ken kamrin, david henann mit, department of mechanical engineering georg koval
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Constructing and verifying a three-dimensional nonlocal granular rheology. Ken Kamrin, David Henann MIT, Department of Mechanical Engineering Georg Koval National Institute of Applied Sciences (INSA), Strasbourg. Stage 1: Local modeling. Static zones and flowing zones. - PowerPoint PPT PresentationTRANSCRIPT
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: