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Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl Shear banding in granular materials Stefan Luding PART/DCT, TUDelft, NL Thanks to: M. Lätzel, M.-K. Müller (Stuttgart), R. P. Behringer, J. Geng, D. Howell (Duke), M. van Hecke, D. Fenistein (Leiden)

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Introduction Results DEM Micro-Macro (global) Micro-Macro (local) Outlook - Anisotropy - Rotations Single particle Contacts Many particle simulation Continuum Theory Contents

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Single particle Contacts Many particle simulation Continuum Theory Contents E x p e r i m e n t s Introduction Results DEM Micro-Macro (global) Micro-Macro (local) Outlook - Anisotropy - Rotations

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Material behavior of granular media Non-Newtonian Flow behavior under slow shear inhomogeneityrotations Yield-surface Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl

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Material behavior of granular media 3D 3-dimensional modeling of shear and sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl P-wave shape and speed Universal shear zones

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Sound propagation in granular media 3-dimensional modeling of sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl P-wave shape and speed

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Applications Granular media (powder, sand, …) Colloidal systems (+fluid) Atomistic systems in confined geometries Many particle systems (traffic, pedestrians) Industrial scale applications ?

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Why ? `Many particle simulations work for small systems only (10 4 - 10 6 ) Industrial scale applications rely on FEM FEM relies on continuum mechanics + constitutive relations `Micro-macro can provide those ! Homogenization/Averaging

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Introduction Results DEM Contacts Many particle simulation Contents

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Equations of motion Overlap Forces and torques: Normal Contacts Many particle simulation Discrete particle model

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Contact force measurement (PIA)

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Hysteresis (plastic deformation)

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- (too) simple - piecewise linear - easy to implement Contact model

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- (really too) simple - linear - very easy to implement Linear Contact model

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- simple - non-linear - easy to implement Hertz Contact model

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Sound Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl

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Sound 3-dimensional modeling of sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl P-wave shape and speed

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potential energyrotationsdisplacements Micro simulation of shear bands

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Strain Control (top): Initial position and final position z 0 and z f Stress-control (right): With wall mass: m x, friction x and (i)Bulk material force F x (t) (ii)External force –p x z(t), and (iii)Frictional force x x(t) Model system bi-axial box

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zz =9.1% zz =4.2% zz =1.1% zz =0.0% Simulation results

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Bi-axial box

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Introduction Results DEM Micro-macro (global) Contacts Many particle simulation Continuum Theory Contents

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Bi-axial compression with p x =const.

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geometrical friction angle kc/k20½124kc/k20½124 p x 100 zz 183 234 264 310 336 p x 500 zz 798 853 915 941 972 c 11 32 40 60 71 f min macro cohesion Cohesion – no friction

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Sliding contact points: - Static Coulomb friction - Dynamic Coulomb friction Tangential contact model

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- Static friction - Dynamic friction project into tangential plane compute test force sticking: sliding: Tangential contact model - spring - dashpot before contact static dynamic static

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k c = 0 and µ = 0.5 Internal friction angle Total friction angle Friction – no cohesion

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Bi-axial box rotations

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Direction, amplitude, anti-symmetric (!) stress Rotations (local)

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Rolling (mimic roughness or steady contact necks) - Static rolling resistance - Dynamic resistance Tangential contact model

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Silo Flow with friction +rolling friction

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Silo Flow with friction

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Tangential contact model Torsion (large contact area) - Static torsion resistance - Dynamic resistance

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DEM simulation macroscopic material behaviour Micro: Disks (in 2-D) Particle size a = [0.5 : 1.5] mm Stiffness k 2 = 10 5 Nm -1 and k 1 =k 2 /2 Cohesion k c = 0, ½, 1, 2, 4 k 2 Friction µ = 0 Macro: Youngs Modulus E k 1 Poisson ratio 0.66 Friction angle 13° (µ=0) and 31° (µ=0.5) Dilatancy angle 5° (p=500) and 11° (p=100) Cohesion: + Bi-axial box setup Summary: micro-macro (global)

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An-isotropy

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Stiffness tensor vertical horizontal shear anisotropy

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An-isotropy Structure changes with deformation Different stiffness: More stiff against shear Less stiff perpendicular Evolution with time:

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Stiffness tensor vertical horizontal shear shear1 shear2 shear3

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From virtual work … For each single contact … Þ Stress tensor Þ Stiffness matrix C (elastic) - Normal contacts - Tangential springs Deformations (2D): - Isotropic compression V - Deviatoric strain (=shear) D, Þ Stress changes - Isotropic V - Deviatoric D, Local micro-macro transition

