Textile Structure
how cloth is put together
Concept

Measurement — where it appears

A number that came from an instrument rather than from an argument, which this collection marks as such wherever one appears. Every result computed from one is reported over its spread, because a number quoted to three figures out of a constant known to one is a number pretending.

Named by 25 essays across 7 fields — each of them below, with the objects they name alongside it.

The number the drape test does not record. A 150 mm specimen over a 90 mm pedestal, for fabrics from limp to stiff. The curve is the fold count the buckling argument predicts — three quarters of a power of the specimen's radius over its own bending length — and it runs from 2 folds to 11. The number beside each mark is the drape coefficient the same specimen would report, which moves by a few points across the whole range.

The nodes a drape test throws away

A drape test lays a circular specimen over a pedestal, photographs the shadow and reports one number. The specimen also falls into a definite number of folds, which is a buckling mode set by the fabric's own bending length — and the standard method observes it, does not record it, and reports the number it is least sensitive to.

mechanics · Buckling A ratio that will not hold still. The exchange rate between the two directions for 4 cloths, against how far each has been extended along the warp. Every curve is above one half everywhere it is drawn, and every curve rises — so a single number for a cloth's ratio has to name the state it was read at, which a material's ratio does not.

A cloth's Poisson ratio is not a material's

The ratio of a cloth's contraction to its extension has a name everywhere else in mechanics, and it breaks every rule the name comes with — above one half on all eight cloths measured, doubling across a four per cent span, and not reciprocal between the two directions.

mechanics · Tensile One weft fault at three periods, a fraction of a millimetre apart. A 260 mm slice of a 1500 mm cloth at 22 picks per centimetre, 220 picks deep, with a thick place recurring along the weft. Each pick takes a whole width of yarn, so the marks land 35 to a pick across the full width and step sideways by the remainder of the width divided by the fault's period. On the left that remainder is zero and every thick place in the piece falls in the same columns — a warp-way stripe made entirely by a weft fault. In the middle the period is five hundredths of a millimetre longer and the same fault draws steep diagonals. On the right it is a third of a millimetre longer again, the marks land nowhere near each other, and the fault reads as texture. Nothing about the yarn distinguishes the three; the cloth's width does. The slice is drawn rather than the whole width because thirty-five marks a pick fill a panel solid at any step but zero, which is a true picture of a dense pattern and a useless one of its structure.

A slub finds the width of the cloth

A thick place recurring along a weft yarn does not make a bar. It makes diagonals — and when the cloth's width happens to be a whole number of fault periods, it makes stripes down the piece instead, from a fault that is entirely in the weft.

pattern · Moire The presser foot sinks 1.8 µm into a 2/2 twill. A thickness gauge presses a flat foot onto the cloth at 1 kPa and reads the gap. It does not read the geometric thickness. The foot sinks until the area it is touching can carry the load, and that is 2.57% of the plan at a depth of 1.8 µm — so a 2/2 twill in sheeting whose outside stands 381.6 µm apart measures 379.8 µm. How far the foot sinks is a property of the draft, because the bearing area near the top is, and a weave with plateaux stops the foot in a fraction of the distance a plain weave lets it travel. The transverse stiffness used here is fitted to measured fabric thickness rather than predicted: across its published range the reading moves between 377.0 µm and 380.6 µm.

A thickness gauge reads the draft

A presser foot does not stop at the top of a cloth. It sinks until the area it is touching can carry the load, and how far that is depends on the shape of the bearing curve near the top — which is a property of the weave. So there is a weave term inside a measurement nobody thinks of as a weave measurement, and it is worth about one per cent.

cloth · Contact What a tear asks for is the weakest of a few. A tensile test pulls every thread in the width and averages them. A tear pulls the handful in the triangle at the tip of the cut, and what lets it move is whichever of those is weakest — so the quantity that governs is the minimum of a small sample, and two things follow that no mean can show. Its expectation is below the mean: at CV 15% the weakest of 4 is 0.853 of the mean thread, and the weakest of 40 is 0.718. And its scatter is enormous compared with a tensile test's: 10.9% against the 0.75% a mean of four hundred threads would show. A tear strength that varies by a tenth between specimens of the same cloth is not a badly run test. It is the only answer an extreme of four can give.

