The modulus a knit has instead of one

The initial slope of a load–extension curve is a modulus, and for most fabrics it is the yarn’s own tensile modulus with the geometry divided out. For a knit it is a different quantity altogether, and the difference is visible before any arithmetic: whatever number comes out, the yarn’s tensile modulus is not in it.

The value for a 20 tex cotton jersey is three and a half newtons per metre of fabric per unit strain. The same yarn laid straight and parallel at the same spacing is a hundred and fourteen thousand.

The dimensions decide the form #

Before any solve, the shape of the answer is fixed. The only material quantity in the model is a bending rigidity, in newton millimetres squared; the only lengths are the loop length and the two spacings, which are all proportional to the loop length.

A force per unit width per unit strain has the dimensions of a force over a length, so it can only be the rigidity divided by a length cubed, times a pure number. The pure number is what the solve supplies, and for a relaxed jersey it is about a hundred and sixty.

That is worth stopping on. The whole of a knitted fabric’s extensional stiffness is one material property and one length, and the length is the loop length — the quantity a knitter sets directly at the machine.

Which material property #

Bending, not tension. And that is not a small distinction, because the two are known to different accuracies and depend on different things.

A yarn’s tensile modulus is well characterised: it is the fibre’s modulus times a translation efficiency times the obliquity of the twist, all of which this collection computes, and it is reproducible to a few per cent. A yarn’s bending rigidity is a bracket a hundred and thirty wide, because whether its fibres slide over one another is a friction question nobody has solved.

So a knit’s extensional modulus is far less predictable than a woven cloth’s, and for a reason that has nothing to do with knitting.

The comparison, made properly #

The honest comparison is not against the yarn’s modulus but against a fabric made of the same yarn doing the simplest possible thing.

Lay the yarn out straight and parallel at the same wale spacing — one yarn per wale, all of them along the direction of pull. Each carries its own axial stiffness, which this collection computes from the fibre’s modulus, the twist’s obliquity and the count. Per metre of width that comes to a hundred and fourteen thousand newtons per unit strain.

The knitted fabric of the same yarn at the same spacing gives three and a half. A factor of thirty-two thousand, four and a half decades, from nothing but the yarn having been looped instead of laid.

Where the factor comes from #

It is a ratio of a bending stiffness to a tensile one, and those differ by the square of a slenderness.

Stretching a rod of length ℓ costs EA per unit strain; bending the same rod through a comparable angle over the same length costs about EI/ℓ², and I over A is the square of a radius of gyration — a fraction of the yarn’s diameter. So the ratio of the two is of order (ℓ/d)², which for a loop of twenty-one diameters is four hundred and forty.

The remaining factor of seventy is the arrangement: a looped yarn is not being bent through one angle over its whole length, and the reconfiguration is far more efficient than a single bend. Both factors are geometric, and neither has any material in it.

What that explains #

The oldest observation about knitted fabric, which is that it stretches.

It also explains why knitting a fibre changes its usefulness more than weaving does. A cotton that stretches three per cent before breaking makes a woven cloth that stretches a few per cent and a knitted one that stretches a hundred, and the fibre did not change.

The comparison is worth doing properly, because the woven cloth is not a control. A woven cloth stretches too, and for a related reason: its crimp straightens, which is also a shape change rather than a fibre extension. The difference is how much shape there is to spend. A woven thread’s crimp is a few per cent of its own length and it is exhausted almost immediately; a knitted loop is nine tenths free run and can be pulled into something like a straight line before the yarn is asked for anything. So the two fabrics are on the same mechanism at opposite ends of it, and the factor between them is a ratio of available shape rather than a difference in kind.

Which is why the slenderness ratio in the paragraph above is the number to hold on to. Converting tension into bending buys a factor of the ratio squared, and for a yarn a few hundredths of a millimetre across bent over a loop a few millimetres long that ratio is in the tens. Squared, it is the thousandfold difference between a fibre’s modulus and a fabric’s — which nothing about the chemistry of the fibre could have supplied, and which no finishing process can take away.

What the number is not #

It is not a Young’s modulus and it should not be quoted as one, for the same reason a cloth’s stiffness has no lower bound in a useful sense. A fabric has no well-defined thickness to divide by — a thickness is a maximum rather than a mean — so dividing a force per unit width by a thickness manufactures a stress with an arbitrary denominator.

The right unit is a force per unit width per unit strain, in newtons per metre, and it is the unit fabric testing already uses for exactly this reason. A modulus in pascals for a knitted fabric is a number with a made-up length in it.

The bracket it inherits #

Three and a half newtons per metre is the figure at the free end of the rigidity bracket. At the coherent end it would be a hundred and thirty times larger, which is four hundred and fifty — still three decades under the straight yarn, so the conclusion survives the whole bracket and the value does not.

