The section that changes both stiffnesses

The cleanest result on the torsion ladder is that a yarn’s two rigidities are in a fixed ratio — twice the shear modulus over the tensile one — however unknown either of them is. The derivation is one line and it has one assumption in it: the section is a circle.

A circle’s polar second moment is exactly twice its flexural one, so the diameters, the fibre counts and the packing factors all cancel and the ratio comes out as a material property.

The other half of this work has just established that a yarn in a knitted fabric is not circular, and has said how far from it. So the assumption is checkable rather than a warning, and checking it changes the result.

Three constants rather than two #

A circular rod has two elastic constants for the deformations that do not stretch it: one bending rigidity, because the two bending directions are equivalent, and one torsional rigidity.

A flattened rod has three. Bending about the long axis of the section is easy, because most of the material is close to that axis. Bending about the short axis is hard, because the material is spread far from it. And twisting is a third thing.

So the question “what is the ratio of a yarn’s torsional rigidity to its bending rigidity” stops having a single answer the moment the section stops being round. There are two answers, and which one applies depends on which way the thread is being bent.

What holds regardless #

One relation survives the flattening exactly and it is worth stating first, because it is the guard on the arithmetic.

The polar second moment is the sum of the two flexural ones. That is the perpendicular axis theorem and it holds for any plane section whatever — a circle, an ellipse, a racetrack, a lobed shape or a random blob.

So a flattened thread’s torsional rigidity is still the sum of its two bending rigidities times the appropriate moduli, and any arithmetic that violates that has a slip in it rather than a finding. It is checked to a part in a million million across every flattening computed here.

The two multipliers #

Hold the area — a squashed yarn rearranges rather than compacting, at least until it runs out of air — and write the section as an ellipse whose short axis is a fraction f of the round diameter.

The easy bending rigidity, about the long axis, falls to exactly f times the round one. The hard one, about the short axis, rises to 1/f. And the polar moment becomes their sum, which is (f + 1/f)/2 times the round one.

So the multipliers on the stiffness ratio are

easy: (f + 1/f)/(2f) and hard: (f + 1/f)·f/2

At a flattening of 0.78 — what a relaxed jersey’s geometry demands — those are 1.32 and 0.80.

Which direction a knitted loop bends in #

The multipliers are only useful if it is known which one applies, and for a knitted loop it is.

A yarn in a knitted fabric is flattened through the fabric’s thickness — that is what the flattening is, because the clearance the fabric leaves is a clearance in that direction. So the section’s long axis lies in the fabric’s plane.

And a knitted loop’s yarn bends in the fabric’s plane, overwhelmingly. The loop is a plane curve whose plane is the fabric’s turned a dozen degrees, so its curvature vector lies nearly in the fabric.

Bending in the plane about an axis normal to it is bending about the section’s short axis, which is the hard direction — so a knitted loop bends the hard way, its bending rigidity is 1/f times the round value, and the stiffness ratio is multiplied by 0.80 rather than 1.32.

That is the opposite of what a first reading suggests, and getting it right requires being careful about which axis is which.

So the ratio is a lower bound, in one direction and not the other #

Putting it together: for a yarn in a knitted fabric bending the way the loop bends it, the ratio C/B is about 0.80 times 2G/E, not 2G/E.

For a cotton that is 0.20 rather than 0.25.

That is a twenty per cent correction to the number the whole torsion ladder is built on, and it runs in the direction that makes the torsion less important rather than more. It does not change any of the ladder’s qualitative conclusions — the balance point moves from 0.200 to 0.167, still nowhere near two thirds; the snarl radius moves by twenty-five per cent, still about three millimetres — and it changes every quoted figure by that amount.

And a woven thread bends the other way #

The same argument applied to a woven cloth gives the opposite answer, which is a useful check that the reasoning is about geometry rather than about knitting.

A thread in a woven cloth is also flattened through the cloth’s thickness, so its long axis is also in the cloth’s plane. But a woven thread’s bending is crimp — it goes up and down through the thickness — so its curvature vector lies in the cloth’s plane and it bends about the section’s long axis.

That is the easy direction. So a woven thread’s bending rigidity is f times the round value rather than 1/f, and its stiffness ratio is multiplied by 1.32.

The same flattening makes a woven thread easier to bend and a knitted one harder, because the two bend in perpendicular directions. That is not a small distinction and it has never been drawn in this collection.

What it does to a cloth’s drape #

The woven half has an immediate consequence, because a cloth’s bending rigidity is what decides how it hangs.

This collection computes a cloth’s rigidity from its threads’ — the yarn’s own bending rigidity times the length of yarn per unit area times a factor for the angles the yarn lies at. Every one of those calculations has used a round section.

A flattening of 0.78 multiplies the yarn’s rigidity in the relevant direction by 0.78, so a cloth’s computed rigidity falls by twenty-two per cent, and its drape — which goes as the cube root of the rigidity over the weight in the usual formulation — moves correspondingly.

That is a systematic correction in one direction to every drape figure this collection has produced, and it explains part of a gap the collection has recorded: computed rigidities sit above measured ones for real cloths, and a flattening correction closes a fifth of it.

The one case where the correction vanishes #

There is a construction where none of this applies and it is worth naming, because it is the control.

A filament yarn that has been heat set into a round section, or a monofilament, is round and stays round. Its section is not deformed by its neighbours because it has no fibres to rearrange, and the flattening the fabric demands has to be found somewhere else — a longer loop, a wider spacing, or a fabric that does not relax as far.

So a monofilament knitted fabric should show the uncorrected ratio, and should also show the dimensional differences that come from not being able to flatten. Both are measurable and neither has been looked for.

Why the two directions matter more than the magnitude #

The twenty per cent is the least interesting thing on this rung, and it is worth saying why.

A twenty per cent correction to a quantity that is otherwise known exactly would be important. This quantity is not otherwise known exactly: the shear modulus it is built from has a factor of two on it, so a twenty per cent geometric correction is well inside the material uncertainty.

What is not inside the material uncertainty is the asymmetry. A yarn in cloth has two bending rigidities that differ by a factor of 1/f² — 1.64 at a flattening of 0.78, and more at tighter constructions — and every calculation in this collection has used one number for both.

That matters wherever a yarn bends in two directions at once, and there is one obvious place: a yarn being drawn into a knitted loop bends in the fabric’s plane at the loop’s sides and out of it at the interlacing. The first is the hard direction and the second is the easy one, and treating them as equal is a real approximation that nobody has priced.

The one number that does not move #

A pleasing consequence of the perpendicular axis theorem is worth extracting because it says something about which quantities are robust.

The polar second moment is the sum of the two flexural ones. So while each bending rigidity moves — one up, one down — and their ratio moves a great deal, the torsional rigidity moves very little: (f + 1/f)/2 is 1.03 at a flattening of 0.78, which is three per cent.

A flattened yarn is almost exactly as hard to twist as a round one of the same area. That is not obvious, it falls straight out of the theorem, and it means every torsional result on this collection’s other ladder is nearly unaffected by the flattening even though the ratio it is expressed through is not.

So the snarl threshold, the snarl radius and the fold balance all move only through the bending rigidity, and each of them by a different amount depending on which power of B it carries. Working that out for each is arithmetic and is worth doing before anybody quotes those numbers to three figures.

What was counted, and how #

The section arithmetic is exact and is elementary: an ellipse of semi-axes a and b has flexural second moments πab³/4 and πa³b/4, and a polar moment that is their sum.

Holding the area means ab is fixed, so writing the flattening as b/a and normalising to the round section of the same area gives the two multipliers directly.

Three checks run and all three could fail. The polar moment is the sum of the two flexural ones, at every flattening, to a part in a million million — the perpendicular axis theorem, which would catch an algebraic slip. A round section leaves the ratio exactly where the circular derivation put it, which is the degenerate case. And the easy-direction multiplier exceeds 1.3 at the flattening a knitted fabric demands, which is the finding stated as a threshold so that it fails if the flattening result ever moves.

What a designer would do with the asymmetry #

There is a design consequence and it is the kind that is invisible until the arithmetic is done.

A yarn whose section is deliberately flattened — a tape yarn, a flat filament ribbon, a slit film — is at an extreme of this arithmetic rather than outside it. Its flattening is not 0.78 but 0.1 or less, so its two bending rigidities differ by a factor of a hundred.

