L3 · Chapter 5

With friction, how shear jams looser grains

Sandbox Physics · 2026-10-29

Real sand has friction. Friction gives each contact a second direction in which it can carry force, so a packing can stand on fewer contacts. That brings a phenomenon frictionless disks lack: below the frictionless jamming density, shearing the grains in one direction can build anisotropic force chains that carry load. Experiments on photoelastic disks have measured this transition and the lowest packing fraction at which shear can jam.

A frictional contact: normal spring, tangential spring and a ceiling

The discrete-element model of Silbert and co-workers puts two springs on each contact. The normal one is the push-only spring of Chapter 1, plus damping so that collisions lose energy. The tangential one records how far the two surfaces have slid past each other since they touched, with a stiffness of 2/7 of the normal one.

The tangential force cannot grow without limit. Once it exceeds the friction coefficient $\mu$ times the normal force, the tangential spring is cut back so the force sits exactly at the limit; that step stands for the surfaces starting to slip. The tangential force also exerts a torque about the centre, so the disks rotate.

|f_t|\le\mu f_n,\qquad m\equiv\frac{|f_t|}{\mu f_n}\in[0,1]

The Coulomb condition. A contact with

m=1

is fully mobilised: its friction is used up, and any extra tangential load makes it slide. Shundyak et al. use

m

to measure how far each contact is from slipping.

Source: L. E. Silbert · K. Shundyak

Friction lets fewer contacts hold

Chapter 2 already counted it: with friction each contact has two unknowns and each disk three equations, so the isostatic contact number drops from the frictionless $2d=4$ to $d+1=3$ . Fewer contacts can balance, so frictional grains can carry load at lower packing fractions.

A fully mobilised contact is not free, though: its tangential force is pinned at $\mu f_n$ , which adds an equation. Shundyak and co-workers generalise the count to $n_m\le z-(d+1)$ , with $n_m$ the mean number of fully mobilised contacts per particle. Their slowly built 2D frictional packings lie close to this line; at small pressure and friction, about half the contacts are fully mobilised.

n_m\;\le\;z-z_{\mathrm{iso}}^{\mu},\qquad z_{\mathrm{iso}}^{\mu}=d+1

Eq. (1) of Shundyak et al. Packings at equality are marginally stable; packings below the line have constraints to spare and are more stable.

Source: K. Shundyak

Shear jamming: fragile and shear-jammed states

Bi and co-workers reported in Nature in 2011 that frictional photoelastic disks below the isotropic jamming density can be jammed from a stress-free state by shear alone. Shear first pushes the grains into fragile states that carry load only in certain directions; further shear makes the force network anisotropic and connected, a shear-jammed state. We read only the abstract of that paper.

Zhao and co-workers gave a practical test in 2019. A fragile state supports only loads compatible with the shear that made it; a shear-jammed state also resists the reverse. So shear forward to some strain, then reverse, and see whether the pressure drops below the noise.

Source: D. Bi · Y. Zhao

Worked example: the packing fraction of the Zhao experiment from its grain count

The shear cell of Zhao et al. is an annulus with inner radius 8.7 cm and outer radius 35.5 cm, of area $\pi(35.5^2-8.7^2)=3.72\times10^{3}\,\mathrm{cm^2}$ . The disks have diameters 1.59 cm and 1.27 cm in a 1 : 3 number ratio of large to small, and a friction coefficient of 0.9.

The model: out of every 4 disks, 1 is large and 3 are small, so the mean area is $\big[\pi(0.795)^2+3\pi(0.635)^2\big]/4=1.446\,\mathrm{cm^2}$ . The packing fraction is the grain count times the mean area divided by the annulus area.

With the paper’s range of 1447 to 2101 grains: $1447\times1.446/3721=0.562$ and $2101\times1.446/3721=0.817$ . That matches the stated $0.56<\phi<0.82$ , so we have read the geometry and the mixture correctly.

The paper’s finding: at $\phi=0.74$ no amount of forward shear produced a shear-jammed state; at $\phi=0.75$ it did. The lowest packing fraction that shear-jams is therefore $\phi_{SJ}=0.745\pm0.005$ . Fitting a straight line to the strain needed to reach steady flow and extrapolating it to zero strain gives $\phi_0=0.84\pm0.02$ , which the authors note is close to the frictionless isotropic jamming density; Chapter 4 also finds a value near 0.84.

Source: Y. Zhao

Checking the count in simulation, then shearing

We wrote frictional 2D disks with the model above and, as Shundyak and co-workers did, inflated a dilute gas of them to a target pressure with the same slow radius equation. With 128 disks, 8 random starts and three pressures, extrapolated in $p^{1/2}$ to zero pressure: at friction coefficient 0.001, $z=3.93\pm0.01$ ; at 10, $z=3.03\pm0.01$ . The paper gives $3.98\pm0.02$ and $3.00\pm0.02$ . The high-friction end agrees; the low-friction end is about 0.05 low, inside the 0.06 tolerance we set beforehand, and the finite-size correction for 128 disks alone is about 0.03. The share of rattlers rises from 4.6% to 11.9%, where the paper reports about 4% to 12%.

At every measured pressure the average of $n_m$ over the eight starts stays below $z-3$ , so the counting bound holds; but at the lowest pressure, $10^{-4}k_{\mathrm n}$ , the low-friction packings still sit about 0.17 above the line, not as close as in the paper.

