Dense granular rheology: from stationary flow to unstable sliding

1
Dense granular rheologyfrom stationary flow to
unstable sliding

• Takahiro Hatano (Universite de Tokyo)

1. numerical simulation rheology at the jamming
density dynamical heterogeneity 2. experiment
using rheometer negative shear-rate dependence
correlation length
2
The essence of granular materials

• The interaction between grains is dissipative.
• No thermal motion (T 0)
• ? A liquid state is a driven state
  nonequilibrium

rheology?
3
predicting granular flow?
important to industries and solid earth science.
before
after
collapse of Mt. St. Helens (1980)
Huge mass flow -gt granular rheology
4
granular rheology in the jamming diagram
T (temperature)
L. Berthier, a talk in UCGP
S (shear stress)
glass transitions
jammed
granular media (T0)
rheology
point J
1/ (density)
Olsson Teitel 2007 PRL TH 2008 JPSJ
Approaching point J, the length and time scales
seem to diverge. the nature of divergence and
what mechanism?
5
our model
an assembly of frictionless inelastic particles
(without attractive force)
1. point J is unique. 2. rheology still not clear.
normal vec.
Force only normal direction
diameter
i
j
elastic damping
relative velocity
repulsive force
mass
overlap length
linear
6
a computational system
dynamics SLLOD eq.
diameter
i
ensuring uniform shear flow
j
mass
Lees-Edwards Bound. Cond.
bidisperse
units
d 1 (length) m1 (mass) m /? 1 (time)
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rheology at various densities
(TH, Otsuki, Sasa, 2006)
(shear stress)
1. At lower densities Bagnolds scaling
Yield stress
2. At higher densities Yield stress
Higher density
3. At the threshold density?
(shear rate)
is suggested
8
power-law rheology at the jamming density
N80,000
(TH 2008 arXiv)
(shear rate)
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dynamical heterogeneity in sheared granular matter
1. four-point correlation function
2. dynamical susceptibility
decreasing function of time!
DH can be detected in instantaneous vel. field.
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dynamical heterogeneity in sheared granular matter
3. two-point correlation function
DH develops at lower shear rates
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dynamical heterogeneity in sheared granular matter
(deformed by 50 )
large velocity
small velocity
intermediate
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system-size dependence of rheology
(TH 2008)
strong system-size dependence at lower shear
rates. (smaller system ? higher stress)
hardening due to finite-size effect
N50
correlation length comparable to system size (?)
N80,000
13
finite-size scaling
data collapse
correlation length
consistent with the exponent via two-point
correlation function
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large stress fluctuations in small systems
shear stress
0
time
if (correlation length of DH) gt (system
size), cooperative rearrangement does not work
? brittle-like behavior
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Gutenberg-Richters law in granular matter
events
elastic energy
energy drop distribution
sudden energy drop
(microfracture, acoustic emission)
distribution of
3 orders of magnitude
b1.4(1) for 2D and 3D
cf. b 1.0(2) for earthquakes
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fluctuations of shear stress
0.01
NOTE no finite-size effect
Poissonian
Gaussian fluctuation
0
20
40
20
40
strain
strain
(system size the same)
growth of dynamical heterogeneity ? large stress
fluctuation
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fluctuations and dynamical heterogeneity
dynamical heterogeneity unit of stress
relaxation?
must be proportional to
n of independent units
2D
3D
However,
for 2D 3D
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fluctuations and dynamical heterogeneity
shear stress
deformed by 5
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change in DH ?? shear stress fluctuation?
rapidly changing
unchanged
slow change
. This remains open.
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partial summary (I)
1. Power-law rheology at the jamming density. 2.
Growing correlation length 3. Dynamical
susceptibility a decreasing function of time ?
DH in instantaneous velocity field. 4. Hardening
due to finite-size effect (correlation length) gt
(system size) ? Gutenberg-Richters law
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part II Negative shear rate dependence in
slowly-sheared granular matter
collaborator Osamu Kuwano
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a constitutive law for (fast) granular flow
(P. Jop et al. JFM2005) (da Cruz et al. PRE 2005)
I 0.3
m the mass, d the diameter, P the normal
pressure
Succeed in describing fast flow in experiments
NOTE Derivation from a kinetic theory is yet
difficult. (e.g. Mitarai Nakanishi 2005 PRL)
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creeplike slow flow?
exponential velocity profile
(T. S. Komatsu et al. PRL 2001)
Constitutive laws using the inertial number
fail to describe the exponential profile.
This profile means negative shear-rate dependence!
(of shear stress)
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Negative shear-rate dependence
stationary force balance
y
(sum of traction) (body force)
z
if density ? is constant,
(1)
if a constitutive law is creep-like
a constitutive law
(2)
using (1),
(1) (2) ?
C must be negative!
(3)
gives velocity profile
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Direct observation of negative shear-rate
dependence

• One needs
• to measure shear stress at very slow shear
  rates.
• to monitor velocity profile
• (because of potential shear-banding).

Sample holder
A commercial rheometer (AR 2000ex, TA
instruments)
layer thickness 1mm 3mm
25mm
quartz glass (see-through)
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observation of shear flow in a rheometer
1 mm/sec
velocity profile
glass beads (monodisperse) diameter 0.2 mm
no shear banding
(depth)
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A preliminary result
flow-induced ordering (volume also changes)
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mechanism of negative rate-dependence
SY Yield stress
A gt 0
yield stress is time-dependent
t duration of contact time (before sliding)
Healing effect
V
stationary sliding
true contact area
D
?
D width of true contact area
?
can be negative
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partial summary (part II)
1. Granular rheology shows negative
rate-dependence at lower shear rates. 2. This
explains exponential velocity profile in
heap. 3. Healing effect is essential. (cannot
be obtained using MD simulations).
cf. simulation down to
(TH, 2007 PRE)
positive dependence everywhere
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summary
1. Power-law rheology at the jamming density. 2.
Growing correlation length 3. Hardening due to
finite-size effect (correlation length of DH) gt
(system size) ? Gutenberg-Richters law 4.
Negative shear-rate dependence in slow
shear healing effect is essential