What wetting does to the bending limit

There is a limit on how sharply a thread can be bent that has nothing to do with what it is made of. A rod of radius r bent to a centre-line radius below r is occupying its own space, and at exactly r the inside of the bend has closed.

A relaxed knitted loop’s tightest bend sits at 2.04 yarn radii — twice the limit — and the ratio falls as the fabric tightens.

Wetting a fabric tightens it, in the only sense the geometry recognises.

The substitution #

A wet cotton yarn is about a tenth thicker than a dry one, and its loop length is unchanged: wetting swells a fibre across its axis and barely along it, so a yarn grows in diameter and not in length.

A yarn’s diameter goes as the square root of its count, so a diameter ten per cent larger is the diameter of a dry yarn of 1.21 times the count. Every number here is computed that way, with the substitution stated.

What it does to the ratio #

The bend radius a loop takes is set by the loop’s geometry, which depends on the ratio of the yarn’s diameter to the loop length. A thicker yarn in the same loop makes a tighter fabric and a sharper crest.

For a twenty tex cotton at a three and a half millimetre loop, the ratio of the tightest bend radius to the yarn’s own radius goes from 2.04 dry to 1.89 wet at a ten per cent swelling, and to 1.77 at twenty per cent.

So wetting moves a fabric about a tenth of the way from where it sits towards the limit at one — and a fabric that was already tight moves from a less comfortable place.

The tight case #

The tightest jerseys anybody knits sit at a ratio of about 1.67 dry, at a two point eight millimetre loop — a tightness factor near the top of what a machine will make.

Wet, the same fabric sits at about 1.55, and at the top of the swelling range at 1.45.

That is more than half the way from a relaxed fabric’s two to the hard limit at one, and it is a fabric that is being asked to accept its yarn passing very close to its own space.

Which is why a tight knit is finished carefully #

The arithmetic gives a mechanism for something the trade knows as a handling rule.

A very tight knitted fabric is difficult to wet-process. It resists penetration, it takes up dye unevenly, and it is prone to permanent creasing and to a hard, boardy handle after drying.

The usual explanations are about access — a tight fabric is hard for liquor to get into — and they are right and incomplete.

This adds a second: a tight fabric’s yarn is being bent close to its own geometric limit, and wetting moves it closer. At that point the fibres on the inside of every crest are being compressed hard and the section is being forced to rearrange, and a fabric in that condition dried under restraint will hold whatever it was forced into.

A creased tight knit is a fabric whose yarn was bent past what it could accommodate while wet, and that is a different failure from a fabric that was merely folded.

And why a swelling limit exists at all #

Following it to the end gives a genuine ceiling.

The ratio falls with the tightness factor and with the swelling. Setting it to one and solving says how tight a fabric can be while wet, and the answer is a tightness factor of about eighteen for a cotton at the working swelling — against about twenty dry.

Eighteen is inside the range the trade reaches. The tightest technical knits and the finest sportswear fabrics are quoted there.

So the prediction is that a fabric at the top of the achievable tightness range should behave differently wet from one a little below it — not gradually worse, but qualitatively so, because its yarn is at a limit rather than approaching one.

The mitigation #

A bundle is not a rod and the limit is softer than the arithmetic says, and it is worth being clear about how much.

A solid rod bent to its own radius has its inside material in contact with itself and nowhere to go. A bundle has somewhere: the fibres on the inside of the bend migrate outwards past their neighbours, and the section becomes kidney-shaped rather than failing.

So a real yarn can be bent somewhat past the geometric limit, by an amount that depends on how much room its fibres have to rearrange — which is set by the packing factor.

A wet yarn is a special case here and it cuts both ways. Its fibres are swollen, so there is less void space and less room to rearrange, which makes the limit harder. And water lubricates the fibres against one another, which makes rearranging easier.

Which dominates is not computed here and it is the honest limit on the whole rung.

What a dry fabric does not have to worry about #

The rung is about a wet fabric and the comparison with a dry one is worth stating, because it explains why nobody has noticed.

A dry relaxed jersey sits at 2.04, which is comfortably clear of the limit. Nothing in ordinary dry handling moves it: pulling the fabric straightens its loops and raises the ratio, pressing it flattens the yarn and changes the section rather than the centre line, and neither approaches one.

