A cloth cannot carry a push

Everything this site says about a fabric’s mechanics rests on one model: a net of inextensible threads, free to rotate where they cross. It is the right model for the bias, for shear locking, for drape over a surface, and for the whole argument that a woven cloth is a mechanism rather than a material.

It is exactly the wrong model for the question in this essay, and the way it fails is instructive.

Push a cloth from two opposite edges. The net says: a thread cannot carry compression at all, because a thread has no bending stiffness in this model, so the net leaves the plane immediately — and it leaves the plane into a wrinkle of any wavelength whatever, because every wavelength costs the same, which is nothing. The net model answers the question with an infinite set of answers, all of them free.

The competition the net leaves out #

Two things resist a wrinkle and they resist different wrinkles.

Bending costs energy. A sharply curved fabric stores more than a gently curved one, and curvature goes as one over the wavelength squared, so short wrinkles are expensive and long ones are cheap. Bending alone would give one enormous wrinkle across the whole width.

The tension across the sheet resists the excursion. Leaving the plane makes the material path longer, and a sheet held taut in the other direction has to do work against that tension to go anywhere. That work goes as the amplitude squared over the wavelength, so long wrinkles are expensive and short ones are cheap. Tension alone would give an infinitely fine crinkle.

Neither wins. The wavelength that costs least is the compromise between them, and Cerda and Mahadevan’s 2003 result gives it in closed form:

λ = 2π ( B L² / T )^¼

for a sheet of bending rigidity B, spanning L across a tension T per unit width.

This is the one place on this site where a stiffness enters a shape calculation. Everywhere else the answer is what is geometrically available; here the answer is what is cheapest, and cheapness needs a modulus.

The quarter power is the whole content #

The exponent is what makes the result useful, and it says the wavelength is remarkably insensitive to everything.

Four times the tension shortens the wrinkle by exactly √2 — twenty-nine per cent. Sixteen times the rigidity doubles it. Twice the span multiplies it by √2, because the span enters as a square under a fourth root and the two exponents cancel to a half.

The site asserts all three as relations rather than as values, which is the house rule and which caught a mistake here: the first version of the span assertion claimed that twice the span doubles the wavelength, which is the exponent read carelessly, and it failed on the correct arithmetic. The claim now written down is the one that follows from the formula.

A wrinkle is hard to change. That is the practical reading of a quarter power. A tailor who pulls a seam four times harder gets a wrinkle spacing that is seventy-one per cent of what it was; a fabric that is sixteen times stiffer wrinkles at twice the spacing. Everything about a wrinkle’s size is stubborn, which is why the trade’s remedies are about the wrinkle’s presence — take the compression out, ease the seam, cut on the grain — and never about its wavelength.

What was counted, and how #

The rigidity is not a new measurement, and that is the part of this essay that connects it to the rest of the site.

The cantilever test hangs a strip of cloth over an edge and measures how far it reaches before its tip droops to a stated angle. The site has modelled it since an earlier essay here, and what it returns is a bending length — a length with the fabric’s own weight already divided out — and, given an areal density, a flexural rigidity.

So the chain runs: droop angle → bending length → rigidity → wrinkle wavelength. Four steps, one measurement, and the measurement is one anybody with a ruler and a piece of card can make.

The range across four ordinary fabrics is under three to one at any tension, which is the quarter power again: the rigidities span a factor of fifty and the wavelengths a factor of under three.

And the table has a consequence a reader can check on any garment. A stiff fabric wrinkles in a few broad folds and a limp one in many fine ones, and the count across a fixed width goes as the fourth root of the rigidity upside down — so a suiting gives one or two across a 300 mm panel and a voile gives four or five. That is exactly what the eye reports and it is not usually connected to a number.

Two fabrics, one number apart #

The table above hides something worth pulling out: the 80 gram voile and the 120 gram shirting have nearly the same rigidity and wrinkle at nearly the same wavelength, while the two 120 gram cloths differ by a factor of eight in rigidity and by seventy per cent in wavelength.

