Textile Structure
how cloth is put together
Concept

Capstan — where it appears

Euler's result that a line wrapped around a surface multiplies tension by e to the friction times the wrap angle, whatever the radius. Its exponential is what makes a gripped length compound rather than add, and the linear account is its tangent at the origin.

Named by 28 essays across 6 fields — each of them below, with the objects they name alongside it.

What a tuft's wrap is worth at μ = 0.3. The two classical ways of binding a cut pile into its ground, with the holding force each provides as a multiple of the tension applied to the free end. The model is the capstan equation and the friction coefficient is measured rather than derived, so the ratio between the two matters and neither absolute number should be quoted alone.

How a tuft is held

A V-fastened tuft and a W-fastened tuft are both attached, and the integrity criterion returns the same verdict for both. One of them is specified for hotel corridors and the other sheds. What separates them is not a topological property at all — it is an angle, and it needs a coefficient somebody has to measure.

fancy · Pile The ratchet a wool fibre is. A fibre with its scales, and the two strokes of one cycle of agitation. The push is the same in both directions; the distance is not, because the scales resist tip-first motion more than root-first. Every cycle therefore nets a displacement in one direction, and no amount of further agitation undoes it.

The ratchet that makes wool felt

A wool fibre is covered in scales pointing one way, so it slides more easily root-first than tip-first. Agitate it and the motion is symmetric while the result is not — every cycle nets a displacement in one direction, and no amount of further agitation undoes it.

finishing · Felting The force at a crossing. One warp end of a sheeting riding over three picks, with the weave angle Peirce's geometry solves for at that construction: 36.8°. An end held at 0.50 N presses each pick it crosses with 0.599 N, which is twice the tension times the sine of the angle and has no material constant in it. What the drawing cannot show is the relaxed case: a cloth with no tension in it still holds its threads together, and what does the holding then is the yarn's own resistance to being bent, which needs an elastica this site does not have.

Every crossing is a force

A thread arrives at a crossing at an angle and leaves at its negative, so the two pulls have transverse parts that add. The force pressing one thread onto another is twice the tension times the sine of the weave angle — and for an ordinary sheeting that is more than the tension in the thread itself.

cloth · Contact Where a thread stops sliding and starts breaking. A pick of sheeting gripped over a length of cloth, drawn one crossing at a time. The resistance is 0.30 times the 0.599 N each crossing presses with, so it rises with the length held; the breaking load of 3.74 N does not. The two are equal at 7.4 mm. What the drawing cannot show is that μ is a range rather than a constant, so the mark is a band and its position is exactly inversely proportional to the friction.

A thread is held one crossing at a time

Grip a thread over a length of cloth and its resistance to being pulled out rises with that length, because it is held at every crossing it makes. Its own breaking load does not rise at all. The two curves cross, and the length at which they cross turns out to be a seam allowance, a frayed edge and a tuft's anchorage — three rules of thumb with one number under them.

cloth · Contact A thread withdrawn from a plain weave and from a satin. One pick of a sheeting being pulled from 1.60 mm of cloth, in a plain weave and in a five-end satin, at a friction coefficient of 0.30. The drawn width of each thread is the tension in it at that point, to a common scale; the faint rules are where the thread actually turns, which is at its interlacings and nowhere else. A plain weave turns 2.80 times per millimetre and a satin 1.12, so the plain weave's tension compounds 2.78-fold over this length against the satin's 1.46. What the drawing cannot show is that the wrap angle is taken from a plain-weave geometry in both panels: a satin's crimp is genuinely smaller, so its real turns are gentler than these and its grip weaker still.

A thread is gripped where it turns

The arithmetic this site has used for fraying, seam slippage and tuft anchorage counts every crossing a thread makes as a grip and adds them up. A thread lying flat on the surface of a satin presses on nothing at all, and a thread that is gripped is gripped by a friction that compounds along its length rather than adding. Both corrections were recorded as missing and both are here.

cloth · Contact What holds a pick in. The holding force on one weft, as a multiple of the tension applied to its free end. For a plain weave it is the capstan on twice the weave angle, which Peirce's geometry gives at each sett and which falls towards nothing as the cloth opens out. For a leno it is the capstan on a half-turn, which the sett does not enter at all.

