What a cuff presses with

A cuff has to grip. That is its whole job: it goes over a hand, closes on a wrist, and stays there while the sleeve above it does not. So it must be pressing on the wrist with something, and the something ought to be computable, because everything a knitted fabric pulls back with has been computable on this ladder for several rungs.

The number is much smaller than anybody would guess, and the reason it is small is the reason a cuff is comfortable.

The arithmetic #

A band wrapped round a limb is a hoop under tension, and a hoop under tension presses inwards with a pressure equal to its tension per unit length divided by its radius. That is Laplace’s law and it is exact for a thin band.

The tension comes from the fabric’s own load–extension curve, which this collection computes from the loop’s bending with the relaxed shape as the yarn’s natural one. At a doubling of its width — which is roughly what a cuff does to admit a hand — a 20 tex cotton knitted at a 3.5 mm loop pulls back with 3.4 newtons per metre of band height.

A wrist is about thirty millimetres in radius. Divide:

113 pascals, which is 0.85 millimetres of mercury.

It is worth being explicit about which fabric that curve belongs to: single jersey, because that is the fabric this collection solves. A rib gets most of its first hundred per cent of width from folding rather than from the loop reconfiguring, which costs nothing at all — so a real rib cuff at a doubling of its width pulls back with less than the figure above, not more.

That makes the number a ceiling rather than an estimate, which is the useful direction for the argument being made.

What that is next to #

Compression garments are specified in millimetres of mercury and the classes are not close to this.

A cuff is a fiftieth of the lightest thing anybody calls compression, and a twenty-fifth of ordinary support hosiery. It is not a weak compression garment; it is not a compression garment at all.

The gap is a fact about elastane rather than about the rib. A medical stocking gets its pressure from a filament that is still pulling at three times its own length; a rib gets its pressure from a loop straightening, and a loop runs out of shape to give long before it runs out of extension. So the two are not the same mechanism at different strengths. They are different mechanisms whose outputs happen to be quoted in the same unit, and the unit is what makes the comparison look like a ranking.

Which is why the comparison is worth drawing anyway. A number with no scale beside it invites the reader to supply one, and the scale a reader supplies for “pressure from a garment” is the medical one, because that is the only one with published figures. Putting the two on one axis is the fastest way to establish that a cuff’s job is not compression at all: it is to stay where it is put, and a fiftieth of the lightest medical class is ample for that, because the thing it has to beat is the weight of a sleeve.

Why the wrist and not the arm #

The radius matters as much as the fabric and it is worth fixing the one used above, because a band’s pressure is not a property of the band.

Thirty millimetres is a wrist measured across the narrow way, which is where a cuff sits and is the smallest radius an arm offers. A forearm a hand’s breadth further up is nearer forty, and a bicep nearer fifty — so the same band held at the same stretch there would press two thirds and half as hard.

The same band slid up the arm is not at the same stretch. A thicker limb opens it further, and a knitted band’s tension rises far faster than its stretch, so a tube of one size presses the thicker part of a limb harder, not the thinner. A cuff is at the wrist because that is where a sleeve ends, and the pressure above is simply what it presses there.

Which is the answer to the question, not a failure to answer it #

The instinct is to check the arithmetic, and the arithmetic is worth checking. But the conclusion is right and it is what a cuff is for.

A cuff has to grip a wrist without marking it, without impeding the blood in it, and without becoming uncomfortable over a day. Every one of those requires the pressure to be low. What a cuff needs is not force but friction against a surface it is lightly held to, and a light hold with a large contact area is exactly what eight tenths of a millimetre of mercury over the whole circumference of a wrist provides.

A garment that gripped at twenty millimetres of mercury would be a medical device and would be uncomfortable to wear for pleasure, which is why medical stockings are prescribed rather than bought casually.

Where a compression garment gets its pressure #

The curve on this page explains it, and the explanation is structural rather than material.

A knitted fabric’s load–extension curve is soft for a very long way and then abruptly stiff. The soft part is the loop reconfiguring at constant yarn length; the stiff part is the yarn running out of slack, where the straight line between two interlacings reaches the thread between them.

A cuff works in the soft part, at extensions between a fifth and a doubling. The pressure there is under a millimetre of mercury throughout.

A compression garment has to work in the stiff part. Following the curve up: 1.2 mmHg at 138 per cent extension, 2.2 at 185, 4.9 at 231, and 28.9 at 277 — where the fabric is very nearly at its geometric limit. Class I is only reached at two hundred and seventy-seven per cent extension.

