Textile Structure
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A tube of one size presses the calf harder than the ankle

A band presses a limb with its tension over the limb's radius, so it is easy to conclude that a band grips hardest where the limb is thinnest. That is true of a band held at one tension, and no knitted tube is. A tube knitted to one size is stretched further wherever the leg is thicker, and its tension rises faster than the radius does: an elastic tube pressing an ankle at twenty millimetres of mercury presses the calf at thirty-nine, and a cotton jersey tube presses its calf three and a half times as hard as its ankle. A stocking graduated the other way has to pull hardest where it presses less.
16 min read 6 figures Measured, not claimedA mechanism, not a materialStructure before fibre

Worth reading first: What a cuff presses with · What a knit gives when it is pulled · An inflated cylinder wants an unbalanced cloth.

What a cuff presses with is a hoop tension over a radius, and the calculation gave a knitted band 0.85 millimetres of mercury on a wrist. It then drew a rule from the radius alone: the same band presses a thicker limb less, so a band grips best where the limb is thinnest, and a garment knitted to one tension would press hardest at the ankle.

The arithmetic behind the rule is right and the rule is not, because it holds the tension fixed. No knitted tube is at one tension. A tube is knitted to one size, a leg is not one size, and wherever the leg is thicker the tube is stretched further and pulls back harder. Whether the extra pull outweighs the larger radius is a question with an exact answer, and for every band a stocking could be made of the answer is that it does.

So a tube of one size presses the calf harder than the ankle, and a stocking that presses the ankle hardest has to be shaped to do it. That turns out to mean something stranger than a narrower ankle: a graduated stocking must pull its calf harder than its ankle while pressing it less.

The pressure a tube puts on one station of a leg

Laplace’s law is the whole of the mechanics, exactly as it was for a cuff. A band wrapped round a limb presses inwards with its tension per unit height divided by the limb’s radius.

What changes along a leg is both terms. The radius is the leg’s own, a circumference over two pi. The tension is the band’s at the stretch it is at, and the stretch is the leg’s circumference over the tube’s knitted circumference, less one. A leg twice as far round at the calf as the tube is knitted opens it by a hundred per cent there; the same tube at an ankle only a little larger than itself is barely stretched.

Take an illustrative leg: twenty-two centimetres round at the ankle, twenty-nine at the lower calf, thirty-six at the calf and thirty-four below the knee. The figures are round numbers for one leg, not a survey, and every result below either compares two of them or holds for any leg that is thicker at the calf than at the ankle.

For the band, take an elastic one that pulls back in proportion to its stretch, sized so that a tube knitted 157 millimetres round — stretched forty per cent over the ankle — presses the ankle at twenty millimetres of mercury, inside the first medical compression class. That fixes its stiffness at 2.33 newtons per centimetre of height for each unit of stretch, and nothing else about it is assumed.

An elastic tube of one size, up the leg

Pull that tube up the leg and read the pressure at each station.

A tube of an elastic band knitted 157 mm round, up one leg. A tube knitted 157 mm round from an elastic band, pulled up an illustrative leg and read at four stations. At each the tube is stretched by the leg's circumference over its own, pulls back with the band's tension at that stretch, and presses with that tension over the leg's radius: ankle 22 cm, stretched 40%, 20.0 mmHg; lower calf 29 cm, stretched 85%, 32.1 mmHg; calf 36 cm, stretched 129%, 39.4 mmHg; below the knee 34 cm, stretched 116%, 37.6 mmHg. The calf is pressed 1.97 times as hard as the ankle. What the bars cannot show is the leg's own give, which a firm tube flattens and which changes the radius the law divides by.
Fig. 1 An elastic tube knitted 157 mm round, pressing 20 mmHg at a 22 cm ankle, read up an illustrative leg. At the lower calf it is stretched 85% and presses 32.1 mmHg; at the 36 cm calf, stretched 129%, it presses 39.4; below the knee, 37.6. The calf is pressed nearly twice as hard as the ankle.

At the lower calf the tube is stretched 85 per cent, pulls with 1.97 newtons per centimetre and presses 32.1 millimetres of mercury. At the calf it is stretched 129 per cent, pulls with 3.01 newtons and presses 39.4. Below the knee, a little narrower again, 37.6.

The calf is pressed 1.97 times as hard as the ankle. The calf’s radius is 1.64 times the ankle’s, which on its own would cut the pressure to 61 per cent; the calf’s tension is 3.23 times the ankle’s, because the tube is stretched more than three times as far there. The tension wins by a factor of two.

