The stiffness with no lower bound

The rung that gave a yarn a bending stiffness found that the stiffness is a bracket rather than a number, and treated that as the difficulty. It is not the difficulty. A bracket of four hundred is inconvenient, and it is closed: both of its ends are derivations, no arrangement of fibres falls outside them, and two fabric measurements taken for other reasons narrow it to a factor of three.

The rung below this one introduced a second stiffness — how hard a yarn is to squash out of round — and it needs the same treatment. This rung gives it the same treatment and gets a different kind of answer.

The upper bound is the same argument #

If the fibres cannot move at all, the section is a rod of fibre material and its shear modulus is the material’s, scaled down by the packing factor because the rest of the section is air. That is a genuine upper bound: nothing is stiffer than solid, and the packing correction only lowers it.

For cotton the arithmetic is short. The axial modulus is 8 GPa; the transverse modulus of an oriented cellulosic is perhaps a fifth of that, so 1.6 GPa; a shear modulus is that over 2(1 + ν), which at ν = 0.3 gives 615 MPa; and multiplying by a packing factor of 0.6 leaves 369 N/mm².

Two of those three steps rest on numbers this site does not have. The anisotropy ratio and Poisson’s ratio for a cotton fibre are measurements, and not good ones. It does not matter, and saying why is the point of the paragraph: what an upper bound has to do here is be of the right order, and three hundred and something megapascals is three orders of magnitude above anything a yarn measures. Refining it would refine a number that is already useless.

The lower bound is nothing #

The bending bracket’s lower end is the sum of the fibres’, and it is a real bound because bending the yarn bends the fibres whether they slide or not. A fibre following the yarn’s curve is bent, and that costs something, and nothing about how freely it slides changes it.

Squashing the section is not like that. If the fibres may slide past one another without resistance, then changing the section’s shape at constant area requires no fibre to bend at all: they rearrange, each staying the length it was and following the yarn’s own long wavelength as before. A bundle of loose fibres inside a sheath is, in cross-section, a fluid. It has a bulk stiffness and no shape stiffness whatever.

So the free-slip bound is not small. It is exactly nought, and the bracket is infinite.

Which is the whole reason nobody could compute it #

That single fact explains a long silence in this collection. Kemp’s racetrack section has taken an aspect ratio as an input since it was written; the function signature has carried it as a free parameter; and the honest note beside it has always said that the two section models disagree and that the disagreement is the point.

None of that was laziness and no better derivation was available. A quantity whose lower bound is zero cannot be estimated from its bounds. Every route this site has used to pin a material property down — take the two limiting arrangements, derive both, and check that the answer is somewhere between — fails here at the first step, and it fails structurally rather than for want of care.

There is exactly one thing to do with a constant like that, which is to measure it. And the only thing available to measure is the cloth.

The measurement that was not taken for the purpose #

A fabric’s thickness is the most routinely measured thing about it after its weight. It is a specification line on every technical cloth, it is what a gauge under a stated light pressure reports, and nobody has ever measured one in order to find out how hard a yarn is to squash.

Peirce’s circular geometry predicts it, and over-predicts it, every time.

The over-prediction is not a rounding error and it is not in one direction by accident. A round section is the thickest a thread can be for its area, so a geometry built on round sections is the thickest cloth those threads can make. Real cloth is thinner because its threads are not round, and how much thinner says how flat they are.

Reading that backwards through the energy of the rung below gives the pressure each cloth is carrying, and the pressures come out between 0.19 and 0.85 newtons per crossing — a spread of 4.6 across a table whose yarn counts run from 10 tex to 60 and whose crossings own areas differing by a factor of eleven. Eight independent inversions landing inside one order of magnitude is the check; a model that needed a different constant for each cloth would have produced eight scattered answers and would have been a fit.

And the aspect ratios it implies #

Between 1.56 and 2.33, warp and weft.

That number is worth dwelling on because nothing in the derivation was arranged to produce it. Published cross-sections of ordinary cotton fabrics show flattening ratios in the range one and a half to two and a half, and have done since people began embedding cloth in resin and photographing it. The model was built from a shape energy, a bending energy and a load, fitted to one constant against one measurement — thickness — and it returns the flattening that microscopy sees.

What the constant comes out at #

Working backwards from the thicknesses at the contact force a loom’s own warp tension supplies gives a transverse shear modulus of about 4 N/mm², which is 4 megapascals.

