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
OUTLINE
• Introduction– Overview– Important issues for force propagation– Models
• Experimental approach
• Exploration or order/disorder and friction
• Conclusion
Friction and frictional indeterminacy
NFi |||
NFi ||| Condition for static friction:
Multiple contacts => indeterminacy
Note: 5 contacts => 10 unknown forcecomponents.
3 particles => 9 constraints
Frictional indeterminacy => history dependence
Stress balance
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
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)
Features of elasto-plastic models
Conserve mass:
(Energy: lost by friction)
Conserve momentum:
0)(/ ii vt
ijji Tdtdv /
Concept of yield and rate-independence
shear stress, normal stress
Stable up to yield surface
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
OSL model
xzzzxx phemonological parameters
)]()([2
),( czxczxF
zxzz
q-model (e.g. in 2D)
q’s chosen from uniform distribution on [0,1]
Predicts force distributions ~ exp(-F/Fo)
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
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:
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
Anisotropic elastic lattice model
Expect progagation along lattice directionsLinear widening with depth
Schematic of greens function apparatus
Measuring forces by photoelasticity
Diametrically opposed forces on a disk
A gradient technique to obtain grain-scale forces
calibration
Disks-single response
Before-after
disk response mean
Large variance of distribution
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
Pentagon response
Elastic response, point force on a semi-infinite sheet
In Cartesian coordinates:
0 r
r
Frr
cos2
pii zxz ])/(1(/[1 2 2,1p
Example: solid photoelastic sheet
Moment test
22 ])/(1[
12),(
zxz
Fzxzz
dxzxxW zz ),(22
zzW )(
(See Reydellet and Clement, PRL, 2001)
Pentagons, width vs. depth
Variance of particle diameters to distinguish disorder
Spectra of particle density
Bidisperse responses vs. A
Weakly bi-disperse: two-peak structure remains
Bidisperse, data
Rectangular packing reduces contact disorder
Hexagonal vs. square packing
Hexagonal vs. square, data
Square packs, varying friction
Data for rectangular packings
Fits to convection-diffusion model
Variation on CD model--CW
Fits to CD- and CW models
Non-normal response, disks, various angles
Non-normal response vs. angle of applied force
Non-normal responses, pentagons
Non-normal response, pentagons, rescaled
Creating a texture by shearing
Evolution of force network– 5 degree deformation
Force correlation function
Correlation functions along specific directions
Response in textured system
Response, textured system, data
Fabric in textured system
“Fabric” from strong network
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
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