A jersey gets taller before it gets shorter

Pull almost anything along one axis and it gets narrower along the other. That is what a positive Poisson ratio means, it is true of nearly every material anybody handles, and it is what a woven cloth does emphatically — pulling one system straight bends the other further, and the cloth narrows as it lengthens.

A plain knit does the opposite for the first eighty per cent of its extension. Pulled along its courses it grows along its wales, by about two per cent, and only then turns round.

What is being computed #

Nothing is imposed on the second dimension. At each wale spacing the course spacing is whatever minimises the loop’s bending, which is a one-dimensional minimisation the model can do at every point, and the transverse response comes out of the same solve as the force.

That is worth insisting on, because the usual alternative is to assume a Poisson ratio and use it. A model that assumes one cannot discover that it changes sign.

The number #

The expansion reaches two per cent at about thirty-five per cent extension and holds there to about fifty. Expressed as a Poisson ratio at twenty per cent extension it is minus nought point zero seven, which is small, negative and not a constant — it varies over the whole range and passes through zero.

Past eighty-one per cent the fabric contracts, and it does so fast: twenty-six per cent contraction at a hundred and eighty-five per cent extension, and eighty-eight per cent by the time the yarn runs straight.

Why the sign is negative at first #

The loop has two ways to give a course-wise extension and they pull the wale spacing in opposite directions.

It can open — the head widening and the legs splaying — which takes yarn from the vertical parts of the loop and puts it into the horizontal, and would shorten the course spacing. Or it can straighten its tightest bends, which costs the least energy of anything available and does not need any height at all.

At small extension the second is much cheaper, because the loop’s tightest bends are precisely where the bending energy is concentrated and where the marginal cost of moving is highest. Relieving them lets the whole loop relax a little in every direction, and the course spacing goes up rather than down.

Why it turns round #

Because the cheap moves run out. Once the tightest bends have been relieved there is nothing left to straighten without taking yarn from somewhere, and the only place left is the loop’s height.

From there on the geometry dominates: the straight line between two interlacings is half a wale across and a loop height down, and its length cannot exceed the yarn spanning it. Widening the wale therefore has to be paid for in height, at an exchange rate that gets worse as the fabric approaches the ceiling.

The crossover at eighty-one per cent is where the two effects balance, and it is a property of the loop’s shape rather than of anything material.

Auxetic, and not in the usual sense #

A material with a negative Poisson ratio is a curiosity, usually engineered, usually with a re-entrant cellular structure built for the purpose.

A jersey is not that. It has no re-entrant cells, nothing was engineered, and the effect is not a material property at all — it is a configuration effect in a structure that has two degrees of freedom and is free to choose one of them. Change the loop length and the crossover moves; change the fibre and it does not move at all.

That is the sharpest possible statement of the difference between a structural effect and a material one, and it is the sort of case this collection exists to point at.

The exchange rate, stated as a rate #

The curve is easier to use as a slope than as a displacement, and the slope is the fabric’s Poisson behaviour at each point.

Over the first thirty-five per cent the exchange rate is about minus seven parts in a hundred — the fabric gains seven units of height for every hundred it gains in width. Between fifty and eighty it falls through zero. Past a hundred it is strongly positive and rising, reaching about minus one in the conventional sign at the ceiling, where a fabric that doubles again loses nearly all of its height.

Three regimes in one curve, with no material parameter anywhere in the explanation.

What a second axis does #

A garment is rarely pulled along one axis, and the model can be asked about the other one directly rather than by inference.

Imposing the course spacing instead and letting the wale spacing find its own minimum gives the mirror problem, and it does not mirror: the fabric is stiffer along its wales and its transverse response there is positive from the start. The asymmetry comes from the loop’s own shape — its legs run more nearly along the wales than across them — and it is the same asymmetry behind the ratio of the two bending rigidities.

So a knit is not a material with two Poisson ratios. It is a structure whose transverse response depends on which way it is pulled, how far, and what the other dimension is allowed to do, and any single number quoted for it is a number quoted without its conditions.

What it is worth knowing for #

Three things, and the third is the one that bites in practice.

A strip test measures neither the free response nor a restrained one: the jaws restrain the transverse dimension at the ends and not in the middle, so the measured stiffness is somewhere between the two and depends on the aspect ratio of the specimen. That is a known problem in fabric testing and this rung says which way the error goes for a knit.

A garment stretched over a body is under biaxial rather than uniaxial extension, and the transverse response decides how much of the second axis’s stretch is free. A fabric that expands transversely under small extension is a fabric that fits more easily than a naive calculation suggests.

