Recovery is measured and nothing predicts it

This collection has been careful, for four fields, about which of its numbers come out of a geometry and which come out of a measurement. A yarn’s bending rigidity is a bracket whose two ends are both computed, and the interesting question is where in the bracket a real yarn sits. A diameter comes from a count by conservation of volume at a stated packing. A crimp comes from a closure condition. Even the friction coefficient, which is unambiguously a measurement, is used only in ways that survive its being a range.

Elastic recovery is not like any of those. It is the fraction of an imposed strain a fibre returns when the load comes off, it is the input that decides how much of a fabric’s deformation is permanent, and there is nothing in this collection or anywhere near it that predicts it. It has to be looked up.

That would be unremarkable if the tables were good. They are thin in three specific ways, and this rung is about all three, because a ladder that rests on a measurement should say plainly how much measurement there is.

What the number is, exactly #

The convention matters more here than the values, and it is worth stating before anything else.

A specimen is extended to a stated strain, held briefly, and unloaded. The recovery is the strain returned divided by the strain imposed, read immediately. Cotton returns 74 per cent of a two per cent strain under that convention, and 45 per cent of a five per cent one.

Three things are excluded by it and all three are large.

The delayed recovery is excluded. Leave the specimen and it goes on returning, for minutes and then for hours. The total recovery after a rest is larger than the immediate figure for every fibre in the table, and larger by different amounts for different fibres — which means the ordering by immediate recovery is not necessarily the ordering by total recovery, and this collection cannot say where the two orderings differ because it has one of them.

The conditioning is excluded. Recovery depends on moisture, and for the hygroscopic fibres it depends on it strongly. Every figure here is at standard conditions, and a wool fibre in a steamy room is a different fibre in this respect. Hanging a suit in a bathroom is not folklore; it is a change to the one input this ladder cannot compute.

And the load history is excluded. The convention is one extension from rest. A fibre that has been cycled recovers differently from one that has not, and every fabric anybody wears has been cycled.

Why the tables stop where they do, and what to do about it #

The measured strains are two per cent, five, and occasionally eight or ten. Nothing below one. That is a sensible range for a fibre-testing laboratory, whose specimens break somewhere between two per cent and forty, and it is the wrong range for a fabric.

A woven cloth just past its interchange budget is at a thread strain of a few hundredths of a per cent. A cloth two points past its budget is at two per cent, and that is the bottom of the measured range. So for most of the strains this collection actually needs, the answer is not in the table.

The easy response is to extend the lowest measured segment downwards. It is quietly disastrous. Cotton’s two points, extended linearly to a tenth of a per cent, give a recovery of 1.03 — a fibre returning more than it was given. Extended to eight per cent they give 0.16, which is not obviously absurd and is not supported by anything.

What the measurement does support is a bracket, and only because recovery falls monotonically with strain for every fibre in the table. Below the lowest measured point the recovery is between that point’s value and one. That is a wide interval near the bottom and a useful one, because it is honest and because the important qualitative claim — a small enough strain is nearly all returned — survives it.

The monotonicity is asserted rather than assumed, and 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 two the other way round.

The story that turns out to have nothing in it #

Textile writing offers an explanation for why some fibres recover and others do not, and it is a good story. A fibre whose stress–strain curve bends over early has a long non-linear range; a strain taken into that range is accommodated by something other than stretching bonds; and what was accommodated can be given back.

The story is testable here, because this collection carries a tenacity and a modulus for every fibre, and dividing the first by the second gives the strain at which a linear fibre of that strength would break. Comparing that with the measured breaking extension gives a number for how far from a spring each fibre is.

Wool and cotton say the story loudly. Wool’s measured extension is 5.7 times its own linear prediction, and it returns 69 per cent of a five per cent strain. Cotton’s is 1.2 times, and it returns 45.

Over six fibres, tau is −0.20. That is not a weak version of the story: it is the absence of one, and the fibres that kill it sit at opposite ends of the range. Nylon is nearly linear and recovers best of all. Viscose bends over as far as wool and recovers worst of anything in the table.

The assertion written into the machinery is on the refutation rather than on the value: the correlation must stay well under a half, which it does by a wide margin and would have to pass before any essay here was entitled to tell the story. An assertion written the other way — that tau is −0.2 — would be a claim about the six fibres that happen to be in the table rather than about the finding.

