1. Forces and Fluctuations in Dense Granular Materials
R.P. Behringer, Dept. of Physics, Duke University
Discussant: C.S. O’Hern, SEAS, Dept. of Physics, Yale University
Quantitative characterization of force chains!

2. Where it all began…
Temporal
fluctuations
Duke University
1994

3. Spatial fluctuations

4. Anisotropy How does anisotropy affect the ‘isotropic’ jamming picture?
New critical points, related critical
points, controlled by point J,…?
What is the coupling between
friction and anisotropy, e.g. more
anisotropic fabrics?
Isotropic
compression
Shear
fabric

5. What are the differences between isotropic
and anisotropic jammed systems?

6. Difference I Different contact networks or fabrics Isotropic
compression
Shear
Different contact networks or fabrics

7. Difference II Broad normal force distributions Isotropic Compression
Shear
Broad normal force distributions
Snoeier et al. PRL 92, (2004)
Tighe et al. Phys. Rev. E, 72, (2005)

8. Difference III Shear Compression
Both directions equivalent
Chain direction
Direction 
normal
To chains
Local ‘pressure’ correlations are anisotropic longer ranged

9. Difference III Simulations of Jammed Packings Isotropic compression
Pure shear
Simulations of Jammed Packings
Chakraborty, Henkes, Lois, O’Hern, Majmudar, Behringer, 2008

10. Difference IV Distribution of jamming onsets P(JA) changes,
e.g. shear dilation

11. Two ways to measure distribution P(JA)V
Quasistatic shear at
constant =0.85 1’ 3’ 3’ 5’
1 3 5
Dynamics from one inherent
structure to another
1’ 3’ 5’
Associated jammed states V
Quasistatic shear at
constant ‘pressure = 0’
1 4  7
Dynamics from one jammed
state to another

12. Quasistatic shear at constant =0.85
distribution of jamming onsets
N~100
no shear
shear
JA
Distribution of jamming
onsets shifts, i.e. shear
dilation

13. Shear at constant ‘Pressure=0’
Number of jamming onsets
depends on shear because
fabrics are different
Does packing fraction even matter
at all? Or is the family or fabric
the more important variable even
if 0?
V=0
strain
JA 
family/fabric

14. Point J is not really a single point?
RCP-RLP ?
unjammed
jammed
1/
Points JAF
Point JA
Point J
For frictionless isotropic jammed packings, there is an infinite number of
J-points, but they all have same J when N  and similar fabrics
For frictionless anisotropic jammed packings, there is a different infinite
set of JA-points with possibly different JA that depend on the fabric. What happens
when N  ?
For frictional jammed packings, there is an infinite set of jammed
packings over a finite range RLP to RCP when N.

15. Questions 1. How do we understand anisotropy?
2. Do anisotropic ‘critical’ points exist that
‘control’ response of anisotropic systems to
deformation, and how do they relate to
‘isotropic’ jamming framework?
3. What is the density of states as a function
of the fabric?
Gao, Lois, O’Hern, 2008.