A cloth gives back less than it took

Pull a strip of shirting between two hands, let it go, and it goes back. Pull harder and it goes back nearly. Pull hard enough and it does not: there is a slack patch where the hands were, and it stays. Every reader of this collection has done the experiment, and nothing in the collection so far can say where between the second case and the third the change happened.

That is not an oversight in one place. It is a property of every deformation this site has built. The trellis returns to square. The locus is a curve a cloth can be walked along in either direction and back again. The relaxed construction is a minimum a fabric finds again when it is disturbed. The wet state is an equilibrium. Every one of them is a statement about where a cloth can be, and every one of them is symmetric in time.

Cloth is not. A garment washed five times is not the garment that was cut, a crease outlives the hanger, a tensioned awning is slack by morning, and a knee keeps the shape of the knee.

The claim #

A woven cloth has two routes to an extension, and only the second of them is the fibre’s.

The first is crimp interchange, which this collection has had since its second field: the warp straightens, the weft takes on the crimp the warp gave up, and the cloth is longer with no thread anywhere in it having changed length. That route has a definite end — the locus stops, either because the warp has gone straight or because the weft has run out of room to accept more crimp — and up to that end the cloth is a mechanism, which is to say it comes back.

Past that end, nothing in the geometry can move. The shape is frozen, and a frozen shape stretched by a hundredth is a similar shape in which every length has grown by a hundredth, including the thread’s own. So beyond the end of the locus, a cloth’s strain and its threads’ strain are the same number, exactly, with no factor between them.

A fibre returns a measured fraction of a strain and no more. So the fraction of a cloth’s extension that comes back is

$$\text{returned} = \varepsilon - (\varepsilon - \varepsilon_{\text{jam}})\,(1 - r)$$

for any strain past the end of the interchange, and everything, everything, turns on the fact that ε_jam is a number about the construction.

The argument #

The end of the interchange is what this site calls the extension bound, and it has been computed here since the tensile ladder was built. It is worth restating what fixes it, because the answer is not what a section drawing suggests.

The obvious bound is the warp’s own crimp: a warp thread 8.97 per cent longer than the cloth it crosses can, in principle, give all 8.97 per cent up. But it can only give it up to the weft, and a weft thread has a limit on how much crimp it can accept — its own straight run vanishes, and at that point the cloth cannot go further whatever the warp still had in hand.

For six of the eight cloths in this site’s table it is the weft that stops the cloth. The overstatement is large: the sheeting’s warp has 14.61 per cent of crimp to surrender and the cloth reaches 4.03 per cent, a factor of 3.63.

The bounds run from 1.91 per cent for the cheesecloth to 6.59 for the muslin. That range — two to seven per cent — is the whole of the interval inside which a fabric is a mechanism. It is a small interval, and a striking one: whatever is different between a scrim and a heavy canvas, it is not this.

What the fibre contributes, and it is a measurement #

Elastic recovery — the fraction of an imposed strain a fibre returns when the load comes off — is not predicted by anything. It is a statement about what a polymer does, this collection models no polymer, and the honest treatment is a table with a convention attached. The convention used throughout here is the standard one: extend to a stated strain, unload, read the strain returned immediately.

Cotton returns 74 per cent of a two per cent strain and 45 per cent of a five per cent one. Wool returns 99 and 69. Nylon returns everything at two per cent and 89 at four and at eight.

It is tempting to look for the reason in the shape of the stress–strain curve — the story being that a fibre whose curve bends over has somewhere to put a strain and gives it back. Wool and cotton say it loudly: wool’s measured breaking extension is nearly six times the strain a linear fibre of its own tenacity and modulus would break at, and it recovers well; cotton’s is barely above its own prediction, and it does not.

Over six fibres there is no correlation at all. That matters here for a reason beyond its own interest: it establishes that recovery is a measurement and stays one, which is precisely what makes the geometric half of the split worth computing. One half of a cloth’s memory can be calculated and the other cannot, and knowing which is which is most of the value.

