A knit bends more easily along its courses

Everybody calls a knit the soft fabric. Measured in extension it is, by three decades. Measured in bending it is not: a plain jersey computed the same way as this collection’s woven cloths lands inside their band, at about one and a half micronewton metres per unit width against one and a half to four and a half for the cloths.

What distinguishes it is not the magnitude. It is that a knit has a direction and a balanced plain weave does not.

How it is computed #

Bending a fabric about an axis in its plane gives every element of yarn inside it a curvature equal to the fabric’s times the square of the cosine of the angle between them, which is Euler’s theorem applied to a curve lying in a bent surface. The energy goes as the square of that, so the rigidity per unit width is

$$G \;=\; B \cdot L_A \cdot \langle \cos^4\psi\rangle$$

with B the yarn’s own bending rigidity, L_A the length of yarn per unit area, and ψ the angle each element makes with the direction being bent.

Every one of those three comes from somewhere already established. The rigidity is the free bound of the yarn’s own bracket. The yarn per unit area is the loop length over the cell. And the angle distribution is read straight off the solved loop, which is the only part that needed the machinery.

The two numbers #

For a 20 tex cotton jersey at a three-and-a-half millimetre loop, the fourth-power cosine averages come to nought point five four along the courses and nought point two five along the wales.

The yarn runs more nearly along the courses than across them, so bending the fabric about its wale axis — rolling it up along a course — engages more yarn and costs more: three point four micronewton metres. Bending it about its course axis, which rolls the top and bottom edges, costs one point six.

Two point two to one, and the soft direction is the one a jersey actually rolls in.

Which is a coincidence worth not over-reading #

A jersey curls at every edge, and the edges roll in different directions with different vigour. That the softer bending axis is the one the top and bottom edges roll about is consistent and it is not an explanation.

Why stockinette curls established what actually drives it: an asymmetry through the fabric’s thickness that has no counterpart in a woven cloth, and that essay was careful to say it computes no moment and predicts no radius. Nothing here changes that. A rigidity is the denominator of a curl radius, and this rung supplies it while the numerator stays missing.

What can be said is the scaling. If the driving moment is known, the radius follows, and the rigidity’s dependence on loop length and count is now available: both axes scale as the yarn’s rigidity times the yarn per unit area, which is the inverse of the loop length once the spacings are put in.

Why it lands among the woven cloths #

The result is initially surprising and stops being so once the three factors are separated.

A knit’s yarn per unit area is high — it has a great deal of thread in a small cell, which is why it is opaque — and that pushes the rigidity up. Its angle factor is low, because the yarn wanders in every direction rather than running straight along the axis, and that pushes it down. The two nearly cancel, and what is left is a fabric of comparable stiffness to a woven cloth of the same yarn.

So the intuition that a knit is limp comes from somewhere else, and it is worth saying where: it comes from shear and from extension, and a knit gives up its extension for three decades less force than a woven cloth while giving up its shear for almost nothing at all.

The one that is not in the model #

Shear. A fabric bent into a double curvature has to shear, and a knit shears at almost no load because its loops rotate. Nothing on this ladder computes that, and it is the single largest reason a knitted garment hangs differently from a woven one of the same rigidity.

That is stated here rather than buried because the figure above invites the wrong conclusion. Two fabrics with the same bending rigidity have the same drape coefficient in a test that measures bending, and behave entirely differently over a shoulder.

The three factors, separately #

Putting numbers on each makes the near-cancellation legible rather than asserted.

For the jersey: the yarn’s rigidity is nine hundred and forty micronewton millimetres squared at the free bound; the yarn per unit area is six point eight millimetres of thread per square millimetre of fabric; and the angle factors are nought point five four and nought point two five.

For a poplin of the same yarn: the rigidity is the same number, and the threads per unit width are two point eight per millimetre with an angle factor close to one, because a woven thread runs nearly straight along its own direction.

So the knit has two and a half times the yarn per unit area and half the angle factor in its stiff direction, and lands slightly above the poplin. The near-cancellation is a coincidence of these particular constructions rather than a law, and a knit at a much longer loop would fall below.

