Support for DEM Simulation
in Chrono
Jonathan A. Fleischmann, Ph.D.
Currently: Assistant Scientist, Simulation-Based Engineering Laboratory (SBEL)
University of Wisconsin-Madison
(As of August 2015: Assistant Professor, Mechanical Engineering Department
Marquette University)
Presented at the Machine-Ground Interaction Consortium (MaGIC), May 13, 2015
Granular Materials
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Granular Materials: Macroscale
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Granular Materials: Macroscale (Triaxial)
Triaxial Test
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From Desrues and Chambon (2002)
Granular Materials: Macroscale (Triaxial)
σ1
σ2
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σ3
Friction work (black = 40 J)
From Lade and Duncan (1973)
Granular Materials: Macroscale (Triaxial)
σ1
σ2
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σ3
Friction work (black = 30 J)
From Lade and Duncan (1973)
Granular Materials: Macroscale (Yield)
Mohr-Coulomb
π-plane
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Drucker-Prager
𝜎𝑖 positive in compression
From Lade and Duncan (1973)
Granular Materials: Macroscale (Yield)
“Uniaxial Compression”: 𝜎1 > 𝜎2 = 𝜎3
Mohr-Coulomb
π-plane
Drucker-Prager
“Uniaxial Extension”: 𝜎1 < 𝜎2 = 𝜎3
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𝜎𝑖 positive in compression
From Lade and Duncan (1973)
Granular Materials: Macroscale (Yield)
ϕ = 24°
𝜇𝑚𝑎𝑐𝑟𝑜 = tan ϕ ≈ 0.45
(𝜇 is the inter-particle
friction coefficient)
𝜖1 (positive in compression)
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Granular Materials: Macroscale (Yield)
ϕ = 33°
𝜇𝑚𝑎𝑐𝑟𝑜 = tan ϕ ≈ 0.65
(𝜇 is the inter-particle
friction coefficient)
𝜖1 (positive in compression)
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Granular Materials: Macroscale (Direct Shear)
Direct Shear Test
ϕ  30
tan ϕ = 𝜇𝑚𝑎𝑐𝑟𝑜 =
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𝑆ℎ𝑒𝑎𝑟 𝑆𝑡𝑟𝑒𝑠𝑠 𝑜𝑛 𝑆ℎ𝑒𝑎𝑟 𝑃𝑙𝑎𝑛𝑒
𝑁𝑜𝑟𝑚𝑎𝑙 𝑆𝑡𝑟𝑒𝑠𝑠 𝑜𝑛 𝑆ℎ𝑒𝑎𝑟 𝑃𝑙𝑎𝑛𝑒
=
𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑆ℎ𝑒𝑎𝑟𝑖𝑛𝑔 𝐹𝑜𝑟𝑐𝑒
𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝐿𝑜𝑎𝑑
Granular Materials: Macroscale (Direct Shear)
50,000 spherical particles
constant velocity
(in 𝑥-direction)
constant normal stress
(in 𝑦-direction)
(ASTM C 778-06)
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Granular Materials: Macroscale (Direct Shear)
ϕ  30°
(3-D)
ϕ  24°
(2-D)
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Granular Materials: Microscale (DEM-Penalty)
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Granular Materials: Microscale (DEM-Penalty)
𝜹𝒕
𝒏
𝑭𝒏 = 𝑘𝑛 𝛿𝑛 𝒏 − 𝛾𝑛 𝑚eff 𝒗𝒏
Coulomb friction: If 𝑭𝒕 > 𝜇 𝑭𝒏
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Projected onto
contact plane at
each time step.
𝑭𝒕 = −𝑘𝑡 𝜹𝒕 − 𝛾𝑡 𝑚eff 𝒗𝒕
then scale 𝜹𝒕
so that 𝑭𝒕 = 𝜇 𝑭𝒏
Granular Materials: Microscale (DEM-Penalty)
𝒗𝒕
True history: The vector 𝜹𝒕 = 𝜹∗𝒕 − 𝜹∗𝒕 ∙ 𝒏 𝒏
where 𝜹∗𝒕 =  𝒗𝒕 ∆𝑡 as long as the pair remains
𝒏
in contact (must be stored from one time step
to the next).
