A thread has a second stiffness

A thread resists two things. It resists being bent, and every force this collection has computed came from that; and it resists being twisted, and nothing here has ever used it.

The omission is not an oversight that went unnoticed. It is written down, in as many words, at the end of the ladder that solved a knitted loop: a thread’s torsional rigidity is of the same order as its bending rigidity, the twist a loop demands is of order a turn a stitch, and so the torsional energy is somewhere between a few per cent of the bending energy and rather more than all of it — with no honest way to narrow it from there. That was the first thing to disbelieve in the whole account.

Narrowing it is what this rung does, and the answer arrives from an unexpected direction.

What the second stiffness is #

Take a thread and hold both ends. There are two independent things that can be done to it without stretching it.

The first is to bend it: to give its centre line a curvature. The energy of that is one half the bending rigidity times the square of the curvature, integrated along the thread, and it is the functional every solve in this collection minimises.

The second is to twist it: to rotate the material about its own axis, so that a mark painted along the thread winds round it. The energy of that is one half the torsional rigidity times the square of the rate at which the mark winds, integrated along the thread in exactly the same way.

Those two together are the whole elastic energy of a thin rod that does not stretch. There is no third term for a rod of circular section, because a circular section has no preferred direction to bend about and so the two bending directions are one quantity.

So a thread has two constants, and the collection has been carrying one.

Twist is not the curve’s business #

The distinction that has to be made before anything else is between the twist a material carries and the torsion a curve has, because they share a word and are not the same quantity.

A space curve has a torsion: the rate at which its osculating plane turns about its own tangent. It is a property of the centre line and nothing else. A straight line has no osculating plane at all, so it has no torsion; a curve lying in a plane has an osculating plane that never turns, so its torsion is zero everywhere.

The picture above is a straight rod, so its torsion is zero. It is carrying a turn and a half of twist. The two quantities are independent, and the one a torsional rigidity resists is the second.

That matters here for a specific reason. This collection’s own account of a knitted loop ends by proving that the solved loop is a plane curve after all — the fabric’s own plane, turned about a dozen degrees. A reader who has confused the two quantities will conclude from that sentence that the loop carries no twist. It does not follow. A plane curve has no torsion and can carry as much twist as anybody puts into it, and a spun yarn arrives with a great deal already in it.

The bracket, which is the problem this collection already has #

A yarn’s bending rigidity is not a number. It is a range, and a wide one, for a reason that is structural rather than experimental.

If the fibres in a yarn may slide past one another freely, each bends about its own axis and nothing couples them, so the bundle is as stiff as the sum of its fibres. If they cannot slide at all, the yarn is a solid rod of its own outside diameter. The second is stiffer than the first by the fibre count over the square of the packing factor — three hundred and twenty-seven fold for an ordinary cotton at twenty tex — and no amount of care about the fibre narrows it, because the uncertainty is not about the fibre.

Every force in this collection is linear in that constant, so every force in this collection inherits the bracket. The convention has been to compute at the free bound and to say so.

The obvious expectation is that a second constant means a second bracket, and therefore a worse position rather than a better one: two unknown numbers where there was one.

The two brackets are one bracket #

They are not two unknown numbers. The same argument that brackets the bending rigidity brackets the torsional one, and it brackets it by exactly the same factor.

Free, the fibres each resist a twist about their own axes, and the bundle’s torsional rigidity is the sum of the fibres’. Coherent, the yarn twists as a solid rod of its own outside diameter. The ratio between those two is the fourth power of the yarn’s diameter over the fibre count times the fourth power of the fibre’s — and the site’s own volume arithmetic says the yarn’s diameter is the fibre’s times the square root of the count over the packing factor, so the fourth powers reduce to the count over the packing factor squared.

Which is the bending bracket, term for term. Not approximately: identically, and it is checked across counts, fibres and packing factors to a part in a million million, because an identity that holds at one point is arithmetic and an identity that holds everywhere is a derivation.

And so the ratio survives #

If the two brackets are the same factor, the ratio of the two rigidities is the same at both ends of it.

At the free bound, the ratio is the fibre’s own torsional rigidity over its own bending rigidity. At the coherent bound, it is the yarn’s over the yarn’s. Both are the shear modulus times the polar second moment, over the tensile modulus times the flexural one — and for a circular section the polar second moment is exactly twice the flexural one, so both are

C/B = 2G/E

with the diameters, the fibre counts and the packing factors all cancelling.

That is the finding of this rung, and it is worth stating as plainly as it can be:

Neither stiffness is knowable to better than a factor of three hundred, and their ratio is knowable to whatever the fibre’s own shear modulus is knowable to.

Every result later on this ladder that depends only on the ratio is therefore a result this collection can state without its usual apology, and there turn out to be more of those than anybody expected: how hard a fold has to be twisted to balance its own torque, how large a snarl is, and what fraction of a loop’s energy the twist can possibly be.

The table, and the row that checks it #

A shear modulus is a worse-known quantity than a tensile one, and the reason is experimental rather than theoretical: a fibre is a few micrometres across, and twisting one against a torque of nanonewton-millimetres is a harder measurement than pulling it. So the table carries a working value and a range, exactly as the modulus table beside it does.

What makes the table trustworthy is not the care taken over any row. It is that one row’s answer is known before any measurement.

Glass is an isotropic solid. A drawn glass filament has no preferred direction in it, so its shear modulus is not independent of its tensile one at all: it is E over twice one plus Poisson’s ratio, and at glass’s ratio of 0.2 that makes 2G/E exactly five sixths. The table says 0.833. If it ever stops saying that, something has drifted.

Every other row sits below the isotropic value, and the ordering is the subject rather than an accident. A textile fibre is stiff along its axis because its molecules are drawn out along it, and the same orientation that raises the tensile modulus leaves the shear modulus to be carried by whatever holds one chain to the next — which is much weaker than the chain. So the more oriented the fibre, the lower the ratio: aramid, the most oriented thing in the table, is twenty times softer in torsion relative to its bending than the glass beside it.

That is a real prediction about materials, made by a table with one control in it, and it is the kind of thing that would have been invisible if the ratio had been quoted as “about two thirds” and left there.

What this says about the loop, immediately #

The ladder that solved a knitted loop priced the out-of-plane excursion at a relief of 2.71 per cent of the bending energy and said that torsion was the item to disbelieve, on the grounds that its rigidity was “of the same order” as the bending one.

For a cotton it is a quarter of it, not the same order. For a polyester it is a sixth. For a flax it is a twelfth. The one common textile fibre where “the same order” is right is wool, at four fifths — which is a keratin, and much less oriented than a cellulose or a drawn polyester.

That does not settle how much energy the twist carries, because that depends on how much twist the loop actually demands, which is a question for further along the ladder. It does settle the multiplier, and it settles it in a direction that makes the earlier account better rather than worse: for the fibre this collection computes in by default, a turn of twist costs a quarter of what the same amount of bending would.

A reader who wants the ordering the other way up — the softest fibre in torsion at the top rather than the bottom — should note that it is the same ordering as orientation, and that orientation is what a spinner buys when a filament is drawn. A fibre is made stiff along its axis on purpose. Nobody sets out to make one soft in torsion, and every process that does the first does the second.

Why the ratio is bracket-free and the stiffnesses are not #

It is worth being explicit about why one quantity escapes and the other two do not, because the pattern recurs.

The bracket exists because the arrangement of the fibres is unknown: whether they may slide is a question about friction, migration and setting, and it is not answerable from a count and a packing factor. Both rigidities depend on the arrangement, and they depend on it in the same way, because sliding is sliding — a fibre that can slide when the yarn bends can slide when it twists.

The ratio depends on the section rather than on the arrangement. Whatever the fibres are doing, they are doing it inside a circle, and the polar second moment of a circle is twice its flexural second moment. That is geometry and it does not care what is inside.

So the general rule this rung establishes, which will be used again: a quantity that depends only on the shape of the section survives the bracket, and a quantity that depends on how the fibres are behaving does not.

What a reader should take from the ordering #

The table is not the point of this rung and the ordering in it is.

A collection that wanted a single working number for torsion could have taken the isotropic value and moved on, and a great deal of engineering does exactly that. The cost of doing so here would have been large and quiet. Two thirds is the answer for a material with no direction in it, and every textile fibre except glass has a direction in it — that is what a fibre is. Using two thirds for a flax would overstate its torsional rigidity by a factor of eight, and the overstatement would propagate silently into every torque this ladder computes, because a torque is a rigidity times a twist and the twist would have been right.

