Textile Structure
how cloth is put together
Series

Pile — the series

12 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A V-fastened tuft. A cut pile bound into its ground by V fastening, drawn in section. The pile end passes beneath 3 of the 6 ground picks and wraps 1 half-turn around them in all. The integrity criterion says the tuft is attached; how hard it is held is a different question with a different model behind it.

    Pile is a third thread system

    Velvet, corduroy, plush and carpet all carry threads that are not woven into the ground in the ordinary sense. Two of the three systems make a perfectly good matrix and the third is not in it — so the encoding this whole site rests on has nothing to say about the part of the fabric a hand actually touches.

    part 1 · fancy
  2. What a tuft's wrap is worth at μ = 0.3. The two classical ways of binding a cut pile into its ground, with the holding force each provides as a multiple of the tension applied to the free end. The model is the capstan equation and the friction coefficient is measured rather than derived, so the ratio between the two matters and neither absolute number should be quoted alone.

    How a tuft is held

    A V-fastened tuft and a W-fastened tuft are both attached, and the integrity criterion returns the same verdict for both. One of them is specified for hotel corridors and the other sheds. What separates them is not a topological property at all — it is an angle, and it needs a coefficient somebody has to measure.

    part 2 · fancy
  3. Corduroy: floats of 6 ends, cut. 3 wales of an extra weft floating over 6 warp ends each, cut at their midpoints so that each half stands away from the ground. The pile height is half a float's length and nothing else decides it, so a finer sett at the same float count gives a shorter pile; the wale spacing is the float plus its binding ends.

    Corduroy is a cut float

    An extra weft floats over some ends, a knife runs along the cloth, and each half of the float stands up. So the pile height is exactly half the float's length — which makes corduroy the one pile fabric whose surface is decided entirely by a quantity this site has been counting since its first essay.

    part 3 · fancy
  4. Terry at a let-off of 5 to one. Towelling has two warp beams. The ground warp is held at ordinary tension; the pile warp is let off 5 times as fast, and the excess has nowhere to go but up. The loop height follows from the ratio and the pick spacing, and no float length appears in it.

    Terry needs two beams

    A towel's pile is not woven longer. It is fed longer — from a second warp beam let off several times faster than the first — so the loop height follows from a ratio and a pick spacing, and the float length that decides a corduroy does not appear in it anywhere.

    part 4 · fancy
  5. One cloth, before the knife. Double plush: two ground cloths woven face to face with a pile warp shuttling between them. Uncut, the pile is the only thing either ground touches the other through, and the integrity criterion says one cloth. Cut down the middle, every pile end is severed and the same criterion says two. The whole manufacture is the deliberate destruction of an integrity the criterion otherwise exists to confirm.

    Velvet is cut apart

    Two ground cloths are woven face to face with a pile warp shuttling between them, and while it does the whole thing is one cloth — the criterion says so, and the pile is the only reason it is true. Then a knife runs down the middle and the same criterion says two. The manufacture is the deliberate destruction of cloth integrity.

    part 5 · fancy
  6. What holds a tuft in, in newtons. The withdrawal force of a V-fastened and a W-fastened tuft over the reported range of yarn-on-yarn friction, on a duck ground with 1.00 N in each pick. Each is the friction at the wraps, with each wrap's share dragged around every wrap between it and the pulled end — a capstan series rather than a single factor. W runs from 0.79 to 6.84 N and V from 0.15 to 0.41. Both are an order of magnitude below what a carpet is specified at, which is a finding about carpets rather than about the model. What the chart cannot show is the backing, which is where the rest of a tufted carpet's anchorage comes from.

    What holds a tuft in, in newtons

    The pile ladder computed a tuft's anchorage as a capstan ratio and said, correctly, that a ratio was all it could offer. A ratio multiplies a tension and there was no tension anywhere on this site. There is one now — and the answer, put beside what a carpet is actually specified at, falls short by a factor of three.

    part 6 · fancy
  7. A 5.0 mm cotton pile standing and crushed. A pile tuft standing 5.0 mm proud and the same tuft pressed flat, which turns it through a right angle over its own length and so bends it to a radius of 3.18 mm. Its fibres are strained 0.187% — an order of magnitude below the smallest strain anybody has measured a recovery at, so this file declines to say what fraction comes back. The consequence is that a crushed carpet is not held down by its fibres: what keeps a pile flat is the tufts leaning on one another and the friction where they touch. Below 0.469 mm the fibre does enter its measured range, which is the difference between a carpet and a velvet and has nothing to do with what either is made of. What the drawing cannot show is the neighbouring tufts, which are the mechanism.

    A crushed pile is not held down by its fibres

    A carpet flattened under a foot has bent its tufts through a right angle, which sounds severe and is not: the fibres in a five-millimetre pile are strained under two tenths of a per cent, an order of magnitude below the smallest strain anybody has measured a recovery at. What holds a pile down is the tufts leaning on one another.

    part 7 · fancy
  8. The pressure a 30° twist puts on its own fibres. A twisted yarn squeezes itself. Every fibre under tension at a radius pulls inward with sin²θ of its tension per unit length, and integrating outward to the surface — where the pressure is zero by definition, because there is nothing outside to push against — gives p(r) = ½σ(cos²θ(r) − cos²α) in closed form. At a surface angle of 30° the pressure on the axis is 12.5 per cent of the core fibre's own axial stress and the mean over the section is 5.65 per cent of it. The shading is that closed form and the curve is the same function plotted; the shape is what matters, and its one uncompromising feature is the zero at the edge. The outermost fibres are held by nothing, which is why a yarn is hairy, why a surface fibre is the one that comes away on a finger, and why singeing changes a yarn's behaviour out of all proportion to the mass it removes.

    A tuft is set so it cannot untwist

    A cut pile tuft has a free end, and at a free end the pressure holding the twist is zero. So the twist runs out over a computable length, the tuft opens, and a carpet loses its appearance long before it loses any material.

    part 8 · fancy
  9. Three kinds of surface, and only one of them starts open. The bearing curves of a 2/2 twill in sheeting, of a terry loop pile, and of a cut corduroy pile of 30 tex, over the first six per cent of a cloth's thickness. The woven curve opens as the square root of the depth and the loop pile's does the same, because a loop's top is a curved thread like any other. The cut pile does not open: it is a flat line at 5.2% of the plan, from a depth of nothing, because a blade severed every tuft in one pass and left every end in one plane. That is not a larger contact area, it is a different kind of contact area — one that does not vanish as the load goes to zero, which is a property no woven surface has.

    A pile is the only surface with no crowns

    Every other fabric touches on the tops of curved threads, so its contact vanishes as the load goes to zero and the pressure on what is touching rises without limit. A cut pile touches on flat ends left in one plane by a blade — so its contact is finite at no load at all, its pressure concentration is bounded, and it is the only fabric whose abrasion mass loss is an honest measure of its damage.

    part 9 · fancy
  10. Hair, nap and pile are one construction at three settings. 12 mm of sheeting at one pair of scales, carrying each of the three protruding surfaces a cloth can have — 50 pixels to the millimetre along the cloth and 29 off it. They are the same object, fibre standing off a cloth with a density, a length and a hold, and they differ by 9-fold in density and 4.8-fold in length. What actually separates them is the third line under each name: hairs are held by whatever the twist happened to leave, a nap by the float it was pulled from, and a pile by a W or a V through three picks. Only the last two were chosen. The pile's tufts are drawn in the float colour because that is what a woven pile is, and they are all one length because a blade cut them; the hairs and the nap are drawn as measured quantities and their lengths come from the population's own exponential. Each panel clips to the room it has, and the densest of the three is drawn at the model's own number rather than thinned.

    Hair, nap and pile are one construction

    There are three surfaces made of fibre standing off a cloth, and they have been arrived at from three different directions and in three different fields. They are the same object at three settings, and what separates them is not what they are but how much of them anybody decided.

    part 10 · fancy
  11. The fibre pieces in one cut tuft. A cut-pile tuft 3 mm tall on a 1 mm base, spun from a 25 mm staple, with a seeded sample of its fibre pieces drawn along the yarn: 16 pass under the binding pick and 2 lie wholly in one leg, between a fibre end and the cut tip. Across a whole pile the loose share is 21.9 per cent of the pieces and 7.0 per cent of the fibre. What the drawing cannot show is the twist, which holds the loose pieces by friction and is what they escape from.

    A blade leaves loose fibre in every tuft

    The account of hair, nap and pile ended on a clean claim: a blade collapses a population's length to one value, so a cut pile has no tail and nothing in it reaches past the rest. The tips are one length. The fibres are not. A tuft cut from staple yarn is a length of yarn with fibre ends scattered along it, and every fibre end that lands in a leg leaves a piece between it and the blade that nothing in the draft holds — a fifth of the pieces in an ordinary wool carpet, none in a filament one.

    part 11 · fancy
  12. Which loose pieces in a cut tuft are free at once. A cut-pile tuft 12 mm tall on a 2 mm base, spun from a 90 mm staple, with a seeded sample of its fibre pieces drawn along the yarn and the tip's untwisted run-out, 5.6 mm, shaded down each leg. 22 pieces pass under the binding pick; of the 9 that lie wholly in one leg, 3 lie wholly inside the run-out, where the twist presses on nothing, and are free at once; the other 6 reach down into twisted yarn and are held until wear opens the twist further. What the drawing cannot show is the run-out's edge, which is a fall in pressure rather than a line.

    A cut pile sheds in two stages

    Every cut tuft of staple yarn holds loose fibre pieces that nothing anchors but the twist, and at the tip the twist holds nothing: it runs out over a length the grip arithmetic already gives, five and a half millimetres in a wool carpet yarn. A loose piece lying wholly inside that run-out is free the day the carpet is laid; one reaching below it is held until walking opens the tip's twist. So the reservoir drains in two stages, and the pile height decides the split. A velvet sheds everything it will ever shed at once. A twelve-millimetre wool carpet sheds a fifth of its reservoir at once and four fifths later. A shag sheds almost nothing at first and fourteen per cent of its fibre eventually.

    part 12 · fancy

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