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**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

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**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 ?

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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”

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**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

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**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)

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**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

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**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

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**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.

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**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)

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**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)

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**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)

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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.

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**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.

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“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

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**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.

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**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

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**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

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**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

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**Always satisfied for frictionless grains**

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

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**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

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**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

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**Persistence of patterns**

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

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**Persistence of patterns**

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

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Measure of rigidity

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**Dual to Spring Networks**

Not an elastic solid. Failures see in overlap.

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**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

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**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.

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**Failures: mini avalanches**

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**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

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