Published yarn transverse compression moduli at packing factors near 0.6 lie between about 1 and 10 MPa. So the value the cloth demands is inside the range yarn measurements report, and it did not have to be: nothing in the arithmetic constrained it to be within three orders of magnitude of a real measurement, and an infinite bracket offered no help at all.

It is a fitted constant and it is named as one everywhere it is used. Every result in this ladder that depends on it is reported at more than one value of it, and the ones that do not depend on it — the flattening threshold, the roundness of a balanced relaxed cloth, the exact shortening of a crossover — are the ones worth having.

The two brackets are not the same shape of thing #

Setting them side by side makes the difference structural rather than numerical.

The bending bracket’s width is n/φ² — the fibre count over the square of the packing factor — and three consequences follow that the previous rung drew out: a coarser yarn’s bracket is wider in exact proportion to its count, a finer fibre widens it at the same count, and a better measurement of the modulus does not narrow it at all, because the modulus cancels. Every one of those is a statement about a ratio of two derived quantities, and it is available because there are two derived quantities to take a ratio of.

The transverse bracket has one derived quantity. There is no ratio to take, no dependence on the count to notice, and nothing for a better fibre measurement to fail to improve. The three consequences are not merely unavailable; they do not have analogues.

That asymmetry is not a quirk of these two properties. It says which of a material’s constants can be reasoned about and which have to be looked up, and the test is a single question asked of the free limit: when the elements are free to move, does the deformation still deform anything? Bending a bundle bends its fibres however freely they slide. Shearing a section’s shape does not. One survives the limit with a mechanism and the other does not, and no amount of care about the fibre changes which.

What was counted, and how #

The upper bound uses three fibre constants and each is stated: an axial modulus of 8 GPa with a reported range of 5 to 12, an anisotropy ratio of a fifth, and a Poisson’s ratio of 0.3. The last two are not this site’s and are not defended; what they are for is an order of magnitude.

The lower bound uses nothing at all, which is its whole content, and is asserted as an exact zero rather than as a small number. That is deliberate. A lower bound written as “very small” gets quietly replaced by a value in a later edit and then a result starts depending on it, so the gate checks that the free bound is 0 and that the bending bracket is finite, for nine yarns across three fibres and three counts, and would fail on either change.

The inversion is a bisection on the pressure, at fourteen steps over a range of three newtons, which resolves the answer to about two parts in ten thousand of a newton. It is run against a table of thicknesses that are trade figures for cloths of these constructions rather than measurements of these particular fabrics — the same caveat this site’s cloth table carries, for the same reason. What is being read out of them is an ordering and an order of magnitude, and neither would survive being quoted to three figures.

The over-prediction assertion is written as a relation. Every cloth over-predicted, and the smallest over-prediction still substantial. A version that asserted “by between 36 and 81 per cent” would be an assertion about the table of measurements rather than about the model, which is the commonest way an assertion goes bad here and has now been caught four separate times.

Which of a yarn’s stiffnesses have floors #

The test the essay proposes — ask whether the free-slip limit still deforms anything — is worth running across the whole list, because the answer turns out to be the same for every deformation a yarn undergoes except one.

Bending has a floor. A fibre following the yarn’s curve is bent whether it slides or not, so the sum of the fibres’ own rigidities is a genuine lower bound. That is the bracket the rung below computes.

Extension has a floor. Stretching the yarn lengthens every fibre’s helical path, so every fibre is strained however freely it slides. The free-slip bound is the sum of the fibres’ moduli weighted by the cosine squared of their helix angles, which is where the twist angle enters yarn strength at all.

Torsion has a floor. Twisting further changes each fibre’s helix angle, which extends and bends it. Nothing is free.

Compaction has a floor, and the floor has a name. Squeezing a yarn to a smaller area forces fibres to bend around one another at their contacts, and that is exactly van Wyk’s compression law — which is therefore not an empirical fit but the free-slip lower bound for compaction, arrived at from the same argument sixty years earlier and never labelled as one.

Shape change at constant area has no floor, for the reason the essay gives: an area-preserving change of section can be made by fibres translating sideways past one another with no fibre bending, extending or twisting at all.

So the list has one exception and it is the exception. Of every way a yarn can be deformed, exactly one can be accomplished by rigid rearrangement, and it is the one this collection has to fit a constant for. That is a satisfying place for the difficulty to be: not spread over the model, but concentrated at the single deformation the geometry permits for free.

Which makes the constant a yarn’s rather than a fibre’s #

If the floor is zero and the real value is not, then the real value is entirely supplied by whatever stops the fibres sliding — and in a spun yarn that is friction under the radial pressure that twist creates.

