Deformation of Sediments via Grain-Scale Simulations: Variational Algorithm Ran Holtzman, Dmitriy Silin, Tad Patzek U.C. Berkeley

2 Motivation Why micromechanics? –Mechanics of granular matter is controlled by interaction of discrete grains Why numerical simulations? –Enable micromechanical analysis, unavailable from experiments (restricted to 2D or a single grain pair) Existing models: –Spatially-averaged solutions (EMT 1 ) –Dynamic grain-scale simulations (DEM 2 ) 1 – Duffy & Mindlin, – Cundall & Strack, 1979

3 Our Model of Granular Matter 3D heterogeneous, disordered pack Spherical grains, differ in size & properties Bounded by a rigid container (imposing boundary conditions) Contact forces & moments  macroscopic stress

4 Variational Algorithm Quasi-static model: sequence of static equilibrium configurations Equilibrium: minimal-work path Moduli: fit stress-strain to Hooke’s law:

5 P P h Hertz (1882) Normal Compression grain i grain j

6 Q Q Shear

7 Frustrated Rotation Q Q

8 Torsion M tor

9 Challenges in Modeling Friction Loads depend on normal force and load history 1 Implementing M-D theory 1 - cumbersome for multiple contacts Simplified models –Ignoring frictional loads (zero tangential stiffness) –Ignoring partial slip (fixed stiffness) 2 –Simplified treatment of partial slip (variable stiffness) – Mindlin & Deresiewicz (1953)3 – Walton & Braun (1986) 2 – Jenkins & Strack (1993)4 – Vu-Quoc & Zhang (1999)

10 Linearized Formulation Incremental loading, small perturbations Shear increment decoupled from normal components k utut Q 0(proj) Q0Q0 initial current Q u ||Q||=  P QQ QQ

11 Predicted Moduli vs. Experiments

12 Predicted Moduli vs. Experiments

13 Summary Quasi-static grain-scale simulations of a deforming sediment Physically-based model, no calibration used Macroscopic moduli match experimental data Application: effect of dissociation on hydrate- bearing sediments

14 Extensions Add cement, angular grains, and pore constituents that interact with the solid grains Statistical and qualitative analysis of microscopic parameters – e.g. force chains Reduce computing time by using parallel computing

15 Thank You! Funded by the assistant secretary for fossil energy, office of Natural Gas and Petroleum Technology, N.E.T.L. D.O.E. Contract #DE-FC26-05NT42664