What holds a pick in

A plain weave grips its weft by the crimp, and the crimp's wrap angle falls towards nothing as the cloth opens out. A leno crossing is half a turn whatever the sett. So the comparison has an exact answer that does not depend on the friction coefficient at all — and no plain weave can ever reach a leno's grip.

fancy · Leno What holds a thread in a seam. A cloth thread inside a seam allowance, drawn past the crossings that grip it, with the capstan's factor at each. The tension falls by that factor at every crossing, so the grip is exponential in the crossing count and the count is what the weave decides. Beyond the thread's own strength the thread breaks rather than slides, and the rest of the allowance holds nothing.

What holds a thread in a seam

A seam fails in two quite different ways, and the one the trade worries about is not the stitches breaking. It is the cloth's own threads sliding out of the weave beside the seam — and what resists that is friction at the crossings, accumulating multiplicatively, so a satin gives a seam a thousandth of the grip a plain weave gives it.

applied · Seams A leno and the open plain weave it is not. Two warp ends and the picks they hold. On the left the doup end passes under its partner between picks and comes up the other side; on the right it never crosses, which is an open plain weave at the same sett. The layer count under each panel is computed by the same criterion that decides every other draft, and it returns the same verdict for both.

The criterion cannot see friction

This site's central check is exact, decidable in linear time, and structurally incapable of distinguishing a carpet from a fabric that sheds. That is not a defect to be repaired — a separation exists or it does not, and there is no margin in it — and the fancy weaves are precisely the constructions that live in the gap.

fancy · Integrity A tear breaks threads or pulls them out. The grip a cloth has on a thread at the tip of a tear, against the sett, with the thread's own strength drawn across. Below the line the thread slides and the yarns group, which is the trade's explanation of why a loose weave tears well; above it the thread breaks where it is and grouping never happens. The essay that asked it could not compute this: it needed a friction, and the fancy weaves supplied one.

A tear stops where the grip is

This collection asked whether a loose weave tears better and had to answer that the geometry could not say — it supplies a grip count and a slack, and neither is a force. The fancy weaves brought a friction. With it the question has an answer — a cloth's threads slide up to a computable sett and break above it — and a ripstop grid has a bound with no free parameter in it.

applied · Damage What holds a tuft in, in newtons. The withdrawal force of a V-fastened and a W-fastened tuft over the reported range of yarn-on-yarn friction, on a duck ground with 1.00 N in each pick. Each is the friction at the wraps, with each wrap's share dragged around every wrap between it and the pulled end — a capstan series rather than a single factor. W runs from 0.79 to 6.84 N and V from 0.15 to 0.41. Both are an order of magnitude below what a carpet is specified at, which is a finding about carpets rather than about the model. What the chart cannot show is the backing, which is where the rest of a tufted carpet's anchorage comes from.

What holds a tuft in, in newtons

The pile ladder computed a tuft's anchorage as a capstan ratio and said, correctly, that a ratio was all it could offer. A ratio multiplies a tension and there was no tension anywhere on this site. There is one now — and the answer, put beside what a carpet is actually specified at, falls short by a factor of three.

fancy · Pile How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

The criterion gets a force

This site's integrity criterion decides exactly whether a draft describes one cloth, and has one standing limitation: it says a tuft bound under one pick and a tuft bound under three are both attached, and it is right, and one of those is a carpet while the other sheds. What separates them needs a normal force in a fabric that is not under tension — the number the rung that computed it recorded as unavailable, and the one a thickness gauge now supplies.

fancy · Integrity The size of the capstan correction. The ratio of the capstan crossover to the crossover a sum of independent contacts gives, for every cloth in the table at a friction coefficient of 0.30. It is exactly ln(1 + z)/z, where z is the thread's breaking load times the wrap angle, over the contact force — a quantity with no friction coefficient in it at all. That is why the earlier result that μ·L* is exactly constant survives this correction to twelve figures: μ was only ever in the factor outside the logarithm. The correction is largest for the duck, whose coarse strong yarn makes z large, and smallest for the batiste. What the rows cannot show is that a real cut edge frays at a friction nobody measured on that particular cloth.

