Twist decides where in the bracket

There is an unfinished piece of business in this collection and it has been unfinished since the essay that opened it.

A yarn’s bending rigidity cannot be computed. It can only be bracketed, between the fibres bending independently — the sum of their own rigidities — and the whole yarn bending as a solid rod of its own diameter. The ratio between the two bounds is the fibre count over the square of the packing factor, which for an ordinary 20 tex cotton is three hundred and twenty-seven.

That essay said, in its own words, that twist is the mechanism by which a real yarn moves from one end of the bracket towards the other, that the amount of the move is a friction problem, and that it did not pretend to solve it. Every calculation this collection has since made from a yarn’s stiffness has used the lower bound and said it was using a lower bound.

The friction problem is now solvable, and the answer is that using the lower bound was right.

The claim #

A yarn bends as a rod at gentle curvatures and as a bundle at sharp ones, and every bend a cloth imposes is on the bundle side by more than an order of magnitude.

• The crossover is a curvature, not a property. The coherent bound demands an axial force in the outer fibre that has to be built up by shear, and the shear available is friction under the twist’s own radial pressure.
• For an ordinary shirting yarn the crossover is at a bending radius of about a hundred millimetres. A thread bent by its own crimp in a cloth has a radius of a quarter of one.
• Pushing the contact efficiency to one — twenty times the honest value — still leaves the cloth’s bend nineteen times sharper than the crossover. The conclusion survives the fitted number by an order of magnitude, which is the only kind of conclusion a fitted number should be allowed to support.

So the collection’s habit of using the free bound is not a convention adopted for want of anything better. It is the right bound, and the bracket is a bracket only for the bends a cloth never makes.

The model, and what it demands #

Take a yarn bent to a curvature κ and ask what the coherent state requires.

In a solid rod under bending, the outer material carries a tensile stress proportional to its distance from the neutral axis: E·κ·y. In a bundle of fibres, the outermost fibre has to carry an axial force of E·κ·y·A_f if the bundle is to behave as a rod — and there is nothing to give it that force except shear from its neighbours.

The shear available is friction. The pressure is the twist’s own, which this collection computes in closed form, and it acts over the length a fibre keeps its place. That length is one twist pitch: a fibre that does not migrate returns to where it started after one turn.

Equating the two gives a curvature:

$$\kappa_c = \frac{4\,\mu\,\eta\,\bar{p}\,\ell_{\text{pitch}}}{E\,d_f\,y}.$$

Gentler than κ_c, the fibres cannot slide and the yarn is a rod. Sharper, they slide and it is a bundle.

The numbers, and how far apart they are #

For a 20 tex cotton at 800 turns a metre, an ordinary contact efficiency and a residual fibre stress of five per cent of breaking:

A thread in a plain-weave shirting is bent round its crossings to a radius of the order of a quarter of a millimetre — its own diameter and a half. The margin is between nineteen and seven hundred and sixty-eight, depending on the contact efficiency, and the smallest of those is at a value nobody would defend.

The comparison is not close and that is the point. A result that survived its fitted parameter by ten per cent would be a coincidence; one that survives by more than an order of magnitude at the extreme end of the range is a result.

A thread in cloth bends as a bundle of loose fibres, at every twist and every friction.

Which twist does move it, and where that matters #

The crossover does move with twist and it moves the right way: a harder-twisted yarn presses its own fibres together harder, so it stays coherent to a sharper bend — 192 mm of radius at 400 turns a metre against 93 at 1,400.

That is asserted rather than observed, because a model of bending that did not depend on twist would not be a model of this mechanism at all. But the movement is a factor of two over a range of twists spanning three and a half times, and the gap to be closed is a factor of four hundred.

So twist decides where in the bracket a yarn is, and the answer is “at the bottom” for every twist anybody uses.

There is one place where the crossover is reachable and it is worth naming, because it is where the bracket earns its keep. A yarn bent to a hundred-millimetre radius is a yarn in a fabric being draped over something large, or hanging from a table edge, and that is exactly the measurement a cantilever bending test makes. The bending length of a fabric is measured at curvatures near the crossover, so a bending test may be measuring a yarn that is partly coherent, while the cloth’s own crimp bends it as a bundle.

If so, a cantilever test is not measuring the quantity that governs how a cloth behaves at its crossings — which would explain a long-standing awkwardness in the subject, where measured fabric rigidities are hard to reconcile with the rigidities the constructions imply.

What the result licenses #

Three things this collection has been doing conditionally can now be done outright.

Using the free bound. Every calculation that has quoted a rigidity as a lower bound and carried the qualification can drop the qualification for anything at cloth curvatures. The stated bound is the value, to within whatever the free-bending model itself is worth.

Reading the fibre count as the whole story. The free bound is n times one fibre’s rigidity, so the fibre count is not merely one end of a bracket — it is the quantity that decides a thread’s stiffness in cloth. A yarn’s bending rigidity in a fabric is its fibres’ rigidity times how many there are, and neither the packing nor the twist enters at all.

And explaining why a cloth is soft. A knit is soft because it bends, and a woven cloth is soft because its threads bend at their crossings; both arguments have used the free bound. The reason a fabric of stiff fibres is not stiff is that its threads are three hundred times less stiff than their own material would suggest, and this essay is why that factor is available in full rather than in part.

The other bracket, and why this one closed #

It is worth putting this beside the collection’s other stiffness bracket, because the two are settled quite differently and the difference is instructive.

The obliquity bracket — between a yarn whose fibres keep their radius and one whose fibres migrate — is 1.16 wide at an ordinary twist, and it does not close. Nothing in this collection can say where a real yarn sits in it, because migration is a property of how the yarn was spun and is not deducible from a construction.

The bending bracket is three hundred and twenty-seven wide and it closes completely, because the question it turns on is not a property of the yarn at all. It is a comparison between two curvatures — one the twist supplies and one the cloth imposes — and both are computable from things a construction states.

A wide bracket that closes is worth more than a narrow one that does not, and the reason is the shape of the question rather than the size of the gap. That is worth carrying, because the instinct when facing an uncomputable quantity is to narrow the range; the more productive move is to find a comparison in which the quantity’s size stops mattering.

What was counted, and how #

The margin is asserted, not the crossover. The check is that every combination of twist and contact efficiency leaves the cloth’s own bend at least ten times sharper than the crossover, and it reports the worst case — which is nineteen-fold, at a contact efficiency of one. Asserting the crossover itself would be asserting a value that carries a fitted number.

And the crossover is asserted to move with twist, because a model in which it did not would not be about twist. The two assertions together are the shape this collection uses whenever a fitted parameter is present: assert the ordering and assert the margin, never the value.

The pressure is the same closed form the grip argument uses, called rather than restated, and checked against its own integral at four angles.

Why the margin is so enormous, and what that means #

Nineteen at the worst and seven hundred and sixty-eight at the honest value is a very large range to be safe over, and it is worth understanding where the size comes from, because a margin that large usually means two quantities are being compared across scales rather than across a mechanism.

They are. The crossover radius is set by how far a fibre’s own friction can transmit an axial force, which is a length built out of a twist pitch and a fibre diameter — millimetres. The cloth’s bend is set by the yarn’s own diameter, which is tenths of a millimetre. So the two lengths differ by two orders of magnitude before any friction, pressure or modulus enters, and everything the model computes is a correction to that.

That is why the fitted contact efficiency cannot rescue the coherent case. Moving η by a factor of twenty moves the crossover by a factor of twenty, and the gap is four hundred. The bracket does not close because a parameter was pinned down; it closes because the two quantities were never the same size, and no plausible value of anything moves them into contact.

