A tow is not a yarn

A spun yarn is round because twist makes it round. The fibres in it follow helical paths, and a helix under tension pulls the bundle towards a circle and holds it there — which is why a diameter is a sensible thing to give a yarn at all, and why every geometric model on this site has been allowed to work with one.

A reinforcement tow has no twist worth speaking of. Nothing is pulling it round. It arrives on the package as a ribbon, and it stays a ribbon: a 400 tex glass tow spread at eight to one is 1.39 millimetres wide and 0.174 thick.

That single change moves four quantities this site computes, in different directions and by different factors, and every one of them is a number some earlier essay quoted for a round yarn.

The jam is the width, and it moves the wrong way #

The closest a cloth can be set is decided by what has to fit between neighbouring threads. For a round yarn that is the diameter, and this site has computed a jammed sett from it since the foundation: the closure condition puts the threads a little under two diameters apart, so a 400 tex glass yarn — 0.554 mm across as a circle — jams at ten and a half per centimetre.

Flatten the same tow to eight to one and it is 1.39 mm wide. Neighbours touch when the spacing reaches that, so the jam falls to seven and a fifth per centimetre. Spreading the tow makes the cloth less closely settable, in threads, and the first reading of that is the wrong one.

The sett is the wrong variable. What a reinforcement is bought for is fibre per unit thickness, and threads per centimetre is a proxy for it that stops working the moment the section changes. Read the same two cloths at their own jams:

Fewer threads, more fibre — because each one covers nearly twice as much of the surface and the cloth they make is a third as thick. The tow’s aspect ratio does more to a laminate than its count, its sett or its weave, and it is the one quantity of the four that is not a decision made at the loom.

The crimp falls, and it falls quadratically #

The crimp of a woven thread is the excess length it carries because it goes over and under, and how far it has to rise is the thickness of the thread it crosses.

A flat tow is thin, so its partner does not rise far. At eight to one the plain-weave crimp of that 400 tex tow is 0.46 per cent, and at twelve to one it is 0.31. The same count as a round yarn, at five threads per centimetre, carries 4.6 per cent — ten times as much — and at its own jam it carries eighteen.

The scaling is the useful part. Crimp goes very nearly as the square of the slope, and the slope goes as the amplitude over the wavelength, so halving a partner’s thickness quarters the crimp. That is why the stiffness knock-down of a spread-tow fabric is a fraction of a per cent while a fabric of round yarns would be unusable as a reinforcement: the two are the same geometry a factor of ten apart in one length.

The cover reaches one, which is why the channel closes so sharply #

Cover is the fraction of the surface a system of threads hides, and for a round yarn it is the sett times the diameter — the quantity this site derived the trade’s constant from when it asked where the cover factor comes from.

For a flat tow it is the sett times the width, and at the jam it reaches one exactly: neighbours touching, no gap at all, a surface fully covered by one system before the other has laid a thread. A round yarn at its own jam covers 58 per cent, because the closure condition holds its neighbours nearly two diameters apart while the section that hides the surface is only one diameter wide.

The consequence is the one the previous rung needed. A covered surface leaves no channel, and a reinforcement with no channel cannot be infused — so with a flat tow the sett that maximises the fibre content is very nearly the sett that makes the fabric unfillable, and the two bounds sit close together for a geometric reason rather than a coincidental one.

The thickness is the denominator of everything #

The fourth quantity has the most leverage of the four, and it has it because it is a division rather than a multiplication.

Two flat tows stack to twice a tow’s thickness, and for a spread tow that is small. Every quantity that divides by the cloth’s thickness therefore rises together: the fibre volume fraction, the stiffness per unit thickness, the number of plies a laminate of a given thickness can carry. A part built from twelve-to-one fabric has four plies where the same thickness of four-to-one fabric has two, and four thin plies of alternating direction behave better under load than two thick ones for reasons that belong to laminate theory rather than to cloth.

The same denominator explains why the fabric is difficult to handle. A tow 0.14 mm thick has almost nothing holding it together across its width; it is a ribbon of parallel filaments with a sizing on them. A spread tow that has drifted sideways at the loom leaves a gap that no later operation closes, and a spread-tow fabric that has been folded has a crease that is a line of misaligned fibre. The property being bought and the property being lost are the same number read twice.

Four quantities, four exponents, one input #

The four changes are all consequences of one number, and putting the exponents side by side says which of them is worth chasing.

Hold the tow’s area fixed and let its aspect ratio be k. Then the width goes as the square root of k and the thickness as one over that square root, because the two multiply to a constant. Everything else follows:

Those are exact rather than fitted, and the table above is a reading of them: at four, eight and twelve to one the fibre content is 32, 45 and 55 per cent, which are in the ratio of the square roots of four, eight and twelve to a tenth of a point. The crimp at a fixed sett goes 0.46 per cent at eight and 0.31 at twelve, which is exactly the ratio of eight to twelve.

So the crimp responds twice as strongly as anything else. Doubling the aspect ratio halves the crimp and raises the fibre content by only forty-one per cent, which is why spreading is such a large win for stiffness and such a moderate one for content — and why the first pass of any spreading operation is worth so much more than the third.

The square roots also say where the effort stops paying. Doubling the fibre content needs four times the aspect ratio, from eight to thirty-two, which is a ribbon 2.8 mm wide and 0.09 thick for a 400 tex tow. Nothing about the geometry forbids it; what forbids it is that a ribbon that thin has almost no cohesion across its width and will not survive a loom. The bound on spreading is a handling bound, and the arithmetic says how expensive it is to approach it: the returns fall as a square root while the fragility rises directly.

