Textile Structure
how cloth is put together
Concept

Fraying — where it appears

Threads working loose from a cut edge, which stops where the grip on them beats their own strength. Its width is a crossover length, so it is a fixed-force question rather than a fixed-length one and the interlacing ordering is exact.

Named by 12 essays across 5 fields — each of them below, with the objects they name alongside it.

One break, two outcomes. The same single break in a knit and in a weave. In the knit nothing holds the loop above the break, so the failure climbs the wale; in the weave every other thread is still held by the threads crossing it, and one thread comes loose.

Ravel, fray and run

Cut a woven cloth and one thread comes loose. Break one loop in a knit and every loop above it follows. The two structures fail in opposite ways, and the reason is topology rather than strength.

knits · Knit Milling, on the nonwoven's own scale. Migrated fibre accumulating with agitation, measured against the stick-percolation threshold this site uses for nonwovens. Above the line the fibre network holds on its own, which is why a milled cloth can be cut without fraying — the weave is no longer the only thing keeping it together.

Milling holds the cloth a second time

This site's integrity criterion returns exactly the same answer for a melton as for the loose twill it was woven as, and it is right both times. What has changed is that a second network now holds the fabric together, made of migrated fibre rather than of thread crossings — and it is the one that decides whether a cut edge frays.

finishing · Felting 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 A cloth does not mind a hole. Strength left after a hole, against the width of the hole, for a woven cloth and for a film of the same width. The cloth's threads carry their own load, so it loses exactly the threads the hole removes and the loss is linear. The film is a continuum, so a hole of any size at all costs it the stress concentration factor — quoted from Kirsch's elasticity solution, not derived here — and the size of the hole does not enter.

A cloth does not mind a hole

A hole in a film costs it two thirds of its strength whatever the hole's size, because a continuum concentrates stress at an edge. A cloth's threads carry their own load and hand almost nothing to their neighbours, so a hole costs exactly the threads it removes — a loss that is linear in the hole, independent of the sett, and zero for a slit along the load.

applied · Damage Which seams slip and which break. For each cloth, the grip a 10 mm seam allowance offers divided by the thread's own breaking load. Above one the fabric or the thread gives first and the seam holds until it does; below one the threads slide out and the seam opens with the cloth intact. voile, batiste, poplin, sheeting break; cheesecloth, muslin, duck, filter slip. The allowance that would save every cloth in the table is 74 mm, which is set by the openest of them alone. What the chart cannot show is the stitching itself, which has its own strength and its own way of cutting the threads it passes through.

A seam slips before it breaks

A sewn seam fails in one of two ways and the trade names them separately: the threads pull out of the cloth beside the stitching, or something breaks. Which one a given cloth does is decided by whether its seam allowance clears the length at which grip beats strength — and at the ordinary ten millimetres, the table divides four against four.

applied · Seams 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 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 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 woven crossing, and the number that never changes. A warp end and a weft pick at 6% crimp, drawn with the thickness expanded three times so the interlacing can be seen. Each goes over its neighbour and comes back; neither passes through the other. The Gauss linking integral over the pair, closed far outside the crossing, returns 0.0000. It returns that at every crimp and for every weave, because crimp moves a thread up and down across its neighbour and a curve that goes over and comes back has done nothing a linking number can see.

A woven cloth is not linked at all

Every thread in every woven cloth passes over its neighbours and comes back. None of them passes through. So the linking number of any two threads in any weave is zero, at any crimp, permanently — and almost everything a cloth does that a knitted fabric does not follows from that one number being nought.

weaves · 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 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

Named alongside it

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

FrictionCapstanContact forceFloatInterlacingSeam slippageCloth integrityPull-outSettSpecificationCrossover lengthConnectivity

All concepts