A knitted tube does it more steeply

The elastic band pulls in proportion to its stretch. A knit does not: what a knit gives when it is pulled is soft for a long way and then abruptly stiff, as its loops run out of shape and start straightening the yarn itself.

Take a tube of the same cotton jersey the cuff was computed for — 20 tex at a 3.5-millimetre loop — knitted 110 millimetres round, so that it is opened to double its width at the ankle, the working stretch of a cuff.

A tube of a cotton jersey knitted 110 mm round, up one leg. A tube knitted 110 mm round from a cotton jersey, pulled up an illustrative leg and read at four stations. At each the tube is stretched by the leg's circumference over its own, pulls back with the band's tension at that stretch, and presses with that tension over the leg's radius: ankle 22 cm, stretched 100%, 0.695 mmHg; lower calf 29 cm, stretched 164%, 1.08 mmHg; calf 36 cm, stretched 227%, 2.39 mmHg; below the knee 34 cm, stretched 209%, 1.77 mmHg. The calf is pressed 3.45 times as hard as the ankle. What the bars cannot show is the leg's own give, which a firm tube flattens and which changes the radius the law divides by.
Fig. 2 A cotton jersey tube knitted 110 mm round, doubled at the ankle, up the same leg. It presses 0.695 mmHg at the ankle, 1.08 at the lower calf, 2.39 at the calf stretched 227%, and 1.77 below the knee: the calf 3.4 times the ankle, because the jersey is well into the stiff part of its curve there.

It presses 0.695 millimetres of mercury at the ankle, the cuff’s order of magnitude. At the calf it is stretched 227 per cent, well into the stiff part of its curve, and it presses 2.39 — 3.4 times the ankle. The absolute pressures are a knit’s, far below any compression class, as the cuff essay found. The direction is the elastic tube’s, and steeper.

So the rule the cuff essay drew points the wrong way for the tube in both fabrics. It would take a band of a third kind to make it right.

The only band the rule describes

That third kind is a band that pulls with the same tension however far it is stretched.

Such a band, at a newton per centimetre, presses the ankle at 21.4 millimetres of mercury, the calf at 13.1 and below the knee at 13.9. Its pressure falls exactly as one over the circumference, and it does press the ankle hardest. It is the band the rule describes and the only one: a band held at one tension is not an elastic band at all, but something like a strap drawn through a buckle, or a weight hung on a cord.

Pressure against the limb's circumference for three bands of one knitted size each. For three tubes, each of one knitted circumference, the pressure on a limb against the limb's circumference, as a multiple of the pressure at a 220 mm ankle, on a logarithmic scale. elastic band: 1.97 times at a 360 mm calf; cotton jersey: 3.4 times at a 360 mm calf; equal tension at any stretch: 0.61 times at a 360 mm calf. Only the band pulling equally at every stretch presses the thinner limb harder, falling as one over the circumference; an elastic band rises and the jersey, whose tension stiffens, rises far faster. What the curves cannot show is where each band's own pressures sit, which differ by orders of magnitude and are divided out here.
Fig. 3 Pressure against limb circumference for one knitted size of each band, as a multiple of its pressure at a 220 mm ankle, on a logarithmic scale. A band at equal tension falls to 0.61 of its ankle pressure by a 360 mm calf; the elastic tube rises to 1.97 times; the jersey to 3.4 times and on towards ten as it approaches its jam.

On one axis the three separate cleanly. Relative to its own pressure at a 22-centimetre ankle, the equal-tension band falls to 0.61 at the calf, the elastic tube rises to 1.97, and the jersey rises to 3.4 and turns sharply upwards as it nears the stretch at which its loops jam. The cuff essay’s rule is the bottom line on this chart, and no knitted fabric lies on it.

The steepness that decides it

There is one number that sorts every band, and it is worth having because real elastomers do not all pull in simple proportion.

At a fixed knitted size the pressure goes as the tension over the circumference, and the stretch ratio goes as the circumference. So the pressure’s sensitivity to the limb’s circumference is the tension’s sensitivity to the stretch ratio, less one — both measured as proportional changes. Where a one per cent larger stretch ratio raises the tension by more than one per cent, a thicker limb is pressed harder; where by less, a thinner one.

A band that pulls in proportion to its stretch has that sensitivity equal to one plus the stretch, over the stretch: 3.5 at forty per cent, 2 at a doubling, 1.5 at a tripling. It approaches one from above and never reaches it. The jersey’s is above three at forty per cent, has its lowest value, a little over two, near a doubling, and climbs past seven as the yarn begins to straighten. The equal-tension band’s is nought.