The general form is worth carrying because it is the cheapest kind of result there is. When a model has an unknown parameter and a comparison to make, the first move is not to estimate the parameter but to compute the comparison at both ends of the parameter’s honest range. Either the answer changes — in which case the parameter is the whole question and the model has said so — or it does not, in which case the parameter never mattered and the model has said that too. This collection’s obliquity bracket is the first case and this one is the second, and both were settled by the same one-line procedure.

The uncomfortable half is that the procedure only works when a range is available. A parameter with no honest range is a parameter that cannot be swept, and the two brackets on this site that stay open are both of that kind.

There is a third case worth naming, because it is the one that looks like the second and is not. A comparison can survive a parameter’s whole range and still be wrong, if the parameter enters both sides of it — a sweep that moves the crossover and the cloth’s curvature together would show a constant margin and would be measuring nothing. The check that separates the two is whether the parameter appears on both sides, and here it appears only on one: the contact efficiency is a property of the yarn’s interior and the cloth’s bend radius is a property of its geometry, and no value of the first reaches the second.

What a stiff cloth is made of, then #

If a thread’s rigidity in cloth is its fibres’ rigidity times how many there are, then everything a designer can do about a fabric’s stiffness runs through a short list, and the list is different from the usual one.

The fibre’s own modulus, which is a material choice and is known to a factor of two at best.

The fibre’s diameter to the fourth power, which is the largest lever by far: a fibre twice as coarse is sixteen times as stiff, and the yarn made of it at the same count has a quarter as many fibres, so the thread is four times stiffer. Coarse fibre makes stiff cloth, and it is the reason a jute sacking and a cotton lawn are different objects for reasons that have nothing to do with the weave — and the reason a cloth’s drape is a fibre choice before it is a construction one.

The count, linearly, since a coarser yarn is more fibres — which is the same n that sets the evenness floor, pulling the other way as it always does.

And nothing else in the yarn. Not the twist, not the packing, not the folding — because none of them enters the free bound.

That is a strong claim and it is worth flagging as the essay’s most falsifiable one: two yarns of the same count and fibre, twisted differently, should bend the same in a cloth. A weaver’s experience is that a hard-twisted yarn makes a firmer fabric, and if that is a bending effect rather than a friction-at-the-crossings effect, this conclusion is wrong. The arithmetic says it is the second.

Where the model stops #

The residual stress is a guess and it is the weakest input. The pressure in a relaxed yarn comes from whatever tension the fibres retained from spinning, and nothing here computes it; five per cent of the fibre’s breaking stress is a plausible figure and the margin is quoted at it. Because the margin is so large, the conclusion survives an order of magnitude of error in either direction — which is the only reason a guess is admissible here at all.

The shear-transfer length is taken as one twist pitch, on the argument that a non-migrating fibre returns to its own radius after one turn. A migrating fibre does not, and migration would lengthen the transfer and raise the crossover. It would have to raise it by four hundredfold to change the conclusion.

Pure bending is assumed, and a thread at a crossing is not in pure bending. It is bent, compressed and sheared at once, and the compression at a crossing raises the local pressure enormously — which would make the fibres more coherent exactly where the bend is sharpest. That is the one mechanism identified here that runs against the conclusion, and it is not quantified.

And nothing here is a bending model. The free bound is the sum of the fibres’ rigidities, which assumes they bend independently about their own axes with no interaction at all; the real free case has some interaction. What has been settled is which bound to use, not whether the bound is exactly right.

Where the ladder goes next #

Into the measurement that sits near the crossover. If a cantilever test bends a fabric at radii where the fibres may not be sliding, then the fabric rigidity it reports and the thread rigidity a construction implies are different quantities — and reconciling them is a piece of work this collection has the pieces for and has not done.

And sideways into the compression at a crossing, where the pressure is not the twist’s own but the cloth’s. A flattened thread is a record of a force, and that force presses the fibres together at exactly the place where the bend is sharpest — the one term identified above as running the other way, and the one worth pricing next.