Relaxation is the crimp coming back

The commonest explanation of why a cotton shirt shrinks is that the fibres shrink. It is nearly always wrong, and the way it is wrong is instructive: it locates the effect in the material when the effect is in the structure.

A cotton fibre does not get shorter when it is wetted. It gets fatter — it swells across its axis by twenty per cent or more and along its axis by about one per cent — and if the shirt were made of straight fibres laid parallel it would barely change length at all. What makes the shirt shrink is that its threads are not straight.

The one line of arithmetic #

A thread that goes over and under is longer than the cloth it crosses. This site calls that excess the crimp and has computed it since the foundation:

c = (thread length ÷ cloth length) − 1

Turn it round and it says something more useful. A thread of length l lying at crimp c spans

p = l / (1 + c)

of cloth. The thread length is fixed — a yarn does not lengthen or shorten when a fabric relaxes, and this is the assumption everything rests on — so if the crimp changes from c₁ to c₂ the cloth’s dimension changes by

shrinkage = (c₂ − c₁) / (1 + c₂)

and that is the whole of relaxation shrinkage. Everything else in this essay is about where the two crimps come from.

Note what the formula does not contain. No fibre property. No count, no twist, no fibre modulus, no swelling coefficient. Two crimps and nothing else. A cloth is a machine for storing length in bends, and shrinkage is the machine giving the length back.

Where the relaxed crimp comes from #

The relaxed state is the one this site can already solve. Peirce’s geometry takes two yarn diameters and two thread spacings, imposes the closure condition h₁ + h₂ = D — the two systems between them fill the thickness of the cloth — and returns the crimps and weave angles that satisfy it.

That closure condition is a statement about a fabric in equilibrium with itself, which is exactly what a relaxed cloth is. So Peirce answers the question what crimp does this cloth want?, and the answer for a balanced plain weave at a spacing of 0.42 in a yarn of diameter 0.2 is 12.79 per cent in each direction.

The equations are one condition short, as yarn.js says at length: they do not decide how the crimp divides between warp and weft, so a ratio has to be supplied. Taking it as one — an even division — is what “balanced” means here, and it is an input rather than a result.

Where the loom crimp comes from, and why it is not solved #

The first attempt at this essay solved the loom state with Peirce as well, at a crimp ratio reflecting the warp’s tension, and then relaxed from it. The result was wrong in a way worth recording, because it looked entirely plausible.

With the diameters fixed, Peirce’s closure condition makes crimp a conserved quantity. h₁ + h₂ = D says the two crimp heights add to a constant, so crimp taken up by the warp is crimp given back by the weft. Solve two states that way and one direction shrinks while the other grows. That is a real effect — it is crimp interchange, and pulling a fabric warpwise really does make it narrower — but it is not what a wash does. A wash shrinks a cloth in both directions.

The mistake was applying an equilibrium condition to a state that is not in equilibrium. A cloth on a loom is under load, its thickness is not settled, and h₁ + h₂ = D has no business being imposed on it.

So the loom state is taken for what it is: the relaxed cloth with a stated fraction of each crimp pulled out of it. The warp’s is pulled out by the let-off and take-up, which act on it continuously through weaving. The weft’s is pulled out by the temples, which hold the cloth to the reed width as it is formed. Both fractions are inputs, both depend on the machine and the setting, and no number in this field is quoted without them.

What comes out #

At seventy per cent of the warp crimp removed and twenty-five per cent of the weft’s:

The area falls by 10.55 per cent. These are not measurements of a real fabric and are not offered as any; they are what the model says about a cloth whose relaxed crimp is 12.8 per cent and whose loom took out the stated fractions. What makes them worth printing is that they are the right size — published relaxation shrinkages for cotton shirtings run from three to eight per cent warpwise and one to three per cent weftwise, and a model landing outside that range would be evidence that something was wrong with it.

The two mechanisms, separated #

There is a second thing happening in a wash and the arithmetic above deliberately leaves it out, because leaving it out is what shows the first mechanism clearly.

Wet a cotton fibre and it swells across its axis by a fifth or more. In a cloth that means D — the sum of the two diameters, which is the thickness the two systems between them must fill — goes up. And since the relaxed crimp is whatever satisfies h₁ + h₂ = D, a larger D means more crimp in both systems, so both dimensions have further to fall.

The two mechanisms are separable and the model separates them. Turn the swelling off and the relaxation is pure crimp recovery, bounded by the crimp the dry cloth wants. Turn it on and the ceiling itself moves upward: at a twenty per cent swelling the relaxed crimp rises from 12.79 to 20.26 per cent, the warpwise shrinkage from 7.94 to 11.79, and the area from 10.6 to 15.5.

