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