A tear asks fewer threads than a pull

A strip test averages a hundred threads and a tear interrogates four. That difference alone accounts for the two things everybody knows about tear testing — that it reads low, and that it scatters — without anything about the cloth being different between the two tests.

applied · Damage The crossings under a 25 mm² presser foot. A 25 mm² foot on a muslin covers 12 ends and 11 picks, which is 132 crossings — and a first guess treats those as 132 chances of finding a thick place. They are not independent chances. Every crossing along one end shares that end's diameter, so the largest crossing is the largest end plus the largest pick, and the number of tries is 23: the threads. Each cell here is shaded by its own two diameters, and the darkest is at the meeting of the darkest row and the darkest column, which is what that identity looks like. The gauge rests on it and reads 0.431 mm, against 0.342 for the cloth's mean crossing — 26% over.

A thickness is a maximum, not a mean

A presser foot rests on whatever is highest beneath it, so the thickness of a fabric is an extreme value — and an extreme grows with how much cloth is asked. The standard specifies the foot's area because the foot's area is in the answer.

mechanics · Compression The same fault in the warp and in the weft. A 50 m piece 1500 mm wide, with a 3-thread fault in each direction. The width is drawn 9.3 times over scale so that the piece is a rectangle rather than a line, and the two faults are drawn as marks rather than at their own widths, which at this scale are a fifth of a pixel. They have the same cause size — 3 threads — and they condemn 0.063 m² and 0.0020 m² respectively, a ratio of 31 to one, because a warp fault runs the length of the piece and a weft fault runs its width. That ratio is the aspect ratio of the piece and nothing else, so it is a property of how cloth is made rather than of what went wrong. It is why a broken end stops the loom and a mispick often does not, and why the two faults are priced by every grading scheme as though they were different kinds of thing.

A missing end is a fault the length of the piece

A broken end and a mispick are the same size of accident — one thread — and they condemn areas that differ by a factor of thirty. The ratio is the aspect ratio of the piece and nothing else, which makes it a fact about how cloth is made rather than about what went wrong.

applied · Faults A yarn's diameter is a contour, not a length. The fraction of the space beside a 20 tex cotton yarn that is occupied by hair, against height, grossed up from the modelled population by the measured split between the long and short populations. At the yarn's own surface it is 8.9% — the layer is almost entirely gap — and it falls away exponentially from there. Every instrument that reports a yarn diameter is picking a contour of this curve, and the contours are far apart: a threshold that needs half the space filled is never met at all, one that needs a twentieth is met at 355 µm, and a hair counter triggers out at 2353 µm. The earlier model gave the layer a single thickness of 25 µm, which is a fair description of where most of the material is and wrong about its extent by more than a decade. What the curve cannot say is which contour any particular instrument uses, which is a fact about the instrument.

A yarn has a diameter for every instrument

Conservation of volume gives a yarn one diameter and every other route gives a different one. The disagreement is not experimental scatter: a yarn's outside is a coverage that falls away exponentially, and each instrument stops at whatever contour of it will trigger the instrument.

cloth · Hair layer One hairiness reading does not fix the other. Every yarn on this curve has exactly the same total protruding fibre length — the quantity an integrating hairiness meter reports — and they differ in how that length is distributed. The count of hairs at least three millimetres long runs from 170 to 4354 per hundred metres, a factor of 26, across decay lengths real yarns actually have. The two instruments read two functionals of one population: the first moment N₀λ and the tail N₀e^(−3/λ). A correlation between them can exist only if λ is fixed across the yarns being compared, and λ is a fibre property, so it is not. That is the whole of why the trade's two hairiness numbers have never agreed, and it is arithmetic rather than instrumentation. What the figure cannot show is which of the two predicts anything: the tail does, because pilling, prickle and a printed edge all need reach.