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Constitutive model with rotations

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+ successful tool – few parameters - microscopic foundations ? - extensions & parameter identification Continuum Theory deformation - rotations cyclic deformations - creep Hypoplastic FEM model

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density vs. pressure & friction vs. density Micro-macro transition

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Granular media are: - compressible - inhomogeneous (force-chains) - (almost always) an-isotropic and - micro-polar (rotations) Summary – part 1: global view

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Introduction Results Micro-macro (global) Micro-macro (local) Single particle Contacts Many particle simulation Continuum Theory Contents

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Global average vs. Local average Global + Experimentally accessible data - Wall effects - Averaging over inhomogeneities Local - Difficult to compare to experiment + Averages away from the walls + Average over `similar volume elements

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potential energyrotationsdisplacements Micro informations: shear bands

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Direction, amplitude, anti-symmetric (!) stress Rotations (local)

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Ring geometry (Behringer et al.)

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Ring geometry

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2D shear cell – energy

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2D shear cell – force chains

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2D shear cell – shear band

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Ring geometry

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Averaging Formalism Any quantity: In averaging volume:

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Averaging Formalism Any quantity: - Scalar - Vector - Tensor

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Averaging Density Any quantity: - Scalar: Density/volume fraction

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Density profile Global volume fraction

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Any quantity: - Scalar - Vector – velocity density Averaging Velocity

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Velocity field exponential

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Velocity gradient exponential:

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Velocity gradient exponential:

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Velocity distribution

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Averaging Fabric Any quantity: - Scalar: contacts - Vector: normal - Contact distribution

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Fabric tensor contact probability … centerwall

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Fabric tensor (trace) contact number density

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Macro (contact density)

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Fabric tensor (deviator) an-isotropy (!)

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Averaging Stress Any quantity: - Scalar - Vector - Tensor: Stress

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Stress tensor (static) shear stress

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Stress tensor (dynamic) exponential

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Stress equilibrium (1) acceleration:

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Stress equilibrium (2) ? ?

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Averaging Deformations Deformation: - Scalar - Vector - Tensor: Deformation

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Macro (bulk modulus)

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Macro (shear modulus)

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Constitutive model – no rotations

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Averaging Rotations Deformation: - Scalar - Vector: Spin density - Tensor

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Rotations – spin density eigen-rotation:

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Spin distribution

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Velocity-spin distribution high dens.low dens. sim. exp.

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Macro (torque stiffness)

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Deformations (2D): - Isotropic V - Deviatoric (=shear) D, - Mean rotation (=slip) * - Difference rot. (=rolling) ** Þ Stress changes ??? - Isotropic V - Deviatoric D, - Asymmetric A An-isotropy and rotations

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Constitutive model with rotations

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Granular media are: - compressible - inhomogeneous (force-chains) - (almost always) an-isotropic - micro-polar (rotations) Summary Granular media are … … interesting

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Open issues Accounting for inhomogeneities (temperature) Structure evolution with deformation (time) Micropolar continuum theory …

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The End

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Contact force measurement (PIA)

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Hysteresis (plastic deformation)

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- (too) simple - piecewise linear - easy to implement Contact model

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- (really too) simple - linear - very easy to implement Linear Contact model

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- simple - non-linear - easy to implement Hertz Contact model

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Sound Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl

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Sound 3-dimensional modeling of sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl P-wave shape and speed

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Sliding contact points: - Static Coulomb friction - Dynamic Coulomb friction Tangential contact model

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- Static friction - Dynamic friction project into tangential plane compute test force sticking: sliding: Tangential contact model - spring - dashpot before contact static dynamic static

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Bi-axial box rotations

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Direction, amplitude, anti-symmetric (!) stress Rotations (local)

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Rolling (mimic roughness or steady contact necks) - Static rolling resistance - Dynamic resistance Tangential contact model

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Silo Flow with friction +rolling friction

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Silo Flow with friction

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Tangential contact model Torsion (large contact area) - Static torsion resistance - Dynamic resistance

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+ successful tool – few parameters - microscopic foundations ? - extensions & parameter identification Continuum Theory deformation - rotations cyclic deformations - creep Hypoplastic FEM model

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density vs. pressure & friction vs. density Micro-macro transition

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From virtual work … For each single contact … Þ Stress tensor Þ Stiffness matrix C (elastic) - Normal contacts - Tangential springs Deformations (2D): - Isotropic compression V - Deviatoric strain (=shear) D, Þ Stress changes - Isotropic V - Deviatoric D, Local micro-macro transition

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