A straight fibre cannot share the load

There is a picture of a twisted yarn that everybody carries and that is wrong in a specific and consequential way. In the picture, the fibres are helices: each one starts at some radius and stays there, winding steadily along the yarn like the strands of a rope.

A yarn like that would fall apart. Not because the fibres would slip — that is a separate problem, and it has its own rung — but because a fibre that stays at its own radius cannot take its share of a load.

The claim #

A twisted yarn’s strength lies between two exactly computable factors, and the ratio between them is 2/(cos α(1 + cos α)) — nothing at no twist, sixteen per cent at a shirting warp’s angle and forty-eight per cent at a crepe’s.

• The affine model, in which every fibre keeps its radius, gives the classical cos²α. It is a genuine lower bound: no arrangement realises less than the fibres strained proportionally to their own inclination.
• The equal-tension model, in which every fibre migrates freely between core and surface, gives 2cos α/(1 + cos α). It is a genuine upper bound: nothing does better than every fibre carrying its full share.
• Real yarns are between. What migration is worth is the whole gap, and it is why a spun yarn is made of fibres that wander rather than of fibres that lie still.

This is a bracket in exactly the sense the bending rigidity is a bracket: two bounds that can be derived and no third calculation that can be. The site’s habit in that situation is to compute both ends and refuse to interpolate, and it applies here.

The affine model, and the gradient it has #

Take a yarn of radius R at a surface helix angle α. A fibre at a fraction u of that radius lies at a helix angle θ with tan θ = u tan α: the angle grows from nothing on the axis to α at the surface.

Now extend the yarn by ε. If every fibre stays at its radius — which is what “affine” means, the fibres deforming with the body they are in — then a fibre at angle θ is stretched along its own length by

$$\varepsilon_f = \varepsilon \cos^2\theta.$$

One cosine because the fibre’s length changes by only the axial component of the yarn’s extension, and a second because the yarn contracts laterally as it extends, drawing the fibre inward.

The consequence is a gradient. The fibre on the axis takes the whole of ε; the fibre at the surface takes cos²α of it. At 25° — an ordinary warp twist — that is a ratio of 1.22 between the two, and it does not depend on how hard the yarn is pulled.

So the fibres do not break together. As the yarn is extended, the core reaches its breaking extension first, while the surface fibres are still at 82 per cent of theirs and carrying correspondingly less than they could. Integrating the axial contributions over the section gives the yarn’s stress as

$$\sigma_y = \varphi E \varepsilon \cos^2\alpha,$$

the packing factor times the fibre modulus times the strain times cos²α — Gégauff’s rule, arriving as the result of an integral rather than as an assertion.

The equal-tension model, and what it needs #

Now suppose instead that each fibre wanders between the core and the surface as it runs along the yarn — which is what a real spun fibre does, and which is called migration.

A fibre that spends time at every radius has, over any useful length, the same mean strain as every other fibre. So they all carry the same tension, and they all reach breaking together. Resolving that common tension along the yarn’s axis and averaging over the section gives

$$\text{obliquity} = \frac{2\cos\alpha}{1 + \cos\alpha}.$$

The two expressions are not obviously related and they come from the same integral with different things held constant — the strain in the first case, the tension in the second. Dividing one by the other simplifies remarkably:

$$\frac{\text{equal tension}}{\text{affine}} = \frac{2}{\cos\alpha\,(1 + \cos\alpha)}.$$

At zero twist it is exactly one, which is the check that matters most: a yarn with no twist has no gradient and nothing to gain from migration, and a formula that did not close there would be wrong.

What migration buys, in three currencies #

Strength, which is the ratio above: 2.3 per cent at 10°, 15.8 at 25°, 47.8 at 40°.

Extension. In the affine model the yarn breaks when the core reaches the fibre’s breaking extension, which happens at a yarn strain of exactly ε_b. In the equal-tension model every fibre is at the section’s mean of cos²θ, so the yarn must be strained further before any fibre reaches its limit — by a factor of tan²α ÷ ln(1 + tan²α), which is 1.105 at 25° and 1.321 at 40°. A migrating yarn stretches ten to thirty per cent further before it breaks, and it does so without any fibre being asked to stretch further than it can.

