Flattening is free and impossible

A knitted fabric’s own arrangement requires its yarn to be squashed to about four fifths of its round diameter, at rest, everywhere. That is a demand. Whether the fabric can meet it is a question about what the deformation costs, and the answer is the most extreme bracket in this collection.

At one end it costs exactly nothing. At the other it costs thirty-six times the whole bending energy of a stitch.

Why the lower bound is exactly nought #

This is the part that makes the bracket different from every other one in the collection, and it is worth deriving rather than asserting.

Bending a yarn is unavoidable work. However freely the fibres slide, each of them has to bend when the yarn bends, because each of them follows the yarn’s own path. So the bending rigidity’s lower bound is the sum of the fibres’ — a small number, but a positive one.

Changing a yarn’s shape at constant area is different. If the fibres may slide past one another without resistance, nothing at all opposes it: the fibres simply rearrange within the new outline, each staying the length it was and bending only over the long wavelength of the yarn itself.

A bundle of loose fibres in a sheath is a fluid in cross-section. Its resistance to a change of shape is not small. It is nought.

So this bracket has no floor, and a quantity bounded below by zero cannot be estimated from its bounds at all. That is why the flattening has been a swept parameter on this site since its second phase, and it is why no amount of care about the fibre would ever have fixed it, as the racetrack’s own aspect ratio has shown for a long time.

And why the upper bound is enormous #

At the other end the fibres cannot move at all, and the section is a rod of fibre material with air in the gaps.

Its shear modulus is the fibre’s, scaled by the packing factor because the rest is air, and squashing it to four fifths is a real elastic deformation of a real solid. The energy comes out at thirty-six times the whole bending energy of one stitch.

A yarn like that could not be knitted into this fabric. The fabric would have to give somewhere else — a longer loop, a wider course spacing, or a different structure entirely — and it does not: Munden measured the spacings, and they are what they are.

What the fabric does #

It flattens.

That is not a deduction from the model; it is what every micrograph of every fabric ever cut shows. A yarn in cloth is not round, and it is not round by an amount that is easy to see with an ordinary microscope.

So the fabric is reading the free end of the lateral bracket. The fibres slide, the section rearranges, and the deformation the arrangement demands is supplied at a cost the fabric does not notice.

That is a fourth independent observation in one phase landing at the same end of this collection’s oldest bracket, and the four are worth listing together because they come from four unrelated phenomena.

Four readings of one bracket #

A slack yarn snarls. The tension a twisted thread needs to stay straight is 1.9 metres of its own weight at the free bound and 616 at the coherent one. Anybody who has let go of a piece of thread has settled it.

A knot holds. A yarn bent to a radius of its own diameter has spent five times its breaking strain if it bends as a solid section, and a third of it if its fibres bend individually. Ropes are knotted and knots hold at about half strength, so the fibres bend individually.

A yarn in a fabric is flat. Which is this rung.

And the collection’s own earliest check. When the bending bracket was first written, the free bound was found to land inside the band of real cloths for every construction in the site’s own table while the coherent bound landed two to three orders above all of them.

Four phenomena, four independent arguments, one answer. That is a stronger position than the collection has held on any other bracketed quantity.

Which of the two the bracket actually decides #

A subtlety worth stating, because the four readings above are not all readings of the same quantity.

The snarl and the knot read the bending bracket — how stiff a yarn is to bend, which is bounded below by the sum of its fibres’ bending rigidities and is a positive number.

The flattening reads the lateral bracket — how hard a yarn is to squash out of round, whose lower bound is exactly nought.

Those are two different brackets on two different deformations, and what they have in common is the same physical question: can the fibres slide? A yarn whose fibres slide is at the free end of both, and a yarn whose fibres cannot is at the coherent end of both.

So the four observations are not four measurements of one number. They are four measurements of one fact about the yarn, made through four different consequences of it, and that is a better position than four measurements of one number would be — because a single number can be right by accident and a consistent picture across four deformations cannot.

What sliding costs elsewhere #

The picture has a consequence that runs the other way and is worth following, because it is the price of the answer.

