A loop bends at twice its own radius

There is a limit on how tightly a thread can be bent that has nothing to do with how stiff it is, how strong it is, or what it is made of.

A rod of radius r cannot be bent to a centre-line radius below r. At exactly r the inside of the bend has closed on itself; below it, the rod is occupying its own space. That is geometry, it applies to a steel bar and to a cotton yarn equally, and it is the hardest limit in the subject.

A relaxed knitted loop’s tightest bend is at 2.04 yarn radii.

The number #

For a twenty tex cotton at a three and a half millimetre loop, fully relaxed: the maximum curvature anywhere on the solved half period is 6.01 per millimetre, so the tightest radius is 0.166 millimetres.

The yarn’s diameter is 0.167 millimetres and its radius is 0.0835.

So the tightest bend is at 1.99 diameters of curvature — which is to say, the centre line’s radius of curvature is almost exactly one yarn diameter, or two yarn radii.

That is a coincidence worth not treating as one. It says the fabric sits at exactly half the tightest bend its yarn can physically take.

Why it is not a coincidence #

It is not exactly two, and how far it is from two is the interesting part.

Across the loop lengths a jersey is knitted at, the ratio runs from 1.67 at a two point eight millimetre loop to 2.46 at four and a half. A tighter fabric bends its yarn harder relative to the yarn’s own size, which is what anybody would expect.

So the fabric is not sitting at a special value; it is sitting on a curve, and the curve passes near two over the range fabrics are actually made in.

What makes that worth noticing is the other end. At a ratio of one, the yarn would be bent to its own radius and the inside of the bend would have closed. The tightest fabric anybody knits is at 1.67, which is two thirds of the way from comfortable to impossible.

The ceiling this implies #

Extrapolating the curve gives a hard limit on how tight a knitted fabric can be, and it is a limit nothing else in this collection supplies.

The ratio falls with the loop length. Running it down, it would reach one at a loop length of about 1.9 millimetres for a twenty tex cotton — a tightness factor of about twenty-four, against the thirteen to seventeen the trade quotes for an ordinary jersey and the eighteen or nineteen for a very tight one.

Below that loop length there is no fabric. Not a difficult fabric, not an expensive one: no arrangement of that much yarn in that much space exists, because the yarn would have to pass through itself at every crest.

That is a ceiling on tightness factor with no material constant in it, and this collection has not had one.

What the existing limits are #

Two other limits on knitted tightness are already in circulation and it is worth putting the new one beside them.

The machine. A knitting machine cannot draw a loop shorter than its own needle geometry allows, and that is a real and binding constraint on most machines.

The yarn’s strength. Drawing a tight loop requires a tension, and a yarn drawn too tight breaks. That is the practical limit in most mills and it is why very tight fabrics are made from strong yarns.

And now the geometry. At a tightness factor of about twenty-four, no arrangement exists at all.

The three are in that order — the machine binds first, then the yarn, then the geometry — which is why nobody has run into the third. But it is the only one that is a property of the fabric rather than of the equipment or the material, and it is the only one that cannot be engineered around.

And the constraint bites twice #

There is a second reason the ratio matters and it is about the section rather than the centre line.

The hard limit — a centre-line radius equal to the yarn’s radius — assumes a round yarn. A yarn flattened through the fabric’s thickness has a smaller extent in that direction, so if it is bent about the axis it is flattened against, the limit is easier.

But a knitted loop is flattened through the fabric’s thickness and bends in the fabric’s plane, which is about the section’s other axis — the one flattening makes longer. So the flattening makes the bending limit harder, not easier.

At a flattening of 0.78 the yarn’s extent in the bending direction is 1.28 diameters rather than one, so the effective ratio is 2.04 divided by 1.28, which is 1.59.

That is much closer to the limit than the round calculation says, and at the tight end it is 1.30.

So the two contact constraints this ladder has found are not independent. The flattening the fabric demands makes the bending limit tighter, and the ceiling on tightness factor is correspondingly lower than the round extrapolation gives.

What this does to the ceiling #

Redoing the extrapolation with the flattening included brings the ceiling down from about twenty-four to about twenty.

Twenty is not comfortably above practice. The trade’s very tight jerseys are at eighteen or nineteen, and there are fabrics — some technical knits, some fine gauge sportswear — that are quoted above twenty, well past what an ordinary jersey reaches.

So the ceiling is not a remote theoretical limit that nobody approaches. It is at or just above the tightest fabrics anybody makes, which is exactly where one would expect a hard geometric constraint to sit if the trade has been pushing against it for a century without knowing what it was.

That is a claim rather than a demonstration, and it is the kind that would be settled by asking whether the tightest achievable jersey depends on the yarn’s strength or on nothing at all.

Why the crest is where it happens #

The tightest bend is at the crest of a loop, and that is worth explaining because it is not the only candidate.

A knitted half period runs from a crest to a trough. Along the way it curves continuously, and the curvature is largest where the curve turns most sharply — which is at its two ends, where the thread reverses direction along the course.

The middle of the half period, where the thread is running down the wale, is nearly straight: nine tenths of a loop is free run, and the free run carries almost no curvature at all.

So a knitted loop concentrates its bending into two short regions and spends the rest of its length doing nothing. That is why the maximum curvature is so much larger than the average, and it is why a bending limit bites at all: an evenly bent loop of the same length would be nowhere near it.

It is also why the crest is where two things go wrong at once. It is the sharpest bend and it is where the two half periods overlap, and both are consequences of the thread reversing direction in a short arc.

What was counted, and how #

The maximum curvature is reported by the solve itself, as part of the same output that gives the loop’s energy and forces. It is not a separate calculation and it has been available since the loop was first solved.

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

The ratio is computed at each of the loop lengths the collection sweeps, and it is monotone in the loop length, which is what makes the extrapolation defensible over the small range it is taken across.

The flattening correction uses the demanded flattening from the same fabric, and the direction of the correction — harder rather than easier — follows from which axis the loop bends about, which is settled by the loop being a plane curve in the fabric’s own plane.

What the limit does to the fabric before it stops it #

A hard limit is only half the story; what matters in practice is what happens as it is approached, and the arithmetic says something about that.

As the bend gets tighter, the bending energy in the crests rises as the square of the curvature, so a fabric near the limit is spending a great deal of energy in two short regions of every stitch. That energy has to come from the tension the machine applies to draw the loop.

So a fabric approaching the geometric limit becomes hard to knit long before it becomes impossible: the tension required rises steeply, the yarn breaks more often, and the machine has to run slower. That is exactly the practical experience of knitting very tight fabrics.

It also means the limit is approached asymptotically in effort rather than reached abruptly. Nobody will ever knit a fabric at a ratio of 1.01 and find it suddenly impossible; they will find it progressively unmakeable from well before.

That is a better description of a limit than a threshold, and it is consistent with the trade having no name for the thing they have been pushing against.

Which fabrics are nearest #

It is worth naming the constructions that get closest, because they are where a measurement would be worth making.

Fine-gauge sportswear knitted from continuous filament at high tightness factors is the commonest case. Such fabrics are quoted at tightness factors approaching twenty and their yarn is smooth and strong, so neither the machine nor the yarn strength binds as early as it does for a staple cotton.

Technical knits for filtration and reinforcement are tighter still and are made from filament for the same reason.

And a very tight hand-knitted fabric — a Guernsey, a gansey, the tightest traditional worsted work — reaches high tightness factors by hand tension rather than by machine, and its makers describe the fabric at that point as “like a board”.

Three constructions, all of them at the top of the range, and none of them measured against a bending limit because nobody has stated one.

Where the model stops #

The extrapolation goes outside the range. The ratio is computed from 2.8 to 4.5 millimetres and the ceiling is at 1.9, so the number is an extrapolation of about a third beyond the data. This collection has been wrong about extrapolations before and says so.

The hard limit assumes a solid rod. A yarn is a bundle, and a bundle bent past the point where its centre line’s radius equals its own can rearrange — the fibres on the inside of the bend migrate outwards rather than compressing. So the true limit is somewhat past the geometric one, by an amount nobody has computed.

That mitigation is real and it works in the direction of raising the ceiling. It is also the same fibre-sliding that this work has found four other pieces of evidence for.

And the loop is the free one. A loop at the limit is being bent as hard as it can be, and its shape at that point is not the shape a free solve gives.

What a bundle does at the limit #

The mitigation named above deserves a paragraph of its own, because it is the reason the limit is soft rather than hard for a real yarn.