There is one exception to the square roots and it is the reason the table above stops where it does. The cover of a single system goes as the sett times the width, and at the jam those two are reciprocal, so the cover is one at every aspect ratio — it does not scale at all. That is the quantity a fabricator would most like to move and the only one spreading cannot touch: a jammed tow fabric is a closed surface whether its ribbons are wide or narrow, and the channel a resin needs has to be bought by opening the sett rather than by changing the section.

So the spreading decision and the sett decision act on genuinely different things. Spreading raises the fibre content and lowers the crimp at any sett; the sett decides whether there is a path through. The first is a property of the tow and the second is a property of the cloth, and a preform specification that names an areal weight and a weave has named neither.

Which is the more useful way to read the whole table: it is not four consequences of one number, it is one number deciding three of the four and being unable to touch the fourth.

The spun-yarn world knew this and called it something else #

None of the above is new to a weaver. It is the flattening this site’s own section models have disagreed about since its earliest essays, taken further than either was proposed for.

The disagreement is worth recalling because it makes a general point about models. Peirce’s circle and Kemp’s racetrack, given the same yarn and the same closure condition, predict jammed setts within an eighth of each other and cloth thicknesses a third apart. Anyone comparing them on the sett — which is what a weaver is thinking about — would conclude that the choice of section model is academic. Anyone dividing by the thickness would find that it is the whole of the answer.

A reinforcement is the case where the thickness is the quantity of interest, so the model that looked academic becomes the one that decides. That is the ordinary way an idealisation gets tested: not by being refuted, but by being carried into a trade that cares about its other output. The same movement takes the cover factor from a shirting’s marketing number to a filter’s specification, and the float from a lustre to a stiffness.

What none of these figures can show is the fibre inside the tow. Every drawing here treats a tow as a solid body of stated area, and a tow is a few thousand filaments that can migrate, twist, and be unevenly spread across their own width. The packing factor is the one number standing in for all of that, and it is stated everywhere rather than assumed, because it is doing more work here than in any other field on this site.

What was counted, and how #

The section model is the flat limit of Kemp’s racetrack, which is already on this site and already run against Peirce’s circle. Here it is written as a rectangle of width w and thickness t with w · t equal to the tow’s own area, and the area conservation is asserted rather than assumed — a section model that changed the amount of yarn in the thread would be comparing two different cloths and reporting the difference as geometry.

The tow’s area comes from the count and the packing factor by the same arithmetic that gives a spun yarn its diameter: mass per unit length divided by density is an area, and dividing by the packing factor accounts for the air. Reinforcement tows pack at 0.65 to 0.8 rather than the 0.6 of a ring-spun cotton, so the packing factor is stated at every use and never defaulted quietly.

One comparison in the code is there only to keep the shape and the aspect ratio separate: the round section of the same area is 12.8 per cent thicker than a square of it, at every count and every fibre, because both go as the root of the area. That factor is the shape alone. Everything else in the table above is the aspect ratio.

And the width jam is a refusal rather than a warning. A tow set wider than its own spacing is a specification no loom can weave, so the fibre-volume routine throws on it and names the jammed sett in the message. It is the same refusal assertBelowJam has made for round yarns since its earliest essays, with the width in place of the diameter.

Where the model stops #

A real tow is not a rectangle and its width is not constant. It spreads where it crosses and necks between crossings, so the width that decides the jam is not quite the width that decides the cover, and neither is quite the width on the package. The rectangle is a bound with the right area.

Nothing here says what sets the aspect ratio. It is decided by the sizing on the fibre, by the tension history through the loom, by the reed, and by whether the tow has been deliberately spread — none of which is geometry. The aspect ratio is an input to every figure in this field and is stated in each of them.

And the flat section has a mechanical cost this arithmetic does not price. A wide thin tow buckles out of plane more easily than a round yarn of the same count, which is why a spread-tow fabric is fragile in the hand and why the trade tapes its edges. The section model knows about area and nothing about bending.

One caution about the table, since it invites a comparison it does not license. The two columns are the same count of the same fibre at their own jams, which is the honest way to compare two sections; they are not two fabrics anybody would put to the same use. A cloth of round 400 tex glass yarns at ten per centimetre would be a heavy, thick, low-Vf fabric with no application in a laminate — it is in the table as the geometric baseline the flat tow is being measured against, not as a competitor.

Who found it, and when #

Kemp’s racetrack is from 1958 and was written for a flattened spun yarn — a cotton in a closely set cloth, squashed a little in the thickness — with a flattening ratio of about two. It was not proposed for a reinforcement tow, and using it at eight to one is an extrapolation by a factor of four in the one parameter it has.

That is worth saying plainly because the model’s own history says nothing about whether it holds there. What can be said is that the extrapolation is dimensionally honest — the area is conserved, the flat portion behaves as a flat portion should, and the limit as the flattening rises is a rectangle rather than something that stops making sense — and that the numbers it gives for a modern spread-tow fabric are the numbers the datasheets carry.

There is a second historical thread and it runs the other way. Spread-tow fabrics are a deliberate exploitation of exactly the arithmetic above: the trade found that spreading a tow raised the fibre content and lowered the crimp, and built machinery to do it on purpose — pneumatic spreaders that widen a tow before it reaches the loom. The geometry was not discovered by the model; it was discovered by people making fabric, and the model’s job is to say how far the effect goes and what else moves with it. Four quantities, one input, and a table that a datasheet does not carry.

Where the ladder goes next #

Three rungs of this field have now treated a preform as a fabric with a thickness, which is exactly what the binary matrix cannot describe: a weave is a repeat in two directions and a preform has a top and a bottom. The compound-cloths field recorded three-dimensional weaving as a possible fifth escape from the encoding and left it open. The next rung closes it, and the answer is not the one the question expected.