That is why a fabric goes on shrinking after its first wash, and why the shrinkage measured dry and the shrinkage measured wet are two different quantities that get quoted interchangeably. A dry relaxation gives back the crimp the cloth had; a wet one gives back the crimp the swollen cloth wants, and then the fibre dries and leaves the cloth in the smaller state.

Why the direction of the argument matters #

It would be possible to run this backwards: measure a fabric’s shrinkage, deduce the crimp difference, and report that. Mills do something like it. The reason this essay runs forwards instead is that the forward direction makes a prediction and the backward direction makes a description.

Run forwards, the model says: given a weave, a sett and a yarn diameter, here is the crimp the cloth wants, and given a loom tension, here is what it will do. Every one of those inputs is known before the fabric exists. A designer choosing between a plain weave and a 2/2 twill at the same sett can be told which will move more in the wash, and by roughly how much, without weaving either.

Run backwards, the model says: this piece shrank by six per cent, therefore its crimp changed by so much. That is arithmetic on a measurement and it predicts nothing about the next fabric.

The forward direction is only available because the relaxed crimp is computable rather than measurable — which is what Peirce’s geometry buys, and which is why this site has kept a solver rather than a table since the foundation.

What was counted, and how #

Two states, one thread, and a consistency check that is not a formality.

The relaxed state is peirce() at the given spacings, and its solution is fed back through assertPeirceConsistent, which re-runs the model’s own equations forwards and requires the residual below 10⁻⁹. That check has been in yarn.js since the foundation and it is the reason a solver failure here would be loud rather than quiet.

The loom state is arithmetic: each crimp multiplied by (1 − tension), each spacing recovered as l/(1 + c).

Then the shrinkage is computed twice, once from the crimps and once from the spacings, and the two are asserted equal to within 10⁻¹². That is the check worth having. The two routes share the definition of crimp and nothing else, so a sign error, an index swap or a confusion between p₁ and p₂ shows up as a disagreement rather than as a plausible number.

That last one is not hypothetical. The span a thread’s own repeat covers is measured across the threads it crosses, so for the warp it is the weft spacing; crossing the two over produces a cloth that grows when it is relaxed, and it did, and the two-route check is what found it.

Where the model stops #

The thread length is conserved and real threads do not quite conserve it. A yarn under repeated wetting and drying consolidates: its twist settles, its fibres migrate slightly, and it can shorten by a fraction of a per cent on its own account. That is a real contribution to shrinkage and it is not here.

The relaxed crimp ratio is an input. How the crimp finally divides between warp and weft depends on the two yarns’ bending stiffnesses, which Peirce’s geometry knows nothing about — a stiff warp and a limp weft do not divide it evenly. The essays here take a ratio and say so.

One relaxation, not a sequence. Real fabric loses shrinkage in instalments and settles asymptotically. The states here are equilibria and the path between them is not modelled.

And the whole calculation is for a plain weave. Peirce’s geometry is a plain-weave geometry: it assumes every crossing is a crossing. A twill has floats, its threads bend less often, and its crimp is lower — which is why a twill relaxes less and why the ceiling essay matters more for a satin than for a sheeting.

Why the effect is so much larger than it looks #

Twelve per cent crimp gives eleven per cent of length back. That seems modest until it is compared with the deformations a fabric will tolerate in other ways.

A cotton yarn breaks at about seven per cent extension. A woven cloth pulled along a thread direction extends by a few per cent before the yarns themselves begin to take load, and every bit of that extension is crimp being pulled out — which is the same mechanism read the other way. So the length available in the crimp is comparable to the entire extensibility of the fabric in its own directions, and vastly larger than anything the fibre contributes.

This is why finishing is a serious industry rather than a detail. A structure that stores ten per cent of its length in bends will give that length up to whatever asks for it first, and if the finisher does not ask, the customer’s washing machine will.

What a designer can actually control #

Three things, and it is worth separating them because two are cheap and one is not.

The weave. Fewer interlacings mean less crimp mean less to give back. A 2/2 twill at the same sett and yarn has appreciably lower crimp than a plain weave, and a satin lower still. This is free — it is decided when the cloth is designed — and it is the reason a satin-back fabric behaves differently in the wash from the plain weave beside it.

The sett. Crimp rises with closeness, as this site computed at the foundation: crowd the threads and each must rise the same height over a shorter run, which is a steeper angle and more length spent. An open cloth has less crimp and less relaxation shrinkage, and also less of everything else.