Whether a tube presses a thicker limb harder: the steepness of its tension. The elasticity of a band's tension with respect to its stretch ratio, d ln T / d ln(1 + ε), against the stretch, for an elastic band and for a cotton jersey. A tube of one knitted size presses a thicker limb harder exactly where this is above one. The elastic band's is (1 + ε)/ε, above one at every stretch; the jersey's is 3.6 at forty per cent, 2.3 at a doubling and more beyond, drawn to a cap of 8. A band pulling equally at every stretch would lie on the axis at nought. What the plot cannot show is an elastomer with a plateau in its curve, whose elasticity could fall below one over part of its range.
Fig. 4 The proportional sensitivity of each band’s tension to its stretch ratio, against stretch. Wherever it is above the dashed line at one, a tube of one size presses a thicker limb harder. The elastic band’s falls towards one and never reaches it; the jersey’s is lowest, a little over two, near a doubling and climbs steeply towards its jam; a band at equal tension would lie at nought.

The criterion also names what would reverse the result. An elastomer with a flat stretch of curve — a range where more stretch buys almost no more tension — would press a thinner limb harder while it was working across that range. Neither band here has one, and whether a particular covered elastane does is a measurement, not something this arithmetic supplies.

A graduated tube presses the calf less and pulls it harder

A medical compression stocking is specified the other way round from what a tube of one size does: highest at the ankle, lower up the leg. Laplace’s law says what that costs, and the answer does not depend on the band.

The tension a station needs is its pressure times its radius. So a station’s tension against the ankle’s is its pressure’s fraction of the ankle’s, times its circumference over the ankle’s. Graduate the tube to press 85 per cent of the ankle’s twenty millimetres at the lower calf, 70 at the calf and 60 below the knee — an illustrative graduation, not a standard’s — and the tensions come out at 1.12, 1.15 and 0.93 times the ankle’s.

The tension a graduated tube needs up the leg, against the ankle's. A tube graduated to press 100%, 85%, 70%, 60% of the ankle's 20 mmHg at the ankle, lower calf, calf and below the knee of the illustrative leg, with the tension each station needs as a multiple of the ankle's. Laplace's law asks for pressure times radius, so ankle 1.00 times, lower calf 1.12 times, calf 1.15 times, below the knee 0.93 times: the calf, pressed at 70% of the ankle, is pulled 1.15 times as hard and stretched 46% against the ankle's 40%. What the bars cannot show is how the tube is knitted to those circumferences, which is a question about the machine.
Fig. 5 The tension each station of a graduated elastic tube needs, as a multiple of the ankle’s, for a graduation of 100, 85, 70 and 60 per cent of 20 mmHg. The lower calf and the calf, pressed less than the ankle, need 1.12 and 1.15 times its tension and are stretched further; only below the knee does the tension fall below the ankle’s. The tube is knitted 157, 200, 247 and 248 mm round.

The calf is pressed at seven tenths of the ankle and pulled harder than it, stretched 46 per cent against the ankle’s 40. The tube that does this is knitted 157 millimetres round at the ankle, 200 at the lower calf, 247 at the calf and 248 below the knee — an almost cylindrical tube from the calf upwards, and a strongly conical one below.

That is a surprise about garments rather than about fabric. The obvious picture of graduated compression is a stocking that squeezes the ankle tight and relaxes up the leg. The fabric does not relax up the leg: over the lower leg it works harder the higher it goes, and the graduation is carried entirely by the leg getting wider faster than the pull increases.

The break-even is the ratio of the leg’s own circumferences

Where the calf’s tension equals the ankle’s is fixed by the leg and nothing else. The calf needs less pull than the ankle only if its pressure is a smaller fraction of the ankle’s than the ankle’s circumference is of the calf’s.

The calf's tension against the steepness of the graduation. How hard a tube must pull at a 360 mm calf against a 220 mm ankle, as the calf's pressure runs from 30 to 100 per cent of the ankle's. The line is the pressure fraction times 1.636, the ratio of the circumferences, so the calf needs less tension than the ankle only below 61.1%; at 70% it needs 1.15 times as much, and a tube pressing both equally needs 1.64 times. What the line cannot show is any leg but this one, and the break-even is that leg's own ratio of circumferences.
Fig. 6 The calf’s tension against the ankle’s, against how steeply its pressure is graduated, for a 22 cm ankle and a 36 cm calf. The line is the pressure fraction times 1.64, so it crosses equal tension at 61.1%; a graduation to 70% needs 1.15 times the ankle’s tension, and a tube pressing both equally needs 1.64 times.