How far a cut edge frays

A seam allowance, a fray width and a tuft's bound length are the same number wearing three hats, and the earlier estimate of it was four times too long. Correcting it moves the whole table across the boundary an ordinary allowance sits on — from four cloths holding and four slipping, to all eight holding — and turns a specification argument into a different one.

applied · Seams How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

The relaxed cloth's contact force

How hard two threads press on each other in a cloth that is not being pulled is the number this site's own integrity criterion has needed since its first essays, and the route to it was an elastica nobody had. A thickness gauge supplies it instead — because a cloth's thickness is a record of how flat its threads are, and how flat they are is a record of how hard they are pressed.

mechanics · Compression The crossover length in four weaves. The gripped length at which a pick of a sheeting breaks rather than slides, in four weaves at a friction coefficient of 0.30. It is a millimetre or two for a plain weave and 8.0 mm for an eight-end satin, which is what a cut edge of each does. The interlacing rate appears in the answer only as a factor outside the logarithm, so the ratios between the four are exactly the ratios of their interlacing rates — in this model and in the sum of independent contacts alike. The rung below expected the float correction to change that ordering; it does not. It changes the size, by a factor of 3.7. What the rows cannot show is that all four use one cloth's crimp, so a satin's genuinely gentler turns are not in them.

A float presses on nothing

The rung below expected the float correction to change how a satin's hold compares with a plain weave's, by something like the ratio of their interlacing rates. It does not change the comparison at all — the interlacing rate leaves the answer outside the logarithm and divides straight out of any ratio. What it changes is the absolute answer, by a factor of four, for every weave alike.

weaves · Float A V-fastened tuft. A cut pile bound into its ground by V fastening, drawn in section. The pile end passes beneath 3 of the 6 ground picks and wraps 1 half-turn around them in all. The integrity criterion says the tuft is attached; how hard it is held is a different question with a different model behind it.

A chenille is a yarn that is already a fabric

A chenille yarn is a woven gauze cut into strips. Every structural question this collection asks about cloth can be asked about it — does it hang together, how many systems has it, what holds the pile in — and the answers are the same answers one level down.

fancy · Fancy yarn The pressure a 23° twist puts on its own fibres. A twisted yarn squeezes itself. Every fibre under tension at a radius pulls inward with sin²θ of its tension per unit length, and integrating outward to the surface — where the pressure is zero by definition, because there is nothing outside to push against — gives p(r) = ½σ(cos²θ(r) − cos²α) in closed form. At a surface angle of 23° the pressure on the axis is 7.6 per cent of the core fibre's own axial stress and the mean over the section is 3.61 per cent of it. The shading is that closed form and the curve is the same function plotted; the shape is what matters, and its one uncompromising feature is the zero at the edge. The outermost fibres are held by nothing, which is why a yarn is hairy, why a surface fibre is the one that comes away on a finger, and why singeing changes a yarn's behaviour out of all proportion to the mass it removes.

What grips the end of a fibre

A twisted yarn squeezes itself, and the squeeze holds the fibre ends in. Write the slip and the break out side by side and the fibre's own strength cancels — and so does the load on the yarn — leaving a gripped length that depends on fineness, friction and the twist and on nothing else.

mechanics · Migration A knitted interlacing, and the number that makes it a fabric. Two loops idealised to rings, one drawn through the other, which is what a needle does. The Gauss linking integral returns 1. A knitted fabric of n wales has that between every pair of adjacent courses n times over, and it is the whole of why a knitted fabric can be made from one thread and taken apart by pulling it.

