A loop is set and not sprung

The model of the previous rung says a jersey should spread, and jerseys do not. The disagreement is not small and it is not in the numbers: the sign is wrong, in both directions at once, and every published relaxation state runs the wrong way up the slope.

The thing that is wrong is one assumption, and it is buried so deep it does not look like an assumption at all. The yarn was taken to be straight.

What was assumed #

An elastica minimises the integral of the square of its curvature. Written that way, the shape with no energy is the shape with no curvature — a straight rod — and every bend is a departure from it costing energy and pushing to be undone.

That is right for a steel wire and it is right for a yarn on a bobbin. It is not obviously right for a yarn that has spent a night in a hot damp fabric and then been dried in that shape, and the whole of the finishing field on this site is about what such treatments do.

What set is #

A fibre held in a shape while its internal bonds are broken and re-formed comes out of the treatment with that shape as its own. In wool the bonds are disulphide and hydrogen; in cotton they are hydrogen bonds between cellulose chains; in a thermoplastic it is the polymer relaxing above its glass transition and vitrifying in the new configuration. The mechanisms differ and the effect is the same.

This collection has used it repeatedly and never in the loop. A wet cloth is set closer than it was woven is that effect in a woven fabric’s crimp. A tuft is set so it cannot untwist is the same thing in a pile. Which fibres crease and why there are two answers is the same thing being complained about rather than exploited.

How it enters the arithmetic #

By one argument, and nothing else in the machinery changes.

The energy becomes the integral of the square of the curvature minus its natural value — the departure from the shape the yarn wants rather than the departure from straight. In the expansion the solver uses, that is a subtraction inside a quadratic form, so the problem stays quadratic, the constraints stay untouched, and the solve is as fast as it was.

At the natural shape the energy is zero and both forces are zero, to machine precision. That is checked rather than assumed, because it is the one property the change has to have.

The set fraction #

A real yarn is not fully set and not unset. Write σ for how far the setting has gone: the natural curvature is σ times the relaxed loop’s own, and every force scales as one minus σ.

That single parameter brackets the whole ladder. At σ equal to zero the loop presses with forty millinewtons and the fabric wants to spread with forty-eight newtons a metre. At σ equal to one it presses with nothing, wants nothing, and sits exactly where it is with no friction required at all.

Every real relaxed knit is somewhere between, and the honest thing is to say which end each result is quoted at and which way the error goes.

What a fibre does when it is set #

The mechanism is worth a section because the fibres this collection deals with set by three different routes and the differences show up in the fabric.

Wool has the richest set of routes and sets most completely, which is why a wool knit is dimensionally stable once it has been properly relaxed and why it felts if the treatment goes too far. Cotton sets by hydrogen bonding and therefore sets whenever it is wetted and dried, and un-sets whenever it is wetted again — which is precisely why a cotton knit’s dimensions are quoted per treatment and why it keeps moving for several washes. Polyester barely sets at all below its own setting temperature, and a polyester knit that has not been heat-set is the least stable fabric in ordinary use.

Three fibres, three degrees of setting, and the ranking matches the trade’s own ranking of dimensional stability without anything being fitted.

Why setting explains the direction #

Friction can hold a fabric anywhere. It cannot say which way a fabric moves when it is released, because friction has no preferred direction — it opposes whatever is happening.

Setting does have a direction, and it is the observed one. Wet relaxation and tumbling are precisely the treatments that set a yarn: hot, wet, and mechanically worked, which is the recipe for breaking and re-forming the bonds that hold a fibre’s shape. So the more completely a fabric is relaxed, the more completely its yarn has been set, and a set yarn holds the fabric wherever the setting found it — which is a smaller, tighter fabric, because the fabric contracts under the same treatment for reasons of swelling and yarn relaxation that have nothing to do with this slope.

Friction explains that a knit can be anywhere; setting explains where it ends up. Neither alone accounts for the three states, and together they do.

What it cannot fix #

It is worth being precise about what the change buys, because it is easy to read it as making the earlier result go away.

It does not. The gradient of the unset energy is real and it is what an unset yarn would do. A jersey knitted from a fibre that does not set at all — a fully drawn polyester at room temperature, a glass filament, an aramid — has no σ to speak of, and it really is a fabric held together against its own springing by friction. Such fabrics are notoriously unstable dimensionally and notoriously prone to spirality, both of which are what a fabric sitting on a slope does.

So the earlier rung’s result applies exactly to the fabrics it applies to, and this rung says what is different about the ones it does not.

What a set yarn is still good for #

Setting the natural shape to the loop does not make the fabric inert. Everything away from the relaxed state still costs energy, and that is where the useful quantities are.

