Textile Structure
how cloth is put together
Setting and geometry

A weight fixes the fibre and not the drape

Every plain cotton cloth of 150 grams a square metre contains the same fibre, and the count decides only how it is arranged. Across the counts that can make that weight, thickness rises threefold and cover falls in step, so their product holds still. The bending length does something stranger: at the bound a woven yarn actually sits near, it depends on neither the count nor the weight, only on the fibre.
17 min read 6 figures Measured, not claimedExactly this many

Worth reading first: What a fabric weighs · A yarn's stiffness is a bracket, not a number.

A hundred and fifty grams a square metre is a specification. What a fabric weighs showed how little of a cloth it specifies: the weight is a sum of setts times counts times one plus the crimp, one equation in four unknowns, and a hundred and fifty grams can be an open coarse cloth or a close fine one with cover factors nearly three times apart. That essay ended by naming what it had not computed. Two cloths of one weight can differ twofold in thickness, and thickness is what decides warmth, bulk and how a garment sits.

Thickness needs the cloth’s section rather than its count, and computing it turns up a result that essay did not anticipate. A weight line splits the properties of a cloth into two kinds. Some move with the count, steadily and by large factors. Others do not move with the count at all, and one of the ones that does not move is the one a buyer is most likely to think the weight describes: how stiffly the cloth bends.

What the count moves at 150 gramsPlain cotton cloths that all weigh 150 g/m², from 20 tex to 200 tex, with sett and crimp solved together. Thickness, which is two yarn diameters, rises 3.16 times across the line; the cover factor of each thread system falls 2.75 times; their product with one plus the crimp is the same number at every count, because the weight has fixed the volume of fibre and the count only decides whether it is laid out flat or stacked up.At 150 g/m² the count trades thickness for cover, 3.16 times one against 2.75 times theotherevery point is a plain cotton cloth weighing 150 g/m², its crimp solved with its sett; each line is aratio to the 20 tex cloth, the finest that can be woven012350100150200count, texratio to the 20 tex cloththickness ×3.16cover ×0.36cover × thickness× (1 + crimp): 1sett and crimp solved together; thickness from Peirce's section150 g/m² · cotton
Fig. 1 Balanced plain cotton cloths that all weigh 150 g/m², from the finest count that can make that weight as one cloth, 20 tex, to 200 tex. Each is solved for the sett and the crimp together. Thickness rises 3.16 times across the line and the cover factor of each thread system falls 2.75 times; the flat line is their product with one plus the crimp, which is the same at every count. What the chart cannot show is the fibre, which is the one thing every point on it shares in exactly the same amount.

The claim

At a fixed areal weight the count trades thickness against cover and leaves their product alone. Across the plain cotton cloths that can weigh 150 grams, thickness runs from 0.33 mm at 20 tex to 1.06 mm at 200 tex, the cover factor of each thread system from 0.54 to 0.19, and cover times thickness times one plus the crimp is one number throughout.

The bending length at the free bound does not depend on the count, and does not depend on the weight either. It is 12.7 mm at 20 tex and 13.3 mm at 200 tex, and the half-millimetre between them is the crimp. It comes out of the fibre’s modulus, its diameter and its density and out of nothing else. The count controls how far a cloth can stiffen above that floor, in proportion to itself, and the weight controls neither.

Why the crimp has to be solved, not assumed

The first essay on weight held the crimp at a stated seven per cent, and for cover that is harmless. For thickness it is not, because the crimp depends on how close the threads sit, and how close they sit depends on the count.

The weight fixes sett × count × (1 + crimp). A finer yarn is set closer in proportion to its count, but its diameter shrinks only as the square root of its count, so relative to its own diameter a finer yarn sits closer to its neighbours. Closer threads bend more sharply around each other, which is what Peirce’s geometry computes. More of each thread’s length goes into the bends, which puts more grams into each end, which lowers the sett the weight needs, which eases the bending. The sett and the crimp have to be found together.

The loop has one root. Higher crimp lowers the sett, and a lower sett gives the geometry less crimp, so the gap between the crimp assumed and the crimp returned falls steadily and crosses zero once. It is found by bisection. A sett past the jam counts as needing more crimp, and a root that turns out to be the jam itself — where the geometry stops answering rather than agreeing — is refused, because a cloth that only exists at the jam is not a cloth at that weight.

