FR27 · Compression and unloading

Jamming Transition: Squeezing Soft Disks

At what packing fraction does a box of loose disks suddenly push back? Loose soft disks in a box push on nothing until, at one packing fraction, they lock into a network that balances every force. That happens when the contacts just outnumber the force-balance equations. Squeeze further and, for spring-like contacts, the pressure grows in proportion to the extra squeezing. For 128 mixed-size disks in a walled box, the onset is near a packing fraction of 0.79 to 0.81.

Squeeze 128 soft disks in a box and watch force chains appear the moment they jam. Pressure, contacts per disk and wall forces are read from the same packing.

Packing fraction $0.795$

Wall pressure $0\,\varepsilon/d^{2}$

Contacts per disk $0$

Contacts beyond isostatic $0$

Squeeze the box

Deepest packing fraction

0.86

\phi_{\max}

Large-to-small diameter ratio

1.4

d_{\mathrm L}/d_{\mathrm S}

Number of disks

128

N

Contact law exponent

2

\alpha

Drag the corner of the bright walls to squeeze or release the box yourself.

Random start

Seed of the random start

2727

Model, units and sources

$d=1,\quad \varepsilon=1,\quad m=1,\quad [p]=\varepsilon/d^{2}$

Compressive stress is positive. Each step rescales the box and every centre by the same factor, then FIRE relaxes the packing until the largest net force is below $10^{-10}\,\varepsilon/d$ . Pressure below $10^{-6}\,\varepsilon/d^{2}$ counts as unloaded.

O’Hern, Silbert, Liu & Nagel (2003), Eq. 1, Table I · Goodrich, Liu & Nagel (2012), Eq. 3 · Guénolé et al. (2020), FIRE 2.0

Derive where jamming starts and how pressure grows past it

Model assumptions

Frictionless two-dimensional disks, half of diameter 1 and half of diameter 1.4, repel only while they overlap, with a harmonic or Hertzian law. After every small change of box size the packing is relaxed to force balance, so there is no inertia, gravity, friction or rotation. Rattlers are disks with fewer than three load-bearing contacts. The walled box and the small number of disks shift the onset; the value does not apply to real sand.

Wall pressure along the cycle

Contacts per load-bearing disk

Contact forces in the shown packing

More numbers and how they are computed

Contact stress over wall pressure: $0$
Stored energy per disk: $0$
Rattlers: $0$
Load-bearing disks: $0$
Mean local six-fold order: $0$
Onset packing fraction of this cycle: $0$
Net work by the walls over the cycle: $0$
Energy lost in relaxation over the cycle: $0$
Relaxation iterations of this step: $0$
Disks: $0$
Seed: $0$

The dashed contact line is the contact number the same load-bearing disks would have with exactly $2N'+1$ contacts, keeping their wall contacts. Net wall work is a trapezoid integral of the pressure along each affine step; on a closed cycle it should match the relaxation loss to a few percent.

Physics tutorial

A pile that learns to push back

BackgroundLoose disks in a box can slide past each other, so the walls feel nothing. Squeeze the box a little more and, at one packing fraction, the disks lock into a network that pushes on every wall.

Why it mattersSand, beads and foam all switch from flowing to holding a load. Frictionless disks are the cleanest version: one contact law, no gravity, no friction. They show how load is carried by a sparse network of force chains, and why that network is as sparse as force balance allows.

Start with the essentials

Focus question: At what packing fraction does a box of loose disks suddenly push back?
One-sentence intuition: Jamming happens when there are just enough contacts to balance every force. At that point the pressure starts from zero and grows linearly for harmonic contacts, while the extra contacts beyond the isostatic count grow from zero too. Unloading undoes it, but the jammed state depends on how it was reached.

Core mathematical model

Contact law

V(r)=\frac{\varepsilon}{\alpha}\left(1-\frac{r}{\sigma_{ij}}\right)^{\alpha},\qquad r<\sigma_{ij}=R_i+R_j

Disks repel only while they overlap. $\alpha=2$ is a harmonic spring; $\alpha=5/2$ is the Hertzian law used by O’Hern et al. (2003).

Counting contacts

N_{\mathrm{iso}}=2N'+1\ \ (\text{walled box}),\qquad N_{\mathrm{iso}}=2N'-1\ \ (\text{periodic cell})

Each of the $N'$ load-bearing disks has two force-balance equations. A packing that carries positive pressure needs one contact more than the unknowns it must fix (Goodrich, Liu and Nagel 2012).

Stress from contacts

\sigma_{ab}=\frac{1}{A}\sum_{\text{contacts}} b_a f_b,\qquad p=\tfrac12\left(\sigma_{xx}+\sigma_{yy}\right)

The branch vector b runs between centres, or from a wall to a centre. At force balance this internal estimate equals the force per length on each wall, which the Lab checks at every step.

Scaling above jamming

p\propto(\phi-\phi_c)^{\alpha-1},\qquad Z-Z_c\propto(\phi-\phi_c)^{1/2}

O’Hern et al. (2003) report these exponents for large periodic packings. With 128 disks the pressure law is reproduced; the contact exponent comes out steeper, a finite-size effect.

Common difficulties

Jammed means packed tight

Typical misconceptionA jammed packing has no free space left.

Better mental modelAt jamming about a fifth of the box is still empty, and a few disks still rattle in cages. What changes is that a connected network can balance every force.

Force is shared evenly

Typical misconceptionEvery disk carries about the same load.

Better mental modelForces are spread unevenly. At the peak of the default cycle about a quarter of the contacts carry less than half the mean force, and the strongest carry about three times the mean. The force histogram below the stage shows the spread.

Run the experiment

01
Watch the onset
Play Load through jamming and watch the box corner move in.
What to observe: The walls feel nothing until the first chains span the box. Then the pressure curve leaves zero and keeps rising.
02
Count the spare contacts
Choose Barely jammed and compare the contacts beyond isostatic with Deep squeeze.
What to observe: At the peak of Barely jammed only about ten contacts are spare and four disks rattle. At the peak of Deep squeeze about sixty are spare and no disk rattles.
03
Squeeze it yourself
Drag the corner handle of the moving walls inward, then back out.
What to observe: Your path appears in green on the pressure chart. Going back out, the load usually vanishes at a higher packing fraction than where it first appeared on the way in.