If a yarn’s fibres slide freely, then everything that depends on them not sliding is at its weakest. A yarn’s bending rigidity is at its lower bound, three hundred-fold below what a coherent section would give. Its torsional rigidity likewise, because the two carry the same bracket. Its ability to hold a shape — to be set, to remember a crease, to resist recovery — is correspondingly poor.

That is consistent with what textile yarns actually do. They are floppy, they take a set poorly unless the fibre itself is set, and their mechanical properties are dominated by friction between fibres rather than by the fibres’ own stiffness.

It also explains why setting matters so much. Setting is the process of moving a yarn away from the free end of its own bracket — which is why a loop is set and not sprung, and every finishing process that improves a fabric’s stability — steaming, resin, heat setting, milling — is doing that.

The working value, and where it comes from #

Between the two bounds sits a number this collection uses when one is unavoidable, and its provenance is worth stating because it is not a measurement of a yarn.

It was fitted to fabric thickness: a cloth’s measured thickness is well below what a circular geometry predicts, the gap is the flattening, and the pressure that would produce that flattening at loom tension implies a lateral rigidity of a few newtons per square millimetre.

At that value, squashing a yarn to four fifths costs about four tenths of a stitch’s bending energy. Not free, and not prohibitive: a real cost, comparable to a real quantity, which is what a working value should look like.

That number is an input to this rung rather than an output of it, and the rung’s own contribution is to say that the fabric’s behaviour is consistent with it and would not be consistent with a value ten times higher.

Why a fabric is not the same as a yarn under a press #

A distinction worth drawing, because the two situations are often conflated and they read the bracket differently.

Put a yarn under a flat press and squash it. The fibres have somewhere to go — sideways, into the space the press is not occupying — so the deformation is nearly free until the fibres run out of room, at which point it becomes a compaction and gets very expensive very quickly. That is the shape of every yarn compression curve ever measured.

Put a yarn in a fabric and the situation is different in one respect: the yarn is squashed by its own neighbours, which are themselves being squashed. There is no rigid platen and no unlimited sideways room, because the sideways direction is occupied by more yarn.

So a fabric’s flattening is a mutual accommodation rather than an imposed deformation, and the energy computed here — a single yarn deformed to a stated section — is an upper bound on what it costs. The real cost is lower, because the neighbours are giving too.

That does not change the conclusion. Nought is nought and thirty-six is a great deal, and a correction that lowers the cost pushes further in the direction the answer already goes.

What was counted, and how #

The energy of the deformation uses this collection’s own shape-strain law — a lateral rigidity times the square of the logarithm of the flattening, per unit length — rather than a second law written for this rung, because two laws for one deformation is two places for it to be wrong.

It is evaluated at three rigidities: the free bound, which the collection’s own function returns as exactly nought; the working value fitted to woven thickness; and the coherent bound, which is the fibre’s shear modulus scaled by the packing factor.

The comparison is against the whole bending energy of one stitch, which is the collection’s own solved number and is the natural scale: it is what the fabric is already spending to exist.

The flattening it is evaluated at is the fabric’s own demanded value, from the previous rungs, rather than an assumed one.

What the bracket says about a filament yarn #

The argument has a corollary for a class of yarn this collection rarely computes, and it is a real prediction.

A continuous filament yarn has no staple ends and its filaments run the whole length. They can still slide past one another sideways, so the lateral bracket’s lower bound is still nought and the flattening is still free.

But a fully drawn, heat-set, textured filament yarn is a different object: its filaments have been set into a shape and are held against one another by that shape rather than by friction alone. It sits further from the free end of the bracket than a spun staple yarn does.

So the prediction is: a filament yarn in a knitted fabric should be less flattened than a staple yarn of the same count and construction — because the demand is the same, the cost is higher, and the fabric has to find the difference somewhere else.

Where it would find it is the interesting part. The alternatives are a longer loop, a wider spacing, or a fabric that simply does not relax as far. All three are measurable, and the last is the one the trade actually observes: filament knits are dimensionally different from staple knits at the same nominal construction, and the usual explanation is the fibre’s elastic recovery.