The crimp a weight line solves for. Crimp, per cent, solved with the sett along the lines of plain cotton cloths at 100, 150, 250 g/m², each from the finest count that can make that weight to 200 tex. At 100 g/m² it runs from 15.2% at 10 tex to 0.9% at 200 tex; At 150 g/m² it runs from 17.0% at 20 tex to 1.9% at 200 tex; At 250 g/m² it runs from 18.8% at 50 tex to 5.2% at 200 tex.
Fig. 2 The crimp each plain cotton cloth settles at, along three weight lines. Each line starts at the finest count that can make its weight as one cloth, and there the crimp is highest: 15.2 per cent at 10 tex for 100 g/m², 17.0 per cent at 20 tex for 150 and 18.8 per cent at 50 tex for 250. Coarser counts sit further apart relative to their diameter and crimp less, down to under two per cent at 200 tex on the two lighter lines. What the chart cannot show is the count just finer than each line’s first point, where no crimp makes the weight without the threads jamming.

The finest end of each line is where this matters most. At 150 grams a 20 tex cloth has to carry 17 per cent crimp. Holding it at seven per cent would have called for nine per cent more ends than the cloth actually has, and put it past the jam. That is why the weavable line at 150 grams starts at 20 tex, and why a 10 or 15 tex yarn cannot make a 150-gram plain cloth at all. It has to be set so close to carry the grams that its threads jam before they get there.

Thickness is two diameters

A balanced plain cloth in Peirce’s geometry is exactly as thick as two of its yarn diameters, whatever its sett. The closure condition says the two crimp heights add up to one sum of diameters. A balanced cloth splits that sum evenly, and a diameter sits on top of each crimp height. So the thickness is 2d, and d goes as the square root of the count.

That makes a weight line’s thickness a square-root curve, and nothing about the sett or the weight enters it. The 0.33 mm of the 20 tex cloth and the 1.06 mm of the 200 tex cloth are the two yarns’ diameters doubled, and their ratio is 10\sqrt{10}.

The cover factor goes the other way by nearly the same square root. Each thread system’s cover is sett times diameter. Sett goes as one over the count and diameter as the root of it, so cover goes as one over the root of the count. The two exponents cancel in the product: cover times thickness is sett times two diameters squared, and sett times a diameter squared is sett times count, which is the weight. The crimp is the only thing left in the way, and multiplying by one plus the crimp takes it out.

So the product in the first figure is not a fit; it is the weight, rearranged. The weight has fixed a volume of fibre per square metre, and the count decides whether that volume is laid flat or stacked up. A fine yarn lays it out as a thin sheet that covers most of the area. A coarse one piles the same volume into a thick open lattice. Thread count quotes neither.

The ratios do not quite mirror each other — thickness 3.16, cover 2.75 — and the difference is the crimp again. The finest cloth carries 17 per cent crimp and the coarsest under 2 per cent, so the finer cloth has more of its yarn tied up in bends and less left to cover with.

How much of a cloth is fibre

A thickness and a weight together give a density, and a density divided by the fibre’s gives the share of the cloth’s volume that is fibre rather than air. How dense a knitted fabric is found a jersey about a quarter fibre. A woven cloth at one weight is anywhere in a wide range, and the count decides where.

How much of a cloth is fibre. Share of the cloth's volume that is fibre along the lines of plain cotton cloths at 100, 150, 250 g/m², each from the finest count that can make that weight to 200 tex. At 100 g/m² it runs from 0.28 at 10 tex to 0.06 at 200 tex; At 150 g/m² it runs from 0.30 at 20 tex to 0.09 at 200 tex; At 250 g/m² it runs from 0.31 at 50 tex to 0.16 at 200 tex.
Fig. 3 The fraction of each cloth’s volume that is fibre, along three weight lines of plain cotton. At 150 g/m² it falls from 0.30 at 20 tex to 0.09 at 200 tex. A heavier line sits higher at the same count, because it holds more yarn in a cloth of the same thickness, and it starts at a coarser count. What the chart cannot show is the fibre’s own share of each yarn, which is held at 0.6 throughout, so the curves are the cloth’s openness and not the yarn’s.

At 150 grams the fine end of the line is 30 per cent fibre and the coarse end 9 per cent: the same grams in a third of the density. The reason is the same pair of square roots. Thickness grows as the root of the count and the weight does not grow at all, so the density falls as one over the root of the count.

Both ends have a use. The dense thin cloth is the one that stops wind, since its cover factor is high and there is little open area for air to go through. The open thick one holds more still air per square metre, which is what makes a knit warm, but only if something stops that air moving. A weight on a label says which of these a cloth is no better than a thread count does.

