A double cloth is only softer if its yarn is set

Five essays have taken a two-layer cloth apart at the loom: the same draft finished four ways, where a stitch may go, what an interchange joins, what the beams cost, and what the loom allows. None of them asked what the cloth is like to hold, and the trade has a confident answer: weight without stiffness. A heavy single cloth needs heavy yarn and is stiff; two light cloths of the same total weight are said to drape like the light cloths they are.

That answer rests on a picture of paper: a stack of thin sheets bends far more easily than a board of the same thickness. The picture is right about the stack and the board. The comparison a buyer is making is a different one — a double cloth against a single cloth of the same yarn — and there the answer depends on something the picture leaves out.

It depends on the yarn. A yarn’s bending stiffness is not a number but a bracket two and a half orders of magnitude wide, between fibres free to slide past each other and fibres locked into a solid rod, and where a cloth’s yarn sits in that bracket decides the whole question. At the free end, dividing the same yarn into layers changes the cloth’s bending rigidity by exactly nothing. At the locked end, dividing it by count halves the rigidity. And joining the layers firmly multiplies it by five.

Two ways to divide the same yarn #

The comparison has to hold the yarn fixed: the same mass of yarn in every square centimetre, which is what “a double cloth of the same weight” means. Take a single cloth of 20 tex cotton set at 48 ends a centimetre, which covers four fifths of its width with yarn. There are two ways to put that yarn into two layers.

By count. Each layer keeps the 48 ends a centimetre and takes a yarn of half the count, 10 tex. A yarn’s diameter goes as the square root of its count, so each yarn is 0.12 millimetres across instead of 0.17, each layer covers 57 per cent of its width instead of 80, and the two layers together stand √2 times as thick as the single cloth.

By sett. Each layer keeps the 20 tex yarn and takes half the ends, 24 a centimetre. Each yarn is as thick as before, each layer covers 40 per cent of its width, and the two layers stand twice as thick.

Both carry exactly the single cloth’s yarn per area. Both are thicker. The question is what each does to stiffness.

At the free bound nothing moves #

When a cloth bends, its bending rigidity per unit width is, to first order, the number of yarns across that width times the stiffness of one yarn. At the free end of the bracket a yarn’s stiffness is the number of fibres in it times the stiffness of one fibre — each fibre bends about its own axis and slides past its neighbours. Multiply the two together and the cloth’s rigidity is the number of fibres across its width times one fibre’s stiffness.

Nothing in that product knows how the fibres are grouped. Twenty-tex yarn at 48 ends, ten-tex yarn at 48 ends in each of two layers, and twenty-tex yarn at 24 ends in each of two layers all put the same number of fibres across every centimetre of width, because each carries the same mass of the same fibre. So all three have exactly the same rigidity at the free bound, and so would any division into any number of layers by any combination of count and sett.

That is an identity, not an approximation, and it is where four unrelated observations place an ordinary unset yarn: a slack yarn snarls, a knot holds, a yarn flattens in a fabric, and a cloth’s own thickness agrees. For a double cloth woven from unset yarn and left free to slide between its layers, the drape advantage is zero. It is exactly as stiff as the single cloth of the same weight, and twice as thick.

At the locked bound, dividing the count halves the rigidity #

A set yarn is somewhere else. Heat-setting, resin finishing and the pressure of a finished cloth press its fibres together until sliding costs something, and in the limit the yarn bends as a solid rod, with a stiffness that goes as the fourth power of its diameter — the square of its count.

Then the grouping matters. Divided by count, each layer’s yarn has half the count and a quarter of the stiffness, at the same number of ends: each layer is a quarter as stiff as the single cloth, and the two together are half as stiff. Divided by sett, each layer’s yarn is the same, at half the ends: each layer is half as stiff, and the two together are exactly as stiff as the single cloth again.

So the trade’s claim is true in one of four cases. A double cloth of set, finer yarn at the single cloth’s sett, with its layers free to slide, drapes at half the stiffness. The same cloth in unset yarn does not drape better at all, and neither does a double cloth that keeps the yarn and halves the sett, whatever the yarn.

Fused layers are a different cloth #

Everything above assumed the layers slide freely over one another, which is what makes a stack of paper flexible. A double cloth is not a stack of loose sheets. It is joined — by stitches, by interchanges, by the friction of two layers pressed together — and a join resists the layers sliding.

The limit of that is a double cloth whose layers cannot slip at all, and then the two layers bend as one section. A yarn in the upper layer is half a layer’s thickness above the middle of the stack and a yarn in the lower layer the same distance below, so each adds to its own stiffness the stiffness of its cross-section carried at that distance. For two layers of solid yarns touching, that multiplies the sliding stiffness by five: the fused finer-yarn double cloth is two and a half times as stiff as the single cloth, and the fused half-sett one five times.

Between sliding and fused lies every stitched double cloth, and the bracket between them is wide — a factor of five at two layers, nearly twelve at three. Where a given cloth sits in it depends on how stiffly its stitches resist slip, which is the stitch yarn’s own stretch, how many stitches there are and how far apart. How close the stitches must be to stop the layers blistering in a fold is already worked out as a length, twice the bend radius times a yarn diameter over a layer’s thickness; dividing by count shrinks the diameter and the thickness together and leaves that length unchanged. What the length does not say is how much stiffness the stitching adds while doing it.

What the division buys is thickness #

If a double cloth of the same weight is not, in general, softer, what it certainly is is thicker: √2 times divided by count, twice divided by sett. And thickness is what a cloth is warm with. Warmth is a thickness of still air, and a double cloth holds a layer of it between its two faces that a single cloth of the same weight has nowhere to put. That, rather than drape, is the property layering reliably delivers at no cost in yarn.

Openness is where the two divisions part. Divided by count, each layer leaves 43 per cent of its width open against the single cloth’s 20, but two such layers lying at random over one another leave, on average, only the product of their gaps straight through: 19 per cent, a shade less than the single cloth. Dividing by count keeps the stack as closed as the single cloth and makes it √2 as thick. Divided by sett, each layer leaves 60 per cent open and the stack 36 — nearly twice as open as the single cloth, so its extra thickness is thickness a draught can blow through.

More layers push both ways further #

Carried on to three and four layers divided by count, the pattern holds and widens. The thickness grows as the square root of the number of layers. The rigidity at the free bound stays exactly where it was. At the locked bound it falls as one over the number of layers — a third at three, a quarter at four. And fused, it climbs, because every added layer sits further from the middle of the stack: nearly four times the single cloth at three layers, over five at four.

So a many-layered cloth of fine, set yarn with free layers is the most flexible way to carry a weight of yarn, and the same cloth fused is among the stiffest. Between them is a factor that grows with every layer, which is why the stitching plan of a multi-layer construction decides its hand more than its layer count does.

The same identity one level down #

The free-bound identity is not special to layers. It says that a cloth’s rigidity at that end is the fibres across its width times one fibre’s stiffness, and so no way of grouping the fibres — into yarns, into plied yarns, into groups of threads working as one, into layers — can move it. Layering is simply the grouping that happens to be sold on its drape.

The same arithmetic has already turned up inside a single cloth. A group of threads working together covers like one thick thread and bends like the separate threads it is made of, and the reason is the same: what it covers is set by its outline, and what it bends is set by what is inside the outline and whether it can slide. A double cloth is that result turned on its side. Its outline in section is twice as tall, which is why it is warmer, and what bends is still the same fibres, which is why it is no softer.

It also says where the trade’s claim came from, and why it is not simply wrong. A cloth that has been finished — scoured, pressed, set, resin-treated — has had its yarns pushed towards the locked end of their bracket, and at that end grouping matters a great deal. The double cloths a buyer handles are finished cloths. So the claim is a true report of what finished double cloths of fine yarn do, generalised into a property of layering, which is exactly the step the arithmetic says cannot be taken.

