A stiff cloth's wrinkle is deeper and no steeper

A cloth cannot carry a push found how far apart a wrinkle’s crests lie, and stopped there. Its formula has a bending rigidity, a span and a tension in it and no word about how much the cloth was pushed, so it gives a shirting and a shirting pushed three times as far the same spacing. Its figure draws a depth and says in its own caption that nobody computed it.

The depth is not hard to compute, because cloth does not stretch. A panel shortened by some share of its length still has all its cloth, and whatever no longer fits in the plane has to leave it. That fixes the shape, and the shape has a property worth having before the numbers: it can be scaled, and scaling leaves its angles alone.

So stiffness decides how big the wrinkle is and has no say at all in how steep it is. A heavy coating and a chiffon carrying one surplus have the same slope, to every figure the arithmetic can print.

The surplus is the only load the shape needs #

Call the share of the cloth’s own length that no longer fits the surplus, and write it $\varepsilon$. Three millimetres of extra cloth in a 300 mm panel is a surplus of one per cent, however it got there: a seam that drew up, a lining that shrank less than its shell, a sleeve bent at the elbow.

A sheet with a surplus and no other instruction takes the shape that costs least to bend, which is Euler’s elastica. Its modulus $k$ is the sine of half its steepest slope $\theta_0$, and the surplus fixes it through the two complete elliptic integrals:

$$\varepsilon = 2\left(1 - \frac{E(k)}{K(k)}\right), \qquad k = \sin\frac{\theta_0}{2}.$$

Nothing else is in that line. There is no rigidity in it, no tension, no width and no weight, because a uniform enlargement of a curve keeps every angle and keeps the ratio of its length to its span. Solving it gives the slope directly from the surplus, with the cloth already gone.

For small surpluses the integrals simplify and the two results a reader can carry are

$$\theta_0 \approx 2\sqrt{\varepsilon}, \qquad A \approx \frac{\lambda}{\pi}\sqrt{\varepsilon},$$

with $A$ the depth above the mean line and $\lambda$ the spacing. That second form is Cerda and Mahadevan’s. Spacing comes from the quarter-power law already in hand, so a depth now has every part it needs, and the rigidity reaches it only through $\lambda$.

What the table says, at one surplus #

Three cloths span most of what a garment is made from: a chiffon of 60 g/m² with a bending length of 6 mm, a shirting of 120 g/m² at 20 mm, and a coating of 300 g/m² at 36 mm. The cantilever test gives each one’s rigidity from those two numbers, and the rigidities run from 0.13 to 137 micronewton metres, a factor of about a thousand.

A thousandfold range of rigidity becomes a factor of 5.7 in depth and a factor of one in slope. The fourth root does the compressing, as it did for the spacing, and the last column is not compressed but absent.

The depth is large for the cause. A shirting holding three millimetres of surplus across 300 mm rises 4.1 mm above its mean line and falls 4.1 mm below it, which is 8.2 mm from crest to trough. That amplification is why so small a mismatch is so easy to see: the cloth has converted a length nobody could detect with a rule into a height anybody can see with a raking lamp.

A square root spends its depth first #

The curve’s shape matters to anybody trying to remove a wrinkle rather than describe one, and it matters unkindly.

Four times the surplus is twice the depth, so the relation runs the other way too: taking out half a wrinkle’s depth takes out three quarters of its surplus. A shirting at 4.1 mm has 3 mm of surplus across the panel. Bringing it to 2 mm means removing more than two of those three millimetres. And the last, smallest wrinkle is the most expensive per millimetre of depth, because there the curve is steepest: a hundredth of a per cent of surplus, three hundredths of a millimetre in the whole panel, still stands 0.4 mm proud on the shirting.

The same square root is at work in a seersucker, where it is why doubling the difference between two beams buys only forty-one per cent more relief. That cloth fixes its spacing with a stripe rather than a tension, and the arithmetic does not care which fixed it. What the seersucker essay does not ask is what stays the same when the spacing changes, and that question has the more useful answer.

The slope has no cloth in it #

Take the elastica’s line again and notice that it has been solved already, for every cloth anybody will ever weave.

The two forms are close over the whole useful range. At ten per cent the exact slope is 36.6° against 36.2° from the short form; even at thirty per cent they differ by only 2.1°. The depth forms part sooner, by 7.6 per cent at a surplus of ten per cent, because depth depends on the shape’s whole length as well as its steepest point.

The result reaches the relief cloths without any further work. A seersucker’s slack stripe, fed thirty per cent faster than its neighbour, holds about 23 per cent of its own length as surplus, and the elastica gives 56°. A honeycomb’s crimp difference of about two per cent gives 16°. Those two numbers are what the eye receives as a blister and a soft cell, and neither needed the cloth’s rigidity, its sett or its fibre to reach them.

Pulling harder trades one for the other #

Tension is the usual remedy for a wrinkle, and it is worth seeing exactly what it buys.