Why a knit runs and a weave frays

The two fabrics fail in two ways and everybody knows which is which. This collection has described both accurately for eighteen phases without being able to say what causes them, and the cause turns out to be one integer each: nought for a cloth, one per wale for a knit.

cloth · Topology A yarn is pressed on part of its length and free on the rest. The distance from each point of one course to the nearest point of the course below, for a 20 tex cotton jersey at a 3.5 mm loop, along two wales. It runs from 0.780 diameters at the worst to 3.81 at the freest, and 20% of the length is inside one diameter of its neighbour. The line at one diameter is where a round yarn would begin to overlap. What the profile says is that a single flattening ratio is an average: the section a yarn takes changes along its own length, which every racetrack section this collection has drawn assumes it does not.

A fabric is a population of contacts

Only a fifth of a knitted fabric's yarn is inside a diameter of its neighbour. So a fabric's friction lives in a fifth of its length, and every calculation this collection makes about withdrawal, slippage and fraying has assumed it lives everywhere.

cloth · Contact A jacquard harness 130 cm wide under a 150 cm fall. A jacquard harness in front elevation, to scale: hooks spread over 40 cm, cords fanning down 150 cm to a comber board 130 cm wide, and hanging straight from the board to their mails. The centre cord is vertical and lifts its mail by the whole 10 cm hook lift; the edge cord leans 16.7° and lifts its mail 9.60 cm, 96.0% as far, and its bend at the board adds 9% to the hook's load at a friction coefficient of 0.3. What the drawing cannot show is the tie that decides which hook feeds which hole, which in a real mount is not the simple spread assumed here.

A jacquard harness needs three half-spans of height

A jacquard has no front shaft and no back one, so every end takes the same shed — as long as every cord hangs straight. Across the width they cannot: the hooks sit in a machine a few tens of centimetres wide and the comber board is as wide as the cloth, so an edge cord leans and its mail rises by the difference of two hypotenuses rather than by the hook's lift. Holding the edge shed within five per cent of the centre's takes a fall of about three times the edge cord's sideways reach, and that is a height a room has to have.

fancy · Jacquard Where a loop's bending actually is. Half a stitch — from the crown of a needle loop's head to the bottom of the next sinker loop — with the curvature at every station drawn as a spine standing off the curve, in units of one over the yarn diameter. The centre line is drawn as a line rather than at the yarn's own width here, because the subject of this figure is the curvature and a yarn drawn at true width covers its own spines. The peak is 1.18, at 28% of the way along, and the curvature varies smoothly from one end to the other with no jump anywhere. That smoothness is the whole point: Peirce's construction joins an arc of constant curvature to a straight line of none, so its bending moment steps at the join, and a step in moment is a point force no thread can carry. An elastica has no steps in it, which is why its forces exist at all and his do not.

A force is what an energy does when a crossing moves

The force a thread presses its neighbour with is the rate its bending energy changes as the crossing is displaced. Solved as a constrained minimisation, that number arrives with the shape rather than after it — and the same force, recovered a second time from the curve's own equilibrium, agrees to a tenth of a per cent.

mechanics · Elastica A doup end crossing every pick at 1 mm. One leno pair of 0.25 mm ends over 2 crossing intervals, with picks 1 mm apart and the doup end changing sides every pick, drawn to scale in plan and in section. The standard end runs 1.00 mm per interval; the doup end, passing from 0.250 mm to one side, down 0.250 mm under its partner and up to the other side, runs 1.225 mm — 22.5% more, 11.8% of it from the sideways travel alone. Over a 100 m piece that is 22.5 m of warp, and on a shared beam a pick spacing of slack in 4.4 mm of cloth. What the drawing cannot show is the crimp both ends share over and under the picks, which the comparison cancels.