So a compression garment is not a knitted fabric stretched further #

Nobody makes a compression stocking by knitting an ordinary rib and pulling it very hard, and this is why. At the extension where the pressure would be right, the fabric is at the edge of its own geometry: its yarn is nearly straight between interlacings, it has almost no give left, and a limb that swelled by a further two per cent would be squeezed by a great deal more.

What compression garments actually do is change the curve. They knit in an elastomeric yarn — a bare or covered elastane — whose own tension supplies the hoop force, and the knitted structure is there to carry it and to spread it. The loop’s bending stops being the mechanism.

That is why this rung’s number is worth having even though it describes no compression garment: it says what a fabric can supply without an elastomer, and the answer is a fiftieth of what is needed. The elastane is not an improvement to the mechanism; it is the mechanism.

Why the soft part is soft #

The shape of the curve is not an accident of this fabric and it is worth naming its cause, because it is the reason a knitted band is the right thing for the job at all.

Extending a knit at constant yarn length does not stretch anything. It reconfigures the loop: the wale spacing grows, the course spacing shrinks, the yarn redistributes its bending between crowns and free run, and the total length of thread involved does not change by a part in a thousand. That costs a bending energy and bending is cheap.

Extending it past the point where the straight line between two interlacings reaches the thread between them is a different operation entirely, and it costs the yarn’s tensile modulus, which is four and a half decades stiffer.

So the curve is not one mechanism getting harder. It is two mechanisms in sequence, and the corner between them is geometric.

The comparison that makes it legible #

There is a second pressure in the same fabric and it is two orders of magnitude larger, which is the clearest way to see how small the grip is.

The same jersey resists being squashed with about fifteen kilopascals — 113 millimetres of mercury — over the area a stitch occupies. That is the through-thickness component of its own contact force, and it is what stops the fabric’s two faces closing under a finger.

So the fabric presses the wrist with 0.85 mmHg and resists a finger with 113. A knitted fabric is a hundred and thirty times stiffer through its thickness than it is round a limb, and the reason is the mechanism: through the thickness a loop is being bent harder, and round the limb it is only being reconfigured.

Both numbers come out of the same solved shape, differentiated in different directions.

Why the radius matters as much as the fabric #

The pressure is a tension over a radius, so the same band at the same stretch gives different pressures on different parts of a body.

At a doubling, on a thirty-millimetre wrist, 0.85 mmHg. On a fifty-millimetre calf at the same doubling, half that. On a ten-millimetre finger — a knitted glove finger — three times as much, at 2.5 mmHg.

A garment is not at the same stretch everywhere, and that reverses the direction on a leg. A tube knitted to one size is stretched further wherever the limb is thicker, so it presses the calf harder than the ankle, and a compression stocking’s graduated specification, higher at the ankle than at the calf, has to be knitted into the stocking’s shape rather than left to the leg.

None of that is a property of the fabric. It is the geometry a fabric is put on, and it means a band’s specification is meaningless without the radius it was written for.

What the number depends on #

Three things, and the ordering is instructive.

The extension, hugely. The curve spans a factor of five hundred between rest and the jam, so a band knitted two per cent smaller and one knitted twenty per cent smaller are different garments.

The loop length, strongly. The fabric’s initial stiffness is a bending rigidity over a length cubed, and the length is the loop’s — so a tighter knit is stiffer and grips harder at the same extension.

The fibre, weakly and only through the bending stiffness, which is a bracket a hundred and thirty wide anyway.

The dominance of the first is why a cuff’s specification is a dimension rather than a fabric. A knitter setting a cuff chooses how many stitches fewer than the sleeve, and that choice is worth more than any change to the yarn.

The bracket, which does not cancel here #

Every force in this account is a yarn’s bending stiffness over a length squared, and a spun yarn’s bending stiffness is a band running from the case where its fibres slide freely to the case where they cannot. The two ends differ by a factor of a hundred and thirty.

The numbers above are at the free end, which is the lower one, so the pressures are lower bounds. At the coherent end a cuff would press a hundred and thirty times harder, which would put it at 110 mmHg and is plainly absurd.

That absurdity is itself informative: it is evidence that a spun yarn in a relaxed knit is near the free end of its bracket, which is what this collection’s own comparison against measured fabric rigidities already concluded. A cuff that pressed like a blood-pressure cuff would have been noticed.

What friction has to do #

A pressure of 0.85 millimetres of mercury holds a sleeve up because of friction, and it is worth checking that it can.

The band’s normal force on the wrist is the pressure times the contact area. For a forty-millimetre-tall cuff on a thirty-millimetre-radius wrist that is about 7,500 square millimetres of contact, so the total normal force is about 0.85 newtons. At a coefficient of friction between fabric and skin of around 0.4 — this collection’s own yarn-on-yarn figures are 0.2 to 0.5, and skin is not far off — the cuff can resist about 0.34 newtons of pull before it slides.

