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![Page 1: 1 Tessellations and granular materials Niels P. Kruyt Department of Mechanical Engineering University of Twente n.p.kruyt@utwente.nl](https://reader036.fdocuments.net/reader036/viewer/2022062404/551a7cba550346e0158b47bd/html5/thumbnails/1.jpg)
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Tessellations andgranular materials
Niels P. Kruyt
Department of Mechanical Engineering University of Twente
[email protected] www.ts.ctw.utwente.nl/kruyt/
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Overview
• University of Twente
• Split personality
• Granular materials– Micromechanics
• Tessellations
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Location Enschede
Enschede
Delft
Eindhoven
Leiden
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Split personality
Science:granular materials
Engineering:turbomachines
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Turbomachines
• CFD methods• Optimisation methods• Inverse-design methods• PIV measurements
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What are granular materials?
• Grains– natural– biological– man-made
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Applications of granular materials
• geotechnical engineering• geophysical flows• bulk solids engineering • chemical process engineering• mining• gas and oil production• food-processing industry• agriculture• pharmaceutical industry
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Features
• elasticity• frictional• plasticity• dilatancy• anisotropy
• viscous• multi-phase• cohesion• segregation
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Fluid-like behaviour
• Fluidised beds
• Collisions
• Kinetic theory
• Inelasticity
• Clustering
Deen, Department of Chemical Engineering, University of Twente
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Solid-like behaviour
• Frictional
• Pressure-dependent
• Elasticity
• Plasticity
• Dilatancy
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Continuum mechanics
• Stress tensor
• Strain tensor
dAd nf
nfd
dA
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Continuum mechanics
• Stress tensor
• Strain tensor
xu dd
xd0u
uu d0
j
iij x
u
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Constitutive relations
• Description of material behaviour
• Relation between stress and strain (rate)
• Elastic
• Plastic
• Viscous
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Categories of constitutive relations
• Continuum theories– phenomenological; elasto-plasticity
• Micromechanical theories– relation with microstructure and particle
properties
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Micromechanics
• Relations: discrete continuum
Discrete ContinuumHomogenisation
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Tool: Discrete Element Method
• Particle interaction• Newton’s laws
• Patience
• Simple model at micro-level
• Complex behaviour at macro-level
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Particle interaction
• Elasticity• Friction• Damping
qppq UUu
)(function pqpq uf
Interaction at contacts!
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Mixing in rotating cylinder
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From discrete information stress and strain
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Micromechanical constitutive relations
Relative displace-
ment Force
Stress Strain
Constitutive relation
Microscopic level(contact)
Macroscopic level(continuum)
Averaging
Averaging
Localisation
Localisation
TE
SS
ELL
AT
ION
S
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Objective
• Expression for strain tensor in terms of relative displacement at contacts
pqpq UUu + +
pUqU
qp
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Average strain tensor
j
iij x
u
dVx
u
VdV
V V j
i
V
ijij
11
Definition of strain
Average strain
Average strain is determined by displacements at boundary!
dBnuV B
jiij 1
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Approach
• Strain expression: – averaging of compatibility equations– displacement of line segment
• Tessellation: network of contacts
• Compatibility equations
• Averaging
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Tessellation: network of contacts
QUESTION 1:Fast algorithm for determining tiles?
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Compatibility equations
S
RSi
pi
qi
pqi UU
0
pqpq UUu
0 R
S
RS uu
0
addccbba
dacdbcab
S
RS
UUUUUUUU
uuuuu
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Averaging of compatibility equations (1)
0
R
Ri
S
RSi
Rj uuY
0 S
Ri
Rj
R S
RSi
Rj uYuY
Ri
Si
RSi YYg
RiSi
RSj
SRi
Sj
RSi
Rj YYuuYuY
0 B
ijCc
ci
cj uYug
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0 B
ij
Cc
ci
cj dsu
ds
dxug
Averaging of compatibility equations (2)
jt
0 B
ijCc
ci
cj uYug
0 B
ij
Cc
ci
cj ds
ds
duxug
0 B
ijCc
ci
cj dsunuh
ni
Bti
ijA
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Summary for strain
• Formulation in relative displacements• Tessellation of network of contacts• Averaging of compatibility equations
Cc
ci
cj
A j
iij uh
AdA
x
u
A
11
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Expressions for stress and strain
Cc
ci
cj
A j
iij uh
AdA
x
u
A
11
Cc
cj
ci
A
ijij flA
dAA
11
Cc
cj
ciij hl
AI
1
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Micromechanically-based constitutive relations
Relative displace-
ment Force
Stress Strain
Constitutive relation
Microscopic level(contact)
Macroscopic level(continuum)
Averaging
Averaging
Localisation
Localisation
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Tessellation (3D)
• Delaunay tessellation
• Edges– physical contacts– virtual contacts
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Bagi’s strain expression
ej
Ee
eiij Hu
V
1
Complex geometrical quantity;complementary area vector
Set of edges
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Use of Bagi’s expression
• Correctness
• Investigation deformation
• DEM simulation of triaxial test
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Triaxial test
• Imposed deformation in X-direction
• Constant lateral stresses
1 1
0
0
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Triaxial test (2D version)
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Response
01
01
22
3
p
q
Dilation
Compression
Imposed deformation
She
ar s
tren
gth
Vol
ume
chan
ge
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Orientational averaging
nεu l
nl0u
uu 0
Average over edges with same orientation!
uu
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Edge distribution function
EDGES CONTACTS
Induced geometrical anisotropy shear strength
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Average relative displacements
• Normal component
2cos),( 0 nn aau
Fourier coefficients
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Evolution of Fourier coefficients
• Relative to uniform-strain assumption!
Edges
QUESTION 2: why?
Contacts;tangential
Contacts;normal
Uniform strain
Imposed deformation
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Dual behaviour
• Stress– particles contacts
• Strain/deformation– voids– contacts tangential
• No simple localisation assumption!
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Tessellation in 3D
• Contact-based: polyhedral cells
• QUESTION 3: algorithm?
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Summary
• Granular materials– Micromechanics
• Tessellations description of deformation
• Bagi’s expression reproduces macroscopic strain
• Isotropy in edge orientations• Anisotropy in contact orientations• Uniform strain for edges• Non-uniform strain for contacts
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Co-workers
• 2D tessellations– L. Rothenburg
Department of Civil EngineeringUniversity of WaterlooCanada
• 3D tessellations– O. Durán & S. Luding
Department of Mechanical EngineeringUniversity of TwenteNetherlands
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Questions
• To audience– Q1: fast contact-based tessellation in 2D?– Q2: why uniform strain for edges?– Q3: contact-based tessellation in 3D?
• To presenter