Stiffness has two bounds, and the weight fixes one of them

Everything so far moves with the count. Bending does not, at least not at the bound that matters, and the reason is worth following closely.

A yarn’s bending rigidity is a bracket, not a number. At the free bound the fibres slide past one another, and the yarn is as stiff as its fibres bent separately: the number of fibres in its section times one fibre’s rigidity. At the coherent bound the fibres are locked and the yarn bends like a solid rod of its own diameter, which is stiffer by a factor equal to the fibre count divided by the square of the packing. For a thread bent in a woven cloth, whether fibres can slide decides where it sits, and the answer found there was that it sits near the free end.

A cloth’s bending rigidity per unit width is its yarn’s rigidity times its sett. Take the free bound: sett, times fibres per yarn, times one fibre’s rigidity. Fibres per yarn is the count divided by the fibre’s own linear density. So the free rigidity per unit width is sett times count, times a fibre constant. Sett times count is the weight with the crimp taken out. The count has cancelled.

So at 150 grams every cloth on the line has the same free rigidity per unit width, apart from its crimp: 3.00 µN·m at 20 tex, 3.45 at 200 tex. The whole 15 per cent between them is the finer cloth’s greater crimp, which has put more of its fibre into the bends and less across the width.

The coherent bound is sett times the fourth power of a diameter, and a diameter to the fourth is a count squared. Sett times count squared is weight times count. So at one weight the coherent rigidity grows in proportion to the count: 981 µN·m at 20 tex, 11,265 at 200.

The bending length has no weight in it

A cloth’s bending length, the quantity a cantilever test reports, is the cube root of its rigidity over its weight. The free rigidity is the weight times a fibre constant, so the weight cancels a second time:

free bending length=(E df232 ρ g (1+c))1/3\text{free bending length} = \left(\frac{E\,d_f^2}{32\,\rho\,g\,(1 + c)}\right)^{1/3}

with EE the fibre’s modulus, dfd_f its diameter, ρ\rho its density, gg gravity and cc the crimp. There is no count in it, no weight and no sett. For cotton that is 13 mm, and for every plain cotton cloth that can weigh 150 grams it lies between 12.7 and 13.3.

Bending length at 150 grams. The bending length of plain cotton cloths along the weight line at 150 g/m², computed from each yarn's two bounds on bending rigidity. At the free bound, where the fibres slide past each other, every cloth has a bending length between 12.7 and 13.3 mm, and what little movement there is comes from the crimp. At the coherent bound, where the yarn bends as a solid rod, the 150 g/m² line runs from 87 mm at 20 tex to 197 mm at 200 tex. Real cotton cloths measure rigidities that give 9 to 34 mm at this weight.
Fig. 4 The bending length of each plain cotton cloth on the 150 g/m² line, at both bounds on its yarn’s rigidity. The solid line is the free bound, 12.7 mm at 20 tex and 13.3 mm at 200. The dashed line is the coherent bound, which rises from 87 mm to 197 mm, roughly as the cube root of the count. The shaded band is the rigidity real cotton cloths measure, 1 to 60 µN·m, read as a bending length at this weight: 8.8 to 34 mm. What the chart cannot show is where in that band any particular cloth sits, which is decided by how freely its fibres slide and not by anything on this axis.

The free line lies inside the band of what real cloths measure, and the coherent line lies far above it at every count. That agrees with what four ordinary observations of cloth had already said about where yarn sits in the bracket. It also means the figure is not only about a lower limit. Real cloths start near the free line and rise some way towards the coherent one. The free line says where they start, and on this line it says they all start in the same place.

The coherent length has a weight-free form too. Its rigidity is weight times count, divided by the weight, so what is left is the count. The coherent bending length belongs to the count and the free one to the fibre, and neither belongs to the weight.

Bending length at three weights. The bending length of plain cotton cloths along the weight lines at 100, 150, 250 g/m², computed from each yarn's two bounds on bending rigidity. At the free bound, where the fibres slide past each other, every cloth has a bending length between 12.6 and 13.3 mm, and what little movement there is comes from the crimp. At the coherent bound, where the yarn bends as a solid rod, the 100 g/m² line runs from 70 mm at 10 tex to 198 mm at 200 tex, and the 150 g/m² line runs from 87 mm at 20 tex to 197 mm at 200 tex, and the 250 g/m² line runs from 118 mm at 50 tex to 195 mm at 200 tex.
Fig. 5 Bending lengths at both bounds along three weight lines of plain cotton, 100, 150 and 250 g/m². The three free lines lie on top of one another between 12.6 and 13.3 mm. The three coherent lines lie on top of one another too, and separate by only the few per cent their different crimps put between them. A cloth two and a half times heavier than another, of the same count, has the same bending length at either bound. What the chart cannot show is the drape of a heavier cloth under its own weight over a longer span, which does depend on the weight, because the bending length is a ratio in which the weight appears on both sides.

That last point is where this sits against experience, and it needs care. A heavy cloth does hang differently from a light one. But a bending length is the length over which a strip’s own weight bends it by a fixed amount, so a heavier cloth that is proportionally stiffer bends by the same amount over the same length. Heavier cloths are usually stiffer out of proportion, because heavier cloths are usually made of coarser yarns, harder twisted and more heavily finished, all of which push them towards the coherent end. The weight is associated with the stiffness. It is not the cause of it.

What the fibre decides

If the free bending length is a fibre property, then fibres should rank by it, and the ranking is not the one a modulus table suggests.

Bending length by fibre. Seven fibres, each made into a balanced plain cloth of 50 tex yarn weighing 150 g/m². cotton: free 13.1 mm, coherent 122 mm; wool: free 14.8 mm, coherent 97 mm; silk: free 12.6 mm, coherent 132 mm; flax: free 31.1 mm, coherent 241 mm; polyester: free 15.3 mm, coherent 149 mm; nylon: free 11.3 mm, coherent 106 mm; viscose: free 13.1 mm, coherent 122 mm. At the free bound the bending length is set by the fibre's modulus, its diameter squared and its density, with the crimp as the only property of the cloth left in it.
Fig. 6 Seven fibres, each made into a balanced plain cloth of 50 tex yarn weighing 150 g/m², with the free bending length drawn as a bar and the coherent one given beside it. Flax, at 60 GPa, is stiffest at 31 mm. Wool, at a modulus of only 3 GPa, comes out stiffer than cotton at 8 GPa, because its fibre is nearly twice as thick. Nylon, with the same modulus as wool and a finer fibre, is the most supple at 11.3 mm. What the chart cannot show is a microfibre, whose diameter the fibre table does not carry, and which the same arithmetic would put, at half the diameter, at about two thirds of its ordinary counterpart.

The modulus enters under a cube root, and so does the fibre diameter squared. Wool’s modulus is less than half of cotton’s, but its fibre is 22 µm across against cotton’s 12, and 22 squared is more than three times 12 squared. Its free bending length comes out at 14.8 mm against cotton’s 13.1. Nylon has wool’s modulus on a 14 µm fibre and comes out at 11.3. Polyester, stiffer than cotton on a fibre of the same diameter, is 15.3.

The diameter term is the one worth carrying away, because it is the one a fibre producer can change. Halve a fibre’s diameter and its free bending length falls by the cube root of four, to 63 per cent, whatever the count and whatever the weight. That is the arithmetic behind the softness of microfibre cloths. It says the softness comes from the fibre and cannot be had by spinning an ordinary fibre finer or weaving it lighter.

What was computed, and how

Every cloth is a balanced plain weave of one yarn in both directions, with the crimp shared equally between warp and weft, at a yarn packing of 0.6. Its sett and crimp are solved together by bisection on the crimp, with Peirce’s circular-section geometry returning the crimp for each trial sett. A cloth whose only solution is the jam is left off its line. Every cloth is weighed again from its solved sett and crimp and must come back at its line’s weight to a millionth.

Four identities are checked on every cloth to nine figures. Thickness is exactly two diameters. Cover times thickness times one plus the crimp is one number along a line. Free rigidity times one plus the crimp is one number along a line, and coherent rigidity times one plus the crimp over the count is another. The free bending length equals its closed form above, computed from the fibre table alone. A fifth check takes the same count at 100 and 250 grams and requires both bending lengths to agree once the crimp is taken out, since the claim that the weight cancels is the claim a single line cannot test.

The fibre moduli, fineness and densities are the collection’s own table, and the measured band of cloth rigidities is the one the bracket essay put its two bounds against.

Where the model stops

The section is circular. Real yarns flatten in a cloth, and a flattened yarn makes a thinner cloth with more cover. Flattening shifts both ends of the thickness line down without changing that a weight fixes a volume. It would bend the square roots, though, since a coarse yarn flattens more than a fine one.

