What grips the end of a fibre

A staple fibre is a few centimetres long and its two ends are loose. Nothing is tied, nothing is glued, and the yarn it is in is a metre or a kilometre long. Pull that yarn and something has to stop the fibre from simply sliding out of the assembly.

What stops it is the twist, and the mechanism is one this collection already knows in another setting: a body under tension that is wound round something presses on what it is wound round, and the press produces friction. A thread in a cloth is gripped where it turns for the same reason.

The difference is that a fibre in a yarn is not wound round anything in particular. It is wound round the rest of the yarn, and the rest of the yarn is wound round it. The whole assembly grips itself, and the pressure that does it can be written down.

The claim #

Whether a fibre in a twisted yarn breaks or slides out does not depend on how strong the fibre is, or on how hard the yarn is pulled. It depends on the fibre’s fineness, the friction, and the twist angle.

The critical length — the depth of embedment at which slipping and breaking cost the same — comes out as

$$L_c = \frac{d_f}{4\,\mu\,\eta\,P(\alpha)},$$

with d_f the fibre’s diameter, μ the friction coefficient, η a contact efficiency, and P a pure function of the twist angle. The fibre’s breaking stress appears on both sides of the comparison and cancels; so does the yarn’s tension.

Two consequences follow immediately, and the second is what makes the argument worth having:

• Below about five degrees of surface angle, no staple anybody spins is long enough to be gripped. A yarn twisted that softly has essentially no strength, whatever it is made of. That is a statement about geometry, not about cotton.
• A finer fibre is gripped in a shorter length, in exact proportion to its diameter — which is a second reason, quite separate from the evenness floor, that fine fibres make better yarns.

The pressure, in closed form #

Consider a fibre at radius r under axial tension, lying at helix angle θ(r). Because it is curved around the axis, its tension has a component pulling inward: per unit length, sin²θ of the tension divided by r.

Add up the inward pull of everything outside a radius r and the result is the pressure at r. The integral runs from r out to the surface, where the pressure is zero because nothing lies outside, and it evaluates:

$$p(r) = \tfrac{1}{2}\,\sigma\,\left(\cos^2\theta(r) - \cos^2\alpha\right),$$

with σ the axial stress in the core fibre. On the axis, where θ = 0, that is ½σ sin²α. At the surface it is nothing.

The area-weighted mean over the section reduces as well:

$$\bar{p} = \tfrac{1}{2}\,\sigma\left(\frac{\ln(1 + \tan^2\alpha)}{\tan^2\alpha} - \cos^2\alpha\right),$$

which at 23° is 2.9 per cent of the fibres’ axial stress. Both forms were reduced by hand, so both are checked against their own defining integrals by quadrature rather than believed.

The shape is more informative than either number. The pressure goes as sin²α at low twist, so it is quadratic in the angle and therefore quadratic in the twist: doubling the turns per metre roughly quadruples the grip. That steepness is the whole reason a small amount of twist changes a bundle of fibres from something that falls apart into something that can be wound onto a bobbin.

The cancellation #

Now write out the two ways a fibre can fail to carry its load, over an embedded length L.

It slips if the friction over that length is beaten:

$$F_{\text{slip}} = \mu\,\eta\,\bar{p}\,\pi d_f L.$$

It breaks if its own section is overloaded:

$$F_{\text{break}} = \sigma_f\,\tfrac{\pi}{4} d_f^2.$$

The comparison is only meaningful at one moment — when the yarn is at the point of breaking — and at that moment the pressure is set by the fibres’ own stress, which is σ_f times a factor from the obliquity. Substitute, and σ_f is on both sides.

It cancels. So does the yarn’s tension, which entered only through the same stress. What is left is a length: a fibre’s diameter over four times the friction, over a pure function of the angle.

This is why a strong fibre does not automatically make a strong yarn. Doubling a fibre’s tenacity doubles what it can carry and doubles the pressure holding its neighbours, so the balance between slipping and breaking is untouched: the yarn gets stronger in exact proportion and its failure mode does not change. Substituting a finer or a longer fibre does change the failure mode, and that is the whole of what a spinner’s fibre selection is doing.