Two hairiness meters read two moments

The trade has two instruments for yarn hairiness and thirty years of failing to predict either from the other. They are not measuring the same thing badly. One reports the first moment of a distribution and the other reports a tail probability, and two functionals of one curve are related only through a parameter neither of them reports.

cloth · Hair layer Two cloths touch on a fraction of what one cloth does. A 2/2 twill in sheeting pressed against a flat plate, and the same cloth pressed against another piece of itself. At an approach of 24.9 µm the single surface is touching 15.87% of the plan and the pair 3.130% — a factor of 5. The reason is that a gap between two rough surfaces is the sum of two depths, so both surfaces have to be near their own maxima at the same place, and the chance of that is the product of two small numbers. The pair's curve is the convolution of the two height distributions, computed exactly on histograms rather than fitted to a Gaussian — because a woven surface is bimodal and nothing about it is Gaussian.

Friction is two surfaces, not one

A gap between two rough bodies is the sum of two depths, so two cloths face to face touch on the convolution of their height distributions rather than on either of them. At the approach a light touch produces that is twenty times less contact than the same cloth against a plate — which is why a fabric's friction against a plate and against another fabric are two different measurements.

finishing · Friction The force at an interlacing, which no longer lies in the fabric. A crossing of a relaxed jersey in section on the left, with the thickness expanded 3×, and on the right the contact force drawn at the angle the solve gives it — which is a true angle, unlike anything in the section beside it. The head of one course and the feet of the next lie one yarn diameter apart through the fabric, 0.167 mm, so the interlacing is not a point in a plane and the force at it is not in one either. It comes out at 38.30 mN a stitch, turned 11.75° out of the fabric: 37.50 mN along the wales, which is what friction has to hold, and 7.81 mN through the thickness, which is what holds the two faces apart. Divided by the area a stitch occupies the second is 15.1 kPa, or 113 mmHg — a pressure, and the quantity a compression measurement reports.

What a thickness gauge reads on a knit

The structure says two yarn diameters and the gauge says more, and the gap is not an error in either. A gauge lands on the highest crowns, through a canopy of protruding fibre, under a load that has already begun to compress both — and it does that on a surface that is nothing but crowns.

cloth · Contact The magnification a cloth will carry. The largest magnification a moiré can be read at, against the irregularity of the cloth making it. The points are measured: a grating whose spacings are drawn from a seeded lognormal is laid against a perfect one, the fringes are found from the phase difference, and the gain is recorded at which their count first departs from what the ideal beat predicts. The product of the irregularity and that gain comes out at 0.82 to 0.84 across every irregularity tried, so the ceiling is 0.84 divided by the coefficient of variation — the solid curve. The dashed curve is the accumulated-wander model, in which position errors random-walk and the ceiling goes as the inverse square; it is wrong by a factor of 42 at a two per cent irregularity. What the plot cannot show is what a cloth's spacing irregularity actually is: it has not been measured, and the curve is therefore a prediction with an unmeasured input.

A moiré is a vernier, and it magnifies the error too

Two gratings a per cent apart in pitch beat at a hundred pitches, so a moiré reads a pitch difference at a hundred times — which is what a vernier is. The magnification is free and its ceiling is not: the fringes split when one thread's own spacing error reaches 0.84 of the pitch difference the beat is built on, so the usable gain is 0.84 divided by the cloth's coefficient of variation, inversely and not inverse-squarely. At an ordinary yarn's spacing irregularity, a moiré carries a magnification of five.

pattern · Moire Two layers in depth, and the beat perspective makes. An eye, a near grid and a far grid of the same pitch, with a ray from the eye to every bar of the far grid and a dot where each ray crosses the near one. The far bars land on the near layer at 8 to every 9, so the two grids drift out of register and back into it every 8 bars: in register the gaps line up and light comes through, half-way between them the far bars sit in the near gaps and block it. That spacing is the distance divided by the gap, times the pitch, and nothing about the threads or the angle between the layers enters it. The gap here is drawn at one eighth of the distance so the bars can be counted; two sheers 50 mm apart seen from 3 m are at a gain of 60. What the drawing cannot show is a real layer's thickness and its own irregular spacing, both of which the arithmetic treats as absent.

Two sheers make a moiré that walks with the viewer

Hang two identical sheer curtains a few centimetres apart and a moiré appears with no angle between them and no difference in their threads. Perspective alone makes the far one look finer. The fringes are p·V/D apart, which means they cover the same angle from every distance; they move one for one with a person walking past, which is the parallax of the horizon; and a far layer stretched by one per cent makes them vanish at exactly one distance, which says which layer is coarser and by how much.

pattern · Moire 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.

