Prickle is a buckling load

The wool trade has one number for whether a fabric can be worn against skin, and it is not a fabric property at all. It is the percentage of fibres in the wool coarser than thirty micrometres. Below about five per cent a garment is comfortable; above about ten it prickles; and the boundary is sharp enough that wool is bought and sold on it.

Thirty micrometres is a measured boundary. It was found by putting fabrics against people’s arms and asking, and it has stayed at thirty for forty years because that is where the measurements put it.

The cloth #

Prickle is not itch and it is not an allergy. It is a mechanical sensation: the nociceptors in the skin — the receptors that report pain rather than touch — are triggered by a small enough contact carrying a large enough force, and a fibre end is exactly that. The measurement that established it put single fibres against skin and found the force at which a subject reported pain rather than pressure.

So the question is a mechanics question: how much force can a protruding fibre end apply before it gives way? And the answer is not a strength. A fibre pressed on its end fails long before it breaks, by buckling.

The claim #

A fibre end pressed against skin is a column held at the cloth and free to slide at its tip, so it carries at most 20.19EI/ℓ² and then lies over. Setting that equal to the measured nociceptor threshold and solving for the diameter gives thirty micrometres at a protrusion of two millimetres — which is where this site’s own hair model puts the top fifth of a wool’s population. Neither number was fitted.

And the corollary that makes the trade’s specification the right one: the load goes as the fourth power of the diameter, so the boundary is a few micrometres wide and it is the coarse tail of the distribution rather than its mean that decides.

Why it is a column and not a beam #

The distinction between two end conditions is doing real work here and is worth setting out, because they differ by a factor of eight.

A hair standing free with nothing at its tip is fixed–free: held where it leaves the yarn, unconstrained at the end. Its buckling load is π²EI/4ℓ², and that is the case the contact ladder uses when a plate is coming down onto a population of hairs, because a hair under a descending plate slides sideways as it goes.

A hair pressed into skin is fixed–pinned: the tip is held laterally, because it has indented the skin and sits in its own small pit. Its buckling load is 20.19EI/ℓ², eight times larger.

Getting that wrong would put the threshold diameter at nineteen micrometres rather than thirty-two, which is well outside the trade’s line and would look like a refutation rather than a confirmation. The two cases are named at every use in this ladder for exactly that reason.

The arithmetic, which is one solve #

Set 20.19·E·(πd⁴/64)/ℓ² equal to the threshold force and solve for d:

d⁴ = 64 P ℓ² / (20.19 E π)

with E a wool fibre’s modulus — three gigapascals, from this site’s own fibre mechanics table, quoted with its range — and P the measured nociceptor threshold.

At a two-millimetre protrusion the answer is thirty-two micrometres, bracketed at twenty-nine to thirty-four by the range of the measured force alone.

Two independent numbers arrive at once and neither was chosen. The thirty-two is the trade’s thirty. And the two millimetres is where this site’s own hair model puts the long end of a wool population: the decay length is fifty-six fibre diameters, which for a twenty-two micrometre wool is 1.2 millimetres, so two millimetres is the top fifth or so of the hairs.

The fourth power, which is why the line is sharp #

The load goes as d⁴ and that is asserted rather than remarked on: over a fifth more diameter the load goes up by 1.85, which the model checks to twelve figures as an exponent rather than as a ratio.

The consequence is that the threshold is a step rather than a slope. A twenty-five micrometre fibre carries under half the load a thirty carries; a thirty-five carries nearly double. So across a ten-micrometre band the buckling force runs over a factor of four, and a population’s members are cleanly sorted into those that prickle and those that do not.

That is what makes a percentage-over-a-threshold the right specification. If the response had been gradual, a mean diameter would have been the natural number to quote. It is not gradual, so what matters is how many fibres are over the line — which is a tail of the distribution, and a tail moves violently for small movements of the mean.

The tail, computed #

Take a wool with a mean diameter of nineteen micrometres and a coefficient of variation of a quarter, which is an ordinary merino. Just over one per cent of its fibres are over the threshold.

Take the same wool four micrometres coarser — twenty-three, still a fine wool by most standards — and seven per cent are over. A four-micrometre move in the mean has moved the prickling share by a factor of six.