Toughness, which is the product of the two and is therefore the currency in which migration is worth the most: at 25° the area under the curve is up by about a quarter.

None of the three is available to a filament yarn, which is why a filament yarn is twisted so little. A continuous filament runs the whole length of the package at whatever radius it was laid down at; it cannot migrate, so every turn of twist costs it the full affine penalty and buys it nothing at all. A spun yarn must be twisted, and pays cos²α to get a grip it cannot do without; a filament yarn has the grip already and can afford to lie straight.

What the earlier curve was missing #

This collection has drawn the strength–twist relation once before, when the helix angle got a page of its own. The falling half of that curve was cos²α, presented as exact — and it is exact, for the affine model, which was the only model in view.

So the earlier essay was right about what it computed and understated what it did not know. That is worth recording as a correction rather than an addition: an exact result computed inside a model is exact inside the model, and the model in that case assumed something about fibre paths that nothing had checked.

Why a rope is not a yarn #

The affine model has one place where it is not an idealisation at all, and putting it there sharpens what migration means.

A rope is made of strands and a strand of yarns, and at every level the members are laid at a fixed radius and stay there: a strand cannot wander into the middle of a rope and back out again, because it is a continuous object metres long being wound by a machine that puts it where it goes. A rope is affine by construction. So a rope pays cos²α in full, with no upper bound available to it, and the whole of its design is a compromise between the helix angle it needs to hold together and the strength that angle costs.

Rope lay angles are correspondingly low — around 20° for the strand helix in an ordinary three-strand rope, and lower still in the wire ropes where nothing needs to grip at all. A cotton yarn is twisted to 23° and a crepe yarn to 40, both far above what a rope would tolerate, and the difference is that a rope’s members do not need holding and a yarn’s fibres do.

The comparison also explains why splicing works and knotting does not. A splice restores the helix; a knot leaves the strands at a different angle over a short length, which is a local change in the obliquity factor and a permanent loss.

The same shape, one field over #

It is worth noticing that this is the second bracket in this collection with the same structure, and that they are brackets for the same reason.

The pattern is worth naming because it recurs whenever a body is made of parts that may or may not move relative to one another. Compute both limits, refuse the interpolation, and say what would decide it. In the bending case what decides it is whether the fibres can slide, and the answer for a thread in cloth turns out to be lopsided. In this case what decides it is migration, and nothing here measures how much of it a given yarn has.

Where the bracket puts a real yarn’s shortfall #

A spun yarn realises far less than the strength of the fibres in it. The trade’s name for the fraction is translation efficiency, it is measured by comparing a yarn’s tenacity with a bundle test on the same fibres, and for an ordinary ring-spun cotton it is between forty and sixty per cent. Roughly half the fibre is not showing up.

The cos²α rule is the usual first explanation offered, and the bracket says it cannot be most of the answer.

At a warp yarn’s 23° the affine model gives 0.847 and the equal-tension model 0.949. So obliquity accounts for between five and fifteen per cent of the loss, and the shortfall is fifty. Even the pessimistic end of the bracket leaves thirty-five points unexplained, and the optimistic end leaves forty-five. Whatever is taking the rest of the fibre’s strength, it is not the angle.

That is worth stating as a correction rather than as a footnote, because the cos²α factor is very often quoted as though it were the explanation — a yarn is weaker than its fibres because its fibres are at an angle, and here is the cosine. The cosine is real, exactly computed, and a minority shareholder.

The three candidates for the rest are all in this collection and none of them is an angle.

The grip, which is the other half of the twist curve. A fibre near the end of its own length is carrying less than its share whatever radius it is at, and the length over which it is not carrying is set by the friction and the twist. At an ordinary cotton staple this is a large fraction of every fibre.

The population, which is the weakest-link argument one level up. A yarn breaks at its thinnest section, and a bundle test averages where a yarn test takes a minimum — so part of the “loss” is the two tests asking different questions of the same distribution. On the numbers in that essay a 500 mm specimen is already twenty-two per cent below its own sections.

And the fibres themselves, which do not break together even at equal strain, because their breaking extensions are a distribution rather than a value. That is the same convexity argument this collection makes about diameters, applied to a strain.

