A stocking's own lengthwise pull smooths its rings

A stepped stocking should step most at the ankle found the saw-tooth a knitting machine leaves on a leg and sized it exactly. Eight equal steps leave a ring of 4.8 millimetres of mercury above the ankle; twenty-four graded steps hold every ring under one. All of that was on a rigid leg, and the essay said so. It left the difference as a thing to measure: a leg smooths a line of pressure over some distance, and that distance decides whether a machine’s step count matters at all.

That distance need not wait for a sensor, though the sensor would still settle it. Soft tissue on bone is a well-studied object, a tube under tension is a simple one, and the two have to agree along the leg. Put them together and the distance has a form, and it is not the form the earlier essay guessed. It expected the smoothing to be set by how deep the tissue is and how stiff. The stiffness is in it; the depth is not, except as a switch; and the other half is the stocking itself.

Why a tube has to bend to pass on a ring #

A stocking on a rigid leg presses what Laplace says, and the saw-tooth in that pressure goes straight into the leg because the leg cannot move. Let the leg move. Where the rigid-leg pressure is a little higher, the tissue sinks a little. So the tube, following it, is no longer straight along the leg: it has a shallow groove in it wherever a ring lies, and a tube under lengthwise tension does not take a groove for nothing.

A taut membrane in a groove pushes outward, and the push is its tension times the groove’s curvature. So part of the rigid-leg ripple goes into bending the tube and does not reach the leg. Write $P_0(q)$ for a ripple of wavenumber $q$ along the leg, $w$ for how far the tissue sinks under it, and the balance of the three is

$$P_0 = \frac{w}{S(q)} + \left(h + Nq^2\right)w,$$

with $N$ the tube’s lengthwise tension, $h$ the small amount of hoop tension the tube loses by being pushed in, and $S(q)$ how far the tissue sinks per unit of pressure at that wavenumber. The leg carries $w/S$. So the share it carries is

$$\frac{p}{P_0} = \frac{1}{1 + S(q)\,(h + Nq^2)}.$$

A long ripple has no curvature and passes whole, which is why the graduation itself, the very slow rise in pressure a stocking is sold on, gets through a soft leg intact. Only the teeth are in question.

What soft tissue does #

$S(q)$ is the published solution for a layer of incompressible material, depth $H$ and shear modulus $G$, bonded to something rigid, under a rippling load:

$$S(q) = \frac{1}{2Gq}\;\frac{\sinh 2x - 2x}{\cosh 2x + 2x^2 + 1},\qquad x = qH.$$

It has two ends. In deep tissue it is $1/2Gq$, and the depth has left the formula: a ripple’s pressure dies away within about a wavelength of the surface, and whatever lies below that is not consulted. In thin tissue it falls to $q^2H^3/3G$. Tissue is nearly incompressible, so it can only sink by flowing sideways, and in a thin layer there is nowhere to flow, so a thin layer is hard to a long ripple however soft it is to the touch.

Put the deep form into the transmission, with the hoop term small, and something simple falls out:

$$\frac{p}{P_0} = \frac{1}{1 + q\ell},\qquad \ell = \frac{N}{2G}.$$

That length is the answer to the earlier essay’s question. It is the stocking’s lengthwise tension over twice the tissue’s shear modulus. It has no depth in it, and no radius, and no count of steps. Soft tissue of calf muscle is around five kilopascals in shear, at the soft end of a large range. A tube pulled up a leg by as much as it is stretched round it carries about a tenth of a newton per millimetre, since the hoop band it was sized as has that at forty per cent. So $\ell$ is about ten millimetres. A stocking pulled up with half the lengthwise stretch smooths half as far.

So depth does not set the length; it sets whether the length applies. At a wavelength of 40 millimetres, a four-centimetre step, 25 millimetres of tissue carries 39 per cent of a ripple, 8 millimetres carries 64, and 3 millimetres carries 92. A step shorter than a few times the tissue’s depth is smoothed by the stocking’s tension and the tissue’s softness; a step longer than that reaches the bone and comes back whole.

Fine steps now pay twice #

The earlier essay found that on a rigid leg the worst ring falls as one over the step count, for a simple reason: the leg’s total change of $1/C$ is fixed and steps divide it. That is the whole of what a step does on a rigid leg. On a soft one it does something else as well: a shorter tooth is a shorter wavelength, and the smoothing bites harder the shorter the wavelength.