The doup end pays for the crossing

A leno holds its picks because its doup end crosses under its partner and comes up on the other side — half a turn at every crossing, whatever the sett. That half turn is also a length. When 0.25 mm ends cross every millimetre, the doup end travels 22.5 per cent further than its partner: twenty-two metres of extra warp over a hundred-metre piece, and a shared beam a pick short within four and a half millimetres of cloth. The extra falls with the square of the crossing interval, which is why a leno is woven from two beams and its crossings are spaced as far apart as the cloth allows.

fancy · Leno The crossed shed, with the back standard 4 shafts behind the doup. A leno's crossing end in section from the fell to the back rest of an ordinary broad loom, in the crossed shed. The doup 300 mm from the fell lifts it 50.0 mm; its back standard 64 mm further back leaves it 60.7 mm down, beneath its partner. Between the two eyes the end climbs 110.7 mm in 64 mm, and the whole path is 70.2 mm longer than the straight line, 5.85% of the free warp, of which 63.8 mm is that one climb. The same end in the open shed, lifted at its back standard, is 7.2 mm longer, and an ordinary end on the doup's shaft 5.5 mm: the crossed shed is 12.7 times the ordinary shed. Vertical scale exaggerated 3 times. What the drawing cannot show is friction at the eyes, which would keep the extra length in the short span between them.

An easer gives back the kink the crossed shed puts in

A leno loom gives its crossing ends slack at every crossed shed, and the obvious reason — the doup carrying its end sideways — is worth a hundredth of an ordinary shed. The length is one heddle further back. In the crossed shed the doup lifts the end on the far side of its partner while the end's own back heddle, a few shafts behind, holds it down, and between those two eyes the end climbs eleven centimetres in six and a half. On a loom that stretches an ordinary end 0.46 per cent, that is 5.85 per cent — nearly thirteen times as much — and no position of the back heddle takes it below four.

fancy · Leno How hard each grouping holds its own threads. The grip a cloth has on one of its own threads inside a 10 mm seam allowance, for six members of the doubled family at 24 threads per centimetre and a friction coefficient of 0.3. Grip accumulates multiplicatively at every crossing — the capstan equation on Peirce's own weave angle — so it is exponential in the crossings, and the crossings are the interlacing rate times the intersections in the allowance. The family's interlacing rate has a closed form, (a + b)/2ab, which is half the sum of the two reciprocals — so the two groupings enter symmetrically and each one saturates. The dashed line is the thread's own strength: a cloth whose grip falls short of it lets the thread slide out rather than break, which is seam slippage. A 2/2 hopsack is below it at this allowance and a 1/4 warp rib is above, on cloths whose firmness differs by an eighth. What the bars cannot show is the friction coefficient, which is measured and is not a constant of cloth; the ordering holds at every value anybody reports and the sizes do not.

What nothing separates comes out together

The doubled family's interlacing rate has a closed form — half the sum of the two groupings' reciprocals — so a seam's grip on its own threads is the exponential of a harmonic mean, and it saturates in each grouping separately. At a ten-millimetre allowance a 2/2 hopsack holds a thread at seventy-four times the applied tension and a 1/4 warp rib at two hundred and eighteen, on cloths whose firmness differs by an eighth.

weaves · Rib weaves A knot, and the tension falling through it. A 20 tex cotton thread wrapped through 1 turn at a bend radius of 1 yarn diameters, with a coefficient of friction of 0.30. The wrap is drawn as a spiral because a thread taken round a pin comes back beside itself rather than onto itself. The marks round it are the fraction of the entry tension still there: 100%, 69%, 47%, 32%, 22%, 15%. The bend has already spent 3.6% of strain at the outside of the thread before any of that happens, against a breaking strain of 6.6%. So the largest total is at the entry, before the knot has done any gripping at all — which is where a knot in a real yarn is observed to break, and why a knot's efficiency is a property of its first bend rather than of the knot.

A knot is nothing but contact

A knot has no fastening in it. Nothing is glued, hooked, sewn or threaded through a hole: a thread is bent round itself until the friction where it presses on itself is more than the load. That makes a knot the purest contact problem in the subject, and the place to look first for what contact does.

applied · Contact A knot, and the tension falling through it. A 20 tex cotton thread wrapped through 1 turn at a bend radius of 1 yarn diameters, with a coefficient of friction of 0.30. The wrap is drawn as a spiral because a thread taken round a pin comes back beside itself rather than onto itself. The marks round it are the fraction of the entry tension still there: 100%, 69%, 47%, 32%, 22%, 15%. The bend has already spent 3.6% of strain at the outside of the thread before any of that happens, against a breaking strain of 6.6%. So the largest total is at the entry, before the knot has done any gripping at all — which is where a knot in a real yarn is observed to break, and why a knot's efficiency is a property of its first bend rather than of the knot.

