Field

Mechanics and drape

Cloth as a mechanism rather than a material. The bias, the angle at which threads jam, and why a flat sheet cannot cover a sphere.

A trellis sheared 30°. The net at an angle, with every thread segment exactly the length it started at. The extension along the diagonal is the bias stretch, and it is a change of shape rather than a change of length.

The bias is a mechanism

Cloth cut at forty-five degrees stretches by a third and springs back, while the threads in it stretch by nothing at all. Almost every explanation given for this is wrong, and the right one is not about elasticity.

How far the bias goes, and where it stops. Extension along the bias against shear angle, with the angle at which the threads jam marked for three settings. The curve is geometry and so is the wall — a more closely set cloth reaches it sooner.

The locking angle

The bias runs out at an angle that yarn diameter and thread spacing decide between them. It is the number behind whether a cloth will go round a curve, and it has nothing to do with how strong the fabric is.

A cloth laid over a sphere. Every thread segment is exactly one pitch and none has stretched. What has changed is the angle at each crossing, and the amount is decided by the surface — a developable one costs nothing and a curved one costs more the further the cloth goes.

Why clothes need darts

A flat cloth goes round a cylinder for nothing and cannot go round a sphere at all without shearing. The amount of shear is decided by the curvature, and when it exceeds what the threads allow, something has to be cut out.

The cantilever test. A strip of cloth pushed out over an edge until its tip has drooped to the stated angle. The overhang at that moment gives the bending length, and cubing it with the mass per unit area gives the flexural rigidity.

Bending stiffness and the drape coefficient

Two standard measurements try to say how a fabric hangs. One measures a length and cubes it; the other measures an area and, on the geometry, turns out to be answering a different question from the one it is asked.

Shear locking in a composite preform

Laying a woven reinforcement over a mould is the bias mechanism doing useful work, and it stops dead at the angle where the threads jam. Where the cloth wrinkles is a geometric prediction with a radius attached.

What the drape coefficient answers to. The drape coefficient against the number of folds, and against how far the hem has come in. Over the range a real specimen shows, the fold count barely moves it; the hem radius moves it across almost its whole range.

A drape coefficient is one number for a directional thing

A fabric bends more easily one way than the other — a factor of two is ordinary. The drape test reports a single percentage, and the quantity that carries the directionality is the fold count, which the coefficient is almost blind to.

Wrinkles 127 mm apart. A 300 mm width of a 120 g/m² cloth whose bending length is 20 mm, held under 5 newtons per metre across it and compressed. It cannot carry the compression in the plane, so it leaves the plane, at a wavelength the bending rigidity and the tension settle between them: 127 mm, which is 2.4 wrinkles across the width. The amplitude is drawn and is not computed — this arithmetic sets the spacing and says nothing about the depth.

A cloth cannot carry a push

The net model that runs this site's mechanics has no bending stiffness at all, so it buckles under any compression whatever, into wrinkles of any wavelength whatever. What picks the wavelength is the competition the net leaves out — and the answer is a quarter power, which is why a wrinkle is so hard to change.

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.

Everywhere a sheeting can go. Every state a sheeting of 28 × 26 threads per centimetre in 25 and 25 tex can reach without a yarn changing length, solved from Peirce's plain-weave geometry. The set is a curve and not a region: 4.03 per cent of extension is available along the warp, and reaching it costs 6.98 per cent of the width.

A cloth extends by moving its crimp

Everybody says the extension available along the warp is the warp's own crimp. On six of eight ordinary cloths it is not — a close sheeting has 14.61 per cent of warp crimp and reaches 4.03 per cent, because the limit lives in the weft.

Everywhere a muslin can go. Every state a muslin of 24 × 22 threads per centimetre in 20 and 20 tex can reach without a yarn changing length, solved from Peirce's plain-weave geometry. The set is a curve and not a region: 6.59 per cent of extension is available along the warp, and reaching it costs 21.82 per cent of the width.

Pulled both ways, only one can give

A cloth at constant thread length has one degree of freedom, so its reachable states are a curve rather than a region. Equal extension in both directions meets that curve at exactly one point — the state the cloth is already in — so the amount available is nought.

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.

The shed in section. A warp line 1200 mm from the fell of the cloth at the left to the back rest at the right, with 8 shafts at 300 mm and every 16 mm behind it. Each shaft lifts its ends in proportion to its own distance from the fell, which is what gives the same 30 mm clear opening at the reed from every shaft. The resulting extension is 0.460 per cent at the front shaft and 0.722 per cent at the back, a ratio of 1.57. Vertical scale exaggerated 4 times.

The shed is an extension

Every model of a finished cloth treats a thread as inextensible, and every one of them is about a fabric that has left the loom. On the loom the warp is stretched thousands of times a minute by the shed itself, and the amount is exact trigonometry with nothing fitted in it: the square of the shed's tangent, times the shaft's own distance from the fell, over twice what is left behind it.

The strain across a harness. The warp strain each shaft of a 24-shaft harness puts into its own ends, from 0.460 per cent at the front to 1.728 per cent at the back. Shafts within a budget of 1.0 per cent are drawn in one colour and those outside it in another; the budget is reached at shaft 13.

The back shaft works hardest

How hard the loom is on a warp end is the product of two numbers from different worlds — how often its column changes sides, which is a property of a binary matrix with no millimetre in it, and the strain of the shaft it happens to be on, which is a property of a machine with no weave in it. Neither knows about the other, and the threading that joins them is a decision nobody makes on structural grounds.

Flexes per end in the same check, ground a pick along. One bar per warp end above the draft of the same check, ground a pick along, each the number of times that end changes sides in one repeat — which is the number of times the shed drags it through its heddle eye. The counts run from 6 to 10, and they are the same numbers that give the cloth its interlacing rate of 0.500 per intersection.

A figure is harder on its warp

Every basic weave flexes all its ends exactly as often as each other — plain, twill, satin and sateen alike, and it is a one-line theorem. Figure one of them on another and that evenness goes, or does not, depending on where the ground weave was started relative to the figure. The same invisible offset that decides whether a fine figure holds together decides, at a coarse one, how unevenly the loom works the warp.

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.

What a shed costs, in newtons. The tension the shed puts into one end at each shaft of a 24-shaft harness, for a 25 tex cotton yarn. The strain is set by the loom's geometry alone; the tension is that strain times the yarn's modulus. The front shaft holds 0.52 N and the back 1.95 N, which is 14 and 52 per cent of the yarn's breaking load. What the chart cannot show is the rest of the warp tension, which the let-off adds on top of all of these and which no geometry decides.

What the shed costs, in newtons

The shed's strain has been computed here and could not be priced: a strain is a length over a length and says nothing about how hard a thread is being pulled. A modulus turns it into a tension — and the back shaft of a twenty-four-shaft harness turns out to hold its ends at half their breaking load, all day, from the geometry alone.

The load–extension curve, computed from a stiffness. The tension in one end of a sheeting against how far the cloth has been extended, computed as the slope of its bending energy along its own constant-thread-length locus. The curve passes through zero at the state of least energy, which is where an unloaded cloth sits, and rises either side of it. What the curve cannot show is what happens after the crimp runs out: past the end of the locus the load is carried by stretching yarn rather than by straightening it, and that is a modulus three orders of magnitude higher and a different figure.

The locus gets a force

This site has drawn the set of states a cloth can reach without stretching any yarn, and has never been able to say which of them it is in or what it would cost to move. Both questions are one derivative of a bending energy — and the answer explains the flat start every fabric's load–extension curve has, which is not slack yarn but a symmetry.

A crossing before and after it is pressed. One warp end of a sheeting riding over three picks, drawn twice to the same scale. Unpressed, both sections are circles and the cloth is 0.388 mm thick. At 0.40 N per crossing the sections flatten to aspect ratios of 1.75 and 1.84, the cloth thins to 0.263 mm, and the warp runs flat for 0.109 mm over each pick before it begins to curve. What the drawing cannot show is why the cloth does not do this by itself: flattening shrinks the arc radius as well as the crimp height, so it costs bending energy, and a relaxed cloth keeps its threads round.

A flattened thread is a record of a force

A yarn in cloth is not round, and this site has modelled the flattening for as long as this collection has run with the amount of it left as a number somebody chose. Give the section a stiffness and ask what the cloth prefers, and the answer is a circle — at every stiffness, for every balanced cloth in the table. Flattening does not happen by itself; it happens because something pressed.

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.

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.

A sheeting's crossing, dry and wetted. One crossing of a sheeting in section at three swellings: dry, at the swelling where its geometry has its last state, and fully wetted at 20%. The closure condition is that the two systems' crimp heights add to the cloth's thickness, and each can supply at most the height it reaches when its straight portion has just vanished. Swelling raises the demand in proportion and the supply more slowly, so the margin closes and then goes negative — at 9.29% for this cloth against cotton's 20%. The bottom panel is drawn as far as the threads reach and no further, because there is no state to draw. What the drawing cannot show is what happens instead, which is that the yarn is compacted.

The swelling a cloth cannot take

Three of the eight cloths in this collection's table have no wet state at all. A thread of fixed length cannot wrap a partner that has grown by a fifth, so the closure condition fails and the geometry has nothing to offer. What happens instead costs megapascals, and the alternative route is not merely dearer but unavailable — the thread would break first.

Cotton and viscose in every column this site carries. The two cellulosic fibres compared across every constant on this site. Both are cellulose at 1.52 g/cm³, both are given a modulus of 8 GPa and a fineness of 1.7 dtex, so a 20 tex yarn of either has the same diameter to the last figure — and therefore the same crimp, the same cover, the same jamming sett and the same bending bracket. Every geometric result on this site is the same number for the two fibres. Water separates them twice: viscose swells 1.75 times as far across and loses half its strength where cotton gains a tenth. What the table cannot show is why, which is a question about how cellulose is arranged inside a fibre and is not in this collection.

Water tells two fibres apart

Cotton and viscose are the same material by every constant this collection carries. Same density, same modulus, same fineness, so the same diameter at every count and the same crimp, cover, jamming sett and bending bracket. A wash separates them by a factor of two, and it is the only thing here that can.

The poplin's two budgets as it is held stretched. What is left of a poplin's interchange in each direction as it is held at more and more warp strain. The warp's budget falls to nothing at 3.93%, which is the point of the curve; the weft's rises, because the crimp the warp gives up is crimp the weft takes on. There is one locus and one position on it, so the two are not two quantities that happen to be related — they are the two distances from one point to the two ends of one curve. The consequence is that a pre-tensioned cloth has almost no warp recovery left and more weft recovery than it started with. What the plot cannot show is that the exchange rate between them is not constant: the curve is not a straight line, and its slope is the Poisson ratio this site computes elsewhere.

A cloth has one budget for two directions

A woven cloth looks as though it carries two independent reserves of free extension, one along the warp and one across the weft. It carries one. There is a single curve of states and a single position on it, so every hundredth spent one way is refunded the other — which means a pre-tensioned cloth has not used its recovery up, it has moved it.

Elastic recovery against strain, for seven fibres. The elastic recovery of seven fibres at the strains it is reported at: extend to a stated strain, unload, read the strain returned immediately. Each fibre's points are joined and the line stops where the measurements stop, which is the point of the figure — a fibre strained past the last point on its own line is a fibre this collection declines to answer for. Nothing here is measured below one per cent of strain, and a woven cloth just past its own interchange budget is at a thread strain of a few hundredths, so the region that matters most for a fabric is the region nobody has reported. Recovery falls monotonically for every fibre, which is what lets the unmeasured region be bracketed between the lowest measured value and one rather than extrapolated. What the plot cannot show is the delayed recovery, which is excluded by the convention and is largest for the fibre with the best reputation for recovering.

Recovery is measured and nothing predicts it

A fibre's stiffness is a bracket this collection can compute the ends of. What fraction of a strain it gives back is not: it has to be looked up, the tables are thin, they stop exactly where a fabric needs them, and the most attractive explanation for the ordering they show turns out to have no signal in it at all.

$What a tensioned sheeting has left of its load. A sheeting pulled to a strain, clamped at that length and left. Its length does not change, and its load does: crossings rearrange locally until the load has fallen to what friction alone can hold, which is 0.0756 N per end and is the same number whatever the cloth was pulled to. So the fraction retained is that floor over the load applied, and it falls — a cloth tensioned to 4.94 per cent keeps 29 per cent of what it was given. Below the resting band's own edge nothing is lost at all, because the cloth was never outside what friction could hold. What the plot cannot show is time: nothing here says how long the rearrangement takes, only where it stops.$

A tensioned cloth loses its load

Clamp a fabric at a fixed length and its tension falls overnight. Nothing crept and nothing flowed: the crossings rearranged locally until the load had dropped to what friction alone can hold, and that level is the same number whatever the cloth was pulled to — so the harder it was tensioned, the smaller the share it keeps.

$Eight fibres through one cloth, and the bracket that hides them. A muslin is 19 per cent fibre and the rest air, so its thermal conductivity is a two-phase mixture, and how the fibre and the air are arranged is not something any amount of knowing the fractions settles. What is settled is Wiener's pair of bounds: heat across a series arrangement sees the harmonic mean and along a parallel one sees the volume average, and every real arrangement lies between. Each bar here is one fibre's whole band, from the series bound at its lowest published conductivity to the parallel bound at its highest. Not one of the eight clears any other, so this model cannot tell wool from nylon in a fabric — and the same cloth at twice the thickness has twice the resistance, exactly. Air's own conductivity is marked, and every band sits within a fifth of it.$

Warmth is a thickness of air

A fabric is a tenth fibre and the rest air, so its thermal conductivity is a mixture of the two — and the pair of bounds that any mixture must lie between comes out narrower than the difference between wool and nylon. The model cannot tell one fibre from another in a cloth. What it can tell, exactly and with no bracket at all, is that twice the thickness is twice the warmth.

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.

$A bundle is weaker than the threads it is made of. Threads pulled in parallel do not break together. The weakest goes first and hands its load to the rest, which are now carrying more than they were, so the bundle's peak load is reached before every thread is at its own strength. With a load per surviving thread of x carried by the fraction that has not yet broken, the bundle's strength per thread is the largest value of x(1 − F(x)) — Daniels' maximum, which for a lognormal at CV 15% is 0.7380 of the mean thread, reached with 93% of the threads still unbroken. A simulated bundle that knows none of that arithmetic sits above the limit at every finite size — its strength is a maximum over the sample it happens to have drawn — and closes on it as the bundle grows: 0.753, 0.744, 0.741 at the last three sizes. The scatter falls the other way, from 14.5% at one thread to 0.7% at 1600.$

A bundle is weaker than its threads

Threads pulled together do not break together. The weakest goes first and hands its load to the rest, so a bundle carries its maximum well before every thread is at its own limit — and the shortfall is a quarter, decided by the spread and by nothing else.

$Where a 20 tex yarn breaks, against how much was clamped. A tensile test clamps a length of yarn and pulls until the thinnest section between the clamps gives. So a yarn's strength is a minimum, and a minimum depends on how many independent tries the sample contains. The tries are not sections — a plane can be taken anywhere — but staple lengths, because two planes closer together than one fibre share most of their fibres. At 28 mm staple a 100 mm specimen holds 3.6 independent tries and a 500 mm one holds 17.9, and the longer test reads 11% lower. The spread is not fitted either: it is the evenness floor at 118 fibres times an index of 1.35, which is 13.4%.$

A yarn breaks at its thinnest place

A tensile test does not measure a yarn. It measures the worst section between the clamps — so evenness and strength are one measurement taken twice, and the number of independent tries in a specimen is set by the length of a fibre.

Where the strain goes, at a 25° twist. Extend a twisted yarn and the fibres in it are not all strained alike. If each stays at its own radius — the affine model — a fibre at helix angle θ takes cos²θ of the yarn's strain, so the fibre on the axis takes all of it and the fibre at the surface takes 82.1% of it. The core reaches its breaking extension first and the yarn cannot realise its own fibres. If instead every fibre migrates between the core and the surface, every fibre has the same mean strain and they all break together. The two are exactly computable — 0.8214 against 0.9509 of what the fibres could give — and the ratio between them, 2/(cos α(1 + cos α)) = 1.158, is what migration is worth. The shading is that arithmetic and the two panels use the same scale.