Twist presses the outer fibres inward; the pressure it generates goes as the square of the twist factor; and the resistance to sliding is that pressure times a friction coefficient. So

the transverse shape stiffness should go as μ times the square of the twist factor, with no floor and no fibre modulus in it at all.

Two consequences follow and both are testable on cloths that already exist.

The fitted 4 MPa is not a property of cotton. It is a property of ordinary ring-spun cotton at ordinary twist, and quoting it without a twist factor is quoting it without one of its two arguments. Everything in this ladder that uses it should be read as applying to yarns near the table’s own twist.

And a hard-twisted yarn should flatten less. Doubling the twist factor quadruples the modulus, so the same contact force produces a rounder section, a thicker cloth and a lower aspect ratio. That is exactly the reputation of voile and crêpe yarns — cloths made from them are thicker, more open and more resilient than their counts suggest — and it has always been attributed to the yarn’s liveliness rather than to its resistance to being squashed.

The prediction has the right shape to be falsified cheaply: weave one cloth construction in three twist levels and measure three thicknesses. A quadratic in the twist factor is a strong claim, and the alternative accounts — that a hard yarn is stiffer in bending, or simply finer — predict much weaker dependences or none.

Where the model stops #

The fitted constant absorbs every modelling error above it. The section is treated as uniform along the thread where a real one is flatter at the crossings; the energy is charged along the whole thread length; the strain measure is a small-strain one used at aspect ratios near two. Each of those is wrong in a direction that changes how much energy a given flattening costs, and the fitted modulus swallows all of them together. So 4 MPa is not a measurement of a yarn. It is the number that makes this model reproduce a thickness, and its agreement with the published range is a consistency check rather than a confirmation.

The bound is zero for a yarn with no twist in it. A real yarn is twisted, twist presses the fibres together, and pressing them together makes sliding cost something — so a real yarn is not at the free bound and never was. What twist does not do is give the bound a floor: the argument that free-sliding fibres have no shape stiffness is exact, and twist moves a yarn away from a bound rather than raising it. The same is true of the bending bracket and was said there.

A wet yarn is a different yarn. Water swells cellulose and lubricates the fibre contacts, and both move the transverse modulus by amounts nobody here has measured. Since half the interesting things that happen to cloth happen wet, that is a real gap and it is the same one the relaxation ladder records.

And there is no compaction in it. Everything above conserves the section’s area. Press hard enough and the fibres run out of room and the yarn densifies instead, which is a cube law in the packing factor and a different regime entirely.

The generalisation #

A bracket is only useful if both of its ends are reachable, and one of them being zero is not a small defect but a change of kind.

The bending case and the transverse case look identical when written down — take the free-slip limit, take the no-slip limit, derive both, and the truth is between. They differ because the free-slip limit of one still has a mechanism in it and the free-slip limit of the other does not. Whether a limiting case retains a mechanism is not visible in the shape of the argument; it has to be asked about the physics, each time.

The pattern recurs anywhere a property is bracketed by an “elements free” and an “elements bonded” pair. A stack of paper has a bending bracket and no shear bracket. A cable bundle resists bending at both limits and resists ovalisation at only one. A granular column has a compression bracket and no tensile one at all. In each, the quantity whose free limit is zero is the one that has to be measured, and — this is the part worth carrying — it is usually the quantity that gets treated as a modelling parameter, precisely because there is no derivation to argue with.

Who found it, and when #

The free-slip and no-slip bounds on a twisted assembly are Platt, Klein and Hamburger’s, from the 1950s, and are set out there for bending and for tension. The transverse case is treated separately in the literature and empirically: yarn compression is measured, fitted, and reported as a modulus or as a pressure-thickness curve, and van Wyk’s 1946 analysis supplies the theory for the compaction regime rather than for the shape-change one.

Kemp’s racetrack of 1958 is a description of the flattened shape and carries no stiffness. Nobody appears to have written down that the shape stiffness has no lower bound, which is unsurprising: it becomes interesting only once something is trying to compute the aspect ratio from first principles, and the trade measures it instead.

Where the ladder goes next #

The next rung spends the constant on a question the rung that computed it recorded as unanswerable. Reading a pressure out of a thickness gives the contact force in a cloth that is not under tension — which is the number an elastica would have supplied and this site does not have.

Sideways, the same constant decides what a calender does, and the finishing field’s account of lustre gains a pressure it always lacked. And the section models the setting field ran side by side for as long as this collection has run turn out not to be rivals at all.