Coarse-grained local and objective
continuum description of 3D
granular flows down an inclined
surface
T. Weinhart, R. Hartkamp,
A.R. Thornton, S. Luding
University of Twente
Dec-04, 2012
Outline
1
Granular chute flows
2
Observations about coarse-graining
3
Stress anisotropy
see also: Weinhart, Hartkamp, Thornton, Luding, sub. Phys. Fluids (2012)
g
z
x
DEM of steady uniform chute flow over fixed-particle base, periodic
in x- and y -direction, varying inclination θ and particle number N.
Part 1:
Observations about coarse-graining
Particle statistics by coarse-graining
Define macroscopic density, using a coarse-graining function W,
Xn
ρ(r) =
mi W(r − ri ).
i=1
Define velocity field V, such that mass balance is exactly satisfied,
Xn
V = p/ρ, where p =
mi vi W(r − ri ).
i=1
8
0
Density ρ of a static system of 5 fixed (×) and 5 free (·) particles
for a Gaussian coarse-graining function of width w = d/8.
Weinhart, Thornton, Luding, Bokhove, GranMat (2012) 14:289
Contact statistics by coarse-graining
Define stress tensor σ = σc + σk → satisfies momentum balance:
Xn
σk = −
mi vi0 vi0 W(r − ri ) with vi0 = vi − V(r),
i=1
Z1
X
c
σ =−
f ij rij W(r − (ri + srij )) ds
contacts {i,j}
0
Z1
X
−
f ik aik W(r − (ri + saik )) ds,
wall contacts {i,k}
0
with branch vector rij = ri − rj , center-contact vector aik = cik − ri .
3
0
Magnitude of (contact) stress |σ|2 for w = d/8.
What is the good coarse-graining scale?
0.55
0.55
ν
ν
0.5
0.5
0.45
0.4
sub-particle scale
particle scale
z = 1.35 d
z = 2.04 d
z = 2.56 d
z = 15.65 d
z = 27.53 d
w = 0.05 d
w = 1d
0.6
0.6
0.45
w = 0.05
w=1
0
5
10
15
z/d
20
25
0.4
30
10−4
10−2
w/d
100
Figure: Volume fraction ν(z) (averaged over x, y , t), varying w
I
two scales (sub-particle w < 0.1d, particle w ≈ 1d) exist
for which coarse-graining results are scale-independent.
I
The larger scale is strictly valid in the bulk (where gradients
are small on the particle scale).
Exception: Kinetic stress σkxx and granular temp. Tg
101
k
σxx
|w=0.05
0.5
0.8
1
k
σxx
|w=1
0.4
k′
σxx
|w=1
0.6
2
0.5
k
σxx
10
0
k∗
σxx
k⋆
σxx
|w=1
0.3
0.7
0.2
0.4
0.3
0.2
0.1
0.1
10−1
0
0
0
5
10
15
20
25
30
z
10−2
10−1
w/d
100
10−2
10−1
w/d
On the particle scale, the kinetic stress
XN
mvix0 vix0 Wi with vi0 = vi − V(r)
σkxx =
i=1
in flow direction is scale dependent (center figure). However,
XN
σk?
=
mvix? vix? Wi with v?i = vi − V(ri )
xx
i=1
is scale independent (right figure), but harder to calculate.
k
2 w2
k?
Instead, we use a correction term: σk0
xx = σxx − ργ̇ 3 ≈ σxx .
100
Oscillations on the sub-particle scale: Volume frac.
0.6
0.3
0.55
0.2
ν̃
ν
0.1
0.5
0
0.45
w = 0.05
w=1
-0.1
0.4
-0.2
0
5
10
15
z/d
I
I
20
25
30
0
2
6
4
8
10
z
Oscillations can be measured by comparing the two scales,
ν|w =0.05 − ν|w =1
ν̃ =
.
ν|w =1
ν̃ can be fitted to
ν̃fit = α cos(2π(z − zw )/L) exp(−(z − zw )/z0 ),
with period L = 0.907, first peak zw = 1.60, rel. amplitude at
first peak α = 0.106, and the 1/e-decay distance z0 /d = 2.58.
30
σxx
σyy
25
σzz
20
−σxz
−σzx
15
σ̃xx
σ̃yy
0.1
σ̃zz
−σ̃xz
−σ̃zx
0.05
σ̃αβ
σαβ
Oscillations on the sub-particle scale: Stress
0
10
-0.05
5
-0.1
0
0
5
10
15
z
20
25
0
2
4
6
8
10
z
I
Similar (weaker) oscillations exist in the non-vertical stresses,
and are in phase with the density oscillations.
I
for σxx : period L = 0.900, first peak zw = 1.63, amplitude at
first peak α = 0.080 and 1/e-decay distance z0 /d = 2.15
I
Note the normal stress anisotropy σxx ≥ σzz ≥ σyy .
Part 2:
Stress Anisotropy
Collinear, viscous stress-strain relationship
For sheared (simple, Newtonian) fluids, the deviatoric
(compressive) stress is collinear with deviatoric strain rate tensor,
∇V + ∇VT
2
with a scalar proportionality constant η, the viscosity.
Here, ∇V = ∂z Vx xz = ˙ D xz.
D
D
σD
N = −2ηS , S =
Collinear, viscous stress-strain relationship
For sheared (simple, Newtonian) fluids, the deviatoric
(compressive) stress is collinear with deviatoric strain rate tensor,
∇V + ∇VT
2
with a scalar proportionality constant η, the viscosity.
Here, ∇V = ∂z Vx xz = ˙ D xz. Thus