That is the pattern throughout this ladder and it is worth naming. The comparisons are robust and the values are not, because a comparison between two things computed the same way cancels the bracket and a value does not.

The anisotropy #

The number above is for pulling along the courses. Pulling along the wales is a different quantity, and the model gives it from the same solve by imposing the other spacing instead.

A jersey is stiffer along its wales than across them, because a loop’s legs run more nearly along the wale direction than across it and straightening them is the cheaper operation. The ratio is not large — under two — and it is far smaller than the anisotropy a woven cloth has between its warp and weft directions when the two systems differ in count or sett.

What the modulus does with the loop length #

As the inverse cube, which is a strong dependence and the most useful thing here for anybody specifying a fabric.

Loosen the loop by twenty per cent and the modulus falls by nearly half. Tighten it by twenty and it nearly doubles. That is a bigger lever than any change of fibre available, and it is the reason a knitter’s tightness factor is a specification rather than a preference.

It is also why two fabrics of the same yarn and the same weight can feel entirely different: weight per unit area is set by the yarn and the stitch density together, and the modulus is set by the loop length alone, so the two can be moved independently.

Beside a woven cloth #

The contrast is the point of computing it, and this collection already has the woven side.

A woven cloth’s initial modulus is of the order of thousands of newtons a metre; a knit’s is a few. Three decades, in fabrics of the same yarn, and the mechanism is the same in both — thread moving at constant length. The difference is entirely how much thread there is to move, which is half the yarn in a knit and four parts in a thousand in a cloth.

That one ratio is doing an enormous amount of work across this ladder, and this is the place it is most visible.

What a fibre change does and does not do #

A useful negative result for anybody choosing a fibre.

Switching a jersey from cotton to polyester at the same count and the same loop length changes the tensile modulus by about half and the bending rigidity by rather more, because the bracket depends on the fibre count as well as the modulus. What it does not change at all is the extension available before the geometric ceiling, because that is a statement about lengths.

So a fibre change moves the vertical scale of the load–extension curve and leaves the horizontal one alone. A knitter wanting more stretch has to change the loop length; a knitter wanting a different feel at the same stretch changes the fibre. Those are separate knobs and the model says they are separate.

The one place the tensile modulus does appear #

At the very end of the curve, and only there. Once the yarn between two interlacings is straight, further extension has to stretch it, and at that point the fabric’s stiffness jumps from three newtons a metre to something of the order of the straight-yarn figure.

The jump is by four decades and it happens over a few per cent of extension. That is what a knitted fabric’s break looks like on a tensile tester: a long soft region, a knee, and then an almost vertical rise to failure. The model reaches the knee and stops, because beyond it the assumption of constant yarn length is false.

The number as a function of tightness #

Since it is a millisecond’s work, the modulus can be plotted against the lever that moves it rather than quoted at one construction.

That is a strong statement and it is worth testing rather than admiring. If the modulus really is a function of the tightness factor and the rigidity alone, then two fabrics of the same tightness factor in different fibres should have moduli in the ratio of their bending rigidities over their loop lengths squared, and nothing else should enter — not the count, not the gauge, not the stitch density.

That is checkable with a tensile tester and two fabrics, and it is the most easily falsifiable claim on this ladder.

What a designer feels, and what this is not #

A caution, because “modulus” and “soft” are not the same word.

The number here is extensional stiffness. A fabric’s handle — the thing a hand judges — is mostly bending and shear at very small loads, and a knit is not especially soft in bending: computed the same way as this collection’s woven cloths, a jersey’s bending rigidity lands in the same band as a poplin’s, at the limp end.

So the sentence a knit is soft needs its object. It is enormously soft in extension and ordinarily soft in bending, and conflating the two is how a fabric that drapes stiffly and stretches easily gets described as though both came from one property. The rung that measures the bending finds the more interesting half is not the magnitude but the direction dependence.

What the picture cannot show #

A modulus. It is a slope at a point, and a slope at a point on a curve that is nearly flat there is the hardest thing to read off a plot. The figures here show the curve; the number is stated.

Nor can any figure show the bracket. Every ordinate on every load–extension curve in this ladder could be multiplied by a hundred and thirty and the drawing would look identical, because the axis would rescale with it.

What the fabric is doing while it obeys it #

It is worth putting the drawn loop beside the number once more, because the modulus is a statement about a shape rather than about a material and the shape is available.

Nothing else is. There is no fibre in the picture, no twist, no gauge and no machine, and none of them appears in the answer except through the one rigidity. That is the same collapse Munden found in the dimensions, turning up in the forces and for the same reason: at a fixed loop length, everything about a plain knit is a function of one ratio.

The difference is that the dimensions were measured to collapse and this is derived to. A measurement that collapses is an invitation to look for a reason; this ladder is the reason arriving sixty-odd years late.

