The stress that stops growing with depth

At its defaults it draws the stress that stops growing with depth. Vertical stress against depth in a silo of radius 0.5 m holding grain of bulk density 1500 kg/m³, with a wall friction coefficient of 0.5 and Janssen's ratio K = 0.5. The straight line is what a liquid of the same density would do — ρgz, with no length in it anywhere. The curve is what grains do: wall friction, mobilised by the sideways stress the grains themselves exert, removes weight from the column at a rate proportional to the stress, so the stress saturates at ρgλ over a screening length λ = R/2μK = 1.00 m. Read off the drawn curve at the 8 m base, the stress is 14.7 kPa against the 117.7 kPa the liquid delivers — 88 per cent of the weight is standing on the walls. Another twenty metres of grain would move the floor's reading by less than a pascal.

granular-column is one function in lib/figures/fluids.js — matter that will not hold a shape, and the forces in it. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

Vertical stress against depth in a silo of radius 0.5 m holding grain of bulk density 1500 kg/m³, with a wall friction coefficient of 0.5 and Janssen's ratio K = 0.5. The straight line is what a liquid of the same density would do — ρgz, with no length in it anywhere. The curve is what grains do: wall friction, mobilised by the sideways stress the grains themselves exert, removes weight from the column at a rate proportional to the stress, so the stress saturates at ρgλ over a screening length λ = R/2μK = 1.00 m. Read off the drawn curve at the 8 m base, the stress is 14.7 kPa against the 117.7 kPa the liquid delivers — 88 per cent of the weight is standing on the walls. Another twenty metres of grain would move the floor's reading by less than a pascal.

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 distance still to run — ½ minus the variance of the normalised load — against the depth of the pile, both logarithmically, from 10 rows to 10000. Every point is exact rather than sampled: the second moments of the lattice satisfy a closed recursion, which is what is iterated here. The dashed line through all 7 has slope −0.471; through the deepest three, −0.494, against the −0.5 of an inverse square root. So the q-model does reach the distribution it is quoted as having, and it reaches it at the rate a random walk closes anything — to halve what is left, dig four times as deep. A pile 10 grains deep has 43 per cent of the steady-state variance and one 100 deep has 79 per cent, which is worth knowing before a closed form is compared against a photograph of a laboratory packing twenty beads high.

The steepest stable slope against the height of the pile, for four cohesions — 0 Pa, 40 Pa, 200 Pa, 1000 Pa — with the same internal friction throughout. Cohesion is a stress and not an angle, so what it buys is a fixed amount of holding force per unit area while the driving force grows with the pile. At a centimetre the cohesive materials stand vertically; by 14 cm for 40 Pa, 71 cm for 200 Pa, 3.5 m for 1000 Pa they are back within a degree of the dry angle of 35°. That is the whole reason a sandcastle works and a sand cliff does not, and the reason is a length: the ratio c/ρg is a height, and above it cohesion is negligible. Damp sand's cohesion comes from the capillary bridges between grains, so the same argument connects a child's bucket to the surface tension of water — and it is why the castle collapses as it dries and also if it is flooded, since either removes the menisci.

How far a layer of grains must rise to slide over the layer beneath it, against how tightly it is interlocked. A dense packing sits in the hollows of the layer below, so shearing it means riding up and over: the material expands before it flows, which is Reynolds's dilatancy. At an interlock angle of 30° the rise is 0.134 of a grain diameter, and the ride-up slope adds to the grain-on-grain friction to give the effective friction the heap actually shows. Two consequences follow that are otherwise puzzling. The angle of repose of a dense sand exceeds the friction angle of its grains, because part of what resists is geometry rather than rubbing. And wet sand goes dry underfoot: pressing on it forces the packing to expand, the pore space grows faster than the water can flow in to fill it, and the surface is left short of water — the same effect that makes a vacuum-packed bag of coffee rigid, run backwards.

A 12-row triangular packing in which every grain weighs the same and passes its whole load to the two grains below it, in a ratio drawn at random for each contact. Nothing is lost: the total load in each row is exactly the number of grains above it, checked here to 1.9e-16. The line widths are the contact forces, and they are uniform nowhere — the load travels in chains with quiet regions between them. In the bottom row the heaviest grain carries 1.91 times the mean and the lightest 0.15 times it, out of a rule containing no heterogeneity at all. This is what Janssen's constant is an average over, and why a silo wall is designed for a pressure it will never see uniformly.

physicscheck asserts something about granular-column that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.