Such a yarn is very easy to bend one way and very hard to bend the other, and a fabric made from it behaves accordingly: it drapes readily about one axis and resists about the other, which is why tape-yarn fabrics have a characteristic stiff, papery hand in one direction and a soft one in the other.

The arithmetic here says that behaviour is not a property of tape yarns as a special case. It is the same formula, evaluated at a flattening an ordinary yarn does not reach, and an ordinary yarn in a tight fabric is a sixth of the way along the same curve.

It also predicts something checkable about tape yarns: their torsional rigidity should be nearly the same as a round yarn of the same area, because the polar moment barely moves. A tape yarn should be very anisotropic in bending and unremarkable in torsion, which is not what anybody’s intuition about a flat ribbon suggests.

Why the collection has not needed this before #

A fair question is why eighteen phases of work on cloth geometry has managed without an anisotropic thread, and the answer is that the collection has mostly computed energies rather than directions.

An energy is an integral of the square of a curvature times a rigidity. If the rigidity is the same in both directions, the integral does not need to know which way the yarn is bending, and every solve on this site has been written that way: one B, one curvature, one integral.

The moment the two rigidities differ, the solve needs to resolve the curvature into two components and weight them separately — which is a different functional and a different solver.

So this rung is a debt on the machinery as well as on the arithmetic, and it is a larger one than it looks. Every energy in this collection is computed as though its thread were isotropic in bending, and the correction is not a multiplier: it depends on how the curvature is distributed between the two directions, which is different for every figure.

Where the model stops #

The section is an ellipse. A yarn’s section is not, and this collection has argued elsewhere that the racetrack is a convenience rather than a measurement. A racetrack’s second moments differ from an ellipse’s of the same area and flattening by a few per cent, which is smaller than the flattening uncertainty and is not computed here.

The area is held. A yarn squashed hard enough loses air, and then the packing factor and the diameter the whole calculation started from are moving too.

The section is uniform along the yarn, and it is not: the flattening is a minimum over a profile, and only a fifth of the yarn is pressed at all. So the correction computed here is an upper bound on the correction that actually applies, and the true figure is somewhere between it and nothing.

That last is a substantial caveat and it cuts the correction down considerably. What survives it is the sign and the asymmetry: a knitted loop bends harder and a woven thread bends easier, whatever the magnitude.

And nothing is re-solved. A loop whose yarn has a different bending rigidity takes a different shape, and every geometry on this site was computed with the round value.

The generalisation #

The rung is an instance of two things and the second is the one worth carrying.

The first is that an assumption stated in a derivation is a debt, and this collection is unusually good about stating them and less good about paying them. The circular-section assumption was written into the ratio’s derivation, in the same essay, as a named caveat — and it stayed a caveat for exactly as long as nobody had a number for the flattening.

The second is about how two ladders built at the same time ought to interact. This collection built a torsion ladder and a contact ladder side by side, and they were planned as separate pieces of work on separate anchors. The contact ladder produced a flattening; the torsion ladder had a caveat that needed one; and neither would have supplied the other if they had been done in separate phases.

Two ladders in one phase can pay each other’s debts, and that is an argument for planning around a pair of related questions rather than around one deep one — which is not how this collection has usually scheduled.

Who found it, and when #

The second moments of an ellipse are elementary and the perpendicular axis theorem is older than anybody’s textbook.

That a flattened thread is anisotropic in bending is well known in the composites literature, where flattened tows are the normal case and their two rigidities are computed as a matter of course.

What is this collection’s own is the pairing: computing the flattening a fabric’s own geometry demands, and then using it to correct a ratio derived on the assumption of a round section — including the observation that a woven thread and a knitted one are corrected in opposite directions.

Where the ladder goes next #

The contact ladder has been about a fabric’s own geometry so far. The next question is what happens when something presses on it from outside, and the most extreme case of that is a thread held only by pressing on itself.

A knot is contact and nothing else, and asking what it is made of turns out to answer a question about strength that has been quoted as a rule of thumb for as long as there have been ropes. A knot is nothing but contact.