Textile Structure
how cloth is put together
Cloth doing a job

A tear asks fewer threads than a pull

A strip test averages a hundred threads and a tear interrogates four. That difference alone accounts for the two things everybody knows about tear testing — that it reads low, and that it scatters — without anything about the cloth being different between the two tests.
15 min read 6 figures A mechanism, not a materialStructure before fibre

Worth reading first: A tear stops where the grip is · A bundle is weaker than its threads · Does a loose weave tear better.

Two things are said about tear testing wherever it is done. It reads low — a fabric’s tear strength is a small fraction of its tensile strength, and nothing in the arithmetic of threads explains why. And it scatters: repeat a tear test five times on one piece of cloth and the results wander by a tenth, which no other fabric test does.

Both are usually explained by the mechanics, and the mechanics are real. A tear is stopped by the grip, the threads at the tip slide and bunch, and a looser weave tears better because more threads share the load. None of that is in dispute.

But there is a simpler thing happening underneath it, and it is arithmetic rather than mechanics: the two tests do not ask the same number of threads.

What a tear asks for is the weakest of a few. A tensile test pulls every thread in the width and averages them. A tear pulls the handful in the triangle at the tip of the cut, and what lets it move is whichever of those is weakest — so the quantity that governs is the minimum of a small sample, and two things follow that no mean can show. Its expectation is below the mean: at CV 15% the weakest of 4 is 0.853 of the mean thread, and the weakest of 40 is 0.718. And its scatter is enormous compared with a tensile test's: 10.9% against the 0.75% a mean of four hundred threads would show. 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.
Fig. 1 The expected weakest of k threads, as a fraction of the mean thread, and the scatter that comes with it. A strip test averages over a hundred threads. A tear interrogates the handful in the triangle at the tip of the cut, and takes whichever of those is weakest.

The claim

A tear is a minimum over a small sample and a strip test is a mean over a large one, and that difference alone predicts both of the tear test’s famous properties.

At a coefficient of variation of fifteen per cent, the weakest of four threads is 0.853 of the mean thread and scatters by 11.1 per cent. The mean of a hundred and twenty threads — the ends in a standard strip — scatters by 1.4 per cent. So the tear test’s scatter is eight times the strip test’s, for a cloth that is perfectly uniform in every other respect, and no laboratory practice will reduce it.

The two tests are not measuring the same quantity with different precision. They are measuring different statistics of the same population.

The argument

A tear propagates thread by thread, and a cloth with a hole in it does not mind until the hole reaches a size the grip cannot bridge. At any moment the load is carried by the few threads in the triangle at the tip of the cut — the del zone, from the shape it makes — and the tear advances when one of them fails.

Which one? Whichever is weakest. So the load at which the tear moves on is the minimum of however many threads that zone holds, and the tear’s progress is a sequence of such minima taken along the cloth.

That gives three predictions at once, and they are the three things that make tear data awkward.

The mean reads low. A minimum is below the mean of what it is drawn from, and the fewer threads it is drawn from the further below. At four threads the deficit is fifteen per cent, at two it is eight, at one there is none.

The scatter is large and irreducible. A mean over n threads scatters as one over the root of n; a minimum over four scatters at nearly the population’s own coefficient of variation, because a minimum of four does almost no averaging.

And the scatter is not the same thing as the deficit. A test can read low and repeat well, or read high and scatter, and which happens is decided by the size of the sample rather than by anything about the fabric.

What a tear asks for is the weakest of a few. A tensile test pulls every thread in the width and averages them. A tear pulls the handful in the triangle at the tip of the cut, and what lets it move is whichever of those is weakest — so the quantity that governs is the minimum of a small sample, and two things follow that no mean can show. Its expectation is below the mean: at CV 25% the weakest of 4 is 0.764 of the mean thread, and the weakest of 40 is 0.574. And its scatter is enormous compared with a tensile test's: 17.9% against the 1.25% a mean of four hundred threads would show. 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.
Fig. 2 The same arithmetic on a much more irregular yarn. The gap between what a tear asks and what a pull asks widens with the spread, because a tear draws from a handful of threads and a pull from thousands — and the weakest of a few is far below the weakest of many.

The prediction this makes that the mechanics do not

Here the arithmetic says something the mechanical account does not, and it is testable with equipment every textile laboratory already has.

The size of the del zone is set by how freely a thread can slide before it breaks — which is friction, sett and weave. A loose, slippery, long-float cloth engages many threads at the tip; a close, high-friction plain weave engages few. That is the established account of why the first sort tears better.

But if the zone is larger, the tear is asking for the minimum of more threads, so its scatter must fall as well. Mean tear strength and repeatability are not independent virtues that a fabric might have separately: they are two readings of the same number.

threads at the tip tear reads scatter
1 1.00 14.9%
2 0.92 12.4%
4 0.85 11.1%
8 0.80 9.3%
20 0.75 7.9%

Read the table the other way and something odd falls out: a larger del zone gives a lower minimum per thread, not a higher one. The mean tear strength rises anyway, because the load is carried by more threads at once and the total is the product of the two — and the arithmetic says the product wins, since the sample size grows faster than the minimum falls. But it says so with a caveat that is not in the usual account: some of the benefit of a loose weave is being spent on the population, not collected by it.

How much a moment exceeds its own mean, at each spread. A quantity going as the kth power of a varying thread does not come out at the kth power of the mean thread. For a lognormal the excess is a single closed form — (1 + CV²) raised to k(k−1)/2 — so the second moment is up by the square of the CV and nothing else, and the fourth is up by that to the sixth. At the fifteen per cent an ordinary staple yarn reaches, the ratios are 1.0225, 1.0690, 1.1428 for k = 2, 3 and 4. The dashed curves are a normal with the same mean and CV, which is what a laboratory quotes and what a diameter cannot actually have, since a normal diameter can be negative: the two laws agree to a few parts in a thousand across the whole range a yarn occupies and part company in the tail, which is exactly where the extremes this family also computes live.
Fig. 3 Why the difference is a moment. Both quantities are order statistics of one population and both rise with the spread, but they rise at different rates — so the ratio between a tear strength and a tensile strength is a statement about the yarn’s evenness rather than about the cloth.

What was counted, and how

The minimum of k draws is computed twice: from the density of the order statistic by quadrature, and by drawing twelve hundred samples of k and taking their minima. The two agree to within the simulation’s own standard error at every k, which is what makes the closed form worth quoting.

The scatter of a mean is not simulated, because it does not need to be: it is the population’s coefficient of variation over the square root of the sample, exactly, and the only judgement in it is what counts as the sample. A fifty millimetre strip at twenty-four ends per centimetre holds a hundred and twenty ends, and a strip test grips every one of them.

What is not computed here is the size of the del zone, and it is the number the whole argument would need to become a prediction rather than a mechanism. It depends on friction, on the float, on the sett and on how far a thread can be drawn before it breaks — quantities this collection has separately and has never combined into a count of threads. That is recorded rather than estimated: putting a plausible number on it would produce a tear strength with a fitted constant in it, which is the one thing this collection refuses.

Why the tensile test cannot see any of this

A strip test engages every thread at once and pulls until enough of them break. It is a bundle, and a bundle has its own population effect — it breaks a quarter below the sum of its threads’ mean strengths.

So both tests are population statistics and neither is the strength of the cloth. But they are different statistics, and the difference has a direction worth stating: the strip test’s answer is stable and biased, the tear test’s is unstable and biased differently. Averaging five tear tests does not converge to what a strip test measures, and no correction factor between them can be constant across fabrics, because the ratio depends on the size of the del zone and therefore on the weave.

A bundle is weaker than the threads it is made of. Threads pulled in parallel do not break together. The weakest goes first and hands its load to the rest, which are now carrying more than they were, so the bundle's peak load is reached before every thread is at its own strength. With a load per surviving thread of x carried by the fraction that has not yet broken, the bundle's strength per thread is the largest value of x(1 − F(x)) — Daniels' maximum, which for a lognormal at CV 15% is 0.7380 of the mean thread, reached with 93% of the threads still unbroken. A simulated bundle that knows none of that arithmetic sits above the limit at every finite size — its strength is a maximum over the sample it happens to have drawn — and closes on it as the bundle grows: 0.753, 0.744, 0.741 at the last three sizes. The scatter falls the other way, from 14.5% at one thread to 0.7% at 1600.
Fig. 4 The other test’s population effect, for comparison. A bundle’s shortfall comes from the load being redistributed as threads break; a tear’s comes from a minimum over a handful. Both are a quarter or so at ordinary yarn evenness, and the coincidence in size is exactly that — the mechanisms have nothing in common.

That is a specific warning against a common practice. Tear and tensile figures are routinely converted into one another by a ratio taken from a table, and the ratio is treated as a property of the fibre. It is a property of the sampling, and two cloths of the same yarn with different weaves will have different ratios.

What it means for a specification

A tolerance on a tear strength has to be at least as wide as the test’s own scatter, and the scatter is a property of the fabric rather than of the laboratory. An eleven per cent standard deviation on single determinations means a five-test mean still carries about five per cent, so a specification demanding agreement to within three per cent is demanding something the population forbids.

And the number of determinations should follow the weave. A close plain weave with a small del zone needs more repeats than an open one to reach the same confidence — which is the opposite of the usual practice, where the number of specimens is fixed by the standard and does not depend on what is being tested.

A bundle of 24 threads, broken one at a time. 24 threads drawn from a lognormal at CV 15%, sorted, and loaded together. Each point is the moment a thread breaks: the load per surviving thread on the abscissa, and what the whole bundle is carrying — that load times the fraction still unbroken — on the ordinate. The bundle's strength is the highest point, 0.747 of a mean thread, reached with 23 of the 24 still whole. After that the curve falls: every further break hands more load to fewer threads and the bundle unloads itself. The dashed line is what the whole distribution gives in the limit of many threads, 0.7380, and this bundle sits above it because its strength is a maximum over the 24 threads it happens to contain rather than over the distribution they came from.
Fig. 5 Twenty-four threads broken in order, with the load the group carries at each break. A tear is this picture with almost all of the threads removed: a handful engaged, the weakest going first, and the tip moving on. The fewer the threads, the more the answer depends on which ones they happened to be.

Reading the scatter backwards

If the size of the del zone cannot be computed, it can at least be measured — and the instrument is the one thing every tear laboratory throws away.

A bundle is weaker than the threads it is made of. Threads pulled in parallel do not break together. The weakest goes first and hands its load to the rest, which are now carrying more than they were, so the bundle's peak load is reached before every thread is at its own strength. With a load per surviving thread of x carried by the fraction that has not yet broken, the bundle's strength per thread is the largest value of x(1 − F(x)) — Daniels' maximum, which for a lognormal at CV 10% is 0.8024 of the mean thread, reached with 95% of the threads still unbroken. A simulated bundle that knows none of that arithmetic sits above the limit at every finite size — its strength is a maximum over the sample it happens to have drawn — and closes on it as the bundle grows: 0.816, 0.808, 0.805 at the last three sizes. The scatter falls the other way, from 9.7% at one thread to 0.5% at 1600.
Fig. 6 A very even yarn, for the other end of the range. The gap between what a tear asks and what a pull asks nearly closes: with little scatter to draw from, the weakest of a few and the weakest of many are almost the same thread. Reading the scatter backwards means using that gap as a measurement of the spread.

The scatter of repeated determinations on one piece of cloth is a function of how many threads the tip engages, and it is a steep one at the small end and a flat one at the large. Inverting the table above:

observed scatter threads at the tip
14–15% 1
12–13% 2
11% 4
9% 8
8% 20

A laboratory that has run a dozen determinations on a fabric of known yarn evenness has both halves of that inversion already: the yarn’s coefficient of variation from its own certificate, and the tear test’s scatter from its own records. What comes out is a count of threads — the quantity that decides how a fabric tears and that nothing in this collection can calculate.