The corner, and what it belongs to #

Put the two halves together and a cloth’s recovery is not a curve. It is two straight pieces with a corner between them.

The corner is at the end of the interchange, and it is worth being exact about whose number that is. Substituting a different fibre’s recovery table into the same geometry moves the whole falling part of the curve — a poplin of wool at just under six per cent returns 99.7 per cent where a cotton one returns 91.2 — and does not move the corner by a hair. The corner is a property of the sett, the two counts and the balance, and of nothing else.

That gives a claim with some bite. Take two cloths of the same yarn at the same strain and their recoveries differ, and the whole of the difference is the construction. The best way to see it is not to compare two rows of the cloth table, because those differ in their counts as well as their setts; it is to hold the yarn fixed and sweep the sett.

At eight ends per centimetre a 20 tex cotton cloth returns three fifths of a five per cent strain. At twenty it returns all of it. The yarn is the same yarn.

What was counted, and how #

Every budget here is the end of a locus sampled at 241 states, each of which has both thread lengths reconstructed from its own geometry and checked against the state the cloth started in. That check is made at every sampled point rather than at the ends, because a deformation path that conserves length at its endpoints and drifts in the middle is exactly the failure a two-state figure cannot show.

The census over the eight cloths is run at one strain and reported as what each keeps.

Where the thread strain falls below the lowest strain anybody has measured a recovery at, the machinery refuses to interpolate down to zero and returns a bracket instead: the recovery is between the lowest measured value and one, because recovery falls monotonically with strain for every fibre in the table, and that monotonicity is asserted rather than assumed. Nylon is the case that makes the assertion worth making — it recovers 89 per cent at four per cent strain and 89 at eight, which is flat and not rising, and a table entered by hand could easily have had those the other way about.

Extending the last measured segment downwards would have been the easy alternative and is quietly disastrous: cotton’s two points extended to a tenth of a per cent give a recovery of 1.03, which is a cloth returning more than it was given.

What is kept depends on the excess and on nothing else #

The recovery expression has a rearrangement in it that says the split more sharply than the curve does, and it is worth writing out because it turns a two-parameter family into one curve.

What comes back is ε − (ε − ε_jam)(1 − r). So what is kept — the permanent set, the slack patch where the hands were — is

$$\varepsilon_{\text{kept}} = (\varepsilon - \varepsilon_{\text{jam}})\,\bigl(1 - r(\varepsilon - \varepsilon_{\text{jam}})\bigr)$$

and the only quantity on the right is the excess of the strain over the budget. The budget does not appear separately. The applied strain does not appear separately. Two cloths of the same fibre stretched the same distance past their own budgets keep exactly the same permanent set, whatever their setts, their counts or their balance.

That is a strong statement and it is checkable against the table this essay already computes. The cheesecloth’s budget is 1.91 per cent; extended to five, its excess is 3.09 and it keeps 1.13 per cent. A muslin, whose budget is 6.59, would have to be taken to 9.68 per cent to have the same excess — and it would then keep 1.13 per cent too, from a strain nearly twice as large.

So the eight curves in the census are one curve, shifted.

The collapse also says which experiments are worth running. Measuring the permanent set of one cloth at several strains measures the fibre, slowly and with a geometric offset in the way; measuring several cloths at one strain measures the offsets, and is the experiment that separates them. Neither measures both, and running the first on a single construction and calling the answer a property of the fabric is how a material’s set curve comes to be quoted with a sett hidden inside it. The geometry contributes a horizontal offset and nothing else, and the falling part of every cloth’s recovery is the fibre’s own set curve read from a different origin. Plotting permanent set against strain gives eight different lines; plotting it against strain minus budget gives one, and that collapse is the cleanest available statement of what belongs to the construction and what belongs to the material.

Why the everyday experience is so equivocal #

The budgets run from 1.91 to 6.59 per cent and the strains a garment meets in wear sit squarely inside that range, which is why the hand experiment at the top of this essay gives such an unsatisfying answer.

