A fibre end prickles only inside a window

Prickle is a buckling load found the wool trade’s thirty micrometres in a column formula, and it ended on two admissions. The protruding length was the weakest input in the calculation, and the answer moved with its square root. And a crop posed a sum the calculation could not do: shortening a hair makes it a stiffer column and so a worse one, while leaving fewer hairs long enough to matter, and nothing said which effect wins.

Both admissions have one cause. The column formula says what a fibre end can carry, and it treats the skin as a wall. Skin is not a wall. It is about the softest thing a fibre end ever presses on, and a tip has to sink into it before it can push on it.

Put that in and a second limit appears, running the opposite way from the first. A fibre end prickles only between two lengths, and below one diameter there is no length between them.

A column has a ceiling on its length #

The first limit is the one already priced. A fibre end held at the cloth and pinned at its tip by the small pit it has made carries at most

$$P_{\text{cr}} = \frac{20.19\,E I}{\ell^2}, \qquad I = \frac{\pi d^4}{64},$$

so to reach a nociceptor’s threshold force $P$, measured at about three quarters of a millinewton, it must be no longer than

$$\ell_{\max} = \sqrt{\frac{20.19\,E I}{P}}.$$

For wool at three gigapascals that is 1.79 millimetres at thirty micrometres, 2.44 at thirty-five and 3.19 at forty. The ceiling rises as the square of the diameter, which is why the earlier account could read a threshold diameter off any stated length. Every hair longer than the ceiling buckles and lies over at a force too small to notice. Nothing in that formula stops a very short hair, so read alone it says the shortest hairs are the most dangerous of all, and that a hair a tenth of a millimetre long, from any fibre at all, should hurt.

Hairs that short exist in vast numbers. A yarn’s surface is a distribution puts most of its population in the short end, and nobody is hurt by them. The formula is missing something.

A soft surface sets a floor #

What it is missing is what the force has to push against. A fibre end on skin is a small flat punch on a soft half-space, and the classical result for that contact, Boussinesq’s, says a punch of radius $a$ pressed to a depth $\delta$ carries $F = 2E^{*}a\delta$. With $a$ half the fibre’s diameter this is simply

$$F = E^{*} d\,\delta,$$

where $E^{*}$ is the effective modulus of the skin. So a tip has to sink a definite distance before it is pushing with force $P$, and a hair cannot sink further than it stands proud of the cloth behind it, which gives the second limit:

$$\ell_{\min} = \frac{P}{E^{*} d}.$$

Indentation of a forearm’s soft tissue is quoted across tens of kilopascals. At thirty kilopascals a thirty-micrometre fibre has to sink 0.83 millimetres, a forty-micrometre fibre 0.63, and a twenty-micrometre fibre 1.25. A hair shorter than that bottoms out against its own cloth first, and the force passes to the crowns, spread over a great deal of yarn and felt as nothing but touch.

So the shortest hairs are harmless for an ordinary reason. They run out of reach before they run out of stiffness.

The two limits cross #

The ceiling rises as $d^2$ and the floor falls as $1/d$, so they must meet. Setting them equal gives

$$d_{*}^{3} = \frac{P/E^{*}}{\sqrt{20.19\,\pi E / (64P)}},$$

which at thirty kilopascals and three gigapascals is 23.2 micrometres, at a length of 1.08 millimetres.

Below that diameter the window is shut. There is no protruding length, however chosen, at which such a fibre can deliver the threshold force, because any length long enough to sink is too long to stand. This is a stronger statement than the earlier one in a specific way: that one said a fibre of a given diameter is safe if its hair is long enough to buckle, and left open how long the hairs really are. This one says the question does not arise.

The protruding length has gone from the answer. It was the weakest input before, carried on the square root, and the floor does not contain it: the length comes out as a result, about a millimetre, rather than going in as an assumption. What has come in instead is the skin’s modulus, which nobody had asked for, and it arrives under a cube root.

A tenfold doubt about skin, from ten to a hundred kilopascals, moves the floor from 33.5 micrometres to 15.6. The range of the measured force moves it, at thirty kilopascals, from 19.0 to 26.8, and the whole range of wool’s modulus from 21.7 to 24.9. That is a band, and the trade’s thirty micrometres sits inside it at the soft end, near fourteen kilopascals. The floor is not a second derivation of thirty. It is a statement that a limit of the right size exists without reference to how long anything is.

