The angle that does not know the size of the heap

A silo does not weigh what it holds, because its walls take the load through friction. This essay is about the same material with no walls at all, where the only thing holding the surface up is the rest of the heap.

The cancellation that makes it an angle #

Take a slab of grains of thickness $h$ lying on a slope of angle $\theta$, and ask what holds it there.

The component of its weight along the slope is $\rho g h \sin\theta$ per unit area, and it pulls the slab downhill. The component pressing it into the slope is $\rho g h \cos\theta$, and friction can supply up to $\mu$ times that. Failure begins when

$$\rho g h \sin\theta > \mu\,\rho g h \cos\theta \quad\Longrightarrow\quad \tan\theta > \mu$$

and every quantity except the friction has cancelled. The thickness of the slab is gone; so are the density, the strength of gravity and the size of the grains.

The cancellation that makes it an angle is the one worth following. Resolve the forces on a block on a slope, along it and across it, and both components carry the same factor of the weight. The weight therefore cancels out of the comparison entirely, and what is left is two dimensionless numbers set against each other: $\tan\theta$ and $\mu$. That is why the answer is an angle rather than a size, and why a grain of sand and a boulder give the same one.

That is why the answer is quoted as an angle. It also predicts things that sound surprising and are true: a heap of the same sand on the Moon has the same slope, a heap a hundred metres high has the same slope as one ten centimetres high, and grinding the grains finer does not change it — provided nothing else changes with the size, which is the qualification the rest of this essay is about.

The same criterion applies to a single block. Tilt the slope and nothing happens until the tangent of the angle reaches the coefficient of friction, at which point it goes. The angle of repose is that experiment performed by a material on itself, with the grains beneath a surface layer playing the part of the incline — which is why the number belongs to the material rather than to any apparatus.

Why there are two angles #

A single number would be tidier than what is observed. Tip a box of sand slowly and it starts to avalanche at one angle; let the avalanche run and it stops at a slope two to five degrees shallower.

The reason is that the friction which resists starting is not the friction which resists continuing. Static friction takes what it needs and can exceed the kinetic value, so the criterion has two forms: $\tan\theta_{\text{start}} = \mu_s$ and $\tan\theta_{\text{stop}} = \mu_k$. Between them lies a band of slopes that are stable if the heap is at rest and unstable if it is moving.

There are two angles because static friction exceeds kinetic friction. A surface sticks, loads up, slips suddenly, and sticks again; a sandpile relaxes the same way, in discrete avalanches rather than by continuous creep. The pile therefore has an angle at which it starts to move and a lower one at which it stops, and the gap between them is the same gap between the two coefficients. The size distribution of those avalanches is what makes a sandpile a standard example of intermittency.

That band is the reason a sandpile is a hysteretic object rather than one with an equilibrium shape. Two piles of the same sand can stand at different slopes and both be stable, and which one a given pile has depends on its history: built by pouring, it sits near the steeper angle; built by draining from below, it sits near the shallower.

Measuring it, and why the number depends on how #

Because the stable slope is a band rather than a value, a measured angle of repose is a measurement of the method as much as of the material — and the standard methods do not agree.

Pouring sand onto a flat surface gives the poured angle, which sits near the steeper edge, because each grain arrives on a slope already at rest and only the last few take part in any avalanche. Draining sand out of a hole in a plate leaves a crater whose walls sit near the shallower edge, because that surface has just finished flowing. Tilting a box until the surface moves gives the starting angle directly. Rotating a half-filled drum gives both: the surface steepens until it avalanches and settles back, and the two extremes are read off the same trace.

The practical consequence is that a quoted angle should always carry its method, and that comparisons across sources are worth less than they look. It is also why engineering standards specify the apparatus in detail: what is being standardised is which edge of the hysteresis band is being reported.

Where the angle turns up #

In every dune and every scree slope. A dune’s lee face stands at the angle of repose and its windward face does not, which is what makes a dune asymmetric and what lets a photograph of one be read for the wind direction. Scree below a cliff sits at the same angle for the same reason, and the material at the bottom is coarser because the larger fragments roll further.

In mine tailings and grain stores, where getting it wrong is expensive. A tailings dam is a heap with water in it, and the cohesion it seems to have can vanish in seconds if the packing is disturbed — the dilatancy argument run backwards, with a loose packing contracting rather than expanding and the pore water taking the load. That is liquefaction, and it is the mechanism behind several of the worst industrial failures on record.

And on other planets, where it is used as a remote measurement. The slopes of Martian dunes and crater walls are measured from orbit and read as angles of repose, which — because the criterion contains no gravity — can be compared directly with terrestrial sand. A slope steeper than the terrestrial angle is evidence of cohesion, and a slope shallower is evidence of recent flow.

What cohesion buys, and what it does not #

Damp sand can be moulded into a wall and dry sand cannot. The obvious reading is that cohesion raises the angle of repose. It does — but the amount it raises it by depends on the size of the pile, and that is the whole story.

Adding a cohesion $c$ changes the criterion to $\rho g h \sin\theta = c + \mu \rho g h\cos\theta$, and the new term is the only one without an $h$ in it. Dividing through, the extra angle it buys goes as $c/\rho g h$ — so the combination $c/\rho g$ is a length, and it is the height below which cohesion dominates and above which it is irrelevant.

For damp sand that length is a few centimetres. A sandcastle is therefore not a demonstration that damp sand is strong; it is a demonstration that a bucket is small.

What cohesion buys is a stress, and it is small. Each pair of touching grains in damp sand holds a little bridge of water whose curved surface pulls them together, with a pressure inside set by the curvature — so the total is of the order of the surface tension divided by the grain size. For sand that is a few hundred pascals, which is enough to hold a vertical face a few centimetres high and nothing more.

The origin of that stress is worth following through, because it connects two subjects. Between two touching grains sits a bridge of water, curved concave outwards, and the pressure inside it is below atmospheric — the same negative pressure that pulls water up a capillary. Each bridge pulls its two grains together with a force of order $\gamma d$, and dividing by the area per contact gives a stress of order $\gamma/d$: about 70 Pa for millimetre sand and 700 for tenth-millimetre sand.

That prediction has two testable consequences and both hold. A finer sand is more cohesive, which is why a sandcastle is built from beach sand and not from gravel. And the cohesion vanishes both when the sand dries — no bridges — and when it is flooded, because a fully saturated packing has no curved surfaces left.

That is also why the effect has a grain size beyond which it disappears. The cohesive stress goes as the inverse of the grain size while the weight to be held goes as its cube, so gravel does not build sandcastles at any moisture content. And too much water is as bad as too little: fill the pores completely and the curved surfaces are gone, along with the pressure that came from them.

The room a packing has to find #

There is a second reason a dense heap resists more than its grains’ friction suggests, and it is geometric rather than frictional.

A dense packing cannot shear without expanding. The grains sit in each other’s hollows, and moving one layer past another means lifting it — so the material must find extra volume before it can move at all. Reynolds named this dilatancy in 1885 and it has consequences out of proportion to its simplicity.

The size of the effect is set by geometry alone. A layer riding over the hollows of the one below rises by $1-\cos\alpha$ of a diameter, where $\alpha$ is how deeply it is nested, and the slope it climbs adds directly to the angle the material appears to have. Nothing about the grains’ material enters, which is why two packings of identical grains — one poured loose, one vibrated dense — have measurably different strengths.

It explains why the angle of repose of a dense sand exceeds the friction angle of its grains: part of the resistance is not rubbing but geometry, and it disappears when the packing is loose. It explains why the same sand poured loose and vibrated dense behaves differently. And it explains the everyday observation that wet sand goes pale and dry under a footstep: the pressure forces the packing to expand, the pore space grows faster than water can flow in to fill it, and the surface is momentarily short of water.

Where the boundary between solid and fluid is #

A heap of sand is a solid below its critical slope and a fluid above it, and it changes over at a stress rather than at a temperature. That is exactly the definition of a yield stress, and the granular case is the extreme member of a family.

Where the boundary between solid and fluid sits is not a sharp line, and granular materials are the standard case in which it is not. Below the yield condition the heap supports a shear stress indefinitely, which is what a solid does; above it the heap flows, and while flowing its resistance depends on how fast it is being sheared, which is what a fluid does. The same material does both, and the transition is set by a stress rather than by a temperature.

The difference between sand and toothpaste is where the threshold comes from. Toothpaste has a yield stress that does not depend on how hard it is pressed; sand’s threshold is proportional to the pressure on it, because friction is. That is why a granular material’s strength is quoted as an angle and a paste’s as a stress, and why a deep pile of sand is not stronger in proportion to its depth — it is stronger exactly in proportion, which is the same statement as the slope being fixed.

A flowing layer also has a profile, and it is not the one a liquid would have. The moving layer at the surface of an avalanche is a few grain diameters deep whatever the size of the pile, and beneath it the material is stationary — so the velocity does not vary smoothly from top to bottom as it would in a sheared liquid. That shallow moving skin is why the angle of repose is a property of a surface rather than of a bulk.

The heap that is not a continuum #

One assumption has been made throughout and it is worth putting under strain, because it is the assumption that separates granular physics from the rest of this site’s fluids.

Every calculation above treats the heap as a material with a density and a friction — a continuum with properties at each point. A heap of sand is not that. The load travels through a sparse network of heavily loaded contacts, most grains carry almost nothing, and the network reorganises whenever anything moves. The pressure in a fluid depends only on depth because a fluid has no contacts to organise; a granular column has both, which is why its pressure saturates with depth and its heap has an angle.

What saves the continuum treatment is averaging over many grains. The slope criterion involves only the mean stress on a surface many grains across, and that mean is well behaved even when the individual contact forces are wildly distributed. So the angle of repose is reproducible to a degree while the force on any given grain is not reproducible at all — and the two facts sit together for the same reason that a gas has a steady pressure made of individual impacts.

The place the continuum picture fails outright is at the scale of a few grains, and it fails in the direction of more strength: a heap only a few grains high can stand steeper than its angle of repose, because a single grain has nowhere to roll to. That is the granular version of a finite-size effect, and it is why laboratory measurements specify a minimum pile size.

The rate at which a heap empties #

The slope criterion says when a heap will move. It says nothing about how fast, and the answer to that question is one of the few genuinely counter-intuitive results in the subject.

Open a hole of diameter $D$ in the bottom of a container of sand and measure the mass flowing out per second. For a liquid, the answer would depend on the depth above the hole — the pressure drives the flow, and a full tank empties faster than a nearly empty one. For a granular material it does not. The rate is constant from full to nearly empty, and it depends on the hole:

$$W = C\rho\sqrt{g}\,(D - kd)^{5/2},$$

with $d$ the grain diameter and $C$ and $k$ near 0.58 and 1.4 for most materials.

The independence from depth is the same fact the neighbouring essay derives for the pressure: the walls carry the weight through friction, so the pressure near the bottom of a deep container has stopped growing and the grains at the orifice have no idea how much is above them.

The exponent follows from a simple picture and is worth deriving because the picture is checkable. Grains arch over the hole and are released from a region roughly one hole-diameter across; having been released, they fall freely through about that distance, so they arrive at a speed of order $\sqrt{gD}$. Multiply that by the area $D^2$ and by the density and the result is $\rho\sqrt{g}\,D^{5/2}$. The correction $kd$ is an empty annulus a grain or so wide around the rim, where a grain’s centre cannot go.

Two consequences are worth having. An hourglass keeps time because of this — a fluid clepsydra runs fast when full and slow when empty and needs a shaped vessel to compensate, while a sand glass needs nothing. And a hole below about six grain diameters does not flow at all: the grains arch permanently rather than transiently, and the hopper jams. That is why a silo discharge is sized in grain diameters rather than in tonnes per hour, and why the standard remedy for a blocked hopper is a vibrator rather than a bigger pump.

The avalanches that run too far #

The criterion at the top of this page has a testable consequence for how far a slide travels, and for large slides the consequence is wrong by a factor of five.

A mass sliding down a slope with friction coefficient $\mu$ and dropping a height $H$ stops when the work done against friction equals the energy released, which puts its total horizontal travel at $L = H/\mu$. So the ratio of drop to runout should equal the friction coefficient — about 0.6 for rock, giving a slide that travels roughly one and a half times as far as it falls.

Small rockfalls obey that. Large ones do not. Heim noticed in the 1880s, and it has been confirmed on hundreds of events since, that the ratio falls systematically as the volume rises: a slide of a hundred million cubic metres has a drop-to-runout ratio nearer 0.1, meaning it travels ten times its fall height rather than one and a half times. The effective friction coefficient of a large rock avalanche is a fifth of the friction coefficient of rock.

Something is lubricating them, and after a century there is still no settled answer as to what. Trapped air was the early candidate and is now largely discounted, for a reason that is a nice piece of evidence: the same volume–runout relation is measured on Mars and on the Moon, from orbital imagery, and the Moon has no air at all. Frictional melting at the base has been found in some deposits and not in others. Acoustic fluidisation — the proposal that vigorous internal vibration momentarily unloads the contacts, so the mass behaves as though its confining pressure were lower than it is — accounts for the volume dependence and is hard to observe directly. And dynamic fragmentation, in which rock breaking apart during the slide generates a dispersive pressure of its own, is the current front-runner.

What makes the problem worth stating here rather than filing under geology is what it demonstrates about the criterion. The angle of repose is a statement about a material at rest, and nothing in it transfers to a material in violent motion. The friction that holds a slope is measured between grains sitting on one another; the friction that stops a moving mass is measured between grains that are colliding, bouncing, breaking and shedding fragments, and there is no reason those should be the same number. That they are the same number for small slides and not for large ones is the clue, and it is the reason the volume dependence is the observation everybody tries to explain.

What the pictures cannot show #

The grains here have no shape. Angularity raises the angle of repose considerably — crushed gravel stands steeper than rounded sand of the same mineral — and none of the arithmetic above contains a shape. What “μ” means for interlocking angular grains is partly friction and partly the dilatancy of the previous section, and separating the two experimentally is difficult.

The slab analysis assumes the failure is a plane. Real slope failures are curved surfaces, and soil mechanics computes them by searching over candidate surfaces rather than by assuming one. The straight-slab criterion is the right leading answer and the wrong detail.

Nothing here is dynamic. The criterion says when a heap starts to move, not how fast it then moves or how far it runs. A large avalanche can travel far past the angle at which it should have stopped, because the flowing layer fluidises and its effective friction drops further.

And the cohesion model is one number. Real damp sand has a cohesion that depends on how much water it holds, non-monotonically: too dry and there are no bridges, too wet and the bridges merge. The optimum is a few per cent by volume, which is the empirical fact every child discovers by experiment.

The ladder from here #

Later rungs on this anchor: the Mohr–Coulomb criterion written properly, as a condition on the stress tensor rather than on a slab; the shear band and why granular deformation localises; the statistics of avalanche sizes and the sense in which a sandpile is critical; segregation, where a shaken mixture of sizes sorts itself rather than mixing; and the jamming transition, which is where the granular case joins the glass and the foam.

The neighbouring ladders are the silo that does not weigh what it holds, which is the same friction acting on a container’s walls, the paste that holds up its own hill, which is the yield-stress fluid whose threshold does not depend on pressure, and the grip that is not a coefficient, which is where the friction number itself comes from.