Why the refutation is worth having #

A negative result about a fibre property is not usually worth an essay. This one is, because of what rests on it.

The whole of this site’s account of fabric memory is a split: part of a cloth’s deformation is a mechanism and can be computed, part is the fibre’s and cannot. The value of the split depends entirely on the second half genuinely being a measurement. If recovery could be predicted from a stress–strain curve, then the fibre half would be computable too, the split would be a convenience rather than a boundary, and the geometric result — that a fabric’s recovery has a corner in it whose position has no fibre in it — would lose most of its force.

So the refutation is load-bearing. Recovery is a measurement, it stays one, and that is why the geometric half is worth computing.

What was counted, and how #

Ten fibres carry a recovery table and seven of them carry a crease-recovery angle as well. The breaking extensions are a separate measured table, carried apart from the tenacity and modulus on purpose, because the two disagree and the disagreement is a finding rather than an inconsistency: for cotton and flax the linear prediction is close, and for wool, silk and the melt-spun filaments it is out by a factor of three to six.

The linearity comparison is run at five per cent, which is the strain the most fibres have a figure at — six of the ten. Running it at two per cent changes the values and not the conclusion.

Kendall’s tau is computed from the concordant and discordant pairs rather than described, and the rank displacement is computed too, so that the refutation names its worst offender rather than gesturing at it.

What a fabric would need instead #

It is worth being concrete about what measurement would actually close the gap, because “the tables are thin” is a complaint and not a specification.

The quantity a fabric ladder needs is recovery at a tenth of a per cent, at half a per cent and at one per cent, for the common fibres, under a convention that states the rest interval. Those are strains a fibre-testing laboratory has no reason to visit: the load is small, the extension is at the limit of what an extensometer resolves, and nothing about the fibre is in danger. They are also the strains a shirt spends its life at, because a woven cloth hands its threads only what its interchange could not absorb, and that remainder is usually small.

The second thing it needs is the split between immediate and delayed recovery as a function of strain, rather than as two numbers at one strain. A fabric is loaded and unloaded thousands of times a day, and whether the delayed part has time to happen between cycles is the difference between a garment that keeps its shape and one that does not. The frictional band this collection computes has exactly the same character — it decides where a cloth comes to rest and says nothing about when — and the two unknowns compound.

Neither of those is a hard measurement. They are simply not the measurements a fibre laboratory was set up to make, and the gap between what is convenient to measure and what a downstream model needs is a familiar shape in every subject.

An extrapolation that respects its own ceiling #

The essay shows a linear extrapolation of cotton’s two points failing absurdly — a recovery above one at a tenth of a per cent — and concludes that the region below the table supports only a bracket. That is right about the linear extrapolation and it is not the only extrapolation available, because the absurdity has an obvious cause: the linear form is fitted to the ratio, and the ratio is the quantity with a ceiling on it.

Fit the recovered strain instead, which has no ceiling of its own and is bounded only by the imposed strain. Cotton returns 1.48 per cent of a two per cent extension and 2.25 of a five, so as a power law

recovered strain ∝ imposed strain to the power 0.46 — very nearly a square root.

That form cannot exceed one in ratio at large strains and it does exceed the imposed strain at small ones, which is where it stops being physical. Where it stops is the useful output: setting recovered equal to imposed gives

full recovery below about 1.1 per cent of strain.

So the two published points, read through a form that respects the bound, say that a cotton fibre returns everything it is given below a strain of about one per cent — and everything above it on a square-root curve. That is a much stronger statement than the bracket, it is consistent with the bracket, and it lands the threshold within the same order as cotton’s own reported proportional limit, which is not an input to it.

It is a fit to two points and should be treated as one. Two points determine a power law exactly, so nothing here is tested; a third measurement at ten per cent would test it, and one below one per cent would test the threshold directly. What the fit buys is a shape with the right asymptotics, which the linear form does not have, and a specific number for the experiment to aim at.

The crease check has a systematic, and it goes one way #

The cross-check against crease recovery compares a fibre’s recovery at its fold strain with the angle a folded fabric opens to, and reports agreement within a factor of two. There is a correction to make before that comparison, and it is not small.

A fold does not impose one strain. It imposes a gradient: zero at the neutral axis of the thread and a maximum at the outside. So the fibres near the middle are at strains where recovery is complete, and only the outermost are at the peak strain the check uses.