Each doubling takes off about seven tenths and not a half. Eight equal steps carry 2.41, sixteen 0.78, thirty-two 0.23, where the rigid leg’s figures are 5.70, 2.93 and 1.45. A tooth’s height halves, and its share carried also falls toward half. In the limit a ripple that passes as one over $q\ell$ falls as one over the square of the steps, and these figures are approaching that.

That reverses part of the earlier reading. It found fine steps mattering at the ankle and being wasted across the calf, and it found graded spacing worth nearly half the steps. Graded spacing is still worth that: at sixteen steps on deep tissue it leaves 0.38 where equal leaves 0.78. But the two questions now have different answers on different kinds of leg. On soft tissue, how many steps matters more than it did, and fewer of them are needed.

The ripple here is measured about the graduation rather than across a step, so its rigid-leg figures run higher than that essay’s rings: 5.70 where it had 4.8, the difference being the target’s own fall over the step. That way a step too fine to leave a tooth does not report the graduation’s slope as a ring. Against that measure the rigid leg needs more than thirty graded steps to bring every ripple under one millimetre of mercury.

How fine a stocking has to be stepped #

The answer the earlier essay’s last paragraph wanted is this table. On deep tissue at an ordinary lengthwise pull, ten graded steps hold every ripple under one millimetre of mercury, against the more than thirty a rigid leg would need. On eight millimetres of tissue it is fourteen, and on three it is sixteen. Pull the stocking up less firmly and the count rises; a tension of 0.03 asks for fourteen to twenty. Pull it firmly and on deep tissue six are enough.

Two practical things follow. A stocking’s lengthwise stretch is part of its specification for this purpose, though it is not normally quoted as one, and a stocking that fits lengthwise by luck is smoothing by luck. A wearer who pulls a stocking up with slack in it has lost part of what a coarse programme was relying on. And since $\ell$ depends on $G$, the same stocking is rougher on stiff oedematous tissue or on a lean leg than on a soft one: a tissue of 15 kilopascals, three times stiffer, has a third of the length. So the patient for whom the stocking is meant may not be the leg the table’s middle row describes.

Where the leg is thin, nothing is smoothed #

That is the favourable half. The earlier essay also found where the worst point on a leg is: the ridge of the shin, at the top of a step, low on the leg, where a band presses hardest because the limb turns. It found the ridge multiplies a ring by about ten, and it warned that a soft leg might be spreading all of this out.

Over the shin it is not.

A ripple of 5.7 on the rigid leg is 2.4 on the calf and 4.5 on the shin, under the same stocking at the same height, simply because the layer under it is thinner. On three millimetres the ripple at a four-centimetre step passes at 92 per cent. So the ridge’s multiplication and the thin tissue arrive in the same place and work the same way. The earlier result that the worst point is on the ridge at the top of a step is stronger than it was: the part of the leg with the largest ring multiplier is also the part that gets the least relief from being soft.

It is why a shin’s worst case does not much improve with more lengthwise pull, either. Multiplying $N$ by three on deep tissue takes the eight-step ripple from 2.41 to 1.39; on three millimetres it goes from 4.50 to only 4.05. The two legs a stocking meets in one fitting, the muscle behind and the bone in front, can differ by a factor of two in the ripple they carry, at one height, under one tube.

A stocking’s tension is not uniform around a section either, and it has already been shown that it bridges hollows. Lengthwise tension can be expected to behave as unevenly, and this flat layer has no way to represent that.

Why a knitted band and a covered one should differ #

The length has $N$ on top, so everything that changes what a tube holds lengthwise changes the smoothing, and the two common constructions change it differently. A covered-elastane stocking is close to linear both ways. A knitted one is not: what a knit gives when it is pulled is soft and then abruptly stiff, and its modulus is not one number. So lengthwise it may hold almost nothing until the wales straighten, which would give it a short smoothing length at light stretch and a long one once it has been drawn up firmly.

That also explains why the problem never appears at the small end. What a cuff presses with is a band a few centimetres deep with little lengthwise pull and a tiny ring across it, and its ring on a soft wrist is the rigid one, near enough. A ripple needs both a stocking’s tension and a stretch of leg to be smoothed along. The machine’s alternative to steps, the loop that shapes a tube, changes $N$ along the leg as it changes girth, and a fashioned edge’s staircase meets the same smoothing at a seam line rather than a ring; neither has been followed through here.