A shirting at 5 N/m is 4.1 mm deep; at 80 N/m, sixteen times as hard, it is 2.0 mm deep with crests half as far apart. That is a real improvement to a finger and none at all to the geometry of the flank. Only removing surplus changes the slope. Tension across a wrinkle does not remove surplus, because the surplus runs along the compression and the tension runs at right angles to it. A wrinkle that disappears when a panel is pulled along its own compression direction has had its surplus taken up, and that is a different pull.

That is one half of what a garment’s fit is up against. A dart removes surplus outright, by cutting it away where a curved body makes it. A pulled panel only moves it into finer waves.

The push it can still carry is small and does not rise #

Before a sheet wrinkles it carries a compression in its plane, and afterwards it carries a much smaller one that stops rising with the push. At small slope the wrinkled sheet’s compressive force per unit width is $B(2\pi/\lambda)^2$, and putting the spacing law into it leaves

$$F = \frac{\sqrt{BT}}{L}.$$

For the shirting under 5 N/m across 300 mm that is 0.023 newtons per metre; for the chiffon 0.0027; for the coating 0.087. The number has no surplus in it, so once the cloth has wrinkled, more shortening costs almost nothing further. The title of the essay that found the spacing can now be given as a figure: under five newtons per metre of tension one way, a shirting carries about a two-hundredth of that the other way before it has stopped resisting.

That also says where the square root’s lower end is. An ordinary woven’s stiffness in its own plane at very small strain, on the crimp it can spend, is of the order of five kilonewtons per metre, taking a low-load extension of about two per cent at a hundred newtons per metre as the guide. So 0.023 N/m of push is a strain near five parts in a million before the sheet buckles, or about 1.4 micrometres across the 300 mm panel. That is a stated guess at an order of magnitude and nothing depends on its second figure. It says that below about a two-thousandth of a per cent a cloth might absorb the shortening in its plane, and that above it all of the surplus has to go out of the plane.

A light reads the slope #

Here is why the split matters beyond being tidy. A wrinkle is mostly seen by the difference in brightness between its two flanks, and for a matte surface that difference comes from Lambert’s cosine: the flank turned towards a lamp receives more light than the flank turned away, by

$$\frac{\cos(\iota - \theta_0)}{\cos(\iota + \theta_0)},$$

where $\iota$ is the angle of the light from the cloth’s normal. The depth is nowhere in this. A matte wrinkle’s contrast is a function of its surplus and its lighting, and two cloths holding one surplus under one lamp show the same contrast, at very different depths.

Raking light is the trick every inspector and photographer of cloth already uses, and the formula says why it works so well. Under a lamp at 75° from the normal the ratio climbs steeply: a surplus of one in a thousand, three tenths of a millimetre across the panel, makes one flank 1.62 times as bright as the other. A difference of a few per cent is near what the eye can resolve on a broad smooth field, so on the arithmetic alone even the smallest surplus that wrinkles at all gives a detectable contrast at raking angles. Past about 1.7 per cent at that angle the dim flank has no direct light left and is simply in shade.

This also says something the ordinary account has the wrong way round. A stiff cloth is said to hide wrinkles, and it does not hide the wrinkles it gets. At the same surplus it has fewer of them and they are larger, and each flank is exactly as steep. It may well look smoother, and there are honest reasons: fewer contrast edges in a given area, flanks spread over a wider patch of the eye’s field where sensitivity to brightness steps is lower, and a stiffer cloth usually picking up less surplus in the first place because it presses a seam harder rather than following it. None of those is a shallower slope, and the formula will not supply one.

And a seam pucker’s usual number turns out to be a better one than it looks. The trade quantifies a puckered seam, when it measures at all rather than comparing against a photograph, by how much the seam line has shortened against the cloth beside it. That is the surplus. It looks crude next to a profile scan of the depth, but it is the number that carries the steepness, so it is the number that should track what a grader sees across different cloths. Whether it does, across cloths of different stiffness at one surplus, is a comparison that photographic grades could settle and nothing here has run.

Why satin shines and a hair layer veiling its highlight are about the specular part of the same light, which this formula deliberately leaves out. A lustrous cloth’s highlight swings by twice the slope, so on a satin the surplus has more to work with. That is a larger effect and it leaves the argument where it was, because it too reads an angle.

The same split, curved round a pedestal #

The nodes a drape test throws away found that the same bending length decides how many folds a draped disc falls into. It ended by naming the depth of those folds as the quantity the drape coefficient is really reading. Half of that turns out to be wrong, and the amplitude arithmetic is what shows it.

A hem at a mean radius $\bar R$ on a specimen of radius $R$ is exactly a compressed sheet closed into a ring: $2\pi R$ of edge has to fit round a circle of $2\pi\bar R$. Write the hem as $r = \bar R(1 + a\cos n\varphi)$ and require its length to match, which the hem model already does, and at moderate amplitude the requirement reads

$$a\,n \approx \sqrt{2\left(\frac{R^2}{\bar R^2} - 1\right)}.$$

The left side is the hem’s steepest slope in plan, and the right side has only the surplus in it. Round a pedestal as across a panel, the surplus sets the slope and the fold count only divides it into depth.

