Textile Structure
how cloth is put together
Theme

The thread: Structure before fibre — page 5

Page 5 of 5 of the essays on this thread.
Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one. Mechanics and drape

Where a torsion model stops

A second stiffness was added because an earlier ladder named its absence as the first thing to disbelieve. It settled four things, refuted one trade explanation, and left the question it was built for exactly where it found it.

 Where a knitted disc is flat. The circumference a knitted disc grows per unit of radius, as a share of the 2π a flat disc needs, against the loop's aspect, for 4 wedges (flat at 1.2732), 10 every 2 rounds (flat at 1.2566). Wedges of short rows fall as the aspect rises and rounds with increases rise. The shaded band is the aspect of plain knit between its dry-relaxed value, 1.2500, and its wet-relaxed value, 1.2927. Knits and other structures

A knitted disc is flat at one shape of loop

A disc lies flat only if its circumference grows by exactly 2π per unit of radius, and a knitted disc's growth is a count of loops in one direction over a count in the other. So it is flat at one value of the loop's aspect and no other. Knitted sideways in wedges of short rows the growth falls as the loop gets wider; knitted outward in rounds it rises. Four wedges and ten increases every two rounds are each flat at an aspect inside the range plain knit moves through when it is washed — and they cross it in opposite directions.

 The wicking front against time, in four rooms. The height of the wicking front in a sheeting-weight strip against time on a logarithmic clock. The fine system, the pores between fibres, rises to a steady height set by the room: 1064 mm at 20 g/m²·h, nine tenths of it after 4.1 h; 501 mm at 100 g/m²·h, nine tenths of it after 53 min; 255 mm at 400 g/m²·h, nine tenths of it after 14 min; 133 mm at 1500 g/m²·h, nine tenths of it after 3.7 min. The coarse system, the holes between yarns, stops at 137 mm after 17 s whatever the room, because gravity stops it rather than drying. Cloth doing a job

A wick reaches its ceiling in the time its cloth takes to dry

A drying cloth lifts water to a steady height and no further, and the question left was how long it takes to get there. The answer has no permeability and no surface tension in it. Where gravity is small the front climbs as the ceiling times the root of one minus a decaying exponential, and the exponential's time is the cloth's own pore water divided by the rate its faces lose water — the time the room would take to dry it. In an ordinary room that is half an hour, and the front is nine tenths of the way up in fifty-three minutes.

 Every count that can make 150 g/m² as plain jersey. Areal weight against count for plain jersey in the fully relaxed state: the shaded wedge is every knit between a tightness factor of 1.3 and 1.6, whose weights are kₛ times the tightness times the root of the count. The rule at 150 g/m² crosses it between 15.8 and 23.9 tex, a ratio of 1.51. A plain woven cloth of the same weight can be made from anything between 20 and at least 200 tex, a range of at least 10 to one against the knit's 1.51. Setting and geometry

A knit's weight nearly names its yarn

A woven cloth's weight is one equation in four unknowns, and a hundred and fifty grams can be woven from anything between twenty tex and two hundred. A plain jersey's weight has the loop in it and nothing else to spare, and the loop is bounded by the yarn it is knitted from. Put the two together and the loop cancels: the weight is a constant times the tightness times the root of the count, so at one weight the count is fixed to within half again — and in each relaxed state it is fixed to a different half.

 The fibre pieces in one cut tuft. A cut-pile tuft 3 mm tall on a 1 mm base, spun from a 25 mm staple, with a seeded sample of its fibre pieces drawn along the yarn: 16 pass under the binding pick and 2 lie wholly in one leg, between a fibre end and the cut tip. Across a whole pile the loose share is 21.9 per cent of the pieces and 7.0 per cent of the fibre. What the drawing cannot show is the twist, which holds the loose pieces by friction and is what they escape from. Compound and figured cloths

A blade leaves loose fibre in every tuft

The account of hair, nap and pile ended on a clean claim: a blade collapses a population's length to one value, so a cut pile has no tail and nothing in it reaches past the rest. The tips are one length. The fibres are not. A tuft cut from staple yarn is a length of yarn with fibre ends scattered along it, and every fibre end that lands in a leg leaves a piece between it and the blade that nothing in the draft holds — a fifth of the pieces in an ordinary wool carpet, none in a filament one.