Three cautions travel with it and none is fatal. The observed scatter includes the instrument’s own, so the inferred k is an underestimate: real scatter is larger, and larger scatter reads as fewer threads. The inversion is insensitive at the top end — nine per cent and eight per cent differ by more than a factor of two in k — so it distinguishes a plain weave from a satin far better than it distinguishes one satin from another. And it assumes the strength population is the one the yarn certificate describes, which is a statement about gauge length as much as about yarn.

What makes it worth doing anyway is that the alternative is nothing. The del zone is visible in a photograph of a tear and has been measured that way, thread by thread, in the literature on tear mechanics; but a count taken from a photograph is a count at one instant of one tear in one fabric, and a count taken from a scatter is an average over every determination the laboratory has ever run.

And the same inversion is a check on the whole argument. If tear scatter had no relation to weave — if a close plain weave and an open satin of the same yarn scattered identically — then the del zone would not be what decides the sample size, and this essay would be wrong in a way that a morning in a testing laboratory could establish.

How many determinations the inversion needs

Reading the del zone off the scatter is proposed above as a use for data a laboratory already has. It is worth pricing, because the number of determinations required decides whether it is a morning’s work or a decade’s accumulation.

An estimate of a standard deviation from n determinations carries a relative error of about one over the root of twice n. The inversion’s steps are not large: eleven per cent against nine is the difference between four threads and eight, so resolving a factor of two in the del zone needs the scatter to about ten per cent, which needs

1 ÷ √(2n) ≈ 0.10, so n ≈ 50 determinations.

Fifty tears on one fabric is not a specification test and it is not out of reach either — a laboratory testing a fabric family over a season accumulates that many without trying, and the inversion needs only that they be on one cloth rather than on one order.

Two things make the estimate worse and one makes it better.

The instrument’s own scatter adds. What is observed is the population’s scatter and the machine’s in quadrature, so the observed figure is always larger and the inferred thread count always smaller. That is a bias with a known direction and no known size, which is the honest position: the inversion returns a lower bound on the del zone, and a laboratory that has characterised its own repeatability on a uniform specimen can subtract it.

And the samples are not independent. A tear running down a piece re-samples the same warp ends, so successive minima are correlated and the effective number of independent determinations is below the number run. That inflates the observed scatter again, in the same direction.

What helps is that the comparison is between fabrics rather than against a number. Two cloths of one yarn tested on one machine share the instrument’s scatter and the yarn’s population, so the ratio of their observed scatters is a clean comparison of their del zones with both nuisances divided out — and a ratio needs far fewer determinations than a value, because it is a comparison of two similar quantities rather than an absolute measurement.

So the practical form is not “measure the del zone” but “rank two fabrics by their tear scatter and the ranking is a ranking of their del zones” — which needs perhaps a dozen determinations apiece, is inside any specification programme, and tests the essay’s central claim without needing the yarn’s certificate at all.

And the strip test cannot be used this way at all. Its scatter is 1.4 per cent from the population, which is below the instrument’s own, so a strip test’s repeatability is a measurement of the machine and carries no information about the cloth. The tear test is the only fabric strength test whose noise is signal, which is an unexpected thing to be able to say about the test everybody complains about.

Where the model stops

The del zone is not computed, as above, so every number here is quoted per k rather than for a named fabric. That is the largest gap in the essay and it is deliberate.

The threads at the tip are not a random sample. They are neighbours, and a yarn’s own variation along its length means neighbouring ends are drawn from different bobbins but the same end’s successive thick and thin places are not independent. As a tear runs down the cloth it re-samples the same warp ends repeatedly, so the sequence of minima is correlated even though each is a minimum of a fresh group across the width.

A minimum is taken over strengths, and the tip does not fail by strength alone. A thread at the tip can slide out rather than break, and which it does is decided by the grip a crossing supplies. Where sliding dominates, the relevant extreme is not the weakest thread but the least gripped one — a different population, with its own spread, which this collection has never measured.

And the strip test’s own scatter is idealised. The 1.4 per cent quoted is the population’s contribution and nothing else: jaw slippage, specimen preparation and the transverse contraction of the strip all add to it, and in practice a strip test scatters by more than that. The comparison with the tear test’s eleven per cent survives comfortably, but the ratio of eight is a ceiling.