The load–extension curve, the modulus, the transverse response, the recovery force and the jamming limit are all differences between a stretched state and the relaxed one, and all of them survive intact — better than intact, because they now start at the right place. An unset model gives a curve that does not pass through zero force at zero extension, which is not a curve anybody can compare with a measurement.

A number the setting does not touch #

It is worth naming the quantities that are indifferent to σ, because they are the ones this ladder can state without a bracket.

The geometry is untouched: the loop’s shape at a given pair of spacings is decided by the loop length, the diameter and the spacings, and the natural curvature enters only the energy. So the slack, the peak curvature, the occupancy and the jamming ceiling are all σ-free.

So does every ratio of two forces computed at the same σ, which is most of the comparisons on this ladder. What σ does touch is any single force in newtons, and any statement about how much energy a fabric is holding.

The bracket, restated #

Two brackets are now stacked, and it is worth keeping them apart.

The first is the yarn’s bending rigidity, which is not a number but a range of a hundred and thirty, decided by how freely the fibres slide. The second is σ, which runs from nought to one and multiplies every force by one minus itself.

They are independent and they compound. A force quoted on this ladder is therefore a force at the free bound with no setting, and it is an upper bound on an upper bound. That sounds worse than it is: ratios between two fabrics computed the same way are untouched by either bracket, and ratios are what nearly every claim here is.

Why σ cannot be measured from the dimensions #

The obvious hope is that the fabric’s own position on the surface pins σ down. It does not, and the reason is a fact about friction.

For the fabric to stay where it is, friction must be able to hold the residual force, which is proportional to one minus σ. Friction available is a coefficient times the contact force, which is also proportional to one minus σ. The factor cancels exactly, and the balance is a pure ratio with no σ in it at all.

So the fabric sitting still is consistent with any degree of setting whatever, and observing that it sits still teaches nothing about σ. That is a negative result and it is worth stating, because the alternative is to fit σ to the observation and then present the fit as a measurement.

What the balance does say is stronger and is the next rung’s subject: along the wales the ratio comes out at twice the friction coefficient and nothing else, which is under one for every fibre in this collection’s table. Friction cannot hold a relaxed knit in that direction at any σ — so σ has to be large, and the setting is a requirement rather than an embellishment.

Why σ cancels at rest and does not cancel everywhere #

The cancellation is exact at the relaxed state and it is a local fact rather than a general one, which means there is a place to look where σ survives.

Write the curvature at a state as its relaxed value κ₀ plus a departure Δ. The energy is then the integral of ((1 − σ)κ₀ + Δ)², which expands into three terms: one in (1 − σ)², one in (1 − σ), and one in Δ² with no σ in it at all.

At the relaxed state Δ vanishes, and everything left carries (1 − σ) — which is the cancellation the essay describes and the reason the fabric sitting still says nothing.

Far from the relaxed state Δ dominates κ₀ entirely, and the energy — and therefore the stiffness — is the σ-free term. So the extensional stiffness a fabric shows near its geometric ceiling is the same number whatever its setting, while the force it carries at rest scales as one minus σ.

Two quantities on one curve, one carrying σ linearly and one not carrying it at all. Their ratio is a measurement of σ, and it needs neither an absolute force nor a bending rigidity.

Which gives the experiment a form #

The recipe is one specimen and one tensile tester.

Clamp a jersey at its own relaxed dimensions and read the force. A fully set fabric reads zero; an unset one reads the forty-eight newtons a metre the model gives; a real one reads (1 − σ) times it.

The obvious objection is that friction absorbs the difference, and along the courses it does. Along the wales it cannot — the next rung’s balance is that friction fails in that direction at any σ, by a factor of nearly two — so a specimen clamped wale-wise at its relaxed course spacing is carrying the whole of its residual force in the jaws, and the reading is (1 − σ) × 48 with nothing subtracted.

And the rigidity bracket divides out. Both the rest force and the high-extension stiffness are proportional to the yarn’s bending rigidity, so taking their ratio removes the factor of a hundred and thirty that this ladder otherwise carries on every force. The measurement returns σ and nothing else.

That is unusually clean for a quantity this collection has just finished saying is unmeasurable. What was unmeasurable was σ from the fabric’s position; σ from the fabric’s forces is a reading.

And it says which fabrics to run it on #

The ratio is most sensitive where 1 − σ is largest, so the experiment should be run on the fabric expected to be least set and the one expected to be most.

An un-heat-set polyester jersey should read close to the unset prediction, since its σ is near zero — and if it does not, the setting story is wrong at its clearest case.

