deformation of sediments via grain-scale simulations: variational algorithm ran holtzman, dmitriy...
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Deformation of Sediments via Grain-Scale Simulations:
Variational Algorithm Ran Holtzman, Dmitriy Silin, Tad Patzek
U.C. Berkeley
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x (mm)y (mm)
z (m
m)
13.85
12.97
12.09
11.21
10.33
9.45
8.57
7.69
6.81
5.93
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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 (EMT1)
– Dynamic grain-scale simulations (DEM2)
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1.5
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2.5
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x (mm)y (mm)
z (m
m)
13.85
12.97
12.09
11.21
10.33
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8.57
7.69
6.81
5.93
1 – Duffy & Mindlin, 1957
2 – Cundall & Strack, 1979
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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
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Variational Algorithm
• Quasi-static model: sequence of static equilibrium
configurations
• Equilibrium: minimal-work path
• Moduli: fit stress-strain to
Hooke’s law:
ijijkkij Gee 2
5
P
P
h
*
1 1 1
ij i jR R R
22
*
111 ji
ij i jE E E
Hertz (1882)
Normal Compression
2321**
3
4hREP
grain i
grain j
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Shear
7
Frustrated Rotation
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Torsion
Mtor
Mtor
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Challenges in Modeling Friction
• Loads depend on normal force and load history1
• Implementing M-D theory1 - 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)3-4
1 – Mindlin & Deresiewicz (1953) 3 – Walton & Braun (1986)
2 – Jenkins & Strack (1993) 4 – Vu-Quoc & Zhang (1999)
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Linearized Formulation• Incremental loading, small perturbations
• Shear increment decoupled from normal components
-1 0 1
x 10-4
-50
-40
-30
-20
-10
0
10
20
30
40
50
Shear Displacement, ut
Sh
ea
r fo
rce
, Q
Mindlin & Deresiewicz (1953)Linear Approximation
k
ut
Q0(proj)Q0
initial
current
Q
u
||Q||=P
122
8
j
j
i
iijij GG
ak
tproj k uQQ )(0
Q
Q
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100
101
102
103
102
103
104
Confining Stress (MPa)
Bu
lk M
od
ulu
s (M
Pa
)
Holtzman(2007)-QuartzDomenico(1977)-SandHoltzman(2007)-GlassDomenico(1977)-GlassMakse(1999)-GlassYin(1993)-GlassDEM-Makse(1999)-GlassEMT-Makse(1999)-Glass
Predicted Moduli vs. Experiments
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100
101
102
103
102
103
104
Confining Stress (MPa)
She
ar M
odul
us (
MP
a)
Holtzman(2007)-QuartzDomenico(1977)-SandHoltzman(2007)-GlassDomenico(1977)-GlassMakse(1999)-GlassYin(1993)-GlassDEM-Makse(1999)-GlassEMT-Makse(1999)-Glass
Predicted Moduli vs. Experiments
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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
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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
1.52
2.53
3.5
1.5
2
2.5
3
3.5
1.5
2
2.5
3
3.5
x (mm)y (mm)
z (m
m)
13.85
12.97
12.09
11.21
10.33
9.45
8.57
7.69
6.81
5.93
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