Cotton sits under its own floor #

A stiffer fibre has a lower floor, but only slightly, because the fibre’s modulus enters more weakly still. Cotton, at eight gigapascals, has a floor of 19.7 micrometres, and a cotton fibre is twelve to fifteen. Its window is shut at every length. The earlier essay reached the same verdict by showing a large margin at a two-millimetre protrusion; here there is no protrusion to be wrong about.

Damp wool works the other way. Halve the modulus, as water does to wool, and the floor rises to 26.7 micrometres while the window narrows at every coarser diameter. On the same cloth the prickling count falls from 1.19 ends a square centimetre to 0.42, which supports the earlier prediction that a damp garment prickles less. It leaves one thing unaccounted for, since wet skin is softer as well and the calculation does not move that.

The hairs that prickle are not the long ones #

A window turns a threshold into a count, because a population of hairs can now be run through it. The hair model puts about a quarter of a hair on each square millimetre of a plain cloth in twenty-tex wool, an exponential in length with a decay of 1.08 millimetres. Fibre diameters are a lognormal population about a mean of twenty-one micrometres with a spread of a quarter. For each length, the share that prickles is the tail above whichever limit is the tighter.

That comes to 1.19 ends a square centimetre, or about a hundred and twenty on a forearm resting on a sleeve, which is the right kind of number for something felt as separate points. The mean prickling hair is 1.23 millimetres long, and the distribution is narrow.

The earlier essay took two millimetres as its length, on the reasoning that this was the top fifth of the population and reach is what matters. The figure says reach is half of it. A hair of two millimetres is a long column, and only the coarsest few per cent can stand at that length. The hairs doing the damage are near the crossing point, where the window is open for the widest range of the fibres present, and those are ordinary hairs rather than the longest ones. They are the same hairs a light touch rides on.

The sum the column could not do #

Now the crop can be weighed. A cropping machine shears a cloth at a stated height, and it does something a flame does not: every hair longer than that height survives as a stub at exactly that height. The tail of the population is not removed. It is moved, and all of it is moved to one length.

Both halves of the earlier uncertainty are real, and they apply at different heights. Cropping at 1.1 millimetres makes this cloth 2.5 times worse, because every long hair that was harmlessly buckling has become a stub in the window. Cropping at 0.6 millimetres removes nearly all of it, because every stub is now too short to sink. The gap between the two is half a millimetre of machine setting.

So the earlier remark that “cutting the hairs shorter makes them worse” was right down to about a millimetre and wrong below it, and its closing suggestion that a raised cloth should be more comfortable than a cropped one turns out to depend entirely on the height. Below about 0.7 millimetres the crop beats the plain cloth and every raising calculated here, at every pressure.

The same shape holds across wools. The nineteen-micrometre wool goes from 0.62 to a peak of 1.6 and the twenty-five from 2.71 to nearly 6, and all three are close to nothing at 0.6. The dangerous height is set by the window’s crossing point and barely depends on the mean, because a coarser wool has more fibres in the window at every height without moving where the window is widest.

A flame cannot make the same mistake #

The difference between shearing and singeing is usually described as one of finish. Singeing is the cheapest change to a surface found that a flame takes hairs and leaves almost all the cloth. Here the difference is one of sign.

A flame reaching down to 1.1 millimetres leaves 0.47 ends where a blade at the same height leaves 2.94, more than six times as many. The flame removes the hairs that were in the window and leaves the ones below it alone, so it can only lower the count. The blade removes nothing, shortening each long hair into the window from above.

That has a practical edge, since the two operations are often treated as interchangeable ways of cleaning a surface. On the arithmetic they agree only at the extremes, where the height is far above the window and nothing happens, or far below it and everything has gone. Across the millimetre that matters they differ by a large factor and in opposite directions from the untreated cloth.

Why the trade’s steep curve belongs to soft skin #

The wool trade’s whole practice rests on the prickling count being steep in the mean diameter. A soft-skin window produces that, and it is worth checking whether a stiffer one could.

It could not. On a surface of three hundred kilopascals, the count between nineteen and twenty-three micrometres barely moves at all, because a surface that stiff lets the abundant short hairs push hard enough, and those are short enough to stand whatever their diameter. The floor has fallen below almost every fibre present, and the tail no longer selects anything.

So the sharpness of the wool trade’s line is evidence about skin. The earlier account explained it with the fourth power of the diameter alone, and on a rigid wall the fourth power is not enough, since a rigid wall lets a short fine hair win. The steepness needs the window, and the window needs a surface soft enough that the shortest hairs are excluded. The measurements of prickle against mean diameter, made for other reasons over forty years, therefore bracket a modulus nobody was trying to measure, and they put it where indentation puts it rather than where the stiffer in-plane measurements of skin do.

