A jersey's drape does not know its loop

A weight fixes the fibre and not the drape found something about woven cloth that looks too tidy to be true. At the free bound — the yarn’s fibres sliding past one another, so the yarn is as limp as its fibres summed — a plain cloth’s bending length has no count in it and no weight: it is the cube root of the fibre’s modulus times its diameter squared, over its density, gravity and one plus the crimp. For every plain cotton cloth that can weigh 150 grams it lies between 12.7 and 13.3 millimetres.

A knit’s weight nearly names its yarn ended by asking whether the same cancellation happens in a jersey. A jersey’s bending is a loop’s rather than a straight thread’s, and its weight is set by a loop rather than by a sett. It does happen, and it goes further than the woven case in one respect that is not obvious: the loop length — and with it the tightness factor and, at a stated count, the weight — cancels at every stiffness the yarn could have, not just at the free bound.

The woven cloth needed the free bound for its cancellation, and its locked-bound line in that figure climbs from 87 millimetres to nearly 200 as the count rises. A jersey’s locked-bound line climbs with the count too — but along a line of fixed count, which is what a knitter moving the stitch cam is on, nothing moves at all.

Two things per millimetre, and one of them is the other #

A bending length is the cube root of the fabric’s rigidity per unit width over its weight per unit area — the length of strip whose own weight bends it by a fixed amount, which is what a cantilever test reports. So the question is how each of those two quantities depends on the loop.

The rigidity is a count of members times a stiffness. Bend a jersey along its wales and the things that bend are the loop legs, two of them to each wale. A knit’s dimensions come from its loop: a relaxed jersey has $k_w/\ell$ wales to the millimetre, with $\ell$ the loop length and $k_w$ a constant for each relaxed state. So the rigidity per millimetre of width is

$$G = \frac{2k_w}{\ell}\,B_\text{leg},$$

where $B_\text{leg}$ is what one leg contributes per millimetre of wale.

The weight is a count of loops times a mass. A jersey carries $k_s/\ell^2$ stitches to the square millimetre and each stitch is one loop length of yarn, so its mass per unit area is

$$W = \frac{k_s}{\ell^2}\cdot \ell \cdot m = \frac{k_s\,m}{\ell},$$

with $m$ the yarn’s mass per length.

The ratio has no loop in it. Both are something per loop divided by the loop length, so

$$\frac{G}{W} = \frac{2k_w\,B_\text{leg}}{k_s\,m},$$

and the bending length $(G/Wg)^{1/3}$ depends on the yarn — its rigidity and its mass per length — and on two constants that belong to the relaxed state. It does not depend on how long the loop is. No assumption about the yarn’s stiffness has been made, so this holds at the free bound, at the locked bound and everywhere between.

What is left when the loop has gone #

The loop was the one freedom a jersey had. With it gone, the tightness factor — the root of the count over the loop — has gone too, because at a fixed count it is just the loop read upside down. And at a fixed count the weight has gone, because a jersey’s weight is $k_s \cdot \mathit{TF}\cdot\sqrt{\text{tex}}$ and the only thing that moves it at a given count is the tightness.

The hero figure is those three statements drawn at once. Across the tightness band from 1.3 to 1.6 — weights from 97 to 239 grams, depending on the count — every line is horizontal. At the locked bound, where the yarn bends as a solid rod, the three counts have three bending lengths: 55 millimetres at 10 tex, 70 at 20, 88 at 40, rising as the cube root of the count because a solid rod’s rigidity per unit mass rises with its cross-section. At the free bound the three are one line.

The count leaves at the free bound for the same reason it left the woven cloth. A yarn whose fibres slide is as stiff as its fibres summed and as heavy as its fibres summed, so its rigidity over its mass per length is one fibre’s rigidity over one fibre’s mass: $E\,d_f^2/16\rho$ for a round fibre of modulus $E$, diameter $d_f$ and density $\rho$. The number of fibres — which is the count — divides out.

So at the free bound a plain jersey’s wale-wise bending length is

$$c = \left(\frac{k_w\,E\,d_f^2}{8\,k_s\,\rho\,g}\;\lambda\right)^{1/3},$$

where $\lambda$ is a factor for the legs’ inclination, taken up below. The fibre, the relaxed state, and nothing else. No count, no loop, no tightness, no weight.

The legs lean, and the lean belongs to the state #

A leg is not parallel to its wale. It climbs one course while it moves half a wale sideways, so it leans at an angle $\varphi$ with $\tan\varphi = (\ell/2k_w)/(\ell/k_c) = k_c/2k_w$. The loop length cancels from that too: the lean of a relaxed jersey’s legs is fixed by its state, 32.0 degrees dry-relaxed, 32.9 wet and 32.6 fully relaxed.

