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
r1
q
r2
 r1'   r1  1
 1 1   r1  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   r1  r2   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 r1  r2   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    r1  r2   q

t collisions 2
1
ˆ q
ˆ  ˆi q
ˆ  ˆj
  f c 1   r1  r2   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 2xxxz 

zzzz 2zzxz 
xzzz 2xzxz 
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  dc  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 ux, z  z h  0
ux,0   x  x
u0, z   0, uw, 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    r1  r2   q

t collisions 2
1
ˆ q
ˆ  ˆi q
ˆ  ˆj
  f c 1   r1  r2   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
d1d2
 
i 1  2 
e
f  , d d


  
1
u1 x, z  
2
4



 

    exp i x      z   
1

f  , 

2
1


2 2
2
d1 d2 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 i1 x     2 z   

2
1
  
 2


0 0
e
 i  r cos   

4


2 2
2
 d d
d1 d2
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
1a, b, z   z
 a  b  1a  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
1a, b, z   z a1
 a  b  1a  b  1
A, B, C, x  1 
hype rge om
AB
AA  1BB  1 2 AA  1A  2BB  1B  2 3
x
x 
x  ...
1 C
1 2  CC  1
1 2  3  CC  1C  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   ux, 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 , j1  u i , j1 
2
2
 2 u i 1, j1  u i 1, j1  u i 1, j1  u i 1, j1 
 u i  2, j  u i 2, j  u i , j 2  u i , j2 
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