The weave is plain and balanced. A twill or a satin at the same weight crimps less and so puts slightly more fibre across the width. The free bending length would move by the cube root of a few per cent. An unbalanced cloth has two bending lengths, one per direction, and its drape is directional.

The free bound is a bound. The claim that the count and the weight both cancel is exact at the free bound and at the coherent one. A real cloth sits between them, and where depends on twist, finish and wear, so its real bending length does depend on the count through the coherent end. What the free bound gives is the part of a cloth’s stiffness that nothing about its construction can change.

Crimp interchange under load is not modelled. Every number here is the cloth at rest.

The generalisation

A quantity that is a sum over the threads in a cloth is fixed by the weight, whatever the arrangement. A quantity that depends on how the threads are grouped moves with the count. The free rigidity is a sum over fibres, so the weight fixes it. The coherent rigidity depends on the fibres being grouped into rods, so the count drives it. Cover and thickness are the two directions a fixed volume can be spread in. A group of threads is one thread for cover and two for bending found the same split one level up, at the scale of threads grouped into a rib rather than fibres grouped into a yarn.

The practical form is a question to ask of any specification: is the property wanted a sum, or a grouping? If a sum, the weight settles it and the rest of the specification is irrelevant to it. If a grouping, the weight says nothing and the count, the twist and the finish say everything.

Who found it

Peirce’s geometry is from 1937, and his paper on the handle of cloth in 1930 defined the bending length as a cube root of rigidity over weight precisely so that the weight would be divided out. The free and coherent bounds on a yarn’s rigidity are Platt, Klein and Hamburger’s, from their work on yarn bending in the 1950s, and Grosberg’s analyses of fabric bending in the 1960s made the frictional picture between them explicit. That a cloth’s weight and its stiffness go together is everyday knowledge in the trade. Solving the weight line with its crimp, and noticing that at the free bound the division leaves only the fibre, is new here.

What comes next

The weight line has been drawn at rest. The obvious next question is what happens to it under wear. Wear loosens fibre contacts and so moves a cloth towards the free bound, and every cloth on a line then converges on the same bending length — which predicts that old cloths of one fibre drape more alike than new ones.

Sideways, the knitted version of the same line has a different geometry and one extra freedom, the loop length, and whether its bending length also sheds the weight is not obvious.

Further out, the volume fixed by a weight is also what a finish has to wet and dry. Drying time should then belong to the weight rather than to the count, which is the opposite of what the thickness line suggests and worth computing before anybody believes either.

Where this goes next

Every cloth at 150 grams. The counts and setts that all weigh 150 g/m² at a crimp of 7 per cent and a balance of 1. Hollow marks are past the jam — arithmetic rather than cloth. Among the 10 that can be woven the cover factor runs from 0.28 to 0.77, a factor of 2.77, and every one of them is the fabric the specification asked for. What a fabric weighs Every fabric is sold by its weight in grams per square metre, and the number is a sum of four products in which no term appears alone. One equation, four unknowns: a hundred and fifty grams describes an open coarse cloth and a close fine one, and the cover factors differ by a factor of nearly three. The two calculations a yarn's stiffness admits. A bundle of 9 fibres bent with the fibres free to slide and with them locked together. Free, the rigidity is the sum of the fibres': 0.00141 N·mm² for a 30 tex cotton yarn. Locked, it is the fourth power of the yarn's own diameter: 0.689. The ratio is the fibre count over the square of the packing factor, 490, and nine fibres are drawn where the yarn has 176. What the drawing cannot show is where a real yarn sits between them, which is a question about friction rather than about fibre. A yarn's stiffness is a bracket, not a number Two calculations are available for how stiff a thread is in bending, and both are exact. One treats the fibres as free to slide and gives the sum of their stiffnesses; the other treats them as locked and gives a solid rod. They differ by the fibre count, which for an ordinary cotton yarn is a factor of five hundred — and no measurement of the fibre narrows it by anything at all. The cantilever test. A strip of cloth pushed out over an edge until its tip has drooped to the stated angle. The overhang at that moment gives the bending length, and cubing it with the mass per unit area gives the flexural rigidity. Bending stiffness and the drape coefficient Two standard measurements try to say how a fabric hangs. One measures a length and cubes it; the other measures an area and, on the geometry, turns out to be answering a different question from the one it is asked.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

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Areal densityBending lengthBending rigidityCloth thicknessCover factorCrimpFibre finenessPeirce's geometrySolid fractionSpecification