1. John Drozd Colin Denniston
Stress Propagation in a Granular Column
In Gravity Driven Granular Flow
John Drozd
Colin Denniston
• 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

2. Outline 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

3. 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.

4. 300 (free fall region) 
250 (fluid region) vz
200 (glass region)
dvz/dt
150 z

5. Hard Sphere Collision Velocity Adjustmentq

6. Stress Tensor Calculation

8. z h h' x w Experimental data from the book:w X'
Experimental data from the book:
“Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials”
By Jacques Duran.

9. Stress Profiles

10. Continuum Mechanics (Two dimensional x-z model)

11. Continuum Mechanics

12. Continuum Mechanics

13. Continuum Mechanics
Choose x and z along principal axes:

14. Continuum Mechanics

15. PDE

16. Nonlinear Density and Biharmonic PDE
For isotropic hard spheres t = r = s = 1:

17. Boundary Conditions z h h' x w X'

18. Stress Profiles

19. Solving terms of order 0 using separation of variables

20. Solving terms of order 1 using Fourier transforms

21. Solving terms of order 1 using Fourier transforms

22. Solving terms of order 1 using Fourier transforms

23. Solving terms of order 1 using Fourier transforms

24. Experimental data from the book:
“Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials”
By Jacques Duran.

25. Hypergeometric function

26. Solving terms of order 1 using Fourier transforms

27. Solving terms 2 and higher

28. Solution

29. Numerical Approach

30. 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.

31. 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.

32. THE END