A leaning member takes a bend partly as bending and partly as twist. Curve the fabric along the wale and a leg at $\varphi$ is bent by $\cos^2\varphi$ of the curvature and twisted by $\sin\varphi\cos\varphi$ of it; per millimetre of wale its stiffness is

$$B_\text{leg} = B\cos^3\varphi + C\sin^2\varphi\cos\varphi,$$

with $B$ the yarn’s bending rigidity and $C$ its twisting rigidity.

The twisting rigidity of a yarn whose fibres slide is not something this account carries, so it is bracketed rather than guessed: nothing at one end, as much as the bending rigidity at the other. That bracket is the width of each bar below.

In cotton, fully relaxed, the wale-wise bending length is 10.13 to 11.36 millimetres, and the woven cloth’s is 13.07. Dry-relaxed it is 10.70 to 11.95.

The state moves it by five or six per cent and in a definite direction: relaxing a jersey makes it slightly more pliable per unit weight. The stitches crowd closer — $k_s$ rises from 19.0 to 23.6, a quarter — and the wales crowd less, $k_w$ rising from 4.0 to 4.3, so the weight rises faster than the number of legs across a unit width. It is a small effect, and it is the only way anything about the fabric’s own history reaches the number.

Bent the other way, nearly the same #

Bend a jersey along its courses and the members that bend are the heads and sinker loops, one yarn to each course, $k_c/\ell$ of them to the millimetre. Treat them as running straight across and the same cancellation happens with $k_c$ in place of $2k_w\lambda$: the loop and the count leave, and the course-wise length at the free bound is 10.36 millimetres fully relaxed.

The two directions are within a few per cent of each other, 10.13 to 11.36 wale-wise against 10.36 course-wise, and that is itself a result of the lean. With straight legs the wale-wise figure would be the larger by the cube root of $2k_w/k_c$, sixteen per cent. The lean takes most of the difference back, because $\cos^3$ of 33 degrees is 0.59.

The straight course-wise member is an upper estimate — heads and sinkers are arcs, and an arc bent across its own plane is softer than a straight rod — so the course-wise figure should be read as a ceiling on that direction. What the pair says is that a plain jersey at the free bound is close to isotropic in bending, which is not what a woven cloth’s two systems at different setts would give.

Against woven cloth, fibre by fibre #

The jersey’s formula and the woven cloth’s differ only in their constants. Divide one by the other and the fibre leaves as well:

$$\frac{c_\text{jersey}}{c_\text{woven}} = \left(\frac{4\,(1+c)\,k_w\,\lambda}{k_s}\right)^{1/3},$$

0.775 with torsion-free legs and a woven crimp of seven per cent, 0.87 with the legs as stiff in twist as in bending.

So a jersey drapes three quarters to seven eighths as far as a woven cloth of the same fibre does, per unit of its own weight, and the ratio is the same for cotton, wool, flax and nylon. Flax’s 31-millimetre woven cloth becomes a 24- to 27-millimetre jersey; nylon’s 11.4 becomes 8.9 to 9.9. The ranking of fibres is exactly the woven ranking — flax stiffest, nylon most supple, wool above cotton because its fibre is nearly twice as thick — and the jersey adds nothing to it and removes nothing from it.

That ratio is where most of a jersey’s reputation for softness would be expected to live, and it is modest. A quarter shorter in bending length is under half the rigidity per unit weight — a cube root turns a factor of 0.47 into 0.78 — which is real, but it is not the order-of-magnitude difference in feel between a jersey and a poplin. The difference in feel has to be somewhere else, and a knit is soft because it bends already says where: a jersey extends for almost nothing because extending a loop bends nothing further, and a hand on cloth reads extension far more than it reads the cantilever.

On the weight line itself #

Put the jersey back on the 150-gram line of the weight essay before it, where the fully relaxed band runs from 15.8 to 23.9 tex, and set it beside the woven line at the same weight.

At the free bound both are flat — the woven line at 12.7 to 13.3 millimetres, its small rise the crimp; the jersey at 10.13 whatever the count. Placed where a bending test and a washing test agree the yarn sits, both rise with the count, because away from the free bound the yarn’s rigidity per unit mass grows with its cross-section. But the jersey’s rise is confined to its band: across 15.8 to 23.9 tex it goes from 22.0 to 23.4 millimetres, a spread of six per cent. Among 150-gram plain jerseys the drape is fixed to six per cent, and the woven cloths of the same weight span 28 to 41 millimetres at the same placement.

That is the drape version of the weight essay’s result. There, a jersey’s weight nearly named its yarn; here, the yarn a weight has nearly named has a drape that the loop does not touch. A plain jersey specified by its weight and its fibre is specified in its drape too, to within the stiffness bracket, and a woven cloth specified the same way is not.

Where a tight jersey’s firmness must come from #

The prediction that the loop is not in the bending length runs into ordinary experience at once. A tightly knitted jersey feels firmer than a loosely knitted one of the same yarn. Knitters know it, and the whole reason the tightness band has a tight end is that the fabric there becomes boardy.