 What flattening costs, against what the loop's bending is worth. The energy of squashing a 20 tex cotton to the 78% the fabric's geometry demands, as a multiple of the whole bending energy of one stitch, at three lateral rigidities. The free bound is exactly zero: a bundle of fibres free to slide resists a change of shape at constant area not at all, so flattening is free and the bracket on it has no floor. The coherent bound is 36 times the loop's bending, so a yarn that could not rearrange could not be knitted into this fabric at all. The fabric flattens, so it is reading the free end — which is the fourth ordinary observation on this site to land at that end of the bracket. Mechanics and drape

Flattening is free and impossible

The fabric demands a flattening and the yarn has to supply it. At one end of this collection's oldest bracket the deformation costs exactly nothing; at the other it costs thirty-six times the whole bending energy of a stitch. The fabric flattens — which is the fourth everyday observation in one phase to land at the same end.

 One gore of a 4-gore knitted ball, laid flat. A gore of 30 stitches from pole to pole, knitted sideways in short rows, drawn flat with its rows stacked at the plain-knit loop's fully relaxed aspect of 1.2791. Turned on the sine, its 10 row pairs turn at stitches 0, 1, 2, 3, 4, 5, 6, 8, 9, 11 from each pole — one stitch apart near the pole and further apart towards the equator — and its edge follows the width a sphere asks for. Turned one stitch every pair, the same gore is a diamond of 15 pairs, 1.50 times as wide at the equator. What the drawing cannot show is the fabric smoothing its own staircase edge. Knits and other structures

A knitted ball's short rows have to slow down

A knitted disc needs one count to meet 2π; a sphere needs a count that follows a sine. Knitted sideways in gores, a ball's short rows must turn one stitch apart at the pole and ever further apart towards the equator — evenly spaced turns knit two flat discs joined at the rim. The pole is the disc again, so gores come in fours; the equator's row pairs have to come out whole, so only some sizes knit round; and a wash moves a four-gore ball from a ruffled pole to a round one.

 What a figured warp pays in tension to share one beam. For six pairings of figure and ground, the difference in warp tension that lets their ends share one beam: the force per end that takes the take-up difference out of the ground's crimp, along that region's own constant-length locus, with the yarn's rigidity at both ends of the band two tests place it in. a damask: satin on its own complement, 0.00% apart: none needed; an eight-end satin figure on a five-end satin ground, 0.31% apart: 0.006 to 0.020 N; a five-end satin figure on a 3/1 twill ground, 0.90% apart: 0.021 to 0.069 N; an eight-end satin figure on a 2/2 twill ground, 3.02% apart: 0.304 to 0.995 N; an eight-end satin figure on a plain ground, 14.15% apart: past what the ground's crimp can give; a 2/2 twill figure on a plain ground, 11.14% apart: past what the ground's crimp can give. The loom's front shaft holds 0.52 N an end. What the chart cannot show is how uneven a warp's tension may be before it shows in the cloth. Compound and figured cloths

A figured warp pays in tension before it needs a beam

Figure and ground take up warp at different rates, and counted against a fixed slack, any difference at all — a third of a per cent — made a round figure need four beams. A real let-off holds tension, not length. Ends that consume more pull harder, and an end pulled harder gives up crimp, until every end consumes alike. For an eight-end satin figure on a five-end satin ground that costs a hundredth of a newton an end, and one beam serves. For a satin on plain it costs more crimp than the ground has, and no tension will do.

 Two ring-dyed faces worn to 5% of their mass. The faces of 3/1 twill and plain woven from the same ring-dyed 50 tex warp and undyed weft, at 24 ends and 16 picks a centimetre, each worn flat by an abrader until 5% of its thread has gone, drawn over two repeats: dyed warp dark, exposed undyed core red, weft pale, holes blank; the 3/1 twill shows 30% of its covered face as exposed core; the plain shows 17% of its covered face as exposed core. A float's crown is a line and is cut along its length; a plain weave's crowns are points and are cut through as small patches. The geometry is the plain-weave solution for these counts and setts with the floats drawn over it, which is more open than a real denim; what the figure cannot show is a denim at its real density. After the loom