A straight fibre cannot share the load

Extend a twisted yarn and its fibres are not all strained alike: the one on the axis takes the whole of it and the one at the surface takes cos²α. So they do not break together, and what a wandering fibre is worth comes out as one expression with nothing fitted in it.

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.

The singles inside a ply are not the singles

Folding leaves each single with a fraction of its own twist, which should ruin its grip on its own fibres. It does not, because the ply's helix presses on the singles exactly as a single's helix presses on its fibres — and across the whole practical range the two very nearly cancel.

Which end of the bracket a thread in cloth is at. A yarn bends as a solid rod while its fibres cannot slide and as a loose bundle once they can, and the crossover is a curvature: the coherent state demands an axial force in the outer fibre that has to be built up by friction under the twist's own radial pressure. The curves are the crossover radius against twist at four contact efficiencies, the top one being 1.0 — the claim that fibres touch along their whole length, which nobody makes. The rule at the bottom is the radius a thread is bent to by its own crimp in a cloth, about 0.25 mm. Every curve is above the rule by at least 19-fold, so a thread in cloth is at the free end of its bracket at every twist and every efficiency, and the collection's habit of using the lower bound is a result rather than a convention.

Twist decides where in the bracket

A yarn's bending rigidity can only be bracketed, and the bracket is three hundred wide. What decides where a yarn sits in it is whether its fibres can slide — and for a thread bent by its own crimp in a cloth, the answer is not close.

$Two exponents in a compression curve. How far a flat plate sinks into plain, 2/2 twill, satin 8 in sheeting, against the pressure it is applying, on logarithmic axes where a power law is a straight line. The measured slopes over the light end of the range are 0.50 for the plain, 0.67 for the 2/2 twill, 0.67 for the satin 8 — against two thirds predicted for any weave carrying a float and one half for a weave carrying none. The prediction is one line of algebra: pressure is a stiffness times a strain times a bearing fraction, the bearing fraction is a square root of depth for a plateau and linear in it for a point, so the pressure is the three-halves power in the first case and the square in the second. Nothing is fitted to produce it; the slopes are measured afterwards and compared. At the heavy end every curve bends, because the crowns have merged and the cloth has stopped being a surface and started being a solid — which is a different regime with a different law, and it belongs to the compaction of a fibre mass rather than to the geometry of an interlacement.$

A cloth compresses along its own bearing curve

A fabric's pressure–thickness curve is always fitted with an empirical power law and the exponent is reported without explanation. It is not empirical. At light loads it is two thirds for any weave carrying a float and one half for a weave carrying none, and the two numbers come out of one line of algebra with nothing fitted in it.

A thickness is a property of the pressure it was measured at. What a gauge reports for sheeting against the pressure it presses with. At 1 kPa it reads 0.382 mm, which is the cloth; at 0.02 kPa it reads 0.555 mm, which is the cloth plus 87 µm of hair on each face. The difference is 31% of the reading and it is not a compressibility: nothing in the fabric has been squashed, the foot has simply stopped in a different place. That is why every thickness standard specifies its pressure to two figures, and why comparing a thickness from one standard with a thickness from another is comparing two different measurements of two different objects. The dashed line is the cloth's own geometric thickness, which no reading below the crossover ever reaches.

A compression curve is two laws in series

A published fabric compression exponent is a fitted number with no derivation attached, and this site has already said why: the range it is fitted over straddles two regimes. One of them turns out not to be in the cloth at all.

Warmth is a canopy, and a canopy grows as a logarithm. Thermal resistance of sheeting against how much its hair population has been multiplied by raising, both faces counted. The unraised cloth is given no still air at all, because its hairs cannot reach one another — n_A λ² is under one and there is no canopy to hold air still. Past the threshold the canopy's depth is λ·ln(n_A λ²), so every doubling of the hair population adds the same depth of nap and no more: the steps here are 434 µm apiece, all the way up. At 256× the nap is 3.07 mm deep and worth 39 times the cloth it grows on, which is the whole reason a flannel is warm and a poplin of the same yarn is not — a canopy is two parts in ten thousand fibre, so its conductivity is air's, while the cloth itself is a quarter fibre. What is claimed is a conduction resistance across a depth of nearly still air; whether the air is still is a question about flow and is not asked here.