A sleeve weighs rather less than that. So the arithmetic closes, with a margin of a few times, and the mechanism is friction rather than compression exactly as it appears to be.

What this does not settle #

Anything about an elastomeric fabric. The moment an elastane is knitted in, the tension is the elastane’s and the loop’s bending is a rounding error.

A rib’s own curve. The extension curve here is single jersey’s, because that is the curve this collection computes. A rib gets most of its width from folding rather than from the loop reconfiguring, so its first hundred per cent of extension is even softer than this and its recovery force is at most this.

Hysteresis. A knitted fabric does not come back along the curve it went out on, because friction at the interlacings dissipates. The pressure on the way off is lower than the pressure on the way on, and none of that is in this model.

And the body. A limb is not a rigid cylinder; it is compliant, and a band pressing on it deforms it, which changes the radius, which changes the pressure. For pressures this small the correction is negligible; for a compression garment it is not.

Why this is a new anchor rather than a rung on an old one #

Pressure on a body is a different question from any of the forces this collection has computed, and the difference is not one of size.

Every other force here is internal: a thread on a thread, a crossing on a crossing, a fabric on itself. This one is a fabric on something that is not fabric, and the quantity that carries it — a tension divided by a radius — belongs to the shape the fabric is wrapped round rather than to the fabric.

That is a class of question with its own arithmetic, and a garment is full of it: what a waistband holds up, what a bandage occludes, what a sock top marks a leg with, what a compression sleeve does to a swelling. All of them are a hoop tension over a radius, and all of them need the fabric’s curve and the body’s geometry together.

What is genuinely new here #

Two numbers and one contrast.

A cuff presses at 0.85 millimetres of mercury, computed from the loop’s own bending with nothing fitted, at a doubling of its width on a wrist.

Class I compression is at 277 per cent extension, which is not a garment anybody would wear — so a compression garment is not this fabric used harder, it is a different mechanism.

And the same fabric resists being squashed with 133 times the pressure it grips with. Both from one solved shape, differentiated two ways.

What the pictures cannot show #

The compression classes are drawn as horizontal bands, which makes them look like targets a fabric could be designed onto. They are specifications for garments made of something else, and the curve on the same axes has no elastane in it at all.

Nor does the curve show hysteresis, which means the picture describes a band being put on and not one being taken off. That is not a small omission for this quantity: the loop model is elastic and reversible by construction, and a real rib pulled over a wrist and released comes back along a lower curve because the yarn has slipped at its crossings. The pressure a cuff exerts after an hour of wear is therefore below the figure computed here, and the collection has no way to say by how much — friction at the crossings is exactly what it does not model.

And nothing in the picture shows the area the pressure acts over. Pressure is force over area and the area here is the band’s own width, so a wide cuff at the same extension presses no harder than a narrow one — the force rises with the width and the area rises with it too. That is the right answer and it is counter-intuitive enough to be worth stating, because a doubled band feels tighter and the reason is that it is harder to stretch in the first place, not that it presses harder once stretched.

What is worth taking away #

A knitted band at a doubling of its width presses a wrist with under a millimetre of mercury — a fiftieth of the lightest medical compression — and the same fabric resists being squashed with a hundred and thirty times that pressure.

Both numbers come out of one solved loop differentiated in two directions, and the contrast between them is the whole account of why a cuff grips comfortably and a compression garment needs an elastane.

The three things that would change the number #

A smaller radius. The pressure is a tension over a radius, so a glove finger at ten millimetres presses three times as hard as a wrist at thirty, from the same fabric at the same extension.

A tighter fabric. The initial stiffness is a bending rigidity over a length cubed, so shortening the loop raises the tension steeply at a given extension.

And an elastomer, which replaces the mechanism rather than improving it.

Nothing else moves it much. The fibre enters only through a bending stiffness that is a wide bracket already, and the structure — rib against jersey — moves it downwards rather than up, because a rib’s first hundred per cent of width is folding rather than loop reconfiguration.

Which rungs this stands on #

The load–extension curve, at what a knit gives when it is pulled, which supplies the tension and is itself computed from the loop’s bending with the relaxed shape as the yarn’s natural one.

The rib’s fold, at a rib pulls back on a force the loop supplies, which is why a cuff’s first hundred per cent costs even less than the curve says.

And the through-thickness force, at the force that holds a knit open, which supplies the contrast that makes 0.85 millimetres of mercury legible as a number.

Laplace’s law is the only thing added, and it is one line.

Where the ladder goes next #

A fabric that presses a body is also a fabric a body presses, and the through-thickness force is the other half of that exchange: what a knit gives up when it is pressed.