This suggests a second mechanism, and the two would be distinguishable by looking at the sections.

A note on what “free” is being claimed #

One clarification, because “free” is a strong word and the claim is narrower than it sounds.

The claim is that the elastic energy of changing a fibre bundle’s cross-sectional shape at constant area is nought when the fibres slide without resistance. It is not a claim that the deformation costs nothing at all.

Sliding fibres past one another does work against friction, and that work is dissipated rather than stored. So the deformation is not free in the sense of costing no energy; it is free in the sense of storing none, which means nothing pushes the yarn back.

That is the distinction that matters for a fabric at rest. A deformation that stores no energy is one the fabric will not undo, and a yarn flattened in a cloth stays flattened when the cloth is relaxed — which is exactly what is observed, and is why a flattening measured on a relaxed fabric is a stable property of it rather than a loading state.

It also means the flattening is hysteretic, like everything else in this collection that depends on fibres sliding: a fabric that has been through a state does not come back to where it was, and what a cloth does not give back is the same mechanism seen at fabric scale.

Where the model stops #

A shape-strain law is a model of a deformation nobody has measured on a yarn. The logarithmic form is defensible and is what the collection uses everywhere, and it is not a measurement.

The area is held. A yarn squashed hard enough loses air rather than changing shape, and at that point the whole framing is wrong: the deformation is a compaction rather than a shear, and it has a different law and a different cost.

And the free bound’s exactness is an idealisation. Fibres in a real yarn are not frictionless, so the true lower bound is small rather than nought. What the argument needs is only that it is far below the coherent one, and that is safe.

The generalisation #

The rung is an instance of something worth doing deliberately, and it is not the same as the usual advice about brackets.

The usual advice is to narrow a bracket. That fails here by construction: a bound of exactly nought cannot be narrowed, and effort spent on it is wasted.

What works instead is to ask what each end of the bracket predicts about behaviour, and then to look at behaviour. The two ends of this one predict a fabric whose yarn is flattened and a fabric that cannot be knitted at all, and those are not two estimates that need reconciling — they are two different worlds, and one of them is the one everybody lives in.

That is a much cheaper way to settle a bracket than measuring, and it is available whenever the two ends differ qualitatively rather than quantitatively. This work has found four such cases and had found none before, which suggests they are commoner than anybody was looking for.

The one measurement that would separate the four readings #

The four observations landing at the free end are consistent, and consistency is not proof. It is worth saying what would distinguish “the fibres slide freely” from “the fibres slide somewhat, and every one of these four phenomena is insensitive to how much”.

The candidate is the snarl threshold, because it is the only one of the four that is a quantity rather than a qualitative alternative. The knot and the flattening are both “possible or not”; the earliest check was a band comparison. The snarl is a number: 1.9 metres of the yarn’s own weight at the free bound.

A yarn sitting a tenth of the way from the free bound towards the coherent one would need about thirty times that — sixty metres — and would not snarl in any hand.

So a careful measurement of the snarl threshold, on a fresh yarn of known count and twist, would place the yarn on the bracket rather than merely at one end of it. That is a better result than any of the four qualitative ones and it needs a reel of thread, a balance and an afternoon.

Nobody has made it, and the reason is that until this work there was no criterion to interpret it with.

Who found it, and when #

The observation that a fibre bundle offers no resistance to a change of section at constant area is elementary and is the reason yarn flattening has always been treated as a boundary condition rather than as an elastic problem.

Peirce’s racetrack of 1937 is the first systematic treatment of a flattened thread in a cloth geometry, and every treatment since has taken the flattening as given.

What is this collection’s own is the pairing of the demand with the cost: computing what the fabric’s arrangement requires, computing what supplying it costs at each end of a bracket with no floor, and reading the fabric’s own behaviour as a measurement of which end it sits at.

Where the ladder goes next #

The flattening result rests on a single number — the closest approach two courses make — and a single number is a minimum over a profile. A yarn is pressed where it crosses and free where it does not, so its section changes along its own length.

That is where a yarn is thinnest, and it is the limitation every number on this ladder so far has shared.