What the numbers are, and the collapse at five degrees #

Put a cotton fibre into the expression and the critical length falls very steeply with the twist.

The interesting row is the first. Below about five degrees the critical length exceeds any staple anybody spins, which means no fibre in the yarn is gripped over enough of its length to be broken: pull the yarn and the fibres slide past one another and it comes apart with nothing broken at all. That is a familiar object — it is a roving, the soft untwisted strand that goes into a spinning frame, and it can be pulled apart with no effort whatever.

The collapse is quadratic, because the pressure goes as sin²α. So the transition from “falls apart in the hand” to “has to be cut” happens over a very small range of twist, which is why spinning works at all: a machine does not have to hit a narrow target, it has to get past a cliff.

The one fitted number, and what it cannot move #

Real fibres do not touch along their whole length. They meet at points and over short arcs, and the fraction of the nominal cylindrical surface that actually carries pressure is a quantity nothing here computes. It enters as η, the contact efficiency, and it scales the critical length inversely.

Every figure and every number in this collection that depends on η says so. What matters is what η can and cannot do.

So the results that stand are these:

• A longer staple wants less twist, because the cohesion factor depends on L_c only through L_c/L. Asserted across the whole tenfold range of η.
• A coarser fibre wants more twist, because L_c is proportional to d_f. Also asserted across the range.
• A filament wants no twist at all, because it has no ends between the ends of the package, so L is effectively infinite and the cohesion factor is one at any angle.

The trade agrees with all three. Long-staple cottons are spun at lower twist factors than Uplands; coarse wools take more turns than fine ones; and a filament yarn is given a few turns a metre for handling and no more.

The friction in it is two numbers, not one #

μ appears once in the expression and it is doing more work than a single symbol suggests. This collection has already found that a fibre has two friction coefficients rather than one — a static value that has to be beaten to start a slide and a lower dynamic one that governs it once it is moving — and the critical length above is computed with a mid-range value that is neither.

The distinction matters here in a specific way. Starting the slide is what decides whether the fibre carries its load; continuing it is what decides what happens afterwards. So a yarn near its break has a static critical length, and a yarn already failing has a shorter, dynamic one — which is a mechanism for the sudden, complete way a soft-spun yarn goes when it goes.

Wool is the extreme case, because its friction is directional: a scale ratchet makes sliding one way easier than the other. A wool fibre is therefore held asymmetrically, and pulling it out root-first and tip-first are different experiments. Nothing here models that, and it is the single largest reason a woollen yarn’s cohesion is not this arithmetic.

The same two-coefficient distinction reappears in the cloth, where it decides how far a cut edge frays: a thread that has started to move keeps moving, and the length that comes away is set by the dynamic value rather than by the static one.

Where the same argument has been made before #

This collection has met this shape twice in the cloth rather than in the yarn, and it is worth putting the three side by side.

The three are the same physics at three scales. A fibre is held in a yarn by the yarn’s twist; a yarn is held in a cloth by the cloth’s crimp; a cloth is held in a seam by the thread’s tension. Each is a capstan problem, each has a characteristic length over which the grip develops, and in each case the interesting question is whether that length is short compared with what is available. In a yarn it usually is, provided there is enough twist; in a seam it usually is not, which is why seams slip.

What was counted, and how #

Both pressure expressions are checked against the integrals they reduce, at four angles and at three radii each, with the worst relative departure below one part in 10¹². The profile is checked at its own defining integral and the mean at an area-weighted integral of the profile, so a sign error in either reduction would show.

The cohesion factor is checked for continuity at its own branch point. It has two branches — one for a critical length shorter than the staple and one for longer — and they must meet, at exactly a half. That value is worth noticing on its own: a yarn none of whose fibres is gripped over its whole length still realises half of them, because a fibre held over part of its length carries a proportionate part of its load.

The two orderings are asserted at three contact efficiencies apart, because an ordering that holds only at the fitted value is a restatement of the fit.

Reading the cohesion factor at its own branch #

The cohesion factor has two branches meeting at a half, and the meeting point is a real construction rather than an algebraic curiosity. It is the yarn in which the critical length is exactly the staple length: every fibre is gripped over the whole of itself and none has any spare length to break in.

