1 Tessellations and granular materials Niels P. Kruyt Department of Mechanical Engineering University of Twente [email protected]

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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

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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

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Expressions for stress and strain

Cc

ci

cj

A j

iij uh

AdA

x

u

A

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Cc

cj

ci

A

ijij flA

dAA

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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