And dimensional measurement of a knit under any tension whatever is measuring two moving quantities. A fabric laid flat and lightly smoothed is somewhere in the first thirty per cent of this curve, and its measured stitch density is therefore a couple of per cent off in a direction that depends on how it was handled.

What it does not mean #

It does not mean the fabric’s area increases. Course-wise extension of thirty-five per cent with two per cent transverse expansion is a large increase in area, which is exactly what happens: a knit’s area is not conserved and nothing in the structure conserves it.

Nor does it mean the fabric is getting less dense in stitches per unit area — it does — or that the yarn is being redistributed unevenly, which it is not: every stitch has the same loop length throughout.

The comparison with a woven cloth #

A woven cloth’s transverse response is crimp interchange and it is emphatically positive: extending one system straightens it, which forces the other to take the whole of the cloth’s thickness, which shortens the cloth in that direction. There is no configuration in which that reverses, because the closure condition ties the two together with a fixed total.

So the sign change is a knitted phenomenon in a specific structural sense: it needs a structure with no closure condition, which is exactly what a knit is and exactly what a woven cloth is not.

What the picture cannot show #

The choice. The figure plots the course spacing that minimises the energy at each extension, and a real fabric under a real test may not be at that minimum — it may be held short of it by friction, or held long of it by the jaws.

So the curve is the free transverse response, and a measured one on a strip will be flatter, in both directions, by an amount friction decides. What can be said is that the free response has a sign change, and a measured one on a wide enough specimen should show one.

What would falsify it #

The prediction is specific enough to fail. A wide, lightly-gripped specimen of plain jersey pulled slowly along its courses should show its wale spacing rise for the first half of its travel, and the rise should be a couple of per cent.

If it falls monotonically from the start, the free response is not what is computed here and the likeliest culprit is the transverse condition rather than the model — a specimen too narrow behaves as though restrained. If it rises by ten per cent rather than two, something in the loop’s energy distribution is wrong.

The measurement needs an optical extensometer rather than a ruler, because two per cent on a course spacing is thirteen micrometres.

Where the crossover sits with construction #

The crossover moves with the loop length, and in the direction a reader can guess once the mechanism is stated.

A tighter fabric has less slack, so its cheap moves run out sooner and its crossover comes earlier. A slacker one has more to relieve and stays in the expanding region longer. Both are consequences of the same ratio that governs everything else on this ladder, and neither depends on the fibre.

That is checkable in the same experiment as the sign change itself, by running it on three fabrics at three loop lengths, and it is a stronger test than the sign alone because it predicts an ordering rather than a fact.

Where the yarn goes #

It is worth following the yarn rather than the fabric for one paragraph, because the accounting is exact and it makes the mechanism concrete.

At small extension the yarn moves out of the bends and into the runs — the tightest curvature is relieved and the arc length that was buying curvature is spent on distance instead. That takes nothing from the loop’s height, which is why the height does not fall.

At large extension there are no bends left to spend and the yarn has to come out of the vertical runs, which is exactly what the height is made of. The two regimes are two different accounts being drawn on, and the crossover is where the first is empty.

A structure with two freedoms #

The general shape of this result is worth extracting, because it turns up elsewhere in this collection and is worth recognising.

Any structure with two independent geometric freedoms and one energy to spend can trade between them, and which way it trades depends on which freedom is cheaper at that configuration rather than on any fixed property. A woven cloth has two freedoms and a closure condition binding them, which reduces the pair to one and removes the choice.

A knit has two freedoms and no binding equation, so the choice is live at every point on the curve, and a sign change is exactly what a live choice looks like when the cheap option runs out.

The same logic predicts where else to look. A leno, a braid and a warp knit all have more freedom than a plain weave, and none of them has been asked this question here. A braid’s third thread system in particular has an obvious candidate freedom and an obvious reason to expect strange transverse behaviour.

What was known before #

That knitted fabrics have odd transverse behaviour is not news to anybody who has stretched one, and negative Poisson ratios have been reported for knitted structures, mostly for deliberately engineered ones — auxetic knits are a small research field with its own designed geometries.

What appears to be missing is the observation that plain jersey does it already, without any design, over a large part of its ordinary working range, as a consequence of a loop having two ways to give and choosing the cheaper one first. That does not need a re-entrant cell or a special stitch. It needs a loop with slack.

The measurement everybody already makes #

There is a standing test that lands squarely in the expanding region and is never read this way.

Every specification of a knitted fabric quotes courses and wales per centimetre, measured on a flat table. The fabric is smoothed by hand before measuring, which puts a small course-wise extension into it, and this rung says that the wale spacing therefore reads high rather than low.

