L3 · Chapter 6

What a packing remembers after it is squeezed and released

Sandbox Physics · 2026-10-29

Squeeze the box, release it back to the starting packing fraction, and the disks are no longer where they began. On the way out, the load vanishes at a higher packing fraction than where it appeared on the way in. Squeeze again and the load appears exactly where it vanished last time; after a few rounds many packings return to precisely the same arrangement every cycle. What the packing remembers is the path it was squeezed along.

The load appears and vanishes at different places

The Lab’s default cycle takes 128 disks from 0.775 to 0.86 and back in 42 steps of about 0.002. On compression the first loaded step is at a packing fraction of 0.789. On unloading there is still load at 0.805; it vanishes at the next step. We ran six seeds with the same protocol, and in every one the load vanished later on unloading: the first loaded step of the first cycle lies between 0.789 and 0.805, and the last loaded step of unloading between 0.805 and 0.813.

Chapter 4 said that a threshold belongs to one configuration. The random start is a poorly packed configuration, so it jams early. While squeezed, the disks rearrange into another configuration with a higher threshold. Unloading measures the threshold of that new configuration.

Source: C. S. O’Hern

Where the walls’ work goes

Each step has two halves. First the box and every centre are scaled, with no disk adjusting; the rise in energy is the work done by the walls. Then the packing relaxes and the energy falls; the model’s relaxation algorithm removes that part, standing in for damping in a real material.

A cycle that starts and ends unjammed stores zero energy at both ends, so the walls’ net work equals the cycle’s total relaxation loss. We checked this with a Simpson integral on a 128-disk cycle from 0.80 to 0.86: the walls’ net work was $1.48\times10^{-2}\,\varepsilon$ and differed from the loss by $-1.2\times10^{-8}\,\varepsilon$ .

The size of the loss depends on the step: smaller steps lose less to smooth elastic deformation in each step. Compare these numbers only at the same step size.

W_{\mathrm{walls}}=\sum_{\text{steps}}\Delta U_{\mathrm{affine}}=\Delta U_{\mathrm{stored}}+\sum_{\text{steps}}\Delta U_{\mathrm{relax}},\qquad W_{\mathrm{walls}}=-\oint p\,dA\;\;\text{over a closed cycle}

The right-hand form is the area enclosed by the loop in the pressure–area plane. Relaxation only lowers the energy, so the loss is non-negative, and so is the walls’ net work over a closed cycle.

Source: J. Guénolé

Squeeze a few times and the packing falls into a fixed cycle

Take one packing between 0.775 and 0.86 five times. For every seed, the load in the second cycle appears exactly at the step where it vanished during the first unloading.

In four of the six seeds, from cycles 2, 2, 3 and 4 respectively, the root-mean-square difference of disk positions at 0.86 from the previous cycle falls below a threshold of $10^{-6}$ small diameters set in advance; most differences are of order $10^{-13}$ , which is rounding error. The packing returns to the same arrangement every cycle. In the other two, the onset, the point where load vanishes and the peak pressure are already identical from cycle to cycle and the loss repeats to three significant figures, but positions in cycle 5 still moved $1.3\times10^{-4}$ and $5.2\times10^{-4}$ diameters from cycle 4. They had not stopped within five cycles.

Once the cycle repeats, the loss per cycle falls from $7.7\times10^{-3}$ – $2.0\times10^{-2}\,\varepsilon$ in the first cycle to $3.0\times10^{-3}$ – $4.9\times10^{-3}\,\varepsilon$ . Part of that loss comes from the finite step, and we have not separated it from the loss in rearrangements. This is the course’s own numerical result, valid for this protocol and these seeds.

Memory under shear: does it hold when the shear reverses?

Zhao and co-workers test shear jamming by the same kind of return trip. Fig. 2(d) shows one test at $\phi=0.781$ : shear forward to a largest strain $\gamma_{\max}$ , then reverse, recording the pressure. Fig. 2(e) plots the lowest pressure during the reversal against $\gamma_{\max}$ .

With too little forward shear, the pressure drops below the 0.3 N/m noise threshold as soon as the shear reverses: the force network remembers only its original direction, a fragile state. Beyond a minimum forward strain $\gamma_{SJ}$ , the pressure stays above the threshold throughout the reversal, and only that counts as shear-jammed. The minimum strain grows as the packing fraction falls. At 0.74 no shear-jammed state appears even after shearing to the steady state; at 0.75 the required strain is about the strain needed to reach the steady state. They fit a formula that diverges at the lower limit, note that the divergence is not clearly visible in their data, and place that limit at $\phi_{SJ}=0.745\pm0.005$ .

