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

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

3 Location Enschede Enschede Delft Eindhoven Leiden

4 Split personality Science: granular materials Engineering: turbomachines

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

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

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

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

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

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

11 Continuum mechanics Stress tensor Strain tensor

12 Continuum mechanics Stress tensor Strain tensor

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

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

15 Micromechanics Relations: discrete  continuum DiscreteContinuum Homogenisation

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

17 Particle interaction Elasticity Friction Damping Interaction at contacts!

18 Mixing in rotating cylinder

19 From discrete information  stress and strain

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

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

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

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

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

25 Compatibility equations

26 Averaging of compatibility equations (1)

27 Averaging of compatibility equations (2) nini B titi

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

29 Expressions for stress and strain

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

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

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

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

34 Triaxial test Imposed deformation in X-direction Constant lateral stresses 11 11 00 00

35 Triaxial test (2D version)

36 Response Dilation Compression Imposed deformation Shear strength Volume change

37 Orientational averaging Average over edges with same orientation!

38 Edge distribution function EDGESCONTACTS Induced geometrical anisotropy  shear strength

39 Average relative displacements Normal component Fourier coefficients

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

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

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

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

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

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