Collision Times and Stress in Gravity Driven Granular Flow
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Transcript Collision Times and Stress in Gravity Driven Granular Flow
Stress Propagation in a Granular Column
In Gravity Driven Granular Flow
John Drozd
Colin Denniston
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bottom sieve
particles at bottom go to top
reflecting left and right walls
periodic or reflecting front
and back walls
3d simulation
Snapshot of 2d simulation from paper:
“Dynamics and stress in gravity-driven
granular flow”
Phys. Rev. E. Vol. 59, No. 3, March 1999
Colin Denniston and Hao Li
Outline
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Granular Matter Definition
Why Study Granular Matter?
Granular Column and Dynamics
Profiles and Stresses From Simulation
Continuum Mechanics
Nonlinear Density
Biharmonic PDE Model
Perturbation Analysis
Numerical Approach
Granular Matter
• Granular matter definition
– Small discrete particles vs. continuum.
• Granular motion
– Energy input and dissipation.
• Granular matter interest
– Biology, engineering, geology,
material science, physics.
300 (free fall region)
250 (fluid region)
vz
dvz/dt
200 (glass region)
150
z
Hard Sphere Collision
Velocity Adjustment
r1
q
r2
r1' r1 1
1 1 r1 q
1
1 1 r q q
r ' r 2
2
2 2
r2 r1
q
r2 r1
velocitydependent
Stress Tensor Calculation
1
1
ij 1 r1 r2 qˆ
t collisions 2
qˆ xˆ 2
qˆ xˆ qˆ yˆ qˆ xˆ qˆ zˆ
2
qˆ yˆ qˆ zˆ
qˆ yˆ
qˆ yˆ qˆ xˆ
2
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
qˆ zˆ
q z q x q z q y
1
1 f c r1 r2 qˆ
2
qˆ xˆ 2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
z
q
x
q
y
q
x
q
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
qˆ yˆ qˆ zˆ
q y
q y q x
2
qˆ zˆ qˆ xˆ
ˆ
ˆ
ˆ
q z
q zˆ q yˆ
z
h
h'
0
X'
w
x
Experimental data from the book:
“Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials”
By Jacques Duran.
Stress Profiles
ij
1
1
ˆ q
ˆ ˆi q
ˆ ˆj
1 r1 r2 q
t collisions 2
1
ˆ q
ˆ ˆi q
ˆ ˆj
f c 1 r1 r2 q
2
z 100
0
3
2
2
4
6
xz
zz
1
0
8
1
10
2
12
3
0
100
200
z
z zz 0
300
400
5
10
15
20
25
30
x
constantslope 0.15554674 x x z
Continuum Mechanics
(Two dimensional x-z model)
i j i jk l u k l
1
u i j ju i i u j
2
i i j i j g j
u xx u zz 2 x z u xz 0
2
z
2
x
Continuum Mechanics
, zz , xz
T
xx
u xx , u zz , u xz
T
U
xxxx
zzxx
xzxx
xxzz 2xxxz
zzzz 2zzxz
xzzz 2xzxz
Continuum Mechanics
U B
B B ij
1
u xx u zz 2 x z u xz 0
2
z
B1j
2
z
2
x
j
B 2 j
2
x
j
2B 3 j x z j 0
Continuum Mechanics
Choose x and z along principal axes:
a c 0
1
c b 0
0 0 d 1 x z
Ez z
c c
Ex x
Ex z Ex
x E z E z
0
0
0
1 x z G
0
Continuum Mechanics
det
b xx c zz c xx a zz 2 2 x z xz 0
d
z zz x xz zz g
2
z
2
z
z xz x xx 0
2
x
2
x
PDE
4
z
t 4x 2r 2x 2z ij 2s 2x 2z g z zz 0
a Ex
t
b Ez
1
ab cc dc c
1 2 z x
2
r
E x
bd
2 G Ez Ex
1
ab cc dc
Ex 1
2
s
x
bd
G 2
Nonlinear Density and
Biharmonic PDE
For isotropic hard spheres t = r = s = 1:
4
z
4x 2 2x 2z ij 2 2x 2z g z zz 0
P zz
0
P
0
Pa
1 c exp
c 1 exp
1
P
Pl
ij ij ij ij 2 ...
