A yarn's stiffness is a bracket, not a number

This collection has been written without a force in it. The matrix counts, the trellis rotates, Peirce’s geometry solves, and every one of those answers is a ratio — a length over a length, a count over a count. The one place the absence is stated outright is in the file that models drape, which says of itself that there is no stiffness anywhere in this file.

This rung supplies the missing quantity, and the first thing it turns up is that the quantity is not a number.

The two calculations, and why there is no third #

The free bound. Suppose the fibres slide past one another without resistance. Then each bends about its own axis, nothing couples them, and the bundle’s rigidity is the sum of the fibres’: n·B_f, with n the number of fibres and B_f the rigidity of one. This is a genuine lower bound. No arrangement of fibres is limper than the fibres themselves.

The coherent bound. Suppose instead that the fibres cannot slide at all. Then the yarn is a rod of its own outside diameter, the whole section resists together, and the rigidity is E·πd⁴/64. This is a genuine upper bound. Nothing is stiffer than a solid section of the same material and outline.

Between them there is nothing to calculate, because what decides where a real yarn sits is how much the fibres actually slide, and that is a friction problem with twist, fibre finish, moisture and pressure in it. It is not a property of the fibre. So the honest output is the pair.

What separates them, exactly #

Take the ratio and something falls out that is worth more than either bound.

A yarn’s diameter comes from its count by conservation of volume at a packing factor φ, which is the route this site has used from its first essay: d_yarn = d_fibre·√(n/φ). Both rigidities go as the fourth power of a diameter, so

**coherent / free = (d_yarn/d_fibre)⁴ / n = n / φ²

The bracket is the fibre count divided by the square of the packing factor. Nothing else survives. The modulus cancels — it multiplies both bounds equally — and so does the fibre’s own diameter, and so does everything about the material.

The machinery asserts that as an identity rather than remarking on it: thirty-six yarns spanning four counts, three fibres and three packing factors, with the worst relative departure at one part in a quadrillion. An identity that holds at one point is arithmetic; one that holds everywhere is a derivation.

Three consequences follow immediately and none of them is obvious.

A coarser yarn has a wider bracket, in exact proportion to its count. The free bound goes as the count and the coherent bound as its square. A 10 tex cotton yarn’s bracket is 163 and a 60 tex yarn’s is 980, and the ratio of those two is the ratio of the counts to fifteen decimal places.

A yarn spun from finer fibre has a wider bracket at the same count. More fibres in the same cross-section is a larger n, and n is the whole of the bracket. So the refinement that makes a yarn smoother, stronger and more uniform makes its stiffness less knowable.

And a better measurement of the fibre does not narrow it at all. Cotton’s modulus is reported between 5 and 12 GPa and everybody who has measured it is right about their own cotton; pinning it down to three figures would move both bounds together and leave the bracket exactly where it is.

A factor of five hundred is not a small problem #

It is worth being blunt about the size of this. For a 30 tex cotton yarn the two bounds are 1.41 × 10⁻³ and 6.89 × 10⁻¹ N·mm², and the second is 490 times the first.

Nothing on this site cares about a factor of two. The cover factor is exact, the float length is a count, the satin census is an enumeration. A quantity known to a factor of five hundred does not belong in the same collection unless something can be done about it.

Something can, and it is not a stiffness test.

The measurement that kills one of the bounds #

A yarn’s rigidity is hard to measure and a cloth’s is not. The cantilever test is a strip of fabric pushed out over an edge until it droops to a fixed angle, and this collection has modelled it since the mechanics field was built: the bending length is the overhang times a factor the geometry supplies, and the flexural rigidity is that length cubed times the mass per unit area.

So the bracket can be asked a question it has to answer. Predict each cloth’s rigidity from its own yarn at both bounds, on the simplest assumption there is — the threads bend independently, so a strip one centimetre wide has the rigidity of the threads crossing that centimetre and nothing is added for the crossings — and put the two numbers beside what such a cloth measures.

The free bound predicts between 1.35 and 4.50 µN·m per unit width across the eight cloths in this site’s table. Ordinary cotton apparel and household cloths measure a few to a few tens of µN·m, so every free-bound prediction lands inside the band of real cloths.

