textiles · friction · a threshold, reproduced

No Fibre Reaches the End

A cotton thread is made of fibres about an inch long, and not one of them runs from end to end. Friction is all that holds it, and twist is what makes the friction grow with the pull. Below a threshold no pull can be carried; above it, any pull can, until fibres snap. Build a random tangle of fibres and watch it lock, with the proof checked in your browser, then set the lock against 3,200 yarn breaks the US Bureau of Standards measured in 1935.

Galileo's question

In the first day of Two New Sciences (1638), Galileo has Salviati ask the thing every spinner knows with their hands and nobody had explained:

“The first question is, How are fibres, each not more than two or three cubits in length, so tightly bound together in the case of a rope one hundred cubits long that great force [violenza] is required to break it?”

“…the very act of twisting causes the threads to bind one another in such a way that when the rope is stretched with a great force the fibres break rather than separate from each other.”

Galileo, Dialogues Concerning Two New Sciences, trans. Crew and de Salvio (1914), p. 8

He had the observation right: at the place where a rope parts, the fibres are snapped, not pulled out. What he did not say is why twisting should make them bind. A spinning-machine maker's manual still states the fact without the mechanism: “Almost all currently used yarns obtain their strength from some kind of twist in the strand” (Rieter's Manual of Spinning, vol. 6). The mechanism turns out to be a threshold, and you can build one.

What one contact can do

Where one fibre bends over another, its tension presses it down on the fibre beneath. Over a small turning angle θ the pressing force is about the tension times θ, the small-angle form of the capstan equation that lets a few turns of rope round a post hold a ship (that layer works it through). Friction then lets tension pass from one fibre to the other, but only up to μ times the pressing force. So a contact can hand across at most a fixed fraction λ ≈ μθ of the tension already in the wrapped fibre.

Now notice what is missing. Double every tension in the yarn and every contact's allowance doubles with it. Nothing in the problem sets a scale. So there are only two possibilities for a given tangle: no pattern of tensions at all obeys every contact's limit, and the yarn slides apart at any load; or one does, and then so does any multiple of it, and the yarn holds any pull until the fibres themselves break. Physicists call the switch between them a percolation transition. Warren, Ball and Goldstein put it this way in 2018, in a paper titled Why clothes don't fall apart, and found where it happens.

A tangle that locks, with the proof in your browser

Below is their abstract yarn, rebuilt here from the paper's description: 20 rows of fibres, 4 laid end to end in each row with a random offset, so that every cross-section is crossed by exactly 20 fibres and every fibre has N contacts. In each column the 20 fibres are paired at random, and at each contact one of the pair (chosen at random) is the one wrapped over the other. The tension it hands across is capped at λ times that wrapped fibre's mean tension across the contact. Fibre ends carry nothing.

Whether a non-zero tension state exists is a linear programming question, and a solver (HiGHS) answered it offline for each tangle, narrowing the threshold to a bracket 0.02% wide. But the page does not ask you to trust the solver. It carries a proof for each side of the threshold, and your browser checks whichever one applies every time you move the slider:

• It locks: the proof is the tension state itself. Check every contact's cap and every force balance, and confirm the tensions are not all zero. Anything that passes can be scaled up without limit.
• It slides: the proof is a list of multipliers, one per constraint, which add the constraints up into a single statement: the sum of all tensions is at most zero. Since tensions cannot be negative, they are all zero. This is Farkas' lemma, and checking it is arithmetic, not optimisation.

A proof of sliding at some grip also proves sliding at every smaller grip, and a proof of locking holds at every larger one, because loosening the caps can only admit more states. So two proofs, one each side of a bracket 2% wide, settle every position of the slider except the sliver between them, which the page reports honestly as too close to call.

Where the lock happens

Warren and colleagues report that for fibres with 30 or more contacts the threshold is “roughly constant at N⟨λ⟩ ≈ 7.3 ± 0.2”, and a mean-field theory they solve with Dawson's integral gives 6.83, which they suggest becomes exact for very long fibres. This model differs from theirs in the random numbers and in giving every contact the same λ (their baseline draws each λ from a distribution of relative width 0.2; they also report the equal case), and it lands close by: So our longest fibres sit a little under their band and above the mean-field limit they should approach, and this page does not know which of the differences accounts for that. The mean-field number reproduces to every digit they print: x₀ = , Λ_c = , Nλ_c = .

