A cloth relaxes until its threads stop pushing

The finishing field is built on a distinction it states carefully and then has to keep restating. A dimension without a state is not a measurement: a cloth’s length and width mean nothing unless it is said whether the cloth is off the loom, dry-relaxed, wet-relaxed or fully relaxed, because the four differ by several per cent and the difference is the crimp coming back.

That framing has the relaxed state as a place — somewhere a cloth arrives at when it is left alone in the right conditions, and stays.

It is not a place. It is a band, and the width of the band is one division.

The claim #

A cloth relaxes until its threads can no longer push hard enough to move a crossing, which is not the same as relaxing to its lowest-energy state.

Formally: the cloth slides along its locus while

|dU/dλ| > F_friction

and stops as soon as that fails. The resting state is anywhere in the interval around the minimum where the slope is shallower than friction, and the width of that interval is friction over stiffness.

Three things follow that the geometry alone cannot say. Relaxation is hysteretic, so a shrinkage figure is a range rather than a value. Agitation helps because it lowers the effective friction, not because it adds energy. And a limper yarn gives a wider band, so the softest cloths are the least dimensionally repeatable — which inverts the usual expectation that a soft cloth is a well-behaved one.

Where the two quantities come from #

Both halves were built for other reasons and neither was built for this.

The energy slope is the load–extension rung’s: the bending energy of the two thread systems, differentiated along the locus, which gives the tension in an end. Near the minimum it is small and it passes through zero, which is why the band is a band rather than a point — a slope that fell away steeply would leave friction with nothing to hold.

The friction is the crossing force times a coefficient: μ·2T·sin θ, the force it takes to make one crossing slide. The tension in that expression is the residual tension in a relaxed cloth, which is small and is stated rather than derived — the weakest input on the page and one every number is proportional to.

The band is where the first is smaller than the second, and it is read off the same curve the crimp-ratio rung minimises and the load–extension rung differentiates. Three rungs, one function, which is worth saying because it is the reason the three agree with each other rather than merely being consistent.

The numbers, and what they say about the yarn #

For a sheeting at an ordinary friction, with the rigidity at the free bound:

Those are large. A real cotton cloth’s relaxation hysteresis is one to three per cent, not seven to eleven, and the discrepancy is the most useful thing on the page.

The band is friction over stiffness, so a band that is too wide means the stiffness is too low. Run the same computation with the yarn a few times stiffer than the free bound and the band comes down into the observed range: at four to five times the free bound it is one to three per cent at every friction in the range.

That is an independent measurement of where a yarn sits in its own stiffness bracket, and it comes from a washing test. It agrees with the other one available — the cantilever, which is a strip of cloth hanging over an edge — and the two are not measurements of the same thing in any obvious sense.

Why the band is not symmetric #

The band in the table runs further below the minimum than above it — roughly two to one — and that asymmetry is not an artefact.

The energy well is steeper on one side than the other, because a cloth’s locus is not symmetric: extending it along the warp runs towards the state where the warp goes straight, and contracting it runs towards the state where the weft jams, and those two ends are different distances away and are approached at different rates.

So a cloth relaxed from the stretched side stops further from the minimum than one relaxed from the compressed side. In practical terms, a cloth that has been held under tension and then released will sit shorter than its own least-energy length by more than a cloth that has been compressed and released will sit longer.

That is exactly the direction the finishing field’s own results run. What comes off the loom is a cloth that has been held under warp tension for the whole of its weaving, and it relaxes a long way; pre-shrinking works by compressing the cloth mechanically and then letting it back, which is the other approach direction. The two processes are not opposite ends of one axis. They arrive at different points of the same band.

Why agitation works, and what kind of thing it is #

Every relaxation procedure in the trade involves mechanical action: a wash cycle rather than a soak, a tumble rather than a hang, a mechanical finish rather than a wet one. The usual explanation is that agitation “helps the cloth relax”, which is true and is not a mechanism.