So the constraint is invisible in every dry operation, and the fabric only comes near it in two circumstances: when it is knitted very tight, and when it is wetted.

The first is a construction limit the trade already has other names for. The second is a process condition that lasts for the length of a dyeing cycle and then goes away.

That is a fair description of why a geometric limit that sits within reach has never been articulated: it binds only transiently, and the failures it causes are attributed to the process rather than to the geometry.

What was counted, and how #

The maximum curvature is reported by the collection’s own loop solve, as part of the same output that gives the energy and forces. It has been available since the loop was first solved and has never been compared against the yarn’s own radius until this ladder.

The yarn’s radius comes from the count, the fibre density and the packing factor by the site’s own volume arithmetic.

The swelling is the collection’s own wet table: a working wet-to-dry yarn diameter ratio of 1.10 with a range of 1.05 to 1.20, arrived at by finding that a yarn’s voids cannot absorb its fibres’ swelling.

The substitution — a diameter ratio s is a count ratio s² — is exact for a diameter and is stated wherever it is used.

Why a fibre’s swelling matters more than a yarn’s #

There is a subtlety about which diameter is doing the work, and it decides how large the effect really is.

The bending limit compares the loop’s centre-line radius with the yarn’s radius, so it is the yarn’s diameter that matters and the substitution above is right.

But the mitigation — the fibres rearranging past the limit — depends on the fibre’s own size and on how much room there is between fibres. A wet yarn’s fibres are individually swollen, so they fill more of the yarn and there is less room to rearrange.

So the wet case is worse than the substitution says: the geometric limit has moved closer and the softening that lets a real yarn exceed it has been reduced.

That is a genuine compounding and it is not computed here. What can be said is that both effects run the same way, so the wet fabric is worse off than the ten per cent suggests, and by an unknown further amount.

What the fibre table predicts #

The effect scales with the fibre’s swelling, and the collection’s table spans a large range.

Viscose swells thirty-five per cent in diameter, so a viscose knit’s ratio moves nearly twice as far as a cotton’s — from 2.04 to about 1.66 at the working figure, which is where a dry cotton’s tightest construction sits.

Cotton at twenty per cent fibre swelling gives the ten per cent yarn figure used above.

Wool at sixteen is a little less.

And polyester at under two per cent barely moves at all.

So the prediction is that viscose knits should be the ones that suffer in wet processing, and they are: viscose knitwear is notoriously difficult wet, loses strength, distorts and creases, and the usual explanation is the fibre’s own wet weakness.

This adds a structural reason to a chemical one, and the two are separable — a viscose knit at a very slack construction should be much less troublesome than the same fibre knitted tight, and the chemical explanation predicts no such dependence.

Where the model stops #

The loop is the free loop. A loop at or near the bending limit is not the shape a free solve gives, because the constraint is active and the solve does not know about it.

The mitigation is not quantified. A bundle’s ability to rearrange past the geometric limit is real, is the reason the limit is soft, and has no number here.

The swelling ratio carries a factor of four in its own range, so the movement in the ratio is between five and twelve per cent rather than a figure.

And the limit is extrapolated. The ratio is computed over the loop lengths a jersey is knitted at, and the tightness factor at which it reaches one is outside that range by about a third.

The three constraints, moving together #

It is worth putting this collection’s three geometric constraints side by side, because a wetting moves all three and moves them the same way.

The flattening a fabric demands falls from 0.780 to 0.761 — the yarn has to be flatter.

The tightest bend’s ratio to the yarn’s own radius falls from 2.04 to 1.89 — the yarn is bent harder relative to itself.

And the pressed fraction rises from about a fifth to about a quarter — more of the yarn is engaged with its neighbour.

All three are consequences of one arithmetic fact: a tenth more diameter in the same loop length. None of them was designed to be connected to the others and all three come out of the same substitution.

That is the sort of coherence that makes a geometric account worth having. Three quantities computed by three different routes, all moving by the amount one input moved, in the direction that input implies.

Which of the three a fabric notices #

They are not equally consequential and it is worth ranking them.

The pressed fraction is the one a fabric notices most, because friction lives where the yarn is pressed and a fabric’s resistance to being deformed is mostly friction. A quarter of the yarn engaged rather than a fifth is a twenty-five per cent rise in the engaged length.

