Ken Kamrin, David HenannMIT, Department of Mechanical Engineering

Georg KovalNational Institute of Applied Sciences (INSA), Strasbourg

Constructing and verifying a three-dimensional nonlocal

granular rheology

Samadani & Kudrolli (2002) Kamrin, Rycroft, & Bazant (2007)

Stage 1: Local modeling.

(Experiment) (Discrete Element Method [DEM])

Static zones and flowing zones.

Inertial Flow Rheology: Simple Shear

• Inertial number:

• Inertial rheology:[Da Cruz et al 2005, Jop et al 2006]

d = mean particle diameters = the density of a grain

packing

fraction

(Show Bagnold connection)

Dense Granular Continuum?

Rycroft, Kamrin, and Bazant (JMPS 2009)

Some evidence for continuum treatment:• Space-averaged fields vary smoothly when coarse-grained at width 5d• Certain deterministic relationships between fields appear in elements 5d wide

(Blue crosshairs= Cauchy stress evecs)

(Orange crosshairs= Strain-rate evecs)

Developed packing fraction (5d scale)

Rycroft, Kamrin, and Bazant (JMPS 2009)

Tall silo

Wide siloTrap-door

Constitutive ProcessConservation of linear and angular momentum require

B Bt

Reference Deformed

Relaxed

Local region(3D version,

Kroner-Lee decomposition)

To close the system:

(1D picture)Dashpot: Inertial rheology [Jop et. al, Nature, 2006]

Spring: Jiang-Liu granular elasticity law [Jiang and Liu, PRL, 2003]

To close, must choose isotropic scalar functions and dp with dp ¸ 0.

“Mandel stress”

Mathematical Form

Definitions:

Enforce: Frame indifference 2nd law of thermodynamics Coaxiality of flow and stress*

Explicit AlgorithmCreate a user material subroutine

Given input: F(t), Fp(t), T(t), and F(t+t)

Compute output: Fp(t+t) and T(t+t)

.

1. Convert T(t) to M(t) using Fe(t)=F(t)Fp(t)-1.

1. Compute Dp(t) from M(t) using the flow rule.

2. Use Dp = Fp Fp-1 to solve for Fp(t+t).

1. Compute Fe(t+t) = F(t+t) Fp(t+t)-1.

1. Polar decompose Fe(t+t) and obtain Ee(t+t) = log Ue(t+t).

1. Compute M(t+t) from Ee(t+t) using the elasticity law.

1. Convert M(t+t) to T(t+t) using Fe(t+t).

Granular Elasticity Law

(Y. Jiang and M. Liu, Phys. Rev.

Lett., 2003)

B is a stiffness modulus and constrains the shear and

compressive stress ratio.

FHertz / 3/2

Under

compression:

Results

• Simulations performed using ABAQUS/Explicit finite element package.

• Constitutive model applied as a user material subroutine (VUMAT).

• All parameters in the model (6 in total) are from the papers of Jop et. al. and Jiang and Liu, except for d:

K. Kamrin, Int. J Plasticity, (2010)

Velocity vector field:

Rough Inclined Chute

Bingham fluid (no elasticity) P. Jop et. al.

Nature (2006)

Elasto-plastic model

Experimental agreement:

x

y

Wide Flat-Bottom Slab Silo

Recall expt:

Cauchy stress field

Eigendirections of stress:

DEM simulation Rycroft, Kamrin, and Bazant (JMPS 2009)

Kamrin, Int. J Plasticity, (2010)

Theory

Flow prediction not “spreading” enough

DEM simulation Rycroft, Kamrin, and Bazant (JMPS 2009)

Log of dp: Kamrin, Int. J Plasticity, (2010)

Theory

Continuum model

Local model predictions are:Qualitatively ok:- Shear band at inner wall- Flow roughly invariant in the vertical directionQuantitatively flawed:- No sharp flow/no-flow interface.- Wall speed slows to zero, local law says shear-band width goes to zero.