0
1
Ex 1
x
s
G 2
2
u zz
Boundary Conditions
0 x w, 0 z h
z ux, z z h 0
ux,0 x x
u0, z 0, uw, z 0
x xz x, z z h constant
z
h
h'
0
X'
w
x
Stress Profiles
ij
1
1
ˆ q
ˆ ˆi q
ˆ ˆj
1 r1 r2 q
t collisions 2
1
ˆ q
ˆ ˆi q
ˆ ˆj
f c 1 r1 r2 q
2
z 100
0
3
2
2
4
6
xz
zz
1
0
8
1
10
2
12
3
0
100
200
z
z zz 0
300
400
5
10
15
20
25
30
x
constantslope 0.15554674 x x z
0
using
Solving terms of order
separation of variables
u
4
z
4x 2 2x 2z u 0 0
2
x
2z
2
x
2
z
0
0
k
k
z h 2 sin k x
u 0k x, z a k sinh
z b k sinh
w
w
w
w
u 0 x, z u 0k x, z
k 1
k h h
b k cosh
w
ak
k h
cosh
w
2
k
sin
x
w
w
bk
k h
sinh
w
Solving terms of order 1 using
Fourier transforms
4z 4x 2 2x 2z u1 2 2x 2z g z u 0 0
f x, z g z u 0 g u u u u x ,z z u 0 x, z
4
4
2 2
z x 2 x z u1 f x, z
u1 x, z 0 as x , z
0
Solving terms of order 1 using
Fourier transforms
1
u1 x, z
2
1
F
2
e
i 1x 2 z
F
2
1
2 2
2
d1d2
i 1 2
e
f , d d
1
u1 x, z
2
4
exp i x z
1
f ,
2
1
2 2
2
d1 d2 d d
2
Solving terms of order 1 using
Fourier transforms
1 cos , 2 sin
x r cos , z r sin
1 x 2 z r cos
1 cos , 2 sin
x r cos , z r sin
exp i1 x 2 z
2
1
2
0 0
e
i r cos
4
2 2
2
d d
d1 d2
Solving terms of order 1 using
Fourier transforms
J 0 z
2
1
2
2
i z cos
e
d ,
0
e i r cos
4
0 0
constant
4
J 0 r
d d 2 r
d r
3
0 r
2
J 0 r
2 r
d r
3
0 r
w
2
2 r 2 4 r J 0 r L 4,1, r r J1 r L 3,0, r
3 b a 3 b a z2
1, , ,
h ype rge om
2 2 2 2 2 2 4
a 1
L Lom m e lS
1a, b, z z
a b 1a b 1
Experimental data from the book:
“Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials”
By Jacques Duran.
Hypergeometric function
3 b a 3 b a z2
1, , ,
hype rge om
2 2 2 2 2 2 4
L Lomme lS
1a, b, z z a1
a b 1a b 1
A, B, C, x 1
hype rge om
AB
AA 1BB 1 2 AA 1A 2BB 1B 2 3
x
x
x ...
1 C
1 2 CC 1
1 2 3 CC 1C 2
Solving terms of order 1 using
Fourier transforms
1
u1 x, z
2
r
,
1
,
4
L
r
J
r
4
r
0
2
r J1 r L 3,0, r
w
r
x z
2
2
f , d d
Solving terms 2 and higher
4
z
2 u 2 f1 x, z
4
x
2
z
2
x
f1 x, z 2 g z u1 , u1 is known
2
z
2
x
4
z
2 u n fn 1 x, z
4
x
2
z
2
x
fn 1 x, z 2 g z u n 1 , u n 1 is known
2
x
2
z
Solution
zz x, z ux, z u 0 x, z u1 x, z u 2 x, z ...
2
Ex 1
x
G 2
z zz x xz zz g
z xz x xx 0
Numerical Approach
u 20u ij 8 u i 1, j u i 1, j u i , j1 u i , j1
2
2
2 u i 1, j1 u i 1, j1 u i 1, j1 u i 1, j1
u i 2, j u i 2, j u i , j 2 u i , j2
Conclusions
• Provided a perturbative analytical treatment
for studying nonlinear stress propagation
• Presented a finite difference numerical scheme
to compare with the analytical solution.
• Future work: To test this pde model by
implementing these methods to get numerical
values of stresses and compare with those
from simulation, and extend the model for the
anisotropic case.
random packing
at early stage
= 2.75
Is there any
difference between
this glass and a solid?
Answer: Look at
Monodisperse grains
crystallization
at later stage
= 4.3
Disorder has a universal effect on
Stresses and Collision Times.
THE END