Why a slack yarn snarls

Take a sewing thread, hold it at both ends, and bring the hands together. Somewhere on the way the thread stops hanging in a curve and jumps into a tight coil, wrapping on itself in a way that will not comb out. Everybody who has sewn has done it, everybody who has spun has watched a yarn do it off the spindle, and the trade word for it is a snarl.

It looks like a mess. It is a buckling — a well posed instability with a criterion, a threshold and a length scale — and because the criterion contains the ratio of a thread’s two stiffnesses and nothing else that is unknown, it turns out to be an instrument.

What the instability is #

A thread carrying an axial torque and held under tension has a straight state available to it. The question is whether that state is a minimum of the energy or a saddle.

Perturb the straight thread into a shallow helix. Bending it costs energy proportional to the bending rigidity and to the square of the curvature. Stretching it against the tension costs energy too, because a helix is longer than the line it wraps. But the torque does work as the thread writhes, because writhing converts twist into writhe and lets the material unwind.

At low torque the two costs beat the gain and the thread stays straight. At high torque the gain wins and the helix grows, and it grows into a self-contacting coil rather than a gentle spiral because nothing stops it.

The threshold is where the three balance, and it is

M = 2·√(B·T)

with M the torque, B the bending rigidity and T the tension. Above that torque the straight thread buckles.

What this collection can do with it #

The formula is not new and the derivation is not this collection’s. What is new here is what can be put into it, because a yarn’s torque is now computable.

A yarn’s twist rate is its turns per metre — six hundred to a thousand for an ordinary cotton — expressed in radians per millimetre. Its torque is the torsional rigidity times that rate. And the torsional rigidity is the bending rigidity times 2G/E, a ratio that is the same at both ends of the stiffness bracket.

So the criterion can be turned round to ask the question a spinner actually has: how much tension does a yarn need to stay straight?

Set the torque equal to the threshold and solve for the tension:

T = M²/(4B) = (C·ω)²/(4B)

with ω the twist rate in radians per millimetre, and with the ratio C/B doing all the work.

Everything in that is known except B, and B appears once rather than twice — because M goes as C and the threshold as the square root of B, so the tension goes as C² over B, which carries the bracket a single time rather than squared.

The number #

For a twenty tex cotton at eight hundred turns a metre, at the free end of the stiffness bracket, the answer is 0.37 millinewtons.

That is thirty-eight milligrams of force, which is not a number anybody has an intuition for. Converted into the form the observation actually takes — the length of the yarn’s own weight that would supply it — it is 1.9 metres.

So: a twenty tex cotton at ordinary twist snarls unless about two metres of itself hangs below the slack point. That is a statement anybody can check without an instrument, and it is right. A short slack loop between two hands snarls immediately. A metre of thread hanging from a reel does not, and a thread hanging from a high shelf certainly does not.

The other end of the bracket #

The same arithmetic at the coherent bound — fibres unable to slide, the yarn twisting as a solid rod of its own diameter — gives 121 millinewtons, which is twelve grams, which is 616 metres of the yarn’s own weight.

Six hundred metres. A yarn like that would snarl on any reel, in any hand, at any length anybody has ever handled, and it would be impossible to sew with.

So the observation discriminates, and it discriminates by a factor of three hundred. Anybody who has let go of a twisted thread has performed an experiment whose answer is that a yarn’s bending rigidity is at the free end of its own bracket.

Why that is worth something #

This collection has been computing at the free bound for its whole life and saying so. The justification recorded when the bracket was first written was a measurement check: the free bound lands inside the band of real cloths for every construction in the site’s own table, and the coherent bound lands two to three orders above all of them.

That is a good argument and it is an argument from cloth. The snarl is an argument from a thread, and it is entirely independent: no cloth, no sett, no crimp, no packing factor beyond the one in the diameter.

Two independent observations landing on the same end of a bracket is worth more than either. And there turn out to be two more — a knot’s efficiency and the flattening a knitted fabric demands — which makes four readings from four unrelated phenomena, all agreeing.

Why it snarls rather than spiralling #

A detail worth understanding, because it explains why a snarl is a nuisance rather than a curiosity.

