Illustrated Physics
physics, drawn
Concept

Force chains — where it appears

The filamentary paths through a granular packing along which most of the load is carried, leaving neighbouring grains almost unstressed. They are why a silo's walls take the weight of what it holds and why a granular material's response to a load is intermittent.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The slope a heap of grains settles at. The steepest slope a cohesionless heap can hold, against the friction between its grains. A slab of thickness h on a slope is pushed down it by the weight's along-slope component and held by friction acting on the weight's across-slope component, and both are proportional to the same ρgh — so the thickness cancels and the criterion is tan θ = μ. Checked here at thicknesses of 2 mm, 50 mm, 2000 mm, the ratio of driving to holding stress differs by 2.2e-16, which is zero. That is why the quantity is an angle: nothing about the size of the pile, the size of the grains, the density or the strength of gravity survives into it, and a heap of sand on the Moon stands at the same slope as one on Earth. The marked materials are glass beads at 24°, dry sand at 33°, crushed gravel at 40°. The band between the two curves is the hysteresis: a slope steeper than 31.0° will keep flowing once started, and one shallower than 35.0° will not start — so a pile has a range of stable angles rather than one, and which it is found at depends on how it was built.

The angle that does not know the size of the heap

Pour sand and it makes a cone with a definite slope, and pouring more makes a larger cone with the same slope. The reason the answer is an angle rather than a length is a cancellation — the force pulling a surface layer downhill and the friction holding it back are both proportional to its weight, so everything about the size of the pile divides out — and everything that puts a length back in is a story about cohesion.

fluids · Granular matter The hourglass that keeps time. Discharge against how much is left above the opening, for grain and for liquid through the same 40 mm hole, each as a fraction of its own rate at a full hopper. The grain rate is a horizontal line: it does not appear in Beverloo's law at all, because the pressure at the outlet does not depend on the head — the walls carry the weight, which is what Janssen's argument establishes, so the grains at the opening are pushed by their immediate neighbours and by nothing else. The liquid falls as the square root of the head and is down to 32 per cent by the time a tenth is left. That is why an hourglass keeps time and a water clock does not, and why the water clocks that worked were built with a float and an overflow to hold the head constant.

The hourglass that keeps time

Grain leaves a hopper at a rate that does not depend on how much is above it, and that goes as the orifice to the five-halves power rather than the one half a liquid gives. Both facts follow from the same thing: the weight is carried by the walls, not by the grains at the opening.

fluids · Granular matter A hopper that runs in bursts. Grains discharged against time, on a logarithmic scale, from a silo of 1 mm glass beads through outlets 3.5, 4.0, 4.4 grain diameters wide; each time the outlet jams it is cleared after 2 s (a tap on the silo) and the flow restarts. Avalanche sizes are drawn at random from an exponential distribution with each outlet's mean. At 3.5 diameters the mean avalanche is 53 grains and the outlet jammed 57 times in two minutes, passing 2,537 grains; at 4.0 diameters the mean avalanche is 1,000 grains and the outlet jammed 37 times in two minutes, passing 33,568 grains; at 4.4 diameters the mean avalanche is 45,821 grains and the outlet jammed 3 times in two minutes, passing 116,983 grains. While it flows, each outlet runs at its full Beverloo rate; a narrower outlet does not flow more slowly, it stops more often, and the flat stretches are the waits.

The outlet that jams by chance

A hopper whose outlet is a few grains wide does not flow slowly. It flows at the full rate and then stops dead, when a handful of grains wedge into an arch across the opening, and it flows again only when something knocks the arch down. The number of grains that pass between one jam and the next is random and exponentially distributed, as if every grain leaving had the same small chance of being the last. Its average rises so steeply with the opening's size that it looks like a threshold, and whether there is one is still argued: two laws agree wherever an outlet can be watched, and they part only after millions of grains.

fluids · Granular matter

Named alongside it

The objects these essays reach for when they reach for this one.

FrictionGranular matterJammingAngle of reposeArchingCloggingCohesionCritical pointDilatancyDimensional analysisDischargeExponential distribution

All concepts