A float fades late and hard, a crossing early and soft

Wear a ring-dyed face flat and the white arrives where the cut first passes the ring at a crown, which is the same depth for every weave. What differs is how much cloth has gone by then and how fast the white spreads after. A plain weave's crowns are points, cut through by almost nothing: it shows white after a fifth of a per cent of its thread and never shows much. A twill's floats are lines, cut along their length: a 3/1 twill shows nothing until more than one per cent has gone, then whitens faster than any plain weave can, 39 per cent of its face by a tenth worn against plain's 24. Denim's high-contrast fade is a float's; chambray's soft one is a crossing's.

 What flattening a section does to the ratio of the two stiffnesses. C/B is 2G/E for a circular section, because a circle's polar second moment is exactly twice its flexural one. A flattened section has two different flexural moments — easy about the long axis, hard about the short one — and the polar moment is still their sum, which is the perpendicular axis theorem and holds for any section whatever. So a flattened thread has three constants rather than two, and the multiplier on C/B depends on which way it is being bent. At the 0.78 a knitted fabric's own geometry demands, the easy direction multiplies the ratio by 1.322 and the hard one by 0.804. A knitted loop bends in the easy direction, so the collection's headline ratio is a lower bound for a yarn in cloth. Mechanics and drape

The section that changes both stiffnesses

A thread's two rigidities are in the ratio 2G/E, and that is a fact about a circular section: a circle's polar second moment is exactly twice its flexural one. A yarn in cloth is not circular, so a yarn in cloth has three constants rather than two — and the ratio a whole ladder rests on is a lower bound.

 A jersey's bending length against its tightness. The wale-wise bending length of plain jersey in the fully relaxed state at 10, 20, 40 tex, across the whole tightness band from 1.3 to 1.6, at the two ends of the yarn's stiffness bracket. At the free bound every count and every tightness gives 10.13 mm. At the locked bound the lines separate by count — 55 mm at 10 tex, 70 mm at 20 tex, 88 mm at 40 tex — and are still flat. The weights along the lines run from 97 to 239 g/m². What the plot cannot show is friction between the loops, which is where a tight jersey's firmness has to come from. Setting and geometry

A jersey's drape does not know its loop

A woven cloth's bending length at the free bound turned out to hold neither its count nor its weight — only the fibre. A plain jersey goes further. Its stiffness per width and its weight per area are both a number of loops per millimetre times something about one loop, so the loop cancels at every stiffness the yarn could have; at the free bound the count cancels too, and what is left is the fibre and which relaxed state the fabric is in.

 A ball knitted in rounds, count by count. The stitch count of each round of a ball knitted outward from one pole in 40 rounds, from the pole to the equator, in the fully relaxed state, against the count a sphere asks for. Increasing 5 at a time it reaches 65 at the equator, increasing on rounds 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 14, 16; Increasing 10 at a time it reaches 60 at the equator, increasing on rounds 3, 5, 8, 11, 15. What the chart cannot show is where round the ball the increases fall, which spirals or stacks them and is a choice of the pattern. Knits and other structures

A ball knitted in rounds washes the other way

A ball can be knitted sideways in gores or outward from one pole in rounds, and both have to follow the same sine. But a gore counts its meridian in stitches and its circumference in rows, and a round counts them the other way, so the loop's aspect sits underneath one construction and on top of the other. A wash raises the aspect — and closes a gored ball's ruffled poles while it opens a round-knitted ball's.

 A crease along a 3/1 twill's floats and one across them, worn alike. A strip of 3/1 twill of ring-dyed 50 tex warp and undyed weft folded round a 2 mm radius, the ridge drawn across the page as a dashed line, worn by a flat abrader 50 µm into the ridge: above, with the ridge running along the warp and its floats; below, running across them. Dyed warp dark, bared core red, weft pale. crease along the warp: 83% of the crease's length white, in runs 1.72 mm long on average, the longest 2.08 mm; crease across the warp: 31% of the crease's length white, in runs 0.15 mm long on average, the longest 0.17 mm. On a twill the white along the warp lies in lines as long as a float's crown, across it in dashes one end wide; on a plain weave, with no floats, both are short. What the drawing cannot show is the fibre a real crease tears loose, which blurs every edge here. After the loom