What the anisotropy is worth #

Two point two to one is a real and usable number, and it has a use in two places.

Roll and curl. A fabric with an anisotropic rigidity and an isotropic driving moment curls preferentially about its soft axis. Whatever the curl moment turns out to be, it will act more strongly where the fabric resists less.

Cutting and handling. A knit fed through a machine along its wales is being bent about its course axis, which is its soft direction, and one fed along its courses is being bent about its stiff one. That is a factor of two in the tension needed to lay it flat, and it is a familiar practical asymmetry with an arithmetic under it.

Against a woven cloth’s anisotropy #

A woven cloth’s is a different quantity with a different cause and is often larger.

A cloth bent along its warp is held up by warp threads at the warp sett; bent along its weft, by weft threads at the weft sett. So its anisotropy is the ratio of the two systems’ rigidities times the ratio of their setts, and a cloth with a fine dense warp and a coarse open weft can easily be five to one — which is a construction rather than a property.

The knit’s is not that. Both directions are the same yarn at the same length per area, and the whole of the ratio is the angle distribution of one solved loop. That makes it a structural constant rather than a construction choice: change the count, the fibre or the gauge and it does not move.

What moves it #

Only the tightness factor, and not much.

A tighter fabric’s loop is more nearly closed and its yarn points in more directions, which flattens the angle distribution and brings the two figures together. A slacker one’s loop is more open and its legs are longer and more nearly vertical, which spreads them. The ratio moves by about a fifth across the whole knittable band, which is small enough that quoting “about two to one” is honest for any plain jersey.

Both rigidities themselves move a great deal more, because the yarn per unit area goes as the inverse square of the loop length while the angles barely change.

What the picture cannot show #

A rigidity. Every drawing here is of a shape; the rigidity is an integral over the shape against a direction, and there is no way to see it.

The angle distribution is visible, in the sense that a reader can look at the drawn loop and see that more of it runs across the fabric than along it. What is not visible is the fourth power, which weights the nearly-parallel parts of the loop enormously and the perpendicular parts not at all — so a reader’s eye, which weights by length, gets the direction right and the size badly wrong.

The number a specification should carry #

A short recommendation, because this is one of the few places on the ladder where the arithmetic points at a practice rather than at a fabric.

So a bending rigidity quoted for a knit without its relaxation state is a dimension without a state wearing different units, and the same objection applies. Three numbers are needed and two are usually given: the value, its direction, and the state the fabric was in.

The bracket, once more #

Both figures are at the free end of the yarn’s bending bracket. At the coherent end both would be a hundred and thirty times larger, and both would be far outside the band of any real fabric — which is the check this collection already ran on the woven side and which the knit passes for the same reason.

That the knit lands inside the measured band at the free bound is a small piece of evidence that the free bound is the right end to quote, arrived at independently of the woven cloths that established it. It is not strong evidence, because nothing here has been measured against a knitted fabric, and it is worth exactly what it is.

Why this is the number a drape test sees #

Bending rigidity is what a cantilever test measures and what the drape coefficient depends on, and both are usually quoted for a fabric as though it had one.

A fifth is comfortably measurable and is routinely averaged away, because the standard asks for both directions and reports a mean. That is the right thing to do for a woven cloth whose two directions differ for construction reasons and the wrong thing for a knit, whose two directions differ for a structural reason that is the same in every plain jersey ever made.

The recommendation is narrow and cheap: report a knit’s bending length in both directions and their ratio, because the ratio is a property of the structure and the mean is a property of nothing.

What the standard’s average is an average of #

A cantilever standard asks for both directions and reports a mean, and for a knit the two figures differ by a factor the fabric cannot help. It is worth asking what the mean is good for, because the answer is that it is good for less than either of the numbers it came from.

The two rigidities are 3.4 and 1.6 micronewton metres, so the arithmetic mean is 2.5 and the geometric mean is 2.33 — seven per cent apart, which sounds negligible and is systematic.