Since λ ≈ μθ at each contact, Nλ is μ times Θ, the total angle a fibre turns through along its length. The lock needs μΘ of about 7. For cotton on cotton the paper takes μ ≈ 0.3 to 0.4, so a fibre must turn through something like 18 to 23 radians: three or four full turns, spread along an inch of fibre.

Twist is what makes a fibre turn

A straight fibre turns through nothing, so pulling it presses on nothing. Twist winds each fibre into a helix, and a helix of radius ρ whose fibre makes angle θ with the axis bends by sin²θ/ρ per unit length. That is exact geometry. Two consequences follow, and the second is not obvious.

First, turning grows with twist, slowly at first (as the square of the angle) and then faster. Second, at the same angle a thinner yarn's fibres turn more, because the helix is tighter. Spinners have long set twist by the rule that turns per inch should go as the square root of the cotton count, T = M√C, which Schiefer and Taft trace to Joseph Köchlin, who presented it to the Société Industrielle de Mulhouse on 28 November 1828. That rule gives every count the same surface angle for the same multiplier M. It does not give every count the same turning, and the lock cares about turning.

The shaded band is where μΘ reaches 7 for the chosen friction: its left edge counts a fibre as staying at the yarn's surface, where it turns most; its right edge averages the turning over the whole cross-section, as a fibre that migrates evenly in and out would feel it, which is (2/R)(1 − α/tan α) per unit length at surface angle α. The yarn radius comes from Peirce's rule for cotton, d = 1/(28√C) inches (the US Army's fabric-design handbook, AMCP 706-300, derives 27.8 from Peirce's density and notes that it is often rounded to 28), and the fibre length is the classer's staple, 1 inch for the 10s and 1⁵⁄₁₆ for the 80s.

Read it for what it is: a back-of-envelope with three soft inputs, not a fit. With μ anywhere from 0.3 to 0.4 it puts the coarse yarn's lock between multipliers of about 3.0 and 4.2, which is where the measured strength climbs and turns over. For the fine yarn it puts the lock below 2.1, under anything Schiefer and Taft spun; their weakest 80s twist, 2.48, already held 79% of the best. Schiefer and Taft knew the pattern from the literature (“maximum strength of fine yarns is obtained at lower twist multipliers than in coarse yarn”) and gave two reasons: fine yarns are usually spun from longer staple, and “the longer fibers require less twist to increase the resistance of fiber slippage”; and a fine yarn exposes more of its fibres at the surface. The turning count contains their first reason, since Θ grows with fibre length, and adds one they did not give: at the same angle, a thinner yarn's helix is tighter. Here the radius does most of the work (the 80s yarn is 2.8 times thinner, its fibres only 1.3 times longer). What the estimate cannot do is place the peak, because the peak is where the lock's gains meet the cost of twist.

What twist costs

A twisted fibre pulls at an angle to the yarn, so stretching the yarn stretches the fibre less. C. Gégauff wrote the relation down at Mulhouse in 1907: fibre strain equals yarn strain times cos²θ. Carry it one step (our arithmetic, which the check repeats): if every fibre is fully gripped and they all break at the same strain, the most-stretched fibres are the straight ones at the core, and when they go the yarn is carrying cos² of its surface angle times what the same fibres would carry pulled straight, because the average of cos⁴θ over the cross-section is exactly cos²α. So if tilt were the only price, and grip only improved with twist, as the lock says it should, strength past the peak could fall no faster than cos² of the surface angle. The dashed lines below draw that bound from each measured peak. A point below its line is twist costing something tilt does not explain.

Galileo's other speaker saw this end of the curve too: “ropes sometimes break not by a lengthwise pull but by excessive twisting.” The 1925 Bureau of Standards paper says the decrease “can be continued until the yarn breaks under no longitudinal strain, but merely from the shearing action of the twisting.”

