The flattening nobody fitted

There is a number in this collection that has been a free parameter since its second phase, and it is not a small one: how flat a yarn in a fabric is.

Every section drawing on this site has a flattening in it. Every thickness computed from a geometry depends on one. Every jamming condition, every cover calculation, every bearing curve. The number has always been an input — swept, assumed, or taken from whatever a micrograph seemed to show — because the mechanical route to predicting it is closed.

The knitted geometry predicts it.

Why the mechanical route is closed #

It is worth being clear about why nobody has predicted this before, because it is not for want of trying.

Flattening a yarn is a change of section at constant area. What resists it is the yarn’s transverse rigidity, and this collection computed the two bounds on that several ladders ago.

The coherent bound — the fibres unable to move — is the shear modulus of the fibre material scaled by the packing factor, and it is a number.

The free bound — the fibres sliding freely — is exactly nought. A bundle of loose fibres in a sheath is a fluid in cross-section: nothing at all resists a change of shape at constant area, because the fibres simply rearrange, each staying the length it was.

So the bracket is not wide. It is infinite, and a quantity bounded below by zero cannot be estimated from its bounds at all. That is a much worse position than the bending bracket’s factor of three hundred, and it is why the flattening has always been measured rather than derived.

The geometric route #

If the mechanics cannot supply the number, the arrangement can, and the argument is short.

Solve the fabric. Lay two adjacent courses one course spacing apart, which is what the model says a fabric is. Ask how close the two curves come.

They come to 0.780 of a yarn diameter. A round yarn of that diameter cannot occupy the arrangement; a yarn flattened to 0.780 through the fabric’s thickness can.

So the fabric’s own geometry names the flattening, and it names it without any reference to what the yarn is made of, how stiff it is, or whether its fibres slide.

Why it is the same number for every fabric #

One measurement is an anecdote, and the answer is only interesting if it is a property of knitted fabric rather than of one arithmetical example.

Compute it across five loop lengths, three relaxation states and three counts — eighteen fabrics — and the values run from 0.707 to 0.839.

That is not a constant, and it should not be: a tighter fabric packs its courses closer and needs a flatter yarn. What matters is that the eighteen fall on one curve against the tightness factor, which is the model’s only dimensionless group.

Nothing about the fibre, the count or the packing factor enters except through that group. Two fabrics with the same tightness factor demand the same flattening whatever they are made of, and that is what makes the result a structural requirement rather than an accident.

Against what the trade reports #

A prediction is worth what it can be checked against, and there is something to check against here, though it is not a clean data set.

Micrographs of yarn in woven cloth commonly show flattening ratios between 0.7 and 0.9, with the tighter constructions at the lower end. Peirce’s own racetrack work and the flattened-thread geometries that followed it use figures in that band, and this collection’s own thickness arithmetic has swept from 0.6 to 1.0 for the same reason, which is why a jammed warp’s thickness has always carried an assumed section.

0.707 to 0.839 is inside that band, and its ordering matches: tighter fabrics flatter.

That is agreement rather than confirmation, and it is worth saying which. The trade’s figures are for woven cloth and the prediction is for knitted; the trade’s figures are measured on sections that have been embedded and cut, which is a process that changes what it measures; and nobody has published a systematic set against the tightness factor.

What would confirm it is a set of knitted sections across the tightness range, and that is a measurement anybody with a microtome could make in a week.

What it says about the swept parameter #

The practical consequence is for every calculation in this collection that has a flattening in it, and there are a good many.

They have been quoted with a band, because the input had a band. If the flattening is a function of the tightness factor, those bands collapse to a line — and more importantly, the direction of the dependence is now known, which the band never supplied.

That matters most where a comparison is being made between two fabrics of different tightness. Sweeping a flattening independently for each is the honest thing to do with an unknown parameter and it discards the fact that the two are not independent: a tighter fabric is flatter, so its thickness falls faster than a constant-section calculation says.

Why the tightness factor is the right variable #

It is worth saying why one dimensionless group rather than several, because it is a fact about the model rather than a fitting choice.

The solved loop depends on the yarn’s diameter, the loop length, the wale spacing and the course spacing. The spacings are proportional to the loop length by Munden’s constants, so the whole geometry is a function of the ratio of the diameter to the loop length and of nothing else — the model’s only dimensionless group.