The coherent bound predicts between 245 and 4,409. The stiffest woven cloth anybody reports — a heavy canvas, a fused interlining — is around 500. Six of the eight predictions are past that, and all eight are above the band of ordinary cloths. A voile made of coherent-bound yarn would bend like denim; a duck made of it would bend like sheet card.

The second measurement, which is not a stiffness test either #

The first narrowing leaves the yarn somewhere between one and about eighteen times the free bound, which is better than five hundred and is still not good. The second comes from a completely different place: how repeatable a cloth’s relaxed dimensions are.

The argument is developed on its own rung and the short form is this. A cloth slides along its own locus while the bending energy it can release exceeds what friction takes to move a crossing, so it stops in a band rather than at a point, and the band’s width is friction over stiffness. Feed the free bound in and the band comes out at eleven per cent of length. A washing test finds one to three.

That is a discrepancy in the direction that pins the answer: the yarn must be stiffer than the free bound, and by an amount the band width solves for. At an ordinary friction the observed one-to-three-per-cent band puts the yarn between four and twelve times the free bound.

The two intervals overlap, and that overlap is the result of this rung. A strip of cloth hanging over an edge and a piece of cloth relaxing in water are not measurements of the same thing in any obvious sense, and neither was taken to answer a question about yarn. Both say a few times the free bound.

What was counted, and how #

Four material constants are used and each is quoted as what it is.

The fibre’s fineness in decitex gives its diameter by the same volume arithmetic a yarn’s count does, with the packing factor set to one because a fibre is not a bundle. Cotton at 1.7 dtex comes out at 11.9 µm, which is what a cotton fibre is.

The fibre’s modulus in GPa gives its bending rigidity through the second moment of a circle, πd⁴/64. Cotton’s working value here is 8 GPa against a reported range of 5 to 12, and every result above that depends on it is reported as a range or is one where it cancels.

The packing factor is 0.6, the site’s standing value for a ring-spun yarn, and the same one Peirce’s own diameter rule turns out to assume.

The friction coefficient is a range, 0.2 to 0.4 for cotton on cotton, and never a value. It is quoted from the same table the tuft argument has used since the pile ladder was built, moved into one file so there are not two.

That last point is worth its own sentence. Until this rung the site carried exactly one kind of material constant — fibre densities — and called them, in the source, the only material constants on this site. A density is exact: a mass over a volume, four figures, no disagreement about cotton anywhere. Everything added here is different in kind. A modulus is a slope taken off a curve that is not straight, at a strain nobody agrees about, on a fibre whose maturity varies along the boll. The site has acquired constants that are ranges, and the discipline that follows is that a result must either survive the range or be reported as an interval.

Two assertions guard the arithmetic. The identity that the bracket is n/φ² is checked across counts, fibres and packing factors at a tolerance of one part in a trillion, because both sides are closed forms of the same quantity and anything looser would mean a route through the diameter had picked up an approximation. And the bracket’s ordering across fibres is required to be exactly the ordering by fineness, which would fail immediately if a modulus had leaked into the ratio.

A folded yarn has three bounds, and only one of them is wide #

The bracket is derived for a single yarn, and the collection’s other ladders spend a good deal of time on folded ones. Working the same argument through a two-fold yarn turns up an extra rung, and where it sits says which part of the uncertainty is actually the problem.

A folded yarn is a hierarchy: fibres inside singles, singles inside a fold. Each level can slide or not, so there are three limiting arrangements rather than two.

Everything free. Every fibre bends about its own axis: 2n·B_f for a two-fold of n-fibre singles.

Fibres locked, singles free. Each single is a rod and the two slide against one another: twice a single’s coherent rigidity, which is 2n²B_f/φ².

Everything locked. The fold is one rod of √2 times a single’s diameter, so its rigidity is four times a single’s: 4n²B_f/φ².

The two gaps are wildly unequal, and that is the finding.

The fibre level is worth the whole fibre count — the step from the first bound to the second is n/φ², which is the same several hundred the single yarn’s bracket was.

The fold level is worth exactly two. The step from the second to the third is a factor of two, at any count, any fibre and any packing, because it is the ratio of a doubled second moment to two undoubled ones and nothing else survives.

So plying does not make a yarn’s stiffness less knowable in any way that matters. The uncertainty was already a factor of several hundred at the fibre level and folding adds a factor of two on top of it. A reader who expected a hierarchy to compound its uncertainties multiplicatively — three hundred times three hundred — would be wrong by two orders of magnitude, and the reason is that the fold has two members while the single has hundreds.