On this leg that ratio is 61.1 per cent. A calf graduated to seven tenths is above it and pulled 1.15 times as hard; a calf pressed equally with the ankle would need 1.64 times the tension; and only a graduation steeper than 61 per cent lets the calf pull less than the ankle. A leg with a slender ankle and a heavy calf has a lower break-even, so on that leg a gentle graduation costs more pull, not less.

A stocking is a pressure vessel turned inside out

The same law has already appeared from the other side. An inflated cylinder wants an unbalanced cloth because a pressure inside a tube is carried as a hoop tension of pressure times radius — the same product, with the pressure pushing out rather than in — and the angle a hose wants is that hoop tension and its axial partner resolved into a single braided direction.

A vessel of changing radius under one internal pressure therefore needs the most hoop tension where it is widest, which is why a long balloon’s wall is under the greatest tension at its fattest part. A graduated stocking is that vessel turned inside out: it applies a pressure rather than containing one, and it applies nearly the same pressure along a limb whose radius changes, so its hoop tension has to follow the radius exactly as the balloon’s does. The only freedom a stocking has that a balloon does not is to let the pressure fall, and until it falls faster than the radius grows the tension goes up the leg with the leg.

Why a tube pressed hardest at the calf stays up

A band pressing a cone harder at its wide end might be expected to slide towards the narrow one. It is pushed that way, and it does not move.

The pressure on a tapering surface has a component along the taper, towards the narrow end, equal to the normal force times the slope of the surface. Friction resists with the normal force times its coefficient. So a band on a taper holds still whenever the slope of the radius along the limb is smaller than the friction coefficient, whatever the pressure. A leg widening from 22 to 36 centimetres round over an illustrative thirty centimetres of height has a radius slope of 0.074; the cuff essay took fabric on skin at about 0.4. The taper is a fifth of what it would take, which is why a stocking stays where it is put until walking works it down, and why that working-down is a question about movement and not about pressure.

What the number depends on, for a tube

For a cuff, the extension dominated, then the loop length, then the fibre. For a tube on a leg one more quantity enters ahead of all of them: the leg’s own taper, because the ratio of the calf’s pressure to the ankle’s is set by how much further round the calf is than the tube is knitted.

For the elastic tube, knitting it smaller raises every pressure and flattens the ratio towards the calf, because the stretches at ankle and calf draw proportionally closer; knitting it only a little smaller than the ankle makes the ankle nearly slack and the ratio very large. For the jersey, the stretch at the calf decides where on its curve the calf sits, and a tube knitted so that the calf reaches the stiff part presses the calf enormously harder than an ankle still in the soft one. The tube’s knitted shape is the specification, as the cuff essay found a cuff’s knitted size was, and a pressure quoted without the circumferences it was read at cannot be reproduced.

A rib does not rescue the rule either. A rib pulls back on a force the loop supplies and takes its first stretch by folding, which costs almost nothing, so a rib tube’s tension stays near zero longer and then rises. A curve that starts flatter and stiffens later is more sensitive to stretch once it pulls at all, not less, so a rib tube of one size should press the calf harder still — an argument from the shape of the curve, since a rib’s own curve is not among the ones computed here.

Shaping a tube is fashioning in the round

If a tube of one size presses the wrong way, a stocking has to be made in the shape the graduation needs, and making a knitted fabric a shape is a problem that flat garments already have. Clothes need darts because a flat cloth cannot lie on a doubly curved body without somewhere to lose its surplus, and a fully fashioned panel narrows by dropping stitches at its edge, which gives an edge whose angle is quantised by the stitch and the course.

A graduated stocking is the same problem with no edge. Its circumference has to grow from 157 millimetres to 247 over the lower leg and then hold, and it cannot drop stitches round a seamless tube without making a seam’s worth of disturbance. What it can do is change how far each stitch stretches — its loop length, or the tension of an elastomer laid into it — so the shape is fashioned into the fabric’s stiffness rather than into its stitch count. The tube stays one needle count and behaves as a tube of changing size.

Tensions, stretches, and one law

Every pressure here is Laplace’s law at a station: the band’s tension per unit height at the stretch that station imposes, over the limb’s radius. The stretch is the station’s circumference over the tube’s knitted circumference, less one. The elastic band’s tension is its stiffness times its stretch; the jersey’s is read from the same computed load–extension curve the cuff used, interpolated between its solved points; the equal-tension band’s is constant. The graduated tube’s tensions are each station’s pressure times its radius, and its knitted circumferences follow from the elastic band’s stiffness.