That is a surprising thing for a cloth specification to know, and it is a comparison and no more. The fit would need a count of prickle events against mean diameter under known conditions, which is what the sensory measurements record only as a proportion of subjects.

A light garment rides on its hairs #

The calculation so far has let the skin come all the way to the crowns, and it has to be checked that the skin gets there. A column with a pinned tip carries a good deal, and a light touch never reaches the crowns found that a plate can be held off a cloth by its hairs alone at low pressures. So the height of the skin above the crowns is solved rather than assumed: every hair taller than that height pushes back with the smaller of its column load and its sinking force, and the sum over the population has to equal the garment’s pressure.

At a hundredth of a kilopascal the skin floats 1.24 millimetres above the cloth, and hardly anything prickles. At 0.05 it is 0.21 millimetres up with 0.69 ends a square centimetre. By 0.07 kilopascals it has reached the crowns and the count has stopped rising, since everything that can prickle already is, and the cloth takes any further load itself.

Seven hundredths of a kilopascal separates two ways of wearing a cloth. A two-hundred-gram cloth under its own weight presses at two pascals, thirty-five times less, so where it merely hangs against the skin it is held well off by its hairs and almost nothing prickles. A cuff, a collar, a fitted sleeve or an arm resting on a table presses far harder, up to a kilopascal and more, and there the hair bed is long since saturated.

So pressing harder adds no prickle once the crowns are reached, and the whole of the response lies below that point. The difference that matters is between draped and held. That fits the ordinary experience that a wool garment prickles at its neck and cuffs rather than across a loose back, and it gives a mechanism for it. It does not prove the mechanism, since those are also the places that move most against the skin, and the calculation has nothing to say about movement.

Raising adds prickle, then holds it off #

A raised cloth has many more hairs, and the same calculation applies with the population multiplied. Raising moves the surface onto the hairs showed that a napped cloth presents its hairs rather than its yarn to anything touching it, and the prickle follows from that in two stages.

While the skin still reaches the crowns, more hairs means proportionally more prickling hairs, and four times the hairs is 4.74 ends a square centimetre instead of 1.19. Past that point the hairs begin to carry the skin. It rests higher, only the hairs taller than its height touch it, and those are increasingly the long columns that buckle harmlessly. At 0.3 kilopascals the count falls to 3.35 at eight times, 1.97 at sixteen and 1.14 at thirty-two.

At the same pressure a raised wool has to reach about thirty-two times the hair population to get back to where the unraised cloth began. Under a lighter garment the turn comes much sooner: at 0.1 kilopascals the count peaks at 1.7 near one and a half times and has already fallen to 0.48 at eight. A heavily raised, lightly worn cloth is better than the plain one, and a moderately raised one, pressed on, is several times worse.

This bears on the same prediction once more. Hair, nap and pile are one construction sets those surfaces on one continuum, and the prickle runs along it non-monotonically. Comfort against skin does not improve steadily as a surface gets hairier or as it gets cleaner. It is worst in the middle of each.

What the spread still does #

None of this removes the earlier finding that a threshold is a measure of a tail. A cloth is a population and not a thread is still the right frame, and the spread moves the count about as the mean does: at twenty-one micrometres a spread of 16 per cent gives 0.62 ends a square centimetre, 24 per cent gives 1.19 and 32 per cent gives 1.67.

What changes is how to read the statistic. The share over thirty micrometres is a proxy for the share inside a window whose lower edge is set by skin and force, and it happens to fall near where the floor falls. A specification written as a percentage over thirty is therefore doing the right kind of thing for a slightly different reason from the one given, and it has one property the argument predicts: it should be blind to finishing. Two fabrics of one wool with the same share over thirty can differ by a large factor on a crop alone, and the specification cannot say so.

This is the same statistic-against-mechanism gap as a bundle breaking at its weakest fibre and a pill surviving on its strongest anchor. Those are extremes of a sample. This is a two-sided selection on a joint distribution, with the extreme replaced by a window, and the window’s position decides which part of the population counts.

The model named #

The fibre end is a straight uniform cylinder standing upright from the crown surface, a fixed–pinned column of its own length, with a flat tip on an elastic half-space. Its force is the smaller of its buckling load and $E^{*}d$ times its sinking depth. The population is the hair model’s exponential in length, with its density and decay scaled with the mean diameter, and a lognormal in diameter independent of length.

