ken kamrin, david henann mit, department of mechanical engineering georg koval national institute of...
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![Page 1: Ken Kamrin, David Henann MIT, Department of Mechanical Engineering Georg Koval National Institute of Applied Sciences (INSA), Strasbourg Constructing and](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ed95503460f94be7132/html5/thumbnails/1.jpg)
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
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Samadani & Kudrolli (2002) Kamrin, Rycroft, & Bazant (2007)
Stage 1: Local modeling.
(Experiment) (Discrete Element Method [DEM])
Static zones and flowing zones.
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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)
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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)
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Developed packing fraction (5d scale)
Rycroft, Kamrin, and Bazant (JMPS 2009)
Tall silo
Wide siloTrap-door
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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]
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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*
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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).
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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:
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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)
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Velocity vector field:
Rough Inclined Chute
Bingham fluid (no elasticity) P. Jop et. al.
Nature (2006)
Elasto-plastic model
Experimental agreement:
x
y
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Wide Flat-Bottom Slab Silo
Recall expt:
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Cauchy stress field
Eigendirections of stress:
DEM simulation Rycroft, Kamrin, and Bazant (JMPS 2009)
Kamrin, Int. J Plasticity, (2010)
Theory
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Flow prediction not “spreading” enough
DEM simulation Rycroft, Kamrin, and Bazant (JMPS 2009)
Log of dp: Kamrin, Int. J Plasticity, (2010)
Theory
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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
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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
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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
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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
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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”
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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:
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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:
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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.
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Same model, new geometry: Vertical Chute
G
Pwall
DEM(symbols)
Theory(lines)
Velocity fields for 3 different gravity values:
Kamrin & Koval
PRL (2012)
G
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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)
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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:
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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
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Flow in the split-bottom Couette cell – surface
flowThe normalized revolution-rate:
van Hecke (2003, 2004, 2006)
Viewed from the top down:
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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
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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:
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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
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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
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Nonlocal model in the split-bottom Couette cell
- Torque
32
Norm
aliz
ed
torq
ue
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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)
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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.
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Annular shear Shear with gravity
Split-bottom cell
Suppose we try different power laws in the cooperativity length formula:
More on the cooperativity length
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• 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:
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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
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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
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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)]
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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)
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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: