The heap that becomes a solid

A material that flows and a material that does not are usually different substances. Sand is both, and the change between them is startlingly small: pour it and it flows like a liquid, tap the container a few times and press, and it holds a load without deforming at all.

The density changes by a per cent or two across that transition. The grains do not change at all. What changes is a count, and the number it has to cross is fixed by the dimension of space.

The count #

Each grain in the plane has two coordinates. A pack of $N$ grains therefore has $2N$ numbers describing where everything is, and any motion of the pack is a choice of $2N$ small changes to them.

Each contact between two grains forbids one motion: they may not approach or separate along the line joining their centres. A contact is therefore one equation in those $2N$ unknowns. With $N$ grains each touching $Z$ others, the number of contacts is $NZ/2$ — halved because each is shared.

So the free motions number $2N - NZ/2$, and they run out when $Z$ reaches 4. In three dimensions the same argument gives 6. Twice the dimension, in both cases, and this is Maxwell’s condition; a network sitting exactly at it is called isostatic.

It is worth noticing what that number does not depend on. It contains no material property: not the stiffness of the grains, not their friction, not their size, not the strength of whatever holds them together. It contains no load either. The threshold is a statement about a network of constraints, and the same statement holds for a pack of glass beads, a pack of steel balls and a pile of gravel — which is why the angle a heap of any material settles at is so nearly the same for materials with nothing else in common.

The word “free” is doing exact work here. A free motion is one that changes no contact separation, so no grain has to be compressed and no energy is required. A pack with even one of them is not a solid: apply any load with a component along it and the pack rearranges rather than resisting, however hard the grains are.

Counting is not measuring #

The count above is an average, and averages can hide things. A pack could sit at $Z = 4$ overall with one region over-braced and another still floppy, and the count would not notice.

So none of the numbers in this essay comes from the count. Each is the rank of the pack’s rigidity matrix — one row per contact, holding the direction along which that contact forbids relative motion — obtained by elimination on the actual matrix. The free motions are the number of coordinates minus that rank, which is a statement about the arrangement in front of it rather than about a typical arrangement.

That distinction is the same one a four-legged table forces, and the machinery is identical: three equations and four unknowns there, $2N$ coordinates and $NZ/2$ constraints here.

The check that the matrix is the right matrix is worth stating, because a sign error in a contact vector produces a matrix of full rank and a figure reporting a rigid pack at every coordination. Translating or turning the whole pack changes no contact separation, so those three motions must be in the null space whatever the contacts are — and they are, to a part in a thousand billion, every time the figure is drawn.

A mechanism, found rather than designed #

The most direct way to see what “free motion” means is to draw one.

An arbitrary push was applied to the grains, and then every component the contacts forbid was removed from it, along with the three rigid-body motions. What remains stretches no contact — to eight parts in a hundred million billion — so the grains slide past one another without any of them being compressed.

That is what it means for the motion to cost nothing. There is no barrier to overcome, no elastic energy stored, no threshold force. A pack with a mechanism yields to an infinitesimal load in the direction of the mechanism, which is the operational definition of a fluid.

The arrows are largest where the network is thinnest, which is the sense in which a granular material fails at its weakest connection rather than at its weakest grain. It is also why the strength of a pack is so much more variable than the strength of the material it is made of.

Above the threshold, and what is still not determined #

Crossing the threshold makes the pack rigid. It does not make the forces in it knowable.

A contact is redundant when its row of the rigidity matrix is already spanned by the others: removing it does not add a free motion, and no balance of forces or moments determines its force. Above the isostatic point the redundancy grows, and in a fully connected triangular patch of 37 grains it reaches 19 contacts.

That is the arithmetic behind a fact every silo designer knows and no elementary statics course explains. Fill two identical silos identically and the chains of heavily loaded grains run in different places. The load sharing depends on how the pack was built — on the packing history, on tiny differences in grain shape, on whether the container was tapped — exactly as the load sharing between four table legs depends on which one is a hundredth of a millimetre short.

The silo’s walls carrying most of the weight is the same indeterminacy seen from outside, and the enormous variability of measured wall pressures — enough that silo codes carry safety factors of several — is the redundancy showing itself in practice.

What the transition is not #

Two natural descriptions of the same phenomenon are wrong, and both are worth stating because both are widely used.

Everything about a pack quoted as a property of the material — a packing fraction, a wall pressure, a yield stress — is really a property of how the pack was made. Pour it, tap it, or shear it and the same grains arrive at different states, all of them rigid and none of them the same. That is what the transition is not: it is not a phase transition to a unique state, and there is no equilibrium argument that picks one arrangement out of the many available.

It is not a density threshold. Random close packing of equal spheres sits near a fraction of 0.64, and that number is quoted so often that it is easy to take for the criterion. It is not: the criterion is the contact count, and density predicts it only for packings prepared in one particular way. Vibrate a pack and it gains contacts at nearly constant density; that is why tapping a jar of sand makes it firm rather than smaller.

And it is not a phase transition in the thermodynamic sense. There is no temperature here worth the name — the grains are far too large for thermal motion to move them, and the pack does not explore configurations on its own. What happens at the threshold is a geometric and mechanical change in a system with no thermodynamics at all, and borrowing the vocabulary of phase transitions for it has been productive and occasionally misleading.

The friction that changes the number #

The count above assumes each contact forbids exactly one motion, and that is true only if the grains are frictionless.

A frictional contact resists sliding as well as approach, so each one removes more freedoms than a frictionless contact does — and the number of contacts a pack needs in order to be rigid falls accordingly. That is why the isostatic count depends on whether friction is present: frictionless spheres need six contacts each and frictional ones need four, and real sand sits between the two because its friction is neither infinite nor absent.

A frictional contact also resists sliding, and it can transmit a torque. Counting those extra constraints — and the extra rotational freedom each grain then has — gives an isostatic number of 3 in the plane rather than 4, and 4 in three dimensions rather than 6. So a rough pack jams at a lower coordination than a smooth one, which is why sand piles up and ball bearings do not.

Real packs sit between the two limits, because friction is neither absent nor infinite: a contact is only sliding-constrained while the tangential force is below the friction limit, and near the threshold some contacts are at that limit and some are not. The effective isostatic number is therefore not a constant of the material but depends on the loading, which is one reason the transition in a real granular material is broad where the idealised one is sharp.

The angle a heap settles at is the same friction seen at the moment the pack gives way, and a paste with a yield stress is what the same threshold looks like when the grains are small enough for surface forces to hold them together.

What it feels like from outside #

The count is invisible from outside the pack, and what is visible instead is a set of behaviours that look like several different things and are one.

A pack above the jamming threshold behaves like a material with a finite intercept: nothing flows until the stress reaches a yield value, and above it the resistance rises with the rate. That is what the transition feels like from outside — not a change in stiffness but the appearance of a yield stress where there was none, which is the mechanical signature of the count having been crossed.

A pack below the threshold has no yield stress: any shear, however small, produces flow. A pack above it has one, because a mechanism has to be created before anything can move and creating one costs energy. Measured on a rheometer, that is the difference between a liquid and a plastic solid, and it is the same difference the rank computation reports.

The sound speed does the same thing. A pack with free motions has soft modes at zero frequency, so a compressional disturbance is carried by whichever chains happen to be loaded and the speed measured is low, erratic and history-dependent. Above the threshold the pack has a genuine elastic modulus and a well-defined speed — though it still rises with the confining pressure in a way no ordinary solid’s does, because squeezing the pack adds contacts.

And the memory is the most peculiar consequence. Because the redundancy leaves the force network undetermined, a pack remembers how it was made: a column pressed from above and a column pressed from the side, at the same density and the same coordination, respond differently to the next load. There is nothing in the grains that stores this. It is stored in which of the available force networks the pack settled into, and it is erased by any rearrangement large enough to remake the contacts.

Why the same count keeps appearing #

Constraint counting is not a granular technique. It is a general statement about when a set of equations pins a set of unknowns, and it turns up wherever a network has to be rigid.

A glass network obeys it: the rigidity of a chalcogenide glass depends on the average number of bonds per atom, and the compositions where that average crosses the isostatic value are anomalously stable — which is a design rule in the glass industry rather than a curiosity.

A protein obeys it: rigidity analysis of a folded structure, counting covalent bonds and hydrogen bonds as constraints, identifies which regions are rigid and which are hinges, and it does so from the count rather than from a simulation.

A folded sheet obeys it, and the counting is the whole of the design method: a crease pattern is rigid or has a mechanism according to how the constraints at each vertex compare with the freedoms, which is why some patterns fold flat along exactly one path and others are floppy.

A truss obeys it, and the engineering vocabulary is older than the physics: a determinate truss is isostatic, a mechanism is under-braced, and a redundant one is over-braced. The vocabulary was invented for bridges and applies unchanged to sand.

The rule and its correction #

The counting rule was Maxwell’s, from an 1864 paper on the stiffness of frames, and it was corrected a hundred and fourteen years later in a way that explains exactly why this essay computes ranks instead of counting contacts.

Maxwell’s statement is the one used above: a frame is just rigid when the number of bars equals the number of joint coordinates minus the rigid-body motions. Too few bars and it has a mechanism; too many and it is redundant.

Calladine pointed out in 1978 that the count does not give either quantity. It gives their difference:

$$m - s = dN - \tfrac{1}{2}d(d+1) - c,$$

where $m$ is the number of independent mechanisms and $s$ the number of independent states of self-stress — sets of bar tensions that balance at every joint with no external load applied.

That single change matters because a structure can have both at once. A frame satisfying Maxwell’s just-rigid count exactly can have one mechanism and one state of self-stress, and the count reports zero because they cancel. No amount of care with the arithmetic finds them; only the rank of the matrix does, and the rank of its transpose gives the other.

The two figures in this essay are therefore the two halves of that theorem, computed separately. The free motions are $m$, obtained from the null space of the rigidity matrix; the redundant contacts are $s$, obtained from the null space of its transpose. Their difference is what Maxwell’s count would have predicted, and their individual values are what the count cannot supply.

There is a mechanical consequence and it is the more useful half. A state of self-stress can rigidify a mechanism. If a structure has a motion that changes no bar length to first order but would stretch a bar at second order, then pretensioning that bar makes the motion cost energy, and the structure becomes rigid despite being a mechanism by count.

Every prestressed structure works this way. A bicycle wheel is a mechanism if its spokes are slack and a stiff wheel if they are tensioned; the spokes never take compression, and the wheel’s rigidity comes entirely from the state of self-stress. So does a tensegrity sculpture, a cable-stayed roof, a tent, a drum skin and a spider’s web — all of them structures that counting says should collapse and that stand because they are pulled tight against themselves.

The gripper made of coffee #

Prestress is what turns the last section into an everyday demonstration, and there is one in most kitchens.

A brick of vacuum-packed ground coffee is rigid. It can be dropped on a floor without deforming, and it rings when it is tapped. The grains inside it are the same powder they were before packing, at very nearly the same density, and nothing has been added.

What has been added is an atmosphere of external pressure squeezing the whole pack. That does two things at once, and both are on this page. It pushes grains into contact that were not touching, raising the coordination past the threshold; and it puts the resulting network into a state of self-stress, so that the motions which would otherwise cost nothing now work against a pretension. The brick is rigid for exactly the reason a bicycle wheel is.

Cut the bag open and the brick becomes a powder within a second, with no rearrangement worth the name — the pressure is released, the contacts open, and the mechanisms return.

The effect has been made into a robot’s hand. Fill a latex membrane with ground coffee, press it onto an object of any shape so that the loose grains flow around the contours, and then evacuate the membrane. The pack jams, the shape is frozen, and the resulting rigid mould grips the object well enough to lift it — with no fingers, no joints, no sensors and no knowledge of what the object is.

A universal gripper built on that principle was demonstrated in 2010 and picks up objects it has never encountered without any programming at all, which is a thing no articulated hand does easily. Three effects contribute: the geometric interlocking of the frozen shape, friction against the surface, and — where the membrane seals against a smooth object — outright suction.

What makes it worth including here is that the design variable is the one this essay is about. The grip strength is set by how far past the jamming threshold the vacuum drives the pack, and the release is instantaneous because crossing back below the threshold takes no rearrangement at all.

What the pictures cannot show #

The grains are discs and the lattice is regular. A real pack is disordered, its grains are not round, and its contacts are not all at the same distance. What survives that generalisation is the count and the rank; what does not is the crispness of the threshold, which in a disordered pack is smeared by finite size and by the distribution of contact numbers.

The patch is 37 grains. That is small enough that a third of them are on the boundary with fewer neighbours than they would have in a bulk pack, which is why the measured threshold sits at 4.11 rather than at 4. The direction of that correction is understood and its size is a statement about the sample.

Nothing here is loaded. The rigidity matrix says which motions cost energy; it says nothing about how much, which needs the contact stiffnesses, nor about which contacts exist under load, which needs the deformations. A pack near the threshold has a shear modulus that vanishes as the excess coordination, and that scaling is a different measurement from this one.

And the mechanism drawn is a snapshot. Moving along it breaks contacts and makes new ones, so it describes the configuration in front of it rather than a trajectory. What a pack actually does under load is a sequence of such rearrangements, and following that sequence is a simulation rather than a rank computation.

The ladder from here #

Later rungs on this anchor: the shear modulus’s vanishing at the jamming point and the exponent it vanishes with; the distribution of contact forces, which is exponential at large force and has a puzzling excess at small; the difference between jamming by compression and jamming by shear; and dilatancy, which is the requirement that a dense pack expand before it can rearrange, and which is why wet sand goes dry underfoot.

The neighbouring ladders are the silo that does not weigh what it holds, where the indeterminacy is measured on a scale, the table statics cannot settle, which is the same count with four unknowns instead of seventy-four, and the paste that holds up its own hill, where the threshold is crossed by surface forces rather than by contacts.