What a buyer can check #

The distinction is testable without a laboratory. A cantilever bending test is a strip pushed off the edge of a table until its tip droops to a fixed angle; the overhang at that moment, cubed, is proportional to the rigidity. Measure a strip of a double cloth, then separate a strip of the same cloth into its two layers and measure each.

If the two layers’ rigidities add up to about the whole cloth’s, the layers slide and the cloth sits at its sliding bound; the stitching adds nothing to its stiffness. If they add up to much less, the stitching is carrying the layers towards the fused bound, and the cloth is stiffer than its layers would be loose. And comparing either against a single cloth of the same weight and yarn says where the yarn sits in its own bracket — which, for a cloth that has been finished, is the number that decides whether the double construction bought any drape at all.

Because the rigidity goes as the cube of the overhang, the differences are easy to see: a factor of two in rigidity is a quarter again in overhang, which on a strip of ordinary length is a couple of centimetres at the table’s edge.

What was counted, and how #

The yarn’s diameter is the volume arithmetic of its count, fibre density and packing; the ends per millimetre follow from the single cloth’s cover. The two bounds on a yarn’s bending stiffness are the ones the yarn-stiffness essays use: the fibre count times one fibre’s stiffness, and a solid rod of the yarn’s diameter. A cloth’s rigidity per unit width is taken as the ends per millimetre times one yarn’s stiffness, summed over the layers.

Fused layers are taken as rows of solid yarns stacked a yarn diameter apart, each contributing its own second moment and its area carried at its distance from the middle of the stack, which gives a factor of one plus four-thirds of (k² − 1) over the sliding stack. The gap straight through a stack is the product of the layers’ gaps, which is the mean over every registration and not the value at any one.

Across cotton, wool and polyester, at 10, 20 and 40 tex and at one to six layers, both divisions were confirmed to carry exactly the single cloth’s yarn per area; the free-bound rigidity to equal the single cloth’s to twelve figures; the locked-bound rigidity to be one over the layer count divided by count and exactly one divided by sett; the thickness to grow as √k and k; and the fused rigidity to carry the stack factor.

Where the model stops #

A cloth is not a row of independent yarns. A woven cloth’s bending rigidity includes the crossing yarns, the friction at every crossing and the crimp that lets a yarn straighten rather than bend; the ends-times-yarn product is the first term of that, not the whole. The identity at the free bound is exact for the first term and says nothing about the rest, which may itself differ between a single cloth and a double one.

The layers are taken as touching rows a yarn diameter thick. Crimp thickens every layer by an amount its weave decides, and a real double cloth has its layers interlaced where it is stitched and apart where it is not. Both change the thickness and the fused factor, and neither changes the free-bound identity.

A yarn is at one bound or the other. A real yarn is in between, and where depends on its twist, its finish and how it has been handled. The finding is that the free end and the locked end give opposite answers to the question the trade answers confidently, and that the answer for a given cloth is a measurement of its yarn.

And warmth is not thickness alone. An open stack loses heat by air moving through it, and the gap through the stack is a mean over registrations in one thread direction only.

Still open: where a stitched double cloth sits between sliding and fused #

The bracket has two measurable ends and a measurable middle. Cut a strip from a stitched double cloth and measure its bending length on a cantilever; unpick the stitches over the strip’s length and measure again; then bond the two layers with a thin adhesive film and measure a third time. The three numbers place the cloth between its own sliding and fused bounds, and the same strips woven in an unset and a set version of one yarn place the yarn.

That is an afternoon’s work with a cantilever tester and it would settle, for any one cloth, whether its double construction is buying drape or only thickness. Until it is done, a double cloth’s drape is a claim about its yarn and its stitching, not about its layers.

Who worked it out #

The bending rigidity of a woven cloth as yarns times yarn stiffness is Peirce’s, from his work on the handle of cloth in 1930, and the bracket on a yarn’s stiffness between free and locked fibres is as old as the mechanics of twisted structures. The stiffening of a stack by joining its layers is the parallel-axis theorem, and the whole theory of sandwich and composite beams rests on it. That a double cloth drapes better than a single one of the same weight is trade knowledge, and the finding here is how narrow the conditions are under which it is true.