Where a yarn is thinnest

A yarn in a fabric is pressed where it crosses and free where it does not, so its section changes along its own length. Every flattening this collection has ever quoted is a single number for a profile that runs from four fifths of a diameter to nearly four.

cloth · Contact How much of a yarn has to hang before it stops snarling. The tension a 20 tex cotton needs to stay straight, at each twist level, expressed as the length of the yarn's own weight that would supply it. The two curves are the two ends of the stiffness bracket: the fibres free to slide, and the section coherent. At 800 turns a metre they are 1.9 metres and 616. Anybody who has let go of a twisted yarn knows which of those is right, which makes this one of very few places where the bracket can be closed from the everyday end. Both rise as the square of the twist, because the torque does and the criterion is quadratic in it.

How much yarn has to hang

The tension a thread needs to stay straight, converted into the only unit anybody has an intuition for: the length of the yarn's own weight. One bound says two metres and the other says six hundred, and everybody who has handled thread already knows which.

setting · Twist A slack twisted yarn takes a coil of one size. A 20 tex cotton at 800 turns a metre. Its own torque is 1.177 µN·mm at the free bound, and a rod under tension T is stable while its torque stays under 2·√(B·T). Below that tension the straight state stops being a minimum and the thread wraps on itself at a radius of 1.59 mm. That radius is 2B/M, which is 2/(r·ω) once the torque is written out: it depends on the twist and on the ratio of the two stiffnesses, and on neither stiffness itself.

A snarl comes in one size

The radius a twisted thread coils to is twice its bending rigidity over its torque. Write the torque out and the bending rigidity cancels completely, leaving a number that depends on the twist and on the ratio of two stiffnesses — and on nothing else about the yarn at all.

setting · Twist The flattening a knitted fabric asks for, against its own tightness. Every point is a solved fabric — 18 of them, over five loop lengths, three relaxation states and three counts — and the vertical axis is the closest its adjacent courses come, in diameters. The values run from 0.707 to 0.839 and they fall on one curve against the tightness factor, which is the model's only dimensionless group. That is what makes this a structural requirement rather than an arithmetical accident of one example: a tighter fabric demands a flatter yarn, by an amount its own tightness decides, and nothing about the fibre or the count enters except through that group.

A flattening that follows the tightness factor

Eighteen solved fabrics — five loop lengths, three relaxation states, three counts — and the flattening each one's geometry demands falls on a single curve against one dimensionless group. Nothing about the fibre or the count survives except through that group, which is what turns an arithmetical result into a structural requirement.

knits · Contact The section the fabric asks for, beside the one the model drew. A 24.2 tex cotton yarn in cross-section, at 900 times life size. The circle is what every solve in this collection has assumed: a diameter of 0.184 mm from the count and the packing factor. The ellipse has the same area and a minor axis of 76% of it, which is the closest the fabric's own adjacent courses come to one another — 0.140 mm. Nothing about that number was fitted to a fabric. It falls out of a solve that knew nothing about flattening, and it lands inside the range the trade reports for yarn in cloth, which this collection has carried as a free parameter ever since it first put a number on a sett.

The diameter that does need a state

A fabric dimension quoted without its relaxation state is not a measurement. The rule was applied to the two plan dimensions and then to the thickness, and it was never applied to the one dimension that is not the fabric's at all — the yarn's.

finishing · State What a knot costs, and what decides it. Knot efficiency against the radius the thread is bent to, for 5 fibres at 20 tex, computed at the free end of the stiffness bracket — each fibre bending about its own axis, so the strain at the outside of the bend is half a fibre diameter over the radius. At one yarn diameter of radius, cotton keeps 46%, which is the rule of thumb that a knot halves a rope's strength. What separates the curves is not the knot and not the friction: it is the fibre's own breaking strain, which is how much the bend is allowed to spend before there is nothing left.