A thread with no ends inside it

A filament yarn (silk, rayon, nylon) is made of fibres that already run the whole length. There is nothing for twist to lock, and the rising half of the curve should all but vanish. It does. In 1931 the Bureau twisted six rayons, up to 100 turns per inch, and found that strength “is not materially changed with an increase in twist up to about 20 turns per inch, but that it decreases rapidly with increase in twist beyond 20 turns.” A 1952 study of nylon for the US Air Force printed the numbers:

A hundred breaks at every twist

Schiefer and Taft did not only print averages. For every twist they printed how many of their 100 breaks fell in each band of strength, which means the whole of their evidence is on the page, and it can be added up again.

Forty-odd thousand years of it

Among the oldest thread anyone has found is a scrap 6.2 mm long lying on a stone flake at Abri du Maras in southern France, in a layer dated between 41,000 and 52,000 years ago, when Neanderthals lived there. Hardy and colleagues (2020) read its structure from photomicrographs: “3 bundles of fibres with S-twist which were then plied together with a Z-twist to form a 3-ply cord”, the fibres probably from the inner bark of a conifer.

The second twist runs the other way, and that is not decoration. A twisted strand stores torque: let it go and it untwists. Plied the opposite way, a strand can only untwist by winding the ply tighter, so the two torques meet and the cord holds its form. The paper's own gloss is that the yarns are plied “in the opposite direction to prevent unravelling”, and the 2018 paper makes the same point in the caption to its photograph of a modern cotton sewing thread: “The composite 3-ply structure prevents untwisting under load.”

Try it with your hands

Pull a wisp from a cotton-wool ball and draw it out between two fingers: the fibres slide past each other and it parts with almost no resistance. Pinch one end, roll the other between finger and thumb until it tightens, and pull again. The same fibres now resist. Then twist a length of string until it starts to kink, fold it in half and let go: it plies itself, in the opposite direction, and holds.

Sources

1. P. B. Warren, R. C. Ball and R. E. Goldstein, “Why clothes don't fall apart: tension transmission in staple yarns”, Physical Review Letters 120, 158001 (2018); arXiv:1804.07606.
2. H. F. Schiefer and D. H. Taft, “Mechanical properties of cotton yarns”, Journal of Research of the National Bureau of Standards 15, 237 (1935), RP826; scan at the Internet Archive. Tables 2 and 4.
3. F. R. McGowan, C. W. Schoffstall and A. A. Mercier, “Effect of twist on the physical properties of a number 7s yarn”, Technologic Papers of the Bureau of Standards 19, No. 278 (1925); scan. Table 1.
4. H. A. Hamm and R. S. Cleveland, “Relation between the twist and certain properties of rayon yarns”, Bureau of Standards Journal of Research 7, 617 (1931); scan.
5. Georgia Institute of Technology, “A study of the effect of twist on the properties of synthetic filament yarns”, WADC Technical Report 52-55 (1952), Table XVII; scan.
6. G. Galilei, Dialogues Concerning Two New Sciences (1638), trans. H. Crew and A. de Salvio (1914); scan.
7. B. L. Hardy et al., “Direct evidence of Neanderthal fibre technology and its cognitive and behavioral implications”, Scientific Reports 10, 4889 (2020); doi:10.1038/s41598-020-61839-w.
8. G. Seguin and J. Crassous, “Twist-controlled force amplification and spinning tension transition in yarn”, Physical Review Letters 128, 078002 (2022); arXiv:2110.04206. A second, experimental route to the same switch, on model yarns of 1 mm strings.
9. US Army Materiel Command, Engineering Design Handbook: Fabric Design, AMCP 706-300 (1972), appendix A (Peirce's diameter rule). Gégauff (1907) as cited by Aychilie et al., Materials 15, 8887 (2022). Rieter, The Rieter Manual of Spinning, vol. 6 (2016).

Two things we could not check and so do not claim: Köchlin's 1828 paper itself (the date and the name are Schiefer and Taft's, citing Brüggemann 1933), and the exact constants of the Hearle, Grosberg and Backer strength formula.