The arithmetic says that firmness is not bending. Every member the rigidity counts is a yarn bent about its own axis, and every one of those counts scales with the loop exactly as the weight does. A tighter jersey has more legs across a centimetre, and each carries proportionally more weight: the ratio does not move.

What does change with the tightness is contact. In a tight jersey the loops are pressed against each other at their interlocking points, and pressed yarns resist sliding past one another — which is exactly the thing whose absence defines the free bound. What friction has to hold in a relaxed knit found that the wale-wise balance collapses to a condition on the friction coefficient alone, and that no yarn in this account meets it; the fabric is held by something the loop model does not contain. So a tight jersey’s firmness has to act through where in the bracket its yarn effectively sits — its fibres, pressed together, sliding less — rather than through the geometry of its loops. The bending length of a jersey moves with its tightness only by climbing its bracket, and the bracket is four hundred wide.

Along that curve the bending length runs from 10.1 millimetres at the free bound to 69.8 at the locked one, and a yarn placed where a cantilever and a washing band agree sits at 16.1 to 22.8. A tight jersey reading boardy is a jersey whose yarn has been pushed to the right along it — by nothing geometric, since the geometry has been shown to cancel, and so by the loops’ contacts making the fibres slide less.

That turns a vague observation into a specific one. Two jerseys of one yarn at tightness factors of 1.3 and 1.6, cut and hung as cantilevers, should give bending lengths whose ratio measures how far pressing the loops together moves the yarn up its bracket — a number this account has no other way to obtain.

How it was computed, and what was required of it #

The rigidity per width and the weight per area are computed separately and divided, rather than from the closed form. The yarn’s two bounds are the ones every stiffness in these essays comes from: the free bound as the fibre count times one fibre’s rigidity, and the locked bound as a solid rod of the yarn’s own diameter at a packing factor of 0.6. A place in the bracket interpolates between them on the logarithmic scale the bracket has to be read on.

The geometry is Munden’s: $k_c$, $k_w$ and $k_s$ for each relaxed state, as published, including the dry-relaxed set whose $k_s$ differs from $k_c k_w$ by five per cent. The weight uses $k_s$ and the rigidity uses $k_w$ or $k_c$, so that discrepancy enters the answer as it stands rather than being tidied away.

The cancellations are required, not observed. Across counts from 8 to 60 tex, loops at both ends and the middle of the tightness band, three places in the bracket, both bending directions and all three states — ninety combinations — the bending length is required not to move with the loop by more than one part in a billion. At the free bound it is required to equal the closed form to the same precision, count and all. And at the locked bound the length at 40 tex is required to exceed the length at 10 tex by more than a fifth, so that a computation which returned one number for everything could not pass.

What the loop model cannot say #

The legs are straight lines at a single angle. A real leg is a curved segment that changes its lean along its length and is pressed into a neighbour at both ends; the bracket on the twisting rigidity covers some of what that does, and nothing here covers the rest.

The twisting rigidity is bracketed, not known. A yarn whose fibres slide has a twisting stiffness of the same kind as its bending one — its fibres summed — and it is somewhere between nothing and its bending rigidity; the answer moves across that bracket by twelve per cent in bending length.

And the fabric curls. A single jersey’s edges roll, because its two faces are different surfaces and its yarn still carries torque. A cantilever test on jersey is notoriously hard to make for that reason: the strip rolls before it bends. Every bending length here is for a strip that lies flat and bends only under its weight, which is the quantity the drape of a hanging garment depends on and not always the quantity an instrument can read off one.

Who found which part #

The constants are Munden’s, from 1959: a relaxed plain jersey’s courses, wales and stitches per unit length are constants over the loop length, and a knit’s dimensions are therefore its loop’s. The free-bound cancellation is the woven result’s, carried across: at the free bound a yarn is its fibres summed in both rigidity and mass, so the fibre count leaves any ratio of the two.

What is new here is that the loop cancels at every stiffness. It is Munden’s result applied twice — once to the members bearing a bend and once to the mass they carry — and the two applications cancel each other. The consequence that a tight jersey’s firmness is not bending, and so must be friction lifting the yarn up its bracket, follows from nothing but that.

Still open: how far pressing the loops together climbs the bracket #

The argument above names one number it cannot compute: how far up its stiffness bracket a jersey’s yarn is pushed by its loops being pressed together.

It has a measurement that isolates it. Two jerseys of one yarn, knitted at the two ends of the tightness band and relaxed alike, have the same bending length on every geometric account here. Any difference between their cantilever readings is contact and nothing else, and with the bracket’s two ends computed from the yarn, the ratio of the two bending lengths cubed is the ratio of the two effective rigidities — a place in the bracket for each.

And it has a prediction for its direction and a bound for its size. The tight jersey must read longer, and it cannot read longer than the locked bound allows: 70 millimetres for a 20-tex cotton against the free bound’s 10. Somewhere between those is the number that would say whether a tightness factor is a geometric specification at all, or, in bending, only a specification of how hard the fibres are pressed.