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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
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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
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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.
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300 (free fall region) 250 (fluid region) vz 200 (glass region) dvz/dt 150 z
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Hard Sphere Collision Velocity Adjustment
q
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Stress Tensor Calculation
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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.
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Stress Profiles
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Continuum Mechanics (Two dimensional x-z model)
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Continuum Mechanics
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Continuum Mechanics
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Continuum Mechanics Choose x and z along principal axes:
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Continuum Mechanics
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PDE
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Nonlinear Density and Biharmonic PDE
For isotropic hard spheres t = r = s = 1:
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Boundary Conditions z h h' x w X'
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Stress Profiles
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Solving terms of order 0 using separation of variables
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Solving terms of order 1 using Fourier transforms
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Solving terms of order 1 using Fourier transforms
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Solving terms of order 1 using Fourier transforms
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Solving terms of order 1 using Fourier transforms
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Experimental data from the book:
“Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials” By Jacques Duran.
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Hypergeometric function
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Solving terms of order 1 using Fourier transforms
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Solving terms 2 and higher
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Solution
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Numerical Approach
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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.
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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.
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THE END
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