That ratio is the tightness factor divided by a constant that depends on the fibre density and the packing factor. So two fabrics with the same ratio have the same loop, to a scale factor — and a scale factor does not change a distance measured in diameters.

The flattening is a distance measured in diameters. So it can only depend on that ratio, and the sweep is a check on the arithmetic rather than a discovery.

What the sweep does establish is that the dependence is smooth and monotone over the range fabrics are actually made in, which is not guaranteed by anything and is what makes a single curve usable.

What a flattened yarn does to everything else #

A prediction is only interesting if something depends on it, and a good deal does.

Thickness. A knitted fabric’s thickness is two yarn diameters, measured through the fabric — so if the yarn is flattened to 0.780 through the fabric, the thickness is 0.780 of what a round yarn would give. That is 0.260 millimetres rather than 0.334, which is a twenty-two per cent correction to this collection’s most falsifiable knitted result and moves it towards, rather than away from, what a gauge reads.

Cover. A flattened yarn is wider than a round one of the same area, by one over the flattening, so a fabric’s cover is higher than a round-yarn calculation gives. At 0.780 the yarn is 1.28 times as wide, which is not a small correction to an opacity or an air permeability.

Bending. A flattened section has two different flexural rigidities, easy about the long axis and hard about the short one. A knitted loop bends in the easy direction, so its bending rigidity is lower than the round calculation gives — by exactly the flattening, as it happens, which is a tidy result and is worked out separately.

And contact. A flattened yarn has a wider contact patch at every crossing, so the pressure is lower and the friction is higher for the same normal force.

Four consequences, all of them in the same direction, none of them previously computable because the input was a swept parameter.

Why this was worth more than a better bracket #

There is a temptation, when a bracket is wide, to try to narrow it — to measure the transverse rigidity more carefully, to argue about the packing factor, to find a better experiment.

None of that would have worked here, and the reason is structural rather than practical. The lower bound is zero for a reason, and it is a good reason: a bundle of fibres that can slide has no shape memory in cross-section. No refinement of measurement moves a bound that is exactly nought.

So the effort was never going to succeed, and recognising that is what made the geometric route worth trying. The general shape is: when a bracket has no floor, stop trying to narrow it and look for a different question whose answer implies the number.

This collection has one other quantity with that structure. The fraction of a loop’s natural curvature that setting has taken cancels out of every balance the dimensions can be put into, so it cannot be measured from the dimensions at all. Whether some other arrangement implies it has never been asked.

What was counted, and how #

Each of the eighteen fabrics is solved independently: a relaxed loop from the count, fibre and loop length; a three-dimensional segment from that; a course assembled from the segment; and a second course displaced by one course spacing.

The approach is the minimum distance from any point of one course to any point of the other, computed point to point over a hundred and twenty samples per half period.

Three relaxation states are used because the collection’s own constants supply three, and the states move the spacings by about ten per cent — enough to test whether the curve is a curve.

The result is reported as a ratio to the yarn’s own computed diameter, so the count cancels twice: once in the tightness factor and once in the units.

Where the model stops #

It predicts a single flattening for a yarn that is not uniformly flattened. A yarn is pressed where it crosses and free where it does not, so the section changes along its own length, and one ratio is an average. That is where a yarn is thinnest and it is a real limitation on the number quoted here.

The section is assumed to hold its area. A yarn that is squashed hard enough loses air rather than changing shape at constant area, and at that point the ellipse is the wrong idealisation and the packing factor is moving.

And the direction of flattening is assumed. The prediction is a distance through the fabric’s thickness, so the ellipse’s short axis is normal to the fabric. That is right for the interlacing and it is not obviously right everywhere along the yarn.

Nothing here is re-solved under the constraint. The loop is the free one, and a loop that has been flattened is a slightly different loop with a slightly different bending rigidity in each direction.

What the number does not settle #

Three things a reader might reasonably expect from a flattening prediction and will not get here.