Velocity field: x

z

G.D.R. Midi, Euro. Phys. Journ. E, (2004)

(Wid

th o

f sh

ear

band

)/d

vwall

DEM data (Koval

et al PRE 2009)

Continuum

model

Flow prediction not “spreading” enough

Past nonlocal granular flow approaches:

- Self-activated process (Pouliquen & Forterre 2009)- Cosserat continuum (Mohan et.al 2002)- Ginzberg-Landau order parameter (Aronson & Tsimring 2001)

Stage 2: Fixing the problem.Accounting for size-effects

R/d

• Inertial number:

• Local granular rheology:

Fix Vwall , Vary R/d

Vwall

Fix R/d, Vary Vwall

R

2RPout

Vwall

I(x,y), (x,y)

Koval et al. (PRE 2009)

Fill with grains of diameter = d.

d and Pout held fixed

[Da Cruz et al 2005, Jop et al 2006]

d = particle diameters = the density of a grain

R/d

Deviating from a local flow law

Nonlocal add-on:

Nonlocal Fluidity ModelExisting theory for emulsions Extend to granular mediaGoyon et al., Nature (2007); Bocquet et al., PRL (2009)

Nonlocal law:

Define the “granular fluidity” :

Local granular flow law:

Local flow law (Herschel-Bulkley):

Define the “fluidity” (inverse viscosity):

Where bulk contribution vanishes, we can still flow even though we’re under the “yield criterion”.

Micro-level length-scale proportional to

Physical Picture

Bocquet et al. (PRL 2009)

1) A local plastic rearrangement in box j causes an elastic redistribution of stress in material nearby.

2) The stress field perturbation from the redistribution can induce material in box i nearby to undergo a plastic rearrangement.

Kinetic Elasto-Plastic (KEP) model

Basic idea: Flow induces flow. Plastic dynamics are spatially cooperative.

Granular Fluidity = g = “Relative susceptibility to flow”

Cooperativity LengthTheoretical form for emulsions is

[Bocquet et al PRL 2009]

Switching

A=0.68

Direct tests: From steady-flow DEM data in the 3 geometries

(annular shear, vertical chute, shear w/ gravity):

Only new material constant is the nonlocal amplitude A.

Local law constants all carry over.

For our 2D DEM disks, we find:

Extract » from DEM using:

M. van Hecke, Cond. Mat. (2009)

Possible meaning of diverging length-scale

c

c+

Lois and Carlson, EPL, (2007)

Staron et al, PRL (2002)

Pre-avalanche zone sizes for inclined plane flow:

Validating the nonlocal modelCheck predictions against DEM data from Koval

et al. (PRE 2009) in the annular couette cell.

R

2RPout

Vwall

I(x,y), (x,y)

DEM(symbols)

NonlocalRheology(lines)

LocalRheology(- - -)

Fix Vwall , Vary R/d

R/d

Vary Vwall , Fix R/d

Vwall

R/d

Kamrin & Koval PRL (2012)

Wall slip? Perhaps part of joint BC.

Same model, new geometry: Vertical Chute

G

Pwall

DEM(symbols)

Theory(lines)

Velocity fields for 3 different gravity values:

Kamrin & Koval

PRL (2012)

G

Same model, new geometry: Shear with gravity

Pwall Vwall

G H

y

DEM(symbols)

Theory(lines)

GVelocity fields for 3 different gravity values:

Kamrin & Koval

PRL (2012)

Nonlocal rheology in 3D

2 local material parameters: 2 grain parameters: 1 new nonlocal amplitude:

Local rheology (standard inertial flow relation): Cooperativity length:

++

Jop et al., JFM (2005)

For glass beads:

Our calibration for glass beads: A=0.48

Solved simultaneously alongside Newton’s laws:

No existing continuum model has been able to rationalize flow fields in this

geometry.

Flow in the split-bottom Couette cell

Schematic:

grains

Inner section(fixed) Outer section

rotatessplit

van Hecke (2003, 2004, 2006)

Filled with grains:

H

Flow in the split-bottom Couette cell – surface

flowThe normalized revolution-rate:

van Hecke (2003, 2004, 2006)

Viewed from the top down:

Rescaled:

All shallow flow data

collapses onto a

near-perfect error

function!

