The following topics were brought up by Peter Sollich and Anael Lemaitre on Tuesday August 23 for discussion. A recording of that discussion can be found at http://online.kitp.ucsb.edu/online/earthq05/discussion/
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?)

## Constitutive modelling of slip and plasticity

The following topics were brought up by Peter Sollich and Anael Lemaitre on Tuesday August 23 for discussion. A recording of that discussion can be found at http://online.kitp.ucsb.edu/online/earthq05/discussion/

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 includingsolvent 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 effectivetemperature 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 thisis 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 andstress propagation)

- Hebraud & Lequeux's "mode coupling" theory (non-spatial; unusual

[amplitude-dependent] linear response; predicts single possible exponentp=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?)