The instability produces a helix, and a helix of growing amplitude eventually touches itself. Once it does, the contact holds it: the two strands press together, friction between them resists sliding, and the coil is stable at whatever it has reached rather than relaxing back.

So the failure is hysteretic. A yarn that has snarled does not un-snarl when the tension is restored, and it has to be picked apart. That is why the sewing-thread version is annoying and why a spinner’s yarn that has snarled on the bobbin is a fault rather than a state.

It also means the observation is one-sided as a measurement: the tension at which a snarl forms is the criterion above, and the tension at which one comes out is much higher and depends on friction. Anybody using the observation as an instrument has to watch it form rather than watch it clear.

The square law #

The threshold tension goes as the square of the twist, exactly, because the torque goes as the twist and the criterion is quadratic in the torque.

That is a strong statement and it is checkable against practice. A yarn at double the twist needs four times the tension to stay straight, so twist factors are not a linear nuisance: doubling a yarn’s twist for strength quadruples its liveliness.

It also explains why the trade’s twist limits are as sharp as they are. A voile yarn at a twist factor around fifty is a manageable thing; a crepe yarn at ninety is a different animal entirely and has to be handled wet, steamed, or under tension at every stage — and ninety over fifty squared is a factor of three and a bit in how much tension it needs.

The coil comes in one size, and that size is bracket-free #

The most surprising result on this rung, and the one worth carrying furthest.

The radius the instability picks is 2B/M. Substitute M = C·ω and the bending rigidity cancels completely:

R = 2B/(C·ω) = 2/(r·ω)

with r the stiffness ratio. The radius does not depend on how stiff the yarn is at all — only on how hard it has been twisted, and on a ratio that is the same at both ends of the bracket.

For a cotton at eight hundred turns a metre that is 1.6 millimetres, so a snarl is about three millimetres across. At the coherent bound it is also 1.6 millimetres. It is 1.6 millimetres for any cotton yarn of any count at that twist.

That is a prediction anybody can check with a ruler, and it is a snarl coming in one size.

Why a folded yarn snarls less #

The rule the trade uses to avoid all of this is to fold the yarn, and the criterion says why it works.

A folded yarn is two or more singles twisted together in the opposite hand. The folding takes twist out of each single — one turn of fold per turn removed, near enough — so each single’s own residual torque falls. That is what folding untwists, and this collection derived it several ladders ago.

What the criterion adds is the size of the benefit. The threshold tension goes as the square of the residual twist, so a fold that removes seventy per cent of a single’s twist reduces the tension it needs by a factor of eleven — from two metres of its own weight to under twenty centimetres.

That is why sewing thread is folded, why a hard-twisted single is almost unusable as a sewing thread whatever its strength, and why a folded yarn that has been folded too little or too much is lively again in the appropriate direction.

It also sets up the question the next two rungs are about: at what fold ratio is the residual torque exactly zero? That is a torque balance, it is computable from the two stiffnesses, and the answer it gives turns out not to be the answer the trade uses.

The direction the snarl takes #

A snarl has a handedness, and the handedness is the yarn’s.

A Z-twisted single snarls in one sense and an S-twisted one in the other, because the writhe the coil supplies has to have the sign that lets the material unwind. So a snarl is a visible readout of a yarn’s twist direction, and anybody sorting unmarked yarn can use it.

That is a small practical point and it carries a larger one: the whole mechanism is chiral, and chirality is what the twist–writhe trade is made of. A structure that has no handedness cannot make the trade at all, which is exactly the finding on the knitted side of this work — a jersey’s course has no writhe, so it has nothing to offer a twisted yarn.

The contrast is worth holding: a free thread snarls readily because a free thread can writhe as much as it likes, and a thread in a fabric cannot, because the fabric decides its path. That is why a yarn that snarls on the reel does not snarl in the cloth, and why the effect it has in the cloth is a lean rather than a coil.

What was counted, and how #

The criterion is Greenhill’s and is quoted rather than derived: it comes from linearising the rod equations about the straight state and asking when a helical perturbation costs nothing.

Everything put into it is the collection’s own. The torsional rigidity is the bending rigidity times 2G/E, with the shear modulus from the site’s own fibre table and its range carried. The bending rigidity is the collection’s free bound, unchanged. The conversion into metres of the yarn’s own weight is a linear density and a gravitational acceleration.

Two checks run on the arithmetic and both could fail. An untwisted thread needs no tension, which is the degenerate case and would catch a sign or an offset. And the threshold rises as the square of the twist, checked as an exponent across four twist levels rather than at a point, because a relation that holds at one pair of points holds for any monotone function.

Where the model stops #

The criterion is for a uniform rod under uniform tension. A yarn hanging under its own weight has a tension that varies along it, so the snarl forms where the tension first falls below threshold, which is at the top of the slack rather than uniformly. The metres quoted are therefore a scale rather than a precise length.

A yarn is not torsionally elastic. The criterion assumes the torque is proportional to the twist rate through a constant. A spun yarn’s fibres slip, so its torque relaxes over minutes and hours: a yarn left on a bobbin overnight is less lively in the morning, and none of that is here. It is why a steamed yarn does not snarl, and steaming is the trade’s whole answer to this problem.

And the contact that stops the coil growing is not modelled. The criterion says when the straight state fails, not what the coil settles at. The radius quoted is the wavelength the instability picks at onset, and a fully formed snarl is a self-contacting plectoneme whose geometry is a contact problem.

The generalisation #

The habit here is worth naming because this collection has been slow to it.

An everyday nuisance is often a threshold, and a threshold reads a constant. The snarl was, until this rung, a thing that happens; it is now a measurement of a quantity that no amount of laboratory care had settled, made by anybody with a reel of thread.

The reason it works is that a threshold is a comparison rather than a value. Nobody has to measure the tension at which a thread snarls to a per cent. The two ends of the bracket predict 1.9 metres and 616 metres, and the observation only has to tell those apart — which it does at a glance, with no instrument, from something that has been happening in front of everybody for as long as there has been thread.

That pattern should be looked for elsewhere in this collection, and there are candidates. The tension at which a cloth’s threads slip past one another at a seam is a threshold. The pressure at which a pile collapses is a threshold. The load at which a preform locks in shear is a threshold, and this collection has computed it without asking what an observation of it would read. So is the point at which a cloth stops being able to shrink.

The two pictures above and below this sentence separate the two things the criterion depends on: the twist, which moves both the threshold and the coil, and the count, which moves only the threshold.

Putting the whole twist range on one plot makes the discrimination visible in a way a pair of numbers does not.

What a spinner does about it #

The trade’s answers to liveliness are worth listing against the criterion, because each attacks a different term and the criterion says which.

Fold the yarn, which removes residual twist and therefore torque. That is the largest lever, because the threshold goes as the square.

Steam or damp it, which lets the fibres slide and relax the torque away without changing the twist at all. This is the one the criterion cannot describe, because the criterion assumes a constant torsional rigidity, and steaming is the yarn ceasing to be elastic in torsion.

Keep it under tension at every stage, which raises T directly, and is why yarn is wound, warped and woven under controlled tension rather than allowed to go slack.

Or use less twist, which is free and costs strength, and is why soft-twisted wefts exist.

Three of those four are visible in the criterion as terms. The fourth is the one that works best and is entirely outside it, which is a fair summary of how far a rod model gets with a spun yarn.

Who found it, and when #

The stability of a twisted rod under tension was settled by Greenhill in 1883, in a paper about the shafts of steamships, and the criterion carries his name. Love’s treatise gives the derivation in the form used here.

The application to yarn liveliness is old in the trade and mostly qualitative: everybody knew that more twist means more liveliness and that tension and steaming suppress it.

What is this collection’s own is putting its own numbers into it — a torsional rigidity that did not exist here until this ladder, and a stiffness ratio that survives the bracket — and reading the observation backwards as a measurement of which end of that bracket a yarn sits at.

Where the ladder goes next #

Two directions, and both are checks rather than extensions.

The first is the length: how much yarn has to hang puts the two bounds side by side across the whole twist range and asks what a person with a reel actually sees.

The second is the size: the coil’s radius is independent of both stiffnesses, so it is the sharpest available test of the model and the easiest to make. A snarl comes in one size, and the size is a ruler’s worth of evidence about a quantity that has resisted everything else.