FR27 · Friction and shear

Shear Jamming: Frictional Grains That Lock Without Squeezing

Can shearing alone make loose frictional grains carry load? Yes, if the grains have friction and are not too loose. Shear at fixed area squeezes grains along one diagonal; with friction, the contacts closed there hold sideways load too, so three contacts per grain suffice instead of four. For 128 model disks at packing fraction 0.81 and friction 0.5, all four test packings carried load within 0.05 strain; frictionless ones still had not by strain 0.3.

Shear 128 frictional disks below the jamming density and watch force chains form along the compressive diagonal, or never form once friction is off.

Shear strain $0.042$

Pressure $0.00116\,k_{\mathrm n}$

Contacts per load-bearing disk $3.38$

Share of contacts sliding $0.0299$

Shear loose frictional grains

Packing fraction

0.81

\phi

Friction coefficient

0.5

\mu

Strain per second of playback

0.03

\Delta\gamma/\Delta t

Amber lines are contact forces, thicker for stronger ones; violet rings mark contacts sliding at the friction limit. Drag the handle on the top plate to shear by hand.

Random start

Seed of the random start

1

Model, units and sources

$d=1,\quad m=1,\quad k_{\mathrm n}=1,\quad k_{\mathrm t}=\tfrac{2}{7}k_{\mathrm n},\quad [p]=[\tau]=k_{\mathrm n}$

Each disk has the mass of its area, with the small disk as unit mass, and the moment of inertia of a uniform disk. The normal damping is $\gamma_{\mathrm n}=1$ and the time step $0.04$ . A strain step is relaxed until every net force and torque per radius is below $10^{-6}\,k_{\mathrm n}d$ and every speed below $10^{-5}$ , or for at most 40000 steps; the count of steps that hit the limit is listed under the data. Contacts carrying less than a thousandth of the mean force are not counted, and disks with fewer than two counted contacts are rattlers.

A run counts as jammed once a relaxed step has a pressure above $10^{-5}\,k_{\mathrm n}$ . Below that pressure the force lines and the force rose fade out, and at a tenth of it they are no longer drawn. Zhao et al. (2019) call a state shear-jammed only if its pressure stays above the noise while the shear is reversed; otherwise it is fragile.

Silbert et al. (2001), Eqs. 2–9 · Shundyak, van Hecke & van Saarloos (2007) · Zhao et al. (2019)

Work through why friction lets shear jam a looser packing

Model assumptions

Two-dimensional disks, half of diameter 1 and half of diameter 1.4, fill a periodic sheared cell at a packing fraction below the point where the same disks jam without friction, about 0.84. Contacts follow the linear spring-dashpot model of Silbert et al. (2001): a push-only normal spring with damping, and a tangential spring two sevenths as stiff that is cut back whenever its force would exceed the friction coefficient times the normal force, so disks can stick, slide and roll. Each step shears the cell by 0.003 and lets the disks come to rest under a small numerical drag, so the shear is slow and quasistatic. Every run starts from a frictionless relaxed packing whose pressure is below a hundred-millionth of the contact stiffness. Gently prepared with the same contacts, as in Shundyak et al. (2007), these disks reach 3.93 contacts each at very low friction and 3.03 at high friction, against the published 3.98 and 3.00. The shear-jamming densities measured on photoelastic disks belong to a different grain and are not reproduced here.

Pressure appears from shear alone

Contacts per disk land between the two isostatic counts

The load runs along the compressive diagonal

More numbers and how they are computed

Shear stress now: $0$
Shear stress over pressure: $0$
Contact fabric anisotropy: $0$
Direction contacts favour: $0$
Load-bearing disks: $0$
Share of rattlers: $0$
Strain at which it jammed: $0$
Lowest pressure since the reversal: $0$
Strain steps so far: $0$
Steps that hit the relaxation limit: $0$
Seed: $0$

Stress is the contact sum $\sigma_{\alpha\beta}=\frac{1}{A}\sum_{c} b_{\alpha}f_{\beta}$ , symmetrised, with compression positive; pressure is half its trace and the shear stress resists the imposed shear. The fabric is $R=\langle\mathbf n\mathbf n\rangle$ over the load-bearing contacts; its anisotropy is the difference of its eigenvalues, and its angle is that of the major axis, where $-45^{\circ}$ is the compressive diagonal of this shear.

