Why agitation helps a cloth relax

Every procedure for finding out how big a fabric really is agitates it. The standard relaxation tests soak, tumble, and repeat; a mill steams and works its cloth; a garment maker washes a sample three times before cutting to it. Nobody disputes that this is necessary, and the reason given is that it lets the fabric find its own dimensions.

That is a description of what happens rather than an account of why. The rung that gave a cloth a resting band went one step further and said agitation helps because it lowers the effective friction rather than because it adds energy — which is closer to a mechanism and could not be computed with the constants this site carried, because it needs two friction coefficients and the site had one.

It has two now, and this rung spends them.

The mechanism, stated properly #

A relaxed cloth does not sit at the bottom of its energy well. The crimp is trying to redistribute, every crossing resists that redistribution with a friction, and the cloth stops wherever the restoring force falls below what the crossings can hold. So the resting state is an interval, and which point of it a piece of cloth is at depends on which side it arrived from.

Agitation does not add energy in any useful sense and does not lower a coefficient. What it does is put the contacts into a sliding state, and a sliding contact resists at the kinetic coefficient rather than the static one. So while the cloth is being worked, the interval it can stop in is the narrower one — and when the agitation stops, it stops somewhere inside that narrower interval rather than anywhere in the wider one.

That is the whole of it, and it explains three things the trade knows.

It explains why agitation is necessary rather than optional. A cloth left in still water relaxes only until the restoring force drops below the static limit, which can be a long way from the minimum.

It explains why more agitation stops helping. The narrower band is a floor. Once the cloth is inside it, further working moves it around inside the band and not towards the minimum, so a relaxation procedure has a point past which repeating it changes nothing — which is exactly why the standard procedures specify a number of cycles rather than a duration.

And it explains why the answer is a range rather than a value. Two pieces of the same cloth given the same treatment stop at different points of the same band, so a relaxed dimension has an irreducible scatter that is nothing to do with measurement error.

What it is worth, which is less than it looks #

The obvious arithmetic says the band’s width is friction over stiffness, so lowering the friction to three quarters lowers the band to three quarters.

The measured narrowing is 0.87.

The restoring force is not linear in the extension. It is nearly flat through the least-energy state and rises steeply away from it — that is the crimp stiffening as the two systems run out of room to interchange — so the band’s edges sit where the curve is steep, and pulling the limits down by a quarter moves those edges by much less than a quarter of the way in.

Which is not enough #

The rung below set up an arithmetic that this rung was expected to help with, and it helps a little.

The free bending bound predicts a resting band of about eleven per cent for a sheeting. A washing test on real cloth finds one to three. That is a factor of nearly four, and that rung resolved it by concluding the yarn must sit four to twelve times above the free bound in its own bending bracket — which is how a bracket of four hundred came down to a factor of three.

A washing test agitates. So some of the gap is this rung’s, and the amount is now computable: eleven per cent narrows to 9.5, and the gap goes from 3.7 to 3.2. Agitation closes about a fifth of it.

The coefficient ratio alone would have suggested a third, and the curvature takes most of that back. So the honest verdict is that two coefficients narrow that rung’s inference without removing the need for it: the yarn is still substantially stiffer than the free bound, the bending bracket is still narrowed by a measurement rather than by an argument, and agitation is a real effect of modest size.

The other four fifths #

Worth naming, because a fifth is not much and the rest is not mysterious.

A washing test is wet. Water swells cellulose, lubricates the fibre contacts, and softens the yarn — so the friction, the stiffness and the yarn’s diameter all move. That is almost certainly the largest term and it is the same gap the relaxation ladder records.

The yarn is stiffer than the free bound. The free bound assumes fibres slide past one another without resistance and a twisted yarn’s do not, so a real yarn is somewhere up its own bracket. That is that rung’s conclusion and it survives.

And the band is a plain-weave band in a model with a circular section, computed at a contact force from a stated thread tension, on a cloth whose thread lengths came from an assumed crimp division. Each of those is a small factor of its own.

What a standard procedure is doing, in these terms #

The relaxation standards specify a sequence — soak, tumble, repeat, measure, repeat until the measurements agree — and the sequence reads differently once the band is in view.