Warmth is mostly the hairs

This collection established that a fabric's thermal resistance is its thickness and not its fibre, and that twice the thickness is twice the warmth. It never asked what the thickness was made of. On a raised cloth almost none of it is cloth.

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 1.00, 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 thread between two crossings is an elastica

Every crossing force Peirce's thread path can give comes from a tension, and a cloth on a table has none. What presses its threads together there is their own unwillingness to be bent — and recovering that needs a shape nobody had, because a path assembled from an arc and a straight line has a bending moment that jumps.

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.

How hard a relaxed fabric presses on itself. The normal force at one crossing of a relaxed cloth, against the force at one interlacing of a relaxed jersey. The woven figures were recovered by inverting a thickness measurement through a compression energy; the knitted one comes from a solved shape and no measurement at all, so the two are genuinely independent rather than two readings of one number. Every cloth in the table presses harder than the knit — by between 5 and 22 times — and the knitted figure is an upper bound besides. One ratio is behind a list of differences usually explained separately: which fabric gives up a fibre end, which pills, which frays, which lets a seam slip.

What a loop presses with

A knitted loop hangs on the loop below it and presses on it with a force nobody has been able to state. Solved from the loop's own bending it comes to about forty millinewtons a stitch — an order of magnitude under a woven crossing's, by two independent routes that have nothing in common.

What a loop model still cannot say

A planar rod with a natural curvature and point contacts gets a knit's forces, its modulus and its extension. It does not get torsion, it does not get the third dimension the interlacing actually needs, and it does not stop adjacent courses passing through one another — which is why its extension ceiling sits three times beyond any jersey.

What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve.

The modulus a knit has instead of one

A fabric's stiffness in extension is usually its yarn's modulus with the geometry taken out. A knit's is not: dimensionally it can only be a bending rigidity over a length cubed, and the same yarn laid straight and parallel is thirty thousand times stiffer than the fabric made of it.

How hard a fabric is to bend, per unit width. Bending rigidity in micronewton metres per unit width, computed the same way for eight woven cloths and for a jersey in each of its two directions: the yarn's own rigidity, times the length of yarn per unit area, times the fourth power of the cosine of the angle each element makes with the bending direction. The knit lands inside the band of the woven cloths rather than below it, which is worth knowing because a knit is usually called the softer fabric. It is not softer to bend; it is softer to stretch, by three decades. What it does have is direction: 2.17 to one between its two axes, where a balanced plain weave is near one, and the soft axis is the one a jersey rolls about at its top and bottom edges.

A knit bends more easily along its courses

A jersey is not a soft fabric to bend. Computed the same way as this collection's woven cloths it lands inside their band, at the limp end — and the useful number is not the magnitude but the direction: two point two to one between its two axes, with the soft one being the axis it rolls about.

What friction has to hold in a relaxed knit

A knit's bending energy slopes away from the fabric everybody measures, so something is holding it there. Along its courses friction holds comfortably. Along its wales the driving force is exactly the contact force, so the whole balance collapses to one condition — the friction coefficient must exceed a half — and no yarn in this collection reaches it.

A knitted loop is a plane curve in a plane that is not the fabric's. A relaxed 20 tex jersey at a 3.5 mm loop, seen from the end of a course. Each course is a straight line because it is one: a half period leaves and arrives along the course direction, so the plane it bends in contains that direction, and a plane curve seen along a line in its own plane projects to a segment. The upper panel expands the thickness 3× so the arrangement can be seen and no angle may be measured off it; the lower panel is the same fabric at one scale, where the tilt is what it really is. The angle is 11.75°: a climb of one yarn diameter, 0.167 mm, against a drop of a course spacing and a diameter, 0.803 mm. Successive courses overlap by exactly one diameter, which is the interlacing, and the fabric is therefore 0.334 mm thick — two yarn diameters, with nothing fitted.

A loop is a plane curve in another plane

A knitted loop was solved as a flat curve because two curves in one plane cannot pass through one another and a loop must. Letting it out of the plane turns out to change nothing about its shape: the loop is still planar, and its plane is the fabric's turned through twelve degrees.

