A flattened thread is a record of a force

Cut a woven cloth across and look at a thread where it crosses another, and it is not round. It is squashed — wider across the cloth than it is thick through it, by something between a half again and three times, depending on what the cloth is and what has been done to it. Every microscope section of a fabric ever published shows this, and no model on this site has ever explained it.

The models have described it. Kemp’s racetrack section — a rectangle with semicircular ends, at the same area as the circle it replaces — has been available here since the setting field was built, running alongside Peirce’s circular section as an alternative account of the same fabric. Which of them a figure uses changes the thickness by a fifth and the cover by more, and the site’s standing answer to which is right has been that they disagree and that the disagreement is the point.

That answer was always incomplete, and the reason has a name. The racetrack takes an aspect ratio as an input. Nothing in this collection computes it. It is a number somebody chose — and every quantity that depends on a section, from the sett a weave can reach to the thickness a gauge reads, has been carrying that choice unexamined.

The energy the previous rung said was missing #

The rung that gave a yarn a bending stiffness ended by naming its own missing companion: a yarn flattens where it crosses, the racetrack models the flattening, and there is no stiffness in that model either. This rung supplies it.

The shape change is area-preserving, so the natural way to write it is as a strain. A circle taken to an aspect ratio f at constant area is stretched by √f one way and by 1/√f the other, so its principal logarithmic strains are ±½ ln f. Plane-strain pure shear at small strain stores 2Gε² per unit volume, and the volume per unit length of thread is the section’s area, so the energy in a unit length is

$$u_c = 2 G \cdot \frac{\pi d^2}{4} \cdot \left(\tfrac{1}{2}\ln f\right)^2 = \frac{\pi}{8} G d^2 \ln^2 f$$

with one material constant and nothing fitted. It is zero at f = 1, as it must be, and it depends on the log of the aspect ratio rather than on the aspect ratio, which is worth noticing: a yarn flattened to twice its thickness is exactly as far from round as one flattened to half would be, and the natural coordinate of a squashed section is not the ratio but its logarithm.

What the geometry has to become #

Peirce’s two equations describe a thread that runs straight, turns through a circular arc of radius D/2 around the thread it crosses, and runs straight again. A thread crossing a flattened thread does one more thing first: it runs horizontally along the flat top before it starts to curve. Writing F for that flat width — (f − 1)·b of the thread being crossed — the equations become

$$p = F + (l - D\theta - F)\cos\theta + D\sin\theta$$

$$h = \phantom{F + {}} (l - D\theta - F)\sin\theta + D(1 - \cos\theta)$$

with D now the sum of the two flattened thicknesses rather than of the two diameters. Set F to nothing and b back to d and Peirce’s equations return term by term, which is checked against this site’s own circular solver at every cloth in the table rather than asserted: the worst departure is two parts in ten thousand million million.

That flat run is the reason to expect flattening to pay. It covers ground at no cost in height at all. A thread crossing a flattened partner spans more cloth per unit length than one crossing a round partner, so it needs less crimp, and less crimp is less bending.

The result, which is the other way round #

Hold the two thread lengths where the loom left them, let the two aspect ratios and the division of the thickness vary, and add the two energies. The sum is least at an aspect ratio of exactly one.

The flat run does help. It is simply not enough, and what defeats it is a term nobody would look for. Peirce’s arc has radius D/2, so its curvature is 2/D, and the bending energy of one modular length is 2Bθ/D. Squashing the threads shrinks D. The angle falls a little; the curvature rises faster. A flattened crossing is a sharper crossing, and the thread pays for the sharpness in exactly the coin it was trying to save.

So flattening costs bending as well as compression, and a cloth with nothing pressing it will not do it.

Which changes what a racetrack section is #

The two sections are not rival descriptions of one fabric. They are descriptions of one fabric at two moments in its history.

Peirce’s circle is the cloth relaxed. Kemp’s racetrack is the cloth pressed — on the loom by its own warp tension, at the fell by the reed, between the bowls of a calender, under the foot of a sewing machine. The aspect ratio is not a modelling choice at all. It is a measurement of a force that was applied, carried in the shape of the thread after the force has gone.

That reframes a great deal of this collection’s own hedging. The site has said, honestly and repeatedly, that the two models disagree and that the model matters most for the quantity nobody thought to ask about. It could not say which was right because the question was badly put: both are right, about different cloths, and what separates those cloths is a pressure that no figure here had a way to name.