Weighting the recovery across the section — each fibre contributing to the recovered curvature in proportion to its distance from the axis — and using the square-root form above, cotton at a peak fold strain of five per cent gives a section-averaged recovery of

0.60, against 0.45 at the peak.

A third higher, and the correction is always upward, because recovery falls with strain and every fibre inside the outermost is at a lower one.

So the prediction to compare with a crease-recovery angle is not 180° times the peak-strain recovery but 180° times the section average — 108° rather than 81° for that cotton. The check as run is systematically low by about a third, and applying the correction moves every prediction the same way rather than scattering them.

That turns a loose agreement into a real test. The essay notes that its errors have both signs, which is what stopped the comparison being a validation; a systematic shift of a third moves the low ones onto the measurements and leaves the high ones high, so afterwards the residual signs mean something. A cross-check with a known bias removed is worth more than one with a wider tolerance, and this bias is computable from the same table the check already uses.

The one place a recovery figure can be checked #

There is one independent test available, and it is worth using because a table with no cross-check is a table.

The crease recovery angle is a fabric measurement — a specimen folded through 180°, loaded, released, and the angle it opens to. If a fibre returns a fraction r of a strain, and a fold imposes a curvature proportional to strain, then the angle recovered should be 180° times r, with no sett and no weave anywhere in the prediction. For the three fibres whose recovery is measured at the strain their own tightest fold imposes, the prediction lands within a factor of two of the reported angle in every case, and inside the reported range for one of them.

That is closer than a prediction with no fabric in it has any right to be, and it is not a validation, because the errors have both signs. What it does establish is that the recovery figures are the right size. A table that was wrong by a factor of two would show up here, and it does not.

Where the model stops #

Ten fibres is not a sample. A correlation absent over six points is weak evidence about fibres in general, and the honest statement is that the story is unsupported here rather than that it is false everywhere. What would settle it is recovery figures for twenty or thirty fibres at a common strain, which do not appear to exist in one place.

No range is carried. Every recovery figure here is a single reported value, because the sources report single values, and this collection’s habit of carrying an interval wherever one exists cannot be followed. That is a real weakness and it is the largest one in this ladder: a result that depends on cotton returning 45 per cent rather than 55 has no error bar to be checked against.

There is no mechanism anywhere in it. Why a polymer returns a strain is a question about chains and cross-links and crystallinity, and this collection models none of them. What is used here is a number and its convention, and everything downstream is arithmetic on geometry.

And the immediate-recovery convention misplaces exactly one fibre. Wool, taken at its own tightest fold, comes out looking like a fibre that does not return — and it has the best crease recovery of any fibre in the table. The reason is the convention rather than the argument, and it is worth having as a standing reminder that a measurement’s definition is part of the measurement.

The generalisation #

A model whose inputs are all computed and one measured should be built so that the measured one enters in one place and can be varied. Every result in this ladder can be re-run against a different recovery table by substitution, which is how the claim that a cloth’s recovery corner is fibre-free was established at all: swap the table, and see what moves.

The second lesson is about attractive explanations. A story that explains the two examples it was invented on will usually survive being told, and rarely survives being ranked. The way to test one is not to find a third example that fits but to order everything available by both quantities and count inversions, which takes a few lines and is decisive in either direction.

Who found it, and when #

Elastic recovery tables for fibres are from the standard physical-properties literature of the 1950s and 1960s, and the immediate-recovery convention is theirs. Breaking extensions are older still and are the first thing measured about any new fibre.

The linear-break comparison — tenacity over specific modulus, against the measured extension — is a standard way of saying how non-linear a fibre is, though it is more usually run the other way round to check a modulus. Using it as a candidate explanation for recovery and then rejecting it appears to be this collection’s own, and the result is unsurprising once stated: recovery is about unloading and the breaking extension is about loading, and there is no reason the two should be ordered alike.

Where the ladder goes next #

The fibre’s own numbers are needed again at a fold rather than an extension, and there they land in a place nobody arranged: the sharpest fold a cloth can make strains its fibres by exactly the amount that separates the fibres which crease from the ones that do not.

Sideways, the geometric half of the split has its own ladder, beginning with what a cloth returns and why the corner belongs to the sett, and the frictional half — which is neither fibre nor geometry — turns up as the load a tensioned cloth is left with.