1. Shear-Induced Rigidity in Athermal Materials
Fundamental issue: How do we predict collective properties of athermal systems such as dry grains and non-brownian suspensions. Systems that change configuration only under external driving.
Set up the stat mech and show that it is a hard problem because of all the constraints that need to be satisfied.
Then propose that in a dual space the problem could look like a spring network, albeit one in which the spring strength is not predetermined. This community has focussed on rigidity of fiber networks, and maybe that will come in handy for understanding the granular problem.
Bulbul Chakraborty
Brandeis University
September 2014
Sumantra Sarkar, Dapeng Bi
Bob Behringer, Jie Zhang, Jie Ren, Joshua Dijksman

2. Shear-Jamming & Shear-Thickening
Dry grains at low densities undergo a transition from unjammed to fragile to shear-jammed solely through changing the shape of a 2D box (no change in density)
Athermal (non-Brownian) suspensions undergo a discontinuous shear- thickening transition (DST) as shear rate is increased.
Experiments and/or simulations show that the driving leads to a proliferation of frictional contacts. The qualitative reason is that constant volume frustrates dilatancy
Simulations of DST show that the transition does not occur if grains are frictionless
What type of solids are these ? How do we understand the transition ?

3. Shear-Jamming
“Jamming apparently occurs because the particles form “force chains” along the compressional direction. Even for spherical particles the lubrication films cannot prevent contacts; once these arise, an array or network of force chains can support the shear stress indefinitely. By this criterion, the material is a solid. In this Letter, we propose some simple models of jammed systems like this, whose solidity stems directly from the applied stress itself”.
Cates et al: Phys Rev. Lett 81, 1841 (1998)
So if we just look at how the strong force-network evolves and grows at a constant density, we can see there are pretty much three regimes:
As strain is increased
1) unjammed: the network is small
2) fragile: it’s percolated in the direction of compression, same as Cate’s example. It can support load in one direction but no the other.
3)Finally, shear jammed, percolation in all directions. This is a more robust state as it can support all loads.
1D Force Network
2D Force Network
“Spectator Particles”

4. Shear-induced Solidification in a Model Granular System
No Thermal Fluctuations
Purely Repulsive, contact Interactions, friction
States controlled by driving at the boundaries
Structure emerges that supports further shearing: a solid
Nature of this solid: that self-assembles to support the load that creates it. Marginal
Force networks well defined: preserve identity. grains can be moving in and out. Patterns easy to visualize. Difficult to quantify. Topological measures. Lou. Suggest another perspective that can help us understand jamming and the nature of this jammed solid

5. Shear-Jamming Experiments (Quasistatic Forward + Cyclic Shear)
Isostatic Point of frictionless grains: given a geometry, all forces can be determined.
So if we just look at how the strong force-network evolves and grows at a constant density, we can see there are pretty much three regimes:
As strain is increased
1) unjammed: the network is small
2) fragile: it’s percolated in the direction of compression, same as Cate’s example. It can support load in one direction but no the other.
3)Finally, shear jammed, percolation in all directions. This is a more robust state as it can support all loads.
Max Bi, Jie Zhang, BC & Bob Behringer Nature (2011)

6. Conditions of Static, Mechanical Equilibrium
(In DST, need to include the velocity-dependent forces)
This complicates the isostatic story
Number of equations
= N d (d+1)/2
For infinite friction coefficient, isostatic number is d + 1 (there are d unknowns at every contact
For arbitrary friction coefficients, need to know how many contacts are fully mobilized, and forces have to satisfy the inequality constraint
Local force & torque balance satisfied for every grain
Imposed stresses determine sum of stresses over all grains
Friction law on each contact
Positivity of all forces

7. F4 F3 F1 F1 F2 Vector fields enforce force balance constraint
Imposing the conditions through gauge potentials (2D)
Ball & Blumenfeld (2002), Henkes, Bi, & BC ( ), DeGuili (2011--)
Vector fields enforce force balance constraint
Additional scalar field enforces torque balance
There is a relation between the two F3 F2 F1 F4
(0,0)
(F1)
(F1+
F2)
(F1+F2+
F3)
Looking at the vector fields
We refer to them as heights: like a vector height field familiar in the context of groundstates of some frustrated magnet
Here the fields are continuous F1

8. F4 F3 F1 F1 F2 loops enclosing voids scalar potential height vector
Imposing the conditions through gauge potentials (2D)
Ball & Blumenfeld (2002), Henkes, Bi, & BC ( ), DeGuili (2011--)
loops enclosing voids F3 F2 F1 F4
(0,0)
(F1)
(F1+
F2)
(F1+F2+
F3)
height vector
scalar potential F1
Gauge potentials: irrelevant additive constant. Any set of these fields satisfy force and torque balance. There are constraints relating the two potentials.