The mechanism is in the inequality. A cloth stops when the energy it can release is less than what friction takes to move a crossing, so anything that reduces the friction lets it go further — and mechanical action does exactly that. A crossing that is already moving has a lower resistance than one that is at rest, which is the ordinary distinction between static and kinetic friction, and vibration removes the static component altogether.

Two predictions follow from putting it that way rather than the vague way.

Agitation should narrow the band rather than shift it. It lets a cloth reach nearer the minimum from whichever side it came, so the two approach directions converge — which is why a fully relaxed state is defined by a procedure that includes agitation and why it is the only one of the four states that is reproducible.

And more agitation should stop helping. Once friction is effectively removed, the cloth is at the minimum and there is nowhere further to go. The finishing field’s own relaxation figures show exactly that shape: successive wash cycles give a diminishing return that flattens rather than continuing.

What a shrinkage specification is really saying #

A shrinkage figure on a label is a single number with a tolerance, and this rung says the tolerance is not a measurement uncertainty.

The scatter is in the cloth. Two pieces of the same fabric relaxed by the same procedure will stop at different points of the same band, because they arrived from slightly different tensions and were handled slightly differently. The band is several per cent wide at the free bound and one to three at a realistic stiffness, and that is the same order as the tolerance a shrinkage specification carries.

So a specification of “3% maximum shrinkage” is not a claim about a quantity known to a tenth of a per cent. It is a claim that the top of the band is below three, and the width of the band is set by two things a finisher can move: the friction, through the finish, and the yarn’s stiffness, through the count and the fibre.

That gives a practical statement the geometry could not: a softer finish makes a cloth’s dimensions less predictable, in exact proportion. It is the same μ that decides how far a cut edge frays and whether a seam slips, so a finisher choosing softness is choosing three things at once and cannot separate them.

The band is why two mills disagree about one cloth #

The hysteresis has a consequence for how a fabric is bought and sold, and it explains a familiar and otherwise mysterious argument.

Two laboratories test the same cloth for shrinkage and report figures a per cent or two apart. Both are competent, both follow the same standard, and neither is wrong — because the standard specifies a procedure and a procedure does not fix which point of the band a specimen stops at. The two specimens arrived at the wash from different histories: different tension on the roll, different handling, different time since the last relaxation. Each stopped where its own approach direction and its own friction put it, and the difference between them is the band.

That reframes what a round-robin between laboratories is measuring. A round-robin is meant to expose differences in method, and here it will find a scatter no method can remove — so a standard that tightens its procedure will not narrow the result past the band’s width, and effort spent tightening it past that point buys nothing.

The one thing that does narrow it is agitation, for the reason above: it lowers the friction, so it lowers the band’s width, so it makes the two specimens converge. That is why the fully relaxed state is the reproducible one and the others are not, and it is a stronger statement than “agitation helps a cloth relax” — it says agitation is the only term in the expression a test procedure can reach.

And it gives a check anybody can run. The scatter between laboratories should be larger for the softer cloth, because friction over stiffness is larger there, and it should be smaller for a firm one. That is a prediction about a quantity every testing house already has in its records and nobody has read that way: the reproducibility of a shrinkage test should correlate with the limpness of the fabric, and if it does not, the friction term is not doing what this rung says it does.

That is the kind of prediction this rung is good for, and it is worth noticing that it needs no new measurement at all — only a re-reading of records already kept for another purpose.

What was counted, and how #

The energy and its slope are the load–extension rung’s, unchanged: a Peirce state at the cloth’s construction, its constant-thread-length locus at 641 samples with both thread lengths reconstructed at every point, and the closed-form bending energy 2(B₁θ₁ + B₂θ₂)/D at each.

The friction force is μ times the contact force at the least-energy state, and the band is every sampled state whose slope is below it.

