A yarn breaks at its thinnest place

Ask what a yarn’s strength is and the answer arrives as a number with a unit — so many newtons per tex, printed on a certificate beside the count and the twist. It looks like a material property, in the way a density is.

It is not one. A tensile test clamps half a metre of yarn and pulls until something gives, and what gives is the thinnest section between the clamps. The result is a minimum, and a minimum is not a property of the material at all: it is a property of the material and how much of it was asked.

This collection has met that shape twice already. A warp jams at its thickest pair of neighbours and a thickness gauge reads the highest crossing under its foot, both of which grow with the sample. A yarn’s strength is the same argument with the sign reversed, and it shrinks with the sample.

The claim #

A yarn’s tensile strength is the minimum over the independent tries in the specimen, and the number of independent tries is the gauge length divided by the staple length.

Three things come out of it and only the first is well known.

• A longer specimen tests weaker. At an ordinary 20 tex ring-spun cotton, a 500 mm test reads eleven per cent below a 100 mm one, and a five-metre length reads twenty-one per cent below. The trade knows this and calls it the gauge-length dependence; it is usually described as a testing artefact to be standardised away.
• The amount is computable from the evenness and the staple, with nothing fitted. The spread comes from the fibre count and the spinning index; the sample size comes from the staple length; the rest is an order statistic.
• So evenness and strength are one measurement. They are the same distribution read at its middle and at its lower tail, and a spinner improving the index improves both by the same act.

Why the tries are staple lengths #

The tempting version of the argument is wrong and it is worth seeing why, because the correction is the interesting part.

A yarn contains as many cross-sections as one cares to name — a plane can be taken anywhere — so if each section were an independent draw from the mass distribution, the minimum over a specimen would be the minimum over an unbounded number of draws, and every yarn would have zero strength. Real yarns do not, so the sections cannot be independent.

They are not, and the reason is physical rather than statistical. A fibre spans a staple length. Two planes a millimetre apart in a 28 mm cotton share nearly all of their fibres, so their masses are almost the same number; two planes 30 mm apart share almost none, so their masses are almost independent. The correlation length of the mass along a yarn is the fibre length.

$$N_{\text{independent}} \approx \frac{L_{\text{gauge}}}{L_{\text{staple}}}.$$

This is why the argument belongs beside the fibre count rather than beside the other extremes. Every other extreme in this collection counts objects — threads across a width, crossings under a foot. This one counts fibre lengths along a thread, and the fibre’s length is the only thing that decides how many there are.

What it predicts, and against what #

Take a 20 tex cotton ring yarn. The fibre count is 118, so the evenness floor is 9.9 per cent; at an ordinary index of 1.35 the yarn’s coefficient of variation is 13.4 per cent. The staple is 28 mm.

The number to check against practice is the middle one: a 500 mm test reads eleven per cent below a 100 mm test on the same yarn, and the reported gauge-length effect for cotton yarns is of exactly that order. Nothing was fitted to obtain it. The spread came from a count, the sample size came from a staple, and the order statistic is arithmetic.

The prediction that would be easiest to falsify is the fibre-length one, and it is sharp: two yarns of the same count and the same evenness but different staple lengths must show different gauge-length effects, the longer-stapled one showing less. A 38 mm Sea Island and a 22 mm Upland spun to the same count and index differ by a factor of 1.7 in the number of tries per metre, which is a difference the test can see.

The other extreme, which is a different shape #

There is a second way an assembly of threads is weaker than its threads, and this collection has already priced it: a bundle pulled together does not break all at once, because the weakest thread goes first and hands its load to the survivors, who may or may not carry it.

The two are worth putting side by side because they are the two ways an assembly can be arranged and they behave quite differently.

A series extreme has no floor and a parallel one does. Add more length to a specimen and the strength keeps falling, slowly and without limit, because there is always a thinner section somewhere. Add more threads to a bundle and the strength per thread settles: the survival curve has a peak and adding members samples it better rather than moving it.

That difference is the reason the two numbers are quoted differently and the reason neither is a material property. A yarn tenacity needs a gauge length beside it; a bundle strength needs a thread count beside it; and a fabric strength, which is both at once, needs both.

The artefact that is not an artefact #

The gauge-length dependence is normally introduced as a nuisance. Test methods specify a gauge — 500 mm for one standard, 250 for another — so that laboratories can compare results, and the specification is presented as a convention chosen for convenience, like the size of a presser foot.

It is not a convention about the test. It is a statement about which question is being asked, and there are two questions.

What is the strength of a section of this yarn? That is a property of the yarn’s construction: its fibre count, its obliquity, its grip. The shortest possible specimen measures it, and the number a certificate ought to carry if it wants to be a material property.

What is the strength of a length of this yarn in use? That is the question a weaver has, and it is a different number, because a warp end is hundreds of metres long and every one of its independent tries is a chance to break. The right gauge for that question is the one that matches the use, and no standard length matches every use.

Standardising the gauge does not remove the dependence; it fixes it at one value and stops anybody looking at it. And what is thrown away is informative: the slope of strength against gauge length is a measurement of the yarn’s irregularity, obtained from a tensile tester rather than from an evenness tester, and it is available to anybody willing to run the same yarn at two gauges.

That is the sharpest practical consequence in this essay. Two instruments that appear to measure unrelated properties are measuring the same distribution, and either can be calibrated against the other.

Two gauges give the irregularity, and here is the conversion #

The claim that a tensile tester can measure evenness is worth more than an aphorism, because the conversion is short enough to write down and closes back on the number it started from.

For a lognormal population the expected minimum of N draws sits below the mean by roughly the spread times the extreme-value coefficient a(N) = √(2 ln N), which is the standard asymptotic and is accurate to a few per cent once N is past about five. So the strength at a gauge is approximately the section strength times 1 − CV·a(N), and the difference between two gauges is

(S₁ − S₂) / S₁ = CV × [a(N₂) − a(N₁)].

Everything on the right but the spread is arithmetic. Run the 20 tex yarn at 100 mm and at 500 mm: the tries are 3.6 and 17.9, so a(N) is 1.60 and 2.40, and the bracket is 0.80. The measured drop between those gauges is eleven per cent. Divide, and the spread comes out at 13.8 per cent against the 13.4 the whole page was computed from.

That agreement is the point of the exercise. Two tensile tests at different gauges recovered the evenness to within four per cent of an independent measurement, using no evenness tester, no fitted constant and no calibration — only the staple length, which is measured anyway on every bale.

Three practical notes, because the conversion has sharp edges.

It needs the staple length and is sensitive to it. The tries enter through a logarithm, so a ten per cent error in the staple is a small error in a(N) — but if the staple is wrong by a factor of two, the bracket moves by about a fifth. That is the accuracy the method has, and it is better than it sounds because the staple is one of the best-measured quantities in the trade.

The gauges must be far apart. The bracket is a difference of two square roots of logarithms, so gauges of 200 and 250 mm give a bracket of 0.14 and divide the measurement error by nothing. A factor of five between the gauges is the minimum worth attempting and a factor of twenty-five is better.

And it measures the spread over the length that matters to the yarn, which is not what an evenness tester measures. A capacitive tester reads over a centimetre or less; this conversion reads over the correlation length, which is the staple. The two agree here because the correlation is treated as all-or-nothing at one length, and a real yarn’s variance at the staple scale is a little below its variance at the centimetre scale. So the tensile route should read slightly low, and it reads high — which is the residual worth chasing rather than the agreement worth celebrating.

The general form of it is worth stating separately. An extreme is a spread read at the tail, so any instrument that measures an extreme is an instrument that measures a spread, provided the sample size is known. The sample size is the part that is usually missing, and here it is a fibre length.

The scatter, which moves the other way #

There is a second thing a longer specimen does and it is the reason the long gauges were adopted.