Weight predicts nothing here. Rigidity goes as the areal density times the cube of the bending length, so the bending length dominates: a factor of two in bending length is a factor of eight in rigidity, and a factor of two in weight is a factor of two. Two cloths of the same weight can be a stiff poplin and a limp lawn, and the cantilever test separates them in about a minute where the scales do not separate them at all.

That is the same argument the site makes about what a weight does not carry, reaching a mechanical property this time rather than a geometric one. A specification quoting grams per square metre and a fibre says nothing about how the cloth will fold.

What sets the bending length is the structure #

The cantilever test measures a fabric and does not explain it, and the site has the pieces of an explanation elsewhere.

A cloth bends by two mechanisms and the balance between them is what makes bending lengths vary so widely. The threads themselves bend, which costs whatever the yarn’s own flexural rigidity is — and yarns are soft, because a yarn is a bundle of fibres that can slide over one another. And the threads must slide at their crossings, because bending a cloth makes the two faces different lengths and something has to take up the difference; that costs friction rather than elasticity.

The second mechanism is why a more closely set cloth is stiffer than an open one of the same yarn, why a resin finish stiffens a fabric enormously without changing a yarn, and why bending stiffness has a hysteresis that a true elastic rigidity would not. It is also why the site’s directional drape essay finds the bias so much limper: on the bias the crossings can rotate instead of sliding, which is the cheapest of the three.

None of that is computed here. What is computed is what follows once a bending length is in hand, and the bending length is measured.

Where the tension comes from, which is not obvious #

The formula needs a tension across the sheet and a wrinkling fabric does not obviously have one.

In the cases where a fabric wrinkles it usually does. A curtain hangs under its own weight, so the tension across it at any height is the weight of the cloth below — which is why a curtain’s folds are broader at the top and finer at the hem, a thing everybody has seen and nobody attributes to a quarter power. A garment over a body is held by seams and by the body’s curvature, and the tension is whatever the fit supplies. A sheet dragged across a table is tensioned by the drag itself.

The case with genuinely no tension is a cloth lying loose, and a cloth lying loose does not wrinkle at a wavelength — it crumples, which is a different and much harder problem with sharp folds and points in it and no single length scale at all.

So the model’s domain is the tensioned sheet, and its boundary is stated as a refusal: asked for a wavelength at zero tension, the arithmetic declines and says that with none the net model applies and every wavelength costs the same. That is not a numerical guard. It is the model naming the point at which it hands the question back to the one this site normally uses.

The exponent is what makes the arithmetic worth having #

A quarter power is a weak dependence and it would be easy to read that as a reason not to compute it — if nothing much moves the answer, why work the answer out. The reverse is true, and the reason is worth stating because it applies wherever an exponent is small.

A weak dependence makes a prediction robust. Every input to this arithmetic is uncertain: the bending length is measured to a few per cent, the tension across a garment panel is guessed, the span is a boundary condition somebody chose. Under a linear law those uncertainties would pass through undiminished and the answer would be worth nothing. Under a fourth root a factor of two in the tension is nineteen per cent in the answer, and a factor of ten is forty-four.

So the wavelength is one of the few mechanical quantities on this site that can be predicted from a badly known input and still be right. A tailor’s estimate of a panel’s tension, wrong by a factor of three, gives a wrinkle spacing wrong by a quarter — which is inside what anybody could measure on a real garment anyway.

The same weakness is what makes the measurement useless in reverse. Reading a wrinkle spacing and inverting it to get a rigidity multiplies the error by four, so a spacing known to ten per cent gives a rigidity known to a factor of one and a half. That is the mirror of the drape test’s fold count, where a three-quarter power makes the reading a usable stiffness measurement — and the two exponents are why the disc is the instrument and the panel is not.

A steep law predicts poorly and measures well; a shallow one predicts well and measures badly. The two shapes in this anchor sit on opposite sides of that, from one measured length, and knowing which is which is the difference between a calculation worth doing and a calculation worth inverting.

