, sigma-sigma_yield ~ (shear rate)^p
  [but is this just a fit or more fundamental?]
- stress overshoots (shear startup), transients for/change of shear rate
- history dependence (repeated deformation: work hardening), aging
- normal stresses (or differences), dilatancy

(b) Localization/shear banding:
- when stable, what type (e.g. shear rate=0 except for narrow region
[=slip], or two [or more?] bands with nonzero shear rates, or one band with
shear rate=0)
- when dynamically accessible (nature of instability)
- stick-slip transition/intermittency/link to localization and fracture
- aging? [could occur in unsheared or weakly sheared zones]

(c) Variation with important "external" control parameters, in addition to
shear rate or stress:
- volume fraction
- temperature
- normal stress/pressure for earthquakes?


Physical basis of models

- are non-spatial models adequate at least for bulk flows, i.e. are
rearrangements/catastrophes sufficiently uniform and interactions
sufficiently long-ranged [to get mean-field behaviour] or short-ranged [to
avoid strong spatial correlations]?
- nature of dissipative events; localized for small perturbations but
extended (cooperative) in steady shear?
- nature of state variables [area of contact/granular temperature/density]
- how _many_ state variables do we need (1 or 2 [rate-state, STZ], or a
whole function [SGR])?
- can we get away with similar models for different materials? e.g.:
* hard materials (granulars; dilatancy is important?) vs soft ones
(colloids/emulsions; basic constituents are compressible, system including
solvent is incompressible)
* thermal (atomic/molecular glasses) vs athermal ones (colloids, emulsions
[most], granulars)
* type of local dissipation (friction in granulars, solvent flows in
colloids, pore fluid in earthquakes...)


How can experiments constrain/discriminate between models?

- various types of shear banding/localization?
- occurrence of stick-slip?
- can experiments constrain the choice of state variable?


Existing modelling approaches [review on demand]

- single state variable: rate-state, Bonn et al's viscosity bifurcation;
Ajdari/Lequeux et al's fluidity models (non-spatial, normally single
relaxation timescale or ill-defined linear response)

- continuum: damage rheology, phase field (good at localization, less good
at broad linear response?)

- mesoscopic/coarse-grained models:
* STZ and variations, e.g.  granular (mainly without spatial information,
but can be extended; good for deformation history, and with effective
temperature maybe also be aging, but single relaxation timescale)
* SGR (non-spatial; can be extended, but coupling between regions not
clear/unique; has aging effects and broad relaxation time spectrum but this
is put in by hand via rho(E)))
* Picard's stress propagation (similar to earthquake models by Obukhov et
al; not convected [yet] and so far explored mainly for localization and
stress propagation)
* Hebraud & Lequeux's "mode coupling" theory (non-spatial; unusual
[amplitude-dependent] linear response; predicts single possible exponent
p=1/5 for flow curve)


Other open questions:

- regeneration of dissipative zones in systems where kT=0?
- for localization: what lengthscales do we need (size of STZ/SGR-element?);
how do we put these into continuum theories?
- in non-spatial models, what assumptions on homogeneity (stress? strain?
strain rate?)