It does not say the yarn is an ellipse. The prediction is a distance — how much room the fabric leaves through its own thickness — and any section whose extent in that direction is 0.780 of a diameter satisfies it. An ellipse is drawn because it is the simplest such section that holds its area; a racetrack, a lens or a lobed shape would do as well, and this collection has argued elsewhere that the racetrack is a convenience rather than a measurement.

It does not say the fabric is at equilibrium. The prediction says the fabric’s measured spacings and its solved paths are only mutually consistent if the section is squashed. It does not say the fabric arrived there by minimising anything, and the loop is still solved as though its yarn were round.

And it does not close the transverse bracket. The rigidity’s lower bound is still exactly nought and always will be. What has been supplied is the deformation, not the constant that resists it, and the two are only connected through a force nobody has measured.

That is a narrower claim than “the flattening is predicted”, and it is the claim that is actually supported.

The generalisation #

The transferable point is about where a parameter can come from, and it inverts the usual order.

The usual order is: measure the material, predict the structure. That is what a transverse rigidity is for, and here it fails completely because the rigidity’s lower bound is zero.

The order that worked is the reverse: measure the structure, and let it name the material parameter. The fabric’s arrangement is known — Munden measured the spacings, the solve produces the paths — and the arrangement is only consistent if the section is a particular shape. So the structure predicts the section.

That is a route worth looking for elsewhere in this collection, and there are candidates. A woven cloth’s thickness is measured routinely and is well below what a circular geometry predicts; the gap has been read as evidence that the threads are flattened, and the amount has been fitted rather than derived. The same argument applied to a woven crossing would produce a prediction instead, and it is asked directly two rungs along.

The profile above and the cost below are the two things the single ratio hides: that the pressing is local, and that it is nearly free.

What a measurement would have to be careful about #

If somebody sets out to check this on sections, three things will decide whether the answer means anything, and they are worth stating because they are the usual reasons such measurements disagree.

The specimen must not be embedded under pressure. A section is cut from a fabric set in resin, and a resin that shrinks or a fabric that is compressed while it cures measures the mounting rather than the cloth. That is the commonest reason published flattening figures scatter.

The section must be normal to the yarn, not to the fabric. A yarn in a knitted loop runs at every angle to the fabric’s plane, so a cut normal to the fabric slices most of the yarn obliquely and reports an ellipse that is an artefact of the cutting angle. The correction is a cosine and it is large.

And the state must be stated. The prediction differs by two per cent between dry-relaxed and fully relaxed, which is small — but the dimensions differ by ten per cent, and the tightness factor the prediction is read against moves with them.

Get those three right and the measurement is a week’s work with a microtome. Get any of them wrong and the result is a number that could be anything, which is roughly the state of the published literature.

The prediction stated as a formula #

For anybody who wants to use it rather than argue about it, the curve is worth writing out in the form a calculation would want.

The flattening depends on one group — the yarn’s diameter over its loop length — and over the range fabrics are made in, from a tightness factor of about nine to about eighteen, it falls close to linearly from 0.84 to 0.71.

So a working form is: flattening ≈ 0.96 − 0.014 × tightness factor, over that range, for a plain jersey.

That is a fitted line through computed points rather than a derivation, and it is offered as a convenience. The computed points are what the claim rests on; the line is what somebody would put in a spreadsheet.

Two cautions come with it. It is for plain jersey — a rib’s courses are arranged differently and nothing here computes them. And it is for the relaxed states; a fabric on the machine or under load is not described by it.

Who found it, and when #

Peirce replaced the circular thread section with a racetrack in 1937, and everybody who has computed a cloth geometry since has carried a flattening parameter and fitted or assumed it.

Munden’s relaxation constants, which supply the spacings this prediction rests on, are from 1959 and have been the input to every knitted geometry since.

What is this collection’s own is the combination: asking a solved fabric whether it fits together, finding that it does not, and reading the shortfall as a prediction of the parameter everybody else has fitted.

Where the ladder goes next #

Two directions.

The first is to make the curve rather than the point the result: the flattening depends on the tightness factor and on nothing else, which is worth establishing carefully because it is what makes the prediction usable. That is a flattening that follows the tightness factor.

The second is to ask what the fabric pays for it, and the answer is a fourth reading of this collection’s oldest bracket: flattening is free at one end and impossible at the other, and the fabric flattens.