Computational GSH Talk-note at PITPC, 2013 Beijing Changsha China 410083
Computational GSH Talk-note at KITPC, 2013 Beijing
GSH = a hydrodynamic theory for granular matter Computational GSH = visualizing its theoretical equations with PC etc.
Why another continuum-mechanic theory?
Outline
Introduction
Background
Completing physical foundation
Unifying complex behaviors
Characterizing material feature
Governing equations for all macroscopic properties: Static stress, sound wave, etc.
Validity range is limited.
constitutive model = equation of motion for effective stress
In GSH, the stress is determined by
Thermodynamic identities
different strategies
Only mechanic quantities are involved
Unsuitable for treating statics, sound waves, etc.
Model and model parameters depend on experimental situations
It is worth to develop a broadly applicable
physical theory of continuous mechanics
Background
Conservation principles
Scale separation
All macroscopic behaviors are governed by a closed set of equations.
Mesoscopic size-effects are not described by the GSH:
Force chains
However some size effects may be accounted, at least partially,
by inclusion of spatial gradient terms.
It is assumed that the separation holds. This implies that:
Elasticity
When grain density > the loosest packing one, elasticity may be present.
It may be strong dissipative. (Then leads to a strong plasticity)
The dissipative flux can be determined by the Onsager relation, and is sensitive to void ratio and granular temperature (transient elasticity).
elastic strain:
In idea elasticity X=0, the two strains are same:
With the deformation rate:
Two stage irreversibility
I>0 is due to inelastic collisions among grains.
a dissipative process
Thermodynamics of mixture
Total differential form:
For an inclusion of water and gas, only the two concentrations are added in the thermodynamics.
For the grain + water + gas mixture,
mass variables are : a total mass and two concentrations
Conservations
(force balance)
Total mass:
Material relations
thermodynamic potential, e.g. free energy
transport coefficients
Due to the background, only these are needed to be studied.
They specify all material characters.
Engineering / GSH what’s the difference?
Though different objects, in certain approximations C could be derived from GSH
no inverse
A starting model of free energy
f0 can be obtained from volume fractions and free energies of gas, water , and grain material.
Transport coefficients
Most coefficients can be set to zero (especially off-diagonal ones)
The coefficient of mass diffusions can be learnt, e.g. by Darcy’s experiment.
Notes in computation
As GSH is nonlinear, having a large number of equations, difficulties may arise:
State boundary
State boundary: thermodynamic instability
GSH elastic energy has an instability boundary (also the van der Waals real-gas theory does)
The instability does exist for granular solid (e.g. explaining the Coulomb yield)
What DYNAMICS as the boundary is approached and after?
Frequently numerical computation breaks down around it. How to treat this?
Probably there are some dynamic modes having a divergent behavior?
The energy conservation
Breaking of conservation due to rounding error
This may cause divergence in numeric computations.
We suggest to consider in numeric code taking the energy conservation explicitly.
(Usually it is not so)
Q
Energy inputted = Heat production
Boundary and initial conditions
They are usually dependent on preparations.
Incomplete
Silo
stress
Note: for a fix preparation, the boundary and initial conditions remain unchanged.
Ensure that experiment is repeatable.
sandpile
An analysis of preparation dynamics is clearly a hard job.
Is a boundary condition rather than a question.
Examples
GSH can be helpful.
Among the four quantities:
Measuring the other two
The experiment is convenient for studying the material models and parameters.
(Allowing many testing variants!)
Drained, uniform axial compression at given radial stress:
To measure
down: simulation by Thorton & Antony
R. Soc. A356(1998)p2763
Logarithm-power
Kolymbas D. International Journal for Numerical and Analytical Methods in Geomechanics (2011). doi:10.1002/nag.1051;
constitutive modeling
friction angle
Bulk elastic strain has two decay times:
Similar reason as the softening.
Drained, constant volume, oscillating axial strain
Measure stress evolutions: q(t), P(t)
Stresses relax to zero!
Int. J. Numer. Anal.Meth. Geomech.
(2009) (submitted)
Undrained Exp.
Strain-control experiment.
Transition between Bagnold and CS: MiDi group
Volume = const.
Inertial number
Compaction
Mechanical explanation (not a statistic one)
Summary
all macroscopic
Insufficient, especially for initial and boundary conditions
Distributed in different samples, not a single one.
Conflicting
Thanks !