Adding the load, which closes it #

If a flattened section is evidence of a load, the load belongs in the arithmetic. Adding the work it does as the cloth thins gives

$$\Phi(h_1, f_1, f_2) = U_{\text{bend}} + U_{\text{comp}} + N \cdot t$$

with N the normal force at one crossing and t the cloth’s thickness. That is an ordinary Legendre transform: the cloth minimises Φ at constant load in exactly the way it minimises the energy at constant thickness. It gives the circle back at N = 0, which is the only real test of an implementation like this one, and it has one property that is worth the whole derivation.

The onset of flattening has no compression constant in it.

The compression energy is quadratic in ln f, so its slope at a round section is zero. The bending term’s slope there is not zero, and neither is the load’s. So the balance at the onset is between two terms that both know nothing about how hard the yarn is to squash, and the threshold pressure is

$$N_c = \frac{\partial U_{\text{bend}}/\partial f}{-\,\partial t/\partial f}\bigg|_{f=1}$$

with G cancelled before the arithmetic starts. In a file whose entire subject is a constant nobody can bound, there is one number that does not need it.

A few thousandths of a newton is a very small threshold. A warp end at one per cent strain in a 25 tex cotton carries about 1.1 N, and it presses on each pick it crosses with about 1.3 N — four hundred times the threshold. So this is not a rule that keeps cloth round. It is a rule that says nothing keeps cloth round once it is on a loom, and that the flattening seen in every published section is not a property of fabric but a receipt.

What was counted, and how #

The compression energy has one material constant and it is stated rather than derived: a transverse shear modulus for the yarn, in newtons per square millimetre, which is a megapascal exactly. The next rung is about where that number can possibly come from, and the answer is not encouraging.

Everything else is enumerated. The section arithmetic is checked to conserve area at every aspect ratio tried, to one part in a million million, because a “racetrack” that quietly changed the yarn’s packing factor would be modelling a different physical process and would still look like a racetrack. The flattened geometry is checked against this site’s own circular solver at f = 1, at every cloth in the table, and agrees to machine precision.

The minimisation is a coarse-to-fine scan and not a gradient method, and that is a decision rather than laziness. The feasible region has holes in it: a state can be refused because a thread cannot reach the height asked of it, or because two threads would have to overlap, and a gradient method walking into a hole has no way to tell that from a boundary. Three rounds of a thirteen-by-thirteen grid over the two aspect ratios, each zooming on the best cell of the last, with a scan and a golden-section refinement in the thickness division inside each.

That refinement matters more than it looks. The results below turn on derivatives of the least energy with respect to the aspect ratio, and a bare scan quantises the least energy by roughly the curvature times the square of the sample spacing — which is the same order as the difference being taken. The refinement takes the quantisation out of the difference entirely, and the threshold above is Richardson-extrapolated from two step sizes because a one-sided difference has an error linear in the step. Halving the step four times moves the answer by three parts in a thousand.

And the roundness claim is exactly true only where the counts are equal.

Seven of the eight cloths in this site’s table have equal counts in warp and weft, and all seven come out at an aspect ratio of exactly one — at three transverse moduli two decades apart and at both ends of the bending bracket. The eighth is the poplin, whose warp is finer than its weft and set half as far again apart, and it does not. Its bending energy falls as the warp flattens, so nothing opposes the flattening but the compression term, and the relaxed aspect ratio is then set by the transverse modulus alone.

That is a genuinely uncomfortable result and it is stated rather than smoothed. At 4 N/mm² the poplin settles at 1.02, which no fabric analysis could detect. At 0.3 it settles at 1.45, which one could. An unbalanced cloth’s relaxed section is not determined by geometry, and every figure on this site that draws one with a circular section is right for seven cloths and making an assumption about the eighth.

One corner where the answer is not round, and it is not a fabric #

Running the roundness check across the whole bending bracket rather than at its free end turns up a case worth recording, because it is a limitation of the model rather than of the arithmetic.

At the coherent bending bound — fibres locked, rigidity four hundred times the free bound’s — combined with a soft transverse modulus, the energy acquires a second basin far out in the aspect ratio, and the four most openly set cloths in the table fall into it. The round section is still a local minimum; a lower one has appeared elsewhere.

