Tessellations and granular materials Niels P Kruyt Department

Tessellations and granular materials Niels P. Kruyt Department of Mechanical Engineering University of Twente n. p. kruyt@utwente. nl www. ts. ctw. utwente. nl/kruyt/ 1

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

Location Enschede Leiden Delft Eindhoven 3

Split personality Science: granular materials Engineering: turbomachines 4

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

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

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 7

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

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

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

Continuum mechanics • Stress tensor • Strain tensor 11

Continuum mechanics • Stress tensor • Strain tensor 12

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

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

Micromechanics • Relations: discrete « continuum Discrete Homogenisation Continuum 15

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

Particle interaction • Elasticity • Friction • Damping Interaction at contacts! 17

Mixing in rotating cylinder 18

From discrete information ® stress and strain 19

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

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

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

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

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

Compatibility equations 25

Averaging of compatibility equations (1) 26

Averaging of compatibility equations (2) ti ni B 27

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

Expressions for stress and strain 29

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

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

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

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

Triaxial test • Imposed deformation in X-direction • Constant lateral stresses s 0 e 1 s 0 34

Triaxial test (2 D version) 35

Shear strength Volume change Response Dilation Compression Imposed deformation 36

Orientational averaging Average over edges with same orientation! 37

Edge distribution function EDGES CONTACTS Induced geometrical anisotropy ® shear strength 38

Average relative displacements • Normal component Fourier coefficients 39

Evolution of Fourier coefficients • Relative to uniform-strain assumption! Contacts; tangential Uniform strain Edges QUESTION 2: why? Contacts; normal Imposed deformation 40

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

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

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 43

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

Questions • To audience – Q 1: fast contact-based tessellation in 2 D? – Q 2: why uniform strain for edges? – Q 3: contact-based tessellation in 3 D? • To presenter 45