Textile Structure
how cloth is put together
Concept

Buckling load — where it appears

The force a slender column carries before it gives way sideways, going as the fourth power of its diameter and the inverse square of its length. For a protruding fibre it sets both what a hair layer can carry and the diameter at which a fibre end is felt as prickle.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

At 0.1 kilopascals the plate is standing on hair. A flat foot pressed onto sheeting at 0.1 kPa, over 3 mm of cloth. It stops 25 µm above the cloth's own crowns, because that is where the hairs it is bending can carry the load: of the 10 hairs drawn, 9 reach higher than the foot and are laid over under it, and the rest are untouched. The vertical scale is set by the approach and not by the layer — 5188 pixels to the millimetre off the cloth against 192 along it — because the foot's height above the crowns is tens of micrometres and the layer it stands in is more than a millimetre, so a picture at one scale shows the second and not the first. The thickness reported is 0.432 mm against 0.382 mm for the cloth itself, so 12% of the reading is a population and not a fabric. The hair layer carries up to 0.23 kPa before the foot reaches the crowns at all. A laid-over hair is drawn as two straight segments where a real one is an elastica: the corner is a convenience and the height it turns at is the measurement. What the picture cannot show is that a bent hair leans on its neighbours, which the arithmetic behind it does not know either.

A light touch never reaches the crowns

A plain weave touches at points and every other cloth touches along lines, and the difference is an exponent rather than a factor. It is a real result about a real surface, and at the pressures a fabric is actually touched at, nothing ever reaches that surface.

cloth · Contact A thickness is a property of the pressure it was measured at. What a gauge reports for sheeting against the pressure it presses with. At 1 kPa it reads 0.382 mm, which is the cloth; at 0.02 kPa it reads 0.555 mm, which is the cloth plus 87 µm of hair on each face. The difference is 31% of the reading and it is not a compressibility: nothing in the fabric has been squashed, the foot has simply stopped in a different place. That is why every thickness standard specifies its pressure to two figures, and why comparing a thickness from one standard with a thickness from another is comparing two different measurements of two different objects. The dashed line is the cloth's own geometric thickness, which no reading below the crossover ever reaches.

A compression curve is two laws in series

A published fabric compression exponent is a fitted number with no derivation attached, and this site has already said why: the range it is fitted over straddles two regimes. One of them turns out not to be in the cloth at all.

mechanics · Compression Thirty micrometres is a buckling load. The wool fibre diameter at which a protruding end reaches the measured threshold for prickle, against how far it protrudes. A fibre end pressed against skin is a column held at the cloth and free to slide at its tip, so it buckles at 20.19EI/ℓ² and carries no more load than that; below the threshold it bends away and is felt as touch, above it the load stands and is felt as pain. At a two-millimetre protrusion — which the hair model behind these figures puts in the top fifth of a wool's population — the threshold diameter is 32 µm, bracketed at 29–34 by the range of the measured force. Neither the thirty micrometres nor the two millimetres was put in. The band is the force bracket, and the fourth power in I = πd⁴/64 is what makes the boundary sharp: at a mean of 21 µm and a spread of 24%, 3.2% of the fibres are over it, and it is that few per cent that decides whether a garment can be worn.

Prickle is a buckling load

The wool trade specifies comfort against skin by the percentage of fibres coarser than thirty micrometres, and the thirty is a measured boundary with no derivation attached. It is a column formula: solve for the diameter at which a protruding fibre end stops bending away and starts standing its ground, and thirty micrometres falls out.

applied · Prickle Where an air arch first wrinkles under snow over the whole span. A semicircular air arch 4 m across, pinned at both feet, of a tube 200 mm across at 50 kPa, under snow over the whole span; beside it, unrolled from the left foot to the right, the share of the tube's wrinkling budget πpr³/2 spent at each section when the first wrinkle forms, with the thrust's part, N·r/2, shaded darker. The first wrinkle forms 25° above the right foot, at 167 N per metre of span, where the moment of 60.2 N·m compresses the inside of the curve and the thrust is 336 N; there the thrust has taken 21 per cent of the budget. Judged by its moment alone the arch would carry 218.

An air arch pays for its thrust out of its pressure

An inflated tube bent into an arch was expected to start with its inside wall pulled differently from its outside, and to wrinkle where that difference and the load's moment combined worst. It does not: a curved tube is pulled along its length at exactly the straight tube's pr/2, all the way round. What the arch spends its pressure on instead is the thing every arch exists to make — its own thrust — which takes a fifth of the wrinkling budget at the haunch of a shelter arch under snow and more than half on a tight one.

applied · Membrane A fibre end prickles only inside a window. The lengths at which a wool fibre end can press on skin with the measured threshold force, against its diameter. Above the rising curve the end is too long a column and lies over before it reaches the force. Below the falling curve it is too short to sink far enough into a skin of 30 kPa to reach it. The two meet at 23.2 µm and 1.08 mm, and no fibre finer than that prickles at any length. A 30 µm fibre prickles between 0.83 and 1.79 mm.

A fibre end prickles only inside a window

A fibre end pressed against skin has two ways to fail to hurt. Too long, and the column lies over before it reaches the force a nerve answers to. Too short, and the tip cannot sink far enough into something as soft as skin to push that hard. The two limits run opposite ways with diameter and they cross, so a fibre below one diameter cannot prickle at any length at all — and that diameter has no protruding length in it. It also settles a sum left unweighed: a crop at 1.1 mm makes a wool cloth two and a half times worse, and one at 0.6 mm removes the prickle.

applied · Prickle

Named alongside it

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

Hair layerBearing curveCanopy criterionCloth thicknessContact pressureContact thresholdFibre diameterPopulationPrickleProtrusion lengthWoolBeam

All concepts