The same contacts in a periodic shear cell at packing fraction 0.81, below the frictionless jamming point of these disks near 0.84: at friction coefficient 0.5, all four packings, each starting below $10^{-8}k_{\mathrm n}$ , pushed the pressure past $10^{-5}k_{\mathrm n}$ within 0.05 strain, with the loaded contacts favouring the compressive diagonal. Without friction none did by 0.3. Reversing the shear at 0.15, both packings still loaded let their pressure fall below the threshold: by the criterion of Zhao et al., they are fragile.

Source: K. Shundyak · Y. Zhao

These numbers do not transfer directly

The 0.745 belongs to one kind of 2D photoelastic disk: one friction coefficient, one size ratio, one base friction and one shear step. Another grain or three dimensions would give another number. The course’s compression Lab is frictionless: with no tangential forces its count stays at 2d, and it shows isotropic jamming. The shear-jamming Lab has friction, but its disks are not photoelastic disks, so the packing fraction at which it starts to carry load should not be set against 0.745 either.

Source: Y. Zhao · K. Shundyak

Try it in the Lab

Use frictionless disks as the control

Predict first: if the Lab’s disks had a friction coefficient of 0.9, would the onset packing fraction rise or fall? Justify it with this chapter’s count.
Play the default preset and note the onset packing fraction and the contacts per load-bearing disk near the onset.
Set these against the 0.745 of Zhao et al. and this chapter’s count. Explain why frictionless disks need more contacts to carry load, and why the two numbers cannot be compared directly.

Make grains carry load by shear alone

Predict first: packing fraction 0.81 is below the frictionless jamming point. Will the pressure leave zero under shear? Is the answer the same at friction coefficient 0 and 0.5?
Play the default preset. Note the strain at which the pressure first passes the threshold and which diagonal the long lobe of the rose points along. Then compare Frictionless grains just slide.
Choose Reverse: the chains let go. See whether the pressure falls below the threshold after the turn, and use the criterion of Zhao et al. to say which kind of state it was.

Exercises with explained answers

A 2D frictional packing has a mean contact number $z=3.4$ . By the generalised count of Shundyak et al., at most how many fully mobilised contacts can each particle have on average?
Show the answer
$n_m\le z-(d+1)=3.4-3=0.4$ . Each contact belongs to two particles, so at most $0.4/(3.4/2)\approx24\,\%$ of all contacts are fully mobilised.
In the reverse-shear test of Zhao et al., the pressure dips to 0.2 N/m at its lowest during reverse shear. Is the state shear-jammed?
Show the answer
No. Their criterion is that the pressure stays above the noise threshold of 0.3 N/m throughout reverse shear. 0.2 N/m is below it, so the force network let go under reversal: the signature of a fragile (or unjammed) state.
With the same grains, by how much does the packing fraction of the Zhao experiment change when the grain count goes from 2000 to 2050?
Show the answer
Each grain takes $1.446\,\mathrm{cm^2}$ on average, so 50 more grains raise the packing fraction by $50\times1.446/3721\approx0.019$ . That is much larger than the 0.005 uncertainty on $\phi_{SJ}$ , which is why the experiment scans packing fraction by adjusting the grain count.

Sources and what we recomputed

L. E. Silbert, D. Ertaş, G. S. Grest, T. C. Halsey, D. Levine and S. J. Plimpton, Granular flow down an inclined plane: Bagnold scaling and rheology, arXiv:cond-mat/0105071v1; Physical Review E 64, 051302 (2001)
Version 1, 3 May 2001. Section II: the frictional contact model, a normal spring with damping plus a tangential spring cut off by the Coulomb condition, with the ratio of tangential to normal stiffness of 2/7. Eq. (12): the stress tensor as a sum of branch vector times contact force divided by volume, plus a kinetic part. We use the contact part of Eq. (12) for static packings.
K. Shundyak, M. van Hecke and W. van Saarloos, Force mobilization and generalized isostaticity in jammed packings of frictional grains, arXiv:cond-mat/0610205v1; Physical Review E 75, 010301(R) (2007)
Version 1, 7 October 2006. The opening paragraphs: with friction, the isostatic contact number drops from 2d to d + 1. Eq. (1) and Fig. 1: each fully mobilised contact, where the friction force sits at the Coulomb limit, adds one constraint, so the number of such contacts per particle is at most z minus d + 1. Fig. 2: the distribution of mobilisation splits into a peak at the limit and a flat part below it. We did not reproduce their simulations.
D. Bi, J. Zhang, B. Chakraborty and R. P. Behringer, Jamming by shear, Nature 480, 355–358 (2011)
Abstract only. Frictional photoelastic disks below the isotropic jamming density can be jammed by shear strain alone, through fragile states, into shear-jammed states with anisotropic force networks. We cite the claim as summarised in the abstract and in Zhao et al. (2019); we did not read the full paper or reproduce its data.
Y. Zhao, J. Barés, H. Zheng, J. E. S. Socolar and R. P. Behringer, Shear-jammed, fragile, and steady states in homogeneously strained granular materials, arXiv:1904.10051v2; Physical Review Letters 123, 158001 (2019)
Version 2, 14 August 2019. Fig. 1(b): a Couette cell of 21 rings that shears 1447 to 2101 bidisperse photoelastic disks (diameters 1.59 and 1.27 cm, friction coefficient 0.9) in steps of 0.6 % strain. Figs. 2(d) and 2(e): the reverse-shear test that separates fragile from shear-jammed states, with a pressure noise floor of 0.3 N/m. Text: the smallest packing fraction that shear-jams, 0.745 ± 0.005, and the extrapolated frictionless jamming density 0.84 ± 0.02. Experimental data; we read the figures and did not reproduce them.