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1 Tessellations and granular materials Niels P. Kruyt Department of Mechanical Engineering University of Twente

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2 Overview University of Twente Split personality Granular materials –Micromechanics Tessellations

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3 Location Enschede Enschede Delft Eindhoven Leiden

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4 Split personality Science: granular materials Engineering: turbomachines

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5 Turbomachines CFD methods Optimisation methods Inverse-design methods PIV measurements

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6 What are granular materials? Grains –natural –biological –man-made

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7 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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8 Features elasticity frictional plasticity dilatancy anisotropy viscous multi-phase cohesion segregation

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9 Fluid-like behaviour Fluidised beds Collisions Kinetic theory Inelasticity Clustering Deen, Department of Chemical Engineering, University of Twente

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10 Solid-like behaviour Frictional Pressure-dependent Elasticity Plasticity Dilatancy

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11 Continuum mechanics Stress tensor Strain tensor

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12 Continuum mechanics Stress tensor Strain tensor

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13 Constitutive relations Description of material behaviour Relation between stress and strain (rate) Elastic Plastic Viscous

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14 Categories of constitutive relations Continuum theories –phenomenological; elasto-plasticity Micromechanical theories –relation with microstructure and particle properties

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15 Micromechanics Relations: discrete continuum DiscreteContinuum Homogenisation

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16 Tool: Discrete Element Method Particle interaction Newton’s laws Patience Simple model at micro-level Complex behaviour at macro-level

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17 Particle interaction Elasticity Friction Damping Interaction at contacts!

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18 Mixing in rotating cylinder

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19 From discrete information stress and strain

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20 Micromechanical constitutive relations Relative displace- ment Force Stress Strain Constitutive relation Microscopic level (contact) Macroscopic level (continuum) Averaging Localisation TESSELLATIONS

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21 Objective Expression for strain tensor in terms of relative displacement at contacts + + q p

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22 Average strain tensor Definition of strain Average strain Average strain is determined by displacements at boundary!

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23 Approach Strain expression: –averaging of compatibility equations –displacement of line segment Tessellation: network of contacts Compatibility equations Averaging

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24 Tessellation: network of contacts QUESTION 1: Fast algorithm for determining tiles?

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25 Compatibility equations

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26 Averaging of compatibility equations (1)

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27 Averaging of compatibility equations (2) nini B titi

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28 Summary for strain Formulation in relative displacements Tessellation of network of contacts Averaging of compatibility equations

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29 Expressions for stress and strain

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30 Micromechanically-based constitutive relations Relative displace- ment Force Stress Strain Constitutive relation Microscopic level (contact) Macroscopic level (continuum) Averaging Localisation

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31 Tessellation (3D) Delaunay tessellation Edges –physical contacts –virtual contacts

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32 Bagi’s strain expression Complex geometrical quantity; complementary area vector Set of edges

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33 Use of Bagi’s expression Correctness Investigation deformation DEM simulation of triaxial test

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34 Triaxial test Imposed deformation in X-direction Constant lateral stresses 11 11 00 00

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35 Triaxial test (2D version)

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36 Response Dilation Compression Imposed deformation Shear strength Volume change

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37 Orientational averaging Average over edges with same orientation!

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38 Edge distribution function EDGESCONTACTS Induced geometrical anisotropy shear strength

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39 Average relative displacements Normal component Fourier coefficients

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40 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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41 Dual behaviour Stress –particles contacts Strain/deformation –voids –contacts tangential No simple localisation assumption!

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42 Tessellation in 3D Contact-based: polyhedral cells QUESTION 3: algorithm?

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43 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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44 Co-workers 2D tessellations –L. Rothenburg Department of Civil Engineering University of Waterloo Canada 3D tessellations –O. Durán & S. Luding Department of Mechanical Engineering University of Twente Netherlands

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45 Questions To audience –Q1: fast contact-based tessellation in 2D? –Q2: why uniform strain for edges? –Q3: contact-based tessellation in 3D? To presenter

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