---
title: "Jamming Transition: Squeezing Soft Disks Interactive Lab | Sandbox Physics"
description: "At what packing fraction does a box of loose disks suddenly push back?"
canonical_url: "https://sandboxphysics.com/labs/disk-jamming-compression/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/disk-jamming-compression/"
last_modified: "2026-10-30"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

**Background** Loose 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 matters** Sand, 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 misconception** A jammed packing has no free space left.

**Better mental model** At 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 misconception** Every disk carries about the same load.

**Better mental model** Forces 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

1. 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.

2. 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.

3. 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.

Physical conclusion

A disordered pile carries load once its contacts can balance every force, and that happens all at once.

## Related experiments and articles

Granular jamming · Reading path

### From two disks to the jamming point and its memory

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How two disks push on each other, and how a hundred share a load**](https://sandboxphysics.com/stories/granular-jamming/contacts-and-force-chains/)
2. [Lab **Granular Impact Chain Designer** Compare with a one-dimensional chain](https://sandboxphysics.com/labs/granular-impact-chain/)
3. [Article **How many contacts a jammed packing needs**](https://sandboxphysics.com/stories/granular-jamming/counting-contacts/)
4. [Article **How to get the pressure from contact forces**](https://sandboxphysics.com/stories/granular-jamming/stress-from-contacts/)
5. [Article **Where the jamming point is, and how pressure grows past it**](https://sandboxphysics.com/stories/granular-jamming/the-jamming-point/)
6. [Article **With friction, how shear jams looser grains**](https://sandboxphysics.com/stories/granular-jamming/friction-and-shear-jamming/)
7. [Lab **Shear Jamming: Frictional Grains That Lock Without Squeezing** Make grains carry load by shear alone](https://sandboxphysics.com/labs/frictional-shear-jamming/)
8. [Article **What a packing remembers after it is squeezed and released**](https://sandboxphysics.com/stories/granular-jamming/memory-and-reversal/)
9. [Article **Finding Point J figure by figure in O’Hern et al. (2003)**](https://sandboxphysics.com/stories/granular-jamming/reading-ohern-2003/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)