Pseudo-history: Let the vector 𝜹𝒕 = 𝒗𝒕 ∆𝑡 at
each time step. Is this good enough?? (No!)
𝜹𝒕
Projected onto
contact plane at
each time step.
𝑭𝒕 = −𝑘𝑡 𝜹𝒕 − 𝛾𝑡 𝑚eff 𝒗𝒕
Note: In general (true history) the vectors 𝜹𝒕 and 𝒗𝒕 are not in the same direction.
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Granular Materials: Microscale (DEM-Penalty)
3 1−𝜈
𝑟𝑐 =
4𝐸
2𝐺𝑟𝑐
𝑘𝑛 =
1−𝜈
4𝐺𝑟𝑐
𝑘𝑡 =
2−𝜈
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1/3
2
𝐹𝑛 𝑟
𝑘𝑡 2(1 − 𝜈)
=
𝑘𝑛
2−𝜈
Hertz/Deresiewicz model
(other models available)
Granular Materials: Microscale (DEM-Penalty)
1800 uniform spheres
randomly packed
Particle Diameter:
D = 5 mm
Shear Speed:
1 mm/s
Inter-Particle Coulomb
Friction Coefficient:
µ = 0.5
(Quartz-on-Quartz)
Void Ratio
(dense packing):
e = 0.4
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1800 uniform spheres
randomly packed
𝜇𝑚𝑎𝑐𝑟𝑜 ≈ 2
ϕ = tan−1 𝜇𝑚𝑎𝑐𝑟𝑜 ≈ 63°
Particle Diameter:
D = 5 mm
Shear Speed:
1 mm/s
Inter-Particle Coulomb
Friction Coefficient:
µ = 0.5
(Quartz-on-Quartz)
Void Ratio
(dense packing):
e = 0.4
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𝜇 = 0.5
ϕ𝜇 = tan−1 𝜇 ≈ 26.6°
𝜇𝑚𝑎𝑐𝑟𝑜 ≈ 0.25
ϕ = tan−1 𝜇𝑚𝑎𝑐𝑟𝑜 ≈ 14°
Direct Shear for 5,000 Uniform Glass Beads:
Validation (DEM-Penalty)
5000 uniform spheres
randomly packed
Particle Diameter:
D = 6 mm
Shear Speed:
1 mm/s
Inter-Particle Coulomb
Friction Coefficient:
µ = 0.18
(Glass-on-Glass)
Void Ratio
(loose packing):
e = 0.7
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Direct Shear for 5,000 Uniform Glass Beads:
Validation (DEM-Penalty)
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Uniform Glass Beads
(Randomly Packed)
Physical Experiment
DEM-P Simulation
Number of spheres
 5000
5000
Sphere diameter (m)
0.006
0.006
Sphere density (kg/m3)
2550
2550
Shear speed (m/s)
 0.00002
0.001
Glass-on-glass Coulomb
friction coefficient
0.18
0.18
Glass-on-box Coulomb
friction coefficient
0.13
0.13
Young’s Modulus (Pa)
4(1010)
4(107) and 4(106)
Poisson’s Ratio
0.22
0.22
Packing method
Rainfall (loose)
Rainfall (loose)
Void ratio*
 0.7
0.66 – 0.40*
Direct Shear for 5,000 Uniform Glass Beads:
Validation (DEM-Penalty)
(Physical Experiment: Härtl and Ooi, 2008)
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(Chrono: DEM-P – Tangential History)
Void ratio e  0.7
Void ratio e = 0.66, 0.65, 0.62, 0.60
Young’s modulus E = 4(1010) Pa
Young’s modulus E = 4(107) Pa
Direct Shear for 5,000 Uniform Glass Beads:
Validation (DEM-Penalty)
(Physical Experiment: Härtl and Ooi, 2008)
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(Chrono: DEM-P – Tangential History)
Void ratio e  0.7
Void ratio* e = 0.58, 0.54, 0.48, 0.40
Young’s modulus E = 4(1010) Pa
Young’s modulus E = 4(106) Pa
Granular Materials: DEM-Complementarity
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•
Rigid body dynamics – no “inter-penetration” or local deformation at the contact point
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Complementarity conditions employed to link distance between shapes and normal force
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Friction posed as an optimization problem
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Equations of motion become equilibrium constraints, an appendix to optimization problem
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Differential Variational Inequalities (DVI) discretized to lead to nonlinear complementarity problem
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Relaxation yields cone complementarity problem (CCP)
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Yields matrix equation to be solved at each time step – no Courant-Friedrichs-Lewy (CFL) stability restriction!