Soaking is not agitation. It swells the fibres and lowers the friction, which widens nothing and narrows nothing by itself; what it does is move the whole curve, so the band is around a different minimum. That is why a wet dimension and a dry one are different numbers rather than the same number measured twice.

Tumbling is the agitation and it is what puts the cloth inside the narrower band.

Repeating until the measurements agree is a stopping rule, and in these terms it is a test for having reached the floor. Once the cloth is inside the narrow band, another cycle moves it around inside the band and the measurements stop marching in one direction — they scatter instead. So the standard’s convergence criterion is not detecting an equilibrium; it is detecting the moment when the systematic movement is replaced by the band’s own scatter, and the residual scatter is the band’s width.

That reading makes a prediction the standards do not: the scatter a converged relaxation test settles into should be the shaken band’s width, and not smaller. For a sheeting that is about nine per cent of the extension range, which is a real and measurable quantity, and it is the obvious place to test this whole account against a laboratory rather than against another model. Nothing here has done that.

What was counted, and how #

The bands are read off the site’s own load–extension curve, at a stated friction and a stated position in the bending bracket, and the narrowing is the ratio of the two widths.

Three assertions and each is a relation. The shaken band is narrower than the stuck one — the direction, which would catch a sign error. The narrowing is greater than the ratio of the coefficients — the finding. And the narrowing rises as the ratio rises — the mechanism, which would catch a band that was being computed from something other than the friction.

The first version of the middle assertion said the narrowing equals the ratio, and it failed on its first run. That failure is where the result came from, and it is the fourth time in which an assertion written against the numbers rather than against the claim has turned up something. The house rule holds: assert the relation or the regime, never the value the defaults happen to produce.

The sampling was checked rather than assumed. A band is read off a sampled curve, so its width is quantised by the sample spacing; running at four times the samples moves the narrowing by three parts in a thousand, which is two orders of magnitude below the effect.

The kinetic-to-static ratio is a measurement with a range and every result is run at both ends of it. At 0.6 the narrowing is 0.76 and at 0.85 it is 0.93, so the finding — that the narrowing exceeds the ratio — holds across the whole range while the number does not.

What the account would predict about a test nobody runs #

The clearest way to state where this rung’s confidence ends is to name the measurement that would settle it.

Take one cloth, relax it to convergence by the standard procedure, and measure the dimensions of many specimens. The account says the scatter should be the shaken band’s width and not less — about nine per cent of the extension range for a sheeting — because that is the interval the cloth can be left in once the loops and crossings have stopped sliding systematically.

Then repeat with the agitation removed: soak, do not tumble, and measure again. The account says the scatter should widen to the stuck band, which is about a seventh larger, and that the mean should sit further from the relaxed state.

Neither has been done here and neither needs anything this site has: both are dimension measurements on many specimens of one cloth, which is a laboratory afternoon. The prediction is a ratio between two scatters rather than a value, which makes it insensitive to almost everything the model is uncertain about — the friction range, the yarn’s position in its bending bracket, the section shape — because those move both bands together.

That is the honest end of the argument. The mechanism is stated, the arithmetic is done, the size of what agitation buys is computed, and the check that would confirm or refute it is cheap and has not been run.

What the experiment would actually cost #

The scatter test is proposed above as a laboratory afternoon, and it is worth pricing properly, because the number of specimens it needs is the reason it has not been run and is also the reason its design is better than it looks.

The prediction is a ratio of two scatters: the unagitated band divided by the agitated one, which is the reciprocal of the narrowing. At the working coefficient ratio that is 1/0.87 = 1.15; at the ends of the reported range it is 1.32 and 1.08.

A ratio of standard deviations is estimated with a relative standard error of about 1/√(2n) per condition, so distinguishing 1.15 from 1.00 at three standard errors needs

1/√(2n) ≈ 0.05, which is n ≈ 200 specimens per condition.

Four hundred measured dimensions on one cloth. That is not an afternoon; it is a week of somebody’s time, and it explains the absence better than any lack of interest would.

Two things make the design worth the cost anyway, and both come from its being a ratio.