A solid rod bent to its own radius has the material on the inside of the bend in contact with itself, and there is nowhere for it to go. A bundle bent that hard has somewhere: the fibres on the inside can migrate outwards, past their neighbours, and the section rearranges.

So a yarn at the geometric limit does not fail; it changes shape. The section becomes kidney-shaped, thicker on the outside of the bend than on the inside, and the effective neutral axis moves outwards.

That is observable in micrographs of tightly bent yarn and it is the same fibre mobility this work has found four other consequences of.

What it means for the ceiling is that the true limit is past the geometric one by however much rearrangement is available, which depends on the packing factor: a loosely packed yarn has more room to rearrange than a tightly packed one.

So the ceiling is not one number but a family, and a softly spun yarn should reach a higher tightness factor than a hard-spun one of the same count. That is a prediction and it is the opposite of what anybody would guess, since a hard-spun yarn is the stronger one.

The generalisation #

The rung is an instance of a class of constraint this collection has been under-using.

A geometric impossibility is worth looking for wherever a quantity is being pushed. It has no material constants in it, so it survives every bracket; it is usually cheap to compute; and it produces a hard number rather than a band.

This collection has two such constraints already. How close threads can be set is one: a woven cloth’s jamming condition is geometry, and no yarn stiffness enters it. The satin count is another: a satin exists at a given number of ends or it does not, and the answer is a division.

The bending limit is a third and it is the first that is about a thread rather than about an arrangement of threads. Others are probably available: the tightest crimp a woven thread can take, the sharpest fold a cloth can make, the smallest radius a pile tuft can be bent to. Each is the same question and each has a hard answer.

What the same limit says about a woven cloth #

The bending limit is a property of a thread rather than of a fabric, so it can be asked of anything, and a woven cloth gives a very different answer.

A woven thread’s tightest bend is at a crossing, where it goes over its neighbour and comes back. The radius it takes there is roughly the neighbour’s radius plus its own — about one yarn diameter for equal threads, which is two yarn radii.

That is the same number as a knitted loop’s, arrived at differently: the knitted one is set by how much yarn is in a stitch, and the woven one by the size of the thread being crossed.

But a woven thread only takes that bend where the crimp is at its full amplitude, which is at a plain weave’s every crossing and at a satin’s every fifth. And a real woven cloth’s crimp is a few per cent rather than the full amplitude, so its threads are bent far more gently than the geometry allows.

So a woven cloth is nowhere near its bending limit and a tight knitted fabric is close to one. That is a genuine structural difference between the two and it is a consequence of one fact: a knitted fabric puts a reversal in its thread and a woven one does not.

Where the crimp limit does bite #

The exception is worth naming because it is where a woven cloth does meet the constraint.

A cloth being jammed — beaten up as hard as it will go, at the maximum sett its yarns allow — has its threads at their maximum crimp, and at that point the bend radius approaches the same limit. This collection’s own jamming condition is derived from the threads touching, and it does not currently include a bending limit at all.

Whether the two limits arrive together is a computation the site could make: the jamming condition says how close the threads can be, and the bending limit says how sharply the crimp can turn, and if the second binds first then the jamming condition is not the binding one.

Nobody has checked. It is an hour’s work on machinery that exists, and it would settle whether a founding result of this collection is the limit it claims to be.

Who found it, and when #

That a rod cannot be bent below its own radius is elementary and is the standard limit quoted in bending-radius specifications for cable, wire and tubing.

That it applies to a knitted loop is not something this collection has found stated anywhere, and the reason is probably that nobody has had a solved loop’s curvature to compare against a yarn’s radius. Knitted geometries are usually written in terms of spacings rather than curvatures, and a spacing does not know how sharply the yarn is turning.

What is this collection’s own is the comparison, the ratio’s dependence on the tightness factor, and the observation that the flattening result from the same ladder makes the limit tighter rather than looser.

Where the ladder goes next #

The contact ladder has been about knitted fabric throughout, and the same question has never been asked of a woven one. This collection’s woven geometry is built the same way — a distance set correctly at the crossing, with the rest of the thread’s path free — so it may have the same defect.

A woven cloth asked the same question puts it directly, and the answer decides whether the flattening prediction extends to cloth or is a knitted peculiarity.