Physics tutorial

Jamming by shear alone

BackgroundFrictionless disks below their jamming density can be sheared forever without carrying load: they always find a way to slide past each other. Real grains have friction, and friction lets a contact resist sideways as well as head-on.

Why it mattersSand, powders and seeds are usually handled below the density at which they would jam isotropically. Whether a gentle shear turns such a material solid decides whether a silo, a chute or a stirred bed keeps flowing.

Start with the essentials

Focus question: Can shearing alone make loose frictional grains carry load?
One-sentence intuition: With friction a contact can hold sideways load, so a packing can stand on as few as three contacts per disk instead of four. Shear squeezes the grains along one diagonal and pulls them apart along the other, and the contacts it closes along the squeezed diagonal are enough to hold. Without friction the same grains slide apart and never carry load at this density.

Core mathematical model

Coulomb limit on each contact

|f_{t}|\le\mu f_{n},\qquad f_{t}=-k_{\mathrm t}u_{t}

The tangential spring $u_t$ stores how far the two surfaces have slid since they touched. When $k_{\mathrm t}|u_t|$ would pass $\mu f_n$ , the spring is cut back to the limit and the contact slides (Silbert et al. 2001).

How many contacts can hold

z_{\mathrm{iso}}^{0}=2d=4,\qquad z_{\mathrm{iso}}^{\mu}=d+1=3,\qquad n_{m}\le z-(d+1)

Without friction each contact carries one force and each disk has two balance equations; with friction a contact carries two and a disk has three. A contact sliding at the limit adds back one equation, so the mobilised contacts per disk $n_m$ are bounded (Shundyak et al. 2007).

Stress from the contacts

\sigma_{\alpha\beta}=\frac{1}{A}\sum_{c}b_{\alpha}f_{\beta},\qquad p=\tfrac12\left(\sigma_{xx}+\sigma_{yy}\right)

The sum runs over contacts, with $\mathbf b$ the vector between the two centres and $\mathbf f$ the full contact force, normal plus tangential. Under shear to the right at the top, the compressive diagonal is $-45^{\circ}$ .

Common difficulties

Shear only slides grains past each other

Typical misconceptionShearing at fixed area cannot squeeze a packing, so it cannot jam it.

Better mental modelSimple shear at fixed area still compresses along one diagonal. At packing fraction 0.81 and friction 0.5, all four test packings passed the jamming threshold within 0.05 strain; with friction off, none did by 0.3.

Once jammed by shear, always jammed

Typical misconceptionA packing that carries load after shearing is now a solid in every direction.

Better mental modelReverse the shear and the network built along one diagonal unloads. In the two test packings still loaded at strain 0.15, the pressure fell below the threshold during reverse shear, so by the criterion of Zhao et al. these states are fragile, not shear-jammed.

Run the experiment

01
Watch load appear
Open Shear builds force chains and watch the pressure chart and the rose.
What to observe: The pressure starts at zero and climbs as the shear goes on, and the rose grows a long lobe along the dashed diagonal from upper left to lower right: the load runs along the squeezed diagonal.
02
Take friction away
Choose Frictionless grains just slide.
What to observe: The disks shear just as far, but the pressure stays near zero and no force chains appear.
03
Shear back
Choose Reverse: the chains let go.
What to observe: Within about 0.06 of reverse strain the pressure falls below the threshold and the rose empties; after about 0.22 a new lobe grows along the other diagonal.

Shear Jamming: Frictional Grains That Lock Without Squeezing

Jamming by shear alone

Core mathematical model

Coulomb limit on each contact

How many contacts can hold

Stress from the contacts

Common difficulties

Shear only slides grains past each other

Once jammed by shear, always jammed

Run the experiment

Watch load appear

Take friction away

Shear back

Related experiments and articles

From two disks to the jamming point and its memory