Where a knot breaks

It breaks at the entry, before the knot has done any gripping at all. Two quantities run along a knot's path and only one of them rises; the other falls from the first millimetre; and their sum is largest where the thread arrives.

applied · Damage Every thread's interlacings in an eight-end satin stripe on a plain ground. An eight-end satin stripe on a plain ground on point paper, with a bar under every end and beside every pick for the share of its crossings at which it changes face. The warp's fewest is 0.25 a crossing against an average of 0.88, and the weft's 0.83 against 0.86; the draft's single firmness number is 0.87. What the bars cannot show is the friction at each crossing, which turns a count into a grip.

A cloth slips at its least-interlaced thread

A weave's firmness is quoted as one number, the interlacings per crossing averaged over the whole repeat. A cloth does not fail on average. A thread pulled through a seam or out of a cut edge is held by its own crossings, the grip is exponential in them, and the thread with fewest goes first. In every four-by-four draft but plain weave some thread interlaces twice a repeat — the fewest possible — whatever the average says, and a satin stripe on a plain ground averages 0.87 while its satin ends grip at a seventh of the average thread.

weaves · Firmness A 100-gram bouclé in section, with the depth to its own load path. A bouclé cloth of 100 grams a square metre in section: the core and binder along the bottom, carrying everything, with the effect thread's loops standing 0.31 millimetres above them. The cloth's surface is the loops and nothing else — its loop envelope is 12.6 times its count diameter — so a rubbing surface meets effect thread first and reaches the core only after that depth. 60.7 per cent of the yarn's mass is effect thread on no load path and 39.3 per cent is core and binder. What the section cannot show is the third dimension: the loops of neighbouring yarns lie between these and are pressed by them.

A bouclé wears from the loops down

A cloth loses its strength before its mass, because a woven cloth's crowns are the very threads that carry the tension. A bouclé inverts it exactly. Its surface is its loops, its loops hang off a core that carries everything, and 61 per cent of its yarn is on no load path at all — so a rub takes three tenths of a millimetre of thread that does nothing before it reaches anything that does, and the cloth is ruined to look at while it is still as strong as it was woven.

fancy · Fancy yarn The crossed shed's three spans at an eye friction of 0.3. A leno's crossing end in the crossed shed on an ordinary broad loom, back standard 4 shafts behind the doup, with both heddle eyes gripping at a coefficient of 0.3 and a background tension of 0.5 N. The end turns 69 degrees at the doup and 64 at the back standard, capstans of 1.44 and 1.40. From fell to doup it settles at 6.95 N, a strain of 5.73%; from doup to back standard it settles at 9.48 N, a strain of 7.97%; from back standard to back rest it settles at 6.78 N, a strain of 5.57%. With frictionless eyes every span would take 7.08 N; an ordinary end at the doup takes 1.02 N and the yarn breaks at 3.74 N on its initial modulus. Vertical scale exaggerated 3 times. What the drawing cannot show is the tension's fall round each eye, which happens over the eye's own few millimetres.

A heddle eye lets the kink through

A leno's crossing end is pulled up at its doup and held down at its back standard, and the length that costs was priced as if both heddle eyes were frictionless. They grip, and gripping ought to trap the kink between them at nearly a hundred per cent strain. It does not come close. A capstan bounds a ratio of tensions, not a difference, and the spans either side are already stretched, so at a coefficient of 0.3 the span between the eyes takes 9.48 newtons against 7.08 with no friction at all — a third more, not fifteen times more — and an easer has to give back 67.9 millimetres rather than 64.4. What friction changes more is when the length is wanted.

fancy · Leno

Named alongside it

The objects these essays reach for when they reach for this one.

FrictionContact forceSpecificationCloth integrityFrayingInterlacingSeam slippageFloatLenoPilePull-outSett

All concepts