Granular Materials: DEM-Complementarity
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Validation: DEM-P vs. DEM-C
• Mass flow rate experiment
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•
•
•
•
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500 µm spheres
Total mass of granular material: 6.38 g
Width of opening: 9.398 mm
Opening speed: 1 mm/s
Maximum opening gap: 2 mm
(DEM-C)
(DEM-P)
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Validation: DEM-P vs. DEM-C
• Mass flow rate experiment
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•
•
•
•
39,000 particles
r = 0.2510-3 m
 = 0.3
2 mm gap size
40 threads
• DEM-C
• DEM-P
• Step-size: 10-4 s
• Step-size: 10-5 s
• Cost per step:
• Collision detection:
• Update:
• Solver:
0.423 s
0.054 s
0.010 s
0.359 s
• Average number contacts: 140,000
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• Cost per step:
• Collision detection:
• Update:
• Solver:
0.054 s
0.026 s
0.010 s
0.016 s
• Average number27contacts: 118,400
Validation: DEM-P vs. DEM-C
• Low-speed impact cratering
• 379,000 spheres
• r = 500 µm
• ρ = 1500 kg/m3
• E = 108 Pa
•  = 0.3
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Validation: DEM-P vs. DEM-C
• Low-speed impact cratering
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𝜁 𝜌𝑏
𝑑=
𝜇 𝜌𝑔
1 2
𝐷𝑏2 3 𝐻1
3
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Validation: DEM-P vs. DEM-C
• Low-speed impact cratering
• 379,030 spheres
• ρ = 700 kg/m3
• h = 0.1 m
• 64 threads
• DEM-C
• DEM-P
• Step-size: 10-4 s
• Step-size: 10-5 s
• Cost per step:
• Collision detection:
• Update:
• Solver:
3.471 s
0.087 s
0.012 s
3.372 s
• Average number contacts: 1.37 million
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• Cost per step:
• Collision detection:
• Update:
• Solver:
0.144 s
0.061 s
0.010 s
0.072 s
• Average number30contacts: 1.08 million
Validation: DEM-P vs. DEM-C
• Direct shear test
1000 uniform spheres
randomly packed
Particle Diameter:
D = 6 mm
Shear Speed:
1 mm/s
Inter-Particle Coulomb
Friction Coefficient:
µ = 0.18
(Glass-on-Glass)
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Void Ratio
(loose packing):
e = 0.7
Validation: DEM-P vs. DEM-C
• Direct shear test
1000 uniform spheres
randomly packed
Particle Diameter:
D = 6 mm
Shear Speed:
1 mm/s
Inter-Particle Coulomb
Friction Coefficient:
µ = 0.18
(Glass-on-Glass)
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Void Ratio
(loose packing):
e = 0.7
Validation: DEM-P vs. DEM-C
• Direct shear test
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•
•
•
•
1,000 particles
r = 3 mm
µ = 0.18
1 mm/s shear speed
10 threads
• DEM-C
• DEM-P
• Step-size: 10-3 s
• Step-size: 10-5 s
• Cost per step:
• Collision detection:
• Update:
• Solver:
0.1615 s
0.0036 s
0.0007 s
0.1571 s
• Average number contacts: 4800
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• Cost per step:
• Collision detection:
• Update:
• Solver:
0.0025 s
0.0008 s
0.0007 s
0.001 s
• Average number33contacts: 4200
Macromechanics: Systematic parameter studies
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Micromechanics: DEM is more than a “black box”
𝑘𝑡
𝛼=
𝑘𝑛
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Uniform Spheres: Microscale  Macroscale
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Preprocessing Tools: Specimen Generation
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Postprocessing Tools: Force Chain Visualization
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Postprocessing Tools: Particle Trajectories
In horizontal
shear plane
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Thank you!
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