Exp data from: Fenistein et al., Nature (2003)

Flow fields on the top surface for five values of H:

Normalized radial coordinate

Flow in the split-bottom Couette cell – surface flow

H=10mmH=30mm

Nonlocal model in the split-bottom Couette cell• Custom-wrote an FEM subroutine to simulate the nonlocal model via ABAQUS User-Element (UEL).• Performed many sims varying H and d.

Flow field on top surfaceExp data from: Fenistein et al., Nature (2003)

Simulated flow field in the plane:

Define: Rc = Location of shear-band centerw = Width of shear-band = (r-Rc)/w = Normalized position

Nonlocal model in the split-bottom Couette cell

Surface flow results for all 22 combinations of H and d tried:

Collapse to error-function flow field Non-diffusive scaling of w with H

The model describes sub-surface flow as well as surface flow.

Nonlocal model in the split-bottom Couette cell - beneath the surface

Exp data from: Fenistein et al., PRL (2004)

Sub-surface shear-band center Sub-surface shear-band width

Nonlocal model in the split-bottom Couette cell

- Torque

32

Norm

aliz

ed

torq

ue

Remove the inner wall and let H increase

The nonlocal model correctly captures the transition from

shallow to deep behavior.

33

Rotation of the top-center point

Exp data from: Fenistein et al., PRL (2006)

Existing 3D wall-shear data using glass beads

Exp data: Losert et al., PRL (2000) Exp data: Siavoshi et al., PRE (2006)

Annular shear flow: Linear shear flow with gravity:

Same exact model and parameters used in split-bottom cases also captures these

flows.

Annular shear Shear with gravity

Split-bottom cell

Suppose we try different power laws in the cooperativity length formula:

More on the cooperativity length

• Synthesized a local elasto-viscoplastic model by uniting the inertial flow rheology with a granular elastic response, and observed certain pros and cons.

• Have proposed a nonlocal extension to the inertial flow law, by extending the theory of nonlocal fluidity to dry granular materials. Introduces only one new experimentally fit material constant.

• Nonlocal model is demonstrated to reproduce the nontrivial size effect and rate-independence seen in slow flowing granular media. Vastly improves flow field prediction.

• Have extended the model to 3D and demonstrated a strong agreement with experiments in many different flow geometries, including the split-bottom cell.

Summary

2D prototype model: K. Kamrin and G. Koval, PRL (2012)3D constitutive relation: D. Henann and K. Kamrin, PNAS (2013)

Relevant papers:

Important to do’s on dry flow modeling

Need more quantitative insights on fluidity BC’s (though much clarified by virtual power argument). BC’s likely responsible for observed nonlocal “thin-body” effects:

Nonlocal model w/ g=0 lower BC

Experiments (Silbert 2001)

“Smaller is stronger” size effect

Important to do’s on dry flow modeling

Focus on silo flow. Can we use our model to predict Beverloo constants that govern silo flow outflow rates? A famous size-effect problem.

Orifice size

Beverloo constants

Particle sizeOutflow rate

Beverloo Correlation for drainage flow:

“Smaller is stronger” size effect

D

Q

D=kd Q

D

Important to do’s on dry flow modeling

Merge a transient model into the steady flow response:

Easiest route:Let this term have transients per a critical-state-like theory

Propose to try: (Anand-Gu form + Rate sensitivity)

Let steady behavior approach inertial flow law

[Rothenburg and Kruut IJSS (2004)]

Important to do’s on dry flow modeling

Non-codirectionality of stress and strain-rate tensors

Recall our usage of codirectionality (i.e. stress-deviator proportional to strain-rate tensor)

However, data shows codirectionality is inexact in shear flows for two reasons:

Slight non-coaxiality:

(Blue) Stress eigen-directions

(Black) Strain-rate eigen-directions

Normal stress difference N2 (for 3D):

(Shear)

Depken et al., EPL (2007)

Important to do’s on dry flow modeling

Material Point Method (MPM): Meshless method for continuum simulation. Useful for very large deformation plasticity problems.

“Phantom” background mesh and a set of material point markers

Great promise for full flow simulation[Z. Wieckowski. (2004)]

Our own preliminary MPM implementation: