Floats and abrasion

“Satin wears badly” is the sort of claim that is repeated because it is nearly true, and it hides a distinction sharp enough to change which cloth a maker chooses.

Two different things happen to a fabric that is rubbed. Thread is removed, slowly, at whatever stands proudest; and at some point a thread is severed, after which a length of it is loose. The first is a rate and the second is a consequence, and the float moves them in opposite directions. A satin is slow to wear and catastrophic when it goes. A plain weave is quick to wear and barely notices.

What the abradant actually touches #

A woven cloth is not flat. A thread that changes face has to climb over the thread it crosses, so the surface is corrugated, and the corrugation has the pitch of the interlacing.

In a plain weave every crossing is an interlacing, so the face is a field of small domes at the pitch of the picks. Something rubbed across it touches the tops of the domes and nothing else. The contact area is a small fraction of the cloth’s area, and the whole of the rubbing load is carried on that fraction.

In an eight-end satin the same end travels seven crossings along the face before descending. Over those seven it is not climbing anything, so it lies very nearly flat, and the abradant rests along its whole length rather than on a point.

The consequence runs against the folk claim. Pressure is load over area, and the satin has the larger area, so at the same load the satin is rubbed at lower pressure than the plain weave. Where abrasion is measured as mass loss per thousand cycles, satins of the same yarn and cover frequently come out ahead, and the standard test methods are perfectly capable of showing it.

Where the trade’s claim is right #

None of that touches the second quantity, and the second quantity is what a garment’s owner experiences.

Wear does not become visible when material has been removed. It becomes visible when a thread is severed, because at that moment a length of thread stops being held. How long that length is has one answer: the float.

Sever a warp end in a plain weave and the loose piece is bounded by the picks on either side of the cut. It is one pick long, it is held at both ends, and it goes nowhere. Sever a warp end in an eight-end satin and the loose piece is bounded by the interlacings on either side, which are seven picks apart, so a length of thread more than four times as long is free to lift.

At each weave’s own densest setting that length runs from a twentieth of an inch in a plain weave to nearly a third of an inch in a twelve-end satin. The float grows by a factor of eleven from the top of that table to the bottom; the freed length grows by a factor of six, because the denser setting a long float permits buys back part of it.

Snagging is the same number arriving from outside #

Abrasion severs a thread from above. A snag lifts one from above without severing it, and the two failures are governed by the same integer.

A float of seven picks presents a length of unattached thread that a fingernail, a zip or a branch can get under. A float of one does not: there is nowhere to insert anything, because the thread is tied down on both sides of every crossing. The catching is a matter of clearance, and clearance is float length times pick spacing — the same product that gave the freed length, arriving from a different direction.

That is why a satin snags in service and a poplin does not, and why the two are usually described as the same defect. They are not the same event, but they have the same cause and, to within the same factor, the same size.

The distinction matters in one practical way. A snag can be worked back: the lifted loop is pulled through to the reverse and the cloth is very nearly as it was, because no thread was cut. A severed float cannot be, and the same length that made the snag recoverable makes the cut conspicuous. So a fabric’s snagging behaviour is an annoyance with a remedy and its abrasion behaviour is not, and a specification that treats them as one number is buying the wrong protection.

What was counted, and how #

The freed length is not an estimate. For each weave the matrix is walked, the longest warp float is taken from the run lengths in the columns, and the picks per inch are computed from the same weave’s own jamming sett — the number of threads a repeat can hold given that every interlacing costs the crossing thread a diameter of room.

Both halves are then asserted while the figure draws. A longer float must always free more thread, or the model has gone wrong somewhere. And the denser setting a long float allows must repay part of the float’s growth and never the whole of it — the freed length must rise, more slowly than the float does. If the setting ever repaid the whole of it, the fragility would be an illusion of comparing weaves at a sett they cannot share, and the figure would say so by throwing.

The second assertion is the one worth having. It is exactly the mistake the standard pass found elsewhere on this site: a comparison run past the point where the two things being compared can both exist.

Where this model stops, and it stops early #

Three things are missing, and the third is the important one.

Nothing here knows about fibre. A staple yarn sheds fibre ends and a filament yarn does not; a nylon float survives rubbing that destroys a cotton one at the same geometry. The whole of this essay is a comparison at constant fibre, and it says nothing about which fibre to choose.

Nothing here knows about finish. A calendered or resin-finished cloth behaves differently from the same cloth greige, and both are the same matrix.

And nothing here knows about friction, which is where the mechanism actually lives. The claim that a satin is rubbed at lower pressure assumes the abradant rests on the flat and does not catch; the moment it catches, the float is lifted and the contact geometry is not the one that was computed. Whether it catches depends on the coefficient of friction between the abradant and the yarn, on how much the float can lift, and on the load — none of which this site models at all.

So the honest statement is narrower than the section above may have sounded. The geometry says the contact area is larger and the freed length is longer. The first of those is a claim about pressure only if nothing catches, and the second holds unconditionally. It is the unconditional one that the trade noticed.

Which thread system pays #

A balanced plain weave shows as much weft as warp, so a rub delivered to it is shared between the two systems and both wear at the same rate.

A satin is not balanced. An eight-end satin puts seven-eighths of its face in warp, so seven-eighths of any rubbing is delivered to the warp and the weft is very nearly untouched. Balance is the mean of the matrix and it is one of the few quantities in this subject that can be read straight off, and it is the number that decides who pays.

Three consequences follow and all three are visible in ordinary cloth.

The warp of a warp-faced fabric is the yarn worth spending money on, because it is the yarn that will be destroyed. Denim is the clearest case: three-quarters warp on the face, and the indigo is dyed on the warp alone — which is why the fabric fades on the face and stays blue at the crossings, and why the fading traces the abrasion pattern so faithfully that it has become a design feature rather than a defect.

The two faces of a warp-faced cloth wear at different rates, so the reverse outlasts the face by a wide margin. A worn satin lining is a fabric whose warp has gone and whose weft is intact.

And the direction of rubbing matters where it does not in a balanced cloth. A rub delivered along the warp travels along the floats; a rub delivered across them crosses a descent every eight picks and finds an edge each time. This is precisely the confound the Martindale test’s continuously turning path was designed to average out, which is a good sign that it is real.

Pilling, which is the same geometry with a different ending #

A staple yarn is a bundle of fibres a couple of centimetres long, held together by twist, and a rub does not only remove material from it. It pulls fibre ends out of the yarn’s body, and those ends, worked over each other, roll into a pill that stays attached by whatever fibre still runs back into the cloth.

Whether that happens depends on room, and room is float length again. A fibre end lifted out of a plain weave’s crown has a crossing thread on either side of it within one pick; a fibre end lifted out of a satin’s float has seven picks of unobstructed surface to be rolled along. The mechanism is not the one this essay computes and the site has no model of fibre migration, so the claim here is bounded: the geometry supplies the clearance, and pilling needs clearance. What it does with it is a fibre property.

The bounded version is still enough to explain the pattern that confuses people. Cotton pills and stops, because the fibre is weak enough that the pill breaks off; polyester pills and keeps the pill, because the fibre is strong enough to hold it. Same weave, same clearance, opposite outcomes — which is a demonstration that the weave supplies the opportunity and the fibre supplies the result, and that a claim about one is not a claim about the other.

The exponent that connects the two halves #

The freed length rises by six across the table while the float rises by eleven, because a long-floated weave can be set closer and the closer setting shortens each pick. That is stated above as a partial repayment; it is worth turning into an exponent, because one exponent then governs three separate failures.

Freed length is the float times the pick spacing, and the pick spacing is the reciprocal of the weave’s own jamming sett. Over the range the table covers, the achievable sett rises roughly as the float to the power a quarter — a weave with four times the float can be set about 1.4 times as close — so

freed length ∝ float^(3/4),

and 11^0.75 is 6.0, which is the ratio the enumeration measured.

Three quantities share it, because all three are the same product.

The freed length after a cut, which is what the tally computes.

The snag clearance, which is the length of unattached thread something can get under, and is the same float times the same spacing.

And the pilling clearance, which is the unobstructed surface a lifted fibre end can be rolled along before it meets a crossing thread.

So a designer shortening a float to fix one of the three fixes all three, by the same three-quarter power, and does not get the benefit a naive reading promises. Halving the float reduces the freed length by forty per cent rather than fifty, because the cloth must then be set more openly and each pick occupies more room. Going from an eight-end satin to a five-end one — the standard compromise — cuts the float from seven to four, a factor of 1.75, and cuts the freed length by 1.75^0.75 = 1.52. A third rather than the four-tenths a reader would expect.

The other half of the ladder runs the opposite way and with an exponent nobody here can supply. The contact area rises with the float, so the pressure falls as its reciprocal, and the rate at which material is removed falls as some power of the pressure. Two curves in opposite directions with an interior crossing is the shape of an optimum: there is a float that maximises the rubbing a cloth survives before a thread is severed, and it is neither the shortest nor the longest.

Where it sits is not computable here, because it depends on the exponent relating pressure to wear rate, which is a friction property and is exactly what this collection does not have. What the geometry supplies is the shape of the problem and one of its two exponents, which is enough to say that the optimum is interior and not enough to say where. That is the same division of labour this ladder has kept throughout: the matrix decides the clearance, and the fibre decides what is done with it.

What a designer does about it #

The responses are the ones the constraint permits, and each gives up something this ladder has already accounted for.

Shorten the float. A five-end satin instead of an eight frees four picks instead of seven, at the cost of a shorter highlight and less lustre.

Put the float where it is not rubbed. A backed or stitched construction carries its long floats on the reverse, where nothing touches them, and presents a firm face to the world. The cloth is heavier and costs two warps, and the long float is still there — but it is out of the way.

Set the cloth denser. A float supported on both sides by neighbouring floats is harder to lift than one standing alone, which is why a well-set satin snags less than a slack one of the same draft. This is the only one of the three that does not shorten the float, and it is the reason a cheap satin snags and an expensive one of the same weave does not.

Who worked it out #

Abrasion is one of the oldest quantities in textile testing and one of the least satisfactory, which is why the literature is so large.

The instruments came first. The Martindale abrasion tester, developed at the Wira laboratories in Leeds in the 1940s, rubs a specimen against a standard abradant in a Lissajous figure so that the direction of rubbing is continuously changing — a design decision made precisely because a directional rub gives an answer that depends on the weave’s direction, which was recognised as a confound before it was understood as a mechanism. The Taber and Accelerotor methods took different routes to the same problem.

What the instruments established quickly is that abrasion resistance is not one property. Mass loss, appearance change and the cycles to a hole rank fabrics differently, and a cloth can be first on one and last on another. That is the distinction this essay is about, arrived at experimentally: the rate and the consequence are different measurements, and the standards ended up specifying which one is being reported rather than trying to reconcile them.

The float’s role was recognised throughout and stated as a rule of thumb — long floats abrade badly — with the two halves not separated. The separation is available for nothing once the cloth is treated as a matrix, because the float length and the interlacing density are both read off it, and they are not the same number.

Where the ladder goes next #

The rung after this is tear, where the trade also has a claim about floats and the geometry turns out to have nothing to say — the opposite situation to this one, and worth the contrast. Then comes the float limit, which is how the whole of this ladder is turned into a constraint a designer can work under.

Sideways from here, the firmness a weave has is the same interlacing count read for a different purpose, and the satin’s own arithmetic decides which float lengths are available at all.

What the pictures here cannot show. No figure on this page shows wear. Each shows a geometry from which a wear behaviour is inferred, and the inference has friction in it that none of them models. A picture of an abraded fabric would be evidence; these are pictures of why the evidence comes out as it does, drawn from drafts rather than from specimens.