A crease across a twill fades in dashes

A garment is not worn flat. It is worn along its folds, where the ridge of a crease stands out and is rubbed first, so the fade follows the crease. On a ring-dyed 3/1 twill a crease running along the warp bares the cores of the floats that lie under its ridge, float by float, and reads as a white line broken only where each float dives: at fifty micrometres of wear, 83 per cent of its length is white, in runs of 1.7 millimetres. The same crease across the warp meets every float at one point and reads as a row of dots, 31 per cent white in dashes of 0.15 millimetres, one end wide. It bares much the same core. A plain weave's creases are dashes whichever way they run.

 What a spin leaves in a cotton load, against the drum's speed. The water left in a 68 mm load of cotton sheeting in a drum of 250 mm radius, as a share of the load's dry weight, against the spin speed. Read with the fibre holding its swollen water, the spin leaves 51% at 1,000 rpm and 41% at 1,400, and no speed takes it below 30%, the volume the fibre grew by. Read with the fibre holding only its regain, as the earlier account of a wet cloth did, the same spin leaves 16% at 1,400 and the floor is 8.5%. What the chart cannot show is the water a drained channel keeps as a film and as rings at the fibre contacts, which the model takes as nothing and which lifts every point on both curves. What cloth is

A spin leaves the water a fibre swelled by

A washing machine's spin is a centrifuge, and it drains a cloth in the order its pores give up water: the holes between the yarns at a few g, the channels inside the yarns only far enough from the drum's wall, and the water inside the fibre never. That last reservoir is not the regain. A soaked cotton fibre holds the volume it swelled by — 30 per cent of its dry weight, three and a half times the regain — and it is the floor under every spin speed there is.

 How high a strip wicks a finite supply, in four rooms. The steady height of the fine pore system's wicking front against the supply of sweat at the strip's foot, both on logarithmic scales, for a sheeting-weight strip in four rooms. Below each room's capacity the height is the supply over twice the evaporation from each face, a line of slope one that no cloth property moves. It meets the room's drying ceiling at the capacity — 43 g/(m·h) at 20 g/m²·h, ceiling 1064 mm; 100 g/(m·h) at 100 g/m²·h, ceiling 501 mm; 204 g/(m·h) at 400 g/m²·h, ceiling 255 mm; 400 g/(m·h) at 1500 g/m²·h, ceiling 133 mm — and past it the strip takes no more. The marks along the bottom are four sweat rates from rest to hard work, fed from a contact patch 50 mm tall. Cloth doing a job

A sweating cloth wicks as high as the room can dry it

Every wicking height so far has had its foot in unlimited water. Skin is not a reservoir: it supplies sweat at a rate, and a cloth fed at a rate stands where the supply equals what its faces lose — the supply over twice the evaporation, with no pore, fibre or thickness in it. The cloth decides only when it has had enough, and it says so all at once: the holes between its yarns stay empty until the fine pores are carrying ninety-nine per cent of what they can.

 One crease at four angles to a 3/1 twill's warp, worn alike. Four strips of 3/1 twill of ring-dyed 50 tex warp and undyed weft, each folded round a 2 mm radius with the ridge drawn across the page, worn 50 µm into the ridge by a flat abrader, 24 mm of crease each. Dyed warp dark, bared core red, weft pale. along the warp (0° from the warp): 80% of the crease white, its longest white run 2.0 mm; along the twill's line (33.7° from the warp): 100% of the crease white, its longest white run 45.0 mm; two degrees off the line (35.7° from the warp): 79% of the crease white, its longest white run 15.7 mm; along the other diagonal (-33.7° from the warp): 66% of the crease white, its longest white run 1.0 mm. Along the twill's line the white never breaks, because the crease meets the crown of every end's float in turn; two degrees off, it runs as long strokes that drift off the line of crowns and come back. What the drawing cannot show is the fibre a real crease tears loose. After the loom

A crease along the twill fades as one line

A crease along the warp of a ring-dyed 3/1 twill fades as broken lines a float long, and across it as dashes one end wide. Turn the crease to the twill's own diagonal and it fades as a single white line with no break in it at all: the fold meets the crown of one end's float, then the next end's one pick along, then the next, and every crown it meets is at the same place in its float. It is the one direction in which the weave and the crease line up — and two degrees off it the line breaks into strokes whose length is a vernier's.