The same reading explains why the wrinkle wavelength has never been a textile measurement despite being an obvious thing to observe. It is easy to see, easy to measure with a ruler, and carries almost no information — a spacing that spans a factor of three across the whole of the ordinary wardrobe is not a discriminating reading of anything. The fold count spans a factor of five and a half over the same range, and it is the one nobody records.

So the two observations a fabric offers for free are on opposite sides of usefulness, and the trade has ignored both — one because it says too little and one because nobody asked what it said.

The relation to the net model is a division of labour #

It would be easy to read this essay as a correction to the rest of the site’s mechanics, and it is not one. The two models answer different questions and neither can answer the other’s.

The net says what is available. Which shapes a cloth can take without stretching a yarn; how far it can shear before it locks; how much of a sphere it can cover before it must be darted. Those are geometric facts and no stiffness enters them.

The stiffness says which of the available shapes is taken. Among all the ways a compressed sheet could leave the plane — and the net says they are all free — the one with the lowest energy is chosen, and choosing needs a modulus.

The site’s habit of asking what a picture cannot show applies to both. A net figure cannot show a wavelength; a wavelength figure cannot show whether the shape it draws is reachable without stretching. This one is: a sinusoidal wrinkle of small amplitude is very nearly inextensible, which is why the arithmetic can treat the excess length as the whole story.

Where the model stops #

The amplitude is not computed. The formula fixes the spacing and says nothing about the depth, which depends on how much compression there is — how much surplus length has to leave the plane — and that is a boundary condition rather than a property. Every figure here draws an amplitude and says so.

One mode, and a real sheet has several. The result is the lowest mode of a sheet with simple boundaries. A real garment panel has seams, darts, curvature and a body under it, and its wrinkles are neither uniform nor sinusoidal.

The fabric is treated as an isotropic plate. It is not: a woven cloth’s bending rigidity is different along the warp, along the weft and on the bias, by factors of several — which is exactly what drape is directional is about. The single rigidity here is a stand-in for a tensor, and the direction a wrinkle runs is decided by which rigidity is lowest as much as by where the compression is.

And nothing here is plastic. A wrinkle in this arithmetic is elastic and disappears when the load does. The wrinkles that matter in clothing are the ones that do not, and a set crease is a fibre-level phenomenon — the same relaxation and setting the finishing field computes — that this model has no term for.

Who found it, and when #

The competition itself is old. Euler’s buckling of a strut is 1744, the wrinkling of a stretched membrane is a nineteenth-century problem, and tension-field theory — the idea that a membrane carries no compression and goes slack instead — is Wagner’s, from 1929, developed for thin aircraft skins.

What was missing until recently is the wavelength. Tension-field theory says a membrane wrinkles and treats the wrinkles as infinitely fine, because a true membrane has no bending stiffness and the theory has no length scale in it. That is the same hole the net model has, arrived at from the other direction.

Cerda and Mahadevan supplied the scale in 2002 and 2003, in a pair of papers whose examples are a stretched sheet, a hanging curtain and skin — and the fabric case is in them explicitly. The result is a scaling law rather than a solution, which is why the prefactor is a matter of the boundary conditions and the exponent is not, and why this essay asserts the exponents and quotes the prefactor.

The textile literature had the ingredients for fifty years before that. Peirce’s cantilever test is 1930 and gives the bending length directly; the drape test is 1950s; the connection between them and a wrinkle spacing is one substitution, and nobody made it because nobody in textiles was asking for a wavelength.

Where the ladder goes next #

There is a wrinkle count that the trade already measures and then discards. The standard drape test lays a circular specimen over a pedestal and reports the shadow it casts as a single number — and the specimen also falls into a definite number of folds, which is a buckling mode set by the same competition. The next rung predicts that count from the bending length and finds that it carries more of the stiffness than the number the test reports.