What leaving the plane costs

Every number the flat loop model produced, beside the same number with the climb in it. Four of the five fall, none moves by four per cent, and the estimate that priced the third dimension beforehand had the sign the wrong way round for a reason worth naming.

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.

The force that holds a knit open

Resolving a knitted loop's contact force out of the fabric's plane leaves a fifth of it pointing through the thickness. That fifth is 7.8 millinewtons a stitch, fifteen kilopascals over the area a stitch occupies, and it is the whole reason a jersey has a thickness rather than a plan.

How little eccentricity a curl needs. The radius a jersey would curl to, against how far the fabric's neutral surface fails to bisect its loops. The model's own loop is bisected exactly — its crest sits half a diameter behind the mid-surface and its trough half a diameter in front — so it carries no curling moment at all, and the curve here is what any departure from that would buy. The moment is proportional to the eccentricity, so one solve settles the whole curve. A radius of three millimetres, which is about what a fine jersey rolls to, needs 5.6% of a yarn diameter — 9.4 micrometres on a yarn 167 micrometres across. That is why curl is easy to see and hard to model: the whole of it lives inside a rounding error on the geometry. Only the course-wise edges are drawn, because bending about the other axis turns the thread's end tangents out of the fabric and a free thread given turned ends buckles clean out of it — 3.2 times the fabric's own thickness, for 37% off its energy. That configuration is not available to a thread with neighbours, so the second moment is refused rather than computed.

How little asymmetry a curl needs

A model with a thickness can finally be asked why stockinette rolls. It answers that it does not — its curling moment is exactly zero, by a symmetry — and the useful part is what that costs to break: five per cent of a yarn diameter buys the whole of the curl anybody has ever seen.

What a knit gives up when it is pressed

The woven half of this collection has had a compression curve for several rungs — a thickness that falls under load, a bearing area that grows, a pressure at every point. The knitted half had a plan and no depth. It has a relaxed thickness and an initial slope now, and the two fabrics turn out to resist for different reasons.

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.

A thread has a second stiffness

Every mechanical number here came from one material constant: how hard a thread is to bend. A thread also resists being twisted, nothing here has ever used that, and the ratio between the two turns out to be the only stiffness number about a yarn that can be known at all.

Twist is not torsion

A curve has a torsion and a material has a twist, they share a word, and only one of them is what a torsional rigidity resists. Getting them the wrong way round would have made this collection conclude that a knitted loop carries no twist, on the strength of a theorem that says nothing of the kind.

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.

Linked: a knitted interlacing: one loop drawn through the next. Two closed curves and the Gauss linking integral taken over them, which returns -1.0002 at 200 segments a curve. A knitted interlacing: one loop drawn through the next. A linking number is an integer, so a value coming back at a few thousandths of one is the discretisation reporting itself rather than a fabric that is slightly linked. The two arrangements are the two ways of making cloth: a knitted fabric's courses link and a woven cloth's threads do not, at any crimp and for ever.

What a closed thread cannot choose

A thread whose ends are held has a quantity it cannot change without breaking: the total number of times its material winds about its own axis, plus the number of times that axis winds about itself. The two can trade, and everything a twisted yarn does when it is let go is that trade happening.

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.

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.

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.

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.

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.

The crest, and the loop that ought to be holding it open. One course of the solved fabric in plan, with the two half periods that meet at a crest marked. They approach to 0.003 mm — 0.018 of a yarn diameter — and run within that of one another for more than a millimetre of arc. The ring drawn between them is the needle loop of the next course, which is what holds them apart in a fabric and what this model does not have: the interlacing was declared a point, and a point holds nothing open. The same omission is what makes the course's writhe zero and its linking number zero, so three of this collection's findings are one defect seen three ways.

What a contact model would have to do

This ladder has measured a fabric that does not fit and priced nothing. The repair is a different class of problem from the one this collection solves, it costs fifteen per cent of the yarn in a stitch, and it buys back four results — which is an unusually good return for a piece of modelling.

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 this collection's thread model now stands

A thread has two stiffnesses and a thickness, and this collection's model has had one stiffness and no thickness. Both were added in one phase, neither reached the question it was built for, and the accounting is worth more than either.

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