Textile Structure
how cloth is put together
Concept

Fibre — where it appears

The single filament a yarn is spun from, whose fineness sets how many are in a cross-section and therefore how uncertain its stiffness is. How many are in a cross-section is what sets the width of this site's bending bracket, which is a fibre count rather than an uncertainty.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

mechanics · Yarn stiffness The energy well, and where the cloth sits in it. The bending energy of a sheeting at every state on its own constant-thread-length locus, plotted against how the crimp divides between the two systems. The minimum is at 1.22 and the value every Peirce solution here is drawn at is 1.00, marked. The well's depth decides how firmly the ratio is settled, which is why an open scrim's measured crimp scatters and a close sheeting's does not. What the plot cannot show is the friction that stops a cloth reaching the bottom, which turns the minimum into a band.

The crimp ratio is not a measurement

Peirce's geometry is two thread systems, four unknowns and three equations. It cannot say how the crimp divides between warp and weft, so every cloth solved so far has been drawn at a ratio somebody chose. Give the threads a stiffness and the missing equation arrives — and for six of the eight cloths in the table it arrives with no material constant in it at all.

setting · Crimp The bracket that closes and the bracket that does not. A 25 tex cotton yarn at a packing factor of 0.6. Its bending rigidity lies between 0.00117 N·mm² — the sum of its fibres', with them free to slide — and 0.478, a solid rod of its own diameter: a factor of 408, and both ends are derivations. Its resistance to being squashed out of round has an upper bound of the same kind, 369 N/mm² for a solid section, and no lower bound at all, because fibres free to slide resist a change of shape with nothing. That is why the aspect ratio of a flattened yarn has been a free parameter here since the setting field was built: a quantity bounded below by zero cannot be estimated from its bounds, and has to be measured.

The stiffness with no lower bound

A yarn's bending rigidity lies between two derivable ends and the ratio is the fibre count — wide, but closed. Its resistance to being squashed out of round has an upper bound of the same kind and a lower bound of exactly nothing, because fibres free to slide resist a change of shape with nothing at all. That is why nobody could ever compute the aspect ratio of a flattened thread.

mechanics · Compression A fifth of a strong fibre buys most of its pilling. How long a pill survives on a wool fabric as nylon is blended into it, relative to the pure wool. A pill is held by several anchor fibres and survives while any of them holds, so its life is set by the strongest anchor it happens to have — and the chance that a pill with 8 anchors has at least one strong one is 1 − (1 − x)^8, which is already 83% at a fifth. The blend therefore gets 83% of the pure strong fibre's pill life while keeping the whole of the weak fibre's fuzz supply, which is the worst of both. Nine tenths of the way arrives by 30%. No average of the two fibres' properties produces this curve: it is a maximum over a small sample, and a maximum is not an average. It is also why a fifteen-per-cent polyamide in a wool knit is notorious, and the arithmetic says the reputation is deserved.

The strong fibre is the one that pills

A pill survives while any one of its anchors holds, which is a maximum over a small sample rather than an average — and a maximum behaves nothing like an average. A fifth of a strong fibre in a blend buys four fifths of the pure strong fibre's pill life while leaving the whole of the weak fibre's fuzz supply in place.

applied · Pilling A 50 tex yarn after 1, 4, 8 dips into a fibre that fills. The cross-section of a 50 tex cotton yarn after 1, 4, 8 twenty-second dips, each fibre shaded by the dye fixed at its distance from the surface, on one scale of shade for all three. The fibre's room for dye is finite and one dip at the bath's strength fills 20% of an empty fibre's, so each dip meets less affinity near the surface than the last and carries its dye further in. after 1 dip: ring 14% of the radius deep, the surface 20% full; after 4 dips: ring 18% of the radius deep, the surface 59% full; after 8 dips: ring 24% of the radius deep, the surface 83% full. With room to spare every dip would leave the first dip's 14%. The share is assumed; nothing here measures it.

A fibre that fills dyes deeper with every dip

Dipped eight times, a ring-dyed yarn is darker than after one dip but no deeper — if its fibres have room for all the dye they are offered. They do not: every dip's dye occupies some of the fibre's room, the next dip meets less affinity near the surface, and dye that is taken up less travels further before it is taken. How much further depends on one number the dip arithmetic never needed — the share of an empty fibre's room one dip fills — and eight dips turn that number into a ring 1.15 times deeper at five per cent and 1.68 times deeper at twenty.

finishing · Dyeing A 50 tex yarn after eight dips, with its pores open and filling. The cross-section of a 50 tex cotton yarn after eight twenty-second dips into a fibre one dip fills by 20%, each fibre shaded by the dye fixed at its distance from the surface, on one scale of shade. open pores: the ring 24% of the radius deep, the surface 83% full; the dark shade's pigment, all in the pores: the ring 22% of the radius deep, the surface 83% full; pores 90% filled at a full fibre: the ring 16% of the radius deep, the surface 83% full. One dip leaves 14%. The middle section is the most a dark denim's indigo could clog its pores, every grain placed in them; the right is a clog no denim's pigment is large enough to make.

Indigo is too little to close the pores it dyes through

Each dip of a fibre that fills carries its dye deeper, because the dye already fixed near the surface leaves less room there. But indigo is fixed as a solid, and some of it sits in the pores the next dip has to travel through, so a heavily dyed skin should also hold the next dip back. The two pull opposite ways, and which wins is decided by how much pigment there is. A dark denim's indigo, every grain of it placed in the pores, fills at most a fifth of them at the surface and takes back about a seventh of the ring's deepening — under two micrometres. The surface fills at the same rate whatever the pores do, so two cross-sections still tell the two effects apart.

finishing · Dyeing The ring across eight dips, for each fibre. The ring's depth after each of eight twenty-second dips, where an empty fibre left in the bath for good would fill by 20% a dip. instant fibre: 14.0% after one and 23.5% after eight; τ = 20 s: 13.0% after one and 19.6% after eight; τ = 100 s: 23.1% after one and 26.1% after eight; τ = 400 s: 35.5% after one and 36.6% after eight. An instant fibre earns its ring by saturating the skin dip by dip. A slow one starts near its final depth, since it never fills its skin.

A slow fibre starts its ring deep

Every account of an indigo dip here lets a cotton fibre take its dye at once. Give the fibre a clock of its own and the ring does not simply run deeper, which is what the question seemed to ask. A clock of a few seconds makes the ring shallower, by nearly a fifth at most, and only past about twenty-six seconds does it run deeper than an instant fibre's. Two sections tell the accounts apart that one cannot: an instant fibre earns its ring dip by dip, from 14 to 23.5 per cent of the radius over eight dips, and a fibre with a hundred-second clock starts at 23 and ends at 26.

finishing · Dyeing

Named alongside it

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

Bending rigidityDenimDye affinityInter fibre porePacking factorRing dyeingSpecificationYarn diameterAbrasionFrictionTwistAreal density

All concepts