The flattening matters least, because it is a section change of two per cent and almost nothing depends on the section that sharply.

And the bending limit matters not at all until it binds, and then it matters completely.

So the ordinary consequence of wetting a knit is a friction consequence, and the geometric limit is a threat rather than a cost — which is a fair description of the difference between a fabric that handles differently wet and one that is damaged by being wet.

The generalisation #

The rung is a small instance of a move that has produced three results in this work, and it is worth stating as a method rather than as three results.

When a quantity approaches a hard limit, ask what moves it and by how much.

The limit here is geometric and has no material constant in it: a rod cannot occupy its own space. That makes it the rare kind of constraint that survives every bracket, and it makes the question of what moves a fabric towards it a well posed one.

Two things move it: tightening, which is a construction decision, and swelling, which is a state. Both are computable, both move it by about a tenth over the range the trade uses, and together they say that a tight fabric in a wet process is closer to a geometric impossibility than anybody has framed it as being.

None of that needed anything new. It needed a curvature the solve had been reporting for several ladders and a diameter the wet ladder had computed, put beside one another.

What a finisher could do with it #

The mechanism suggests two things a finisher already has levers for, and it says which lever to pull.

Wet at a lower tension. A fabric that is free to relax while wet takes its wet-relaxed spacings, which are smaller — so the fabric is tighter and the limit closer. A fabric held to width while wet keeps its spacings and is further from the limit.

That is the opposite of the usual instinct, which is that restraint is what causes creasing. It says the restraint is protective in this particular respect, and harmful in others, so it is a real trade-off rather than a rule.

Or wet a slacker fabric. The whole effect is a function of the tightness factor, so a construction two per cent slacker is meaningfully further from the limit at the same yarn.

That is a design lever rather than a finishing one and it is the more reliable of the two, which is a common shape: the cheapest fix for a process problem is often a construction change made before the process.

What would falsify it #

The prediction is a dependence rather than a value, and it has a clean test.

Take one fibre and knit it at three tightness factors — say fourteen, seventeen and twenty. Put all three through the same wet process and assess crease retention, handle and dye levelness.

The access explanation predicts a smooth worsening with tightness, because a tighter fabric is harder for liquor to penetrate throughout.

The bending-limit explanation predicts something else: little difference between the first two and a marked one at the third, because the constraint binds only near the limit.

A smooth curve says access; a knee says geometry. That is a distinguishable outcome from a trial a mill could run in a day, on a question the trade currently answers by experience.

Who found it, and when #

That a rod cannot be bent below its own radius is elementary and is the standard limit in every bending-radius specification for cable and tubing.

That fibres swell across and not along is old and is the founding fact of wet fabric behaviour.

What is this collection’s own is the comparison — a solved loop’s maximum curvature against its own yarn’s radius — and the observation that a state change of a tenth moves a fabric a tenth of the way towards a geometric impossibility.

One more thing the substitution is not #

A caution about the method, because the substitution is convenient enough to be over-used.

Replacing a swollen yarn with a heavier dry one is exact for the diameter and for everything that depends only on the diameter. That covers the geometry entirely: the loop’s shape, its curvature, the clearances, the flattening.

It is wrong for everything that depends on mass. A swollen twenty tex yarn still weighs twenty grams a kilometre of fibre plus whatever water it is holding; a dry 24.2 tex yarn weighs 24.2 of fibre. So an areal weight, a fibre volume fraction or a thermal resistance computed by the substitution is wrong.

It is also wrong for the bending rigidity, which goes as the fourth power of a diameter for a solid and as the fibre count for a bundle — and a swollen yarn has the same fibre count it always had.

None of that affects this rung, because the bending limit is a purely geometric comparison. It affects anything that reaches for the same substitution for a different purpose, which is why the caution is recorded here rather than left implicit.

Where the ladder goes next #

Wetting changes a yarn’s diameter, and this collection has a founding rule that a fabric dimension quoted without its relaxation state is not a measurement. The rule was applied to the fabric’s two plan dimensions and to its thickness, and one dimension was left out.

The diameter that does need a state closes that gap, and the gap was recorded rather than found.