Textile Structure
how cloth is put together
Concept

Bundle strength — where it appears

The breaking load of threads pulled in parallel, per thread, which is strictly below one thread's mean strength. The weakest breaks first and hands its load to the rest, so the bundle peaks at the largest value of the load times the fraction still unbroken — a shortfall decided by the spread rather than by the strength.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A muslin's warp, drawn at the diameters it actually has. 18 ends of a muslin's warp yarn, drawn at a seeded sample of their own diameters — mean 167 µm, coefficient of variation 15% — and spaced exactly, because the reed does not vary. The threads look even enough. The gaps do not: the widest here is 280 µm and the narrowest 229 µm, against a mean of 250 µm, and their coefficient of variation is 7% — larger than the yarn's, by the ratio of the diameter to the gap. That amplification is the whole reason a close cloth's holes are so much less uniform than its threads, and it gets worse as the cloth is set closer, because the gap in the denominator is the thing being shrunk.

A cloth is a population, not a thread

Every number in this collection was computed from a diameter, and no yarn has one. Putting the distribution back changes some answers by nothing at all, some by a few per cent, and some by a factor — and which of the three happens is decided by one derivative.

cloth · Variation 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.

mechanics · Tensile Where a 20 tex yarn breaks, against how much was clamped. A tensile test clamps a length of yarn and pulls until the thinnest section between the clamps gives. So a yarn's strength is a minimum, and a minimum depends on how many independent tries the sample contains. The tries are not sections — a plane can be taken anywhere — but staple lengths, because two planes closer together than one fibre share most of their fibres. At 28 mm staple a 100 mm specimen holds 3.6 independent tries and a 500 mm one holds 17.9, and the longer test reads 11% lower. The spread is not fitted either: it is the evenness floor at 118 fibres times an index of 1.35, which is 13.4%.

A yarn breaks at its thinnest place

A tensile test does not measure a yarn. It measures the worst section between the clamps — so evenness and strength are one measurement taken twice, and the number of independent tries in a specimen is set by the length of a fibre.

mechanics · Assembly A fifth of a strong fibre buys most of its pilling. How long a pill survives on a wool fabric as nylon is blended into it, relative to the pure wool. A pill is held by several anchor fibres and survives while any of them holds, so its life is set by the strongest anchor it happens to have — and the chance that a pill with 8 anchors has at least one strong one is 1 − (1 − x)^8, which is already 83% at a fifth. The blend therefore gets 83% of the pure strong fibre's pill life while keeping the whole of the weak fibre's fuzz supply, which is the worst of both. Nine tenths of the way arrives by 30%. No average of the two fibres' properties produces this curve: it is a maximum over a small sample, and a maximum is not an average. It is also why a fifteen-per-cent polyamide in a wool knit is notorious, and the arithmetic says the reputation is deserved.

The strong fibre is the one that pills

A pill survives while any one of its anchors holds, which is a maximum over a small sample rather than an average — and a maximum behaves nothing like an average. A fifth of a strong fibre in a blend buys four fifths of the pure strong fibre's pill life while leaving the whole of the weak fibre's fuzz supply in place.

applied · Pilling

Named alongside it

The objects these essays reach for when they reach for this one.

Order statisticPopulationTenacityCoefficient of variationBreaking extensionCanopy criterionConvexityCover factorFibreFibre countGauge lengthHair balance

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