Granular jamming: how loose grains become a solid

Two soft disks with radii

R_i

and

R_j

whose centres are a distance

r

apart. The overlap

\delta=R_i+R_j-r

is exaggerated; the arrows show the equal and opposite push along the line of centres. Original schematic.

From two disks to a jammed packing

The first chapter works out the force between two overlapping disks and shows how a packed box of them carries load along a few strong paths. The second counts how many contacts a packing needs before it can resist at all. The third turns contact forces into a pressure and checks it against the walls. The fourth finds the jamming point from random starts and derives how pressure grows past it. The fifth adds friction and asks how shear alone can jam a packing. The sixth squeezes and releases a packing and follows what it remembers. The seventh shears a jammed packing and tests whether its softest vibrations predict where it slips. The last chapter reads the 2003 paper by O’Hern and co-workers figure by figure.

You need Newton’s laws, Hooke’s law, plane vectors and reading a log–log plot. Each chapter introduces stress tensors, scaling laws and counting arguments when it first needs them. FR27

01 · L1
How two disks push on each other, and how a hundred share a load
Get the contact force from the overlap, solve two disks squeezed between walls, then see why the forces in 128 packed disks follow a few paths.
02 · L2
How many contacts a jammed packing needs
Count unknown forces against force-balance equations, derive the minimum of 2N′ − 1 contacts, and see why rattlers drop out and the mean sits near 4.
03 · L2
How to get the pressure from contact forces
Derive the 2D contact stress, show it equals the force per length on each wall and the energy derivative with area, then match all three in one example.
04 · L3
Where the jamming point is, and how pressure grows past it
Find each packing’s jamming threshold from random starts, watch the spread narrow with system size, then derive and test how pressure and contacts grow past it.
05 · L3
With friction, how shear jams looser grains
Add friction and the Coulomb limit, count why friction lowers the contacts needed to jam, then read how Zhao et al. tell fragile from shear-jammed states.
06 · L3
What a packing remembers after it is squeezed and released
Compare loading and unloading, find where the walls’ work goes, watch repeated squeezing settle into an exact repeat, and see how reversed shear tests jamming.
07 · L4
Predicting where a sheared packing slips next
Shear a jammed packing slowly and it slips in jumps. Derive why the softest mode goes soft first, then test how well it names the disks that move.
08 · L4
Finding Point J figure by figure in O’Hern et al. (2003)
Read the paper figure by figure: how states are made, how thresholds are found and extrapolated, which constants are fitted, which we reproduced.

Experiments along the way

01Squeeze a box of disksCompress 128 soft disks through the jamming point and back, and watch contacts, pressure and the force network appear and vanish.02Send a pulse down a chain of grainsGive the first bead in a line a kick and follow the pulse through Hertz contacts that only push, never pull.03Predict the next slipShear 64 soft disks slowly, watch the softest mode lose its stiffness before each slip, and score how often it named the disks that moved.04Jam loose grains by shearShear 128 frictional disks below the jamming density, watch force chains grow along the squeezed diagonal, then switch friction off or reverse the shear.

The compression experiment simulates frictionless soft disks between walls: every step squeezes the box a little and relaxes the disks until all forces balance. Its pressures and contact counts are simulated; the published fits it is compared with are listed separately in the sources. The shear experiment runs the same disks in a periodic cell; its hit rates are simulated, and the averages recorded on held-out packings are drawn as reference ticks. The shear-jamming experiment adds friction and rolling to the disks; its pressures come from the simulation alone, with no published curve drawn beside them.

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.
C. P. Goodrich, A. J. Liu and S. R. Nagel, Finite-size scaling at the jamming transition, arXiv:1204.4915v3; Physical Review Letters 109, 095704 (2012)
Version 3, 6 June 2012. Eq. (2): the contact count needed to remove every zero mode except global translations. Eq. (3): one more contact is needed for a positive bulk modulus, so a periodic packing of N disks in two dimensions needs at least 2N − 1 contacts. Eqs. (4) and (5): the corresponding minimum contact number. The finite-size crossover where the usual power laws break down at low pressure. We checked the 2N − 1 count on our own threshold packings.
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.
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.
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.
C. E. Maloney and A. Lemaître, Amorphous systems in athermal, quasistatic shear, arXiv:cond-mat/0510677v2; Physical Review E 74, 016118 (2006)
Version 2, 28 October 2005. Section IV: elastic branches end when the energy minimum merges with a saddle. Eqs. (9) and (10): motion along the critical mode grows as the square root of the remaining strain. Fig. 5(b): the lowest eigenvalue of the dynamical matrix falls as the square root of the strain left before the event. We checked the square-root approach on our own sheared packings.
M. L. Manning and A. J. Liu, Vibrational modes identify soft spots in a sheared disordered packing, arXiv:1012.4822v1; Physical Review Letters 107, 108302 (2011)
Version 1, 21 December 2010. 2500 Hertzian disks at packing fraction 0.95, sheared quasistatically. Fig. 2: soft spots built from the 20 particles that move most in each of the 30 lowest modes. Eq. (1): the overlap score between a soft spot and the next rearrangement; the example in Fig. 2 scores 0.64. Fig. 4(a): the score of the best spot against the strain left before the event. Text: soft spots have 6 % fewer contacts and 103 % more free volume, yet these local quantities alone did not find them. Our blind test uses smaller packings and its own thresholds.