A warp breaks in a tail no tensile test reaches

A yarn breaks at its thinnest place read a tensile test as a minimum. The yarn between the clamps is a chain of independent tries, one for every staple length, because two sections closer than a fibre share most of their fibres; the test breaks at the weakest try; and a longer specimen, with more tries in it, tests weaker. At an ordinary 20 tex ring-spun cotton that essay found a 500 mm test reading eleven per cent below a 100 mm one, recovered the yarn’s irregularity from the difference, and ended with a sentence it did not follow up: the gauge that matters is the one that matches the use, and a warp end is hundreds of metres long.

A warp is not hundreds of metres of one end. It is four thousand ends of a thousand metres each, and every one of them passes through the shed. At a 28 millimetre staple that is 143 million independent tries, and an end breaks wherever one of them is weaker than the tension the shed puts on it.

It is the third extreme of one population this account has priced. A warp jams where its threads are thickest and a gauge reads the highest crossing under its foot, both maxima that grow with the sample; a warp breaks at a minimum that shrinks with it, and the sample is the largest any of the three has met.

So the question a weaving shed asks of a yarn is not the question its certificate answers. The certificate gives the weakest of eighteen tries. The loom needs to know how many of a hundred and forty million fall below a line — a probability of about one in a hundred million, eight orders of magnitude into the lower tail of the yarn’s strength. Nothing on the certificate reaches that far, and it turns out that nothing on the certificate constrains it either.

A certificate samples eighteen staple lengths #

A breaking load quoted at a 500 mm gauge is the expected strength of the weakest of 500 ÷ 28 = 17.9 independent staple lengths. The scatter of that test — every test is repeated, and the individual breaks spread by several per cent — is a statement about the same minimum, drawn again and again.

Every number a tensile laboratory produces is of that kind. Change the gauge from 20 millimetres to five metres and the number of tries runs from one to 179, and the breaking load runs from about the mean section’s strength to about two-thirds of it. That window, one try to a couple of hundred, is everything the laboratory ever sees of the distribution: the middle and the near shoulder of its lower half.

The warp lives six decades further out. A single end of a thousand metres is 35,700 tries; the whole warp is 143 million. The laboratory’s data are being asked to predict an event that happens once in a hundred million, from tests that sample events of one in eighteen.

Three laws a laboratory could not tell apart #

The distribution of a section’s strength is not known; it is a model. This account has always read a yarn’s sections as lognormal — a positive quantity with a spread, at the coefficient of variation the evenness floor and an ordinary spinning index give, 13.4 per cent. Two other laws are just as defensible for a strength. A normal is the default of every textbook. A Weibull is the law a weakest-link argument produces for a brittle material, and it is the law fibre strength is usually fitted with.

Give each of the three whatever mean and spread make it reproduce the same two tensile tests, at 100 and at 500 millimetres, exactly. That is what a laboratory would do with those data, and it produces three different yarns: the normal comes out with a spread of 11.4 per cent and the Weibull with 8.9, because a law with a heavier lower tail needs less spread to produce the same fall in strength with gauge.

Across every gauge a tester can clamp, the three are one curve to within five per cent. A single break scatters by more than that, so separating them would take hundreds of breaks at the longest gauges — and even a laboratory that did it would have measured the distribution at about one chance in two hundred, which is still the shoulder rather than the tail.

The same three laws at a hundred million tries #

Carry the same three laws out to the number of tries a warp contains.

The weakest place in the warp is somewhere between a third of the certificate and three-fifths of it, depending on a choice the tensile data cannot make. At one end’s length the three are already fifteen points apart; at the whole warp they are twenty-eight points apart. Nothing about the yarn has changed along the curve — only the amount of it being asked.

The reason is visible in how the curves bend. The lognormal’s lower tail thins fastest: a strength that is a product of many small factors cannot easily reach near zero, so its weakest place falls slowly with the number of tries. The Weibull’s lower tail is the fattest of the three near the bottom, because it is built to describe materials that fail at their worst flaw, and its weakest place keeps falling. The normal sits between them.

What one staple length’s chance of weakness looks like #

The curve that decides a loom’s break rate is simpler than a minimum: the chance that any single staple length of yarn is weaker than a tension.

The shaded band is all a tensile test ever samples, and the three curves are indistinguishable inside it. Below it they fan out, and by the level where a warp’s break rate is decided they are five orders of magnitude apart. A yarn is not described by its breaking load and its spread; it is described by a whole curve, and the part of that curve a loom reads is the part a certificate has never seen.

A break is a place weaker than the tension #

What the shed costs, in newtons turned the loom’s geometry into a tension: on a twenty-four-shaft harness a 25 tex cotton end is held at 14 per cent of its breaking load on the front shaft and 52 per cent on the back, before the let-off adds anything. A weak place that arrives in the shed weaker than that tension breaks on its first lift, because the shed is an extension imposed by the geometry, not a load the yarn can decline. So the expected number of ends a warp will break is the number of its tries weaker than the tension — the tries times the chance per try.

The three laws, fitted to the same two tensile tests, disagree about the same warp by a factor of 375,000. At the back shaft’s tension the Weibull yarn breaks an end every two metres of warp; the normal yarn breaks nineteen ends in the beam; the lognormal yarn weaves the whole beam without breaking one. All three pass the same tensile certificate and show the same gauge-length curve.

That is not a failure of any one law. It is the arithmetic saying that the break rate is a property of the tail and the certificate is a property of the middle, and that no amount of precision about the middle transfers to the tail.

The cliff is robust and its position is not #

Two features of the break curve survive every law, and they are the useful ones.

The count is a cliff. From 45 to 52 per cent of the breaking load — seven points of tension — the expected breaks rise fifteen-fold under the normal, seven-fold under the Weibull and nearly three thousandfold under the lognormal. Under every law, a small increase in tension multiplies the break rate. That is the familiar behaviour of a weaving shed, where a warp that runs cleanly at one tension setting becomes unweavable a few per cent higher, and it has a plain reason: the loom is working in the lower tail of a distribution, where the density of weak places changes by orders of magnitude over a small range of strength.

The cliff sits in the band a loom works in. The tension at which a warp holds one weak place lies between 33 and 60 per cent of the breaking load under the three laws, and the shed’s tensions run from 14 per cent at the front of a deep harness to 52 at the back, with the let-off adding to all of them. The loom is not far from its yarn’s cliff under any of the three. It is operated at the edge of it, and where exactly the edge is decides everything a weaver sees.

What does not survive is the position of the cliff. Twenty-seven points of tension separate the three laws’ one-break tensions, and a weaver who knew only the certificate could not say whether a warp would run at 45 per cent or break constantly at 40.

The back shafts take the breaks #

The shed’s tension rises down the harness, because the back shaft works hardest: each shaft’s lift is proportional to its distance from the fell, so its strain and its tension rise with it. On a cliff, a steady rise in tension is not a steady rise in breaks.

Under every law the breaks concentrate on the back shafts, and how completely they concentrate is decided by the tail. That is the one feature of the break distribution a weaver can measure without knowing the law: a log of breaks by shaft. The harness has a depth noted that a weaver meets the harness’s depth as a rate of breakage on the back shafts, and this is why the rate is so lopsided — the back shafts are further up a cliff than the front ones, and on a cliff a quarter of the tension range can hold nearly all of the events.

It also says what a record of breaks by shaft is worth. The ratio between the back shaft’s breaks and a middle shaft’s is a measurement of the tail’s steepness between those two tensions, which is exactly the quantity the tensile data could not supply.

Evenness moves the tail, not only the middle #

A spinner who improves a yarn’s evenness lowers its coefficient of variation, and the certificate records that as a small rise in breaking load. The loom records something much larger.

Each yarn here is loaded at the same share of its own certificate, so the uneven yarn is not being penalised for a lower breaking load; it has been given a lower tension to match. It still breaks orders of magnitude more often, because its tail is longer relative to its middle. A certificate that improves by a few per cent can mean a warp that breaks a hundred times less, and the spinning index, which the gauge-length essay recovered from two tensile tests, is the certificate’s one number that reaches towards the tail at all.

It is the same logic as a finer yarn is a worse yarn: fewer fibres in a section mean a wider spread, and a wider spread matters far more at the extreme than at the mean.

What a size coat buys #

A warp is sized before it is woven — coated with a film that binds the surface fibres and lays the hairs — and the usual account of what sizing does is that it makes the yarn a little stronger and a lot less hairy. The tail says the first of those is worth more than it sounds.

Suppose a size coat raises every section’s strength by a tenth, which moves the back shaft’s tension from 52 per cent of the breaking load to 47. On the break curve that is a move down the cliff, and it cuts the expected breaks by a factor of four under the Weibull, nearly seven under the normal and 236 under the lognormal. A ten per cent strength gain is a four- to two-hundredfold fall in warp breaks, which is why a weaving mill treats sizing as a condition of weaving at all rather than as a refinement, and why a warp that was sized badly in one set of ends breaks in exactly those ends.

Why the trade counts faults instead of quoting a spread #

If the certificate cannot describe the tail, something else has to, and the trade has long had it. An evenness tester reports, beside the coefficient of variation, a count of imperfections: thin places a stated fraction below the mean mass per kilometre of yarn, thick places above it, and neps. A fault classifier counts rarer and larger defects per hundred kilometres.

Those are counts of events in the tail, reported as rates per length, and they are the right kind of quantity for this arithmetic even where their thresholds are not the loom’s. At the back shaft’s tension the three fitted laws predict weak places at 0.14, 0.005 and 0.0000004 per kilometre of yarn — numbers a count over a few hundred kilometres could tell apart, where no tensile certificate could. A count of rare events per length is a reading of the tail, and a coefficient of variation is not, which is presumably why a spinning mill quotes both.

The same division runs through the cloth. A strip test averages hundreds of threads while a tear asks a handful at once, and so reads further out along the same distribution; and a random error hides where a periodic one shows at the scale of a cloth, while a single rare fault decides a warp.

What the arithmetic adds is the reason the two cannot be converted. A spread and a fault count are independent descriptions of a distribution, one of its middle and one of its tail, and the whole of this essay is that the middle does not fix the tail.

A faster tester is still a short one #

High-speed tensile testers break yarn continuously, tens of thousands of specimens an hour, and are sold on exactly this problem: finding the weak places a warp will meet. At thirty thousand breaks of 500 millimetres an hour a tester samples about half a million staple lengths an hour.

That is a great deal closer to a warp than a certificate is, and it is still three hundred hours short of one warp’s 143 million tries. On the minimum curve it reaches about 10⁵·⁷ tries, where the three laws put the weakest place at 47, 58 and 68 per cent of the breaking load — already twenty points apart, and not yet at the depth where the back shaft works. A fast tester reads further into the tail than a certificate and still stops short of the level where a warp breaks. Its value is that it measures the slope there, which lets the tail be extrapolated from where the tail actually is instead of from the middle.

Extreme-value statistics began with cotton yarn #

The mathematics used here has a textile origin. F. T. Peirce, at the British Cotton Industry Research Association, published the weakest-link account of a yarn’s gauge-length effect in 1926. L. H. C. Tippett, at the same association in the same years, was working on the extremes of samples in connection with the strength of cotton yarn, and with R. A. Fisher in 1928 showed that the largest or smallest of a large sample can approach only three limiting forms. Weibull’s law for the strength of brittle materials followed in 1939.

So the theory that engineers now use for flood heights, material flaws and insurance losses began as the question this essay is asking: how weak is the weakest part of a long piece of cotton? What the theory also established is the caution at the centre of it — that which limiting form applies is decided by the shape of the parent distribution’s tail, and that the tail is the part least constrained by the data one usually has. A yarn’s tensile certificate is the textbook case.

How the numbers were produced #

The yarn is a 20 tex ring-spun cotton at a spinning index of 1.35, so its section strength has a coefficient of variation of 13.4 per cent, from the evenness floor at 118 fibres, exactly as in the gauge-length essay; its staple is 28 millimetres, so a gauge of L holds L ÷ 28 independent tries. The reference law is the lognormal at that spread. The expected minimum of n tries is $\int_0^\infty (1 - F(x))^n\,dx$, computed with $(1-F)^n$ taken through a logarithm so that a hundred million tries do not round $F$ away. The normal and Weibull laws are each given the mean and spread that reproduce the lognormal’s expected breaking loads at 100 and 500 millimetres exactly, by one bisection on the spread against the ratio of the two loads.

A tension is a share of the 500 millimetre breaking load, which all three reproduce, so the three are always compared on two yarns sold on the same certificate. The warp is 4,000 ends of 1,000 metres; the shed’s tensions, 14 per cent at the front of a twenty-four-shaft harness and 52 at the back, are the shed essay’s for a 25 tex yarn, used here as shares of the breaking load and taken as rising evenly between them. The normal distribution’s far tail is computed with a complementary error function accurate to a fraction of its own value, since the usual approximation is accurate only to a fixed amount and means nothing below a probability of a millionth.

Several things are required to hold. The normal tail agrees with its asymptotic series eight standard deviations out to a part in ten thousand. Each law has the mean and spread it was built with, by quadrature. Each fitted law reproduces both tensile tests. The three agree within five per cent at every gauge from 20 millimetres to five metres, and disagree more than a thousandfold on the back shaft’s weak places. Under every law the count rises with the tension and with the spinning index, and the expected weakest falls with the number of tries.

What the arithmetic leaves out #

The tries are independent. A real yarn’s strength is correlated over a staple length and not beyond it, which is the model; but it also carries periodic faults from drafting rollers, and a periodic fault is not an independent try.

The only weak places are the spread’s. A real yarn also carries discrete faults — a slub, a nep, a bad piecing — which are a separate population with its own rate, and a warp’s breaks come from both. That population is what the fault count measures, and a real loom’s breaks may be dominated by it at tensions where the spread alone would give none.

A weak place breaks only by tension. The heddle eye and the reed abrade an end as it is woven, and an abraded place is weaker than it was on the beam; a bundle is weaker than its threads in part because damage accumulates at the same places the load concentrates. And the shed’s tension is not steady: it is a peak once a pick, repeated thousands of times as a place travels from the back rest to the fell, and a fatigue model would lower every one-break tension.

Three laws are not all the laws. They are three reasonable ones, fitted the same way; a real yarn may follow none of them. The point is not that one is right but that the tensile data cannot choose.

Still open: which tail a real warp has #

Everything here reduces to one measurement, and a weaving shed is already making it. A break log that records each warp break by shaft, with the shaft’s tension known from the loom’s geometry, is a sample of the tail at twenty-four tensions at once. On this arithmetic the log of breaks per end on each shaft, plotted against that shaft’s tension, is a direct reading of the lower tail’s slope in the region the loom works in — and the three laws predict three different slopes there, from the Weibull’s gentle rise to the lognormal’s near-vertical wall.

The same log, set against the yarn’s imperfection count, would say how much of a warp’s breakage comes from the spread and how much from discrete faults. The first would move with the tension as the curves above do; the second, a population of slubs and bad piecings much weaker than any tension on the loom, would be nearly flat across the harness. The shape of breaks against shaft number separates the two, and it needs nothing but the loom’s own record and the shed’s arithmetic.