Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, 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)

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

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

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,

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

Sound propagation in granular media 3-dimensional modeling of sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, P-wave shape and speed

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

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

Introduction Results DEM Contacts Many particle simulation Contents

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

Contact force measurement (PIA)

Hysteresis (plastic deformation)

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

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

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

Sound Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding,

Sound 3-dimensional modeling of sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, P-wave shape and speed

potential energyrotationsdisplacements Micro simulation of shear bands

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

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

Bi-axial box

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

Bi-axial compression with p x =const.

geometrical friction angle kc/k20½124kc/k20½124 p x 100 zz p x 500 zz c f min macro cohesion Cohesion – no friction

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

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

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

Bi-axial box rotations

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

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

Silo Flow with friction +rolling friction

Silo Flow with friction

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

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)

An-isotropy

Stiffness tensor vertical horizontal shear anisotropy

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

Stiffness tensor vertical horizontal shear shear1 shear2 shear3

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

Constitutive model with rotations

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

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

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

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

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

potential energyrotationsdisplacements Micro informations: shear bands

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

Ring geometry (Behringer et al.)

Ring geometry

2D shear cell – energy

2D shear cell – force chains

2D shear cell – shear band

Ring geometry

Averaging Formalism Any quantity: In averaging volume:

Averaging Formalism Any quantity: - Scalar - Vector - Tensor

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

Density profile Global volume fraction

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

Velocity field exponential

Velocity gradient exponential:

Velocity gradient exponential:

Velocity distribution

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

Fabric tensor contact probability … centerwall

Fabric tensor (trace) contact number density

Macro (contact density)

Fabric tensor (deviator) an-isotropy (!)

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

Stress tensor (static) shear stress

Stress tensor (dynamic) exponential

Stress equilibrium (1) acceleration:

Stress equilibrium (2) ? ?

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

Macro (bulk modulus)

Macro (shear modulus)

Constitutive model – no rotations

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

Rotations – spin density eigen-rotation:

Spin distribution

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

Macro (torque stiffness)

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

Constitutive model with rotations

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

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

The End

Contact force measurement (PIA)

Hysteresis (plastic deformation)

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

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

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

Sound Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding,

Sound 3-dimensional modeling of sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, P-wave shape and speed

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

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

Bi-axial box rotations

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

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

Silo Flow with friction +rolling friction

Silo Flow with friction

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

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

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

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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