A fabric strained two per cent is inside almost every cloth’s budget and comes back completely. A fabric strained eight per cent is past every cloth’s budget and keeps something. Between three and seven per cent, whether a cloth comes back is decided by which cloth it is — and the difference between two cloths that behave oppositely at the same strain can be nothing but their setts.

That is exactly the range an elbow, a knee or a seat works a fabric through, which is why the same garment can be perfectly recovered at the shoulder and permanently bagged at the knee without anything about the fibre differing between the two places. It is also why a fabric’s reputation for holding its shape is so hard to pin on a material: two mills’ versions of the same cloth in the same yarn, differing only in how closely they set it, sit on opposite sides of the corner at the strain that matters.

Where the model stops #

There is no time in any of this. A fibre’s recovery has an immediate part and a delayed part, and only the first is in the table; a cloth left overnight recovers further, by an amount this collection does not carry for any fibre. Everything below is therefore an upper bound on what is kept.

The frictional set is not included. A cloth’s crossings are held by friction, so even a strain wholly inside the interchange leaves a little behind — the cloth rests in a band rather than at a point, which this collection computes elsewhere and which is deliberately switched off here so that the geometric statement can be made alone. Adding it lifts the flat part of the recovery curve off the ceiling and changes nothing about the corner.

The model is a plain weave, as every Peirce argument here is. A twill’s thread passes several crossings before it turns, so the arc-and-straight construction is different and every number moves; the shape of the argument survives, because the two routes exist in any interlacement.

And the thread strain past the jam is the model’s, not a measurement. The claim that the cloth’s strain and the thread’s are equal there depends on the shape being genuinely frozen. A real cloth past its jam can still find a little room by flattening its sections, which this collection has priced separately and which would give a third route, cheaper than stretching and dearer than interchange.

The generalisation #

The shape of this result has nothing to do with textiles.

When a system has a mechanism and a material in series, the mechanism’s travel is the whole of its reversible range, and the travel is usually a geometric number that nobody has computed. A cloth’s crimp interchange, a chain’s slack, a bolted joint’s clearance, a laminate’s ply rotation, a tendon’s crimped collagen: in each case there is a distance the thing can move by rearranging rather than by straining, that distance comes back for nothing, and past it a material property takes over and does not.

The second half generalises differently and is the sharper lesson. A property that presents as belonging to a material can turn out to belong to the arrangement. “This fabric recovers well” sounds like a claim about a fibre and is mostly a claim about a sett. The test for which it is, here, was substitution: change the fibre’s table and see what moves. The corner did not move at all.

Who found it, and when #

Crimp interchange is Peirce’s, from 1937, and the constant-thread-length locus is implicit in his geometry from the start. The extension available from it has been quoted in textile mechanics ever since, usually as “the crimp available in the direction of pull”, which is the warp-crimp bound and is the overestimate this rung’s first figure is about.

Elastic recovery tables for fibres are from the standard physical-properties literature and go back to the 1950s and before; the convention of an immediate recovery from a stated strain is theirs and is the one thing here that a reader should carry away as being someone else’s measurement.

What appears to belong to this collection is the composition: putting the geometric bound and the measured recovery into one expression, and observing that the resulting curve’s corner is fibre-free. The two halves are old and are usually discussed by different people — the first by fabric mechanics and the second by fibre physics — which may be why the corner between them has not been given a name.

Where the ladder goes next #

The next rung asks the obvious follow-up question and gets a surprising answer: since the budget rises and then falls as a cloth is set closer, there is a best cloth, and the most interchange any cotton cloth can have is 7.29 per cent at a cover factor of 0.4078 — a number with no yarn count in it at all.

Sideways, the same split decides a fold rather than an extension, and a crease turns out to be a fold too sharp for the crimp to supply. It also decides what happens when a cloth is held stretched, which is not that its recovery is used up but that it moves into the other direction.