The loom tension, which is the one that costs. Weaving with less warp tension leaves less crimp to give back, and it also lowers the loom’s productivity, makes the shed less clean and raises the end-breakage rate. The tension is set by what the machine needs, not by what the finished fabric would prefer, and that conflict is the reason compressive shrinkage exists as an industry.

Pricing the three levers against each other #

The three things a designer controls are listed as two cheap and one expensive, and none of them is given a number. All three are computable from the arithmetic already on this page, and the comparison changes which one is worth reaching for.

The weave. A thread only deviates from straight where it changes face, so a weave with a float of f turns once every f + 1 picks against a plain weave’s once every two — and its crimp is 2/(f + 1) of the plain weave’s. Feeding that through the shrinkage formula at the same seventy per cent loom tension:

A factor of 3.7 between the extremes, at the same yarn, the same sett and the same loom, decided entirely by the draft.

The loom tension. The shrinkage is very nearly proportional to the fraction of crimp the loom removes, so halving the warp tension halves the shrinkage — from 7.9 to 4.0 on a plain weave.

Set those two side by side and the comparison is decisive. Going from a plain weave to a three-one twill is worth the same as halving the loom tension, and one of them costs nothing while the other costs productivity, shed quality and end breakages.

Which recasts the finishing problem #

That has a consequence for the operation the whole ladder ends in. A fabric with 7.9 per cent to give back needs a pre-shrinking range working hard and leaves a large residue; one with 2.2 per cent needs very little.

So a satin is cheap to finish and a sheeting is expensive, and the difference is a property of the draft rather than of anything the finisher can influence. That is worth knowing before the cloth is designed, because the finishing cost is usually treated as a fixed overhead per metre and it is not.

It also explains a pattern in what gets sanforised. The cloths carrying the trademark are the plain and near-plain weaves — shirtings, sheetings, drills — and the satins and long-float constructions mostly do not need it. That looks like a market segmentation and it is an arithmetic one.

And the honest caveat on the table #

The comparison is at constant sett, which is what the essay’s own list specifies and is not how the two cloths would actually be made.

A satin can be set sixty per cent closer than a plain weave in the same yarn, and crimp rises with closeness — so a satin woven at its own proper density has more crimp than the table’s constant-sett figure, and the gap narrows.

How far it narrows is not computed here, because it needs the crimp against sett curve run at each weave’s own ceiling, which is a sweep this essay does not do. The direction is certain and the size is not, so the table should be read as the draft’s contribution in isolation rather than as a prediction of what two finished cloths will do.

What survives the caveat is the ordering and the mechanism. Fewer interlacings mean less crimp mean less to give back, at any sett, and the lever is free where the alternative is not.

The same arithmetic run backwards is a measurement #

There is a useful inversion here and mills use it, so it belongs in the essay that sets the arithmetic out.

Crimp is measured destructively: unravel a thread from a marked length of cloth, straighten it under a small standard tension, and the ratio of the two lengths is the crimp. That is a laboratory operation on a sample, and it destroys the sample.

Shrinkage is measured non-destructively, by marking a specimen and relaxing it. And since shrinkage is (c_relaxed − c_loom)/(1 + c_relaxed), a relaxation measurement plus one crimp measurement gives the other crimp — which means a mill can characterise a loom once, destructively, and then infer the loom-state crimp of everything it weaves from a relaxation test alone.

The arithmetic converts a destructive measurement into a repeatable one, which is a common and underrated use for a model: not to predict a quantity nobody can measure, but to make a quantity that is expensive to measure derivable from one that is cheap.

Who found it, and when #

Peirce’s 1937 paper is the source of the geometry, and the crimp bookkeeping is older than that — it is implicit in any fabric analysis, which has always measured a thread’s length by unravelling it and straightening it against the cloth it came from.

What the twentieth century added was the recognition that the loom state is a state, and that a fabric measured in it is not a fabric anybody will ever hold. The standards that require a specimen to be relaxed by a specified procedure before being measured are the institutional form of that recognition, and they exist because dimensions quoted without a state turned out to be a commercial problem.

The mechanism in this essay is not novel and is rarely written down in this form. What is unusual is doing the arithmetic with a solver rather than a rule of thumb, and finding that the two routes to the same number agree — which is a small thing to have checked and is exactly the check that caught the indexing error.

Where the ladder goes next #

The table above has two rows and they are different sizes, which is not an accident of the numbers chosen: the next rung asks why the warp shrinks more and finds the answer in the machine rather than in the cloth.

After that comes the bound — a cloth cannot shrink past its own crimp — and then what an industry does about it, which is to do the shrinking in advance and sell the residue.