Textile Structure
how cloth is put together
Series

Assembly — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 20 tex, counted. The cross-section of a 20 tex cotton yarn, with every fibre in it drawn. The count is a division and nothing else: a 20 tex yarn spun from 0.17 tex fibre has 117.6 fibres crossing any plane through it, and the yarn is 14.0 fibre diameters across because n fibres packed at 0.6 fill a circle √(n/φ) times as wide. The arrangement is drawn on a lattice and is not claimed: real fibres are not on one, they migrate between the core and the surface as they run, and everything this collection says about a yarn's strength turns on their doing so.

    How many fibres make a thread

    Every number in this collection began with a diameter, and a diameter is not a measurement — it is a count of fibres, divided. Once the division is written down, three quantities that had nothing to do with each other turn out to be the same number.

    part 1 · setting
  2. The evenness a cotton yarn cannot be better than. Lay staple fibres down at random and count how many cross a plane: the count is Poisson, its variance is its mean, and the coefficient of variation of the mass per unit length is therefore 1/√n with no material and no machine in it. The lower curve is that floor. At 5 tex there are 29 fibres in the section and the floor is 19.86 per cent; at 5 tex there are 29 and it is 19.86. The upper curve is what a yarn spun at an index of irregularity of 1.35 actually measures, which is the floor times a constant — so the whole shape belongs to the counting and none of it to the spinning. The exponent is exactly −½ and is asserted as such rather than fitted to the curve.

    A finer yarn is a worse yarn

    Fineness is the thing a yarn is priced for, and it is bought with irregularity at a fixed exchange rate. Once the spread is a function of the count, every correction this collection computes becomes a function of the count too — and the cheapest yarn on the shelf is the one the arithmetic describes best.

    part 2 · setting
  3. 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.

    part 3 · mechanics
  4. How likely one staple length of yarn is to be weaker than a tension, under three laws. The probability that one independent try — a staple length of a 20 tex cotton yarn — is weaker than a tension, on a logarithmic scale, with the tension as a fraction of the yarn's 500 mm breaking load. Three laws are drawn, a normal, a lognormal and a Weibull, each fitted to reproduce the same two tensile tests at 100 and 500 mm exactly. Near the breaking load, where a tensile test samples at about one chance in 18, they lie together. At one chance in 1.4×10⁸, where a warp of 4,000 ends by 1,000 metres samples, they are far apart: at the back shaft's 52 per cent the three give normal 1.3×10⁻⁷, lognormal 1.1×10⁻¹¹, Weibull 4.0×10⁻⁶ per try. What the chart cannot show is which of the three a real yarn follows, which no tensile test can decide.

    A warp breaks in a tail no tensile test reaches

    A 500 mm tensile test is the weakest of about eighteen staple lengths of yarn. A warp of four thousand ends by a thousand metres is the weakest of a hundred and forty million, and an end breaks wherever one of them is weaker than the shed's tension. Three strength laws fitted to the same two tensile tests agree within five per cent at every gauge a tester can clamp, and disagree by a factor of 375,000 on how many ends a warp will break at the back shaft.

    part 4 · setting

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