0 0 1
-1 0 0
D
D
˙
˙
 0 0 0  =
R(φ )  0 0 0  RT (φ ),
SD =
2
2
1 0 0
0 0 1
D
D
σD
N = −2ηS , S =
where the first eigenvector
(compressive eigenvalue) is
rotated clockwise by φ = 45◦
in the xz-plane.
z
z
ǫ3
ǫ2
φǫ
ǫ1
Vx
y
x
Non-Newtonian stress-strain relationship
1) Is the “viscosity” really a constant?
2) Are (the orientations of) stress and strain collinear?
3) Is the stress-tensor of the same “shape” as the strain-rate?
Non-Newtonian stress-strain relationship
1) Is the “viscosity” really a constant?
2) Are (the orientations of) stress and strain collinear?
3) Is the stress-tensor of the same “shape” as the strain-rate?
No, No, and No, ...
Stress-strain: 1) viscosity → yield-stress
For granular fluids, the shear stress has a constant, strain-rate
independent (yield-stress) contribution µ := σD /p = µ0 + µ1 (˙ D ).
“Viscosity” can be expressed as η = µ ˙pD , so that:
σD
= −2ηS D

1
= η ˙ D R(φ )  0
0

1
= µ p R(φ )  0
0
0
0
0
0
0
0

0
0  RT (φ )
-1

0
0  RT (φ )
-1
The macroscopic friction
p µ was found to be a function of the
inertial number I = γ̇d ρ/p;
... but which density ρ and pressure/stress should be used?
Stress-strain: 2) Non-collinearity
The deviatoric stress is rotated slightly against the shear rate
tensor,


λ1 0
0
σD = R(φσ )  0 λ2 0  RT (φσ ),
0
0 λ3
with φσ 6= φ (and eigenvalues (λ1 , λ2 , λ3 ) 6= µp(1, 0, −1)).
z
n3
n2 , ǫ2
φσ
ǫ3
φǫ
n1
ǫ1
y
x
Eigendirections of dev. stress,
ni , and dev. strain rate, i .
Stress-strain: 3) “Shape” of stress
2
15
1
λ1
λ2
λ3
10
0
5
λi
[◦ ]
-1
-2
αzy
-3
-5
0
-5
αxz − φǫ
αxy
-4
5
10
15
20
0
-10
25
0
5
10
I
I
20
15
z/d
25
z/d
Left: Clockwise rotation angle αab = − tan−1
first eigendirection of
σD
Right: Eigenvalues of
σD .
n1b
n1a
of the
in the zy -, xy -, and xy -plane.
Data for θ = 28◦ , N = 6000, w /d = 0.05.
Non-collinear, general stress-strain relationship
We propose to an objective description of the deviatoric stress:


1/c
0
0
 RT (φσ ),
Λ12 /c
0
σD = sD? p R(φσ )  0
0
0
(−1 − Λ12 )/c
with c =
q
1 + Λ12 + Λ212 , where the independent variables are
I
the orientation difference ∆φσ = φσ − φ ,
I
the shape factor Λ12 = λ2 /λ1 , and
I
the objective friction
p coefficient
1
∗
?
√
µ := sD = 6p (λ1 − λ2 )2 + (λ2 − λ3 )2 + (λ3 − λ1 )2 .
... and: how are pressure and confining stress σzz related?
-0.16
1
−4.07 I + 0.15
-0.18
-0.19
0
-0.2
Λ12
∆φσ [◦ ]
−0.10 I − 0.20
-0.17
0.5
-0.5
-0.21
-0.22
-1
-0.23
-0.24
-1.5
-0.25
-2
0
0.05
0.1
0.15
0.25
0.2
0.3
0.35
0.4
-0.26
0
0.45
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.25
0.3
0.35
0.4
0.45
I
I
1.1
0.6
1.05
σzz /p
s⋆D
0.55
0.5
0.45
1
0.4
0.35
1.05
0
0.05
0.1
0.15
0.25
0.2
I
0.3
0.35
0.4
0.45
0.95
0
0.05
0.1
0.15
0.2
I
Confirming the fit (good news?)
fit to objective stress model
fit to Pouliquen-Forterre model
0.6
0.6
0.4
λi /p
µ
0.55
0.5
0.2
0
0.45
-0.2
0.4
-0.4
0.35
0
0.05
0.1
0.15
0.2
0.25
I
0.3
0.35
0.4
0.45
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
I
Friction µ and eigenvalues λi , compared to the objective stress
model (blue) and Pouliquen-Forterre model (red line).
Data for θ = 28◦ , w = 1.
0.45
What about other tensors?
Stress (kinetic and contact), and fabric orientation
and anisotropy
∆φ
s⋆D
0.4
0.3
0.2
0.1
4
3
2
1
0
-1
-2
-3
0
σ
σc
-0.1
σk
F
-0.2
Λ12
0.6
0.5
-0.3
-0.4
-4
-5
0
5
10
15
z
20
25
-0.5
5
10
15
20
25
5
z
10
15
20
25
z
I
Magn. of deviatoric comp. sD∗ is larger in σc than in F, σk .
I
Orientation difference ∆φ is smaller in σc than in F, σk .
I
Shape factor Λ12 is larger in σk than in σc , F.
Data for θ = 28◦ , N = 6000, w = 1.
Conclusions
Coarse-graining
I two coarse graining length scales exist
(subparticle and particle scale)
I Exception: kinetic stress has a scale-dependent component
on the particle scale (evaluate velocity at particle-position)
I Oscillations in density and stress near the base (layers)
Stefan Luding, Puerto Varas, 04.12.2012
21/22
Conclusions
Coarse-graining
I two coarse graining length scales exist
(subparticle and particle scale)
I Exception: kinetic stress has a scale-dependent component
on the particle scale (evaluate velocity at particle-position)
I Oscillations in density and stress near the base (layers)
Non-collinear, objective stress-strain rate relation
I Orientation difference ∆φ and shape (Λ12 ) for stress and
strain rate relation; objective friction sD? defined.
I Objective friction and magnitude of orientation difference and
shape factor increase with inertial number (with finite friction
at I → 0), while σzz /p remains constant.
I Objective stress model yields same friction as the
Pouliquen-Forterre model.
Stefan Luding, Puerto Varas, 04.12.2012
21/22
Future Work
I
Reasons for orientation difference ∆φ and non-scalar relation
(Λ12 ) between stress and strain rate need to be understood.
I
Can the fabric help here?
Thank you.
Questions?
Stefan Luding, Puerto Varas, 04.12.2012
22/22