Computational GSH Talk-note at KITPC, 2013 Beijing

1. Computational GSHTalk-note at KITPC, 2013 Beijing Jiang Yimin, Liu Mario Physical department, Central South University Changsha China 410083

2. GSH = a hydrodynamic theory for granular matterComputational GSH = visualizing its theoretical equations with PC etc. Why another continuum-mechanic theory? We consider : grain + water + gas mixture

3. Outline • Introduction • Background • Material relations • Notes for Computations • Examples • Summary

4. Introduction Reasons for a new theory • Completing physical foundation • Unifying complex behaviors • Characterizing material feature

5. Biot theory vesus GSH • Force balance • Mass conservation • Darcy law • Terzaghi effective stress • Constitutive model e.g. saturated soil The Biot is • same • same • explained by mass diffusion • explained by free energy • explained by hydrodynamic model Governing equations for all macroscopic properties: Static stress, sound wave, etc. Validity range is limited.

6. effective stress = total stress – water pressure constitutive model = equation of motion for effective stress different strategies for characterizing material features • In GSH, the stress is determined by • Thermodynamic identities • Expressions of free energy, transport coefficients A traditional physical method

7. Why the constitutive model is not fully convincing ? • 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

8. Background • Scale separation • Elastic strain • Entropies • Thermodynamics of mixture • Onsager dissipative flux-force relation • Conservation principles

9. Scale separation All macroscopic behaviors are governed by a closed set of equations. It is assumed that the separation holds. This implies that: • Mesoscopic size-effects are not described by the GSH: • Force chains • Clogging • etc. However some size effects may be accounted, at least partially, by inclusion of spatial gradient terms.

10. Elasticity When grain density > the loosest packing one, elasticity may be present. elastic strain: With the deformation rate: 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). In idea elasticity X=0, the two strains are same: Their difference is plastic strain.

11. Two stage irreversibility Temperature , granular temperature T , Tg, and corresponded entropies . a dissipative process Rg >0 I*Tg>0 R>0 I>0 is due to inelastic collisions among grains.

12. Thermodynamics of mixture Total differential form: For the grain + water + gas mixture, mass variables are : a total mass and two concentrations For an inclusion of water and gas, only the two concentrations are added in the thermodynamics.

13. Conservations of masses, energy , momentum: (force balance) Here g is gravitational acceleration. Total mass:

14. Material relations All are algebraic relations. (Local properties) • 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? no inverse Though different objects, in certain approximations C could be derived from GSH Experiment data are crucial for revealing the material relations.

15. A starting model of free energy f0 can be obtained from volume fractions and free energies of gas, water , and grain material.

16. 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.

17. Notes in computation As GSH is nonlinear, having a large number of equations, difficulties may arise: • State boundary • Breaking of conservation due to rounding error • Lack of some boundary or initial conditions

18. 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?

19. 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 In case of steady motion, we have the balance: Energy inputted = Heat production

20. Boundary and initial conditions Silo stress • In most cases we need to measure them. • They are usually dependent on preparations. • Incomplete Note: for a fix preparation, the boundary and initial conditions remain unchanged. Ensure that experiment is repeatable. An analysis of preparation dynamics is clearly a hard job. sandpile Is a boundary condition rather than a question. There is a pressure dip or not?

21. Examples • Triaxial dynamics • Stationary movement • Compacting dynamics • etc. Just consider Saturated sample (without gas) Nonsaturated case: the constitutive modeling faces many difficulties. GSH can be helpful.

22. Triaxial dynamics Cylindrical dynamics: measurements most frequently used by engineers. drained or undrained Among the four quantities: Controlling two Measuring the other two undrained And the water pressure Assume uniformity =0 for fully drained dynamics The experiment is convenient for studying the material models and parameters. (Allowing many testing variants!)

23. Drained, uniform axial compression at given radial stress: To measure Softening: approaching to CS has two different decay times: Critical state: Stationary shear deformation Logarithm-power Dashed: up: experiment by Wichtmann, PhD 2005 down: simulation by Thorton & Antony R. Soc. A356(1998)p2763 Rate independent!

24. Kolymbas D. International Journal for Numerical and Analytical Methods in Geomechanics (2011). doi:10.1002/nag.1051; Drained, constant radial stress, : constitutive modeling To measure friction angle

25. Kolymbas D.: constitutive modeling Drained, constant radial strain, oscillating axial stress

26. Constant volume: drained To measure Experiment: usually UNDRAINED Wichtmann, PhD 2005 P Bulk elastic strain has two decay times: Pressure decreasing. Strongly enhanced if water is present. Similar reason as the softening.

27. Drained, constant volume, oscillating axial strain Measure stress evolutions: q(t), P(t) Strain-control experiment. Undrained Exp. Stresses relax to zero! Wichtmann, L., Niemunis, A., Triantafyllidis, T Int. J. Numer. Anal.Meth. Geomech. (2009) (submitted)

28. Stationary movement (rheology) Volume = const. Relationships among pressure, shear stress, shear rate and density • Bagnold scaling : • Critical state (CS) at high density: • Transition between Bagnold and CS: MiDi group rate-independent Logarithm-power Inertial number Bagnold Bagnold CS? low density high density

29. Symbols: simulation by C. S. Campbell, J. Fluid Mech. 465, 261, 2002

30. Compaction A simple GSH computation P=const. Tg pulses: More realistic computations Further numeric ability Tg During a pulse: Densification stops if seismic pressure and P are balanced. t mechanic tapping or flow pulses Mechanical explanation (not a statistic one) stretched exponential law or inverse logarithmic law Ciamarra,Coniglio,Nicodemi PRL 2006 97 158001 (accelaration ratio a/g)

31. Summary A broadly applicable theory of continuous mechanics Under careful examinations all macroscopic properties Free energy Transport coefficients • Main difficulties come from measured data: • Insufficient, especially for initial and boundary conditions • Distributed in different samples, not a single one. • Conflicting Thanks !