 A ball knitted in rounds with its increases stacked in 5 lines. The shape a ball of plain knit takes unstuffed when its increases, 5 at a time, are stacked in 5 lines from pole to pole, with the same stitches in every round as a sphere. Every stitch between two lines is flat knit, so the 5 panels are flat and the one shape they close up into without stretching has 5 flat sides in every round and 5-sided points at its poles: 2.42 sphere radii tall, 2.14 across its ridges and 1.73 across its flats, so 1.13 to 1.40 times as tall as it is wide. The knitted counts of a 40-round ball give 1.13 to 1.40, against 1.02 for the same counts with the increases spread round, whose rounding is its own. What the drawing cannot show is stuffing, which would stretch the panels towards round. Knits and other structures

Stacked increases knit a ball with flat sides

A ball knitted in rounds can put its increases anywhere in each increase round. Stack them in lines from pole to pole and every stitch between two lines has a flat knit's neighbours, so the ball is made of flat panels and all its curvature sits on the lines. Flat panels with straight rows close up without stretching in exactly one way: every round a regular polygon, each pole a point. At five lines that ball is 13 per cent taller than it is wide across its ridges and 40 per cent across its flats, and to make it round the stuffing has to stretch the middle of every panel by π²/4m² — nine per cent at five lines, less than a wash moves a course at ten.

 Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one. Mechanics and drape

A fabric reads its own bracket four ways

A yarn's stiffness is unknown to a factor of three hundred, and no laboratory measurement has closed it. Four unrelated everyday observations — a snarl, a knot, a flattened yarn and a cloth's own thickness — all say the same thing about which end of it a yarn sits at.

 Where an air arch first wrinkles under snow over the whole span. A semicircular air arch 4 m across, pinned at both feet, of a tube 200 mm across at 50 kPa, under snow over the whole span; beside it, unrolled from the left foot to the right, the share of the tube's wrinkling budget πpr³/2 spent at each section when the first wrinkle forms, with the thrust's part, N·r/2, shaded darker. The first wrinkle forms 25° above the right foot, at 167 N per metre of span, where the moment of 60.2 N·m compresses the inside of the curve and the thrust is 336 N; there the thrust has taken 21 per cent of the budget. Judged by its moment alone the arch would carry 218. Cloth doing a job

An air arch pays for its thrust out of its pressure

An inflated tube bent into an arch was expected to start with its inside wall pulled differently from its outside, and to wrinkle where that difference and the load's moment combined worst. It does not: a curved tube is pulled along its length at exactly the straight tube's pr/2, all the way round. What the arch spends its pressure on instead is the thing every arch exists to make — its own thrust — which takes a fifth of the wrinkling budget at the haunch of a shelter arch under snow and more than half on a tight one.

 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. After the loom

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.

 How tall a skin contact can be before its cloth floods. The tallest patch of skin contact whose cloth keeps the holes between its yarns empty, against the sweat rate, on logarithmic axes, for a sheeting-weight cloth in rooms evaporating 20, 100, 400, 1500 g/m²·h from each face. The patch evaporates from its outer face only and hands its surplus to the free cloth above, which spreads it until the fine system's capacity; so a contact floods when it is taller than 98.9% of that capacity over the surplus. Below the room's own evaporation rate no contact of any size floods. In an ordinary room a run floods any contact taller than 247 mm and hard work any taller than 90 mm. Cloth doing a job

Wicking borrows drying area from the cloth that is not touching

A shirt is fed with sweat across the whole of the skin it touches, not at the foot of a strip. Through its own thickness the cloth could pass sweat a hundred thousand times faster than a body makes it, so a patch pressed flat never lacks capacity; what it lacks is drying area, because it can give water to the room only from its outer face. Pressed flat everywhere, it floods as soon as the sweat exceeds the room's evaporation — light work, in an ordinary room. With free cloth above the contact, wicking carries the surplus to faces that are not touching, and the sweat decides how much area it needs: three times the contact's at a run, whatever the contact's size.

All themes