A wet-relaxed and tumbled wool jersey should read close to nothing, since wool sets most completely by the most routes.

And a cotton jersey should read differently before and after a wash, because cotton sets by hydrogen bonding and re-sets every time it is wetted and dried. Running the same specimen twice, with a wet relaxation between, gives two σ values on one fabric — which is a measurement of what a wash does to the setting rather than to the dimensions, and this collection has nothing else that reaches it.

Three predictions, one instrument, and a quantity that until this section had no route to a number at all.

What would measure it #

Something that separates the two, and there are two candidates.

Cut a fabric and watch. A frictionally-held fabric is a fabric in which every loop is loaded; cut a wale and the loops on either side of the cut have lost half their restraint and should move. A set fabric’s cut wale sits there. The displacement to look for is a fraction of a stitch and it happens once.

Heat it. Setting is undone by treatments that give the bonds enough energy to re-form, so a fabric that has been set and is then boiled unloaded should move towards where the unset model says it wants to be. A frictionally-held fabric moves too, so the discriminator is the direction, and the two predictions differ.

Neither has been run here, and both are within reach of anybody with a fabric and a ruler.

The comparison with the woven side #

The woven half of this collection has been assuming an unset yarn all along, and it has been getting away with it for a structural reason rather than a lucky one.

A woven cloth’s crimp is a small departure from straight — a wrap angle of twenty or thirty degrees, over a fraction of a millimetre — and the setting a finishing treatment supplies moves the equilibrium rather than removing the restoring force. The essay on setting a wet cloth is exactly that: a new equilibrium, not an inert fabric.

A knit’s loop is a departure of a hundred and eighty degrees and more, and the yarn has been held in it while hot and wet. The two cases differ in degree by an order of magnitude and it turns them into different problems.

The fabric that is not set at all #

The clearest way to see what σ is doing is to look at a fabric where it is nearly zero, and the trade supplies one.

Unset synthetic knits do exactly that. They grow in wear, they lose their shape at the cuffs and hems, and they come back from a wash in a different size each time — which is why every commercial polyester knit is heat-set before it is cut, and why the setting temperature is specified more tightly than almost anything else in the process. Heat-setting is not a finish applied for handle. It is the operation that gives the fabric a natural shape to return to.

Read the other way, it is a strong argument for this rung: the fabrics that behave as the unset model predicts are precisely the fabrics whose yarn has not been set, and the ones that do not are the ones whose yarn has.

What is genuinely new here #

Not the idea that a yarn sets. That is old, it is the basis of the whole finishing trade, and it is what every relaxation specification is about.

What is new is the observation that the loop model’s zero is the thing set changes, and that the choice of zero is what decides whether the model predicts a fabric that spreads or a fabric that sits. A modelling assumption that looks like a formality — “the yarn is straight when unstressed” — turns out to carry the entire disagreement between the model and every published measurement.

That is worth recording as a general caution as much as a result. The assumptions that do the most damage are the ones nobody writes down because they seem too obvious to need saying.

What the picture cannot show #

Set. There is no drawing of a natural curvature: it is a property of the material’s memory rather than of its shape, and two yarns in identical loops with entirely different σ look identical from every angle.

That is a real limitation on this ladder’s figures rather than a rhetorical one. Every drawn loop here is a shape, and the shapes are the same whichever end of the bracket the forces are quoted at.

Why the recovery is not perfect either #

A set yarn returns to the loop and an unset one returns to straight, and a real one does neither exactly — which is the third thing σ is quietly standing in for.

Recovery is measured and nothing predicts it recorded that on the woven side: a yarn’s return from a bend is neither complete nor characterisable by one number, because part of the deformation is elastic and part is fibres having slid and stayed. A knit inherits all of that, and adds to it a fabric-scale hysteresis from friction at the interlacings.

So a measured load–extension loop on a real jersey has three things in it that this ladder computes one of. The elastic part is here; the frictional part is priced in the next rung; the fibre-scale part is not modelled anywhere on this site and is recorded as absent.

That is why the computed curve is called a lower bound wherever it is used, and why the sentence saying so is repeated rather than left to be inferred.

Where the ladder goes next #

Two ways. One is the friction, which turns out to have the margin to hold whatever is left over, and which explains the three states as a history rather than as three answers to one question.

The other is everything the set origin makes computable, and it is most of the rest of this ladder: what a knit gives when it is pulled, the modulus it has instead of a modulus, the transverse response that changes sign, and the geometric ceiling that turns out to sit three times further out than any jersey ever reaches.