A knot halves a yarn and says why

The rule is quoted for every rope and every knot and derived nowhere. It comes out at forty-six per cent for a cotton — but only if the yarn's fibres bend individually. A yarn bending as a solid section has already spent five times its breaking strain before any load arrives, so it could not be knotted at all.

applied · Tensile What a knot costs, and what decides it. Knot efficiency against the radius the thread is bent to, for 5 fibres at 20 tex, computed at the free end of the stiffness bracket — each fibre bending about its own axis, so the strain at the outside of the bend is half a fibre diameter over the radius. At one yarn diameter of radius, cotton keeps 46%, which is the rule of thumb that a knot halves a rope's strength. What separates the curves is not the knot and not the friction: it is the fibre's own breaking strain, which is how much the bend is allowed to spend before there is nothing left.

Which yarns knot well

A knot's efficiency depends on three things and only one of them is the knot. The other two are the fibre's fineness and its breaking strain, and across this collection's own table those give efficiencies from six per cent to eighty-three at exactly the same bend.

applied · Tensile What the four-point system charges for a fault, and what the cutting room pays. For a single warp fault of each length: the points the four-point system scores it — one up to three inches, two to six, three to nine and four beyond — and the number of 900-millimetre panels it condemns. Below a panel's length every fault condemns exactly one panel while its points run from one to four, so the scheme charges four times as much for a fault that costs the same. Above a panel's length the panels grow without bound and the points stay at four, so the scheme stops charging exactly where the cost starts rising. A ten-metre fault scores 4 and condemns 12 panels. What the chart cannot show is the marker, which decides the panel size and therefore the whole of the second curve.

A grade charges by the length and a cutter pays by the panel

The four-point system scores a fault by how far it runs — one point to three inches, four beyond nine — and caps a linear metre at four points however many faults it holds. A cutting room pays by how many panels the fault lands in. Below a panel's length every fault costs exactly one panel while its score runs from one to four; above it the panels grow without bound and the score does not move at all. Two fifty-metre pieces built to the same 267 points a hundred square metres lose 33 per cent of their panels and 92.

applied · Faults The first 12 metres of a bolt, cut three ways. A bolt of 50 metres carrying 40 faults, with its first 12 metres drawn: the fault positions above, and below them the 900-millimetre panels a cutter gets cut blind, cut with the same rigid tiling slid to its best offset, and cut around the faults with the map in hand. Over the whole bolt the three yield 28, 30 and 40 sound panels of 55. Sliding the tiling buys 2; breaking it buys 12, which is 43 per cent more cloth from the same roll. What the strip cannot show is the width, across which the same argument runs again with a different panel dimension.

A fault map is worth most where the grade is worst

A cutter who knows where the faults are can slide the marker or break it, and only one of those is worth anything: sliding a rigid tiling to its best offset recovers two panels of fifty-five, and letting the tiling break recovers twelve — two panels in five more cloth from the same roll. The gain has a maximum in the middle of the range, because there is nothing to recover on a clean bolt and nothing to be done on a ruined one. And the prediction the grading essay made, that a map is worth most on a bolt whose faults are bunched, is false: bunching leaves clear runs for the blind cutter too.

applied · Faults A fibre's two stiffnesses, as the ratio that survives the bracket. C over B for every fibre in the table, at 20 tex, which is 2G/E and nothing else. The mark beyond each bar is the range the shear modulus is reported over. Glass is the control and is not a measurement: it is a drawn isotropic solid, so its ratio must be 1/(1+ν) and at ν = 0.2 that is 0.833, which is what the table says. Every fibre with molecules drawn out along its axis sits below the isotropic value, and the ordering is the subject rather than an accident — a fibre is stiff along its axis because its molecules are drawn out along it, and the same orientation that raises E leaves G to whatever holds one chain to the next. Aramid, the most oriented thing here, is twenty times softer in torsion relative to its bending than the glass beside it.

The one fibre whose answer is known

A table of measured constants is worth what its worst row is worth, and nobody can tell which row that is. This one has a member whose answer was known before anybody measured it, and the ordering of the other nine turns out to say something about how fibres are made.

mechanics · Torsion 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.

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.

mechanics · Contact 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.

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.

mechanics · Yarn stiffness

Named alongside it

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

SpecificationContactPacking factorPopulationCloth thicknessYarn diameterBending rigidityCoefficient of variationFibre finenessJammingOrder statisticTorsional rigidity

All concepts