Two per cent on a wale spacing is two per cent on the courses-per-centimetre figure, which is the same order as the difference between the relaxation states everybody is careful to specify. Nobody specifies the smoothing, and this is the first argument here that they should.

The tension that puts a fabric in this region is smaller than a hand #

The last section says a fabric on a measuring table has been smoothed into the expanding region and that nobody specifies the smoothing. How far in it goes can be estimated, and the answer is that the region is very much easier to reach than it sounds.

This ladder gives a jersey carrying about 1.5 newtons per metre of width at fifty per cent extension — a fabric so soft that the forces involved are grams rather than newtons. Working backwards along a curve that is nearly linear over its soft region, five per cent of extension takes about 0.15 N/m, which is fifteen grams-force across a metre of cloth.

No hand smoothing a fabric applies less than that. A fabric cannot be flattened on a table, at all, without being taken several per cent into the region this rung is about, and the two spacings it is being measured for have both moved before the ruler is laid down.

Which is a stronger version of the essay’s point than the essay makes. It is not that the smoothing is unspecified; it is that a specification for it would be unmeetable by hand. The standards deal with this by specifying a rest rather than a tension — lay the fabric out, leave it, measure it — and the reason that works is that the frictional band lets it stay wherever it was put rather than springing back. The rest is doing the job a tension specification could not.

Which explains what a hanger does to knitwear #

The same arithmetic, applied to a garment rather than a specimen, gives a familiar result an unfamiliar size.

A jersey at 150 grams per square metre, hung as a panel one metre long, carries its own weight at the top edge: 0.15 kilograms per metre of width, or 1.47 newtons per metre. Which is, to the precision any of this deserves, exactly the tension that corresponds to fifty per cent extension.

A knitted garment on a hanger is at something like half again its own length at the shoulder, under nothing but gravity. That is why knitwear grows on a hanger, it is why the trade says to fold it rather than hang it, and the arithmetic says the effect is not a slow creep under a small load but an immediate elastic response to a load the fabric finds large.

And this rung adds the part nobody says. Fifty per cent extension is inside the expanding region, which runs to eighty-one, so a hanging knitted garment is transversely expanded as well as longitudinally — it gets wider at the shoulders while it gets longer, by up to a couple of per cent.

That is a small effect beside the length change and it is the opposite sign from what anyone would predict from a woven cloth, where hanging narrows. It is testable on any garment and on any hanger, and it is the sign rather than the size that would decide the matter.

How far the crossover moves with construction #

The essay predicts an ordering with loop length and does not size it. The mechanism gives a scaling: the cheap moves are the loop’s tightest bends, and how much extension they can supply scales with the yarn’s diameter against the loop’s length — which is the tightness factor.

So the crossover extension should go roughly as one over the tightness factor. The computed eighty-one per cent belongs to the reference fabric, and that fabric is at K = 12.8 rather than in the middle of the band — √20 over 0.35 centimetres — so the scaling predicts about sixty-one per cent at K = 17 and a hundred and four at K = 10, the second of which is past the geometric ceiling and would mean the expansion never turning round at all on a very slack fabric.

The model can be asked directly rather than scaled, and it disagrees in a useful direction.

The other end of the range is where the scaling is put under real strain, because it is where it predicts the effect running off the end of the fabric’s own travel.

That is a spread of half a point in the size of the effect across the knittable range, against the two per cent the effect itself amounts to — smaller than the scaling suggested, and still an ordering a measurement could check. A measured ordering that ran the other way would be decisive against the mechanism, and that is what a prediction is for.

Two per cent, and why it is not larger #

The size of the expansion deserves an explanation, because a reader who has just been told a fabric does something surprising will reasonably ask why it does so little of it.

The expanding move is a relief of the loop’s tightest bends, and those bends contain about two thirds of the loop’s bending energy but occupy a small fraction of its length. Relieving them buys a large energy saving and a small displacement, so the fabric is very willing to make the move and there is not much of it to make.

The contracting move is the opposite: it takes yarn from the loop’s height, which is most of its length, so it buys little energy and moves a lot. That asymmetry is why the curve is shallow on the left and steep on the right, and it is why the sign change looks like a small wobble on a large descent rather than like two comparable regimes.

A fabric in which the two were comparable would be one whose bending energy was spread evenly along the loop, and no loop with a real yarn diameter in it is.

Where the ladder goes next #

The other direction. Bending a knit rather than stretching it gives a rigidity in each of two directions, and the ratio between them turns out to be the more interesting number — because it says which way a jersey rolls, which is a question this collection has had open since it first drew a curling edge.