That is the ordinary way a wrong constant does damage in this collection: not by making a result absurd, but by making it plausible and wrong by a factor nobody can see. The pair of shape constants taken from two different rows of a relaxation table moved every occupancy by five per cent and sat there for several rungs, because the assertion guarding them checked a relationship that held for any pair.

So the table’s one control does real work. It is not there to validate glass, which nobody is knitting. It is there so that the nine rows around it can be read as a physical ordering rather than as a list of numbers somebody chose.

What was counted, and how #

Nothing here is fitted, and every number came from arithmetic already in the collection plus one new table.

The diameters come from the count, the fibre density and the packing factor, by the same volume arithmetic that has produced every yarn diameter on this site from its earliest work. The fibre count is one count divided by another. The bending bounds are the collection’s own, unchanged. The torsional bounds are the same two arguments applied to the polar second moment instead of the flexural one.

The shear moduli are new, and they are the one thing here that is a measurement rather than a derivation. They are working values with ranges, and the ranges are wide — a factor of two on cotton — which is why every result below that depends on a shear modulus carries that range and says so.

The identity was checked over four counts, four fibres and three packing factors, on both claims: that the torsional bracket equals the bending bracket, and that the ratio equals 2G/E at both ends of it. The worst departure over all thirty-six yarns is a part in ten thousand million million, which is a closed form agreeing with itself.

Where the model stops #

The fibres are treated as straight and parallel to the yarn’s axis. They are not: a spun yarn’s fibres run at a helix angle that rises from nothing at the centre to the surface angle at the outside, and a fibre at an angle is neither bending nor twisting purely about its own axis when the yarn does. That approximation is the collection’s own, made when the bending bracket was written, and it is inherited here rather than introduced. It biases both bounds in the same direction, which is part of why the ratio is more robust than either.

The section is circular. A yarn in a fabric is not — that is what the next ladder along is about — and a flattened section has a different polar second moment relative to its two flexural ones. For a mild flattening the correction is second order; for the four-fifths this collection is about to find that fabrics demand, it is not negligible and it is not computed here.

And a real yarn’s fibres slip differently in torsion and in bending. The free bound assumes frictionless sliding and the coherent bound assumes none, and a real yarn is somewhere between — but not necessarily at the same place between for the two deformations, because a twist shears the whole section while a bend only stretches one side of it. If the two deformations sit at different points in their brackets, the ratio stops being 2G/E. Nothing here bounds that, and it is the honest limit on the whole result.

The generalisation #

The sentence to carry away from this rung is not about torsion. It is about which quantities a bracket can be defeated on.

This collection is full of quantities it cannot pin down: a yarn’s bending rigidity, its lateral rigidity, the fraction of its natural curvature that setting has taken. The instinct has been to sweep them and report a band. This rung is a case where the band was not necessary, and the reason it was not necessary generalises: the unknown cancelled because two quantities shared it.

That is worth looking for elsewhere. Wherever two results in this collection carry the same bracket, their ratio does not, and a ratio may be exactly the quantity a question is about. The crimp balance between two thread systems is a ratio of two things that share the same unknown stiffness; so is the comparison of a knit’s two bending directions. Both are more robust than the numbers they are made of, and neither has been described that way.

Who found it, and when #

Kirchhoff wrote the equations for a thin elastic rod carrying both deformations in 1859, and everything above about a rod’s energy is his. The observation that a circular section’s polar second moment is twice its flexural one is older than that and is arithmetic.

What is this collection’s own is the pairing: that the free-and-coherent bracket this site invented for a yarn’s bending applies unchanged to its twisting, that the two are the same factor, and that the ratio therefore escapes. The bracket itself was written down here several ladders ago and its consequence for torsion was not noticed then because there was no torsion in the model to have a consequence for.

Where the ladder goes next #

Two directions, and both of them use the ratio rather than either stiffness.

The first is the thing a twisted thread does when it is let go, which is to stop being straight — and the tension that stops it doing so turns out to depend on the ratio and on the twist and on nothing else that is unknown. That is why a slack yarn snarls.

The second is the rule the trade uses for how hard to fold a yarn, which has been quoted in this collection as a bracket from practice and derived nowhere. A torque balance is a ratio of two moments and therefore a ratio of the two stiffnesses, so it is computable — and the answer it gives is wrong by a factor of three, which turns out to be the most useful thing on this ladder.