The closed form is good to about three and a half per cent at the deepest hem and better than one at the shallowest, and the hem solver owes nothing to the elastica, so the agreement between them is a check rather than a restatement. Since depth is $a\bar R$, a fold’s depth is roughly the surplus’s slope times the mean radius divided by the number of folds. Put the three-quarter power from the fold count in and a stiffer cloth’s folds are deeper as the three-quarter power of its bending length, at the same mean hem radius.

What the shadow actually measures #

Now the drape coefficient can be split, and it divides sharply. The area inside the hem is $\pi\bar R^2(1 + a^2/2)$: a round disc at the mean radius, plus a term that the folds add. So the depth of the folds enters the number only as a square, and because the slope $a\,n$ is held by the surplus, that term falls as the inverse square of the fold count.

Moving the mean hem from 105 to 140 mm takes the round part from 0.20 to 0.80. At six folds and a 120 mm hem the round part is 0.438 and the folds add 0.016. At twelve they add 0.004. Only at two or three folds is the term large, 0.14 at two folds and 0.064 at three on the same hem, which is the stiff end of the range and the place earlier work saw the coefficient rise.

So the coefficient reads where the hem has come to and not how deep its folds are, except on the stiffest specimens with the fewest folds. That sharpens an earlier statement rather than reversing it: the hem model had already found the mean radius dominating the fold count, and the amplitude is the reason. Depth is a second-order term in the shadow, and the fold’s own slope is pinned by the mean radius. The test has three numbers available — mean radius, fold count, fold depth — of which only two are independent, because the third follows from them. It records a blend of the first with a small correction, and discards the second.

So anybody with a photograph of a drape test can recover the depth with a ruler and a count, and the depth tells them nothing the other two did not.

How the numbers were made #

The surplus shape is solved by bisecting $k$ against the elliptic integrals, which are computed by the arithmetic-geometric mean, so the surplus is met to the precision of a double. The figure’s slopes and the table’s depths come from that solution, multiplied by the spacing from the quarter-power law. The rigidities are the cantilever test’s: area density times gravity times the cube of the bending length.

Four relations are tested rather than read off. Two cloths at one surplus must have identical slopes, compared as exact doubles. The ratio of depths must have the exponent one quarter against the ratio of rigidities, to nine figures. The exact shape must reduce to its small-slope forms at a surplus of one in a million, to one part in a hundred thousand. And the hem, whose solver knows nothing of the elastica, must hold its slope product within ten per cent across four to twelve folds. Inputs that must be refused are refused: a negative surplus, a surplus of the whole length, no tension, and a hem larger than its specimen.

What this leaves out #

The spacing law’s conditions all carry through. It is a thin isotropic sheet held under a uniform tension with a small slope, and cloth is directional, so the rigidity and the tension each have a preferred direction. The depth’s error includes the spacing’s.

A wrinkle here is periodic and evenly deep. Real wrinkles localise: past a surplus of some per cent a sheet tends to gather into a few sharp folds rather than many soft ones, with most of the surplus in them. A fold’s slope is then far above this curve while its neighbourhood’s is below. The elastica applies to one wave and is silent on how many a surplus would rather be, and at large surpluses its answer is the wrong shape for a garment. How far past a few per cent it can be trusted on cloth has not been measured here.

Friction and memory are not in it. A wrinkle that has been worn in has spent some of its surplus in a crease the crimp cannot supply, and the cloth then keeps a shape that no surplus accounts for. Every number here is an elastic state with no history.

The light is a single lamp on a matte surface. Sheen, a hair layer, a weave’s own texture and a real room’s spread of light all reduce the brightness ratio, and the eye’s threshold depends on the size of the patch as well as on the ratio. So the ratios are upper bounds to what is seen, and their use is comparing cloths rather than predicting a sensation.

The picture cannot show the claim it is best at. Three drawn curves can show that three depths differ, and no drawing can show that a slope is the same to every figure. That is a statement about a formula with nothing of the cloth in it, and the figure can only be consistent with it.

Who worked out which part #

Euler’s elastica is from 1744 and the elliptic form is nineteenth-century. Lambert’s cosine law is from 1760. Cerda and Mahadevan’s 2003 account of wrinkling gives both the spacing and, in the same paper, the small-slope depth from inextensibility, and a scaling of this kind is the ordinary route in the physics of thin sheets. Peirce’s bending length of 1930 and the drape apparatus of the 1950s are the textile side, as the essay on the drape test recounts.

What is put together here for cloth is the separation of the two consequences of that one scaling: spacing and depth carry the rigidity, and slope and matte contrast do not; the exact slope for any surplus, with no cloth in it; the same separation on a hem in a drape test; and the reading that follows, that the drape coefficient measures the mean hem radius and carries the fold depth at second order.

Still open: where the hem’s mean radius comes from #

Everything above takes the hem’s mean radius as given, and it has now turned out to be the number that matters. Nothing in this account says what it is. A radial strip of the overhang droops as a cantilever under its own weight, which would put a hem at a computed radius for a given bending length; but a fold stiffens the strip against drooping, as a corrugated sheet is stiffer than a flat one, and the fold’s own depth depends on the radius being sought. Solving those two together for the mean radius would predict a drape coefficient from a bending length, with no fitted constant, and the comparison with published values would be the test of it.