FR27 · Quasistatic shear

Soft Spots: Predicting Where a Sheared Packing Slips

Can the softest vibrations of a jammed packing point to the disks that slip next? Often, but not always. As a jammed packing is sheared slowly, its softest vibration mode loses stiffness just before each sudden slip, and grains that move most in the softest modes are likelier to slip. For 64 soft disks, the six disks named by the two softest modes include on average 26% of the six that move most, against 9% by chance; in about one slip in four they name none.

Shear 64 soft disks slowly, catch each sudden slip, and score how often the softest vibration modes named the disks that moved, against chance.

Shear strain $0.00693$

Softest mode stiffness $0.0024\,\varepsilon/d^{2}$

Soft-mode hit rate $0.5$

Hit rate by chance $0.0938$

Shear the packing

Packing fraction

0.9

\phi

Soft modes used for the prediction

2

N_{\mathrm m}

Number of disks

64

N

Strain per second of playback

0.004

\Delta\gamma/\Delta t

Green rings mark the predicted soft spot before each slip; amber shows which disks actually moved. Drag the handle on the top plate to shear by hand.

Random start

Seed of the random start

24

Model, units and sources

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

Each step shears the cell by $\Delta\gamma=2\times10^{-4}$ and relaxes it with FIRE until the largest net force is below $10^{-11}\,\varepsilon/d$ . A step is reversible when shearing back returns every load-bearing disk within $10^{-3}\,d$ ; otherwise bisection brackets the event to $10^{-9}$ in strain.

The stiffness readout is the lowest eigenvalue of the dynamical matrix of the load-bearing disks, after the two rigid translations. The soft-spot score of a disk is its summed squared displacement in the $N_{\mathrm m}$ softest modes; the six highest scores are the prediction.

Manning & Liu (2011), Fig. 2, Eq. 1 · Maloney & Lemaître (2006), Sec. IV, Fig. 5 · Guénolé et al. (2020), FIRE 2.0

Work through why the softest mode warns of a slip

Model assumptions

Frictionless two-dimensional disks with Hertzian contacts, half of diameter 1 and half of diameter 1.4, fill a periodic sheared cell. After every small strain step the packing relaxes to force balance, so the shear is slow, without inertia, heat or friction. A step that does not come back when sheared back is a rearrangement; bisection then narrows down the strain where it starts. Predictions are read at the start of each smooth stretch of loading, and the six disks that jump furthest define the rearrangement. Sixty-four disks are far fewer than the 2500 used by Manning and Liu (2011), so the hit rates here are a rescoring with our own protocol, not a reproduction.

Stress builds, then drops at each rearrangement

The softest mode loses its stiffness as the slip nears

How often each guess named the disks that moved

More numbers and how they are computed

Rearrangements located: $0$
Shear stress now: $0$
Hit rate of the fewest-contacts guess: $0$
Hit rate of the lowest-pressure guess: $0$
Strain of the latest rearrangement: $0$
Stress drop in the latest rearrangement: $0$
Largest jump in the latest rearrangement: $0$
Stiffness exponent close to the latest event: $0$
Overlap of the critical mode with the movers: $0$
Load-bearing disks: $0$
Energy minimisations so far: $0$
Seed: $0$

A hit fraction is the share of the six predicted disks that are among the six disks with the largest non-affine jump. Chance is the expected share for six disks drawn at random from the load-bearing ones, about $0.094$ . The stiffness exponent is the slope of $\ln\lambda_{\min}$ against $\ln(\gamma_c-\gamma)$ between $3\times10^{-8}$ and $3\times10^{-7}$ ; a saddle-node gives $1/2$ .

Physics tutorial

Listening for the next slip

BackgroundShear a jammed packing slowly and it stores stress smoothly for a while, then a few grains suddenly slip and the stress drops. Between slips the packing sits in one energy minimum that the shear slowly deforms.

Why it mattersGlasses, foams and sand all yield through such local slips. If the weak places can be found before they fail, the failure of a disordered solid becomes partly predictable. The softest vibration modes are a candidate: they are computed from the packing alone, before anything moves.

Start with the essentials

Focus question: Can the softest vibrations of a jammed packing point to the disks that slip next?
One-sentence intuition: A slip happens when the energy minimum the packing sits in disappears. Just before that, one vibration mode loses its stiffness, so the disks that move most in the softest modes are the likeliest to slip. In these small packings the prediction is nearly three times better than chance, and it still names none of the movers in about one slip in four.

Core mathematical model

Stiffness of each mode

H_{i\alpha,j\beta}=\frac{\partial^{2}U}{\partial r_{i\alpha}\,\partial r_{j\beta}},\qquad H\,\mathbf e_{m}=\lambda_{m}\,\mathbf e_{m}

The dynamical matrix of the load-bearing disks (unit masses). Each eigenvalue $\lambda_m$ is the stiffness of one collective motion $\mathbf e_m$ ; the two rigid translations of the periodic cell have zero stiffness and are left out.

Approach to a slip

\lambda_{\min}\propto(\gamma_c-\gamma)^{1/2}

When a minimum merges with a saddle as the strain approaches the critical value, the softest stiffness vanishes as a square root (Maloney and Lemaître 2006). The Lab samples it from a millionth down to a hundred-millionth of strain before each slip.

Soft-spot score

P_i=\sum_{m=1}^{N_{\mathrm m}}\left|\mathbf e_{m,i}\right|^{2}

How much disk i moves in the softest modes, after Manning and Liu (2011). The six disks with the highest score are the prediction, read when the smooth stretch of loading begins.

Scoring a prediction

h=\frac{\left|\mathcal P\cap\mathcal R\right|}{k},\qquad \langle h\rangle_{\text{chance}}=\frac{\left|\mathcal R\right|}{N'}

The share of the $k=6$ predicted disks that are among the six with the largest jump. A random guess among the $N'$ load-bearing disks scores about $0.094$ on average.

Common difficulties

The weakest disk has the fewest contacts

Typical misconceptionDisks with few contacts or low pressure are the ones that will slip.

Better mental modelThose simple local guesses do better than chance, but on 192 test slips the two softest modes named 26% of the movers, against 18% for the fewest-contacts guess and 15% for the lowest-pressure guess. A slip is a collective motion, and the modes see the whole network.

A slip comes without warning

Typical misconceptionNothing measurable changes until the grains move.

Better mental modelWith Hertzian contacts the softest stiffness falls by more than half over the last ten-thousandth of strain in almost every slip. With spring-like contacts it often stays flat and collapses only when a contact opens; the Spring contacts preset shows that case.

Run the experiment

01
Watch one slip
Open Soft spot gives way and watch the first rearrangement.
What to observe: Three of the six green-ringed disks get amber rings: half of the prediction moved. The stress chart shows the drop, and the stiffness chart shows the softest mode falling with slope close to one half as the slip nears.
02
Keep score
Let it run for a dozen slips and watch the hit-rate bars.
What to observe: The soft-mode bar settles well above the chance bar, near the white tick recorded on the test seeds, while single slips range from no hits to most of the six.
03
Compare contact laws
Choose Spring contacts give no warning and look at the stiffness chart before each slip.
What to observe: The softest stiffness stays nearly flat until the slip, because a harmonic contact keeps its full stiffness until it opens.