Two assertions guard the result and one of them is the interesting kind. The band must straddle the minimum rather than sitting to one side of it, which would fail immediately on a sign error and would produce a perfectly plausible-looking interval. And the orderings must hold across both sweeps: more friction is a wider band, a stiffer yarn is a narrower one. The second of those is asserted across the bracket rather than at one point, because it is the ordering the whole stiffness inference rests on.

The residual tension is stated at a fifth of a newton, which is a small tension in a relaxed cloth and is not a measurement. Every band width is proportional to it, and the inference about the yarn’s stiffness is proportional to it too — so the conclusion that a real yarn is a few times the free bound is a conclusion at that stated tension. It is the weakest step in the chain and it is written into the source beside the number.

Where the model stops #

The residual tension is stated rather than derived, as above.

There is no compression term. Threads flatten where they cross and flattening stores energy, and the site’s racetrack section has no stiffness in it. A closely set cloth’s compression energy is not small, and adding it would deepen the well and narrow the band — in the same direction as the stiffness correction, which means the two are not separable by this measurement.

Static and kinetic friction are one coefficient here. The whole account of why agitation works turns on their being different, and the model uses a single μ. Making the argument quantitative rather than directional needs both, and neither is on this site.

And the model has no fibre swelling in it. Wet relaxation is not only mechanical: a cotton fibre swells in water, which changes the yarn diameter and therefore the whole geometry, and mercerising is the extreme case of that. The band computed here is the mechanical part of a process that has a chemical part.

The generalisation #

The result is a statement about any system that settles under friction, and it is one that is routinely got wrong in the direction of over-precision.

A minimum plus friction is not a minimum; it is a set, and the set’s size is friction over curvature. Anything that comes to rest against friction — a mass on a rough incline, a beam settling on its supports, a mechanism with a stiff joint — stops in a band around its equilibrium rather than at it, and the band is not a measurement error. It is a property of the system, and it is the same width whoever measures it.

Two things follow that are worth carrying. The resting state depends on the approach direction, which means “the equilibrium” is an incomplete specification and any procedure that reports one has to say how it got there. And reducing friction reduces the band without moving the equilibrium, which is why a procedure that includes vibration is more reproducible than one that does not — in metrology, in mechanical testing and in a washing machine alike.

The third point is the one this rung uses in the other direction. An observed band width is a measurement of friction over stiffness, so if either is known the other follows. That is how a washing test turned into a statement about a yarn’s bending rigidity, and the general form is that a hysteresis is a measurement rather than a nuisance.

Who found it, and when #

Relaxation shrinkage and its dependence on mechanical action is trade knowledge of long standing and is codified in the wash-test standards; the four-state framework this site uses — loom, dry-relaxed, wet-relaxed, fully relaxed — is Munden’s, from the knitted-fabric work of the 1950s and 1960s, and the whole reason it exists is that dimensions without a state are not reproducible.

The energy formulation of fabric mechanics is Olofsson’s, from 1964, and friction at the crossings as the source of a fabric’s hysteresis belongs to the same tradition; Grosberg’s work on fabric bending hysteresis in the 1960s makes precisely this argument for bending — a fabric’s moment–curvature curve has a frictional plateau because a crossing does not move until the moment exceeds what friction holds it with.

What is done here is to run that argument along the extension locus rather than in bending, and to read the result backwards. Grosberg’s frictional plateau is a nuisance in a bending measurement; the same plateau in a relaxation measurement is a way of weighing a yarn’s stiffness with a washing machine.

Where the ladder goes next #

The missing compression term is the next thing this whole group owes, and it would deepen every well on this site by an amount nobody has computed.

Sideways, the same band read at the crimp ratio rather than at the length says how firmly a cloth’s crimp division is settled — the shallow-welled cloths are the ones whose measured crimp scatters, and the two statements are one curve read twice.

Further out, static against kinetic friction is the missing pair. Every account of why mechanical action helps a cloth relax is an account about the difference between them, and this site has one coefficient.