It is not a fabric. Locking a yarn’s fibres against sliding is what makes it stiff in bending and what makes it stiff in shear — they are the same fact about the same yarn seen twice. A yarn at the coherent bending bound with a transverse modulus of 0.3 N/mm² is one whose fibres are simultaneously locked and free, and no such thing exists.

The model treats the two stiffnesses as independent inputs taken from two separate arguments, and has no way to refuse the combination. Enumerating where it bites: the far basin closes below 3.4 N/mm² for every cloth, against a working value of 4, so the corner sits outside anything this site computes with — and the cloths it catches are the open ones, every one of them more openly set than every cloth it does not catch. An open cloth has long spans to bend across, so its bending energy is the most improved by the flat run a squashed partner offers.

Two constants taken from two arguments, treated as independent, and correlated in the material: that is a shape worth carrying out of this rung, because nothing about either derivation says so.

Where the model stops #

The section is uniform along the thread. A real yarn is flatter where it crosses than between crossings; Kemp’s racetrack is a constant section and so is this. The energy is therefore charged along the whole thread length rather than at the contacts, which overstates the compression and understates how localised the squashing is. Nothing above depends on the difference except the size of the fitted modulus, which absorbs it.

The energy is a small-strain form used at strains that are not small. At an aspect ratio of 3 the log strain is 0.55 and a quadratic energy is being asked to work outside its warrant. Every result past about 2 says so where it is stated, and the calendering figures are past it.

There is no yield and no set anywhere. Everything here is recoverable, so this model says a cloth taken out of a calender springs back to round. Real calendering is hot and does not, which is the subject of a rung in the finishing field and is not in the arithmetic here.

Area is conserved, which is only true up to a point. Flattening at constant area is a shape change; press harder and the fibres run out of anywhere to go sideways and the yarn compacts, its packing factor rising and the air leaving. That is a cube law in a different variable and it is a different regime.

And the model is a plain weave, as every Peirce argument on this site is. A twill’s thread passes over more than one crossing before it turns, so the arc-and-straight construction is a different one and every number would move. The direction of the argument survives, because it turns on a curvature that goes as 1/D, and that is true of any arc.

The generalisation #

The shape of the result has nothing to do with yarn.

When a body can lower one energy by changing its shape, check whether the shape change raises another energy through a length that appears in the first. Here the length is D, the crossing’s own thickness: flattening lowers the crimp the thread must accommodate and simultaneously shortens the radius it must accommodate it over. The second effect wins, and it wins because curvature is an inverse length and energy goes as its square.

That pattern recurs wherever a bent member is being squeezed. A rope bent over a small sheave, a nerve in a narrow foramen, a cable in a tight tray, a stack of laminations pressed at a fold: in each, thinning the member reduces the bend it has to make and sharpens the bend it does make, and which of the two wins is decided by exactly the trade this rung computes.

The second half generalises differently and is the more useful lesson for a reader of this collection. A parameter that a model takes as an input is sometimes not a parameter of the material at all, but a record of the object’s history. The racetrack’s aspect ratio looked like a property of a yarn, was written into a function signature as though it were one, and turns out to be a property of what happened to the cloth. There is no reason to expect a model to announce which of its inputs are of that kind.

Who found it, and when #

Peirce’s geometry is from 1937 and its circular section is explicit in it. Kemp proposed the racetrack in 1958 precisely because real yarns are not round, and Hearle and others carried it through the flattened-thread geometry that this rung’s two equations are; every account of it treats the aspect ratio as something to be measured off a section and supplied.

Van Wyk’s compaction law is from 1946 and is the standard account of what a fibre assembly does when it is genuinely densified rather than merely reshaped.

What is this site’s is the observation that the flattening has an energy and can therefore be asked to justify itself — and the finding that it cannot. Nobody appears to have minimised the two energies together and noticed that the round section wins, which is unsurprising: the question only arises once a figure’s aspect ratio has to be computed rather than chosen, and that requirement is this collection’s habit rather than the trade’s.

Where the ladder goes next #

The next rung asks where the one material constant here could possibly come from, and finds that its lower bound is exactly nothing — so the transverse stiffness is the one quantity on this site that cannot be bracketed at all, and has to be read out of a fabric.

The rung after that reads it, and gets a number back that the rung that computed it recorded as unavailable: the contact force in a cloth that is not under tension.

Sideways, this changes what two other ladders were about. The setting field’s disagreement between Peirce and Kemp is not a disagreement, and the finishing field’s account of calendering acquires the pressure it always lacked.