How two disks push on each other, and how a hundred share a load

Objects, units and directions first

The objects are disks in a plane, half with diameter $d$ and half with diameter $1.4\,d$ . Mixing two sizes stops them from settling into a neat crystal. Lengths are in units of $d$ , energies in units of the contact stiffness $\varepsilon$ , so forces are in $\varepsilon/d$ .

Take disks $i$ and $j$ with radii $R_i$ and $R_j$ and centre distance $r$ . They just touch when $r=\sigma_{ij}=R_i+R_j$ . Soft disks can press slightly into each other; call the overlap $\delta=\sigma_{ij}-r$ . There is a force only when $\delta>0$ .

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.

Source: C. S. O’Hern

More overlap, harder push

The contact energy used by O’Hern and co-workers is non-zero only while disks overlap. With exponent $\alpha=2$ it is a spring; with $\alpha=5/2$ it is a Hertzian contact. Elastic spheres in three dimensions behave that way; using it for disks in two dimensions is a modelling choice.

Minus the derivative with respect to $r$ gives a force along the line of centres that pushes the disks apart. For the harmonic case $f=\varepsilon\delta/\sigma_{ij}^2$ , with a constant stiffness $\varepsilon/\sigma_{ij}^2$ . For the Hertzian case the force grows as $\delta^{3/2}$ , so the contact stiffens as it is pressed.

There is no friction, so the contact force has only this one component along the line of centres. That stays true until friction enters in Chapter 5, and it fixes how many unknowns Chapter 2 has to count.

V(r)=\frac{\varepsilon}{\alpha}\left(1-\frac{r}{\sigma_{ij}}\right)^{\alpha},\qquad f=-\frac{dV}{dr}=\frac{\varepsilon}{\sigma_{ij}}\left(\frac{\delta}{\sigma_{ij}}\right)^{\alpha-1}\quad(\delta>0)

Eq. (1) of O’Hern et al.; Goodrich et al. use the same energy. For

\delta\le0

the energy and the force vanish: only disks that overlap interact.

Source: C. S. O’Hern · C. P. Goodrich

A wall acts like a mirror

The Lab’s box has four flat walls without friction. When a centre is closer to a wall than its radius, $h<R$ , the wall pushes as if the disk had met its own mirror image: the energy is $(\varepsilon/\alpha)(1-h/R)^{\alpha}$ , and the harmonic stiffness is $\varepsilon/R^2$ .

This wall law is our own choice. O’Hern and co-workers used periodic cells without walls; when we compare with their data in Chapter 4, we recompute in a periodic cell.

Worked example: two disks squeezed between walls

Put a small disk ( $R_1=0.5$ ) and a large one ( $R_2=0.7$ ) on one horizontal line between a left and a right wall. Their natural length is $2R_1+2R_2=2.4$ . Bring the walls to $2.39$ apart, so the contacts must absorb $\Delta=0.01$ of overlap in total. Use harmonic contacts.

First the model. Each disk feels only a left and a right force, and to balance they must be equal. So all three contacts carry the same force $f$ : springs in series. Their stiffnesses are $1/R_1^2=4$ , $1/\sigma^2=1/1.44$ and $1/R_2^2=1/0.49$ in units of $\varepsilon/d^2$ .

Then the numbers. The three overlaps add up to $\Delta$ , which gives $f=0.01/2.18=4.59\times10^{-3}\,\varepsilon/d$ . The overlaps are $1.15\times10^{-3}$ , $6.61\times10^{-3}$ and $2.25\times10^{-3}$ , which add back to $0.01$ . The disk–disk contact is the softest and takes about two thirds of the squeeze.

Finally a check. Our code relaxes the same configuration with FIRE and matches this closed form to a relative error of at most $2.2\times10^{-10}$ , for three overlaps and both contact laws.

\Delta=\delta_{1w}+\delta_{12}+\delta_{2w}=\frac{f}{\varepsilon}\left(R_1^2+\sigma^2+R_2^2\right)\quad\Longrightarrow\quad f=\frac{\varepsilon\,\Delta}{R_1^2+\sigma^2+R_2^2}

Compliances of springs in series add. This holds for harmonic contacts. A Hertzian force grows as the overlap to the power 3/2, so that case is solved numerically.

Source: C. S. O’Hern · J. Guénolé

A hundred disks: force follows a few paths

Pack 128 disks into the box and squeeze. Every contact follows the law above, yet the forces differ widely. At the tightest point of the Lab’s default cycle (packing fraction 0.86) there are 303 load-bearing contacts. About 25 % of them carry less than half the mean force, about 6 % carry more than twice the mean, and the strongest carries about 3.1 times the mean. Half of the total contact force runs through 29 % of the contacts.

Draw each contact as a line whose width is proportional to its force, and the strong contacts link into chains that cross the whole box. These are force chains. Fig. 20 of O’Hern et al. draws the force network of a 2D bidisperse packing the same way.

Experiments can see force chains directly. A photoelastic material changes how it transmits polarised light when it is stressed. Zhao and co-workers photograph photoelastic disks between crossed polarisers and read the pressure of each frame from how sharply the brightness varies across the image.

Source: C. S. O’Hern · Y. Zhao

Try it in the Lab

Find the force chains in a packing

Predict first: at the tightest point, will the force histogram cluster around the mean or trail a long tail?
Play the default preset and pause when the box is smallest. Look at the shape the thickest lines make on the stage, then at the force histogram below it.
Switch the contact law from harmonic to Hertzian and repeat. Compare the width of the two histograms, and say which quantity you used to compare them.

Compare with a one-dimensional chain

Open the one-dimensional Hertzian bead chain and watch an impact travel along its single path to the wall.
Return to the 2D packing and count the contacts of a typical load-bearing disk. Explain why force can take a detour in the 2D packing but not in the chain.

Exercises with explained answers

Two equal disks ( $R=0.5$ ) are squeezed between walls with total overlap $\Delta=0.01$ and harmonic contacts. Find the force and the three overlaps.
Show the answer
The compliances add to $0.25+1+0.25=1.5$ , so $f=0.01/1.5=6.67\times10^{-3}\,\varepsilon/d$ . Each wall overlap is $fR^2=1.67\times10^{-3}$ and the disk–disk overlap is $f\sigma^2=6.67\times10^{-3}$ , which add to 0.01.
For a Hertzian contact, by what factor does the force grow when the overlap doubles? And the stiffness, the derivative of force with respect to overlap? What about a harmonic contact?
Show the answer
For Hertz $f\propto\delta^{3/2}$ , so doubling the overlap multiplies the force by $2^{3/2}\approx2.83$ and the stiffness, $\propto\delta^{1/2}$ , by about 1.41. A harmonic contact doubles its force and keeps its stiffness.
Can a frictionless disk with only two contacts carry load? How many contacts does it need at least?
Show the answer
Frictionless contact forces point along the lines of centres, through the disk’s centre. Two such forces cancel only if they are equal and opposite, which means the two contact points sit at the two ends of one diameter. A generic arrangement cannot do that, and the slightest disturbance breaks the balance. Three contacts in different directions can balance a generic load, which is why the Lab calls a disk with fewer than three load-bearing contacts a rattler.
Using the worked example, find the elastic energy stored in the two-disk system in two different ways.
Show the answer
As one series spring: $U=f^2(R_1^2+\sigma^2+R_2^2)/(2\varepsilon)=2.29\times10^{-5}\,\varepsilon$ . Contact by contact, $k\delta^2/2$ gives $2.63\times10^{-6}$ , $1.52\times10^{-5}$ and $5.16\times10^{-6}$ , which also add to $2.29\times10^{-5}\,\varepsilon$ .

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.
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.
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.