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Sumantra Sarkar, Dapeng Bi Bob Behringer, Jie Zhang, Jie Ren, Joshua Dijksman Shear-Induced Rigidity in Athermal Materials Bulbul Chakraborty Brandeis University September 2014

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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) Experiments and/or simulations show that the driving leads to a proliferation of frictional contacts. The qualitative reason is that constant volume frustrates dilatancy Athermal (non-Brownian) suspensions undergo a discontinuous shear- thickening transition (DST) as shear rate is increased. 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) “Spectator Particles” 1D Force Network2D Force Network

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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

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Shear-Jamming Experiments (Quasistatic Forward + Cyclic Shear) Max Bi, Jie Zhang, BC & Bob Behringer Nature (2011) Isostatic Point of frictionless grains: given a geometry, all forces can be determined.

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Conditions of Static, Mechanical Equilibrium (In DST, need to include the velocity-dependent forces) 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 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

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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 F3F3 F2F2 F1F1 F4F4 (0,0) (F 1 ) (F 1 + F 2 ) (F 1 +F 2 + F 3 ) 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 F1F1

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Imposing the conditions through gauge potentials (2D) Ball & Blumenfeld (2002), Henkes, Bi, & BC ( ), DeGuili (2011--) F3F3 F2F2 F1F1 F4F4 (0,0) (F 1 ) (F 1 + F 2 ) (F 1 +F 2 + F 3 ) F1F1 loops enclosing voids height vector scalar potential 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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9 Planar Packings F3F3 F2F2 F4F4 F1F1 Forces on central grain (0,0) (F 1 ) (F 1 + F 2 ) (F 1 +F 2 + F 3 ) height fields (h) live on voids Ball & Blumenfeld (2002) Airy Stress Function 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

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10 Microscopic and coarse-grained stress tensor The microscopic stress tensor (force moment) of grains is defined as (h 4 ) (h 1 ) (h 2 ) (h 3 ) Coarse-grain by summing over a few grains

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A Tour of Shear Jamming Experiments Reynolds Pressure Non-rattler Fraction (Non-Spectators)

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Shear Stress vs Strain 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. Looks deceptively like an elastic regime but is not

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Cyclic Shear Packing Fraction = Small strains: below peak of shear stress curve

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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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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. Rigidity of amorphous solids

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Force Tilings Geometric representation: force tile (Tighe) ★ 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 Height difference across the boundaries: determined by boundary stresses Points: height vectors at a void starting from some arbitrary origin for systems where forces are all repulsive, we have one sheet of tiling force tile Reciprocal space: a tiling of polygons

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What height configurations are allowed ? ★ 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 Local force balance Imposed stresses Are all types of polygons allowed ? Repulsive forces, no friction: No non- convex polygons 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

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

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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 Sarkar et al PRL (2013)

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Persistence of patterns Sarkar et al PRL (2013)

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Measure of rigidity

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Dual to Spring Networks

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Broken Translational Symmetry in Height Space 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 Competition

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Non-convex polygons Forward shear Strain Step Non-rattler fraction Shear Stress #non-convex polygons Strain Step Experimental data

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Failures: mini avalanches

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

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