The minimum of many draws is not only smaller than the minimum of few; it is also less variable. The lower tail of a distribution gets thinner the further out one goes, so a test that samples the worst of a hundred and seventy-nine tries repeats itself far better than one that samples the worst of one. A laboratory running at 500 mm sees tidy numbers; a laboratory running at 20 mm sees enormous scatter and concludes its clamps are slipping.

So the two properties of the test move in opposite directions: the longer the specimen, the lower and the more repeatable the answer. The test that looks best-behaved is the one furthest from the section’s own strength, and the one that would measure the construction is the one nobody trusts.

This is the same trade the tear test loses. A tear asks the weakest of a handful and scatters accordingly, and a tear strength that varies by a tenth between specimens of the same cloth is not a badly run test — it is the only answer an extreme of four can give.

Where it reaches the cloth #

The cloth-level consequences follow directly and two of them explain measurements that otherwise look inconsistent.

A tensile strip test and a tear test disagree about which cloth is stronger, and part of the reason is that a tear asks fewer threads at once — it is an extreme over a handful rather than an average over hundreds. The other part is here: the threads a tear asks are short lengths at the tip of a cut, so each one is being tested at a tiny gauge and is stronger than the certificate says.

A seam is a series of stitch holes, and a seam slips before it breaks partly because each puncture is a thin place with the number of independent tries set by the stitch pitch rather than by the staple.

And a warp is a very long specimen indeed. A weaver’s warp end runs hundreds of metres through the loom and is tensioned the whole way; the relevant number of independent tries is not a test’s eighteen but tens of thousands. That is why warp breakage rates are so much worse than a certificate suggests and why they are quoted per hundred thousand picks rather than as a strength.

What was counted, and how #

The order statistic is integrated and then simulated against itself. The expected minimum of N draws from a lognormal is computed by quadrature over the standard normal, and checked against direct sampling — four hundred replicates at each sample size — with the two required to agree within the simulation’s own standard error rather than within a tolerance somebody chose.

The number of tries is asserted to be the gauge over the staple, and asserted to rise with the gauge, so that a change which broke the correlation-length argument would fail rather than quietly rescale the curve.

The spread is computed rather than supplied. It is the evenness floor at the yarn’s own fibre count times a stated index, so a reader who disagrees with the index can substitute one and re-read every number; a reader who disagrees with the floor has a different argument to make.

Where the model stops #

Strength is taken as proportional to mass in a section, and it is not exactly. Fewer fibres in a thin place means fewer to break, which is the linear part; but a thin place is also usually a place where the twist has concentrated, which changes the obliquity and the grip, and neither effect is included. The direction of the twist correction is not obvious, which is why nothing is claimed about it.

The correlation is treated as all-or-nothing at one staple length. The real correlation falls off gradually over a fibre length, so the effective number of independent tries is a smoothed version of L/staple rather than the integer. This makes the curve slightly too steep at short gauges and the error is second order.

The staple length is a single number and it is a distribution. Every real staple is a mixture of lengths, with the short-fibre fraction a quantity the trade measures separately, and the correlation length ought to be an average weighted by the fibres’ contribution rather than the nominal length.

Nothing here is a strength model. It says how the strength of a specimen relates to the strength of the thinnest section in it, and takes the section’s strength as given. What decides that — obliquity, the grip on the fibre ends, whether the fibres break or slide — is a separate argument further along this ladder, and it is the one with the fitted parameter in it.

And nothing here is about a filament yarn, which has no staple length and therefore no correlation length of this kind. A filament yarn’s gauge-length dependence exists and comes from flaws in the filaments rather than from thin places in an assembly, which is a different distribution with a different tail.

Where the ladder goes next #

Down into the section itself. This essay took the strength of a thin place as given; the next two ask what a section’s strength is, and the answer turns out to have almost nothing to do with the fibre’s own strength. A fibre that keeps its radius cannot share the load, which puts a bracket on what a twisted assembly can realise; and the length of fibre a twist can grip decides whether a fibre in that section breaks or slides past its neighbours.

Sideways, into the cloth. A cloth made of a yarn with this distribution has its own extremes, and they are the jam, the thickness reading and the tear — three questions asked of one population, each over a different amount of cloth.