The skin’s height is found by bisection so that the summed force equals the stated pressure, with the crowns as a floor beneath it. The prickling count is the lognormal tail above the tighter limit, integrated over length, with stubs from a crop added as a single term at the crop height.

What was counted, and how #

At the column’s ceiling the formula must return the threshold force exactly, and at the skin’s floor the punch must, both checked at three diameters. At each of three moduli the two lengths must agree at the floor’s diameter to nine figures, the window must be shut two per cent below it and open two per cent above. A nearly uniform population three per cent under the floor must produce no prickling ends and one ten per cent over must produce some, which tests the population calculation against the closed form rather than against itself. A light touch must ride at exactly the pressure it was given, and a firm one must reach the crowns with the hairs carrying only part of it.

The count must rise and then collapse as a crop comes down, a flame must never exceed the untreated cloth or a blade at the same height, and the steepness in the mean must be at least twice as strong at thirty kilopascals as at three hundred. The model must also refuse a fibre of no diameter, a skin of no stiffness, a garment at no pressure, a crop of no height, a raising below one and a population with no spread.

What the picture cannot show #

Every hair here stands upright. A real hair leaves its yarn at an angle, and an inclined hair is a bending cantilever long before it is a column. A hair inclined at thirty degrees and bent flat, as in the account of a light touch, carries under a tenth of the column load. So the count is of hairs that happen to stand near the normal, an unknown share of those present. It makes every absolute count here an upper bound. It should not move the heights much, since the limits belong to each hair.

The peak in the crop is sharpened by the model. Every stub stands at exactly the crop height, so the whole of the long population arrives in the window at once. A real cropper has a scatter of tenths of a millimetre, and a cloth’s crowns are not one plane, so the 2.5 times is a ceiling and the sharp peak should be a broader hump. The rise and the collapse are the robust part of it.

Skin stops at the crowns. A cropped cloth pressed on skin sits with the crowns as its datum, and a soft skin in fact bulges a little between them, which would give a short hair slightly less sinking room than its height. If skin were to reach deeper into the valleys between crowns as the hairs were removed, the effective height of a stub would rise and the collapse would come at a lower crop height. Its sign would survive.

The fibre’s tip is a flat punch. A cut wool end has a shape, an angled cut has a smaller first contact, and a sharp tip would sink at a lower force and so lower the floor. And skin’s modulus at the scale of a twenty-micrometre punch may not be the bulk indentation modulus, since the outer layer is much stiffer in plane. The bracket is wide because the input is, and the picture draws one line through it.

Nothing here is movement. A garment moving over skin rakes its hairs, and a hair arriving sideways is another loading altogether. That may be where most actual prickle events come from, and it has been left out.

Who found it, and when #

The prickle threshold and the mechanism are Garnsworthy, Kenins and Mayfield’s, from CSIRO in the 1980s: fibres presented singly, forces at which subjects reported pain rather than touch, and the fibre end as the stimulus. Later work in the same tradition, including Naylor and colleagues in the 1990s, tied the count of high-load fibre ends and the mean and spread of diameter to the sensory response and established that fabric structure and finishing matter as well as fibre. The contact mechanics of a flat punch is Boussinesq’s, from 1885, and the numbers for skin belong to a large biomechanical literature whose measurements disagree by orders of magnitude according to method.

The floor diameter, the window as a joint selection on length and diameter, and the crop’s two-sided answer are computed here, and none of them is known to have been tested. They are predictions from a calculation with one poorly known input, and the input is named.

Still open: the crop that makes it worse #

The sharpest test the calculation offers is a piece of wool fabric cut into strips and cropped at five heights from two millimetres down to half a millimetre, at one fibre population, with the prickling load counted on an instrument that presses a probe across the surface and records high-force events, or on a panel of forearms. The prediction is a rise of about double followed by a collapse below about 0.8 millimetres, and it can be separated from any effect of the yarn because the strips come from one cloth. A flame at the same heights is the control, since it should fall throughout.

The same strips would put a number on the skin. The height of the peak moves as the window’s crossing point moves, and that goes with the skin’s modulus, so where the worst crop falls is a measurement of $E^{*}$ at the scale of a fibre end, which the indentation literature cannot supply.

And the fabric’s own hair length, which the earlier essay asked for, is now asked for differently. What decides prickle is the share of the hairs lying between half a millimetre and two, not the longest, and a hair layer is a balance rather than a stock, so wear feeds that band from both directions. Whether a wool garment grows more prickly or less as it wears depends on which way the balance moves through the window, and that has not been computed.