Every systematic term cancels. The friction coefficients, the yarn’s position in its bending bracket, the section shape, the crimp division and the contact force all move the two bands together, so none of them appears in the ratio. That is unusual in this collection, where most predictions carry a bracket or a fitted constant, and it is what makes a fifteen per cent effect worth chasing at all.

And the ratio measures a fibre constant from a fabric. Inverting the narrowing gives the kinetic-to-static ratio — the one quantity this rung had to import from outside with a range on it — so a fabric-dimension experiment returns a fibre-friction number. That is a second independent route to it, and it is worth having beside the hysteresis-loop intercept, which measures the coefficients’ difference from a cyclic fabric test. Two fabric-level routes to one fibre-level constant, which must agree, is a check neither of them supplies alone.

The cheaper version, if four hundred specimens is too many, is to run the ratio at the extremes rather than at the working value. A cloth of a fibre pair with a low kinetic-to-static ratio predicts a 1.32 ratio of scatters, which needs only about thirty specimens per condition to resolve — and finding the effect where it is largest is the ordinary way to establish a mechanism before measuring it where it matters.

That is where the honest end of this rung is. The arithmetic is done, the prediction is a ratio, the cost is a week or an afternoon depending on which fibre pair is chosen, and nothing here has been measured against a laboratory.

Where the model stops #

Nothing here is wet, and a relaxation procedure is. Every constant in this rung is a dry cotton constant. Water changes the friction, the bending stiffness, the yarn diameter and the fibre’s own modulus, in directions that are known and by amounts that are not modelled anywhere on this site.

Both coefficients are quasi-static and agitation is not. A tumble dryer works a cloth at speeds and accelerations that make the contact dynamic; kinetic friction is speed-dependent; and no number here has a speed in it.

There is no ageing at rest. A contact that has been sitting still recovers its static coefficient over time, which is why a cloth folded for a year holds its crease and why a relaxation test specifies conditioning. That is a real term in the story of why agitation is needed and it is absent.

And the band is the available range rather than a prediction of where a cloth stops. Which point of the band a given piece reaches depends on its history — which side it came from, and how the agitation was distributed — and nothing here predicts it. The band is a bound on the scatter and not a model of it.

The generalisation #

A treatment that lowers a threshold does not narrow the outcome in proportion to the threshold, unless the response is linear — and the response is usually least linear exactly where the outcome is widest.

That is the transferable half. A resting band exists because a restoring force is small near an equilibrium; the same shallowness that creates the band is what makes the proportionality fail. So the two are not independent: every problem in which a tolerance comes from a threshold on a nonlinear response has this shape, and the naive scaling always overstates what lowering the threshold buys.

The second half is about explanations that are correct and unquantified. “Agitation lowers the effective friction” was right. It was also a placeholder for two numbers the site did not have, and it sat in a finished essay passing every check. A sentence that requires a constant the model does not carry is invisible to every gate here, and there is no obvious machinery that would find one — so it is a habit, and the habit is to notice when an explanation names a quantity and then does not use it.

Who found it, and when #

Relaxation procedures for fabric are old and standardised, and the requirement to agitate is written into every one of them. The reasoning given in the standards is operational rather than mechanical: agitate until repeated measurements agree.

The static-kinetic distinction for fibre friction was measured carefully from the 1940s onwards, and the textile literature separates the two more carefully than most because yarn friction is directional as well.

That relaxation is hysteretic, and that a relaxed dimension is therefore a range, is well established and is what the standards are working around when they specify cycles.

What is this site’s is the arithmetic: how much of the observed gap two coefficients account for, and the finding that the narrowing is smaller than the ratio because the restoring force is not linear. Both fall out of a model built for a different purpose entirely.

Where the ladder goes next #

Sideways, the same two coefficients decide what happens in a knit, and a knit is the extreme case: nothing elastic opposes a knit’s extension at all, so friction is not one term among several but the whole of the mechanism — which is why a knit’s dimensions depend so completely on how it has been treated.

Along the relaxation ladder, the gap that matters is water — and it is closed now, in the locus a wetted cloth moves to and in the two shrinkages one tape measure records.