Source: Y. Zhao · D. Bi

Worked example: reading a record of repeated cycles

Here are the five cycles of seed 2732 (pressure in $\varepsilon/d^2$ , loss in $\varepsilon$ ). Onset and vanishing point here mean the first and last loaded steps. Cycle 1: onset 0.8054, load vanishes at 0.8094, peak pressure 0.01742, loss $8.98\times10^{-3}$ . Cycle 2: onset 0.8094, vanishes at 0.8114, peak pressure 0.01613, loss $9.74\times10^{-3}$ , peak arrangement shifted 0.173 diameters from the previous cycle. From cycle 3 on: onset and vanishing point both 0.8114, peak pressure 0.01613, loss $4.85\times10^{-3}$ , peak arrangement shifted by $5.5\times10^{-14}$ diameters.

Step one: does each onset equal the previous vanishing point? For cycles 2 and 3 it does, so the arrangement left by unloading is where the next cycle starts. Step two: the shift of the peak arrangement. Cycle 2 still moved 0.173 diameters, so disks were still changing places under compression; cycle 3 moved $10^{-14}$ , which is rounding error, so the arrangement repeats. Step three: the loss. Once repeated, each cycle loses $4.85\times10^{-3}\,\varepsilon$ , or $3.8\times10^{-5}\,\varepsilon$ per disk.

Conclusion: this packing entered a fixed cycle at cycle 3, judged by a shift of the peak arrangement below a threshold of one millionth of a diameter set in advance. For seed 2728 the onset, vanishing point and peak pressure are identical from cycle to cycle while the arrangement still drifts, so the test has to compare positions as well as pressures.

Try it in the Lab

Compare where the load appears and vanishes

Predict first: on unloading, does the load vanish at a higher or lower packing fraction than where it appeared on compression?
Play the default preset, read this cycle’s onset under “More numbers and how they are computed”, then find where the unloading curve on the pressure chart returns to zero. Compare the two.
Pause, then drag the box-corner handle inward past the onset and back out, twice. Watch the curve labelled “your path” on the pressure chart: where does the load appear the second time, compared with where it vanished the first time?

Exercises with explained answers

A cycle starts unjammed and ends unjammed. Why can the walls’ net work never be negative?
Show the answer
The stored energy is zero at both ends, so the walls’ net work equals the total relaxation loss. Relaxation only moves downhill in energy, so each step loses zero or more, and so does the sum.
For seed 2732, the last unloaded step of compression is at 0.8033 and the first loaded step at 0.8054. Where can the true onset be?
Show the answer
Between 0.8033 and 0.8054. To pin it down, redo that interval with a smaller step. The Lab’s onset extends the straight line through the pressures of this step and the next back to zero pressure, and never goes below the previous step.
In the reverse-shear test of Zhao et al., what do you see during reversal if the forward shear stopped below $\gamma_{SJ}$ ?
Show the answer
Somewhere during reversal the pressure drops below the 0.3 N/m noise threshold: the network built by forward shear carries only loads in the same direction and lets go when reversed. If forward shear had raised the pressure above the threshold, the paper calls this a fragile state; if forward shear never built up pressure at all, the packing is unjammed.

Sources and what we recomputed

C. S. O’Hern, L. E. Silbert, A. J. Liu and S. R. Nagel, Jamming at zero temperature and zero applied stress: the epitome of disorder, arXiv:cond-mat/0304421v1; Physical Review E 68, 011306 (2003)
Version 1, 17 April 2003. Eq. (1): the finite-range repulsion between soft spheres. Section II: random infinite-temperature starts, conjugate-gradient minimisation in a periodic cell, and the energy test for an unjammed state. Figs. 5 to 7 and Eq. (3): the fraction of jammed states, the distribution of thresholds and its width, which falls as the number of particles to the power −0.55. Table I: pressure and contact-number exponents and prefactors. Fig. 20: a 2D force network. We recomputed the threshold width, the pressure exponent and the contact-number exponent with our own code; the contact-number exponent did not reproduce at 128 disks.
J. Guénolé, W. G. Nöhring, A. Vaid, F. Houllé, Z. Xie, A. Prakash and E. Bitzek, Assessment and optimization of the fast inertial relaxation engine (FIRE) for energy minimization in atomistic simulations and its implementation in LAMMPS, arXiv:1908.02038v4 (2020)
Version 4, 30 January 2020. Algorithm 2: FIRE 2.0 with a semi-implicit Euler step and the half-step correction after an uphill move. Table 1: the default parameters we use. The relaxation method only; nothing about jamming.
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.