Two consequences follow, and the second is the more useful.

The middle bound is a real state and the others are not. A fibre in a twisted single is pressed against its neighbours over its whole length and does not slide freely; two singles in a fold touch along a helix and slide rather easily. So a real folded yarn sits near its middle bound, which is a place a single yarn has no analogue of — and that is a much better-determined position than anything available for a single.

And it says where a measurement would pay. Narrowing the fibre-level gap is worth a factor of hundreds and needs the two fabric measurements this rung uses. Narrowing the fold-level gap is worth a factor of two and needs nothing, because the answer is already bracketed within it. A folded yarn is the case where this collection’s bracket is nearly a number, and it is the one the essay’s own generalisation about ropes points at: a rope is a hierarchy whose every level has few members, which is why a rope’s bending stiffness is calculable and a yarn’s is not.

Where the model stops #

Twist is not in the bracket. Twist is the mechanism by which a real yarn moves from the free end towards the coherent one — it presses the fibres together and makes sliding cost something — and how far it moves is a friction calculation this rung does not attempt. What twist does enter is the tensile side, where the obliquity factor cos²α is standard and the site already computes α from the count and the turns per metre.

The independent-threads assumption is a floor. Predicting a cloth’s rigidity as the sum of its threads’ assumes the crossings add nothing, and they do not add nothing: threads pressed together at a crossing resist sliding past each other, which is why a cloth stiffens when it is starched without any yarn changing. So the free-bound fabric prediction is a lower bound on a lower bound, and the narrowing above is conservative in the direction that matters.

The measured band is a band for a class of cloth, not a measurement of these eight constructions. Nothing in the argument depends on where in it any one cloth sits; what it is used for is a single question with a decisive answer, which is whether the coherent bound is anywhere near a real cloth.

And there is no plasticity anywhere. A yarn that has been bent hard does not come all the way back, which is why a crease survives, and every energy on this rung is recoverable by construction.

The generalisation #

The shape of the result is not about yarn. A bundle of slender elements has a stiffness bracket whose width is the number of elements, and the position inside it is set by whatever resists shear between them.

That is the same statement for a wire rope, a nerve, a stack of paper, a bundle of drinking straws, a laminated leaf spring and a fibre-reinforced tow — and the engineering of several of those is entirely about controlling where in the bracket the object sits. A leaf spring is deliberately kept near the free end by letting its leaves slide; a laminated beam is deliberately driven to the coherent end by glue. The textile case is unusual only in that nobody chooses: a spun yarn’s position is set by a friction that varies along its own length.

The second half generalises further and is the more useful lesson. When a quantity’s uncertainty is a ratio of several hundred, the way out is not a better measurement of the quantity. It is to find two consequences of it that can be measured on something else, and to take the intersection. Neither the cantilever nor the washing test could have been proposed as a way of measuring yarn stiffness. Both bound it, and their bounds do not fail together.

Who found it, and when #

The two bounds are old and belong to the theory of twisted structures: Platt, Klein and Hamburger set out the mechanics of a twisted assembly in the 1950s, and the no-slip and free-slip limits are named there as the two ends of a range rather than as competing models. Backer and others measured yarn bending in the same decade and found values in the lower part of it, which is the empirical form of the answer this rung reaches by two other routes.

Peirce’s cantilever test is from 1930 and predates all of it. It was designed to give a number for the handle of a cloth, not to constrain a yarn, and its use here is entirely parasitic.

What is this site’s is the arithmetic that makes the bracket a fibre count — which is one line and follows from the volume route to a diameter that this collection has used from its first essay — and the observation that two of its own existing models, built for unrelated reasons, bound the answer from opposite sides.

Where the ladder goes next #

The next rung spends the stiffness on the loom: the strain the shed puts into a warp end was computed here and could not be priced, and with a modulus it becomes a tension that can be put beside the yarn’s breaking load.

Sideways, the bending rigidity is what closes the hole in Peirce’s geometry, what puts a force on the tensile locus, and what says why a knit is soft and a woven is not.

Further out, the missing companion to this rung is a compression stiffness. A yarn flattens where it crosses, the racetrack section is this site’s model of the flattening, and it has no stiffness in it either — so the same bracket problem is waiting there, unposed.