Download - Force Transmission in Granular Materials
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Force Transmission in Granular Materials
R.P. Behringer
Duke University
Support: US NSF, NASA
Collaborators: Junfei Geng, Guillaume Reydellet, Eric Clement,
Stefan Luding
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OUTLINE
• Introduction– Overview– Important issues for force propagation– Models
• Experimental approach
• Exploration or order/disorder and friction
• Conclusion
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Friction and frictional indeterminacy
NFi |||
NFi ||| Condition for static friction:
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Multiple contacts => indeterminacy
Note: 5 contacts => 10 unknown forcecomponents.
3 particles => 9 constraints
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Frictional indeterminacy => history dependence
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Stress balance
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Stress balance, Continued
• Four unknown stress components (2D)• Three balance equations
– Horizontal forces – Vertical forces– Torques
• Need a constitutive equation
0
zxxzxx
0
zxzzxz
zxxz
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Some approaches to describing stresses
• Elasto-plastic models (Elliptic, then hyperbolic)
• Lattice models– Q-model (parabolic in continuum limit)
– 3-leg model (hyperbolic (elliptic) in cont. limit)
– Anisotropic elastic spring model
• OSL model (hyperbolic)
• Telegraph model (hyperbolic)
• Double-Y model (type not known in general)
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Features of elasto-plastic models
Conserve mass:
(Energy: lost by friction)
Conserve momentum:
0)(/ ii vt
ijji Tdtdv /
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Concept of yield and rate-independence
shear stress, normal stress
Stable up to yield surface
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Example of stress-strain relationship for deformation
||/ VkPVPT ijijij
2/)( jiijij vvV
(Strain rate tensor with minus)
22|| ijVV |V| = norm of V
Contrast to a Newtonian fluid:
)()3/2(]3/)([2 VTrVTrVPT ijijij
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OSL model
xzzzxx phemonological parameters
)]()([2
),( czxczxF
zxzz
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q-model (e.g. in 2D)
q’s chosen from uniform distribution on [0,1]
Predicts force distributions ~ exp(-F/Fo)
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Long wavelength description is a diffusion equation
)],(2)1,()1,([),(
jzwjzwjzwz
jzw
2
2
x
wD
z
w
)4/exp(2
),( 2 DzxDz
Fzxzz
Expected stress variation with depth
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Convection-diffusion/3-leg model
0 OO]///[ 22 xDxczO
]}4/)(exp[]4/)({exp[4
1
222 DzczxDzczx
Dz
Fzz
Applies for weak disorder
Expected response to a point force:
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Double-Y model
Assumes Boltzmann equation for force chains
For shallow depths: One or two peaksIntermediate depths: single peak-elastic-likeLargest depths: 2 peaks, propagative, with diffusive widening
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Anisotropic elastic lattice model
Expect progagation along lattice directionsLinear widening with depth
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Schematic of greens function apparatus
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Measuring forces by photoelasticity
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Diametrically opposed forces on a disk
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A gradient technique to obtain grain-scale forces
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calibration
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Disks-single response
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Before-after
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disk response mean
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Large variance of distribution
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Organization of Results
• Strong disorder: pentagons
• Varying order/disorder– Bidisperse disks– Reducing contact number: square packing– Reducing friction
• Comparison to convection-diffusion model
• Non-normal loading: vector/tensor effects
• Effects of texture
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Pentagon response
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Elastic response, point force on a semi-infinite sheet
In Cartesian coordinates:
0 r
r
Frr
cos2
pii zxz ])/(1(/[1 2 2,1p
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Example: solid photoelastic sheet
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Moment test
22 ])/(1[
12),(
zxz
Fzxzz
dxzxxW zz ),(22
zzW )(
(See Reydellet and Clement, PRL, 2001)
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Pentagons, width vs. depth
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Variance of particle diameters to distinguish disorder
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Spectra of particle density
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Bidisperse responses vs. A
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Weakly bi-disperse: two-peak structure remains
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Bidisperse, data
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Rectangular packing reduces contact disorder
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Hexagonal vs. square packing
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Hexagonal vs. square, data
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Square packs, varying friction
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Data for rectangular packings
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Fits to convection-diffusion model
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Variation on CD model--CW
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Fits to CD- and CW models
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Non-normal response, disks, various angles
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Non-normal response vs. angle of applied force
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Non-normal responses, pentagons
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Non-normal response, pentagons, rescaled
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Creating a texture by shearing
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Evolution of force network– 5 degree deformation
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Force correlation function
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Correlation functions along specific directions
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Response in textured system
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Response, textured system, data
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Fabric in textured system
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“Fabric” from strong network
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Conclusions
• Strong effects from order/disorder (spatial and force-contact)
• Ordered systems: propagation along lattice
• Disorderd: roughly elastic response
• Textured systems– Power law correlation along preferred direction– Forces tend toward preferred direction
• Broad distribution of local response