This document proposes a numerical simulation method for granular flow based on micropolar fluid theory. It summarizes existing discrete element and continuum methods, and proposes extending the Hibler rheology model for sea ice by incorporating micro rotations using micropolar fluid theory. Governing equations are presented accounting for mass, momentum, angular momentum conservation and constitutive relations. Parameters like internal friction angle and characteristic length are described. Simulation results on problems like dam breaking show the model can reproduce effects like angle of repose that depend on these parameters. The results were presented at a mechanics conference in 2014.

1. Numerical Simulation of
Granular Flow
Based on Micropolar Fluid Theory
Shin-Ichiro Serizawa/Tomoyuki Ito
(serizawa.shinichiro@gmail.com)

2. Needs of Granular Flow Analysis
 From Natural Phenomena to Industry
 Prediction of Natural Phenomena
 Landslide
 Avalanche
 Pyroclastic flow
 Industry
 Civil Engineering
 Agriculture
 Pharmacy
 Electrophtography

3. Numerical Analysis Method
 Distinct Element Method
 Each Particle Interactions are presented by Cundall Model
 Cundall, P.A., Strack, O.D.L., A Discrete Numerical Model for
Granular Assemblies, Geotechnique, 29-1(1979), 47-65.
 DEM Takes Huge Computing Cost.
Ct
Kt
Kn
Cn
m
rj
ri
mj
mi
Cundall Model

4.  Description of Granular Flow as the
Constitutive Law model is required.
 FEM/FDM and etc. can be applied to solve.
Reduction of Computational Cost

5. Hibler’s Rheology Model
 Numerical Model of the Sea Drift Ices
 Hibler, W. D., III. A Dynamic Thermodynamic Sea
Ice Model, Journal of Physical Oceanography, 9-4
(1979), 815-846.
 Viscosity coefficient h
 Decided by Principle Strain Speed
 If Large then Act as Plastic Flow
 Else if then Act as Viscous flow
 Mohr-Coulomb Yield Criterion
 Internal Friction Angle f  1 2 1 2 sin    f  
max
1 2
Psin
min ,
f
h h
 
 
  
 

6. Hibler’s Rheology Model
 Governing Equations
 Conservation Law
 Momentum
 Constitutive Equation
ij ij ij kk ij
1
P 2
2
  h   
 
    
 
i
ji,j
d
dt
   
v
f
k ,k
d
0
dt

 v

7. Problems of Hibler’s Rheology Model
 The Rotations of Particles are not Considered.
 The Grain Size is not Explicitly Described in
Constitutive Equation.

8. Micoropolar Fluid Theory
 Microstructure in Continuum
 Micro Rotation w
 Characteristic Length d
1
11
2
m32
m31
22
w3
d
12
21
3
v1
v2

9. Micoropolar Fluid Theory
 Governing Equations
 Conservation of Mass
 Momentum
 Angular Momentum
 Constitutive Equation
 Stress
 Coupled Stress
i
ji,j ijk jk i
d
I c
dt
 m     
w
i
ji,j i
d
dt
   
v
f
k ,k
d
0
dt

 v
 ij k,k ij i,j j,i c i,j  m h w  h w h w h w   
  j,i i,j
ij k ,k i
j,i i,j
kj r ijk
v v
2
v v
P v 2
22
h  w   h
    
  
 
 
     
   

10. Model Based on Micropolar Fluid Theory
 Kanatani, K., A Micropolar Continuum Theory
for the Flow of Granular Materials,
International Journal of Engineering Science,
17-4, (1979), 419–432.
 Mitarai, N. Hayakawa H. and Nakanishi, H.,
Collisional Granular Flow as a Micropolar Fluid,
Phys. Rev. Lett. 88, (2002), 174301.

11. Extended Hibler Model
 Viscosity
 Decided by Equivalent Strain Speed
 Micro Rotation Viscosity
 Angular Viscosity
 Pressure Equation
 Pressure is Non Negative value
max
Psin
min ,h

f
h
 
  
 
 
1
2
ij ij1 2 ij ji 2
ij3 ij
e e e e
d k k
2
g g
g
 
  
 
 
1
3
ij ij kk ije    
r f (d ) Const.h  
2
c
1
I
1
d
0
h h h 
max
0
0 0
0
0
P P










 
 

1
3
ij ij kk ijk    

12. Smoothed Particle Hydrodynamics
 Physics Quantities are expressed by Kernel
Function
 Lucy, L. B., A numerical approach to the testing of the fission
hydrodynamics, Astron., J., 82-12 (1977), 1013- 1024.
 Gingold R. A. Monaghan, J. J., Smoothed particle hydrodynamics:
Theory and application to non spherical stars, Mon. Not, Roy. Astron.
Soc., 181 (1977), 375-389.
Approximation by Kernel Functions

13. Example by SPH
 Schematic of Sand Pile Formation
0
0.1
0.2
0.3
0.4
0.5
0 0.25 0.5 0.75 1
Hieght(m)
x (m)
25 particles
50 particles
x
y
xmax

14. Result of Pseudo Viscous Fluid
 Broken Dam
MaxvMinv

15. 0.0 4.91 (/s)-4.91
0.0 4.91 (/s)-4.91
Result of Extended Hibler Model
Velosity:v
Angular Velosity:w
Counter Clockwise
No Rotation
Clockwise
0.0 0.78 (m/s)
0.0 0.78 (m/s)
Velosity:v
Angular Velosity:w
Counter Clockwise
No Rotation
Clockwise

16. Parameters in Constitutive Equations
 Internal Friction Angle : f
 Characteristic Length : d

17. Internal Friction Angle
 Internal Friction Angle and Angle of Repose
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.2 0.4 0.6 0.8 1.0
70
60
45
30
15
y
x
15f  30f  45f  60f  75f 

18. Influence of Parameters in Equations
 Internal Friction Angle : f
 Characteristic Length : d

19. Characteristic Length and Fluidity
 Propagated Front Position
 as Index of Fluidity
0.8
0.9
0.9
1.0
1.0
1x10
-5
1x10
-4
1x10
-3
1x10
-2
1x10
-1
1x10
0
x
max
d
Hibler Model
Radius of Kernel Function:h
xmax

20. Conclusion
 The constitutive model of granular flow based
on micropolar fluid theory is presented.
 Part of granular matter flow with parallel and
rotational motion and the other part do not flow.
 The proposed model can reproduce sand pile unlike
fluid.
 Angle of repose depends on internal friction angle
of granular matter.
 The size of granular matter has an influence on
fluidity.

21.  The result is presented in The 63rd Japan
National Congress of Theoretical and
Applied Mechanics.
 Tokyo Institute of Technology
 2014.9.26