What was known before #

That knitted fabrics are soft in extension, universally. That the softness comes from loop reconfiguration rather than yarn extension, since at least the 1950s. That the tightness factor governs it, empirically, from the trade.

What is new is the form of the answer: a bending rigidity over a length cubed, with a pure number in front that the geometry supplies and no material in it beyond that one rigidity. The form is what turns a set of empirical rules into a single statement, and it is what says which measurement would improve the prediction — a better bending rigidity, not a better tenacity.

The tightness factor cannot carry a modulus on its own #

The claim that everything on this ladder is a function of the tightness factor is right about the dimensionless quantities and cannot be right about this one, and the reason is worth working through because it produces the specification the essay is reaching for.

A modulus in newtons per metre is a dimensional quantity, and the tightness factor is a pure number. So no function of the tightness factor alone can give it; a length has to come from somewhere. Following the dependences through says which length and with what exponent.

The modulus is B/ℓ³. Bending rigidity goes as the fourth power of a diameter and a diameter as the square root of the count, so B ∝ tex². And the tightness factor is √tex over ℓ, so tex = K²ℓ². Substituting,

modulus ∝ K⁴ ℓ.

At a fixed tightness factor, a knit’s extensional modulus is proportional to its loop length — so two fabrics a knitter would call equally tight, one at a two-and-a-half millimetre loop and one at five, differ by a factor of two in stiffness. The coarse-gauge one is the stiffer.

That is not a contradiction of the ladder’s finding; it is the finding stated dimensionally. Every ratio on this ladder is a function of K alone, and every quantity with units in it needs a length as well.

Which gives the specification a second line, and it is one already printed #

Run the same substitution through the areal weight and the two dependences turn out to be the same one.

A knit’s yarn length per unit area is the two shape constants over the loop length, so its mass per unit area is that times the count — which with tex = K²ℓ² comes to

areal mass ∝ K² ℓ.

The same first power of the loop length that the modulus carries. So dividing one by the other removes it:

modulus ÷ areal mass = a function of the tightness factor alone.

That is the clean statement the essay wants and could not have with the tightness factor by itself. A knit’s stiffness per unit weight is set by its tightness factor, and its stiffness is that times its weight. Both numbers are on every specification already — grams per square metre and a tightness factor, or the loop length and count they come from — and putting them together predicts the modulus without a tensile test.

It also settles a familiar comparison. Two jerseys of the same weight and different gauges have the same modulus if and only if they have the same tightness factor, whatever their counts and stitch densities; and two jerseys of the same tightness factor and different weights differ in modulus in exactly the ratio of their weights.

And it says which sweep tests which exponent #

The falsification test proposed above — knit one yarn at five loop lengths — is a fixed-count sweep, and there the tightness factor moves as one over the loop length while the rigidity stays put. So the modulus goes as ℓ⁻³ and the essay’s exponent of three is the right thing to look for.

A fixed-tightness sweep is a different experiment and a harder one, because it requires the count to move as the square of the loop length: a five-millimetre loop needs a yarn four times the tex of a two-and-a-half-millimetre one. There the exponent to look for is plus one, and the two sweeps together are a much stronger test than either.

A model that got the fixed-count exponent right and the fixed-tightness one wrong would have the rigidity’s count dependence wrong — which is exactly the fourth power that this collection carries as a bracket, and which nothing on this ladder has otherwise been able to test.

Where the specification bites #

A garment engineer choosing a knit for a given stretch has three levers and now knows their exponents: the loop length as the inverse cube, the yarn’s bending rigidity linearly, and the structure through the pure number in front.

Of those, only the first is under real control at the machine. The second is a consequence of the fibre and the twist and is known only to within a bracket. The third changes when the structure changes — a rib, an interlock, a tuck — and is not computed here for anything but a plain jersey.

A test that would refute it #

Stating what would break the claim is cheaper than defending it, and there is a clean one.

The claim is that a knit’s initial extensional modulus is proportional to the yarn’s bending rigidity and inversely proportional to the cube of the loop length, with a pure number in front that depends on nothing else. A single fabric cannot test that; a series can.

Knit the same yarn at five loop lengths and measure the initial modulus of each. If the exponent on the loop length comes out near three, the form is right. If it comes out near one, the extension is being resisted by something proportional to a length rather than to a bending — friction at the contacts, most likely, which would say the elastic part is not the part being measured. If it comes out near five, something in the model’s dependence on the spacings is wrong.

The exponent is the discriminator and it does not need any absolute number to be known, which is what makes it the right test when the vertical scale is uncertain by a factor of a hundred and thirty.

Where the ladder goes next #

The transverse behaviour, which is the other half of any extension and which does something a material would not: it changes sign. A jersey pulled along its courses gets slightly taller before it starts to get shorter, and the model says where the crossover is.