Which of the two a downstream calculation wants depends on what the fabric is doing. A strip bent about one axis wants that axis’s own figure and neither mean. A sheet taking a double curvature — a fabric over a shoulder, a drape over a disc, anything that bends two ways at once — has an effective stiffness nearer the geometric mean, because the two curvatures multiply in the plate’s own energy rather than adding.

So the standard reports the average that suits a woven cloth and a drape test wants the other one, and the difference goes one way: the arithmetic mean is never below the geometric, so a knit’s drape stiffness quoted from a standard cantilever mean is systematically overstated. Seven per cent for a jersey; more for any structure with a larger anisotropy.

That is a small effect and it is worth stating because of what it sits beside. The anisotropy itself is a factor of two point two and the standard averages it away entirely, so the loss is not the seven per cent — it is the ratio, which the mean discards and which is the informative quantity.

And the ratio has a second use the mean cannot have: it identifies the fabric. A plain jersey’s two-to-one is a structural constant, the same at every count, every fibre and every gauge, so a measured ratio is a check on what the specimen is rather than a measurement of it. A jersey coming back at three to one has been finished, coated, laminated or is not a plain jersey; one coming back at one to one has been calendered flat or is an interlock.

That makes the cantilever test do something it is never asked to do. Run in two directions and reported as a ratio, it is a structural identification rather than a stiffness measurement — and it costs nothing beyond the test the standard already requires, since both readings are taken and one number is thrown away in the averaging.

What would test it #

The anisotropy, by cantilever, in two directions on the same fabric. That is a standard test, it takes minutes, and the prediction is specific: about two to one, with the soft direction being the one in which the fabric’s wales run along the strip.

The magnitudes are a weaker test because they inherit the bracket. The ratio does not, which is why it is the number worth putting a prediction on.

What a rib and an interlock would do #

The calculation is a plain jersey’s and the two-bed structures would answer differently, in a direction worth predicting even though it is not computed.

A rib in its relaxed folded state should be much stiffer about its wale axis, because the folds act as corrugations and a corrugated sheet is stiff along its corrugations. Opened out under extension it should soften towards a jersey’s figure. That is a prediction with a mechanism and no arithmetic, and it is the sort of thing this ladder cannot compute because the folding is a three-dimensional problem and the model is planar.

An interlock, being two ribs knitted together and symmetric by construction, should be stiffer than either and have a smaller anisotropy. Both are testable in an afternoon with a cantilever and neither has been tested here.

What was known before #

That knitted fabrics are anisotropic in bending is not news; every fabric-mechanics text says a knit should be tested in two directions, and the standards require it.

What appears not to have been said is that a plain jersey’s ratio is a structural constant — the same for every plain jersey of every fibre at every count — because it comes from a loop shape that is itself a function of one dimensionless group. That turns a measurement that has to be made per fabric into a number that can be predicted, and it turns a disagreement between two measurements into evidence about something other than the fabric.

Why the fourth power and not the second #

The exponent is the part of the formula most likely to be doubted, so it is worth a paragraph on its own.

Bending a surface gives a curve lying in it a normal curvature equal to the surface’s principal curvature times the square of the cosine of the angle between them. That is Euler’s theorem and it is the first cosine squared. The energy of a bent rod goes as the square of its curvature, and squaring the first factor gives the second. So the fourth power is two separate squarings, one geometric and one energetic, and neither is a fitted exponent.

Its practical consequence is severe. An element of yarn at forty-five degrees to the bending direction contributes a quarter of what a parallel one does, and one at sixty degrees contributes a sixteenth. So a fabric’s bending rigidity is dominated by the small part of its yarn that happens to run nearly along the direction being bent, and a structure that keeps its yarn off that direction is soft however much of it there is.

That is why a knit’s angle factor is low despite having a great deal of yarn, and it is why the two directions differ by two rather than by the ratio of the yarn lengths pointing each way.

Where the ladder goes next #

The extension ceiling, which the model puts at three hundred per cent and which no jersey reaches. The gap turns out to be a statement about what actually stops a knit stretching, and it is not the yarn.