Twenty-four graded steps, for comparison #

The earlier essay’s most careful design holds every ring under one millimetre of mercury on a rigid leg.

On deep tissue that design is more than needed by a factor of about four, at 0.26 mmHg. It is not wasted, because it is the design that keeps the shin and the ankle in order, and on those it is about right: on five millimetres of tissue the same count leaves 0.42, and on the bone 1.29. A stocking stepped for the calf would be stepped far too coarsely for the shin, and one stepped for the shin spends most of its changes on a calf that cannot feel them. Graded spacing cannot do anything about that, since the same height carries both.

The model named #

The leg and the stocking are the earlier essay’s: the illustrative leg with 220, 290, 360 and 340 mm girths at 0, 12, 24 and 32 cm, the elastic band, and each step knitted to the ideal girth at its own middle. The ripple is the rigid-leg pressure less the smooth graduation it approximates, on half-millimetre samples, and it is extended evenly at both ends of the leg by a cosine series and filtered one wavenumber at a time.

The tissue is one incompressible layer bonded to a rigid base, in plane strain, with its surface solution taken as published. The tube is a membrane with lengthwise tension $N$ and a hoop term of the band’s stiffness over 45 mm squared. The values are 5 kPa in shear and 0.1 N/mm unless a figure says otherwise; both are illustrative, the first a low-end value from a wide published range and the second the hoop band’s own tension at the same strain.

What was counted. Eight step counts from 2 to 32 in both placements, on tissue 3, 5, 8 and 25 mm deep, at lengthwise tensions of 0.03, 0.1 and 0.3 N/mm.

Required of it, not shown: that a very thin layer return the rigid-leg rings; that a tube with no lengthwise tension barely smooth them; that a deep layer’s transmission be exactly $1/(1+q\ell)$ at no hoop term; that no ripple be amplified and a uniform pressure pass untouched; and that a shallower layer carry more of a ripple.

What the flat layer leaves out #

A leg is a tube of tissue on a bone, and this is a slab. The plane-strain layer is good for wavelengths much shorter than the leg’s radius, and at the longest steps here, near ten centimetres, it is being pushed to its limit. It is the short teeth, where it holds, that the argument is about.

One depth is used for the whole leg in every figure. A real one runs from thin at the ankle to deep at the calf, and thin over the shin to deep at the back at one height. The figures bracket a person rather than compute one, and a ripple moving between depths has an edge behaviour no figure here has.

The worst tooth in the first figure is at the ankle’s end, where the leg stops and the series reflects. Stockings continue over a foot, and interior teeth are the fairer reading, at about a quarter of their rigid height against the end tooth’s four tenths.

Tissue is not linear or elastic. It stiffens as it is squeezed and creeps, and a stocking is worn for hours, in which the knit too relaxes for as long as it is allowed. Over a long wear the two relaxations pull in different directions and neither has been taken up. Blood and lymph are not in it either, and they are what a stocking is for. The pressure the tissue carries at its surface is taken as what matters, though what happens to a vein underneath is a further layer down.

And nothing here says what anyone feels. A skin’s receptors and a ring’s visibility are a separate question, and neither can be read from a pressure curve.

Who worked out which part #

The elastic layer on a rigid base is a classical solution in soft-solid mechanics, and that soft tissue under a compression garment moves is a large clinical literature with measured stiffness ranges. Laplace’s law and the stepped stocking’s arithmetic are the earlier essays’. What is added here is the membrane’s lengthwise tension as the term that does the smoothing; the smoothing length $N/2G$, with its depth-free form; the role of depth as a switch rather than a length; the change of the rings’ fall from one over the steps toward one over their square; and the conclusion that the ridge’s multiplier and the thin tissue compound.

Still open: how much tension a stocking’s length actually holds #

Everything here turns on $N$, and the figure used is an analogy, the hoop band’s own tension at the same strain. A real stocking’s lengthwise tension depends on how it was pulled up, how it knit, and how far along the leg it has crept.

It is a measurable quantity with a direct prediction. A strain gauge in a stocking’s wale direction, read on legs of known build, would give $N$ directly. Then the same stocking with its lengthwise stretch deliberately changed should move the ripple on the calf, read with a thin pressure sensor at a step, by the ratio this account predicts, and should leave the ripple on the shin nearly unmoved. That difference between two sites on one leg, moved by one thing, is the test: if the calf and the shin respond alike, it is not the tube’s tension doing the smoothing, and the account above has found the wrong cause.