For the 28 mm cotton drawn above, that yarn is at just under ten degrees of surface angle. The table’s collapse is quadratic in the sine of the angle, so the critical length that is 105 mm at five degrees is a quarter of that at ten, which is 26.7 — a little under the staple. Below ten degrees a cotton yarn of ordinary staple is on the wrong branch, and the cohesion falls in proportion to the pressure rather than gently: at five degrees it realises about an eighth of its fibres.

An eighth is the arithmetic’s description of a roving, and it is worth pausing on how well it matches the object. A roving is not weak in the way a thin yarn is weak. It is soft, it draws out rather than snapping, and pulling it apart leaves two tapered ends with no broken fibres in either of them — which is exactly what a structure realising an eighth of its fibres by sliding should do. The arithmetic did not have that behaviour put into it; it has one length compared against another, and the branch it lands on decides the failure mode.

The other branch behaves quite differently and is where every usable yarn sits. Above the branch point the cohesion is one minus the critical length over twice the staple, so it approaches one slowly: a 28 mm cotton reaches 0.90 at twenty-three degrees and would need something over thirty to reach 0.95. The first nine tenths cost twenty-three degrees and the next twentieth costs nine more, which is the shape of every diminishing return there is.

That asymmetry is what makes the twist decision hard rather than obvious. If cohesion were the only thing at stake a spinner would simply twist harder, since the curve keeps rising; what stops them is that the obliquity is falling the whole time, and the product of the two has a maximum. The maximum sits where it does because the cohesion curve has already flattened while the obliquity curve has not — so the optimum is not a balance between two comparable slopes but a point where one term has stopped paying and the other is still charging.

How good the small-angle form is #

The pressure goes as sin²α at low twist, so the critical length goes as its reciprocal, and that single expression reproduces the table better than a leading-order form has any right to. Taking the five-degree entry as the anchor and scaling by the ratio of squared sines gives 6.8 mm at twenty degrees against an exact 7.1, and 3.2 mm at thirty against an exact 3.5.

A tenth, at the far end of the practical range, and less than that over most of it. Anything a spinner actually sets — a surface angle somewhere between fifteen and twenty-five degrees — is inside a few per cent.

The direction of the error is the part to carry, because it is the unhelpful one. The small-angle form gives a shorter critical length than the exact expression does, so it says the fibre is gripped in less of itself than it really is, and every quantity derived from it is therefore slightly optimistic about the yarn. That is a small effect against the order of magnitude the contact efficiency is uncertain by, and it is the wrong sign to be comfortable ignoring in a place where the two happened to be comparable.

Where the model stops #

The contact efficiency is fitted and it is the largest thing this argument does not know. Nothing here predicts what fraction of a fibre’s surface carries pressure, and the honest range spans an order of magnitude. Every absolute length quoted above should be read as having that uncertainty attached.

The pressure is computed at the moment of breaking and used as though it were the pressure throughout. A yarn at rest has a residual pressure from its own spinning tension which nothing here computes, and a yarn under a small load has less pressure than this argument gives it — so the grip is understated at low loads and the failure mode near the break is what is being described.

The fibres are supposed to be under the tensions the affine model gives them. The pressure integral inherits whatever the obliquity model assumed, and that model is a bracket rather than a value. Using the other end of the bracket raises the pressure and shortens the critical length.

The embedment is supposed uniform and it is not. A fibre’s two ends are at different depths, its path wanders, and the migration that makes the obliquity favourable also carries the fibre through regions of different pressure. Averaging over an assumed uniform distribution of end positions is the crudest step in the derivation.

And there is no fibre-length distribution. Every fibre is given the nominal staple; a real staple is a mixture, and the short-fibre fraction is exactly the part of it this argument is most sensitive to.

Where the ladder goes next #

Straight into the curve both halves belong to. The obliquity falls with twist and the cohesion rises with it, and their product has a maximum which is what a spinner is choosing when they set a twist factor — a curve this collection previously drew with a stated constant in its rising half and can now draw with a derived shape.

And sideways into the folded yarns, where the same pressure appears one level up. A ply’s own helix presses on the singles inside it exactly as a singles’ helix presses on its fibres, which is why the folding twist costs so much less than it looks.