Adding the second of those to the pessimistic end of the bracket gets most of the way to the measured figure without invoking anything else, which suggests the ordering: the extreme is the largest term, the grip is next, and the obliquity is third. That ordering is the opposite of the order in which the three are usually mentioned, and it is a consequence of having computed the obliquity properly enough to see how small it is.

The practical reading is a warning about attribution. A factor that is exactly computable attracts the explanation, because it is the one anybody can write down — and a term that has to be bracketed or measured gets left out of the account even when it is three times larger.

What migration costs, which is not nothing #

If migration is worth this much, why does a yarn not migrate more?

Because migrating is bending. A fibre that runs from the core to the surface and back is following a path with curvature in it, over and above the curvature of its own helix, and bending a fibre costs energy — the same energy this collection prices when a thread bends round a crossing and when a knit’s loop is formed. A fully migrating yarn is a yarn whose every fibre is permanently strained in bending, and the strain does not go away when the load does.

Three consequences follow, and all three are things spinners talk about without connecting them to this.

A migrating yarn is hairier. A fibre on its way to the surface arrives there with an end, and an end at the surface with the pressure holding it falling to zero is an end that stands off.

A migrating yarn is bulkier. The bending strain has to go somewhere and it goes into the packing: a yarn whose fibres are trying to straighten presses outward, which is one of the reasons a packing factor is not a constant.

And the migration is not free to the spinner either. It is produced by the tension differences between core and surface at the twisting point, so it is a consequence of how the yarn was made rather than a property that can be specified. Rotor and air-jet yarns migrate differently from ring-spun ones, which is part of why they realise different fractions of their fibres at the same twist.

What was counted, and how #

Both closed forms are checked against the integrals they reduce, at five angles from 5° to 45°, by Simpson quadrature over twenty thousand intervals, with the worst relative departure at four parts in 10¹⁵. Reducing an area-weighted integral over a section by hand is exactly the step at which a plausible slip survives, and a quadrature is the only check that catches it.

The ratio is checked against the quotient of the two rather than assumed. Its closed form was obtained by a cancellation of three factors, and a cancellation is a second chance to be wrong.

And the bracket is required to close at no twist and to open with the angle — at every step, not merely at the ends. An assertion that a quantity rises between the endpoints of a range is satisfied by a function that wanders in between.

Where the model stops #

Neither model is a real yarn and no third calculation is offered. Migration is a continuum: a real fibre wanders irregularly, spends unequal times at different radii, and does not complete a full excursion in every yarn length. Nothing here computes the degree of migration for a given yarn, and there is no interpolation between the two bounds that could be justified.

The fibres are supposed elastic to break and identical. Real fibres vary in modulus and in breaking extension as much as they vary in fineness, and a distribution of breaking extensions blurs the sharp distinction between “the core breaks first” and “they all break together”. That is the same population argument this collection makes elsewhere, applied one level down, and it is not made here.

The yarn is supposed to have a uniform packing factor from core to surface. It does not: a ring-spun yarn is denser at its centre, which weights the average over the section differently and moves both bounds in the same direction.

Nothing here has any friction in it. Both models assume a fibre is carrying the tension the model gives it, which requires it to be gripped by its neighbours over most of its length. Whether it is, and over what length, is the other half of the problem and is what the whole strength–twist curve turns on.

And nothing here says whether a real yarn migrates enough to matter. The bracket is what can be computed. Measurements of migration exist — a fibre traced through a sectioned yarn — and this collection has none of them.

Where the ladder goes next #

Directly into the grip, because the two halves of a yarn’s strength are the obliquity computed here and the cohesion that has not been computed anywhere in this collection. What holds the end of a fibre turns out to have a surprising cancellation in it — the fibre’s own strength and the load on the yarn both drop out — and with both halves in hand the strength–twist curve can be built rather than sketched.

Sideways, into the folded yarns. A single inside a two-fold yarn is at its residual twist rather than its spun twist, so its obliquity is better and its grip is worse — the two halves moving in opposite directions, which is why a folded yarn’s strength is so much less sensitive to its folding twist than anybody expects.