Textile Structure
how cloth is put together
Series

Friction — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Stick, slip, and the ratio between the two coefficients. The force in a thread as a cloth is agitated, with a static coefficient of 0.300 and a kinetic one of 0.225 — a ratio of 0.75, which is what fibre on fibre measures. The force climbs until it reaches the static limit, the contact breaks away, and while it is sliding it resists only at the kinetic limit. So a cloth that is being shaken can be left anywhere in the narrower band, and a cloth at rest anywhere in the wider one. What the trace cannot show is how much this buys: the band does not narrow in the ratio of the coefficients, because the restoring force stiffens away from the minimum, and the real narrowing is nearer 0.87 than 0.75.

    Two coefficients, not one

    Every friction on this site is a single number, and the account of why a cloth relaxes better when it is agitated depends entirely on there being two. Separating static from kinetic changes what a resting state is: a cloth at rest is held by one coefficient and a cloth being shaken by the other, and the band it can be left in narrows — by less than the ratio, because a cloth's restoring force is not linear in its extension.

    part 1 · cloth
  2. The resting band with two coefficients in it. The range of extensions a sheeting can be left in, at rest and while being agitated, at three ratios of kinetic to static friction. The upper bar of each pair is the stuck band, held by the static coefficient; the lower is the band a cloth being shaken can be left in, held by the kinetic one. Agitation narrows the band but by less than the ratio of the coefficients: at 0.75 the narrowing is 0.868. The restoring force is not linear in the extension, so cutting the friction by a quarter does not move the band's edges by a quarter of the way in. What the bars cannot show is where a given piece of cloth actually stops inside its band, which depends on which side it came from.

    Why agitation helps a cloth relax

    Every standard relaxation procedure agitates: tumble it, wash it, steam it, work it. The explanation given is that agitation lets the fabric find its own dimensions, which is true and is not a mechanism. The mechanism is that a sliding contact resists less than a stuck one — and putting a number on it shows the effect is real, is smaller than the obvious arithmetic suggests, and does not account for what a relaxation procedure achieves.

    part 2 · finishing
  3. Two cloths touch on a fraction of what one cloth does. A 2/2 twill in sheeting pressed against a flat plate, and the same cloth pressed against another piece of itself. At an approach of 24.9 µm the single surface is touching 15.87% of the plan and the pair 3.130% — a factor of 5. The reason is that a gap between two rough surfaces is the sum of two depths, so both surfaces have to be near their own maxima at the same place, and the chance of that is the product of two small numbers. The pair's curve is the convolution of the two height distributions, computed exactly on histograms rather than fitted to a Gaussian — because a woven surface is bimodal and nothing about it is Gaussian.

    Friction is two surfaces, not one

    A gap between two rough bodies is the sum of two depths, so two cloths face to face touch on the convolution of their height distributions rather than on either of them. At the approach a light touch produces that is twenty times less contact than the same cloth against a plate — which is why a fabric's friction against a plate and against another fabric are two different measurements.

    part 3 · finishing
  4. How hard a fabric is to bend, per unit width. Bending rigidity in micronewton metres per unit width, computed the same way for eight woven cloths and for a jersey in each of its two directions: the yarn's own rigidity, times the length of yarn per unit area, times the fourth power of the cosine of the angle each element makes with the bending direction. The knit lands inside the band of the woven cloths rather than below it, which is worth knowing because a knit is usually called the softer fabric. It is not softer to bend; it is softer to stretch, by three decades. What it does have is direction: 2.17 to one between its two axes, where a balanced plain weave is near one, and the soft axis is the one a jersey rolls about at its top and bottom edges.

    What friction has to hold in a relaxed knit

    A knit's bending energy slopes away from the fabric everybody measures, so something is holding it there. Along its courses friction holds comfortably. Along its wales the driving force is exactly the contact force, so the whole balance collapses to one condition — the friction coefficient must exceed a half — and no yarn in this collection reaches it.

    part 4 · mechanics

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