For all three bands the pressure was confirmed to change with circumference in the direction the tension’s proportional sensitivity to stretch, less one, predicts, across limbs from 16 to 38 centimetres round; the elastic band was confirmed to press twenty millimetres of mercury at its sizing; and each graduated station’s tension against the ankle’s was confirmed to equal its pressure fraction times its circumference ratio exactly, with the calf of the seven-tenths graduation pulled harder and stretched further than the ankle.

Where the law is not the whole of it

The leg is rigid and round. A real calf is soft, and a firm tube flattens it where it presses hardest, which lengthens the contact and changes the radius the law divides by; a shin has a ridge of bone where the radius is small and the pressure concentrates. None of that is computed, and it is why medical stockings are measured on a leg-shaped form rather than calculated.

The bands are idealised. The elastic band’s tension is exactly proportional to its stretch, which a real elastomer approximates over a working range; the jersey’s curve has no hysteresis in it, and a knit gives back less than it took, so a tube worn for an hour presses less than a tube just pulled on.

The leg is illustrative. Its four circumferences are round figures, the graduation is chosen to show the break-even rather than copied from a standard, and a different leg moves every pressure. The two results that do not move are the direction — a tube of one size presses the thicker part harder whenever its tension outruns its stretch ratio — and the break-even at the ratio of the circumferences.

And nothing is said about the body’s response, which is what compression is for and which is a question for physiology — nor about the fabric’s own give through its thickness, which a knit gives up when it is pressed and which a stocking at forty millimetres of mercury is pressing on as well as round.

Still open: whether a stepped shape leaves rings on the leg

A machine changes loop length or laid-in tension in steps, course by course, and the graduated tube asks for a circumference that rises smoothly. Between two steps a tube is one size over a stretch of leg that is not, so by the argument here each step is a short tube pressing its thicker end harder — a saw-tooth of pressure up the leg, with a small ring at the top of every step. How large those rings are depends on how many steps a machine takes between ankle and calf and on how far pressure spreads along a soft limb from where it is applied, and neither the step count nor the limb’s spreading is computed here.

What this corrects

The cuff essay’s pressure is right, and so is its contrast with medical compression. Its rule about radius was stated for a band at one tension, and it was carried over to a tube knitted to one size and to a graduated stocking, where it points the wrong way. The correction is one sentence: a knitted band’s tension rises far faster than its stretch ratio, so a tube presses where the limb is thickest unless it is shaped not to.

Where this goes next

What a knitted band presses a limb with. Pressure against extension for a 20 tex cotton band at a 3.5 mm loop, wrapped round a 30 mm radius — a wrist. The pressure is the fabric's own tension per unit width divided by that radius, and the tension is the loop's bending with the relaxed shape as the yarn's natural one, so nothing here is fitted. Over the range a cuff is actually used across it runs from a twentieth of a millimetre of mercury to 0.85. The shaded bands are what a compression garment is specified at, and the curve does not reach the lowest of them until 277 per cent — which is not a cuff, it is a fabric stretched almost to the point where its yarn runs straight. What a cuff presses with A rib cuff holds a sleeve on a wrist, so it must be pressing. Divide its own recovery force by the radius it is wrapped round and the pressure comes out at eight tenths of a millimetre of mercury — a fiftieth of the lightest medical compression, and two orders below what the same fabric resists being squashed with. What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve. What a knit gives when it is pulled How far a knit stretches by rearranging its loops is usually given as a bound rather than a number, because saying more needs a loop with bending stiffness in it. Solved from the loop's own bending, the answer is a curve: soft for a hundred per cent, then stiffening by a factor of eighty as the yarn between two interlacings runs out of ways to be anywhere but straight. What a balanced cloth wastes under pressure. The fraction of a balanced fabric's fibre that is along for the ride, in a stress field of each ratio. A closed cylinder is exactly two to one — the ratio of the two areas the pressure acts on — so a balanced cloth reaches its limit around the circumference with the axial system at half its capacity, and a quarter of the fibre is doing nothing. An inflated cylinder wants an unbalanced cloth Balance is a virtue in almost every other cloth. Under pressure it is a defect with a size — a closed cylinder carries exactly twice the stress around its circumference as along its axis, so a balanced fabric reaches its limit in one direction with a quarter of its fibre doing nothing at all.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Contact pressureElastic recoveryExtensibilityLoad-extensionPressure vesselSpecification