That is the whole reason wool is graded so finely and priced so steeply against diameter, and it is a property of a Gaussian tail rather than of wool. A cloth is a population and not a thread is the essay that set this site up to notice it: a quantity linear in the diameter is unbiased at any spread, and a quantity that is a threshold is not a quantity of the mean at all.

And it says the spread matters as much as the mean. Two wools of identical mean diameter and different variability have different prickling shares, and the more even one is comfortable at a coarser mean.

How many fibres are actually in contact #

A wearer does not meet one fibre; they meet whatever the garment presents at the places it touches, and the count is small enough to be worth stating.

The hair population puts about one and a third hairs on every square millimetre of an ordinary woven cloth, and the ones that reach two millimetres are a fifth of them. A forearm resting on a sleeve is perhaps a hundred square centimetres of contact, so of the order of a few thousand long hairs are in the region — and of those, the prickling share is the fraction of the wool coarser than the threshold.

At one per cent that is tens of fibres; at seven per cent it is hundreds. The sensation is a count of independent events rather than an intensity, which is exactly the character prickle has: it is felt as discrete points rather than as a general roughness, and it comes and goes as a garment moves.

That also explains why the specification is a percentage rather than a maximum. A single very coarse fibre is not a problem; a few hundred of them are. The response is a rate of events and the rate is proportional to the tail’s share.

Why a finer fibre is comfortable and a smoother yarn is not #

There is a second lever and the arithmetic says it is much weaker, which is worth knowing because it is the one that looks obvious.

The threshold diameter goes as the square root of the protrusion length, so halving the hairs’ reach raises the threshold diameter by only forty per cent. A singeing, a cropping or a compact spinning frame moves the length and the count; none of them moves the diameter, and the diameter is the fourth power.

So a comfortable wool garment is made by buying finer wool and not by finishing. That is what the trade does, at considerable expense, and the arithmetic says the expense is unavoidable.

The one exception is a fabric that keeps its hairs away from skin altogether — a lining, a double-faced construction, or a fine fibre on the inside and a coarse one on the outside. All three are used and all three are geometry rather than materials.

Why cotton does not prickle and a coarse wool does #

The comparison across fibres is the model’s cleanest corroboration and it takes one line.

The threshold diameter is a property of the modulus and the force, and it is the same for any fibre at a given protrusion length: about thirty micrometres for a modulus of three gigapascals. What differs between fibres is where their diameters sit relative to it.

A cotton fibre is twelve micrometres and its modulus is nearly three times wool’s, which raises its buckling load by the same factor — so its effective threshold in diameter terms is lower, at about twenty-four micrometres, and a twelve-micrometre cotton is a factor of sixteen in load below it. Cotton cannot prickle, and the margin is enormous.

A coarse wool at thirty-five micrometres is above the line. A carpet wool at forty is far above it, which is why carpet wools are not made into garments and why the distinction between apparel and carpet wool is a diameter and not a variety.

And a coarse synthetic staple prickles exactly like a coarse wool, which the trade knows and which disposes of the folk explanation that wool prickles because of its scales. The scales are what make it felt; they have nothing to do with this.

What was counted, and how #

Three assertions, and the first is the essay’s whole claim.

That the threshold diameter for wool at a two-millimetre protrusion lands between twenty-five and thirty-six micrometres. That range is wide enough to be passed by a model that was roughly right and narrow enough to be failed by one that had the wrong end condition, the wrong modulus by a factor of two, or the wrong power.

That four micrometres on the mean moves the prickling share by more than threefold, which is the statement that the specification has to be a tail rather than a mean.

And that the load goes as the exact fourth power of the diameter, checked as an exponent to twelve figures — which would break if the second moment of area were ever computed for anything but a circle.

Where the model stops #

The protrusion length is an input and the answer goes as its square root. The two millimetres is justified by the hair model rather than measured, and the essay’s headline number moves with it: at one millimetre the threshold is twenty-two micrometres and at three it is thirty-nine.

Skin is not modelled. The nociceptor threshold is a measurement made on people by other people and is the only number in this collection that is about a person. It is quoted with its range and every result is quoted at both ends of it.

A fibre is taken as a straight, uniform, isotropic cylinder. A wool fibre is crimped, elliptical in section and structurally anisotropic; the crimp lowers the buckling load, the ellipse means the second moment depends on which way it is bent, and neither is here.