The generalisation

The number of elements a test engages is part of what the test measures, and two tests that engage different numbers are measuring different statistics however similar their apparatus looks.

The diagnostic question is the one this ladder keeps returning to, in its sharpest form here: how many independent things does this measurement average over? One, and the result is the population itself, with all its spread. A hundred, and the spread is divided by ten. A minimum over four, and the answer is a tail quantity with almost none of the averaging and a systematic offset besides.

The second lesson is that scatter is data. A test that wanders is usually treated as a test being run badly, and the response is to tighten the method. Here the wandering is the population speaking through a small sample, and it carries information — the width of a tear test’s scatter is a measurement of how many threads the del zone holds, which is precisely the quantity nobody can compute. A laboratory with a great deal of tear data already has, buried in its repeatability statistics, the number this essay could not calculate.

Who found it, and when

Tear testing’s scatter has been notorious since it was standardised, and the standards respond to it in the usual way, by requiring more specimens. The weakest-link account of a thread’s strength is Peirce’s, from 1926, and the extension of it to a small group at a crack tip is standard in fracture work on brittle materials, where it goes under the name of weakest-link statistics and is usually attached to Weibull.

What appears not to be written down for cloth is the pairing: that the same geometric change which improves a tear strength must also reduce its scatter, and that the ratio between tear and tensile figures therefore cannot be a constant of the fibre.

Where the ladder goes next

This is the last of the three extremes. The population argument now turns from a fabric’s strength to its appearance, where the question is not how large a deviation is but how it is arranged: a periodic error and a random one of exactly the same size behave completely differently, and the ratio between them is another piece of arithmetic with no fitted constant in it.

Sideways, a tear is the extreme case of the load-sharing question a whole fabric answers more gently, which is why a cloth does not mind a hole until the hole reaches a size the grip cannot bridge.

Where this goes next

A tear breaks threads or pulls them out. The grip a cloth has on a thread at the tip of a tear, against the sett, with the thread's own strength drawn across. Below the line the thread slides and the yarns group, which is the trade's explanation of why a loose weave tears well; above it the thread breaks where it is and grouping never happens. The essay that asked it could not compute this: it needed a friction, and the fancy weaves supplied one. A tear stops where the grip is This collection asked whether a loose weave tears better and had to answer that the geometry could not say — it supplies a grip count and a slack, and neither is a force. The fancy weaves brought a friction. With it the question has an answer — a cloth's threads slide up to a computable sett and break above it — and a ripstop grid has a bound with no free parameter in it. What the geometry says about a tear. A slit in a woven cloth, with the ends ahead of the tip gathered into the group that will break together. How many can gather is the opening divided by the slack in each gap, and the slack is the thread spacing less the thread diameter — an expression with no weave in it at all. Does a loose weave tear better The trade says a twill tears stronger than a plain weave of the same yarn, because fewer interlacings let the threads group. The geometry of that grouping has no weave in it at all — and where the geometry does speak, it predicts the opposite. A bundle is weaker than the threads it is made of. Threads pulled in parallel do not break together. The weakest goes first and hands its load to the rest, which are now carrying more than they were, so the bundle's peak load is reached before every thread is at its own strength. With a load per surviving thread of x carried by the fraction that has not yet broken, the bundle's strength per thread is the largest value of x(1 − F(x)) — Daniels' maximum, which for a lognormal at CV 15% is 0.7380 of the mean thread, reached with 93% of the threads still unbroken. A simulated bundle that knows none of that arithmetic sits above the limit at every finite size — its strength is a maximum over the sample it happens to have drawn — and closes on it as the bundle grows: 0.753, 0.744, 0.741 at the last three sizes. The scatter falls the other way, from 14.5% at one thread to 0.7% at 1600. A bundle is weaker than its threads Threads pulled together do not break together. The weakest goes first and hands its load to the rest, so a bundle carries its maximum well before every thread is at its own limit — and the shortfall is a quarter, decided by the spread and by nothing else.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Coefficient of variationFrictionMeasurementOrder statisticPopulationPropagationStress concentrationTear strength