9. Forces on central grain
Planar Packings
Forces on central grain
Height fields enforce force balance constraint
Torques should also balance out
For isotropic objects the above condition implies that
height field is divergence free
in 2D, a scalar field F4
(F1+F2+
F3) F3
(0,0)
(F1+
F2) F1
(F1) F2
height fields (h)
live on voids
Airy Stress Function
Ball & Blumenfeld (2002)

10. Microscopic and coarse-grained stress tensor
The microscopic stress tensor (force moment) of grains is defined as
(h3)
(h4)
(h2)
Coarse-grain by summing over a few grains
(h1)

11. A Tour of Shear Jamming Experiments
Reynolds Pressure
So if we just look at how the strong force-network evolves and grows at a constant density, we can see there are pretty much three regimes:
As strain is increased
1) unjammed: the network is small
2) fragile: it’s percolated in the direction of compression, same as Cate’s example. It can support load in one direction but no the other.
3)Finally, shear jammed, percolation in all directions. This is a more robust state as it can support all loads.
Non-rattler Fraction (Non-Spectators)

12. Shear Stress vs Strain
So if we just look at how the strong force-network evolves and grows at a constant density, we can see there are pretty much three regimes:
As strain is increased
1) unjammed: the network is small
2) fragile: it’s percolated in the direction of compression, same as Cate’s example. It can support load in one direction but no the other.
3)Finally, shear jammed, percolation in all directions. This is a more robust state as it can support all loads.
Looks deceptively like an elastic regime but is not
Curves for different packing fractions can be scaled: the strain at which the stress peaks, goes to zero as the packing fraction approaches that of the isotropic state.

13. Cyclic Shear Packing Fraction = 0.825
Small strains: below peak of shear stress curve
So if we just look at how the strong force-network evolves and grows at a constant density, we can see there are pretty much three regimes:
As strain is increased
1) unjammed: the network is small
2) fragile: it’s percolated in the direction of compression, same as Cate’s example. It can support load in one direction but no the other.
3)Finally, shear jammed, percolation in all directions. This is a more robust state as it can support all loads.

14. “Jamming apparently occurs because the particles form “force chains” along the compressional direction. Even for spherical particles the lubrication films cannot prevent contacts; once these arise, an array or network of force chains can support the shear stress indefinitely. By this criterion, the material is a solid. In this Letter, we propose some simple models of jammed systems like this, whose solidity stems directly from the applied stress itself”.
Cates et al: Phys Rev. Lett 81, 1841 (1998)
Increasing strain

15. Rigidity of amorphous solids
Permanent density modulations distinguish solids from liquids: a necessary condition for shear rigidity.
Broken translational symmetry (not obvious because structure is not crystalline) Positional correlations : not destroyed by small thermal fluctuations, average survives
Traditionally: energy or entropy gain leads to solidification
The granular story has to be one of CONSTRAINTS: there are no entropic (thermal ) or energetic preferences for any positional patterns.
Fundamentally, fluid to solid: broken translational symmetry is the origin of shear rigidity.

16. Geometric representation: force tile (Tighe)
Force Tilings
Geometric representation: force tile (Tighe)
Well known trick in statistical models where the ensemble has to satisfy some constraints
Can define a gauge potential in 3D also
The contact forces form a closed polygon (Newton’s 2nd law)
edges match up (Newton’s third law)
Vertices given by the heights
Geometry of tile depends on the other constraints

17. Looking at the vector, height fields only
force tile
Looking at the vector, height fields only
Height difference across the boundaries: determined by boundary stresses
Points: height vectors at a void starting from some arbitrary origin
Well known trick in statistical models where the ensemble has to satisfy some constraints
Can define a gauge potential in 3D also
for systems where forces are all repulsive, we have one sheet of tiling
Reciprocal space: a tiling of polygons

18. What height configurations are allowed ?
Local force balance
Imposed stresses .
Are all types of polygons allowed ?
Repulsive forces, no friction: No non-convex polygons
Unlike other known height models: the height fields here are continuous.
With friction: can get non-convex and self-intersecting polygons
With friction: force tiles can be convex and non-convex
Torque balance from 2 large and two small forces 1 2 3 4
Heights are continuous variables
Any configuration allowed
Like an ideal gas
Change bounding box: points can rearrange freely
From force balance constraint, we do not expect any correlations

19. Always satisfied for frictionless grains
Positivity of all forces
Friction law on each contact
Convexity condition
Always satisfied for frictionless grains

20. IF these introduce correlations, then maybe ...
Torque Balance
Friction law on each contact
IF these introduce correlations, then maybe ...
Example: Polygons have to be convex for frictionless grains
Change bounding box: points cannot rearrange freely
Rigidity
Changing bounding box is the analog of straining a spring network
Spring constant is not fixed but depends on configuration

21. Test of persistent pattern in height space
Overlap between two configurations
Grid stretched affinely with bounding box
If height pattern evolves affinely, large overlap
The stress-generated pattern can sustain further loading

22. Persistence of patterns
Strain Step
Not an elastic solid. Failures see in overlap.
Sarkar et al PRL (2013)

23. Persistence of patterns
Not an elastic solid. Failures see in overlap.
Sarkar et al PRL (2013)

24. Measure of rigidity

25. Dual to Spring Networks
Not an elastic solid. Failures see in overlap.

26. Broken Translational Symmetry in Height Space
Competition .
Under shear: Box shape changes and point density increases
Shear favors: non-convex polygons
Increasing number of contacts favors convex polygons
Competition drives transition (much like entropy vs energy ?)
Transition driven by density of points in this reciprocal space
Broken Translational Symmetry in Height Space

27. Non-convex polygons Forward shear
Strain Step
Non-rattler fraction
Shear Stress
#non-convex polygons
Experimental data
Initially: lots of complex polygons. Nearly linear chains. Two large forces that are almost colinear.

28. Failures: mini avalanches

29. Separation of Structure and Stress
Real Space
Real space with forces
Reciprocal Space
Unjammed
Jammed
Zhang et.al., Soft Matter, 2010
Sarkar et.al., PRL, 2013
“Frozen” structure, Large changes in force network