And nothing is wet. Wool’s modulus falls by more than half when it is wet, so the buckling load falls with it and the threshold diameter rises — which predicts that a damp garment prickles less, and that is a testable statement the model can make and this collection cannot check.

What a garment can do that a fibre cannot #

The essay so far is a counsel of buying finer wool, and there is one structural escape worth pricing because it is the one every fine knitwear maker uses.

The threshold’s dependence on the protrusion length is a square root, so length is a weak lever on the diameter threshold. But the number of fibres that reach skin at all is an exponential in the length, and that is a strong lever on the share.

Cropping a fabric — shearing its surface mechanically, at a height well above what a flame reaches — truncates the population at the cropping height. Everything longer than that is gone. So a cropped fabric’s longest hairs are the cropping height rather than the exponential’s tail, and the threshold diameter is evaluated at that height instead.

Cropping at one millimetre rather than two raises the threshold diameter from thirty-two micrometres to twenty-two — which is the wrong direction. A shorter hair is a stiffer column and prickles at a finer diameter.

That is a genuinely counter-intuitive result and it is the one place in this essay where the arithmetic contradicts a plausible intervention. Cutting the hairs shorter makes them worse. What saves a cropped fabric is that there are far fewer of them reaching skin at all, and the model cannot weigh the two effects against each other because it has no model of how many contacts a wearer makes.

The generalisation #

A sensation with a threshold in it is a measurement of a tail, and a tail is a different statistic from a mean.

The transferable form is that whenever a response is a step function of some per-item quantity, the population’s mean is nearly irrelevant and the fraction over the step is everything — and the fraction over a step is exquisitely sensitive to both the mean and the spread. A specification written as a mean will fail to predict; a specification written as a percentage over a threshold will work, and will look arbitrary to anyone who has not found the step.

This collection has now met the pattern three times: a bundle breaking at its weakest fibre, a pill surviving on its strongest anchor, and a garment prickling on its coarsest few. Every one of them is an extreme of a small sample and every one is invisible to an average.

What the model would have to get wrong to be a coincidence #

The agreement is close enough to be worth attacking, because a model with three inputs landing on a two-figure trade number could easily be an accident.

There are exactly three inputs: the fibre’s modulus, the nociceptor threshold force, and the protrusion length. The modulus is a table entry from this site’s own fibre mechanics, quoted with a range of two to four and a half gigapascals, and taken at three. The force is a measurement with a range of a half to one millinewton. The length comes from this site’s hair model and is not adjustable once the model is fixed.

Move any one of them to the end of its range and the threshold moves by a quarter or less, because the fourth root of everything is a very forgiving function. That is the honest reason the agreement is not impressive by itself: a fourth root compresses errors, and a model wrong by a factor of two in the force is wrong by nineteen per cent in the diameter.

What is not compressed is the end condition, which is a factor of eight in the load and therefore a factor of 1.7 in the diameter — enough to be decisive. And what is not compressed is the length’s role in the share, which is exponential.

So the right reading is that the arithmetic confirms the mechanism rather than the number: a column formula with plausible inputs lands where the measurements land, and no other simple mechanism does. A strength-based account would put the threshold ten times higher; a bending-stiffness account with a free tip would put it half as high.

Who found it, and when #

The prickle threshold is Garnsworthy, Kenins and Mayfield’s, from the 1980s at CSIRO: single fibres presented to skin, the force at which subjects reported prickle rather than touch, and the identification of the mechanism as nociceptor rather than as irritation. The thirty-micrometre line and the percentage-over-thirty specification follow from that work and are the wool trade’s own.

The buckling calculation is elementary and is stated in that literature as the mechanism. What does not appear to have been done is the joining: solving the column formula for the diameter, supplying the protrusion length from a hair population rather than assuming it, and showing that the trade’s number falls out of the two with nothing fitted.

Where the ladder goes next #

Prickle is the one place in this ladder where the hair layer meets a person rather than an instrument, and it is where the model’s weakest input — the protrusion length — carries the most weight. Measuring a fabric’s own hair length distribution rather than its yarn’s is the obvious next step and it is the same measurement the knit essay needs for a different reason.

Sideways, the same population doing something nobody minds: a woven filter beats its own rating, where the hairs catch particles far below the cloth’s own opening.